1.

1 General Introduction

Gear transmission systems are an important aspect in many industries. The knowledge and
understanding of the behaviour of gears in mesh including Transmission Error (T.E.), load
sharing ratio, distortion field variation etc., is vitally important for condition monitoring
and system control of gear transmission systems. A large range of literature on the topic of
gear transmissions exists, as demonstrated by the review by Ozguven (Ozguven 1988).
Fundamental publications on gears in mesh can be found in particular by Walker (Walker
1938), Harris (Harris 1958), Niemann (Niemann 1970) and Munro (Munro 1994).
However, the fundamental study of the basic gear transmission unit still needs to be
finalised and with recent software (FEA) advances and fast and cheap PC technology,
many research studies are now aimed at furthering the basic understanding of gears in
mesh. The improved understanding of gear behaviour will increase our capability to
monitor gear transmission systems using measured parameters such as the gear case
vibration signal.

The research into the elastic behaviour of gears in mesh has also highlighted the use of the
gear strain variation as a new method of measuring gear transmission error and the effects
of modified tooth profiles.
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The range of non-metallic gear applications continues to grow fast due to the low cost
(injection moulding), environmental benefits (without external lubrications) and many
advanced features (White 1997; Walton 1998; Walton 1998; Luscher et al. 2000). Recent
developments have sought to minimise the disadvantages of non-metallic gears, for
example, using long-fiber reinforced composite materials (Alagoz 2002). The research has
concentrated on the fundamental mechanism properties of non-metallic gears in mesh using
both numerical analysis and experimental tests.

1.2 Basic Transmission Unit

The basic transmission unit is illustrated in Fig 1.2.1 with parallel shaft involute gears since
this type of drive dominates the field of power transmission.

Figure 1.2.1 The basic transmission unit.

If the bearing supports are rigid enough (in many cases, the gearbox casing was ignored in
the study of transmission error), the model shown here can be taken as the simplest unit for
defining gear transmission error.

In this research, the model for the test rig was build according to Figure 1.2.1, with
different gear materials being used in an attempt to measure accurate data. Numerical
modelling has shown that the use of different spline or keyway methods for coupling the
gear onto the shaft may influence the experimental data.

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One of the complicating factors in the numerical study of gears in mesh is the large number
of assumptions, which must be understood and documented. One of the simplest models
developed in this work involved just a single tooth drive gear in contact with a rigid
surface. In general, the less assumptions made in the model, the more complicated and
difficult the FEA model.

1.3 Torsional Mesh Stiffness

The torsional mesh stiffness of gears in mesh was first defined by Sirichai (Sirichai 1997;
Sirichai et al. 1998; Sirichai 1999). He defined it as the ratio between the torsional load
and the total elastic angular rotation of the input gear hub, where both gears were of pure
involute form. Because the shafts were not taken into account, the nodes on the input gear
hub were coupled torsionally so that the FEA solution obtained the unique displacement
value of the input gear hub which was the total elastic angular rotation. In the general case,
the total angular rotation of the input gear hub will consist of elastic deformation and “rigid
body motion” due to the tooth geometry, run out errors or profile modifications. This
illustrates that accounting for the total angular rotation can be a complex task.

The combined torsional mesh stiffness as defined here is the ratio between the torsional
load and the total angular rotation of the input gear (hub or the shaft). With a constant
input load M, the combined torsional mesh stiffness Km varies as the driven gear rotates
and it has significant variability in a small region between the single-tooth-pair contact and
the double-tooth-pair contact. The development of a torsional mesh stiffness model of
gears in mesh can be used to determine the transmission error throughout the mesh cycle.

1.4 Non-linear problem

As mentioned above, the total angular rotation of the drive gear (pinion) hub represents the
magnitude of angular rotation under a load (torque), mainly due to bulk tooth movements
such as tooth bending and shearing with some rotation occurring at the tooth root. It also
includes gear body distortions. The other part of the rotation amplitude would be that due
to localised Hertzian contact deflection, which is non-linear. This component can be as

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much as 25 percent of the total tooth deformation (Coy 1982). In FEA solutions, this non-
linear problem can be solved by applying various types of contact elements to simulate
different types of contact problems in ANSYS® or by definition of contact bodies (groups
of element) in MSC/MARC®. However, obtaining unique results in the mesh cycle can be
difficult and in some situations solutions will not converge due to the chaotic condition at
the contact(s). Adaptive mesh with contact(s) using quad or brick elements has been found
to be particularly useful in h-method finite element analysis of gears in mesh, where it has
been found to significantly improve the quality of the solution data.

1.5 Solving the Unconstrained Structure

Solving unconstrained body motion is one of the critical stages in numerical simulation of
tooth profile modifications (including tooth tip-relief) of gears in mesh. For example, FEA
solutions for tip-relieved gears in mesh will be subject to unconstrained body motion when
the mesh position is near the centre of the double contact zone. A few options can be
applied to overcome this problem, such as adding weak spring(s) or solving the problem
dynamically. However, when considering avoiding numerical errors and the accuracy of
the final results, the element birth and death option was found to be one of the best methods
for producing consistent results.

1.6 Transmission Error

Transmission error, according to most authors, is considered to be one of the main

contributions to noise and vibration in a geared transmission system. Harris (Harris 1958)
was the first to identify transmission error as a significant contributor to gear dynamics and
the measurement of transmission error was first performed in the 1960’s by the National
Engineering Laboratory and by Gregory, Harris and Munro (Gregory 1963; Gregory 1963;
Gregory 1963). Many others have performed measurements of transmission errors since
that time.
In general, the main components of the total transmission error can be classified into three
parts,
1. First order components: including profile, spacing and run out errors from the
manufacturing process. Geometrical errors in alignment and tooth profile
4
 modifications are also included here and the long modifications typically add “rigid
body motion” into the total transmission error.
2. Higher order components: including the elastic deformation of the local contact
tooth pair, tooth bending, shearing, some rotation about the tooth root and the
deflection of the gear body due to the transmitted load through and transverse to the
gear rotational axis.
3. Higher order dependent components: the relative sliding at the contact(s) truly is the
first order component, however, this component is dependent on the variation of the
higher order components. This special component can also be classified into the
loaded transmission error, in contrast with the other first order components that can
be counted as the unloaded transmission error. These components also include
geometrical errors that may be introduced by static and dynamic elastic deflections
in the supporting bearings and shafts.
The total transmission error always consists of the first and higher order components so
that the relationships between transmission error and noise and vibration are complex.

Gear transmission error is the special case of the total transmission error. For gear
transmission error, the relative model is just a pair of meshing gears where both gears have
rigid axle centres. This has been defined by Welbourn (Welbourn 1972; Welbourn 1979)
as the difference between the actual position of the output gear and the position it would
occupy if the gear drive were perfectly conjugate. The equation for gear transmission error
may be expressed as below,
TE = q g - ( Z )q p , rad, (1.1)

where Z is the gear ratio and q g , p denotes the angular rotation of the input and output gear

in radians respectively. However, gears are sometimes assumed to vibrate only torsionally
but this assumption is incorrect, as any model of gears must allow for lateral movement
perpendicular to the gear axis, Smith 1999. As the shafts cannot be ignored in the T.E.
definitions, Smith (Smith 1983) has defined the total Transmission Error as the difference
between the position that the output shaft of a gear drive would have if the gearbox were
perfect, without errors or deflections, and the actual position of the output shaft. It can be
applied for the case of statical or dynamic, loaded or unloaded gears. The related
measurement of the transmission error is given,
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 g g
TE = q g - ( Z ) q p , rad / s, (1.2)
g
where Z is the gear ratio and q g , p denotes the angular speed of the input and output shaft
that is measured semi-statically. Equations (1.1) and (1.2) both have similar form, but the
use of one or other of the equations for the research may result in significant differences in
the resulting gear fundamental properties. The proposed research here is based on the use
of equation (1.1).

1.7 Layout of the current study

The objectives of this research were the use of numerical approaches to develop theoretical
models of the behaviour of spur gears in mesh.

This thesis consists of 8 chapters and the outline of each chapter is given below:

Chapter 1 This chapter presents a general introduction and describes the significance of
the research work. The concept of the combined torsional mesh stiffness and its
development from previous research (Sirichai 1999) is presented. The definitions of total
transmission error and gear transmission error along with the major differences are
discussed. The components of total transmission error were re-classified with this research.
Finally, the objectives to be achieved and the layout of the study is described.

Chapter 2 Presents a historical review of the study of the literature related to the research.
It contains a significant presentation of relevant and pertinent publications on the subject of
analysis and documents the vast amount of literature on gear mathematical models and
measurements for vibration analysis and noise control.

Chapter 3 Presents some of the basic considerations required for applying finite element
methods to gear modelling. The brief history, development and theory of finite element
analysis are also presented. In particular, the non-linear problems that relate to the solution
convergence and the application of adaptive re-meshing with the contact(s) using quad or
brick elements are discussed.

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Chapter 4 Introduces the involute and fillet tooth profile equations. The equations were
used to generate the profile of the teeth by an APDL (ANSYS Parametric Design
Language) program, so that the CAD/FEA method can be applied to avoid possible
geometry data loss. The stiffness of parts of the basic transmission unit (except the
bearings) were studied. In particular, the variations of the distortion field in a meshing gear
pair were analysed with various boundary conditions. The ratio of local deformation was
defined. The characteristics of local contact deformation with reference to the global
deformation and the transmission error are also outlined.

Chapter 5 A wide range of FEA problems are often solved with 2D assumptions, and one
of the reasons is the computational efficiency and cost. However, when the numerical
analysis involves non-linear factors such as contact, fracture or other extreme load cases,
the 2D assumptions can be restricted in a very narrowed range. A large amount of FEA
calculations were made in this chapter. The comparisons were concentrated on the
torsional stiffness, first maximum principal stress and stress intensity factors that are
obtained under assumptions of plane stress, plane strain and 3D analysis. All models were
considered with thickness variations from 5mm to 300mm. With the conclusion of this
research, errors were found in the literature of previous research studies, especially when
2D assumptions were used with solid gears.

Chapter 6 The combined torsional mesh stiffness was generated with three major
different considerations. First, the individual torsional stiffness was generated with the
models of single and multiple tooth gears mating with the rigid line. Using the later model,
the stiffness can be obtained over a complete mesh cycle, and it can also be derived from
the results of the first model. The individual torsional stiffness can further be used to
generate the combined torsional mesh stiffness. Single tooth gear models with flexible
contact were then analysed, and the results used to generate the combined torsional mesh
stiffness. It has been noted that similar models were used by previous researchers (Kuang
1992; Arafa 1999; Wang 2000) to generate the combined torsional mesh stiffness over a
complete mesh cycle. However, errors were found in the previous methods and a better
solution has been proposed and used in this research. Finally, the FEA of multiple tooth

7
gear models with flexible contact was carried out, involving the use of adaptive mesh with
contact (both 2D and 3D). Solution of the combined torsional mesh stiffness for involute
gears in mesh was given, along with the definition of the handover region and its
characteristics.

Chapter 7 Detailed analysis of static T.E. of spur gears in mesh including tooth profile
modification is presented as the major part of this chapter.

Firstly, the detailed hand over process of involute gears in mesh is analysed over a
complete mesh cycle. Results of most mechanical properties are given. Except for the
static T.E., the results include the combined torsional mesh stiffness, ratio of local
deformation and load sharing ratio, so that more precise details about the hand over region
of recess and approach cases can be obtained.

Secondly, the detailed analysis with tooth profile modifications is given, in which the
history of the research on the topic, current recommendations (standards) and the popular
forms of the modification curves are introduced. The development of loaded TE o.p.c. is
presented and its relationship with the amount of (tip) relief Ca is given by an approximate
formula. Furthermore, the proposed tooth modification curve used in this research is
introduced, and then analysis carried out on the short and the long tooth profile
modifications. Some important results such as the relief starting point S1 and S2 are given.
The characteristics of the static T.E. with centre distance variations and with a tooth root
crack are also presented.

Finally, the results of the gears with a new tooth profile are presented.

Chapter 8 Presents the non-linear FE modelling of the standard involute spur gears in
mesh, in which the polyamide (PA 6) material behaviours are simulated with ANSYS
MELAS option, so that the analyses results of T.E., combined torsional mesh stiffness and
load sharing ratio are expressed as the function of the input load at each temperature as
well as a function of temperature when the input load remains stationary. Meanwhile, the
detailed variations of the contact ratio and the handover regions are also presented. A

8
detailed analysis concerning the load-sharing ratio is presented and a difficulty with the
standard theoretical solution is outlined. It also briefly introduced the analysis of optimal
tipping on non-metallic gears, and the tipping analysis was also involved with the test
gears.

Detailed experimental work is an important aspect of this chapter. The tooth profile
inspections and the load – deflection measurements on a static gear test rig are presented.
The experimental results are not only for the nylon gears but previous tests on the
aluminium gears are also discussed.

Finally, the related error analysis is given which indicates several possibilities for
improving the accuracy of both numerical and experimental results.

Appendix Various analysis of metallic HCRG (High Contact Ratio Gears) and the 30o
pressure angle nylon gears are discussed for future reference. Some customised ANSYS
APDL programs are also presented.

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2.1 Overview

Gears are one of the most critical components in industrial rotating machinery. There is a
vast amount of literature on gear modelling. The objectives in dynamic modelling of gears
has varied from vibration analysis and noise control, to transmission errors and stability
analysis over at least the past five decades. The ultimate goals in gear modelling may be
summarised as the study of the following,

Stress analysis such as bending and contact stresses,

Reduction of surface pitting and scoring,
Transmission efficiency,
Radiated noise,
Loads on the other machine elements of the system especially on bearings and their
stability regions,
Natural frequencies of the system,
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Vibratory motion of the system,
Whirling of rotors,
Reliability, and fatigue life.

The models proposed by several investigators show considerable variations not only in the
effects included, but also in the basic assumptions made. Although it is quite difficult to
group the mathematical models developed in gear dynamics, Ozguven and Houser
(Ozguven 1988; Ozguven 1988) have presented a thorough classification of gear dynamic
mathematical models. In 1990, Houser (Houser 1990) and Zakrajsek et al. (Zakrajsek
1990) outlined the past and current research projects of gear dynamics and gear noise at
Ohio State University’s Research Laboratory and NASA Lewis Research Centre
respectively. Du (Du 1997) also classified various gear dynamic models into groups.

The current literature review also attempts to classify gear dynamic models into groupings
with particular relevance to the research presented in this thesis. It is possible for some
models to be considered in more than one grouping, and so the following classification
seems appropriate.

· Models with Tooth Compliance. There are a very large number of studies that
include the tooth stiffness as the only potential energy storing element in the
system. This group includes single tooth models and tooth pair models. For
single tooth models, the objectives usually are tooth stress analysis. For the
models with a pair of teeth, the focuses mostly are contact stress and mesh
stiffness analysis. That is, the flexibility (torsional and/or transverse) of the
shafts, bearings, etc., is all neglected. In such studies the system is usually
modelled as a single degree of freedom spring-mass system. Some of the
models have also been analysed using the Finite Element Method.
· Models for Gear Dynamics. Such models include the flexibility of the other
elements as well as the tooth compliance. Of particular interest has been the
torsional flexibility of shafts and the lateral flexibility of the bearings and shafts
along the line of action. In some studies, the transverse vibrations of a gear-
carrying shaft are considered in two mutually perpendicular directions, thus

11
 allowing the shaft to whirl. In such models, the torsional vibration of the system
is usually considered.

· Models With A Whole Gearbox. The studies in this group may be viewed as
current and advanced studies and all elements in the system including the gear
casing, are considered in the models. The gearbox may be single stage or multi-
stage.

In the solution of the system equations, numerical techniques have usually been employed.
Although most of the models for which numerical techniques are used are lumped
parameter models, some investigators have introduced continuous system or finite element
models. While closed form solutions are given for some simple mathematical models,
numerical computer solutions have sometimes been preferred for non-linear and more
complicated models, particularly in the earlier studies.

In some studies the main objective has been to find the system natural frequencies and
mode shapes and, therefore, only free vibration analyses are made. However, usually the
dynamic response of the system is analysed for a defined excitation. In most of the studies
the response of the system to forcing due to gear errors and to parametric excitation due to
tooth stiffness variation during the tooth contact cycle is determined. The models
constructed to study the excitations due to gear errors and/or tooth stiffness variation
provide either a transient vibration analysis or a harmonic vibration analysis by first
determining the Fourier series coefficients of the excitation. Some studies also include the
non-linear effect caused by loss of tooth contact or by the friction between meshing teeth.
The excitation is then taken as an impact load and a transient vibration analysis is made.

2.2 Gear Modelling

Numerous mathematical models of gears have been developed for different purposes, the
basic characteristics of each class of dynamic models along with the objectives and
different parameters considered in modelling have been discussed in section 2.1. This

12
section presents a review of papers published in the areas outlined above, including brief
information about the models and the approximations and assumptions made.

2.2.1 Models with Tooth Compliance

The basic characteristic of the models in this group is that the only compliance considered
is due to the gear tooth and that all other elements have been assumed to be perfectly rigid.
The model is either a single tooth model or a tooth pair model. For single tooth models, the
objectives usually are tooth stress analysis. For models with a pair of teeth, the focus is
mostly contact stress and meshing stiffness analysis. The resulting models are either
translation or torsional. With torsional models one can study the torsional vibrations of
gears in mesh, whereas with translation models the tooth of a gear is considered as a
cantilever beam and one can study the forced vibrations of the teeth. In either of these
models, the transmission error excitation is simulated by a displacement excitation at the
gear mesh.

In 1956, Nakada and Utagawa (Nakada 1956) considered varying elasticity of the mating
teeth in their vibratory model. Introducing an equivalent translation vibratory system
simulated the torsional vibrations of two mating gears. The time variation of stiffness was
approximated as a rectangular wave and closed form solutions of piecewise linear
equations were obtained for different damping cases for accurately manufactured gear tooth
profiles. Another mass and equivalent spring model was introduced in 1957 by Zeman
(Zeman 1957). He neglected the variation of stiffness and analysed the transient effects of
periodic profile errors. Harris’s work (Harris 1958) was an important contribution in which
the importance of transmission error in gear trains was discussed and photo-elastic gear
models were used. In his single degree of freedom model, he considered three internal
sources of vibration: manufacturing errors, variation in the tooth stiffness and non-linearity
in tooth stiffness due to the loss of contact. He treated the excitation as periodic and
employed a graphical phase-plane technique for the solution. Harris seems to have been
the first to point out the importance of transmission error by showing that the behaviour of
spur gears at low speeds can be summarised in a set of static transmission error curves. He

13
also appears to have been the first to predict the dynamic instability due to parametric
excitation of the gear mesh.

In 1963, Gregory, et al. (Gregory 1963; Gregory 1963) extended the theoretical analysis of
Harris (Harris 1958) and made comparisons with experimental observations. The torsional
vibratory model of Gregory, et al., included a sinusoidal-type stiffness variation as an
approximation. They treated the excitation as periodic, and solved the equations of motion
analytically for zero damping and on an analogue computer for non-zero damping. The
experimental data (Gregory 1963) and the computational results (Gregory 1963) generally
confirmed Harris’s contention that non-linear effects are insignificant when damping is
more than about 0.07 of critical. It was claimed that when damping is heavy the simple
theory of damped linear motion could be used. Aida, et al. (Aida 1967; Aida 1968; Aida
1969) presented examples of other studies in this area. He modelled the vibration
characteristics of gears by considering the excitation terms due to tooth profile errors and
pitch errors, and by including the variation of teeth mesh stiffness. In the model of Aida, et
al., time varying mesh stiffness and periodic tooth errors were considered, and the model
was used for determining stability regions and steady state gear vibrations. A comparison
with experimental measurements was also made.

Rollinger and Harker (Rollinger 1967) investigated the dynamic instability that may arise
due to varying mesh stiffness. They used a simple single degree of freedom model with an
equivalent mass representing the inertia of the gear and pinion. Mesh stiffness variation
was assumed to be harmonic. The solution of the resulting equation of motion was
obtained by using an analogue computer, and it was shown that the dynamic load may be
reduced by increasing the damping between the gear teeth or by reducing the amount of
stiffness variation.

In 1967, Tordion and Geraldin (Tordian 1967) used an equivalent single degree of freedom
dynamic model to determine the transmission error from experimental measurements of
angular vibrations. They first constructed a torsional multi-degree of freedom model for a
general rotational system with a gear mesh. Then, only the equations of the gears were
considered for obtaining an equivalent single degree of freedom model with constant mesh

14
stiffness and a displacement excitation representing the transmission error. An analogue
computer solution was used to obtain the transmission error from the measured angular
accelerations. The transmission error was proposed to be used as a new concept for
determining the gear quality, rather than individual errors.

In 1973, Wallace and Seireg (Wallace 1973) used a finite element model of a single tooth
to analyse the stress, deformation and fracture in gear teeth when subjected to dynamic
loading. Impulsive loads applied at different points on the tooth surface and moving loads
normal to the tooth profile were studied. In the same year, Wilcox and Coleman (Wilcox
1973) also analysed gear tooth stresses. They developed a new accurate stress formula for
gear teeth based entirely on the finite element method and presented a comparison between
the new formula and the previous one.

In 1978, Remmers (Remmers 1978) presented a damped vibratory model in which the
transmission error of a spur gear was expressed as a Fourier series. He used viscous
damping and constant tooth pair stiffness, and considered the effects of spacing errors,
load, and design contact ratio and profile modifications.

Rebbechi and Crisp (Rebbechi 1981; Rebbechi 1983) considered the material damping of
the gear-wheel shafts, while the compliance of the shafts was neglected. The three-degree
of freedom model was reduced to a two degree of freedom model for the study of the
torsional vibrations of a gear pair, and an uncoupled equation, which gave the tooth
deflection. The other effects included in the model were material damping inherent to the
tooth, perturbations of input and output torque, arbitrary tooth profile errors, time variation
of that error due to deformation, and perturbations of the base circle due to profile errors.
The effects of kinetic sliding friction at the contact point and the sliding velocity on the
dynamics of continuous meshing were also studied (Rebbechi 1983) and in 1996,
Rebbechi, et al. (Rebbechi 1996), obtained measurements of the gear tooth dynamic
friction under various speed and load situations.

In 1985, Wang (Wang 1985) studied the effect of torsional vibration in his model. The
research was focused on the analytical evaluation of gear dynamic factors based on rigid

15
body dynamics and discussed different cases in which the transmission errors have
different effects on the dynamic load. He commented that the transmission errors have a
system wide effect and could be used to analyse rigid-body vibrating gear systems in which
the gear deflection is not considered.

In the late 1980s, Ramamurti and Rao (Ramamurti 1988) presented a new approach to the
stress analysis of spur gear teeth using FEM. Their new approach, with a cyclic system of
gear teeth and with asymmetry of the load on the teeth, allowed computation of the stress
distribution in the adjacent teeth from the analysis of one tooth only. The boundary
conditions imposed between the two adjacent teeth in the conventional FEM were avoided
in their approach.

In 1988, Vijayakar, Busby, and Houser (Vijayakar 1988) used a simplex type algorithm to
impose frictional contact conditions on finite element models. They established the contact
equations with the frictional factor and solved them for known output moment load on the
output gear. In their finite element model, they analysed their problem in two dimensions
and in order to model the involute profile as closely as possible, a special five node linear
transition element was used. In the same year, Ozguven and Houser (Ozguven 1988)
presented a non-linear model of a single degree of freedom system for the dynamic analysis
of a gear pair. In their studies, they developed two methods for calculating the dynamic
mesh and tooth forces, dynamic factors based on stresses, and dynamic transmission error
from measured or calculated loaded static transmission errors. The first method was an
accurate method, which included the time variations of both mesh stiffness and damping.
The second approach was a more approximate method in which the time average of the
mesh stiffness was used.

In 1990, Sundarajan and Young (Sundarajan 1990) developed a three dimensional finite
element substructure method to improve the accuracy of calculation of the gear tooth
contact and fillet stress in large spur and helical gear systems. The finite element analysis
and pre-processing software they developed simplified the data input and reduced the
manual effort involved in the analysis. When some parameters (misalignment for example)
were changed, most of the stiffness matrices were not recalculated. They considered the

16
contact problem by using contact boundary conditions, which meant that the contact or area
was defined in the analysis. One year later, Sundarajan and Amin (Sundarajan 1991)
investigated the finite element analysis of a ring gear and the casing and presented another
finite element computer program to solve this problem.

The contact conditions of gear teeth are very sensitive to the geometry of the contacting
surfaces, which means that the finite element mesh near the contact zone needs to be very
highly refined. However it is not recommended to have a fine mesh everywhere in the
model, in order to reduce the computational requirements. Vijayakar and Houser
(Vijayakar 1993) studied the contact analysis of gears using a combined finite element and
surface integral method. They developed a Contact Analysis Program Package which
supports stress contours, transmission errors, contact pressure distribution and load
distribution calculation. Their approach was based on the assumption that beyond a certain
distance from the contact zone, the finite element method accurately predicted
deformations and the elastic half space method was accurate in predicting relative
displacements of points near the contact zone. Under these assumptions, it was possible to
make predictions of surface displacements that make use of the advantages of both the
finite element method as well as the surface integral approach.

In 1994, a review of the current contact stress and deformation formulations compared to
finite element analysis was given by Gosselin, et al. (Gosselin 1994). They presented an
original approach to meshing line contact for spur gears and point contact for spiral bevel
gear pairs using finite element analysis with contact elements and they then compared the
contact deformation results with recognised analytical formulations. Their results showed
that the contact deformations differ from 20% to 150% between the analytical approaches
and FEM. In the same year, Chen, Litvin, and Shabana (Chen 1994) proposed an approach
for the computerised simulation of mesh and contact of loaded gear drives that enables
determination of the instantaneous contact ellipse, the contact force distributed over the
contact ellipse and the real contact ratio. They also established a finite element model for
the maximum bending stress calculation on a tooth. The friction forces between gear teeth,
the elastic deflection of the body of the gear, the shaft and the bearings were neglected in
their approach and their model.

17
2.2.2 Models for Gear Dynamics

Some of the early mathematical models, in which the stiffness and mass contribution of the
shafts carrying the gears in mesh were ignored, showed good agreement with the
experimental measurements. However, it was realized in the late 1960s and early 1970s
that dynamic models in which the shaft and bearing flexibility were considered would be
necessary for more general models. Unless the stiffness of these elements were relatively
high or low compared to the effective mesh stiffness, the vibration coupling of different
elements cannot be neglected. In general, a high degree of correlation was obtained
between the experimental results and the predictions provided by many of the early single
degree of freedom models. This can be explained by the fact that the experimental rigs
used in such studies satisfied most of the basic assumptions made in the mathematical
model. For example, a very short shaft might be assumed to be rigidly mounted in the
transverse direction. In practical applications however, these assumptions may not always
be satisfied and so one then needs more general models in which the flexibility and mass of
the other elements are considered as well.

The models that could be considered in this group are either torsional models, in which
only the torsional stiffness of the gear-carrying shafts is included, or torsional and
translation models, in which both the torsional and transverse flexibility of the gear-
carrying shafts are considered.

In the early 1960s, Johnson (Johnson 1962) used a receptance coupling technique to
calculate the natural frequencies from the receptance equation obtained by first separately
finding the receptances at the meshing point of each of a pair of general shafts. In the
model, the varying mesh stiffness was replaced by a constant stiffness equal to the mean
value of the varying stiffness and thus a linear system was obtained. His work was one of
the first attempts to use mesh stiffness in coupling the torsional vibration of gear shafts.
Mahalingam (Mahalingam 1968) presented a similar model in 1968, where the formulae
for support receptance at a gear-wheel bearing was developed and then used to study the
effects of gearbox and frame flexibility on the torsional vibration.

18
An important contribution in this area came in 1970 from Kobler, Pratt and Thomson
(Kobler 1970) who concluded from their experimental results that dynamic loads and noise
result primarily from the steady state vibration of the gear system when forced by
transmission errors. They developed a six-degree of freedom dynamic model with four
torsional degrees of freedom and one lateral degree of freedom in the direction of the tooth
force on each shaft. They assumed the tooth mesh stiffness to be constant in their model
and the spectrum analysis of the static transmission error for the single-stage reduction gear
unit used was also given. In 1971, Kasuba (Kasuba 1971) used one and two degree of
freedom models based on his previous work (Kasuba 1961), to determine dynamic load
factors for gears that were heavily loaded. He used a torsional vibratory model, which
considered the torsional stiffness of the shaft. He also argued that the rigidity of the
connection shafts was much lower than the rigidity of the gear teeth in meshing, and then
decoupled the meshing system. The tooth error in mesh was represented by a pure sine
function having the frequency of tooth meshing. In his model the tooth meshing stiffness
was time varying.

In 1972, Wang and Morse (Wang 1972) constructed a torsional model including shaft and
gear web stiffness as well as a constant mesh stiffness. The model was represented by a
spring mass system having many degrees of freedom. The transfer matrix technique was
applied to give the static and dynamic torsional response of a general gear train system. It
was found that the torsional natural frequencies and mode shapes determined from a free
vibration analysis correlated with experimental results at low frequencies. Later, Wang
(Wang 1974) extended this work to the linear and non-linear transient analysis of complex
torsional gear train systems. In this later model he considered the variation of tooth
stiffness, and included gear tooth backlash, linear and non-linear damping elements and
multi-shock loading. Three different numerical methods that can be used in the solution of
non-linear systems that cannot be approximated piecewise linearly were also briefly
discussed in his work.

In the 1980s more and more complicated models were developed in order to include
several other effects and to obtain more accurate predictions, while some simple models
were still developed for the purpose of simplifying dynamic load prediction for gear

19
standards. In 1980, Iida, et al. (Iida 1980) investigated the coupled torsional-flexural
vibration of a shaft in spur geared systems in which they assumed that the output shaft was
flexible in bending and the input shaft was rigid in bending. They derived equations of
motion for a 6-degree-of-freedom (DOF) system where the driving gear had a torsional
DOF while the driven gear had x, y and torsional DOF due to mass imbalance and
geometrical eccentricity. They assumed that the tooth contact was maintained during the
rotation and that the mesh was rigid. Four years later, Iida and Tamura (Iida 1984)
continued to study coupled torsional flexural vibration of geared shaft systems. In that
study, their model consists of three shafts, rather than two shafts, one of them being a
counter shaft.

Neriya, et al. in 1985 (Neriya 1985) also investigated the coupled torsional flexural
vibration of a geared shaft system due to imbalance and geometrical eccentricity. The
difference in the work with respect to Iida, et al. (Iida 1980) was that they used the finite
element method to solve their problem. In their model, there were 6 beam elements for
each of the driving and driven shafts that were coupled at the contact to account for the
tooth flexibility. Their model had 41 degrees of freedom. They solved the free vibration
problem to obtain the natural frequencies and mode shapes. The normal mode analysis was
then employed to obtain the dynamic response of the system under the excitations arising
from the mass imbalance and geometrical eccentricity in gears.

In early studies, the mesh stiffness of teeth was considered to be constant. Iwatsubo and
Kawai (Iwatsubo 1984) studied the coupled lateral and torsional vibrations of geared rotors,
considering mainly the effect of the periodic variation of mesh stiffness and a tooth profile
correction. Their model had two simply supported rotors with a spur gear at the centre of
each rotor. The stability condition of the system was analysed in their study. In the same
year, Iwatsubo, Arii and Kawai (Iwatsubo 1984) analysed the coupled lateral and torsional
vibration of the geared system constructed from a pair of spur gears using the transfer
matrix method. In their research, they considered three cases in the analysis of the free
vibration of the system: 1. The mesh force acting on the contact line was a function of the
rotation of each gear, 2. The mesh force acting on the contact line was a function of the

20
rotation and flexure at each gear, and 3. The system was not coupled by the gears. The
forced vibration caused by the mass imbalance of the gears was also calculated.

A new topic, the computer simulation of the torsional and flexural vibration in drive
systems, was studied by Laschet and Troeder (Laschet 1984). They developed computer
programs and applied simulation techniques to predict and analyse the performance of
gears trains. The distinctive feature of their research was that the backlash of the gears was
considered in their programs and CAD data of the gear geometry could be used in their
programs.

In 1985, Wang (Wang 1985) developed a torsional only vibration model. He focused on an
analytical evaluation of gear dynamic factors based on rigid body dynamics and discussed
different cases in which the transmission errors have different effects upon the dynamic
load. He commented that the transmission error had a system wide effect and could be
used to analyse rigid-body vibrating gear systems in which the gear deflection was not
considered.

Tavares and Prodonoff (Tavares 1986) proposed a new approach for torsional vibration
analysis of gear-branched propulsion systems in 1986. Idle gears in a gear-branched
system were modelled as part of the inertia of the master gear and the finite element
method was used in their approach. In the same year, Umezawa, et al. (Umezawa 1986) set
up a test gearing unit which consisted of an input shaft, countershaft and output shaft. The
gears were placed at arbitrary positions on the shafts in their unit so that the effect of the
countershaft on the bending vibration and on the sound radiation became clear. At almost
the same time, Iida, et al. (Iida 1986) studied a three axis gear system but with some
differences from Umezawa, et al., firstly, because the countershaft was on soft supports and
secondly, the model was a coupled torsional-lateral vibration analytical model.

In 1992, a finite element model of a geared rotor system on flexible bearings was
developed by Kahraman, et al. (Kahraman 1992). The coupling between the torsional and
transverse vibrations of the gears was considered in the model. They applied the
transmission error as excitation at the mesh point to simulate the variable mesh stiffness.

21
They presented three different geared systems as numerical examples and discussed the
effect of bearing compliance on gear dynamics. The assumptions they used were that the
gear mesh was modelled by a pair of rigid disks connected by a spring and a damper with
constant value which represented the average mesh values and tooth separation was not
considered.

Another model presented by Kahraman (Kahraman 1993) was a linear dynamic model of a
helical gear pair. The model considered the shaft and bearing flexibility and the dynamic
coupling among the transverse, torsional, axial, and rocking motions due to the gear mesh.
The natural frequencies and mode shapes were predicted and the forced response due to the
static transmission error was predicted. After the parametric study of the effect of the helix
angle on the free and forced vibration characteristics of a gear pair, the conclusion was
reached that the axial vibrations of a helical gear system could be neglected in predicting
the natural frequencies and the dynamic mesh forces. The assumption for their model was
that the gears were modelled as rigid disks, the clearances and stiffness changes of the
bearings were neglected, and the system was assumed to be symmetrical about the
transverse plane of the gears.

2.2.3 Models with a Whole Gearbox

The research models reviewed in this section are seen as being advanced because
traditional analysis approaches mentioned previously in the gear dynamic area have
concentrated on the internal rotating system and have excluded dynamic effects of the
casing and flexible mounts. The focus of this group is on the dynamic analysis of the
geared rotor system, which includes the gear pair, shafts, rolling element bearings, a motor,
a load, a casing and flexible or rigid mounts.

In 1991, Lim and Singh (Lim 1991) presented a detailed study of the vibration analysis of
complete gearboxes. Their research was based on previous studies including the bearing
stiffness formulation (Lim 1990) and system studies (Lim 1990). They developed linear
time-invariant, discrete dynamic models of an overall box by using lumped parameter and
dynamic finite element techniques. They studied three example cases: case I, a single-stage

22
rotor system with rigid casing and flexible mounts; case II, a spur gear drive system with
rigid casing and flexible mounts; and case III, a high-precision spur gear drive system with
flexible casing and rigid mounts. They used the gear mesh coupling stiffness matrix to
couple the two gears and used the bearing stiffness matrix to link the shafts and casing. In
their finite element model, the gear, pinion, motor and load were simulated as generalised
mass and inertia elements and the gear mesh stiffness matrix and bearing stiffness matrix
were modelled as six-dimensional generalised stiffness matrices. They used the FEM
software ANSYS to analyse their models. They made a parametric study of the effect of
casing mass and mount stiffness on the system natural frequencies. A comparison of the
casing flexural vibrations between the simulation and the experiment was presented.

Choy, et al. (Choy 1991) presented a vibration analysis with the effect of casing motion and
mass imbalance for a multi-stage gear transmission in 1991. In order to investigate the
effect of the casing motion and mass imbalance, four major cases of external excitations
were examined in their study. They employed the modal method to transform the
equations of motion into modal coordinates to solve the uncoupled system. They
concluded that the influence of the casing motion on system vibration was more
pronounced in a stiffer rotor system. In the same year, El-Saeidy Fawzi (El-Saeidy Fawzi
1991) presented an analytical model for simulating the effect of tooth backlash and ball
bearing dead band clearance on the vibration spectrum in a spur gearbox. The contact
between meshing teeth using the time-varying mesh stiffness and mesh-damping factor was
discussed. From their study, they concluded that the backlash and bearing dead band
clearance had a pronounced effect on the vibration spectrum of a gearbox. In this model,
the gearbox casing was assumed to be rigid, therefore, both ends of each shaft had the same
displacements. There was no experimental result to verify the analytical result of this
research.

One year later, Choy, et al. (Choy 1992) continued their study on the multi-stage gear
system. The work presented in that study was the development and application of a
combined approach of using the modal synthesis and finite element methods in analysing
the dynamics of multi-stage gear systems coupled with the gearbox structure or casing. In
their solution procedure, modal equations of motion were developed for each rotor-bearing-

23
gear stage using the transfer matrix method to evaluate the modal parameters, and the
modal characteristics of the gearbox structure were evaluated using a finite element model
in NASTRAN. The modal equations for each rotor stage and the gearbox structure were
coupled through the bearing supports and gear mesh.

After this study, they used their analytical model to predict the dynamic characteristics of a
gear noise rig at the NASA Lewis Research Centre and then used experimental results from
the test rig to verify the analytical model (Choy 1993). Their conclusions were that the
dynamics of the casing can be accurately modelled with a limited amount of analytically
predicted vibration modes, and that the characteristics and trends of the casing vibration
spectra predicted by the analytical model were very similar to those found in the
experimental data.

In 1992, Ong (Ong 1992) described the application of the eigenvalue economisation
method coupled with the frontal solution technique to a vehicle transmission system
comprising an integral bell-housing gearbox, extension housing, drive shaft and rear-axle
assembly. In this study, super-elements were applied using the structural dynamic sub
structuring technique for the finite element analysis. The pinion shafts were represented by
beam elements and the gears and bearings were represented by lumped mass elements in
the analytical model. Finally, a comparison was made between experimental results and
the finite element method, which showed good agreement.

Most analyses of gearboxes appear to be concerned with the dynamic response and
vibration characteristics. In 1994, Sabot and Peret-Liaudet (Sabot 1994) presented another
phase of study, noise analysis of gearboxes. They pointed out that a troublesome part of
the noise within the car or truck cab could be attributed to the gearbox and that this noise
was associated with the vibrations induced by the transmission error which gives rise to
dynamic loads on the teeth, shafts, bearings and casing. They computed the noise radiated
by the gearbox casing using the Rayleigh Integral Formulation in which the acceleration
response of the casing associated with the finite element method calculation was
considered. Their results showed that although the test model was a simplified gearbox,

24
their numerical analysis provided a better understanding of the sound radiation
characteristics of geared transmission systems.

At the same time, Kato, et al. (Kato 1994) developed a simulation method by integrating
finite element vibration analysis and boundary element acoustic analysis for the purpose of
evaluating the sound power radiated from the gearbox and achieved good agreement with
the experimental results. In their model, each shaft was modelled using beam elements and
the mass and rotating inertia of the gear was modelled as lumped masses and added to the
shaft. Each of the rolling element bearings was represented as a spring and damper and the
casing of the gearbox was modelled by a thin shell element in the finite element package
program ISAP-6. Their acoustic analysis in the frequency domain showed that the sound
power at the mesh frequency was greater than the sound power at other frequencies.

There have also been various studies aimed at modelling a single mechanical element, a
complex system containing several gears as well as other mechanical or electrical elements
and other special topics, as indicated by the following grouping;

Load Sharing Ratio (Chabert 1974; Remmers 1978; Drago 1979; Walford 1980; Bahgat
1983; Ozguven 1988; Kuang 1992; Liou 1992; Liou 1996; Litvin 1996; Zhang 1998).
Mesh Stiffness (Gargiulo 1980; Kahraman 1991; Kuang 1992; Brousseau 1994; Daniewicz
1994; Sirichai 1996; Velinsky 1996; Du 1998; Elkholdy 1998; Gosselin 1998; Nadolski
1998).
Bearing Stiffness (Childs 1980; Gargiulo 1980; Bahgat 1981; Smith 1987; Lim 1990; Lim
1990; Kahraman 1991; Lim 1991; Lim 1992).
Shaft In A Gear System (Johnson 1962; Iida 1980; Iida 1984; Neriya 1985; Iida 1986; Lin
1988; Prasad 1992; Zhong-Sheng 1993; Okamoto 1994).
Casing Analysis Of A Gearbox (Drago 1979; Randall 1980; Randall 1984; Mcfadden
1985; Tavares 1986; Houser 1990; Randall 1990; Zakrajsek 1990; Sundarajan 1991; Choy
1992; Inoue 1992; Maruyama 1992; Oswald 1992; Oswald 1992; Perret-Liaudet 1992;
Kato 1994; Kissling 1994; Oswald 1994; Sabot 1994; Zhang 1994; Zhang 1994; Du 1997).
Gear Transmission Error Measurement (Gregory 1963; Tordian 1967; Hayashi 1981;
Houser 1989; Rebbechi 1992; Vinayak 1992; Bard 1994; Barnett 1994; Munro 1994; Velex
1995; Houser 1996; Sweeney 1996).
25
Transmission System (Drago 1979; Ong 1992; Choy 1995; Choy 1996; Choy 1996;
Forrester 1996; Regalado 1998).
Computerized Design And Generation Of Gears (Litvin 1993; Litvin 1993; Blazakis 1994;
Zhang 1994; Litvin 1995; Litvin 1995; Litvin 1996; Litvin 1996; Seol 1996; Seol 1996;
Litvin 1997; Litvin 1998; Litvin 1998; Litvin 1998).
Gear Geometry (Chen 1992; Fujio 1992; Kin 1992; Kubo 1992; Tsai 1992; Donno 1998;
Dooner 1998; Feng 1998; Su 1998).
Gear Noise Design (Aida 1967; Aida 1968; Aida 1969; Kobler 1970; Daly 1979; Drago
1979; Welbourn 1979; Smith 1987; Baron 1988; Houser 1990; Randall 1990; Randall
1990; Zakrajsek 1990; Mark 1992; Maruyama 1992; Oswald 1992; Oswald 1992; Alattass
1994; Lewicki 1994; Oswald 1994; Sabot 1994; Tuma 1994; Zhang 1994; Zhang 1994;
Litvin 1995; Roosmalen 1995; Wang 1996; Cheng 1998; Stadtfeld 1998).
Stress Analysis (Arikan 1989; Arikan 1991; Arikan 1992; Moriwaki 1993; Rao 1993; Baret
1994; Daniewicz 1994; Vijayarangan 1994; Lu 1995; Refaat 1995; Litvin 1996; Kalluri
1998; Kim 1998; Richard 1998).
Modification of Gear Geometry (Walker 1938; Harris 1958; Niemann 1970; Tobe 1976;
Wang 1978; Kishor 1979; Cornell 1981; Kiyono 1981; Terauchi 1981; Lin 1989a; Rosinski
1992; Li 1994; Munro 1994; Palmer 1995; Walton 1995; Walton 1998).
Non-Metallic Gears (Yelle 1981; Terashima 1986; Tsukamoto 1986; Janover 1989;
Walton 1989; Enzmann 1990; Tsukamoto 1990; Breeds 1991; Kral 1991; Baumgart 1992;
Kudinov 1992; Walton 1992; Zhang 1992; Mao 1993; Nabi 1993; Solaro 1993; Tessema
1993; Tessema 1994; Walton 1994; Koffi 1995; Tessema 1995; Walton 1995; Williams
1995; Du 1997; Nitu 1997; Smith 1997; White 1997; Kleiss 2000; Kurokawa 2000;
Luscher 2000; Panhuizen 2000; Bushimata 2001; Wright 2001; Alagoz 2002; Andrei
2002).
High Contact Ratio Gears (Cornell 1978; Townsend 1979; Rosen 1982; Elkholdy 1985;
Barnett 1988; Lee 1991; Lin 1993; Yildirim 1994; Yildirim 1999; Yildirim 1999)

26
3.1 INTRODUCTION

3.1.1 Chapter Overview

This chapter presents some of the basic considerations required for applying finite element
methods to gear modelling in this research. Some features may exceed the capability of the
currently available software resources but the author believes the considerations are
necessary for the research to be continued in the future.

3.1.2 Brief History of Finite Element Analysis

Structural mechanics has been applied in the building industry for the past one hundred
years (Hoff 1956). During that time, the stiffness matrix of bar and beam elements were
derived from the theory of elementary strength of materials. With the development of the
direct stiffness method, the global matrix associated with load and boundary condition was

27
formed. Courant (Courant 1943) in 1943 used an assemblage of triangular elements and
the principle of minimum potential energy to study the St. Venant torsion problem in which
he suggested piecewise polynomial interpolation over triangular sub regions as a way to
obtain approximate numerical solutions. He recognized this approach as a Rayleigh-Ritz
solution of a variation problem. This was the basis of the finite element method, as we
know it today.

With the availability of the digital computer in 1953, stiffness equations in matrix notation
could be solved (Levy 1953). In 1960, the name finite element method was introduced by
Clough (Clough 1960). Soon after that, the method was recognized as a general method of
solution for partial differential equations with its efficacy.

In 1965, Zienkiewicz and Cheung (Cheung 1965) reported that the finite element method
was applicable to all field problems that can be cast into variation form so that the method
received an even broader interpretation. After development for two decades, the
acceptance of the method was worldwide.

Today there are more software options. More recently, finite element packages have been
extended to include non-linear static stress, dynamic stress (vibration), fluid flow, heat
transfer, electrostatics, and FEA based stress and motion analysis capabilities. These
capabilities are frequently combined to perform analyses that consider multiple physical
phenomena, tightly integrated within a CAD interface.

3.2 Energy and Displacement

3.2.1 The Fundamental Equation

In most cases, the finite element method uses the total potential energy of the system
instead of a differential equation to solve engineering analysis problems. The approach
uses the stationary principle of total potential energy which states that among all the
permissible configurations, the actual configuration (u) will make the total potential energy
V(u) stationary, or, mathematically,

28
 d V(u) = 0. (3.1)
In a static analysis,
V(u) = U(u) – W(u), (3.2)
where U is the internal energy which is equal to the strain energy, given by,

òe ×C ×e dW ,
T
U= 1
2 (3.3)

and the external energy W is equal to the work done by an external load q,
W = ò qu d W . (3.4)

Here, e is the strain vector, C is the constitutive matrix and d W is the infinitesimal
volume. If the system is discretized into elements, then we can write,
U= åU
e
e and W= åW
e
e , (3.5)

òe × C e × e e d W e and We = ò q e u e d W e .
T
where, Ue = 1
2 e (3.6)

The nodal displacement vector, u e can also be written as,

ue = H × u e , (3.7)

where H is the shape function (or interpolation function). The strain vector can then be
obtained by direct differentiation to be,
e e = ¶u e =( ¶H ) . u e =B .
ue, (3.8)

where ¶ is an appropriate differential operator and B is the strain-displacement matrix.

Hence, we can write,

U e = 12 u Te × ( B T ×C × B d W e ) × u e = 12 u Te × Ke × ue, (3.8)

and We = ò q e H × u e d W e = f e × u e , (3.9)

where K e is the element stiffness matrix, and f e is the element equivalent force. They

become,

Ke= B T ×C × B d W e ,

and f e = ò qe H d W e . (3.10)

If it is supposed that the element displacement vector u e is related to the system nodal

displacement vector u through the transformation T e , then

29
 ue = Te×u . (3.11)

Equations 3.2, 3.5, 3.8 and 3.9 then give,

V(u) = å (Ue
e - We ) ,

= å 1
2 u Te × Ke × ue - å f × ue,
e e

So that we have V(u) = 1

2 u× (å T Te K e T e ) × u – ( å f e × Te) × u .
e e

The variation principle (3.1) gives,

d V(u) = d u T × (å T Te K e T e ) × u – ( å f e × T e ) × u = 0,
e e

or, written in standard form,

K × u = f, (3.12)

where K= å e
T Te K e T e and f =å fe×Te ,
e

are the system stiffness and the force vector respectively. Equation 3.12 cannot be solved
for system nodal displacement vector u for given load vector f until it has been modified to
account for the boundary conditions. In general elasticity problems, the displacement field
or the set of nodal displacements cannot be found unless enough nodal displacements are
fixed to prevent the structure from moving as a rigid body when external loads are applied.

3.2.2 The Major Limitation On Gear Transmission Error Study

For a gear set, the major components of total transmission error are the tooth geometry
error, imperfect mounting and elastic deformation of the meshing gear pair. In most cases,
for a precision gear set, the elastic deformation is the main component of the transmission
error and the determination of this component is one of the main tasks in the proposed
research. However, other components in some situations can also be significant for the
total transmission error. One of the examples is a meshing gear pair with tip-relieved teeth.
When the applied load is relatively small, the “rigid body motion” (rotation) will be the
main component of the total transmission error. In such a case, trying to use the FEA
program to determine the total displacement (total angular rotation, which includes the
initial rigid body motion and the final elastic deformation) will be very difficult. Contact
30
analysis with initial un-connected bodies is a very difficult problem as the stiffness matrix
K in the fundamental equation 3.12 becomes singular and unsolvable. As mentioned in the
last section, finite elements require at least some stiffness connecting all the elements
together along with sufficient displacement constraints to prevent rigid-body motion.

In ANSYS®, with this case, the program will issue pivot ratio warnings. The program will
continue to try to solve the problem, but it is likely to get either a negative main diagonal
message or a DOF limit exceeded message.
ANSYS® offers the following options to overcome the difficulty in convergence.
1. Build the models in the just-touching position. This option seems the easiest
solution, but it requires the user to know what the position is. If the surfaces on
both bodies are curved or irregular (caused by surface damage, plasticity), this
certainly is difficult.
2. Use imposed displacements to move it into position. Again, the “free” body is
moved into its starting position using a specified displacement value. If the actual
initial contacting positions are not previously determined in advance, this also
causes difficulty.
3. Use weak springs to connect the bodies. This is a interesting option, and is worth a
try in some cases. In this technique, the user would add springs to connect the two
bodies. The suggested spring stiffness would typically be 6 to 8 orders of magnitude
weaker than the contact stiffness. However, care must be taken when very flexible
elements are connected to very rigid elements, as the finite element solution can
render poor solutions for displacement due to round off error, or matrix ill-
conditioning. Ill-conditioning errors typically manifest themselves during the
solution phase of an analysis. A detailed example of this has been given by Lepi
(Lepi 1998).
4. Solve the problem dynamically. In a mesh cycle of a meshing gear pair, if part of
the solution is obtained by static (no gap for contact) analysis and the other part by
dynamic analysis, then the overall solution wouldn’t be so good for the study of
gear transmission error.
The state above is not peculiar to the application of the ANSYS® software. This capability
is missing from many standard FEA programs (Dvorak 1999), and currently is the major

31
limitation for gear transmission error study using finite element analysis, especially with
limited software resources.

A recently developed analysis method that is likely to be one of the keys for solving this
problem is called The Precise Solid Method (PSM) from Precision Analysis Int., and it
promises to improve the relationship between CAD and analysis technologies. The feature
of mesh-less analysis and highly accurate solutions on solids was introduced in the paper
by Kurowski (Kurowski 2000). Some other interesting features were presented by Dvorak
(Dvorak 1999), where he explained that the new method tolerates point boundary
conditions that do not produce stress at the point. Such “dummy” constraints only restrain
motion. The software is also able to display rigid body motion at the beginning of the
solution. However, contact and non-linear analysis is expected in the future. PSM
theoretically can simulate any kind of phenomena that is described with differential
equations as boundary value problems (Kurowski 2000).

3.3 Methods for Accurate FEA Solution

3.3.1 h- and p- Method

The h- method for FEA entered the engineering world in the 1970s, and is a common
approach for solving analysis problems. This method involves the use of lower order
displacement assumptions (linear or quadratic) and increases the number of elements in
areas where the displacement is expected to be a non-linear function. The error between
the exact displacement and the finite element solution decreases with an increasing number
of smaller elements in the chosen areas. Theoretically, when the number of elements used
in a finite element model approaches infinity, the error between the finite element solution
for displacement and the exact displacement is expected to approach zero. (The error
between predicted and exact energy is also expected to approach zero.)

The p-method dealt with the more complete design and was introduced in the early 1990s.
To calculate displacements, the p-method manipulates the polynomial level (p-level) of the
finite element shape functions, which are used to approximate the real solution so that the

32
results obtained are to a user-specified degree of accuracy. The p-method can improve the
results for any mesh automatically. Usually, the elements are mid-noded. In ANSYS® the
range within which the p-level may vary can be controlled locally through the p-element
KEYOPT settings (KEYOPT(1) and KEYOPT(2)) or globally across the entire model with
the command PPRANGE. By default, the p-level range is from 2 to 8. More detailed
discussion about the p-method can be found in Strang (Strang 1973), Babuska (Babuska
1988), and SzabO (SzabÒ 1991).

The p-method solution option offers many benefits over the traditional h-method. The
most convenient benefit is in the ability to obtain good results to a desired level of accuracy
without rigorous user-defined meshing controls. In addition, the p-method adaptive
refinement procedure offers error estimates that are more precise than those of the h-
method, and can be calculated locally as well as globally (stress at a point rather than strain
energy). For example, if a high accuracy solution at a point is needed, such as for fracture
or fatigue assessments, the p-method offers an excellent means for obtaining these results
to the required accuracy.

Modern FEA software can combine the h- and p-adaptive approaches in a third method
called the hp-method or the hp-adaptive solution. Mixing the two technologies into a
single model lets users construct most structures with h-elements to obtain global responses
while areas of particular interest can be modelled with p-elements to improve solution
accuracy. The p-elements tolerate distortions and shapes that render standard or h-elements
unreliable.

3.3.2 Methods for Non-linear Contact Analysis

Finite element analysis of meshing gear pairs will be subject to non-linear contact analysis.
In this situation, the contact is highly non-linear because one or both of the following are
unknown:
· The actual regions of contact are unknown until the problem has been
solved. Depending on the load, material, boundary conditions and other

33
 factors, and surfaces can come into and go out of contact with each other
in a largely unpredictable and abrupt manner.
· Most contact problems need to account for friction. There are several
friction laws and models to choose from, and all are non-linear. Frictional
response can be chaotic, making solution convergence difficult.
From a mathematical point of view, both the system stiffness matrix K and load vector f in
equation 3.12 may be functions of nodal displacement. This requires significant computer
resources to solve. For example, when the (traditional) Newton-Raphson method is
employed to solve such non-linear equations, the analysis involves using a series of linear
approximations with corrections. Each linear approximation requires a separate pass, or
iteration, through the program’s linear equation solver. Each new iteration is about as
expensive as a single linear-analysis solution. (In ANSYS®, the iteration shows up as
equilibrium iteration.)

There are many papers on the application of the h-method to the solution of contact
problems. A comprehensive review can be found by Zhong (Zhong 1992) including many
references. When the h-method is used in contact problems for stress analysis, in most
cases, the mesh is not sufficiently refined to indicate the presence of singularities in the
boundary points of the contact zone (Paczelt 1999). The use of the p-method for the
solution of contact is relatively recent (Lee 1993; Gabbert 1994). In this case, the analysis
is typically high enough (shape function orders) for the singularities to indicate oscillations
in the numerical solutions. In the paper of Lee and Oden (Lee 1993), the hp-method was
used which produced a fine mesh at the contact zone with p=2. In the paper of Gabbert
(Gabbert 1994), special “pNh elements” were used in the contact zone. These elements
incorporate piecewise linear approximations on the contact edges or surfaces which are
blended with the standard basis functions of the p-version elements. However, in both
cases, there are stress discontinuities between the h- and p- elements or in the hNp-
elements. In the paper by Paczelt (Paczelt 1999), a special form of the hp-method was used
that combines the p-extension with a minor iterative modification of the mesh. A very high
accuracy solution was achieved, but the relative model didn’t account for friction.

34
As presented above, for the stress analysis, the p- or hp-method is desired. The h-method,
however, is relatively easy to use and all standard FEA programs support the method. On
the other hand for example, ANSYS®, does not by default support the p- or hp-method in
contact analysis. In choosing the analysis method one also has to consider the type of
gradient output that is desired. If one is analysing a structure for maximum displacement, a
somewhat coarse mesh by h-method might be sufficient. More details of this will be
discussed in the next section.

3.4 Rate of Convergence

3.4.1 The Rate of Displacement and Stress

The displacement based FEA approach is currently the most common, and the term finite
element method is typically used without the modifier displacement based. (There are other
means of developing a finite element solution like the hybrid method that uses both an
assumed displacement field and an assumed stress field. Only the displacement based FEA
is used in this research). The primary unknown in the finite element method is the nodal
displacement. The finite element method generates an approximate solution for
displacement, and then typically uses the displacement to calculate approximate values of
stress and strain that may be called derivative values as secondary unknowns. At any given
point in a structure, an FEA predicted value of displacement could be either too large or too
small, highlighting one of the difficulties in evaluation of convergence. In light of this,
proofs of h-convergence are often stated in terms of norms. Using the norm, a formal proof
of convergence shows that the total potential energy computed by FEA converges to the
exact value as the size of the largest element dimension approaches zero. Convergence in
displacement is anticipated with convergence in energy, but the convergence rates of these
two matrices are not normally the same.

The convergence of displacement (Lepi 1998) can be expressed as,

e2 h2 p +1
displacement convergence º e1 =O h1 , (3.13)

The equation (3.13) states that convergence is defined in terms of the error associated with
the two solutions, e 1 and e 2 . The relative error is a function of the largest element size h 1
35
and h 2 in each of the models, and the order of the interpolation function p. O denotes “of
the order”. Stress convergence may be computed using a similar equation,
h2 p +1- r
stress convergence = O h1 , (3.14)

where r º order of derivative,

and the convergence of stress is controlled by both the order of the interpolation
polynomial and the highest order of the derivative(s) used to define the strain. It should be
noted that the stress is not expected to converge as quickly as the displacement for a given
degree of interpolation. The practical application is that if one is analysing a structure for
maximum displacement, a somewhat coarse mesh might be sufficient, whereas a more
refined mesh would be required for accurate stress analysis (Lepi 1998).

3.4.2 Comparison

The typical component of the basic transmission unit is a pair of meshing gears. A series
of FEM (2D) calculations on a pair of meshing gears is presented here. The results were
used to examine the actual convergence rate of the displacement (angular rotation) and the
stresses (SMX, s 1 max ). Figure 3.4.1 shows a pair of standard involute spur gears in contact

at the pitch point (material: T6 aluminium). While one of the gears is restrained at the hub,
the torque load T is applied on the other gear’s hub. The displacement q on the pinion hub
has a unique value because a coupled equation was used to enable the hub to be rigid in
rotation.

T
q

Figure 3.4. 1 A pair of meshing gears.

Two critical stresses (SMX, s 1 max ) were chosen to be calculated, as follows,

36
 · SMX --- the maximum von Mises stress should occur at the contact point. The von
Mises criterion is best applied (and best understood) when used to predict the onset
of yielding in a structure where the material behaves in a ductile fashion.
· s 1 max (S 1 max ) --- the maximum (principal) tensile stress that is near the root of the
teeth. The maximum principal stress theory suggests that the largest principal
tensile stress is responsible for cleavage associated with (brittle) fracture.
Table 3.1 shows the results from an initial coarse mesh which has only 1802 nodes,
with the refinement gradually increasing the mesh density until the final model (Model
18) has reached 17466 nodes. The displacement and stress values with their positions
for each refinement are also listed.

Free mesh refine at

contact tooth
Nodes
No. UY( q *R) mm ´ 10-3 SMX at Contact (M Pa) s 1 max (M Pa) s 1 max Position
Model1 1802 5.760 51.892 25.767 near root
Model2 2315 5.810 74.592 28.473 near root
Model3 2779 5.954 99.238 30.108 near root
Model4 3392 6.028 143.028 43.667 near contact
Model5 4021 6.106 147.477 34.046 near contact
Model6 4691 6.115 178.136 47.681 near contact
Model7 4986 6.152 177.194 39.344 near contact
Model8 5393 6.207 221.486 51.052 near contact
Model9 6049 6.157 195.002 46.008 near contact
Model10 6385 6.193 207.859 44.708 near contact
Model11 7035 6.240 240.872 55.034 near contact
Model12 7604 6.190 219.308 37.915 near contact
Model13 8241 6.159 231.132 49.985 near contact
Model14 8710 6.194 256.647 34.392 near root
Model15 9417 6.190 236.885 38.184 near contact
Model16 10115 6.170 219.744 37.066 near contact
Model17 12755 6.171 220.416 37.461 near contact

Model18 17466 6.177 250.729 34.763 near root

Table 3.1. FEA calculation results.

Figure 3.4.2 and 3.4.3 clearly show that when the mesh density is just over 5000 nodes, the
displacement has almost certainly converged. The stress at the contact (maximum Van
Mises stress) does not appear to have converged even when the mesh density has reached
8000 nodes.

37
 300

250

200

M Pa
150 SMX at Contact(MPa)

100 Sigma1(MPa)

50

0
0 5000 10000 15000 20000

Mesh Density (No. of nodes)

Figure 3.4. 2 The stress convergence

0.0063

0.0062
UY(θ*R) mm

0.0061

0.006

0.0059 UY

0.0058

0.0057
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000
Mesh Density (No. of nodes)

Figure 3.4. 3 The displacement convergence

Note that s 1 max reaches its highest value close to the largest displacement value, however,

s 1 max jumps to the position near the contact area with increasing mesh density at the contact
tooth, except when some extra refinement was applied to the tooth root on Model14 and
Model 18. Further calculations were carried out with Model 6, with the refinements
applied to the area of the tooth root and the hub. There were two reasons for the further
calculations:
· To verify the position of s 1 max .

· To check the other influences (other than contact) on the displacement convergence.

The results in Table 3.2 show that a refinement on the tooth root (the main part) and the
hub is necessary for determination of the convergence of s 1 max , but that the relative

influence on the displacement convergence is minor.

Refine at roots and hub

Nodes q
UY( *R) mm SMX at Contact (M Pa) s 1 max (M Pa) s 1 max Position
No. (R-hub radia)
Model6 4691 0.006115 178.136 47.681 Near contact
Model6-1 6049 0.006157 195.771 46.36 Near root
Model6-2 6408 0.006157 195.96 46.569 Near root
Model6-3 6563 0.006156 195.871 46.352 Near root
Table 3.2. FEA results of further refinement for specify s 1 max position.

38
The primary requirement for building FEA models of meshing gear pairs has been obtained
from these results. The model below is one of the examples used in the research for
obtaining reliable converged displacement values.

Figure 3.4. 4 Finite element model of a meshing gear pair

Mapped Mesh Nodes No. UY ( q *R) mm SMX at Contact(M Pa) s 1 max at tooth root(M Pa)
2D-183PlaneStress 7600 0.006216 134.86 34.581
Table 3.3. Data for mapped mesh (Figure 3.4.4)

3.5 Modelling

The most time consuming step of finite element analysis is creating the finite element
model. After determining the type of analysis required and the characteristics of the
operating environment, a finite element model must be produced with appropriate analysis
parameters, such as loads, constraints, element choice and a suitable mesh.

3.5.1 CAD vs. FEA

First of all, the CAD/FEA interoperability has to be considered. Generally, there are three
methods:
1) The CAD universal file format method.
2) The “one window away” CAD/FEA method.
3) The “one window” CAD/FEA method.
The first method has been widely used. It requires the CAD solid model to be exported
with a neutral file format, such as IGES, ACIS or Parasolid. The neutral file is then
imported into the FEA system for set up and analysis. The major draw back with this
method is the loss of CAD geometry data that can happen during the translation of the

39
model. The second method is used in multi-processor FEA analysis or in the case when a
single interface is used for multiple CAD packages.

Nowadays, many FEA vendors build analysis capabilities into the CAD solid modeller.
The files no longer need to be translated so that geometry data isn’t lost. However, this
“one window” CAD/FEA method often encounters operating difficulties that are time
consuming to fix or impossible to use for building complex models because the FEA
providers often simplify their one-window versions due to space and interface limitations.
If the file translator schemes worked, it would be an ideal method for quickly building
geometry required for this research. For a reasonably effective analysis, no matter which
method above is used, the geometry must be simplified by deleting features (such as
keyways away from the area of interest) that will not significantly affect the desired results.
This “disfeaturing” process should result in a geometry entity that will define a relatively
easy-to solve mathematical model (when subjected to the loading and boundary
conditions).

3.5.2 Mesh with high order elements

Once the geometrical model is available, the next step in the analysis process is to create
the mesh. The use of an adequate finite element mesh is crucial. An adequate mesh
consists in using proper types of elements and applying a quality mesh (no excessive
distortion).

Both quadrilateral (2D) and hexahedral (3D) elements are better suited for solid elasticity
modelling. They are the most commonly used elements for non-linear analysis. For a
given mesh density with high order, the elements create edges that adapt more closely to
curved surfaces than similarly sized linear elements. They also produce better
mathematical formulations that result in the element being less sensitive to distortion and
poor orientation in the model due to complicated geometry.

However, in the past when contact boundary conditions were present, high-order elements
have been avoided for the reason that the mid-side nodes do not behave the same as the

40
corner nodes. The new 18X series second order elements in ANSYS® (since version 5.6),
provide even higher accuracy for meshes and are particularly suitable for modelling curved
boundaries in contact. They were successfully used in this research. On the down side,
high order elements require much greater computational resources because many of the
nodal DOF’s are wasted, and the problem takes more time to solve. One technique which
could be considered to deal with this involves using some linear elements, with far fewer
nodal DOF’s, in areas of constant stress, then applying the constraint equation on each edge
between the high order element and the linear element as shown in Figure 3.5.1.

8-node quad 4-node quad

constrained to remain co-

linear with shared nodes 1
and 3

Figure 3.5. 1 Joining an 8-node quadrilateral element to a 4-node element.

In MSC/MARC®, the “glue” option can be used to automatically apply this type of
constraint connection. However, the option of automatic constraint is not available for all
FEA packages, especially if the model is to be re-meshed several times. With many
thousands of elements, the specification of the constraint equations for each element pair
may prove tedious, while at the same time introducing another potential source of error.
For 3D modelling, eighteen strain states are needed to completely define linearly varying
strains when polynomial displacement assumptions are used. A 20-node hexahedron can
represent all 18 linear strain states, and in addition, 21 quadratic strain states. However,
due to the limitations of automatic mesh generation with hexahedron elements in ANSYS®,
especially when the geometry is complex, a high order tetrahedron element with 10 nodes
was considered. Despite some disadvantages commonly associated with the 10-node
tetrahedral element (Lepi 1998), it at least represents all the 18 strain states, and produces
results at least as good as the 8-node hexahedron.

It’s important to note that with the p-method, the 10-node tetrahedron forms system
matrices with smaller bandwidth than the 20-node hexahedron. This means that with the p-

41
method, the 10-node tetrahedron provides accuracy comparable to that of the 20-node
hexahedron with a faster solution (Niazy 1997).

A good mesh pattern is one that is as coarse as possible on uninteresting areas, yet as fine
as necessary for accurate results where it encounters contact or high stresses, (Thilo
Trantwein, managing director of ACES Ing GmbH in Stuttgart, Germany,
(www.acesgmbh.de)). There are three areas in a gear that have to be meshed with finer
elements. The first is near the region of contact where the mesh density in the area should
be the highest in the model as has been discussed in 3.4.2. The second area is near the root
of the tooth in contact, where the requirement on the mesh density here is at least fine
enough to show the correct s 1 max in the area. The third area is around the hub where only a

minor refined mesh is required. Over a mesh cycle, the contact from one pair of teeth
changes to two pairs (or vice-versa), which causes a constraint condition change on the
hub. Also, an adequate mesh density is required in the gear body underneath the teeth in
contact.

However, to produce an accurate result one also has to consider mesh quality. Using quad
mesh well-shaped elements, (no excessive distortion), is essential in mapped mesh
situations. Care must be taken when using mesh grading or transition mesh to prevent
numerical problems due to placing very stiff elements directly adjacent to very flexible
larger ones. The stiffness of an element K can be expressed as,

E
Kµ . (3.15)
V
Hence, the stiffness of an element is directly proportional to the elastic modulus and
inversely proportional to size. As recommended by Cook (Cook 1989), the element
stiffness K should not change by more than a factor of three across adjacent elements.

Under this guideline, the model in Figure 3.5.2 would be considered poor.

42
 Figure 3.5. 2 Example of poor mesh transition

Furthermore, considering the particular features of this research involving calculations on

more than 100 different contact points in a mesh cycle, creating a well-shaped boundary-
aligned mesh (on the tooth flank face) is vitally important for obtaining quality results.
One method for obtaining even higher accuracy results without considering manually
created well-shaped boundary-aligned mesh is to use the ANSYS® built in smart mesh
capability in contact areas. This can be used to obtain automatic adaptation re-mesh with
contact at every point in the mesh cycle, as illustrated in Figure 3.5.3.

3.5.3 Element Distortion

Finite element modelling of gears and gears in mesh with 2D assumptions of plane stress or
plane strain is relatively easy. With a few efforts, the requirement for producing all quads
well shaped elements with no excess distortion and well shaped-aligned boundary elements
in mesh can also be met. Combined with mapped meshing, automatic adaptation re-mesh
with contact can produce higher accuracy results for 2D modelling. Figure 3.5.4 displays
four of the 2D models used in this research. For transmission error study throughout the
mesh cycle, all models can produce acceptable results. For stress analysis specially for
finding local detail of the maximum stress in the critical areas, further refinements are
required.

The requirement for 3D modelling of meshing gear pairs involves the use of (high order)
brick elements in this research. Automatic mesh volumes such as gears in mesh with brick
elements are not available currently. With the available technique, the modelling
procedure produces a 2D model then extrudes it into 3D or produces the volume of the
gear, then sweeps it with 2D elements. For controlling the 3D mesh pattern, the mapped
43
mesh for the 2D model of the source area of the gear volume can be used. The other
controlling factor is in deciding how many divisions for the extrusion or the sweeping.
Due to the difficulty in applying different divisions in different areas, the distorted
elements can hardly be avoided. Even when building a huge model, there usually are large
numbers of distorted elements in the low stress area. So 3D mesh using brick elements is
limited with finer mesh. Figure 3.5.5 is an example of mesh adaptation for contact with
brick elements.

Figure 3.5. 3

44
Figure 3.5. 4 2D mesh patterns

45
Figure 3.5. 5 Mesh adaptation with contact using brick elements produced by the 2D
model extrusion or sweeping the 2D pattern through the volume

The critical stage for building the 3D model above is to decide how many divisions are
required for extrusion or sweeping. If not enough divisions are used, distorted elements
with large aspect ratios will be present due to the previous 2D mesh adaptation with contact
that has produced many small elements in the region near the point of contact. Figure 3.5.6
shows the distorted elements by issuing an element check option in ANSYS® when the
division is three.

Figure 3.5. 6 Distorted elements on one of the gears.

In this case, the element check option also provides a shape-testing summary as shown in
Table 3.4,
<<<<<< SHAPE TESTING SUMMARY >>>>>>
Element count 4323 SOLID186
Test Number tested Warning count Error count Warn+Err %
Aspect Ratio 4323 309 0 7.15%
Parallel Deviation 4323 3 0 0.07%
Maximum Angle 4323 0 0 0.00%
Jacobian Ratio 4323 0 0 0.00%
Warping Factor 4323 0 0 0.00%
Any 4323 309 0 7.15%
Table 3.4. Elements shape testing summary.

It can be seen that 309 of a total of 4324 elements have exceeded the aspect ratio warning
limit and they are all located in the high stress area. The other type of distortion, such as
parallel deviation, only occurred in 3 elements and its influence on the accuracy of the FEA
solution is considered minor compared with the one of large aspect ratio. The large aspect
46
ratio can be limited in the high stress area if more divisions are to be used to extrude the 2D
into a 3D model, but a very large model may result with lots of distorted elements in the
low stress area. Figure 3.5.7 shows the extruded 3D model with 13 divisions.

Elements with large aspect

ratio in the relatively low
stress area.

Figure 3.5. 7 Thirteen division model containing 89095 nodes, showing

distorted elements in the low stress area.

The model in Figure 3.5.7 is a very large model for FE analysis using existing PC
hardware. To solve the model in 10 sub steps with a PC PIII – 800 MHz with 1GB Ram
memory, the total run time could be 100 hr+. An alternative way to build the 3D model is
to apply different divisions in different areas as shown in Figure 3.5.8.

Figure 3.5. 8 Illustration of transition of the divisions.

The figure illustrates the number of divisions changing from 2 to 4. The result is a reduced
element distortion in both low and high stress areas, and a reduced model size. Applying
the transition of the division involves combining the extrusion and the swept volume. The
model seems to produce acceptable solutions for displacements. However, when checking
elements, it some times gave messages such as,

*** WARNING *** CP= 17.806 TIME= 00:42:26

ELEMENT 1079071637 DOES NOT EXIST.

47
The warning means that somewhere in the model, “ghost” elements exist, containing
something unpredictable. Further debugging efforts are then needed.

3D modelling of gears or gears in mesh using brick elements has been discussed as above,
where its required to eliminate the distorted elements in high stress areas so that reliable
finite element analysis data can be obtained from the model. Meanwhile one should
beware not to produce a very large model that requires enormous computer time for
solution. Full 3D modelling of gears or gears in mesh is required in most general cases
such as when shafts and bearings are included in the model, also in the cases of crowned
flank face gears and non-through cracked teeth and many other situations that have to be
modelled in 3D.

In a special case, such as in the modelling of two standard involute gears in mesh, when
ignoring the hub centre movement of each gears, the 3D modelling can be applied on one
half of the volumes that are separated by the symmetry plane, as shown in Figure 3.5.9.

Figure 3.5. 9 The symmetry plane of a gear.

3.6 Verifying Results

As a general rule, post analysis checks, such as reasonable displacements, animation and
reaction forces summing to give the applied load, and evidence supports such as closed-
form calculation and experimental testing should always be used to judge if the FEA results
correlate with the response of the physical structure.

A closed-form calculation (Jia 2001) has shown that it is particularly useful for verifying
the results in this research, and experimental testing will be presented in a later chapter.
48
4.1 INTRODUCTION

This chapter introduces the involute and fillet tooth profile equations that are used to
generate the profile of the teeth by an APDL (ANSYS Parametric Design Language)
program, so that the “one window” CAD/FEA method could be applied to avoid the
possible geometry data loss. The stiffness of the basic transmission unit (except the
bearings) was then studied where, in particular, the variations of the distortion field in a
meshing gear pair were analysed with various boundary conditions. The ratio of local
deformation was also defined and the characteristics of the local contact deformation on the
global deformation, related to the transmission error.

4.2 The Use of a Specified Coordinate System

When the need arises for imposing loads or displacement boundary conditions in a
direction that is not aligned with the global coordinate system, a specified coordinate
system, local coordinate system or working plane, may be defined at a desired location.
The nodal coordinate system, which by default is parallel to the global Cartesian coordinate
system in ANSYS®, of selected node(s) can be rotated into the specified coordinate system.
The other way to rotate the nodal coordinate system may be by using known rotation angles
or direction cosine components. Input data that may be interpreted in the rotated nodal
coordinate system includes component values of the following,
49
 · Degree of freedom constraints,
· Forces,
· Master DOF,
· Coupled nodes,
and · Constraint equations.
As shown in Figure 4.2.1, the nodal coordinate system of the nodes on the hub were rotated
into the working plane, defined as cylindrical, which has been moved to the centre of the
hub. The applied boundary conditions (WX i =0 or r i =0) constrain the gear hub radially,

allowing only free rotation. The restrained nodes also couple with the master node, which
means that any constrained nodes would have the same value on WY (or q ) if there were
deformation or rigid body motion of the gear. Input torque can be expressed as the sum of
the applied nodal forces at radius r, where T is the input torque load, n is the total number
of constrained nodes, f i is the tangential nodal force (usually f i = f, f is constant value)

and r is the hub radius.

n
T = å fi × r , (4.1)
i =1

The master node of the hub

nodes coupling set.

Figure 4.2. 1 Load, boundary conditions and coupled set nodes on the gear hub.

For clarity of the loads and boundary conditions in Figure 4.2.1, only the corner nodes are
selected and they are symmetrical to the hub. In the case of problems related to coupled
field analysis, for example heat transfer, loads should not be applied on mid-side nodes.
However, in the proposed research, for computing global displacements, particularly with
non-linear contact, loads are applied on as many hub nodes as possible, including mid-side
nodes while they are necessarily kept symmetrical to the hub. Otherwise, a small point

50
load could cause a very localised deformation that would not correlate with global
measurements. Figure 4.2.2 shows an example of a poorly loaded case.

Figure 4.2. 2 Poorly loaded model.

4.3 Tooth Profile Generation

The finite element model of spur gears in mesh developed in this thesis was initially based
upon test gears that have been used since an experimental investigation by Sirichai
(Sirichai 1996), and the test gear parameters are shown in Table 4.1. The test gears have a
ratio of 1:1. The involute and fillet tooth profile equations used in the finite element model
have been introduced by references (Litvin 1989; Townsend 1991; Kuang 1992). For most
of the other gears, profile equations can be found in references (Litvin 1993; Litvin 1993;
Zhang 1994; Litvin 1995; Litvin 1995; Litvin 1996; Litvin 1996; Seol 1996; Seol 1996;
Litvin 1997; Donno 1998; Feng 1998; Litvin 1998; Litvin 1998; Litvin 1998). This section
shows some of the basic equations used for calculating gear tooth involute profile
coordinates and gear tooth fillet coordinates. Alternative ways (programs) for generating
the involute and fillet tooth profile are also given.

Gear type --------------------------------- Standard involute, full-depth teeth

Material-----------------------------------------------------------------Aluminum
Modulus of elasticity, E --------------------------------------------------------69 Gpa
Poisson’s ratio, n ----------------------------------------------------------------- 0.33
Friction coefficient--------------------------------------------------------------- 0.06
Number of teeth N ------------------------------------------------------------- 23
Pressure angle -------------------------------------------------------------------20 degree
Module Mn ----------------------------------------------------------------------- 6 mm
Pressure angle, deg -------------------------------------------------------------20
Addendum, a (mm) ------------------------------------------------------------Mn
Dedendum, b (mm) -------------------------------------------------------------- 1.25 * Mn
Face width, mm (in.) -----------------------------------------------------------15 mm
Theoretical contact ratio -------------------------------------------------------- 1.59
Theoretical angle of meshing cycle-------------------------------------------24.912 degree
Table 4.1. Test Gear Parameters
51
The following algorithm calculates the coordinates of an involute and fillet tooth profile
based on the tooth profile generating method introduced by (Litvin 1989; Kuang 1992).
The following equations for the generated involute curve AB and fillet curve BC tooth
profile as shown in Figure 4.3.1 are valid for gears conjugate to the counterpart basic rack
where,
· Mn is metric normal module, mm (for spur gear expressed M)
· N is number of teeth
· Addendum, a = a Mn (usually a = 1.0)
· Dedendum, b = b Mn (usually b = 1.25)
· Tip radius, rc = g Mn (usually g = 0.25)
· f is pressure angle
· Addendum modification coefficient X = e/Mn
· e is cutter offset

C
r

Figure 4.3. 1 Coordinate system for generating involute and fillet tooth profiles.

The parametric representation coordinates of the involute profile (curve AB) is,

ìx (q)ü
r ( q) = í ý, (4.1)
î y(q) þ

52
where coordinates of the involute curve are given by equation (4.2) and (4.3),

N ·Mn ì éæ p ö 2X ù üï
x (q) = ísin(q ) - êç q + ÷ cos f + sin f ú cos(q + f ) ý , (4.2)
2 î ëè 2N ø N û ïþ
and

N ·Mn ì éæ p ö 2X ù üï
y(q) = ícos(q ) + êç q + ÷ cos f + sin f ú sin(q + f ) ý . (4.3)
2 î ëè 2N ø N û ïþ
The parameter q , in radians, of the involute curve is limited to the following range,

q min £ q £ q max (4.4)

where the parameters q min and q max of the involute curve are given by equation (4.5) and

(4.6),
2
q min = [U + (V + X ) cot f ] , (4.5)
N
and
1 æ 2X ö p
q max = (2 + N + 2X ) 2 - ( N cos f) 2 - ç1 + ÷ tan f - . (4.6)
N cos f è N ø 2N

The parameters U and V of the involute curve are given by,

ép g ù
U = - ê + (a - g ) tan f + , (4.7)
ë4 cos f úû
and
V =g -a. (4.8)

The parametric representation coordinates of the fillet profile (curve BC) is given by

ìx (q)ü
r ( q) = í ý, (4.9)
î y(q) þ

53
where coordinates of the fillet curve are given by equation (4.10) and (4.11),
x (q) = Mn·( P cos q + Q sin q ) , (4.10)
and
y(q) = Mn·( - P sin q + Q cos q ) . (4.11)
The parameter q of the fillet curve is limited to the following range,
q min £ q £ q max , (4.12)

where the parameters q min and q max of the fillet curve are given by equation (4.13) and

(4.14),
2
q min = [U + (V + X ) cot f ] , (4.13)
N
and
2U
q max = . (4.14)
N
The parameters P, Q, and L of the fillet curve are given by,
g æ Nq ö
P= + çU - ÷, (4.15)
L è 2 ø

2g æ V + X ö N
Q= ×ç ÷ +V + + X , (4.16)
L è 2U - Nq ø 2
and
2
æ V+X ö
L = 1 + 4ç ÷ . (4.17)
è 2 U - Nq ø

As a general method, an AutoLisp, (AUTODESK. 1997), AutoCAD programming

language for generating the points of an involute and fillet tooth profile is given in
Appendix A. It can be copied as plain text then read by AutoCAD to create a series of
points of the curve. Once the profile or entire geometry is created then it can be transferred
to a FEA program.

The ideal method for this research is the “one window” CAD/FEA method. A program
writen with APDL (ANSYS Parametric Design Language) for generating the points of the
curve directly in ANSYS® is rather simple. In this research the program is developed to

54
 /GST,ON
/PREP7
pi=3.1415926 !start input data
x1=0
fi=pi/9
g1=0.25
a1=1
n=23
m=6
nu=200 !200 points in curve AB
nu2=120 !120 points in curve BC
u=-(pi/4+(a1-g1)*tan(fi)+g1/cos(fi)) !user can decide any number in each curve
v=g1-a1 !u, v are parameter
thmin=(u+(v+x1)/tan(fi))*2/n
thmax=((2+n+2*x1)**2-(n*cos(fi))**2)**0.5/(n*cos(fi))-(1+2*x1/n)*tan(fi)- !th stand for theta- q
pi/(2*n)
inc=(thmax-thmin)/nu
!inc-the angel increment
*do,i,1,(nu+1)
!start a loop to produce points for AB
th=thmin+inc*(i-1)
x=(n*m/2)*(sin(th)-((th+pi/(2*n))*cos(fi)+(2*x1*sin(fi))/n)*cos(th+fi))
y=(n*m/2)*(cos(th)+((th+pi/(2*n))*cos(fi)+(2*x1*sin(fi))/n)*sin(th+fi))
k, ,x,y,,
*enddo
thmax2=2*u/n
!thmin2=thmin
inc=abs(thmax2-thmin)/nu2
*do,i,1,(nu2+1)
!start loop to produce points for BC
th=thmin+inc*(i-1)
labc=(1+4*(((v+x1)/(2*u-n*th))**2))**0.5
!labc, pq and qp are parameter L, P and Q
pq=(g1/labc)+(u-n*th/2)
qp=2*(g1/labc)*(v+x1)/(2*u-n*th)+v+(n/2)+x1
x=m*(pq*cos(th)+qp*sin(th))
y=m*(-pq*sin(th)+qp*cos(th))
k, ,x,y,,
*enddo
finish
produce splines through the points to create the geometry area as a 2D solid, as shown in
Table 4.2.
Table 4.2. APDL program for producing involute coordinates.

4.4 Gear Body

Most of the previously published finite element work with gear models has involved only
partial tooth models. In an investigation of gear transmission errors using factors such as
variation combined torsional mash stiffness, the whole body of the gear in mesh must be
modelled. Sirichai (Sirichai 1999) developed the concept from the performance of the
contact element point of view. However, regardless of the local deformation, the effect that
gear body flexibility has on the torsional stiffness has to be studied. The results in this
section will show that the torsional stiffness varies with the gear body due to different
constraints on the dedendum circle. The influences of the gear body on the total torsional
stiffness were normally ignored by partial tooth models. The gear body is taken from the
dedendum circle of the gear, as shown in figure 4.4.1.

55
 Alternative nodal constraint
from tangential movement UY = 0
or all DOF.

Figure 4.4. 1 The model of the gear body with its load and boundary conditions.

As shown, the nodes on the hub were constrained from radial movements. The model used
a coupled set of nodes and equivalent nodal forces were applied. The constrained nodes
on the outer circle are shown in Figure 4.4.1, where they are alternatively constrained from
tangential movement only (UY=0, when the coordinate system is cylindrical), or all DOF.
The FEA results for the stiffness under the application of a torque load are given as shown
in Figure 4.4.2.
695000
Torsional Stiffness (Nm/rad)

690000

685000
K: Constraint UY
K: Constraint ALL DOF

680000
0 20 40 60 80 100 120 140 160
Load (Nm)

Figure 4.4. 2 The variation of the stiffness caused by type of constraint.

The results show a difference in the torsional stiffness with changing constraint from all
DOF to only UY, where the stiffness decreased from 692946.03 Nm/rad to 692452
Nm/rad, an amount of 494.08 Nm/rad. This difference can be ignored considering the
percentage difference is only 0.07%. Further investigation of the model involved
extending the constraint on the outer circle from one tooth space to two teeth space
considering nodes constrained from tangential movement that would be allowed to rotate
about the tooth root if the tooth was present. The model has been re-meshed and the details
are given in the following figure.
56
 Constrained nodes
(tangential movement UY = 0)
covering two teeth spaces.

Figure 4.4. 3 The re-meshed model showing constraints over two teeth space.

The torsional stiffness, as shown from the previous calculation, can be obtained with a
series of input torque loads. The results can be compared as shown in Figure 4.4.4.
800000
Torsional Stiffness (Nm/rad)

600000

400000
ConstraintUY for 2T space

200000 ConstraintUY for 1T space

0
0 20 40 60 80 100 120 140
Input Torque Load (Nm)

Figure 4.4. 4 The variation of the stiffness caused by increasing the constraint space.

The torsional stiffness was calculated to reach 755709.74 Nm/rad with the two teeth space
constraint, which is 63257.74 Nm/rad higher than the stiffness obtained with one tooth
space constraint. The torsional stiffness has thus increased by 8.6%. Such a difference for
partial tooth models is very large and cannot be ignored. To ensure the correct modelling,
this shows that the whole body of the gears in mesh must be modelled.

4.5 The Variations of Distortion Field in the Meshing Gears

For a constant input load, the distortion field in the gear was found to change with meshing
position. The significant change occurs when the meshing of the teeth changes from the
single pair to double pair in contact and vice-versa if the gears are perfect. From FEA
results, the von Mises stress in the gear can be plotted, which reveals the distortion field as

57
the von Mises stress computes the magnitude of stress that tends to distort a body, as given
below,

s mises = 1
2
(s xx
2
( )
- s yy ) + (s yy - s zz ) + (s zz - s xx ) + 3 s xy2 + s yz2 + s zx2 .
2 2
(4.18)

It should be noted that the von Mises stress is always positive, since distortion is
considered neither positive nor negative. The following Figure 4.5.1 and 4.5.2 were
obtained from the FEA results of a 2D plane stress model and a 3D model. It can be seen
that the von Mises stress field in 2D and 3D models are different in extent in both figures.
The 3D element model can represent all 18 linear strain states, which meets the
requirement for completely defining the linear strain variations when polynomial
displacement assumptions are used. So, when predicting the von Mises stress field, a better
global response was obtained with the 3D model.

Figure 4.5. 1 The von Mises stress field for single pair of teeth in contact at the pitch point.

Figure 4.5. 2 The von Mises stress field for double pair of teeth in contact (midpoint).

It has been noted, in both 2D and 3D models, that one of the major differences between the
case of Figure 4.5.1 and Figure 4.5.2 is that the von Mises stress field changes dramatically
in the area around the hub. These changes can be further confirmed by the plots of the
load-reaction forces of the models from the FEA solutions, as shown in the Figure 4.5.3,
58
 Figure 4.5. 3 The load reaction forces for one tooth and two teeth in contact.

The reaction forces around the hub tend to be even when two pairs of teeth are in contact,
and the even loads and boundary conditions around the hubs are believed to be the main
reason. In application, a single keyway is often used to transfer the torque, as shown in
Figure 4.5.4,

Figure 4.5. 4 The distortion field for one tooth and two teeth in contact.

In this case, the von Mises stress field around the hub has only minor changes when the
contact teeth change from one pair to two pairs in mesh.

4.6 Shaft

The shaft is an important component in the gear transmission system. In most cases, the
torsional stiffness of the shaft is lower than the stiffness of the gears. In the case of uneven
input torque load, the shaft could contribute the major part of the system transmission error
59
(or total TE). This research has concentrated on the study of the shaft under the steady
input torque load. The study is trying to find out the properties of the components such as
stresses and strains with varying boundary conditions. The basic FEA model of a shaft is
shown as in Figure 4.6.1.

Figure 4.6. 1 The basic FEA model of a shaft.

The considered boundary conditions are:

· Coupling those nodes in rotation at input end A while applying the equivalent
tangential nodal forces or without the nodes coupling.
· Tangential constraints at output end B can be applied on those nodes that form a full
circle as shown in the Figure 4.6.1, or applied on part of those nodes as shown in
Figure 4.6.2,

Figure 4.6. 2 The partial constraints at output end B.

According to the boundary conditions, the FEA model of the shaft was calculated with four
cases as shown in Figure 4.6.3. So far, it has been found that a minor elastic strain on the
surface of the shaft changes with the varying boundary conditions at output end B. This
indicates that the gear mesh conditions could be monitored by the measurement from a
high order property such as strain from the shaft. A related example will be discussed in
more detail in a later chapter.
60
Figure 4.6. 3 The von Mises stress field in the shaft under 4 types of boundary conditions.
61
4.7 Ratio of Local Deformation

The ratio of local deformation can be defined as,

RT , ANG = q c / q . (4.19)
where q is the total angular rotation of the input gear hub while the output gear has been
fully constrained at its hub where both input and output gears are flexible. q c represents

the angular rotation of the input gear hub which is only due to the local contact deformation
and sliding from the contact tooth. The subscripts T and ANG denote the input torque and
mesh position respectively. For example, with the test gear in this research (Table 4.1),
ANG = 0 o is the meshing gear pair in the position where the contact involves one pair of
teeth and the contact point is at the pitch point. When the mesh position is in the middle of
the zone with two pair of teeth in contact, ANG = 7.83o . Figure 4.5.3 has shown the mesh
positions and the model used for the determination of the value of q . To determine the
value of q c at the position ANG = 0 o , the FEA model as shown in Figure 4.7.1 has been

used.

Figure 4.7. 1 The FE model for evaluation of the effect of the local deformation.

As shown in Figure 4.7.1, the nodes attached to the edge DEF were constrained in all DOF,
while the nodes attached to the edge BC were constrained radially so that the tooth is able
to rotate about the hub centre. Equivalent tangential nodal forces have been applied on
those nodes that attach to the edge BC, while those nodes attached to the edge ABC were
coupled together in rotation about the hub centre as the tooth’s shear and bending
displacements have to be limited. At the mesh position ANG = 7.83o the model for
evaluation of q c was extending to two pair of teeth in contact, as shown in Figure 4.7.2,
62
Figure 4.7. 2 The extended FE model for evaluation of the effect of the local deformation.

The procedures for building this model are basically the same as for the previous one. It
can be seen that the nodes attached to the edge of abc and def remain as one node coupling
set. So, for the input load T = 76.2 Nm, the FEA results can be given as below,
R76.2, 0o = 25% and R76.2, 7.83o = 16.7%.

The results have shown that the effects of the local deformation to the total angular rotation
q are significantly different for the single pair and double pair of teeth in contact.

In the position ANG = 0 o , because the contact is at the pitch point, theoretically there is no
sliding between the contact teeth, therefore q c and q can be considered as pure elastic

deformations. R76.2, 0o is very close to the particular value of 25% which Coy and Chao

(Coy 1982) obtained from a partial tooth model as the ratio of total pure elastic
deformation. However, when the contact is not at the pitch point, both q c and q contain

elastic deformation and rigid body motion. From another point of view, once the sliding in
the contact cannot be ignored, both q c and q will not be considered as pure elastic

deformation. So, the effects of local deformation as the ratio RT , ANG has to be studied in

the entire mesh cycle, the details of which will be discussed in Chapter 6.

In a mesh position, the ratio RT , ANG also changes with the changing of the input torque

load, because the contact area varies with any variation of the input torque load. A series

63
of calculations were taken with the model as shown in Figure 4.7.1 for various input torque
loads from 5Nm to 150Nm, with the major load increment of 15Nm. With the contact area
increasing as shown in Figure 4.7.3, the ratio RT , ANG decreases from 27.2% to 24.3% as

shown in Figure 4.7.4.

Figure 4.7. 3 The contact areas shown by the reaction forces.

0.275
Ratio of Local Deformation

0.27
0.265
0.26
(%)

0.255
0.25
0.245
0.24
0 15 30 45 60 75 90 105 120 135 150
Torque (Nm)

Figure 4.7. 4 RT , ANG as a function of input torque at the pitch point.

The difference of RT , ANG inside the input torque loads was 2.9% with the mesh position in

the pitch point. This difference would be smaller if the mesh position were inside the
double pair zone.

Finally, it has to be mentioned that the FE models for evaluation of the ratio RT , ANG appear

to be suitable for the test gears in this research, and that the ratio should also depend on the
thickness of the gear (or the back up ratio) as well as the material of the gear.
64
5.1 INTRODUCTION

In general, when a gear is subject to a transmitted load, the stress conditions near the areas
of contact and the tooth root, are neither plane stress nor plane strain, but are three-
dimensional. However, most previous FEA models on standard involute gears reduce the
problems to two dimensions and many of them provide acceptable approximations. In two-
dimensional models, at least one of the principal stresses or strains is assumed to be zero,
and the model is either plane stress or plane strain respectively. Examples of models based
on plane stress assumptions include (Muthukumar 1987; Lewicki 1997; Arafa 1999), and
some based on plane strain elements include (Kuang 1992; Filiz 1995; Sirichai 1997;
Sirichai 1999). It should be noted that most of the models didn’t have significant
differences in geometry. So the question remains as to why one would use a plane stress or
a plane strain assumption. There appear to be no obvious criteria for choosing either
approach for 2D gear modelling. On the other hand, as a general guide which is based on
elasticity theory (Arthur 1987; Richards 2001), for FE modelling of a thin plate with in-
plane loads and boundary conditions, the plane stress assumption should be used regardless
of whether the solutions are for displacements or for stress. Once again questions are
raised as to what extent the special structure (a thin gear) will produce results with similar
65
errors to that of the thin plate when plane stress assumptions are used. For gears subject to
contact, fracture and other situations, it is not known if the plane stress or the plane strain
assumption is the most appropriate.

The understanding of these issues is vitally important for ongoing gear transmission error
studies and other gear studies too, so they are the major research objectives discussed in
this chapter.

5.2 Stiffness

One of the aims of this research was to try to find out the effect of changing geometry
(thickness) on the variations of the stiffness in different models (three cases). The first
model was a simple disk as shown in Figure 4.4.1. Twelve different thicknesses varying
from 5mm to 300mm were considered. For each disk thickness, the various loads
including 10, 30, 50, 70, 90, 110 and 130 Nm were applied and for each load, the result of
torsional stiffness as K1, K2 and K3 (K1 – plane stress, K2 – plane strain, K3 – 3D) were
calculated under assumptions of plane stress, plane strain and 3D models respectively.
Figure 5.2.1 shows the variations of these stiffnesses, where it can be observed that K3
varies mainly between K1 and K2. Figure 5.2.2 gives the comparisons of the three curves
by showing the difference between K1, K3 and K2, K3.

As Figure 5.2.3 shows, when the disk has thickness of 10mm and 15mm, the relative error
has a minimum value if the disk is modelled with the plane stress assumption, whereas
there will be a maximum error value if the plane strain assumption was used. When the
thickness is 160mm, the result shows a cross over point, where both 2D assumptions have
the same value of the error. With further increasing disk thickness, the error from the plane
strain assumption decreases and that of plane stress increases. This is in agreement with
the conventional solid elasticity theories.

The second model was based on a pair of meshing gears, each of them simplified with one
tooth only. One of the gears was further simplified as a rigid involute surface as shown in
Figure 5.2.4. Figure 5.2. 1 The variations of the torsional stiffness as function of the thickness.

66
67
 250000

The Torsional Stiffness Difference From

200000

3D Model (Nm/rad.)
150000

100000

K3-K1
50000 K2-K3

0
0 25 50 75 100 125 150 175 200 225 250 275 300 325
Thickness (mm)

Figure 5.2. 2 The absolute errors of model 1.

2.9
(K3-K1)/K3
2.4 (K2-K3)/K3
Relative Error (%)

1.9

1.4

0.9

0.4

-0.1
0 25 50 75 100 125 150 175 200 225 250 275 300 325
Thickness (mm)

Figure 5.2. 3 Relative errors of model 1.

Figure 5.2. 4 Second gear model.

The calculation procedures for the second gear model were identical to those used with the
first model. The results have shown the difference in absolute errors compared with the
first model as shown in figure 5.2.5.
68
 240000

difference from 3D model

The torsional stiffness
190000

(Nm/rad.)
140000

90000
K3 - K1
K2 - K3
40000

-10000
0 25 50 75 100 125 150 175 200 225 250 275 300 325
Thickness (mm)

Figure 5.2. 5 The absolute errors from the second model.

It can be seen that the errors have reduced when the plane stress assumption was used for
the thinner model and the crossover point of the curves was extended from the thickness at
160mm to 210mm. This shows that the absolute errors varying with the thickness are
slower than that of the first model. Similarly, the relative errors can be produced as shown
in Figure 5.2.6.
12
(K3-K1)/K3
10 (K2-K3)/K3
Relative Error (%)

0
0 25 50 75 100 125 150 175 200 225 250 275 300 325
-2

-4 Thickness (m m )

Figure 5.2. 6 Relative errors from the second model.

Figure 5.2.6 has shown the significant difference between the first model and the second
model. Using the plane strain assumption to model a disk the relative errors are less than
3%, but to model a gear with 200mm thickness, the errors are greater than 3% when
compared with the 3D model. Under the plane stress assumption, within the thickness
range from 10mm to 200mm, the relative errors of the gear model are less than 3%.
However, the errors are quite different when the thickness is less than 20mm. These results
are important for gear transmission error (displacement) studies. As an example of the test
gear model (15mm thickness), there are basically 2% errors in solutions of displacements
when the plane stress assumption was used, whereas about 8.3% error will be present if
using the plane strain assumption. In the displacement field, it has been shown that the
gear is different from that of a simple disk.
69
The third model is similar to the previous one, but now a 4mm crack is located in the root
area as shown in Figure 5.2.7.

Figure 5.2. 7 The third model with a tooth crack.

One of the major differences from all previous models is that the crossover point of the
error curves is now located inside the 50mm thickness, as shown in Figure 5.2.8.

350000 K3-K1
K2-K3
different from 3D model's

300000
The torsional stiffness

250000
(Nm/rad.)

200000

150000

100000

50000

0 25 50 75 100 125 150 175 200 225 250 275 300 325
Thickness (m m )

Figure 5.2. 8 The absolute errors of the third model.

Consequently, using the different assumptions can significantly alter the relative errors in
the solutions for displacement, depending up on the thickness, as can be seen in Figure
5.2.9.
9

7
Relative Error (%)

4 (K3-K1)/K3

3 (K2-K3)/K3

0
0 25 50 75 100 125 150 175 200 225 250 275 300 325

Thickness (mm)

Figure 5.2. 9 Relative errors of the third model.

70
Figure 5.2.9 shows that the plane stress assumption can only be used in a very narrow
range if the relative error is not allowed to be greater than 3%. When the thickness is
inside the range from 25mm to 90mm, both 2D assumptions cannot avoid the errors that
are greater than 3%. Meanwhile, the range for the plane strain assumption with lower
errors has increased since the error is less than 3% when the thickness is greater than
90mm. If the torsional stiffness in model 1 is assigned to be K(11) for plane stress, K(12)
for plane strain and K(13) for the 3D model, then the model 2 results will be given by
K(21), K(22) and K(23), and the model 3 results will be K(31), K(32) and K(33) for the
relative stiffness. Further comparison between the model results can then be made as
shown in Figure 5.2.10.
290000
K(13)-K(11)
K(23)-K(21)
Torsional stiffness difference

240000 K(12)-K(13)
from 3D model (Nm/rad)

K(22)-K(23)

190000

140000

90000

40000

-10000
0 50 100 150 200 250 300
Thickness (m m )

Figure 5.2. 10 Comparison of absolute stiffness errors from model 1 and model 2.
(absolute error here is referred to the absolute value of the difference between the amplitudes.)

It can be seen in Figure 5.2.10 that the absolute errors from model 1 and model 2 are of
similar magnitude. Under the plane strain assumption, the absolute errors from both
models show the same trends even though there is a significant difference in the relative
errors. When the tooth thickness is less than 50 mm (about 1/3 of maximum in plane
dimension), the plane stress assumption shows less error in comparison to the 3D model
result. When the thickness is between 5mm and 300mm (which is about the maximum in
plane dimension of the gear), the plane stress result shows less error than the plane strain
result. When the thickness becomes the major dimension of the model, the plane strain
assumption should therefore be chosen if the study is to be based on a 2D assumption.

Similar comparison between model 2 and model 3 is shown in Figure 5.2.11, where a
significant difference in absolute errors is easy to see.
71
 350000
K(33)-K(31)
300000

Torsional stiffness difference from

K(32)-K(33)
K(23)-K(21)
250000
K(22)-K(23)

3D model (Nm/rad.)
200000

150000

100000

50000

0
0 50 100 150 200 250 300
-50000 Thickness (mm)

Figure 5.2. 11 Comparison of absolute stiffness errors from model 2 and model 3.

As mentioned before, due to the through crack in the tooth root, the thickness that is
suitable for plane stress assumption has reduced to about 20mm. Within the thickness,
which is about 1/3 of the major dimension, this shows that the plane stress is still a better
assumption than plane strain. The comparison of the relative errors from all the models has
been made as shown in Figure 5.2.12.
10 M odel 1: (K3-K1)/K3 (K2-K3)/K3
M odel 2: (K3-K1)/K3 (K2-K3)/K3
M odel 3: (K3-K1)/K3 (K2-K3)/K3
8
Relative Error (%)

0
0 50 100 150 200 250 300

-2 Face Width (mm)

-4

Figure 5.2. 12 Comparison of the relative errors of the torsional stiffness.

It has clearly been shown that when using 2D assumptions to model gears for solutions of
the displacements it is hard to avoid errors associated with the plane stress and plane strain
assumptions.

5.3 Stress

Though the results obtained from the stiffness analysis presented above are sufficient for
guiding the following gear transmission error study, theoretically, stresses (the higher order
products yielded from FEA analysis) will verify the important difference between the

72
models and many experts indicate that stress analysis should always be used to give the
most reliable results.

The stress analysis developed here will concentrate on a one-tooth gear model (Figure
5.2.4) and a one-tooth gear model with a crack at the tooth root (Figure 5.2.7). The
analysis procedures are similar to those used for the stiffness analysis, with each model
varying with different thickness from 5mm to 300mm, and all calculations achieved with
an input torque load of 76.2Nm. For T6 aluminium, there is a wide range of input torque
loads that can be applied on the models in the elastic range (Sirichai 1999). 76.2Nm is in
the middle range of the loads for the thinnest gear model (5mm thickness) and with this
load, the comparison with previous research (Sirichai 1999) is relatively easy. The FEA
results for the one tooth gear model are given in Table 5.1.
Load is constant at 76.2Nm.
thickness SINT difference
Plane Stress Plane Strain 3D Stress difference from 3D
(mm) from 3D
S1max(Mpa) SINT S1max(Mpa) SINT S1max(Mpa) SINT 31 32 31 32
5 136.771 202.730 111.890 186.960 141.130 241.830 4.359 29.240 39.100 54.870
10 68.410 101.320 55.967 93.442 73.101 121.540 4.691 17.134 20.220 28.098
15 45.612 67.533 37.317 62.286 49.753 84.045 4.141 12.436 16.512 21.759
20 34.211 50.646 27.990 46.711 36.125 65.892 1.914 8.135 15.246 19.181
30 22.809 33.761 18.661 31.139 23.813 45.081 1.004 5.152 11.320 13.942
45 15.206 22.506 12.441 20.758 15.396 30.097 0.190 2.955 7.591 9.339
60 11.405 16.879 9.331 15.568 11.436 22.489 0.031 2.105 5.610 6.921
80 8.554 12.659 6.999 11.676 8.525 16.818 -0.029 1.526 4.159 5.142
100 6.843 10.127 5.599 9.341 6.803 13.421 -0.040 1.204 3.294 4.080
150 4.562 6.751 3.733 6.227 4.519 8.935 -0.043 0.787 2.184 2.708
200 3.422 5.064 2.800 4.670 3.383 6.697 -0.039 0.583 1.633 2.026
250 2.737 4.051 2.240 3.736 2.702 6.287 -0.035 0.462 2.236 2.551
300 2.281 3.376 1.867 3.114 2.249 5.238 -0.032 0.382 1.862 2.125

Table 5.1. FEA results for one tooth model.

S1max refers to the maximum first principal stress, and SINT refers to the stress intensity
factor. The stress difference between the 3D model and the plane stress model is denoted
by 31 and 32 represents the difference between the 3D model and the plane strain model.
Here we are concerned with the relative errors, and the 3D model has been taken as the
reference. The following table gives the relative errors.
thickness Relative Stress Errors (%) Relative SINT Errors (%)
(mm) for Plane Stress for Plane Strain for Plane Stress for Plane Strain
5 3.09 20.72 16.17 22.69
10 6.42 23.44 16.64 23.12
15 8.32 25.00 19.65 25.89
20 5.30 22.52 23.14 29.11
30 4.22 21.64 25.11 30.93
45 1.23 19.19 25.22 31.03
60 0.27 18.41 24.95 30.78
80 0.34 17.90 24.73 30.57
100 0.59 17.70 24.54 30.40
150 0.95 17.40 24.44 30.31
200 1.15 17.24 24.38 30.26
250 1.31 17.11 35.57 40.58
300 1.44 17.01 35.56 40.56
Table 5.2. The calculated relative errors.
73
 Figure 5.3. 1 The relative errors of one tooth model.

Figure 5.3.1. shows that using the plane strain assumption to predict maximum first
principal stress in a gear has far more error than using the plane stress assumption.
However, when the gear thickness is 15mm, both assumptions have a maximum error given
by 8.3% for the plane stress assumption, and 25% error for the plane strain assumption.
With increasing thickness, the error from the plane strain assumption decreases slowly,
almost linearly, whereas the error from the plane stress assumption increases linearly. In
this case the plane stress assumption shows a lower error for predicting the maximum first
principal stress over a wide range of thickness values of the gear. Figure 5.3.1 also shows
that the 2D assumption is not accurate for predicting the maximum stress intensity factor
because the minimum error for the plane stress assumption is 16.2% and that for the plane
strain assumption is 22.7%. Overall the plane strain assumption gives higher error values
compared with the 3D model. Under both 2D assumptions, SINTmax and S1max occur at the
same physical location in the FE model, and the positions do not change with varying
thickness. The locations for SINTmax and S1max are shown in Figure 5.3.2.

Figure 5.3. 2 The position of SINTmax and S1max under 2D assumptions.

It has been found that in the 3D case, the position of S1max always changes with varying
thickness, and both SINTmax and S1max always occur on the surface. Figure 5.3.3 shows
the change of SINTmax and S1max positions with the varying thickness.
74
 max
max

Figure 5.3. 3
In the model with the 4mm crack at the tooth root (Figure 5.2.7) using the same analysis,
the relative errors of S1max are obtained as shown in Table 5.3, and they were all found to
be located at the crack tip.

75
 S1max (Mpa) Relative Errors (%)
thickness (mm) Plane Stress Plane Strain 3D for Plane Stress for Plane Strain
5 471.330 496.000 566.650 16.82 12.47
10 235.950 248.100 285.350 17.31 13.05
15 157.360 165.410 185.360 15.11 10.76
20 118.050 124.420 136.220 13.34 8.66
30 78.713 82.961 90.938 13.44 8.77
45 52.482 55.314 59.991 12.52 7.80
60 39.364 41.488 44.237 11.02 6.21
80 29.525 31.117 32.910 10.29 5.45
100 23.620 24.895 26.258 10.05 5.19
150 15.748 16.597 17.438 9.69 4.82
200 11.811 12.448 13.050 9.49 4.61
250 9.449 9.959 10.418 9.30 4.41
300 7.874 8.299 8.676 9.24 4.34

Table 5.3. FEA results of one tooth cracked model.

The comparison of the relative errors can be shown in Figure 5.3.4.

20 for Plane Stress

18
Relative Errors of S1max (%)

for Plane Strain

16
14
12
10
8
6
4
2
0
0 25 50 75 100 125 150 175 200 225 250 275 300
Thickness (mm)

Figure 5.3. 4 The relative errors of S1max.

Compared with Figure 5.3.1, this result shows the opposite conclusions than for the model
without the tooth crack. It was found that the results obtained using the plane strain
assumption to predict S1max in a gear with the root crack are better than the results obtained
from using the plane stress assumption. However, when the gear is thinner than 25mm, the
errors from using the 2D assumptions can be significant. For global SINTmax, in the 3D
case, not all SINTmax positions were located at the crack front. When the thickness varies
from 30mm to 60mm, the global SINTmax values were located near the contact point on the
surface. However, the maximum SINT of the crack front, SINTc was still located in the
position of S1max, and the global SINTmax and local SINTc are close given by 51.842 and
51.728 respectively for the 30mm model, and 25.857 and 25.723 respectively for the 60mm
model. Figure 5.3.7 shows the locations of S1max, SINTmax and SINTc (SINTc is in the
crack tip or the maximum value in the crack front) in all models. For the 2D case, SINTmax
was located at the crack tip for plane stress models. However, when using plane strain
assumptions, they are not located at the crack tip but near the contact point. In this case,
the relative error for SINTmax (compared with 3D) was low when the gear is thicker than

76
25mm, but the comparison was not made in the same location. In the same location (crack
tip and crack front), the relative errors are much higher as shown in Figure 5.3.5.
40 for Plane Strain (global,near contact)
for Plane Strain at node 780--crack tip
30

20

10

0
0 25 50 75 100 125 150 175 200 225 250 275 300
Thickness (mm)

Figure 5.3. 5 The relative errors of SINTmax and SINTc.

For the comparison, SINTc was calculated so that the comparison was made for all models
SINT at the crack tip or at the crack front. The results are shown in Table 5.4.

SINTc Relative Errors (%)

thickness (mm) for Plane Stress for Plane Strain 3D for Plane Stress for Plane Strain
5 471.330 224.948 359.280 31.19 37.39
10 235.950 112.525 166.460 41.75 32.40
15 157.360 75.019 106.900 47.20 29.82
20 118.050 56.439 80.555 46.55 29.94
30 78.713 37.633 51.728 52.17 27.25
45 52.482 25.092 34.652 51.46 27.59
60 39.364 18.820 25.723 53.03 26.84
80 29.525 14.116 19.299 52.99 26.86
100 23.620 11.293 15.410 53.28 26.72
150 15.748 7.529 10.259 53.50 26.61
200 11.811 5.647 7.677 53.85 26.45
250 9.449 4.517 6.131 54.11 26.33
300 7.874 3.765 5.102 54.34 26.20

Table 5.4. The calculated relative errors of SINT at the crack tip.

Figure 5.3.6 clearly shows that the calculation of SINT at the crack tip using the plane
strain assumption contains less errors than that obtained using the plane stress assumption,
except for the case where the thickness is very thin (eg. 5mm). Because the comparison is
made at the crack tip or the crack front for these models, these results are in agreement with
common fracture mechanics theories. Except for the analysis of thin plates, the asymptotic
or near-crack-tip behaviour of stress is usually thought to be that of plane strain (ANSYS
2000).
60
Relative Error of SINTc (%)

50

40

30

20
for Plane Stress (global)
10 for Plane Strain at node 780--crack tip

0
0 25 50 75 100 125 150 175 200 225 250 275 300 325
thickness (mm)

Figure 5.3. 6 The relative errors of SINT at the crack tip.

77
Figure
7 5.3.

78
From a general case, Anderson (Anderson 1995) has analysed the state of stress and strain
variations near the crack front (r<<B) through a certain thickness as illustrated in Figure
5.3.8.

Figure 5.3. 8 Schematic variation of transverse stress and strain through the thickness at a
point near the crack tip(Anderson 1995).

At the mid-plane (Z = 0), plane strain conditions exist and s zz = u ( s xx + s yy ). For the

material on the plate surface (Z = ± B), a state of plane stress exists (because there are no
stresses normal to the free surface). There is a region near the plate surface where the
stress state is neither plane stress nor plane strain and this is the region where most of the
S1max results were found in this research.

Figure 5.3.9 is a plot of s zz as a function of z/B and r/B. These results were obtained from
a 3D elastic-plastic finite element analysis performed by Narasimhan and Rosakis
(Narasimhan 1988). Note the transition from plane strain (z = 0) to plane stress as r
increases relative to thickness.

Both cases above illustrate the variations of 2D-closed state in a 3D situation. They show
that when the structure is analysed with fracture mechanics theory, for a certain thickness,
there are differences in the result if 2D analysis is needed, compared with 3D analysis.

79
Figure 5.3. 9 Transverse stress through the thickness as a function of distance from the
crack tip (Narasimhan 1988).

For the varying thickness, the variations through the thickness of stress S1 and SINT in the
critical point should be produced as an important proof for this research. The following
figure (Figure 5.3.10) gives the results for the one tooth model and Figure 5.3.11 shows the
results for the model with a root crack.

It should be noted that the maximum SINT at the crack tip or in the crack front, in the sense
of fracture mechanics, is given by K (total ) and K (total ) ¹ K I + K II + K III . As the current

research is concentrating on the comparison of the global response of the structure, details
of fracture parameters such as K I , K II , K III and others have not been calculated.

5.4 Conclusion

Overall, the plane stress assumption can produce well-matched results with 3D models.
However, care must be taken not only for the model’s geometry, if the model is subject to
fracture. In this instance the assumption might only be valid in a very limited range.
Failure to understand these results may lead to large errors in the analysis results.

80
Figure 5.3. 10

81
 1

Figure 5.3. 11

82
6.1 INTRODUCTION

In recent years, many different procedures have been developed to model the behaviors of
gears in mesh. Examples of this can be seen in references (Tordian 1967; Remmers 1978;
Drago 1979; Gargiulo 1980; Walford 1980; Hayashi 1981; Bahgat 1983; Ozguven 1988;
Kahraman 1991; Kuang 1992; Liou 1992; Rebbechi 1992; Vinayak 1992; Bard 1994;
Brousseau 1994; Daniewicz 1994; Kowalczyk 1994; Munro 1994; Refaat 1994a; Refaat
1995; Velex 1995; Liou 1996; Litvin 1996; Sirichai 1996; Sweeney 1996; Velinsky 1996;
Sirichai 1997; Underhill 1997; Elkholdy 1998; Gosselin 1998; Nadolski 1998; Zhang 1998;
Arafa 1999; Sirichai 1999; Wang 2000). One of the many factors, which can be
investigated, is the torsional mesh stiffness variation as the gear teeth rotate through the
mesh cycle. For prediction of the torsional mesh stiffness, FEA modelling in particular can
encompass three major stages: analysis with partial tooth models, analysis with single tooth
gear models and analysis of multi-teeth gears in a complete mesh cycle. With continuing
software and hardware developments, it is expected that advances with gearing analysis
will involve FEA 4D models associated with coupled field problems and non-linear
materials etc., over the next few years.

83
With the current modelling capability, it is possible to predict the combined torsional mesh
stiffness of two spur gears (multi-teeth) in mesh, where one of the gear hubs is restrained
from rotating, with the other gear hub having a torque input load. The combined torsional
mesh stiffness of two gears in mesh is calculated at each selected position in the mesh
cycle, and the overall FEA solution shows that the combined torsional mesh stiffness varies
with the meshing position as the teeth rotate within the mesh cycle. In particular, the
combined torsional mesh stiffness decreases and increases dramatically as the meshing of
the teeth changes from the double pair of teeth in contact, to the single pair of teeth in
contact and vice-versa (Sirichai 1999).

6.2 The Individual Torsional Stiffness

In order to understand the combined torsional mesh stiffness, the variations of the
individual torsional stiffness for each of the gears in the mesh cycle have to be studied.
However, to predict the individual torsional stiffness for one of the gears in mesh is a rather
complex procedure, due to the non-linear contact. The actual position of the contact(s) is
usually unknown until the solution for both gears in mesh is completed.

A simple strategy for overcoming the difficulties can be developed, with a small torque
input load, where one of the meshing gears can be modelled with rigid elements. In this
case, the solutions for the combined mesh stiffness in the mesh cycle are those given by the
individual torsional mesh stiffness. When the input load is large, there certainly are errors
in this strategy, and the relative error has to be estimated.

6.2.1 The Use of Rigid Elements

The use of rigid elements involves the use of MPC (multipoint nodal constraints) without
the analyst having to write out the constraint equations. This capability is available in
ANSYS® and in many other codes. It’s important to note that the term “rigid element” may
be somewhat misleading. Recent publications have used very stiff elements instead of
using MPC based rigid elements in gear stiffness studies, where the elements are
essentially rigid relative to other elements in the mesh. However, as stated in section 3.2.2,

84
the connection of a very stiff element with a very flexible one is not recommended, due to
the associated numerical problems.

The advantages of using “rigid elements” are that they allow the imposition of common
multipoint constraints without having to actually write out the constraint equations. Since
they are essentially MPC’s, they avoid numerical errors associated with matrix ill
conditioning that very stiff elements would cause (Lepi 1998).

6.2.2 The Individual Torsional Stiffness of a Single Tooth Gear Model

If one of the mating gears is rigid, and the input torque is small, the elastic gear will
produce a combined torsional mesh stiffness that is at least close to the real individual
torsional stiffness with the same input load (note: the combined torsional mesh stiffness is
not constant with input load, as discussed in a later section).

In the 2D case, modelling one of the mating gears as rigid (as for the driven gear in Figure
6.2.1) is as simple as modelling a line segment AB with rigid contact elements with
compatible mesh density. Table 4.1 has given the parameters for the gears in this model.

Figure 6.2. 1 The model for prediction of individual torsional stiffness

(The driven gear is simplified as a rigid line AB).

As shown in section 4.2, the input torque load, boundary conditions and nodal couplings
are applied on the drive gear (pinion) hub. In order not to over constrain the model, line
AB should be left “free” after meshing with rigid line elements. The first solution is
calculated at the mesh position of the pitch point (0 degrees, as shown in Figure 6.2.1), then
the pinion is rotated with an angle increment and the line AB is rotated about O 2 for the
next solution. There were 107 angular increments used here to cover the mesh cycle.
85
Calculations were carried out by using a looping program written with APDL (ANSYS
Parametric Design Language). In ANSYS®, the solution for angular displacement is given
by tangential displacement values under the defined polar coordinate system. Because the
nodes on the drive gear hub were coupled with the master node in rotation, they will all
have the same tangential displacement value UY as the master node. The program
automatically saves each solution (UY) in the file such as PRNSOL.lis. When the looping
is finished, the solution data will be input to Microsoft Excel so that a further calculation
for the stiffness can be done as well as graphical representation of the numerical solutions
produced.

As given, the hub radius was r = 15mm . If Dq denotes the elastic angular rotation of the
pinion hub, then
UY = r * Dq , (6.1)
so, the stiffness can be calculated by,
K = M / Dq = M * r / UY , (6.2)
where M is input torque load.
Figure 6.2.2 and Figure 6.2.3 give the results for this single tooth model.

5.00E-03 UY-0.45Nm
UY-1.8Nm
4.50E-03
UY-4.5Nm
UY-22.5Nm
4.00E-03
UY-45Nm
3.50E-03 UY-76.2Nm

3.00E-03
UY (mm)

2.50E-03

2.00E-03

1.50E-03

1.00E-03

5.00E-04

0.00E+00
-13 -9.75 -6.5 -3.25 0 3.25 6.5 9.75 13
ANG (degree)

Figure 6.2. 2 FEA solution data for displacements under varying load.
86
 500000 K-0.45Nm
K-1.8Nm
K-4.5Nm

Individual Torsional Stiffness (Nm/rad)

450000
K-22.5Nm
K-45Nm
400000
K-76.2Nm

350000

300000

250000

200000
-13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13
ANG (degree)

Figure 6.2. 3 The individual torsional stiffness

From Figure 6.2.3 it is hard to tell there are six series of stiffness data that have been
produced by the series load. Taking the series produced by the minimum load as a basis
for comparison with the one produced by the maximum load, the relative errors can be
calculated as shown in Figure 6.2.4. The maximum error is only 0.0012%. This doesn’t
mean that with increasing input load the model can still produce accurate results, as
previous research (Sirichai 1999) has shown that load vs. stiffness results at positions in the
mesh cycle are slightly non-linear. Using rigid elements may reduce the system
nonlinearity.

0.0015 Relative Errors

Relative Errors (%)

0.001
0.0005
0
-0.0005-15 -10 -5 0 5 10 15

-0.001
-0.0015
Mesh Position (degree)

Figure 6.2. 4 Relative errors of the torsional stiffness between the

case of minimum and maximum load.

The results in Figure 6.2.4 have shown that within a wide range of input load, the model
can produce results for individual torsional stiffness as accurate as that of the minimum
loads.
87
6.2.3 The Individual Torsional Stiffness of a Multi-Tooth Gear Model

In order to obtain the individual torsional stiffness covering a complete mesh cycle, the
rigid line on the driven gear can be extended onto the next two teeth (the minimum can be
just one if there is no damage on the gear teeth). The meshed model is as shown in Figure
6.2.5. The driven gear in the meshed model is presented as three segments of rigid line
elements. Each segment is meshed with a compatible mesh density. The overall mesh
density was obtained with 4790 nodes.

Figure 6.2. 5 The model of meshing with multi-rigid lines.

FEA calculation procedures were the same as those used in previous models. The results
are given in Figure 6.2.6 and 6.2.7.
3.50E-03 UY_0.45Nm UY_1.8Nm
UY_4.5Nm UY_22.5Nm
UY_45Nm UY_76.2Nm
3.00E-03

2.50E-03

2.00E-03

1.50E-03

1.00E-03

5.00E-04

0.00E+00
-15 -12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5 15
Angular Position (pitch point represents 0 degree)

Figure 6.2. 6 FEA solution data for tangential displacements on pinion hub under various load.
88
 Double Zone (theoretical)
600000

Individual Torsional Stiffness

500000

400000
(Nm/rad)
300000
Single Zone K_0.45Nm

(76.2 Nm) K_1.8Nm

200000 K_4.5Nm
K_22.5Nm

Single Zone K_45Nm

100000
(0.45 Nm) K_76.2Nm

0
-15 -12.5 -10 -7.5 -5 -2.5 0 2.5 5 7.5 10 12.5 15
ANG (degree)

Figure 6.2. 7 The individual torsional stiffness produced by various loads.

The major trend for the individual torsional stiffness over a complete mesh cycle is shown
in Figure 6.2.7. It can be seen that the hand over points from single to double teeth contact
move slightly with increasing load, so that the single zone reduces when the input load
increases, while, the double zone is relatively stable. As expected, due to the unmodified
involute profile of the teeth (without tip-relief), the solutions for the hand over points are
difficult particularly for small load. With increasing load, the solutions tend to smooth out
at the hand over points, because for sliding contact the FEA solver handles pressured
sliding (surface to surface contact) better than a tip touching one (point to surface contact),
specially when the load option for the FEA solver was the default “ramped load”. For
example, if the model at the hand over mesh position was solved in eight sub-steps, the first
solution was solved with 1/8 input torque load. For this solution, the point to surface
contact may occur with large sliding, which may lead to a non-converging solution. This
may indicate that when the input torque is small, the choice for “step load” option (one take
all) may be a better option. However, if the solution converged, using the “ramped load”
option, the solutions were usually found to be more accurate.

6.3 The Combined Torsional Mesh Stiffness

6.3.1 Derived From The Individual Torsional Mesh Stiffness

Instead of rigid line elements, now consider there is a single tooth elastic driven gear pair
(ratio 1:1) and its individual torsional stiffness variation in the mesh cycle has a similar

89
curve to the previous results but with reverse variation. The cross over point for the curves
should be the pitch point, as shown in Figure 6.3.1.

Figure 6.3. 1 The torsional stiffness of gear and pinion

At any position in the mesh cycle, gears can be modelled as two torsional springs
connected in series. The system stiffness against input torque, so called the combined
torsional mesh stiffness at each position, can be calculated by the following equation,
K p × Kg
Km = , (6.3)
K p + Kg

where K p is the individual torsional stiffness of the pinion and K g is the individual

torsional stiffness of the gear. K m denotes the combined torsional mesh stiffness and the

result is plotted in Figure 6.3.2.

250000
Combined Torsional Mesh Stiffness

200000

150000
(Nm/rad)

100000

50000
Km
0
-13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13
ANG (degree)

Figure 6.3. 2 The combined torsional mesh stiffness for single pair of teeth in contact.

Similarly, once the individual torsional stiffness of a multi – tooth pinion over a complete
mesh cycle is obtained, the stiffness for the gear can be produced as shown in Figure 6.3.3,
as long as the mating gears are the same.

90
 Figure 6.3. 3 The individual torsional stiffness of pinion and gear in a complete mesh cycle.

Equation (6.3) can be used to calculate the results for the combined torsional mesh stiffness
for different torque loads. Figure 6.3.4 shows the combined torsional mesh stiffness results
for a minimum and a maximum input load.

Double Zone

Single Zone
(76.2 Nm)

Single Zone
(0.45 Nm)

Figure 6.3. 4 The comparison of combined torsional mesh stiffness results produced
by the minimum load and maximum load.

From the results shown in Figure 6.3.4, the mesh position ANG varies over the complete
mesh cycle. The dramatic change of the combined torsional mesh stiffness can be seen to
occur near the hand over points from single to double teeth in mesh and vice versa. The
results tend to be flat over the zone when the single pair of teeth is in contact, which is
identical to the single tooth gear model results shown in Figure 6.3.2. Near the hand over
points, a difference occurs between the stiffness results produced by the different loads.
91
This result has also been shown in the individual torsional stiffness curves shown in Figure
6.2.7. The curves also tend to be symmetrical about the pitch point, with improvement of
the model and its solution.

6.3.2 Models With Flexible Contact

The 2D model in Figure 6.3.5 has been used to calculate the combined torsional mesh
stiffness. After applying the adaptive re-mesh with the mating position (similar to that
shown in Figure 3.4.4), the contact line elements were created in the possible contact areas
with the Real Constants of the materials.

Figure 6.3. 5 The single tooth mating gears.

The mapped mesh associated with the adaptive re-mesh near the contact areas was the main
strategy for obtaining accurate results in this research. One of the major requirements for
the use of mapped mesh was that the model solution convergence could be easily
examined. When using mapped mesh it is essential that a monotonic convergence curve be
obtained. For the comparison, the results obtained with the free mesh model are also
presented. The free mesh results were only calculated with an input load of 76.2 Nm.
Results are given in Figure 6.3.6 and Figure 6.3.7.
0.009

0.008 UY(4.5Nm-
adaptive)
0.007
UY(22.5Nm-
0.006 adaptive)
UY (mm)

0.005
UY(45Nm-
0.004 adaptive)

0.003 UY(76.2Nm-
adaptive)
0.002
UY(76.2Nm-
0.001 free mesh)
0
-13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13
Roll Angle (degree)

Figure 6.3. 6 The (tangential) displacement of pinion hub under various loads.
92
 250000

Combined Torsional Mesh

200000

Stiffness (Nm/rad)
150000

100000

K_4.5Nm K_22.5Nm
50000 K_45Nm K_76.2Nm
K_76.2free

0
-13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13
ANG (degree)

Figure 6.3. 7 The combined torsional mesh stiffness results.

Figure 6.3.7 has shown that the combined torsional mesh stiffness increases slightly with
the increase of the load, mainly due to the increasing amount of area in the contact region.
At the pitch point the stiffness was predicted to increase by 0.02% from the minimum input
load to the load of 76.2 Nm. It would be expected that this would further increase if the
input load continued rising in the elastic range of the structure. On the other hand with free
mesh (Wang 2000), it was hard to produce solutions with such small variations and the
solution error of 0.07% was found at the pitch point compared with the adaptive mesh.
Further comparison, taking into account the stiffness derived from the individual torsional
stiffness of the single tooth gear model (contact with rigid line), is shown in Figure 6.3.8.

250000
Combined Torsional Mesh

200000
Stiffness (Nm/rad)

150000

100000

K_4.5Nm K_22.5Nm
50000 K_45Nm K_76.2Nm
K_76.2free

0
-13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13
ANG (degree)

Figure 6.3. 8 Comparisons of the stiffness with the calculated stiffness.

It can be seen that the calculated stiffness results are significantly different from the
adaptive mesh results, but they are at least as accurate as the free mesh results.
93
6.3.3 The Problems with Single Tooth Gear Models

A number of problems occur when using single tooth models for producing the combined
torsional mesh stiffness over a complete mesh cycle. Three examples of single tooth
models used in previous research by Kuang (Kuang 1992), Arafa (Arafa 1999) and Wang
(Wang 2000) are presented in Figure 6.3.9, 6.3.10 and 6.3.11.

Figure 6.3. 9 Kuang’s model and result (Kuang 1992).

Figure 6.3. 10 Arafa’s model and results (Arafa 1999).

94
Figure 6.3. 11 Previous model of this research and its results, Wang’s model (Wang 2000).

The common error in the above examples is that they incorrectly predict the combined
torsional mesh stiffness in the zone of double pair of teeth in contact, because the stiffness
of the mesh is far too high. Such high stiffness was developed by parallel superposition of
the corresponding single tooth stiffness of the tooth pairs in contact. It was expressed as,
K m = K sA + K sB . (6.4)

The equation above can be illustrated as shown in Figure 6.3.12,

Figure 6.3. 12 The illustration of equation (6.4).

where K sA , K sB stand for the combined torsional mesh stiffness of the single tooth gear

model at mesh position A and B respectively. It can be seen that the error results from each
95
of the two teeth sharing the one gear body, resulting in the high stiffness by parallel
calculation.

Another type of prediction for the combined torsional mesh stiffness when the double tooth
pair are in contact, is calculated by combining the individual torsional mesh stiffness of the
double tooth pair K pA, B (for pinion) and K gA, B (for gear) as springs connected in series,
A,B
where the superscript denotes the contact point of each tooth pair along the path of
contact of the double tooth pair contact zone. This is illustrated as shown in Figure 6.3.13.

K Ag K Ap

K Ap ,B ´ ×K gA ,B
K Ag ,B K Ap ,B K Am ,B =
K Ap ,B + K Ag ,B

K Bg K Bp

Figure 6.3. 13 A parallel spring connected in series (Sirichai 1999).

Once again, the parallel calculation was used for K pA, B and K gA, B which will result in much

higher stiffness. If K pA, B and K gA, B are calculated as results that have been shown in Fig.

6.3.3, there wouldn’t be a large error except at the hand over points. However, in this
section, it is interesting to use the previous results to generate a closed form calculation to
predict the combined torsional mesh stiffness in the double pair zone.

In most cases, gear body stiffness is a significant component of the total torsional stiffness
of a gear. According to the analysis here, the gear body stiffness is 2-4 times higher in
magnitude than a single tooth on its own. In general, the gear body stiffness will not be a
constant value in the single zone or in the double zone and between these two zones there
would be a maximum change in the stiffness. Recall from chapter 4, that the constraint on
the gear body from rotation within one tooth space to two teeth space resulted in the
torsional stiffness increasing by 8.6%. This shows that the torsional stiffness variations of
the gear body in the double zone (or in the single zone) is believed to be smaller than 8.6%.
96
Now, an assumption can be made by assuming the torsional stiffness of the gear body has a
constant value in the double zone of K B = 755709.74 Nm/rad (See section 4.4). The
nominal tooth stiffness can then be calculated in the single tooth pair mesh cycle. The
nominal tooth stiffness can be defined as a partial tooth stiffness, which is connected with
the gear body stiffness in the manner of a series spring in the single tooth mesh zone. The
individual nominal tooth stiffness can then be calculated with the results in Figure 6.2.3. It
also can be calculated as combined nominal tooth stiffness with the results in Figure 6.3.8.
The procedure will be simplified if we choose to calculate the combined nominal tooth
stiffness, where the combined nominal tooth stiffness as expressed below,
K B ´ K m,s
K m= , (6.5)
K B - 2 K m,s

where K m , s represents the combined torsional mesh stiffness of the single tooth gear

model, in a single tooth pair mesh cycle, and K B represents the stiffness of the gear body.

K m , s can be the stiffness derived from the model of the single tooth gear in contact with the

rigid line as shown in Figure 6.3.8. The results are given in Figure 6.3.14.
400000

300000
Stiffness (Nm/rad)

200000

Combined nominal tooth stiffness(origin from one tooth gear with rigid line)
100000
Combined nominal tooth stiffness(from one tooth gear in mesh)
Combined torsional mesh stiffness (derived from one tooth gear with rigid line)
Combined torsional mesh stiffness (for one tooth gear in mesh)

0
-13 -9.75 -6.5 -3.25 0 3.25 6.5 9.75 13

Roll Angle (degree)

Figure 6.3. 14 Combined nominal tooth mesh stiffness.

Here, the combined torsional mesh stiffness of the gears (as shown in Figure 6.3.12, in the
double zone) has been separated as three portions, body-teeth-body, as illustrated in Figure
6.3.15.

97
Figure 6.3. 15 A parallel spring connected for meshing teeth and in series with both gear bodies.

As shown above, the combined torsional mesh stiffness for the double pair of teeth in mesh
can then be calculated as,
A, B
KB ´ K m
Km = A, B
, (6.6)
K B + 2K m
A, B
where K m is calculated as a parallel spring, and K mA and K mB are calculated in the mesh

positions A and B respectively. The combined mesh stiffness K m is plotted as shown in

Figure 6.3.16.
250000
Combined Torsional Mesh Stiffness

200000

150000
(Nm/rad)

100000

50000 Km (from one tooth gear mesh with the rigid line)
Km (from one tooth gear in mesh)
0

-14 -10.5 -7 -3.5 0 3.5 7 10.5 14

ANG (degree)

Figure 6.3. 16 The combined torsional mesh stiffness.

In Figure 6.3.16, the single zone data was taken from Fig. 6.3.8 so that the combined
torsional mesh stiffness in a complete mesh cycle has been produced. Due to the use of two
different data sources, the combined torsional mesh stiffness has an overall 4.5%
difference. With the data sources from the elastic meshing teeth, the result of this

98
calculation is at least as accurate as the one derived from the individual torsional stiffness
that has been shown in Figure 6.3.4, but the calculations here are much easier. The basic
stiffness sharing inside the meshing gear pair as that given in Figure 6.3.15 has been proven
and it can be considered as one of the quick methods to evaluate the basic gear stiffness
property.

6.4 The Complete FE Gear Model with Flexible Contact

The complete FE model of a meshing gear pair with flexible contact is more realistic than
the previous models in simulation of the combined torsional mesh stiffness for gears in
mesh. For undamaged gears in mesh, FEA calculations for the displacements are carried
out over half of a complete mesh cycle. A complete mesh cycle for the test gears is 24.912
degrees. If the FE model contains a tooth crack, FEA calculations will be carried out over
a complete mesh cycle, which must cover the entire damaged tooth.

6.4.1 2D Modelling

Here, the plane stress assumption was used in the 2D FE modelling gears in mesh with
flexible contact as shown in Figure 6.4.1.

Figure 6.4. 1 The loads, boundary conditions, coupling and reaction forces of
the 2D plane stress model.

The ANSYS® eight node quad 183 series elements were chosen to produce a primary
mapped mesh. Within a looping program, adaptive re-mesh with contact was used at each
mesh position. The results were automatically saved to the computer network disk. At the
end of the loop, the program searched for the input load increments for the starting of the
99
next loop. The solutions for the displacements under an input load of 5Nm to150Nm (5Nm
increment) were repeated over the complete mesh cycle, as shown in Figure 6.4.2.

150Nm
0.0008
145Nm
140Nm
135Nm
0.0007 130Nm
125Nm
120Nm
115Nm
0.0006 110Nm
105Nm
100Nm
95Nm
0.0005
90Nm
85Nm
80Nm
0.0004 75Nm
70Nm
65Nm
60Nm
0.0003 55Nm
50Nm
45Nm
40Nm
0.0002 35Nm
30Nm
25Nm
0.0001 20Nm
15Nm
10Nm
5Nm
0
-16 -12 -8 -4 0 4 8 12 16
Roll Angle (degree)

Figure 6.4. 2 The angular rotations of the input gear hub in a complete
mesh cycle under various input loads.

The FEA solution was for q , the angular rotation of the input gear hub in the mesh cycle
under the input load series. It can be seen, from Figure 6.4.2 that the angular rotation
appears to increase linearly in the single zone and double zone with the increase of the
input load. With the results of the displacements from Figure 6.4.2, the combined torsional
mesh stiffness over the complete mesh cycle can be produced for the various input torque
loads, as shown in Figure 6.4.3.

240000

220000
Km (Nm/rad)

200000

180000
Km_5Nm Km_15Nm Km_30Nm Km_45Nm
Km_60Nm Km_75Nm Km_90Nm Km_105Nm
Km_120Nm Km_135Nm Km_150Nm
160000
-16 -12 -8 -4 0 4 8 12 16
R o ll A ngle ( de gre e )

Figure 6.4. 3 The combined torsional mesh stiffness over a complete mesh
cycle produced by various input torque loads.
100
In previous research, the combined torsional mesh stiffness has been assumed to be
constant at each mesh position, so that the combined torsional mesh stiffness was
independent of the input torque load (Du 1997; Sirichai 1997; Sirichai 1999; Svicarevich
1999; Jia 2000). Using the adaptive re-mesh with contact, accurate results have been
obtained as shown in Figure 6.4.3, where it has been shown that the stiffness curve varies
with the input torque load. In these solutions, except for the hand over points, the
maximum stiffness difference in the (common) single and double zone is only about
0.01%. So, for most metallic gears in mesh, this difference might be ignored. It should be
noted however, that the hand over points of the stiffness curve change their positions with
various input torque loads. For the input torque variation from 5Nm to 150Nm, as given in
Figure 6.4.3, the results show nearly one degree of difference.

6.4.2 3D Modelling

Due to the difficulty in solution convergence at the hand over points, the 2D FE model
hasn’t produced enough converged solutions of the stiffness in the hand over points, as
seen in Figure 6.4.3. For more detailed analysis, 3D modelling using brick elements and
adaptive re-mesh with contact was carried out as shown in Figure 6.4.4. where it was
found to be particularly useful for overcoming the difficulty in solution convergence.

Figure 6.4. 4 The details of the 3D model.

101
The elastic angular rotations of the input gear hub q were calculated under a series of input
torque loads. The solutions were for a complete mesh cycle with concentration on the
angular ranges near the hand over points. The results also compared with the previous
solutions (Wang 2000) that were produced by 3D mapped mesh as shown in Figure 6.4.5.

Adaptive152.4Nm
0.0008

MappedMesh152.4Nm

0.0007

Adaptive114.3Nm

0.0006
Adaptive76.2Nm

MappedMesh76.2Nm
0.0005

Adaptive38.1Nm

0.0004
MappedMesh38.1Nm

0.0003
Adaptive19.05Nm

MappedMesh19.05Nm
0.0002

Adaptive5Nm

0.0001
MappedMesh5Nm

0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

ANG (degree)

Figure 6.4. 5 The solutions for displacement produced by 3D models.

It can be seen that the 3D adaptive re-mesh with contact, using brick elements, produced
results much better than that of 3D mapped mesh, especially for high input load and near
the hand over points. It has been found that the hand over from the single zone to the
double zone or vice-versa gradually changes in a narrow region, the hand over region,
covering 0.02 degrees when the input load is 5Nm. The hand over region increases with
the increase of the input load, to cover 1.1 degrees for the input load of 150Nm. The data
produced by the displacement curves in those regions also show a change from convex to
concave curves between the input load of 38.1Nm or less and 76.2Nm or greater.

The combined torsional mesh stiffness then can be given as the raw data shown in Figure
6.4.6 and the simplified stiffness curves under various input loads were produced as shown
in Figure 6.4.7.

102
 250000

240000

230000

220000

Km 5Nm
210000
Km 19.05Nm
Km 38.1Nm
200000
Km 76.2Nm
Km 114.3Nm
190000
Km 152.4Nm

180000
-14 -10.5 -7 -3.5 0 3.5 7 10.5 14
ANG (degree)

Figure 6.4. 6 The torsional mesh stiffness as a function of load.

Figure 6.4. 7 The simplified curves of the combined torsional mesh stiffness.

For the study of the major characteristics of the combined torsional mesh stiffness, it is
necessary to produce the simplified stiffness curve as shown in Figure 6.4.7. The
simplifications include the following aspects,
· The stiffness difference due to the different input load, in the single zone and double
zone, were ignored, because for metallic gears in mesh the differences were less than

103
 0.02% (section 6.3.2, 6.4.1). So common curves such as lk, df and ji, can be developed
as shown in Figure 6.4.7.
· In Figure 6.4.5 the displacements in the hand over regions for the larger loads have been
shown to be concave, so the stiffnesses in these regions are truly non-linear. However,
for metallic gears, a linear approximation is close enough, as can be seen in Figure 6.4.7.
· With increasing input load, the hand over region changes only by the movement of the
bottom hand over point, such as position b moving along the common curve towards a.
The movement distances for the bottom hand over points as a function of input load can
be given as shown in Figure 6.4.8.

Figure 6.4. 8 The hand over regions versus input load.

Theoretically, point b as in Figure 6.4.7 will have the same angular mesh position as the
point l, if the input load is 0 rather than 5Nm. Points i, j, k and l are truly the theoretical
hand over points. While the actual hand over points always fall in the bottom of the
common curve (single zone), they represent when the second pair of teeth come into
contact and start to share the load. Now, for perfect involute gears in mesh, the hand over
region can be defined as the region of the mesh position that is between the actual and
theoretical hand over points.

For a certain load condition, the (variation of) hand over region will be highly reliant on the
material property. It could be very narrow for steel gears in mesh, while for non-metallic
gears in mesh, for example nylon gears under a certain temperature; their hand over regions
can reach or cross each other so that the single zone of the stiffness curve can be
significantly reduced, as discussed further in chapter 8.

104
7.1 INTRODUCTION

The term transmission error is used to describe the difference between the theoretical and
actual input and output angular motion of gears in mesh, as shown in Figure 7.1.1. At low
speed, the angular motions can be represented as angular positions of the input and output
ends, so that the transmission error can be expressed in angular units as shown in equations
(1.1).

Figure 7.1. 1 Illustration of gears in mesh.

105
Transmission error is considered to be one of the main important causes of gear noise and
vibration. Numerous works have been published on gear transmission error measurement
(Gregory 1963; Hayashi 1979; Hayashi 1981; Vinayak 1992; Daniewicz 1994; Munro
1994; Yau 1994; Velex 1995; Munro 1997). Experimentally, rolling gears together with
backlash operating at their proper centre distance have been used for transmission error
testing. The input and output angular motion characteristics of the gears are normally
measured by encoders and associated electronics. The data can be presented in graphical
analogue form or can be further processed by Fast Fourier Transform (FFT) techniques to
aid in pinpointing the source of excitation.

However, the measurement of transmission errors from test rigs has traditionally not shown
a great amount of detail within each meshing cycle. Secondly, the manufacturing
geometrical errors such as spacing errors and run out error can be much greater than the
loaded transmission error. Therefore the results of gear transmission error measurement
from the test rig is greatly influenced by manufacturing irregularities, as noted in previous
test rig results (Gregory 1963; Tordian 1967; Hayashi 1981; Houser 1989; Rebbechi 1992;
Vinayak 1992; Bard 1994; Barnett 1994; Munro 1994; Velex 1995; Houser 1996; Sweeney
1996).

Two types of transmission error are commonly referred to in the literature. The first is the
manufactured transmission error, which can be obtained for unloaded gear sets when they
rotate in single flank contact. The manufactured transmission error is affected most by
profile inaccuracies, spacing errors, and gear tooth run out. Gears that are rigid and that
have perfect involute profiles and no spacing or run out errors should produce a perfectly
straight transmission error trace, which would result in a spectrum with no peaks at discrete
frequencies. It has also been shown that there is a direct relation between the manufactured
transmission error and noise, (Welbourn 1972; Welbourn 1979; Smith 1983; Smith 1987;
Smith 1987; Baron 1988).

The second is the loaded transmission error, which is similar in principle to the
manufacturing transmission error but takes into account tooth bending deflection, shearing

106
displacement and contact deformation due to load. When gears operating at low speed are
loaded, two additional factors contribute to the transmission error (Townsend 1991),

· A constant component due to the mean tooth compliance. This component is of

major significance in choosing appropriate profile modifications, but of much
less significance with regard to its contribution to mesh frequency noise,
and
· A time-varying component that is a function of gear tooth geometry and
torsional mesh stiffness variation, as well as the manufactured transmission
error. This component contributes heavily to mesh frequency noise.

As gears are run at higher speeds, a dynamic component that is a function of the system
dynamics, (Ozguven 1988), must be included with the low speed effects aforementioned.

From previous experimental results, the time varying component of transmission error,
which is periodic at the tooth mesh frequency, has been shown to relate to gear noise
amplitude (Smith 1983; Smith 1987). In fact, it has been shown that the transmission error
of spur gears, which have large changes in torsional mesh stiffness, can be reduced
significantly by applying appropriate profile modifications (Gregory 1963; Welbourn 1979;
Baron 1988; Alattass 1994; Oswald 1994; Litvin 1995; Wang 1996).

The primary purpose of this chapter is to develop the detailed static transmission error over
one completed cycle of mesh, in which the effect of the tooth profile modifications is
included. The method developed for FEA solution of the detailed static transmission error
with tooth profile modifications has general applicability for solutions relating to the
components like profile, spacing and run out errors, and it can also be used for creation of
gears with new tooth profiles. It has been noted that the related components above are all
in the first order, in contrast to the elastic deformations that are higher order components.

For the ongoing research, it is necessary to re-classify the components that contribute to the
total (static) transmission error,

107
 1. First order components: including profile, spacing and run out errors from the
manufacturing process. Geometric errors in alignment and tooth profile
modifications are also classified here and they typically add “rigid body motion”
into the total transmission error.
2. Higher order components: including the elastic deformation of the local tooth
contact, tooth bending, shearing, some rotation about the tooth root and the
deflection of the gear body due to the transmitted load through and transverse to the
gear rotational axis.
3. Higher order dependent components: the relative sliding at the contact(s) is a first
order component, however, this component is dependent on the variation of the
higher order components. This special component can also be classified into the
loaded transmission error, in contrast with the other first order components that can
be counted as unloaded transmission error. These components also include
geometric errors that may be introduced by static and dynamic elastic deflections in
the supporting bearings and shafts.
The higher order components store strain energy and some of the first order components
are likely to initiate the energy release with considerable speed with the potential to cause
system vibration. From the elastic strain energy point of view, the variations of higher
order components can transmit their influence to the whole system easily or cause global
effects, such as to the shafts, within various rotation speed ranges. The first order
components, especially when the load is light, tend to cause a more local effect and
produce noise. It has been shown that there is a direct relationship between noise and most
of the first order components, (Welbourn 1972; Welbourn 1979; Smith 1983; Smith 1987;
Smith 1987; Baron 1988).

7.2 Static Transmission Error for Perfect Involute Gears in Mesh

When gears are unloaded, a pinion and gear with involute profiles should theoretically run
with zero transmission error. However, when gears with involute profiles are loaded, the
combined torsional mesh stiffness of the gears change, as shown in chapter 6, causing
variations in angular rotation at the output gear hub or the shaft. At each particular
meshing position, the angular rotation of the loaded drive gear due to tooth bending,
shearing and contact displacement is calculated in the gear reference frame by restraining
108
the driven gear from rotating. In relation to the drive gear reference frame, it is restrained
from further rotating, while the torque input load and the resulting angular rotation of the
gear is computed. The angular rotation is the static transmission error of gears under load
at low speed, which can be expressed in angular units as shown in equation (1.1). The
procedures above can be illustrated as in Figure 7.2.1, in which the bearings are assumed to
be rigid and shaft bending is negligible.

Figure 7.2. 1 With the gear ratio 1:1, static T.E. is shown as q p - q g .

For perfect involute gears in mesh, with rigid mounting and linear elastic material
properties, the relationship between the input torque T and the angular rotation of the gear
hubs q exists throughout every position of a mesh cycle as shown in equation (7.1),
T = K m ×q , (7.1)
where K m represents the combined torsional mesh stiffness. q represents the relative

angular position difference between the hubs of the mating gears in mesh which is due to
the pure elastic deformation. q remains zero only if the mating gears are perfectly rigid.
So, in a mesh cycle, q will denote the transmission error of the mating gears. In the 2D
modelling example shown in Figure 6.4.1, the transmission error will be the same as in
Figure 6.4.2 and is represented here as a function of torque input loading as shown in
Figure 7.2.2.

109
 150Nm
0.0008
145Nm
140Nm
135Nm
0.0007 130Nm
125Nm
120Nm
115Nm
0.0006 110Nm
105Nm
100Nm
95Nm
0.0005
90Nm
85Nm
80Nm

0.0004 75Nm
70Nm
65Nm
60Nm
0.0003 55Nm
50Nm
45Nm
40Nm
0.0002
35Nm
30Nm
25Nm

0.0001 20Nm
15Nm
10Nm
5Nm
0
-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16
ANG (degree)

Figure 7.2. 2 The transmission error of mating gears over a complete mesh
cycle under various input loads

Further investigations of the transmission error were accomplished using 3D modelling of

the mating gears. Due to the time consuming nature of solving 3D models, the input loads
of 5Nm, 19.05Nm, 38.1Nm, 76.2Nm, 114.3Nm and 152.4Nm were considered. As shown
in Figure 3.13, the 3D modelling was based on half of the volume as separated by
symmetry. The 3D model was mapped meshed with 20 node brick elements containing
35456 nodes. Automatic mesh adaptation with contact was also used to increase the
solution accuracy and the computational efficiency. The adaptive meshed model contained
about 27000 nodes for one pair of teeth in contact and about 28000 nodes for two pairs of
teeth in contact. The 3D results of the transmission error were obtained, as shown in
Figure 7.2.3.

110
 Handover Regions
(double contact) Single Zone
Mapped mesh
(Theoretical) Double Zone (152.4 Nm)

Adaptive mesh
(152.4 Nm)

Pitch Point
Transmission Error (rad)

Adaptive mesh
(114.3 Nm)

Mapped mesh
(76.2 Nm)

Adaptive mesh
(76.2 Nm)

Mapped mesh
(38.1 Nm)

Adaptive mesh
(38.1 Nm)

Mapped mesh (19.05 Nm)

Adaptive mesh
(19.05 Nm)
Map_5 Nm
Adap_5 Nm

Roll Angle (degree)

Figure 7.2. 3 Static T.E. of 3D modelling with adaptive and mapped mesh.

It can be seen in Figure 7.2.3, that the major difference between the two methods is the
results in the hand over region (details about the hand over region can be referred to in
section 6.4.2). The mapped mesh results were observed to be unsymmetrical about the
pitch point (0 degrees), by up to 0.5 degrees and abrupt changes from the single zone to the
double zone and vice-versa were also obtained. The adaptive mesh results produced much
smoother symmetrical curves and more gradual changes in the hand over regions. With the
input loads increasing from 5Nm gradually to 152.4Nm, the following conclusions were
obtained from the adaptive mesh results:
1. The convex nature of the curve in the hand over region gradually changes from
positive to negative convexity.
2. The changes from the double zone to the hand over region (or vice-versa) become
smoother, while the changes from the single zone to the hand over region (or vice-
versa) get sharper as the load increases.

111
 3. It was observed from the results that the length of the single zones were decreased
as the load increased by an amount up to 2 degrees, while the double zones were
relatively stable in their size. The hand over regions were observed to extend into
the single zone region.
The comparisons can also be made between the 2D and 3D modelling results using
adaptive mesh as shown in Figure 7.2.4. The 2D results show similar features as shown for
the 3D results; however, the 2D modelling produced fewer details in the hand over region.
In comparison with the results from 3D modelling with the mapped mesh, the 2D adaptive
mesh results can be seen to be more reasonable. The difference between the 2D and 3D
modelling becomes obvious when the input load is large, with a 4.31% relative error in the
magnitude of the transmission error being found in the single zone with the input load of
152.4 Nm.

2D Results
3D Results
152.4 Nm
Transmission Error (rad)

114.3 Nm

76.2 Nm

38.1 Nm

19.05 Nm

5 Nm

Roll Angle (degree)

Figure 7.2. 4 T.E. results comparison between 2D and 3D modelling
(adaptive meshes applied on both models).

It should be noted from this analysis that 2D modelling with adaptive mesh is able to
produce acceptable results for transmission error studies when compared with 3D
modelling. It can be achieved with greatly reduced computational time when using 2D
analysis.

112
7.3 Detailed Hand Over Process of Involute Gears in Mesh

The analysis has shown that the use of adaptive meshing is able to reveal more details
about the change over between the single zone and the double zone, defined as the hand
over region. However, previous research on gear dynamic models has ignored the effect of
gear flexibility (hand over regions) (Cornell 1978; Kubo 1980; Terauchi 1981; Yang 1985;
Velex 1989; Lin 1989a; Du 1997; Svicarevich 1999; Jia 2000; Howard 2001; Jia 2001)
This means the zone of tooth contact and average tooth mesh stiffness are underestimated,
and the individual tooth load is overstated. The analysis by Lin et al. (Lin 1994) shows that
neglecting this flexibility (hand over regions) results in underestimating resonant speed and
overestimating the maximum dynamic load, the errors are considerable within moderate
gear contact ratio (about 1.6) and will be significant with higher gear contact ratio.

The hand over region was also described by Lin (Lin 1994) as the extended tooth contact
region, as shown in Figure 7.3.1. Lin used the concept of T.E. separation to generate the
separation curves BSr and CSa for the case of recess and approach respectively as shown

in (a), then combined with the double zone T.E. and the reduced single zone T.E. to
produce the final T.E. curve as shown in (b). This work was a considerable progress in the
research field, especially for material with higher modulus under light load conditions.
However, there is a major limitation in use of the separation curves in this case due to the
incompatibility between the separation and the T.E. curves:
· TE separation, also known as the transmission error outside the normal path of
contact, TE o. p. c. (Munro 1999), has been calculated geometrically without

consideration of the elastic deformations. It has been found to be well suited for the
evaluation of the effects on T.E. caused by manufacturing errors, wear and other
geometric errors.
· The T.E. (curve) presented here (Figure 7.3.1) is the load T.E. of the perfect
involute spur gears, and is only depending on the load, boundary conditions and the
material flexibility.
So, the use of the T.E. separation is actually not suitable for describing the loaded T.E.
behaviours.

113
 The gap between the T.E. curve
and the separation curve is
observed. [The gap should be
load dependant in this case.]

Figure 7.3. 1 Static transmission error and separation distance for

gears with contact ratio = 1.64 (Lin 1994).

The handover region defined in this research is valid only when gears are loaded, and
evaluations can only be carried out with numerical methods such as FEA. The following
sections will give detailed features of the handover region.

7.3.1 Ratio of Local Deformation Over a Complete Mesh Cycle

The local deformation of the contact tooth is one of the most complex components of the
meshing gears. The ratio of local deformation represents the influence to the deflection of
the global gear system (total T.E.) by local contact(s). It also represents the character of the
meshing gears in different mesh positions.

In chapter 4 section 7, the ratio of local deformations in two particular mesh positions has
been defined. However, it is necessary to produce a curve for the ratio of local deformation
over a complete mesh cycle to enable us to further explore the mechanism of gears in mesh.
Local deformation q c is represented as shown in Figure 7.3.2, where some details can be

referred to chapter 4.

114
 Figure 7.3. 2 Illustration of the local deformation q c under input torque load T.

Generally, the components of the local deformation q c are elastic contact deformation and

the sliding at the contact points, the latter component caused by rigid body motion. When
the contact is at the pitch point, the amount of sliding can be ignored, and q c may be

considered as pure elastic deformation.

The FEA solutions of the local deformation over a complete mesh cycle were obtained
from 78 points covering a complete mesh cycle with several points occurring in the hand
over region. The local deformation q c was also compared with q , the solutions of the

complete elastic gears in mesh, as shown in Figure 7.3.3 when the input load was 76.2Nm.

0.0004

0.0003
TE (rad)

TE of the elastic gears in mesh

0.0002 The local deformation

0.0001

0
-16 -12 -8 -4 0 4 8 12 16
Roll Angle (degree)

Figure 7.3. 3 The local deformation over a complete mesh cycle.

The curve of the local deformation is noted as being similar to that of the transmission error
of the elastic gears in mesh, and both 2D adaptive mesh results were found to be about
115
4.31% above the corresponding results obtained from the 3D models. As the investigation
here concerns the ratio of local deformation q c / q (see function 4.19), the difference from

the results obtained with 3D models would be minimized. The ratio of local deformation
for the input load 76.2Nm is given as shown in Figure 7.3.4.
30 Single zone
c

25
a1 a

20
Ratio (%)

b b1
15

10 Double zone

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.3. 4 The ratio of local deformation for the input load 76.2Nm.

It can be seen from the results that the ratio in the single zone aa1 averages close to 25%

while in the double zone bb1 the ratio averages approximately 16.7%. The hand over

region ab here is estimated to cover the range from 2.7 0 to 3.750 . The maximum value of
the ratio is 27.6% at c which is located at 3.20 , close to the middle of the hand over region.
The corresponding results for the increased input loads of 114.3Nm and 152.4Nm are given
in Figures 7.3.5 and 7.3.6 respectively. The ratio in the single zone was found to have
decreased to 24.5% and 24% respectively, approximately 1% lower than before the higher
load is used. The hand over region was found to have increased by 0.20 with position a
shifting from 2.7 0 to 2.50 as the load increased.
30
c
25
a
20
Ratio (%)

15 b

10

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.3. 5 The ratio of local deformation for the input load 114.3Nm.

116
 30 c

25
a
20

Ratio (%)
15 b

10

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.3. 6 The ratio of local deformation for the input load 152.4Nm.

An analysis of smaller loads was also conducted with a series of input loads varying from
38.1Nm to 1Nm. The input loads were unevenly spaced here, because the ratio of local
deformation in the hand over region was found to dramatically change within this range of
input load, especially when the input load was about 8Nm. The series of input loads used
here were 38.1Nm, 30Nm, 25Nm, 19.05Nm, 15Nm, 10Nm, 8Nm, 5Nm and 1Nm. The
ratio of local deformation is given for each input load, as shown from Figures 7.3.7 to
7.3.15 respectively, and combined as shown in Figure 7.3.16.
35

30

25
Ratio (%)

20

15

10

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.3. 7 The ratio of local deformation (load 38.1Nm).

35 30

30 25
25
20
Ratio (%)
Ratio (%)

20
15
15
10
10

5 5

0 0
-16 -12 -8 -4 0 4 8 12 16 -16 -12 -8 -4 0 4 8 12 16
A NG (degree) A NG (degree)

Figure 7.3. 8 The ratio of local deformation (load 30Nm). Figure 7.3. 9 The ratio of local deformation (load 25Nm).

117
 35 30

30 25
25
20
Ratio (%)

Ratio (%)
20
15
15
10
10

5 5

0 0
-16 -12 -8 -4 0 4 8 12 16 -16 -12 -8 -4 0 4 8 12 16
A NG (degree) A NG (degree)

Figure 7.3. 10 The ratio of local deformation (load 19.05Nm). Figure 7.3. 11 The ratio of local deformation (load 15Nm).

30 30

25 25

20 20
Ratio (%)

Ratio (%)
15 15

10 10

5 5

0 0
-16 -12 -8 -4 0 4 8 12 16 -16 -12 -8 -4 0 4 8 12 16
A NG (degree) A NG (degree)

Figure 7.3. 12 The ratio of local deformation (load 10Nm). Figure 7.3. 13 The ratio of local deformation (load 8Nm).

30 30

25 25

20 20
Ratio (%)

Ratio (%)

15 15

10 10

5 5

0 0
-16 -12 -8 -4 0 4 8 12 16 -16 -12 -8 -4 0 4 8 12 16
A NG (degree) A NG (degree)

Figure 7.3. 14 The ratio of local deformation (load 5Nm). Figure 7.3. 15 The ratio of local deformation (load 1Nm).

Throughout the results above, the maximum ratio of local deformation of 29% was found
in the hand over region when the input loads were in the region between 38.1Nm and
19.05Nm. While the hand over region decreased with the decrease of the input load, the
hand over region was reduced to 0.50 which was about half of the maximum size of the
hand over region when the input load was 30Nm. Further decreases in the input load from
15Nm to 10Nm, resulted in the maximum ratio of local deformation being reduced from
28% to 26% respectively and the positions remained in the hand over region. When the

118
input load was 8Nm, no significant abrupt changes to the ratio of local deformation were
observed in the hand over region. Finally, when the input load was dropped to 5Nm and
1Nm, the minimum ratio of local deformation of 15.4% and 14.9% were found
respectively. The minimum ratio was found to be located at the same position of 3.50
which was still inside the hand over region. Ignoring the hand over region, the ratio of
local deformation in the single and double zones were relatively steady for the input load of
38.1Nm and less, with average values of 25% for the single zone and 17% for the double
zone.

The overall configuration of the ratio of local deformation with the input loads from 1Nm
to 152.4Nm is given, as shown in Figure 7.20. It has been shown that the abrupt changes to
the deformation ratio as a function of mesh angle mainly occurred in the hand over region.
The variations of the ratio of local deformation were in agreement with the previous
research on the variations of the hand over region as discussed in section 7.2.1.

The large ratio of local deformation indicates that contacts at (or near) the tooth tips consist
of large elastic (or plastic) deformation as well as large sliding. This result may be useful
for further research into the failure analysis of gear flank face conditions. Further study
into the ratio of local deformation may also help to design a suitable tooth tip relief so as to
modify the shape of the gear transmission error.

7.3.2 Detailed Analysis – T.E. and Load Sharing Ratio

Section 7.2 has proven the existence of changes within the hand over regions by using
adaptive meshing and desired results can be achieved with finer level of 3D adaptive
meshing as long as the computational time is not excessive. 2D adaptive mesh models can
produce similar results to that of 3D modelling, but less detail will be observed in the hand
over regions, unless a finer level adaptive mesh is carried out. A finer 2D adaptive mesh
model was conducted here with a mesh density of up to 21000 nodes with the element
dimension near the contact(s) of about 0.033 mm. More realistic stress fields near the
contact(s) should be observed with such a fine mesh, as shown in Figure 7.3.17.

119
Figure 7.3. 16a
120
 Figure 7.3. 17 More realistic stress fields near the contact(s) with a finer adaptive mesh.

The precision FEA results of T.E. (as shown in Figure 7.3.18) are able to provide detailed
variations of the critical points that are positioned at the edge of the single zone for various
input loads. Those critical points include both the recess case (start of the single zone) and
the approach case (end of the single zone). The results listed in Table 7.3.1 have shown
that the recess case is slightly different from that of the approach case as discussed further
here; i) Minor difference between the T.E. values. When the input load is 5 Nm or greater,
the T.E. results for the recess case are slightly larger than for the approach case. When the
input load is 1 Nm, the greater value of T.E. is found at the approach case.

Single zone edge

points Recess Approach
Input load Position (degree) T.E. (rad) Position (degree) T.E. (rad)
152.4 Nm -2.55 8.6856E-04 2.64 8.6505E-04
114.3 Nm -2.75 6.5625E-04 2.8 6.5200E-04
76.2 Nm -2.9 4.4283E-04 2.93 4.4142E-04
38.1 Nm -3.15 2.2461E-04 3.15 2.2435E-04
19.05 Nm -3.3 1.1312E-04 3.3 1.1283E-04
5 Nm -3.5 2.9675E-05 3.5 2.9546E-05
1 Nm -3.635 5.9157E-06 3.63 5.9473E-06
* Position is counted as the rolling distance away from the pitch point.

Table 7.3.1. Details of the single zone edge points under various input loads.

However, the differences between the T.E. of recess and approach for each input load are
very small, as the maximum (relative) difference of 0.7% occurs when the input load is
114.3 Nm. ii) Difference between their sizes. When the input load is 1 Nm, the hand over
region is very small.

121
 Figure 7.3. 18 The precision analysis results of T.E. under various loads.

In the recess case, the hand over region is in between –3.64 to –3.635 degrees as counted
from the pitch point (the pitch point is at 0 degree). While for the approach case it is
between 3.63 to 3.635 degrees. When the input load is 76.2 Nm or greater, the results
show different expansion rates of the hand over regions. When the input load is 152.4 Nm,
there is 0.99 degree in difference between the single zone edge points that makes the T.E.
asymmetrical to the pitch point, and the width of the hand over regions are 1.085 and 0.99
degrees for the recess and the approach case respectively. It should be noted that the width
of the recess hand over region is always wider than that of the approach. Those features
above can be illustrated in Figure 7.3.19, however the T.E. differences cannot be visually
observed.

122
 Figure 7.3. 19 Diagram illustrates Table 7.3.1.

The same FEA calculations can also produce results for the load-sharing ratio, as shown in
Figure 7.3.20.

Figure 7.3. 20 Load sharing ratio under various input loads.

The variation of the load-sharing ratio can be noted in the change over between the single
and the double zone. Over a complete mesh cycle, four domains I, II, III and IV will be
produced by the load sharing curve sweeping with the input load from the minimum (1
Nm) to the maximum (152.4 Nm) load, as shown in Figure 7.3.21.

123
 Figure 7.3. 21 Domains of load sharing ratio under certain range input loads.

Theoretically, domain I = domain III for the approach cases and domain II = domain IV
for the recesses. The double zones remain relatively stable with load, whereas the single
zone results obviously change with load.

7.3.3 Conclusion

The existence of the hand over region is due to the flexibilities of the mating gears, so it has
a general application in geared transmission systems. In the hand over region, the
mechanism properties of the mating gears such as combined torsional mesh stiffness, ratio
of local deformation, load sharing ratio and the T.E. are changing rapidly as shown in
Figure 7.3.22. For metallic gears in mesh, these changes occur in a relatively narrow
range. However, some of the properties are difficult to predict, especially when the
material modulus is high. Approximate expression (T.E.) can be obtained (Lin 1994) as
long as the width of the hand over region is estimated correctly (in such a case, the
difference between the recess and the approach may be ignored). When the material
modulus is low, approximate curves may not be practicable. However, for wider handover
regions, the values of the properties are relatively easier to be predicted by numerical
methods. In general, it’s not desirable to have rapid changes of the mechanism properties.
Modifications to tooth profiles, including tip relief will be more important for material with
higher modulus values.

124
There are some points to note about Figure 7.3.22;
1. The combined torsional mesh stiffness is no longer showing the “common” path
(such as in Figure 6.30) due to the finer meshed FEA model.
2. The ratio of local deformation was produced by a different FEA model (coarse
meshed) so that the hand over regions are slightly different.

7.4 Tooth Profile Modifications (Tip Relief)

It has been discussed in 7.2.1 that the transmission error of the (pure involute) gears
changes rather dramatically in the hand over region with increased input load, where the
tooth tip runs into the mating tooth flank surface with large sliding (called edge or corner
contact). This dramatic change in transmission error is a major factor in subsequent tooth
flank surface spalls and system vibration and noise. For most metallic gears, such as steel
gears, the hand over region would be much narrower than the one shown here for the
aluminium gears. For steel gears the large ratio of local deformation and the abrupt nature
of the changes in the narrower region on the harder surface would give rise to high stress
and rapid failure (spall, crack).

Manufacturing errors can add to these effects so that it is necessary to relieve the tooth tip.
There are different types of tip relief according to various applications and requirements,
but they would have the same effects on the meshing gears:
· Altering the components of the ratio of local deformation by reducing the local
elastic deformation that causes high stresses, and avoiding corner contact.
· In extending the range of the sliding contact, an extra margin of the angular rotation
(of the input gear hub) is gained (specially) in the double zone. In other words,
rigid body motion is added in the double zone. That means it is possible to gain a
smoother curve of the gear transmission error with a certain input load.
If the first point were considered of more importance in the application, then a short tip
relief design would be chosen. If the second point or both were considered of primary
importance then one must choose a long tip relief design. Long and short tip relief will be
discussed in this section.

125
Figure 7.3. 22 The hand over process of involute gears (ratio 1:1) in mesh.
126
7.4.1 The History and Some Questions

Tooth profile modification of (involute) gears has been an accepted practice in gear design
since the late 1930s when the theory behind this was proposed by Walker (Walker 1938).
Figure 7.4.1 has shown his results on a short and a long profile modification.

0.0012

0.0008 Deflection 0.0010 Deflection

0.0006 Curves 0.0008 Curves
0.0006 d c2
0.0004 0.0004
0.0002 0.0002 d c1
0 0 0 0
0.0002 Modification 0.0002
0.0004 Curve 0.0004
Profile Modification
0.0006
Curves
0.0008
y

P c d M P c N d
I1 a I2 I1 a I2
b b

1.0 1.0
Proportion of Full Load

0.9
0.8 Load
0.8 Diagram
Load 0.7
0.6 Diagram 0.6
0.5
0.4 0.4
0.3
0.2 0.2
0.1
0
0

Angular 0.0012
0.001 Velocity 0.0010
0.0008 Variation 0.0008 Angular Velocity
0.0006 0.0006
0.0004 0.0004
Variation
0.0002 0.0002
0 0 0

Figure 7.4. 1 Walker proposed results on a short and a long profile modification(Walker 1938).

The most significant step forward in the understanding of gear kinematics was made in the
late 1950s by Harris (Harris 1958), who proposed the relationship between the T.E. of tip-
relieved gears and different input loads under quasi-static conditions, which was presented
as a series of plots against the gear rotation. The plots are now known as Harris maps and
have become a very powerful tool for clear understanding of gear kinematics and can help
to predict their dynamic behaviour. In the 1960s, Munro and Gregory (Munro 1962;
Gregory 1963) continued work on a back-to-back test rig for measurement of the

127
transmission error by means of optical gratings and considerable progress was made in
understanding spur gear dynamics. Subsequent research has put the design of profile relief
on a proper scientific basis (Niemann 1970; Munro 1990; Munro 1994; Rosinski 1994;
Yildirim 1994; Palmer 1995).

One recent considerable step forward was made by Lin et al. (Lin 1994; Lin 1994), who
came up with an analytical formula for calculation of T.E. and load sharing ratio as shown
in Figure 7.4.2 and presented a thorough analysis of its effects on spur gear transmission.

The profile modification made by Lin was anticipated with Walker’s long profile
modification which starts from the tooth top land and extends to the HPSTC (Highest Point
of Single-Tooth Contact). One of the questions is whether the HPSTC (including the
LPSTC - Lowest Point of Single-Tooth Contact) should be as, conventionally, the fixed
position of the tooth profile that consequently results in the contact ratio independent of the
input load. However, the results achieved with adaptive mesh (see section 7.2.1) have
shown the existence of the hand over region between the single and double zone where the
single zone will be reduced by the extension of the hand over regions with the input load
increasing, so that both the HPSTC and LPSTC will move toward the pitch point resulting
in the increase of the contact ratio.

Lin’s result does show that the single zone changes with the amount of normalized
modification D = 1.25 as shown in Figure 7.4.2. However, the amount of change of the
single zone, according to this research, is load dependent and contributes to the double
zone variations. So that, according to Lin’s calculations, the drop in contact ratio may not
be correct.

128
 AMOUNT OF
DOUBLE SINGLE DOUBLE MODIFI-
CATION. AMOUNT OF
.0010 CONTACT CONTACT CONTACT .0010 MODIFI-
25 25 CATION.
DOUBLE SINGLE DOUBLE
1.25 CONTACT CONTACT CONTACT
1.25
.0008 .0008
TRANSMISSION ERROR. I N.

TRANSMISSION ERROR. I N.
20 1.00 20

.
1.00
.75
TRANSMISSION ERROR.

TRANSMISSION ERROR.
.75
.0006 15 .50 .0006 15 .50

NO NO
PROFILE PROFILE
MODIFI- MODIFI-
.0004 10 CATION. .0004 10 CATION.
DOUBLE SINGLE DOUBLE DOUBLE SINGLE DOUBLE
CONTACT CONTACT CONTACT CONTACT CONTACT CONTACT

.0002 5 .0002 5

0 0 0 0
(a) STATIC TRANSMISSION ERROR. (a) STATIC TRANSMISSION ERROR.

.4 .4
1.25
1.25
TOOTH LOAD, NM/ M.

TOOTH LOAD, NM/ M.

2000 1.00 2000
1.00
TOOTH LOAD, LB/I N.

TOOTH LOAD, LB/I N.

.3 .75 .3 .75
1500 .50 1500
.50
.2 NO .2 NO
1000 PROFILE
1000 PROFILE
MODIFI-
MODIFI-
CATION.
CATION.
.1 500 .1
500

0 0 0 0
10 15 20 25 30 35 10 15 20 25 30 35
ROLL ANGLE. DEG ROLL ANGLE. DEG
(b) SHARED TOOTH LOAD. (b) SHARED TOOTH LOAD.
STATIC TRANSMISSION ERROR AND SHARED TOOTH STATIC TRANSMISSION ERROR AND SHARED TOOTH
LOAD FOR GEAR PAIRS WITH LINEAR TOOTH PROFILE LOAD FOR GEAR PAIRS WITH PARABOLIC TOOTH
MODIFICATION. FULL DESIGN LOAD; LENGTH OF PROFILE MODIFICATION. FULL DESIGN LOAD; LENGTH
MODIFICATION, Ln = 1.00. OF MODIFICATION, Ln = 1.00.

Figure 7.4. 2 Lin’s results of linear and parabolic profile relief (Lin 1994).

7.4.2 The Current Recommended Tip Relief Allowances

It is noted that excessive thinning of the tooth profile could result in a significant reduction
in load carrying capacity, and will result in a reduction in load sharing and this in turn can
lead to a decrease in bending fatigue strength (Walton 1995). The existing standards of
metallic gears such as British Standard (BS 1970), ISO (ISO/DIS 1983) and DIN Standard
(DIN 1986) (Figure 7.4.3) give the maximum amount of tip and flank modifications to
prevent the possible over – tipping. However, there are no precise recommendations for
applying the modifications in all these standards. Below these limits the designer is free to
choose the amount of relief and the form combinations.

129
 Mn p
0.005 Mn = Ca
0.02 MAX Mn p /2 Mn p /2 TIP RELIEF

0.6 MAX
0
20

Mn
2.25 Mn

0.3 Mn = DLn
1.25 Mn
RACK RACK

(a) BS 436: Part 2: 1970 (8). DATUM UHE Pf = 0.38 Mn

Normal Module Mn - 25.4/DP

Pressure angle a - 20o
p Fillet radius Pf - 0.38 Mn
0.02 MAX Tooth depth hp - 2.25 Mn
p /2 p /2 Gear addendum hap - Mn

0.6 MAX
Gear dedendum hfp - 1.25 Mn
1.00

0
20 Pitch P - p Mn
1.25

p
(b) ISO standard (9). r = 0.39 (c) DIN 876 (FEB 86).

Figure 7.4. 3 Tip relief for a metric module of 1 mm on the basic rack.

Figure 7.4.3 has shown that the British Standard (a) and the ISO standard (b) give the same
recommendations for the maximum relief on a basic rack of metric module 1 mm and all
the dimensions are a function of the actual module m as shown in (c) where M n = m.

However, the DIN standard has been shown to be more restrictive on tip relief.

7.4.3 The Form of the Tooth Profile Modification

There are various types of curves that could be used for the shape of the modification of the
gear tooth profile. Linear and parabolic curves are common for tooth profile modifications,
as shown in Figure 7.4.4.

It should be noted that the linear and straight-line relief are not the same as the line OT
(Figure 7.4.4 (b)) represents the pure involute curve. In application, gear companies often
supply gears (of manufacturing standard) with straight-line tip relief due to the small
amount of tip relief modification that is easy to manufacture. The shape of the
modification on the tooth profile can also be an involute curve with a limited minimum
radius under British Standard (revised standard No. 436 – 1940) as shown in Figure 7.4.5.

130
 Figure 7.4. 4 Example of linear and parabolic tooth modification.

Figure 7.4. 5 The basic rack of circular pitch 1 and the involute tip relief (p is the circular pitch).

Each different form of the curve used for tooth profile modification can cause a significant
difference in the gear mesh characteristics, especially when the modification is applied on a
significant part of the original involute profile, such as the example of the transmission
errors shown in Figure 7.4.2. An extreme case given by Smith (Smith 1999) involved the
use of tip and root relief, as indicated in Figure 7.4.6, where the two meet roughly at the
pitch point. Regardless of whether the tip relief form used is linear or parabolic, the gear
(in concept) is no longer an involute one, and some basic properties of the gears in mesh

131
can vary in a complex manner, such as the pressure angle. So, special considerations may
be needed when considering applying a long modification.

Figure 7.4. 6 Tip and root relief applied on a gear (Smith 1999).

In this research, choosing the form of the modified tooth profile is based on the following
considerations.
· Simple and less variation. The form of the modified profile will have little or no
change from the original involute curve. This could also be easier to manufacture.
· Fits the global deformed shape. In previous research, the forms were generated
with detailed consideration of the local tooth deformed shape. The parabolic form
may provide a more suitable deformed shape for the local tooth bending (such as
when the module Mn is large).
FEA of gears in mesh at or near the corner contact was carried out for the study of the
actual tooth deformation characteristics. Figure 7.4.7 shows part of the von Mises stress of
the meshing gear pair, where the material is nylon (PA 6), the input load is 30 Nm and the
temperature is at 23 C 0 (more detailed analysis of nylon gears in mesh can be referred to in
the next chapter).

The global deformations of the tooth are observed, consisting of the tooth rotation about its
root and the tooth movement near the tangential direction of the base circle. In order to
find the local tooth deformation, the original tooth profile (edge before the load) was
rotated and moved to match the deformed tooth. The best match was made in which the
rotation was 1.3 degree at the screen (Figure 7.4.7) geometric centre, as shown in Figure
7.4.8.
132
 Figure 7.4. 7 The von Mises stress of the mating tooth and its edge before the load.

Figure 7.4. 8 Illustration of the major local tooth deformations.

It can be seen that the tooth root deformations (at locations 1 and 2), specially that in
compression, are the major components that cause the tooth (global) rotation. The tooth
local deformations are observed at location 3, 4 and 5, where they are much less than the
global deformations, particularly the tooth bending (shown at location 5) which was small.
Except for the deformation at the contact, the shearing deformation near the tooth top land
is also one of the significant tooth local deformations when corner contact is present.

133
In conclusion, the tooth rotation of the global deformation is the most significant
component when the corner contact is present. It may be worth investigating the use of a
rotated original tooth profile as the form of the modified profile curve, which is simpler
than the linear and parabolic profile relief and so may be easier to manufacture. Such an
involute tooth profile modification will be used in this research, the details of which are
illustrated in Figure 7.4.9.

Figure 7.4. 9 Illustrations of an involute tooth profile modification.

The amount of the modification Ca can be achieved by rolling the section of the original
involute curve ST1 at the relief start point S with an angle a , so that the gears are always

mating with involute profiles. There can also be a round off with a radius of 0.3 to 0.5 mm
at the tooth tip T2 in order to assist the FEA solution convergence if corner contact occurs.

7.4.4 TE o. p. c. and C a Due To Elastic Deformation

The first attempt to understand the geometry of corner contact in detail was made by
Richardson (Richardson 1958) before Harris suggested the concept of TE. He calculated
the value of what would now be termed the transmission error outside the normal path of
contact, TE o. p. c. (i.e. when the contact occurs at the tooth corner) (Munro 1999). In 1962,

Munro (Munro 1962) proposed a different approach for calculation of TE o. p. c. . It involved

134
the use of some approximations that lead to a conveniently simple formulae. In 1975
Seager (Seager 1975) produced some work on analysis of Munro’s (Munro 1962)
developments. More recently, a considerable step forward was made by Lin et al. (Lin
1994) (approach case) and Munro (Munro 1999) (recess case). The paper of Munro’s
presented more results on corner contact theory, bringing all hitherto acquired knowledge
on this subject into systematic order.

TE o. p. c. is geometrically defined as the distance AK (approach case) and DF (recess case)

along the path of contact as shown in Figure 7.4.10.

O2

Driven a1
aE2
rb2

Dq 2 gE2
N2
(recess)
gC2
Dq 2 FE2
(approach)
FC2
sE2 sC2
ra2

D F TE o.p.c. (recess)

E
C
P
B

K
TE o.p.c. (approach)
A

Dq 1
ra1 N1 (approach)
SE1
rb1 aE1
SC1
a1
Driving
O1 Dq1 (recess)

Figure 7.4. 10 Tooth pairs of spur gears in mesh at the beginning (B), end (E) and beyond the
normal path of contact and the definition of the transmission error outside the
normal path of contact, TE o.p.c., in recess and approach (Munro 1999).

135
TE o. p. c. definitions and the calculation formulas (Lin 1994; Munro 1999) are very

important for the study of various cases such as how the profile, wear and alignment errors
lead to the corner contact. It should be noted that the validation of the formulas are limited
within gears containing only first order (imperfect) components that cause rigid body
motions. When gears are loaded, they are elastically deformed, contain higher order and
the higher order dependent components. Loaded TE o. p. c. are no longer calculated with any

existing formulas, as FEA evaluation of loaded TE o. p. c. is now readily possible.

Thus TE o. p. c. can be classified as the unloaded or loaded type, where unlike the unloaded,

the loaded TE o. p. c. is load dependent. A sample analysis of loaded TE o. p. c. is shown in

Figure 7.4.11.

Figure 7.4. 11 Loaded TE o. p. c. in approach case (pure involute gears in

mesh, ratio= 1:1, round off radius at the tips is 0.4 mm).
136
The loaded TE o. p. c. is the distance AB along the theoretical line of contact between the

drive gear tooth profile and the theoretical driven gear tooth profile. Loaded TE o. p. c.

occurs mainly due to the non-unique global torsional deflections between the drive and
driven gear, as explained with Figure 7.4.12.

Figure 7.4. 12 Detailed deflections of the drive and driven gear where
the corner contact (approach) is about to take place.

137
Except for the elastic deformation of the drive gear, the deflected driven gear also leads the
drive gear hub motion, so called the rigid body motion and the higher order dependent
component. Moreover, the pure elastic deformation of each gear is also varying
simultaneously with the mesh position changes, as the individual torsional stiffness (see
chapter 6.2) of each gear is varying asymmetrically. These complex deflection relations
will result in a slight difference between TE o. p. c. ( approach ) and TE o. p. c. ( recess ) , as well as a

slight asymmetry in the T.E. curve (as seen in Figure 7.4.13), if gears were kept rolling in
the same direction.

Figure 7.4. 13 Illustrations of the slight asymmetry in the T.E. curve of nylon gears in
mesh (note: the roll angle increment is 0.50 except the last one that is 0.6520 ).

138
It can be clearly seen that the positions when the (corner) contact take place, (approach
case) and when it finishes (recess case), are asymmetric to the centre of the base pitch
which will become more obvious with heavier load. When the load is very light, the T.E.
curve tends to be symmetric and for zero load, there are no deflections, which should
produce a symmetric T.E. curve – including the case of contact ratio smaller than 1.
Without the concept of loaded TE o. p. c. some confusions could be made in the analysis, as

shown in Figure 7.4.14.

Figure 7.4. 14 Theoretical transmission error curve with corner contact taking
place (no profile relief, no manufacturing errors, zero load
applied, contact ratio smaller than 1) (Munro 1999).

As mentioned before, loaded TE o. p. c. ( approach ) and TE o. p. c. ( recess ) are also slightly different,

but the amount of difference is very small as found in most cases of this research (including
gear ratio 1:1 to 1:2) and technically difficult to be observed. However, the research is
more interested in their maximum values, where the difference between their maximum
values can be represented by their different T.E. values, as shown in Figure 7.4.15, which is
the part of the analysis results in Figure 7.2.3. At this situation, the TE o. p. c. and T.E.

provide different measurements for the same deflections. Figure 7.4.16 illustrates the FEA
result evaluated at the TE o. p. c. ( approach ). max. while the contact is just about to be made at C.

139
 Figure 7.4. 15 Illustration of the T.E. peak values.

In order to eliminate the corner contact that occurs due to the deflections of gears under
load, the amount of tip relief Ca has to be evaluated (geometrically) by the loaded

TE o. p. c. ( approach ). max. , and technically Ca can be expressed by TE max . An approximate

formula for the minimum amount of tip relief can be given by,

2700
cos(200 + )
Ca B rb × TE max × N . (7.2)
min
360 0 rp
2
- rb
2
r
cos(200 - + - arccos b )
N rb rp

Figure 7.4. 16 FEA evaluation of TE o.p.c. (approach). max.

140
141
where rb and rp are the base and the pitch circle radius respectively, N is the number of

teeth, the formula is only for gear ratio 1:1. For gears with large ratio, especially when the
pinion (the smaller one, usually the drive gear) has fewer teeth, there should be some
variations between loaded TE o. p. c. ( approach ) and TE o. p. c. ( recess ) , consequently the relief Ca on

the driven gear teeth (approach case) and pinion teeth (recess case) will be different.
Further development of the formula (7.2) for general cases would require large FEA
calculations.

For the special case (gear ratio 1:1) and this research, formula (7.2) is convenient and its
calculations can be further simplified by ignoring the T.E. variations within the single
contact zone. The value of TE max can be calculated with the single tooth gear model or

even the model of the single tooth gear contact with a rigid line (chapter 6).

This section has discussed the FEA evaluation of TE o. p. c. which is due to deflections, so

called the loaded TE o. p. c. compared to that of the traditional TE o. p. c. which was for the

unloaded case, and calculated with geometrical formulas. Once the loaded TE o. p. c. max. is

known, the required tip relief Ca can be evaluated geometrically (with some

approximations) for preventing corner contact. In general cases, Ca can be calculated from

both loaded and unloaded TE o. p. c. max. . The formulas for calculation of unloaded TE o. p. c.

have been given by Munro (Munro 1999).

7.4.5 Additional Consideration On Tooth Modification

If the gear face width is significant (such as pinion) or the material of the gear(s) has a low
Young’s modulus (such as Nylon), a longitudinal modification may be required, as shown
in Figure 7.4.17. The modification is expected to achieve a unique load w ' (the reaction
force) along the tooth face width. However, the modification due to heavy load could lead
the gear to be a non-standard form and as a result, the line of action will vary along the
shaft. Investigations on the combination effects of T.E., load sharing, contact ratio and so
on have not been found.

142
 Figure 7.4. 17 Elastic deformation of the pinion and longitudinal modification (MAAG 1990).

7.4.6 Harris Mapping - Long and Short Tip Relief

The term “Harris Maps” refers to the diagram that describes the handover process of a
meshing gear pair with the measured and predicted transmission error curves at different
torque levels, as developed by Harris (Harris 1958). At the time, gears were tested under
quasi – static conditions, and the results are shown in Figure 7.4.18.

Harris maps clearly show that for a gear pair having a particular (long) tip relief, there
exists a design load where the transmission error is minimised. Other investigators such as
Gregory et al. (Gregory 1963), Lin et al. (Lin 1994), Munro and Yildirim (Munro 1994),
Kahraman and Blackenship (Kahraman 1999) have shown that the design load concept is
valid under dynamic conditions as well using straight line, linear and other more complex
modifications.
143
 0.003

RELATIVE DISPLACEMENT – in AT P.C.D.

LOAD ON STEEL GEARS – lb/in -FACE

1 PAIR
0.002 4000 (ON 1 D..P.)

3 PAIR THE FIGURES 1, 2 OR 3

2 PAIR 4000 (ON 8 D..P.) INDICATE THE NUMBER OF
0.001 PAIRS OF TEETH IN CONTACT.
0.004

1000 (ON 1 D..P.) 3 3 3

1000 (ON 8 D..P.)
250

LOAD ON STEEL GEARS – lb/in -FACE

0 0 0.003
ROTATION 6000

2 2 2

RELATIVE DISPLACEMENT – in AT P.C.D.

PITCH POINT 4000
0.002
(a) Load – Displacement – Rotation for Uncorrected
Involute Teeth, 20 – deg Pressure Angle, Both Gears 48
Teeth. 0.001 2000
1 1 1 1
NUMBER OF PAIRS OF TEETH IN CONTACT
0.004 0 0
1 2 1 2 1 2 1
RELATIVE DISPLACEMENT – in AT P.C.D.

LOAD ON STEEL GEARS – lb/in -FACE

6000 0.002
DEFLECTION OF
0.003 PAIR b FOR LOAD

APPLIED LOAD 4000 lb/in -FACE

SHOWN BELOW

4000 0.001
PROFILE CORRECTION PROFILE CORRECTION
0.002 TO FACE OF DRIVEN TO FACE OF DRIVEN
TOOTH TOOTH
0
2000
PAIRS OF TEETH –
ON ON
LOADS CARRIED
BY INDIVADUAL

0.001 4000 PAIR b PAIR c

ON a
lb/in -FACE

2000
0 ON d
0 0
ROTATION
b d a c
a c b d
INDIVADUAL PAIRS OF

ROOT OF DRINING TIP OF DRINING

LOADS CARRIED BY

TEETH – lb/in -FACE

TIP OF DRIVEN ROOT OF DRIVEN

4000
(c) Loads and Profile Correction for Constant
ON PAIR b ON PAIR c ON d Angular Velocity at 4000 lb/in. face.

(b) Load – Displacement – Rotation for Gears with

Involute Teeth and Heavy Profile Correction to Give a
Short Period of Transfer of Load from One Pair of Teeth
to the Next.

Figure 7.4. 18 The static transmission errors produced by Harris (Harris 1958).

However, the original works made by Harris (as shown in Figure 7.4.18 b and c) didn’t
show correct T.E. curves above the design load, as the variations with the load have shown
little possibilities for the occurrence of corner contact (or the protrusion of the single zone
T.E.). Five years later, Gregory, Harris and Munro presented their works (Gregory 1963)
in which the diagram of T.E. vs. loads is shown as in Figure 7.4.19.

The developed Harris maps (Figure 7.4.19) were based on Weber’s (Weber 1949) data on
tooth deflection under load.

144
 0.002

3000

0.0015
2500

2000
2 2
0.001 X
1500

1 B 1 1 1000
0.0005
500
A
0 0 0
0
MESHING AT MESHING AT MESHING AT
PITCH POINT PITCH POINT PITCH POINT
OF PAIR A OF PAIR B OF PAIR C

ROTATION

Figure 7.4. 19 Designed static transmission error curves (Gregory 1963).

In 1970, the ideas of long and short tip relief designs were introduced and developed by
Neimann (Niemann 1970) and Munro respectively. Those designs were for the two
extreme load cases. The explanations will refer to the “modern” Harris maps as shown in
Figure 7.4.20 and 7.4.21 as revised by Smith (Smith 1999). Figure 7.4.20 shows the
variations of transmission error with various input load for a long linear tip relief design,
which is aimed at producing minimum noise at the “design load”.

It can be seen that curve n represents the transmission error with zero input load, and the
deflections are rigid body motions in the regions centered at C – the middle of the double
contact zone. When gears are loaded, elastic deflections take place with large amplitudes
in the single zone. For the particular input load, curve d, the elastic deflection of the single
zone catches up to the combined deflections of the double zone. Hence, the overall
transmission error is balanced, but is not zero (the T.E. concept is based on function 1.1,
otherwise, based on function 1.2 the design load T.E. will be zero). As the curve of the
145
transmission error is perfectly straight, it shouldn’t cause vibrations outside the system
throughout the shafts. Curve o is the transmission error when the input load exceeds the
design load, and the deflection in the single zone exceeds that of the double zone. Input
loads in a range near the design load would produce transmission error, such as curve o, but
the transmission error is still smoother than that of pure involute gears with the same input
load. So, (long) tip relief design is not only for obtaining a smooth transmission error in
the design load, but it also has benefits for other input loads that are in a range near the
design load.

Figure 7.4. 20 Harris maps of deflections and long tip relief (Smith 1999).

To have a quiet running gear design at very light load is often a requirement since
machinery often operates under light or cruise condition. This is the requirement for short
tip relief design. In practice, it’s usual to have the relief varying linearly, starting well
above the pitch point so that there is a significant part of the meshing cycle where two
“correct” involutes are in mesh. A short tip relief design gives transmission error with
various input loads as shown in Figure 7.4.21. Again, the hand over regions have been
ignored.

Figure 7.4. 21 Harris map of deflections with short tip relief

146
It can be seen that the pure involute extends for the whole of a base pitch so there is no tip
relief encountered for light loads as for curve n. However, the tip relief still allows for all
deflections (and errors). Palmer and Munro (Palmer 1995) have succeeded in obtaining
good agreement between the predicted and measured transmission error with various input
loads in a test rig.

However, numerical simulations such as FEA modelling of tip relieved gears in mesh have
not been used extensively due to the difficulties in solution convergence when
encountering contact problems, and the limitations in solving unconstrained bodies.
Development of the numerical simulations will add further capability in tip relief design,
opening the door to the creation of new types of gears. In the following section, a new
technique for FEA simulations of tip relieved gears in mesh will be discussed which to
some extent has succeeded in overcoming these problems.

Section notes:
1. The design load of the long tip relief design is referring to the condition where
vibration and noise has to be minimized. It is different from the maximum design
load the drive can take.
2. Both the over load and the full torque load of the long and the short tip relief
designs will not encounter corner contact.

7.4.7 Solving Unconstrained Structures

As a preliminary step in this topic, it is straightforward to add a link element between the
drive and driven gear or to add link elements on the drive gear when the driven gear is fully
constrained at its hub. Once the input torque is loaded, the elastic deformation of the link
element(s) will take place, until the drive gear has touched the driven gear. This will assist
the modelling of the meshing gears when they are running with a tip-relieved section. The
application of the link element is illustrated as in Figure 7.4.22.

It seems that the unconstrained rigid body motion problem has been solved when a solution
check shows that the effect of the link element on whole system stiffness is very small.
However, the overall solution (for a short tip relief) does not satisfy the Harris map at all,
as seen in Figure 7.4.23.
147
 Figure 7.4. 22 Preliminary models of the link element.

152.4 UY_114.3 UY_76.2

0.016 UY_38.1 UY_5Nm

0.014

0.012

0.01
UY (mm)

0.008

0.006

0.004

0.002
s s s s
0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
ANG (degree)

Figure 7.4. 23 Failed solutions for short tip relieved gears in mesh.

According to this type of analysis, extra care must be taken, for any change to the final
tooth deflections can result in a totally different transmission error result. It should be
noted firstly that for this type of connection between the gear body and the link element,
the potential exists for numerical errors of the system matrix. The lower the stiffness of the
link element, the longer the computer time for solution and the harder it will be to obtained
solution convergence. Secondly, a 0.005mm error in the profile at the tooth tip will cause a
significant difference in the shape of the transmission error. So, considering the fact that
the stiffness of the link element cannot be too low, it will always tend to push the mating
teeth away. It can be seen in Figure 7.4.23 that the relief starting point s is located where it
shouldn’t be; it looks rather like a long tip relief. The numerical method approach for
solving such a problem involves the elimination of the numerical error in the system
matrix, which is due to the soft element directly connecting to stiffer elements. One of the
goals to be achieved involves the removal of the link element from the system. As Figure
148
7.4.24 illustrates, the link element can be connected to the master node of the input gear
hub, which means the nodes of the input gear hub have to be coupled in rotation about the
global coordinate system so that the numerical error can be limited within a wide range of
stiffness of the link element. In order to minimise some arbitrary effects, in each position
of the mesh cycle, the length and the orientation of the link element have to remain
constant and tangent to the input gear hub. Because the adaptive re-mesh doesn’t affect
where the master node is located on the gear hub in the mesh cycle, once the length of the
link element is defined, the coordinates of the other end of the link element can be obtained
by,
x1 = OD × cos[900 - q - (a + ANG )] , (7.3)
y1 = - OD × sin[90 0 - q - (a + ANG )] , (7.4)

as illustrated in Figure 7.4.25.

Figure 7.4. 24 The model for solving tip relieved gears in mesh.

Figure 7.4. 25 The coordinates of the link element.

149
As part of the looping program, the generation of the required link element will involve
commands such as those shown in Table 7.1.
CSYS,0 ! Activate the global cartesian coordinate system.
!*
OD=101.1187420807834218975649377962 ! The length of the link element has been defined as
!* 100 mm.
*afun,deg ! Angle expressed in degrees.
ALPHA=atan(12.255/8.6502)
THETA=atan(15/100)
FAI=90-THETA-(ALPHA+ANG(%FNAM%)) ! ANG ( ) is an array parameter and its dimensions of
X1=OD*cos(FAI) meshing position. FNAM is the name of the scalar
Y1=-OD*sin(FAI) parameter to be used as the loop index.
!* ! Create a key point at D ( x 1 , y 1 ) .
!*
K, ,X1,Y1,, ! Generator node at the key point.
NKPT,0, 240
!*
! Generator link element MD with type 2 material
TYPE, 2 properties.
MAT, 2
REAL, 1
ESYS, 0
SECNUM,
TSHAP,LINE
!*
FLST,2,2,1
FITEM,2,8977
FITEM,2,2069
E,P51X

Table 7.1. The program for generating the link element in the mesh cycle.

The link element death option, alternatively, can be achieved if the stiffness of the link
element has been carefully chosen to be related to the input load and issuing a sufficient
number of iterations in the first load step. When the solution is obtained, it has to be
checked to ensure the link element was “killed”. One of the checks is querying the Von
Mises stress of the link element in the postprocessor, as shown in Figure 7.4.26.

Figure 7.4. 26 Query the result of Von Mises stress to ensure the element death.

150
7.4.8 The Analysis of Short Tip Relieved Gears in Mesh

A short tip relief involves the relief starting point well above the pitch point, but close
enough to the actual change over point (or the hand over region) so that a smooth
transmission error in the mesh cycle is obtained when the input load is very light.
According to the amount of the relief, high contact stresses can be avoided when the teeth
change over. In this research, the short tip relief starts at point s, as seen in Figure 7.4.27,
where the pure involute profile section ST1 takes 1/3 of the entire involute curve above the

pitch point. If the pure involute profile section ST1 is rotated by a small angle a then it has

to have a minor extension in order to reach the tooth tip at T2 . The amount of tip relief Ca

is T1T2 which is achieved by removing the metal from area ST1T2 . Finally, the new tip is

rounded off with a 0.5mm radius.

Figure 7.4. 27 The typical short tip relief.

Seven FE models were established for the study of short tip relieved gears in mesh, each of
the models with its own relief angle a . The parameters of each model are listed, as shown
in Table 7.2.
parameters Segment ST1 rotation
Ca Length of Modification angle
Absolute Absolute value Normalized value
Model name value (mm)
Normalized value
DLa (mm) Ln a (degree)
R1thd01 0.005 0.042 2.2 0.61 0.005
R1thd02 0.009 0.075 2.2 0.61 0.009
R1thd03 0.014 0.12 2.2 0.61 0.014
R1thd05 0.023 0.192 2.2 0.61 0.023
R1thd09 0.042 0.35 2.2 0.61 0.042
R1thd16 0.075 0.625 2.2 0.61 0.075
R1thd24 0.114 0.95 2.2 0.61 0.114

Table 7.2. The parameters of the short tip relieved models.

151
It can be seen, the normalised parameters for each model are also given. The
normalisations are referred to the maximum allowance of current standards, such as BS
436: Part2: 1970 (8), for example the normalised relief length Ln can be expressed as,
the actual relief length : DLa
Ln = , (7.5)
- - -

the max . allowable : 0.6 * M

- -

so, the normalised parameters are non-dimensional. Using the normalised parameters can
also be used to check whether current standards are still suitable for some special cases
(such as non-metallic gears).

Over a complete mesh cycle, FEA simulations for each model under various input loads
were carried out with looping programs. The input loads used included 5Nm, 38.1Nm,
76.2Nm, 114.3Nm and 152.4Nm. The results for each model are given in the following
figures from Figure 7.4.28 to 7.4.34 respectively.
5Nm 38.1Nm
R1thd01 76.2Nm 114.3Nm
152.4Nm
The single zone o
0.0008
starts protruding

o
0.0006
TE (rad)

o
0.0004

o
0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.4. 28 The transmission error of the model R1thd01, a = 0.10 .

5Nm 38.1Nm
The single zone starts R1thd02 76.2Nm 114.3Nm
152.4Nm
protruding. o
0.0008

o o
0.0006
TE (rad)

o
o
0.0004
o

ft ft (full torque load)

0.0002 ft

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.4. 29 The transmission error of the model R1thd02, a = 0.20 .

152
 5Nm 38.1Nm
The single zone R1thd03 76.2Nm 114.3Nm
protrudings. 152.4Nm

o
0.0008

o
0.0006

TE (rad)
ft ft
0.0004
ft
The single zone
has been hiden.
0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.4. 30 The transmission error of the model R1thd03, a = 0.30 .

5Nm

The single zone can be

R1thd05 38.1Nm
76.2Nm
114.3Nm
seen when the load is 152.4Nm
high.
o
0.0008

ft ft
0.0006
ft
TE (rad)

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 31 The transmission error of the model R1thd05, a = 0.50 .

5Nm 38.1Nm
R1thd09 76.2Nm 114.3Nm
152.4Nm

0.0008

0.0006
TE (rad)

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.4. 32 The transmission error of the model R1thd09, a = 0.90 .

153
 5Nm 38.1Nm
R1thd16 76.2Nm 114.3Nm
152.4Nm

0.0008

0.0006

TE (rad) 0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.4. 33 The transmission error of the model R1thd16, a = 1.60 .

5Nm 38.1Nm
R1thd24 76.2Nm
152.4Nm
114.3Nm

0.0008

0.0006
TE (rad)

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.4. 34 The transmission error of the model R1thd24, a = 2.40 .

The results above have shown,

· The double zone retains its TE value unaffected by the tip relief within a small
preserved region located in the middle of the double zone. This is for all the cases
and the input loads. Values outside the small region almost linearly increase
towards the single zone. The increasing ratio is dependent on the amount of the
relief a or Ca , and is independent of the input loads.

· The transmission error of the single zone retains its value, unaffected by the tip
relief (ignoring the handover regions).

154
· The double zone T.E. value can not exceed that of the single zone; or the maximum
values of the double zone T.E. will fall into the trend of the single zone data, so that
is looks like the single zone has been extended when the load is increased.
However, the single zone has been hidden by the double zone data.
· Within the range of the input loads, the protruding nature of the single zone can be
clearly seen when the relief angle a is 0.50 or less, in such cases the “protection”
of tip relief does not exist. The full torque design load with short tip relief may be
defined as the input load which prevents the single zone from protruding.
· When the input load is very light, the transmission errors of all the models should
have the same path, as they are only relying on the relief start point s. A
comparison has been made between all the tip relieved models as well as a model
with unmodified teeth (pure involute gears in mesh), for an input load of 5Nm for
each model, the result of which can be seen in Figure 7.4.35.

0.00003

0.000025
TE (rad)

0.00002
preserved
regio ns
T.E. o f pure invo lute gears.
0.000015 alpha=0.1 alpha=0.2
alpha=0.3 alpha=0.5
alpha=0.9 alpha=1.6
alpha=2.4
0.00001
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 35 Comparisons between all the models with an input load of 5Nm.

There are two small differences in the two extreme cases of a equal to 0.10 and
2.40 , that could be due to numerical errors. The rest of the tip-relieved models
appear to remain the same. The existence of the preserved region suggests this type
of tip-relief design may not be suitable for light load or cruise conditions because
the change over appears to be similar to that of pure involute gears and may cause
surface spalls with the machine constantly running.
· A small preserved region in the middle of the double zone was found in all cases.
All the preserved regions are nearly the same, within a 1.50 width. This is the
155
 major difference in comparison with the Harris mapping result. Such a difference is
due to the relief being too short. Further calculations and investigations have been
made on this difference and will be presented in the following section.

7.4.9 Investigation of the Short and Long Tip-relief

If the relief starting point s is in the middle of the involute curve above the pitch point as
seen in Figure 7.4.36, the relief is recognized as part way between the short and long relief.

Figure 7.4. 36 The tip – relieved tooth.

Two FE models have been established for the study of the various types of relief. The
parameters of each model are listed, as shown in Table 7.3.

Ca Length of Segment ST1

Parameters rotation angle
Modification
Model name Absolute Normalized Absolute value Normalized
a (degree)
value (mm) value DLa (mm) value Ln
Rhalf03 0.020 0.167 3.156 0.877 0.3
Rhalf05 0.034 0.28 3.156 0.877 0.5
Table 7.3. The tooth parameters.

FEA solutions for the transmission error of each model are shown in Figure 7.4.37 and
Figure 7.4.38.

156
 5Nm 38.1Nm
Rhalf03 76.2Nm
152.4Nm
114.3Nm

preserved
o regions
0.0008
o
o o
0.0006 o
TE (rad)

nf nft (near the full torque load)

nft
0.0004

d d d
0.0002

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 37 The transmission errors of model Rhalf03, a = 0.30 .

5Nm 38.1Nm
Rhalf05 76.2Nm
152.4Nm
114.3Nm
53.4Nm

preserved regions.
ft
0.0008
ft

0.0006

0.0004

d d
0.0002

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 38 The transmission errors of model Rhalf05, a = 0.50 .

The results have shown long tip relief characteristics similar to that of Harris maps, except
near the preserved region in the middle of the double zone with its “fixed” size 1.50 , which
is similar to the behaviour of short tip relief. Figure 7.4.39 and 7.4.40 compare the results
with that of pure involute gears in mesh and the differences in the preserved region are
almost the same in most cases. The over load curve o in Figure 7.4.37 has shown its
difference with that of Harris map, in this case the “protection” of the tooth tip relief does
not exist due to the appearance of the protruding single zone TE. It has been found from

157
the results of both models that the design load (curve d) rises from 38.1Nm to 53.4Nm with
the increasing amount of 0.20 in the relief angle, meanwhile the full torque curve was
increased from 76.2 Nm to 152.4 Nm.

It has been recognized that the real starting point between the short and the long tip relief
should exist with characteristics such that the preserved regions remain as their minimum
width which may be just a point (zero width). The value in the region should still be the
same as that value in the middle of the double zone of standard involute gears in mesh. If
S1 is used to denote this particular relief starting point, when a tip relief starts beyond the

point S1 ( ST1 > S1T1 ), the preserved region will start to move away from that of the standard

involute gears with the input loads. The further the tip relief starts away from S1 , the wider

the preserved region will be. On the other hand, the shorter the ST1 length, the wider the

preserved regions will be, but in this case the preserved regions will always keep the same
value as that of standard involute gears. S1 can be defined as such a floating point. In

order to find out the floating point, a few models have been built and looping programs
generated for each of them. FEA solutions for each model have been examined, and the
5
results for one particular model with the tip relief starting position ST1 = PT1 , and the
12
tooth parameters listed in Table 7.4 will be presented.

RH03_5Nm RH03_38.1Nm RH03_76.2Nm RH03_114.3Nm RH03_152.4Nm

St_5Nm St_38.1Nm St_76.2Nm St_114.3Nm St_152.4Nm

0.0008

0.0006
TE (rad)

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 39 Comparison TE between standard involute gear model and the model Rhalf03, a = 0.30 .
158
 RH05_5Nm RH05_38.1Nm RH05_76.2Nm RH05_114.3Nm RH05_152.4Nm RH05_53.4Nm
St_5Nm St_38.1Nm St_76.2Nm St_114.3Nm St_152.4Nm

0.0008

0.0006
TE (rad)

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 40 Comparison TE between standard involute gear model and the model Rhalf05, a = 0.50 .

Ca Length of Segment ST1

Parameters rotation angle
Modification
Model name Absolute Normalized Absolute value Normalized
a (degree)
value (mm) value DLa (mm) value Ln
R5out12_03 0.016 0.13 2.676 0.743 0.3
Table 7.4. The tooth parameters.

The FEA solutions for the transmission error compared with that of a standard involute
gear are shown in Figure 7.4.41.
159
 Figure 7.4. 41 Transmission errors of model R5out12_03, a = 0.30 .

The results clearly show that the short tip relief design has reached its limit and this
represents the longest relief that will meet the requirements of the short tip relief design.
5
For this particular gear, the floating point is located at S1 where S1T1 = PT1 . Especially,
12
within the full torque load, the transmission error variations are very close to Harris
predictions for the short tip relief design. Over load curves are above the curve of the full
torque load, where the protruding single zone transmission error can be seen. The
variations of the hand over regions in the over load curves are also observed. When the
160
input load is very light, such as 5 Nm, a much smoother T.E. is achieved compare to Figure
7.4.35. Moreover, there is no edge contact observed with light load which means short tip-
relieved gears will avoid early spalling, especially for gears constantly running with light
load.

7.4.10 The Analysis of Long Tip Relieved Gears in Mesh

Further extension of the relief starting point s from the tooth tip is expected to gain
smoother TE at the design load. The first long tooth relieved gear model was built with the
relief having 2/3 of the involute curve above the pitch point. The tooth details are shown in
Figure 7.4.42 and the parameters are listed in Table 7.5.

Figure 7.4. 42 Long tip relief.

Parameters Ca Length of Modification

Segment ST1
rotation angle
Model name Absolute Normalized Absolute value Normalized
a (degree)
value (mm) value DLa (mm) value Ln
R2thd025 0.022 0.183 4.14 1.15 0.25
Table 7.5. Tooth parameters of model R2thd025, a = 0.250 .

The FEA results with their details are illustrated in Figure 7.4.43.

The results show similar improvements of the TE at the design load when compared with
the results that have been obtained in Figure 7.4.37 and 7.4.38. For the design load curve,
the slight downward changes at the edge of the preserved regions can be seen. The wider

161
preserved regions are also observed due to the longer relief length compared to that of the
model Rhalf03 and Rhalf05 (Figure 7.4.37 and 7.4.38). The comparison of results between
the long tip relief and the standard involute gears in mesh is shown in Figure 7.4.44.
152.4Nm 114.3Nm 86.9Nm 76.2Nm

0.0009 67.3Nm 38.1Nm 19.05Nm 5Nm

o over load preserved

0.0008 regions

o
0.0007
near the full torque
nft
0.0006 nft

0.0005
TE (rad)

d design load d
0.0004

0.0003 h
h
half the design load
0.0002

0.0001

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 43 Transmission errors of model R2thd025, a = 0.250 .

5Nm 19.05Nm 38.1Nm

76.2Nm 114.3Nm 152.4Nm
St_5Nm St_19.05Nm St_38.1Nm
0.0009 St_76.2Nm St_114.3Nm St_152.4Nm

0.0008

0.0007
Preserved regions
0.0006
TE (rad)

0.0005

0.0004

0.0003

0.0002

0.0001

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 44 The comparison of the TE of standard involute gears and the long
tip - relief gears in mesh.

162
The longest tip relief design extends the relief starting point s from the tip to the pitch
point, which is the traditional long tip relief concept (Smith 1999), as shown in Figure
7.4.45. Two relevant models Rp01 and Rp02 have been established with their relief angles
of a = 0.1 and 0.2 degrees respectively. The tooth parameters are listed in Table 7.6.

Figure 7.4. 45 The maximum length of the tip relief.

Parameters Segment ST1

Ca Length of Modification rotation angle
Absolute Normalized Absolute value Normalized
Model name value (mm) value DLa (mm) value Ln a (degree)
Rp01 0.013 0.11 6 1.67 0.1
Rp02 0.026 0.22 6 1.67 0.2
Table 7.6. The parameters of the tip relieved tooth.

The FEA solutions for the transmission error have been obtained, and the results for each
model compared with that of standard involute gears in mesh, as shown in Figure 7.4.46
and 7.4.47 for a = 0.10 and a = 0.20 respectively.
5Nm 38.2Nm 76.2Nm 114.3Nm
152.4Nm St_5Nm St_38.1Nm St_76.2Nm
St_114.3Nm St_152.4Nm
0.0009
o
0.0008
o
0.0007
o
0.0006 o
TE (rad)

0.0005
d (o) d (o) d (o)
0.0004

0.0003

0.0002

0.0001

0
s'
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 46 Transmission errors of the model Rp01, a = 0.10 .

163
 0.001 5Nm 38.2Nm 76.2Nm 114.3Nm 152.4Nm
St_5Nm St_38.1Nm St_76.2Nm St_114.3Nm St_152.4Nm

0.0009
o o
0.0008

0.0007
o
o
0.0006
TE (rad)

0.0005

0.0004

0.0003

0.0002

0.0001

0
s'
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 47 Transmission errors of the model Rp02, a = 0.20 .

The results from both models show that neither can meet the requirements of a smooth T.E.
at the design load while providing sufficient protections for the tooth flank face. The
design load for Rp01 shows an over load condition in the single zone T.E. especially with
the hand over region increasing. With the tooth tip relief increasing as shown in Figure
7.4.47, the single zone T.E. rises rapidly during the hand over region before the “design
load” has been reached. It should be noted that the relief start point of the T.E. s’ does not
match the actual relief starting point which is at the pitch point. The location of s’ is
dependent on a , usually the larger a , the closer to the actual relief starting point will be
observed. Those results, however, are valuable since they have shown the variations of the
preserved region and reveal the important changes to the T.E. It might be expected that a
critical tip relief starting point S 2 could be found, where the design load T.E. will be

smooth. A relevant model has been built with its tooth relief as shown in Figure 7.4.48 and
the tooth parameters are listed in Table 7.7.

Parameters Segment ST1

Ca Length of Modification rotation angle
Absolute Normalized Absolute value Normalized
Model name value (mm) value DLa (mm) value Ln a (degree)
R5out6_025 0.026 0.217 5.124 1.423 0.25
Table 7.7. The tooth relief parameters.
164
 Figure 7.4. 48 Details of the tooth relief.

FEA solutions for the transmission errors have been obtained as shown in Figure 7.4.49.
Two loops of the programs for the input loads 133.35Nm and 138.35Nm were used, and
then the design load of 133.35Nm was found. As seen from the T.E. calculations, the T.E.
now appears to be very smooth with only a minor ripple. The small ripple means that the
critical relief starting point S2 has not been precisely reached yet, but the starting point of

the tip relief must be very close. With the design load curve getting smoother, it should be
noted that the over load curve is coming down and is very close to the design load. If the
preserved region extends to its maximum, which is in the double zone width, the protruding
section of the T.E. in the single zone will be present for any input load greater than the
design load. Here the single zone and the double zone refer back to that of the standard
involute gears in mesh.

Tip relief designs have been discussed, in which it can be seen that the tooth modifications
are small but the influence to the gear transmission error is significant. For the tooth tip-
relief modifications, two critical relief-starting points S1 and S 2 have been found which

effect the variations of the double zone preserved regions that have been defined in this
section. The concept of the full torque load and the over load torque have been clarified in
the sense of tooth flank surface protection. Specially, the full torque load should be taken
into account by gear designers to avoid early spall failure (pitting or crack) on the gear
flank face.

165
 5Nm 19.05Nm 38.1Nm
d 76.2Nm 114.3Nm 133.35Nm
138.35Nm 152.4Nm
0.0009
o o Preserved regions
0.0008

0.0007

0.0006

TE (rad)
0.0005

0.0004

0.0003

0.0002

0.0001

-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.4. 49 Transmission errors of the model R5out6_025.

7.5 The Effect of Modifying the Centre Distance

The previous methods used in the analysis of tip-relieved gears in mesh can also be used to
investigate the effect of modifying the centre distance of the gears in mesh. Firstly, a fixed
increment of 0.2 mm along the line of original centre distance was considered, and the FEA
solutions for the transmission errors under various input loads were obtained. The results
over a complete mesh cycle are shown in Figure 7.5.1 as a function of input load.
C+02_5Nm C+02_38.1Nm C+02_76.2Nm C+02_114.3Nm
C+02_152.4Nm St_5Nm St_38.1Nm St_76.2Nm
St_114.3Nm St_152.4Nm
0.002

0.0018

0.0016

0.0014

0.0012
TE (rad)

0.001

0.0008

0.0006

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.5. 1 The transmission errors with modified centre distance.

It should be noted there is a group of data in the bottom of the figure, which is for the
meshing gears without any modifications. It can be noted that the shapes of the
transmission error curves between the two groups are slightly changed in their hand over

166
regions, but the major difference between the two groups is in their absolute T.E. values.
For the maximum input load, the difference (relative error) was 227.5% in the single zone
and 269% in the double zone. When the input load was 5Nm, the difference (relative error)
was up to 3947%. Concerning the relative errors it should be noted that if a specific
geometry error (misalignment or eccentricity, etc.) were present, the relative transmission
error would tend to an infinite value when the input load tends to zero. For an input load of
76.2Nm, considering 0.2mm as the maximum change of the centre distance, FEA solutions
for the transmission error of the meshing gears with various increments decreasing to
0.005mm have been obtained, as shown in Figure 7.5.2.
Increment=0.2mm Increment=0.15mm Increment=0.1mm
Increment=0.05mm Increment=0.01mm Increment=0.005mm
Increment=0mm
0.0016

0.0014

0.0012

0.001
TE (rad)

0.0008

0.0006

0.0004

0.0002
Input load = 76.2 Nm
0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.5. 2 T.E. of various centre distance modifications.

The relative errors can be found from,

(TEINC - TEST ) / TEST , (7.5)

where TEINC is the transmission error with a certain increment of the centre distance and

TEST is the transmission error of standard involute gears in mesh (without misalignment or

tip-relief). The relative errors of each increment are given, as shown in Figure 7.5.3.

The results have shown an almost linear increase in the relative errors with the centre
distance increment from 0.05mm to 0.2mm. For the small centre distance increment of
0.01mm, the relative errors were found to be 17.5% in the double zone and 12.2% in the
single zone. For the minimum increment of 0.005mm, the relative errors were found to be
4.7% (maximum) in the single zone and 0.8% (minimum) in the double zone. Under
normal operating conditions, the running gear pair can have such small variations in their
centre distances. The centre distance variation will not be a fixed value so that considering
167
the combination of possible profile errors and other types of misalignments, the
experimental measurement of gear transmission error becomes extremely difficult.
400 Increment=0.2mm Increment=0.15mm Increment=0.1mm
Increment=0.05mm Increment=0.01mm Increment=0.005mm
350

Relative Errors (%)

300

250

200

150

100

50

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.5. 3 The relative errors as a function of centre distance.

7.6 T.E. in the Presence of a Tooth Root Crack

In order to improve the vibration condition monitoring of gear systems, detailed knowledge
of the vibration generation mechanisms for various types of failure are required (Du 1997).
One of the recent modelling approaches (Howard 2001) involves the use of coupled
torsional and transverse motions of the shafts, along with the changes to the tooth bending
stiffness as the teeth go through the mesh point. The effect the single tooth crack has on the
frequency spectrum and on the common diagnostic functions of the resulting gearbox
component vibrations can be obtained, where the tooth stiffness is modelled using finite
element analysis to ascertain the changes to the teeth stiffness as a function of gear rotation.
The details of the finite element model near the root crack of 3.5 mm are shown in Figure
7.6.1.

Figure 7.6. 1 The details of the tooth crack of 3.5 mm.

It should be noted that the all-quad element model is no longer possible due to the necessity
of using the triangle elements to mesh the singularity in the region near the crack front.
Right at the crack tip (2D), 10 special triangular elements were used with their mid-side
168
nodes located at ¼ of their adjacent edges from the crack tip (Anderson 1995). The LEFM
(linear elastic fracture mechanics) assumption was used in the analysis. The solutions for
the transmission error were obtained, as shown in Figure 7.6.2.
C35_5Nm C35_38.1Nm C35_76.2Nm C35_114.3Nm
C35_152.4Nm St_5Nm St_38.1Nm St_76.2Nm
0.001 St_114.3Nm St_152.4Nm

0.0009

0.0008

0.0007

0.0006
T.E. (rad)

0.0005

0.0004

0.0003

0.0002

0.0001

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.6. 2 T.E. of the involute gear in mesh with a localized tooth crack.

The LEFM assumption is valid only as long as the non-linear material deformation is
confined to a small region surrounding the crack tip. In many materials, it is virtually
impossible to characterise the fracture behaviour with LEFM (Anderson 1995), and an
alternative gear model with a (single) root crack is required. An elastic-plastic fracture
mechanics model of the cracked tooth behaviour under various load conditions should be
carried out so that the crack tip opening displacement (CTOD) or the J contour integral can
be evaluated. The associated parameters such as the component of plastic mouth opening
displacement and the hinge point can be obtained and used to determine the modified tooth
profile which is due to the plastic deformation at the crack tip. Once the modified tooth
profile is known, the analysis for the transmission error can be carried out using the method
that has been used for tip relieved gears in mesh. However, detailed analysis with elastic-
plastic fracture mechanics applies to the cracked gear(s), and the analytical or experimental
analysis are currently not included in this research. As a preliminary study on the topic, the
evaluations of the transmission error with the crack tip plasticity will be investigated with
assumptions, the details of which are shown in Figure 7.6.3.

169
 Figure 7.6. 3 The tooth profile modification due to the crack front plasticity.

It can be seen that the hinge point is assumed to occur at the crack tip. This is because
compared with the structure dimension, the distance from the crack tip is very small. The
major part of the tooth, especially the mating side profiles above the crack, rigidly rotate
about the hinge point where the hinge model assumption is used (Anderson 1995). In what
extent the hinge model is suitable for the cracked tooth is a critical task and it may be
investigated in the future. V p represents the plastic component of the mouth opening

displacement which is illustrated in Figure 7.6.4.

Figure 7.6. 4 The crack mouth opening displacement.

FEA solutions for the transmission error were carried out using the more realistic models of
tip-relieved gears in mesh. Model Rhalf05 (see Figure 7.4.36 and Table 7.3) was used
under the consideration of profile modification due to a 3.5mm root crack. The crack
orientation angle was 67.4 degrees between the crack and the original tooth symmetry line

170
T3T4 (see Figure 7.6.3). According to Lewicki (Lewicki 1997) the angle greater than 45

degrees is realistic for this model. It has been assumed that the plastic component of the
crack mouth opening displacement was 0.005mm, which results in the tooth rotation about
the hinge point of 0.082 degrees. The transmission errors under various input loads are
presented in Figure 7.6.5, and the results are also compared with that of the un-cracked
gear model over a complete mesh cycle.
0.001 M0005RH_5Nm M0005RH_38.1Nm M0005RH_53.4Nm M0005RH_76.2Nm
M0005RH_114.3Nm RH05_5Nm RH05_38.1Nm RH05_53.4Nm
RH05_76.2Nm RH05_114.3Nm
0.0009

0.0008

0.0007

0.0006
T.E. (rad)

0.0005

0.0004

d d d
0.0003

0.0002

0.0001

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.6. 5 T.E. of cracked gears in mesh with permanent tooth deformation.

The results have shown that the maximum change in T.E. of 2 ´10-4 (rad) can be found in
each curve even at the design load. The abrupt starting and ending positions, however, in
this case also rely on the relief starting point.

FEA solutions for the transmission error have produced results for the tip-relieved gears in
mesh with the root crack under the assumption of LEFM (linear elastic fracture
mechanics). The results can also be compared with those of the un-cracked gears in mesh
over a complete mesh cycle as seen in Figure 7.6.6.

It should be noted that when the input load is smaller than the design load (53.4 Nm) the
transmission error path is actually smoother than that of the un-cracked gears in mesh. So
far, the transmission error behaviour of the meshing gear pair has been investigated under

171
three considerations with the same root crack, (i) the standard involute meshing gears
analysed with the LEFM assumption, (ii) the analysis of tip relieved gears, and (iii) the tip
relieved gears analysed with its permanent deformation due to the plasticity at the crack tip.
Over a complete mesh cycle, the transmission error behaviours are significantly different to
each other, as shown in Figure 7.6.7.

For the modelling of tip-relieved gears in mesh using the LEFM assumption, the results
should be valid for light load conditions. An example can be used to highlight the
diagnostic consequences, as shown in Figure 7.6.6. When the input load is less than the
design load, the transmission error curves may be smoother than that of the un-cracked
gears in mesh. In this case, gear diagnostic techniques using the gear case vibration may
fail to detect the root crack. This may provide a clue for gear diagnosis. If the design load
is light, so that the LEFM assumption is valid, the gear diagnosis through the gear case
vibrations may fail to detect the root cracks due to the gears constantly running within their
design load (cruise conditions). With the hinge model assumption, the effect of the 0.005
mm plastic mouth opening displacement on the transmission error becomes significant, but
the question is to what extent the hinge model is suitable for the cracked tooth. The
validation of the hinge model as applied to the cracked tooth should depend on the crack
length, the orientation as well as the gear thickness (back up ratio) amongst other factors.
For a certain plastic mouth opening displacement, it has been suspected that the tooth may
rigidly rotate about the hinge point. In that case, the actual rotation angle may be less than
what it was, due to the elasticity of the surrounding material outside the plastic zone. From
another point of view, if the actual rotation angle is greater than what it was, the gear
failure diagnostic task will be made easier. If the actual rotation angle, as suspected, is less
than what it was, the actual transmission error curve under the certain input load will fall in
between the curves produced by the LEFM assumption and the hinge model assumption.
There will be a domain in the mesh cycle for each input load that is between the curves
produced with the above two assumptions, as shown in Figure 7.6.8. The domain size to
some extent represents the amount of maximum plastic deformation when the hinge model
is used. It also contains rigid body motions, – the sliding component due to the rotated
tooth profile. So, in this case the domain represents both possibilities.

172
 C35_5Nm C35_38.1Nm C35_53.4Nm C35_76.2Nm
C35_114.3Nm C35_152.4Nm RH05_5Nm RH05_38.1Nm
RH05_53.4Nm RH05_76.2Nm RH05_114.3Nm RH05_152.4Nm

0.001

0.0009

0.0008

0.0007

0.0006

0.0005

0.0004
d d
0.0003 d

0.0002

0.0001

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.6. 6 T.E. of the tip-relieved gears in mesh with LEFM assumption.

St_C35_5Nm St_C35_38.1Nm St_C35_76.2Nm St_C35_114.3Nm

E_RHC35_5Nm E_RHC35_38.1Nm E_RHC35_76.2Nm E_RHC35_114.3Nm
P_RHC35_5Nm P_RHC35_38.1Nm P_RHC35_76.2Nm P_RHC35_114.3Nm

tip relieved
9.00E-04 gear__LEFM
tip relieved gear
8.00E-04
Vp = 0.005 mm
standard involute
gear__LEFM
7.00E-04

6.00E-04
T.E. (rad)

5.00E-04

4.00E-04

3.00E-04

2.00E-04

1.00E-04

0.00E+00

-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.6. 7 T.E. of the cracked gears under different assumptions.

173
 Figure 7.6. 8 The domains for the possible T.E. path.

The actual T.E. path can be estimated within the major trend of the variations, as shown in
Figure 7.6.9.

Figure 7.6. 9 Sample of an estimated T.E. path.

As shown, the estimated T.E. path was taken to be the mean of the two assumptions, for the
input load of 76.2 Nm. The different curves could be used for gear failure (root crack)
detection and diagnosis through gear case vibration and would give results with different
reliabilities and also analysis with different difficulties.

174
7.7 New Tooth Profiles for Reducing Gear Vibration Below the Design Load

Based on the knowledge of existing gear transmission error studies, it may be possible to
design gears with new tooth profiles having the following features:
· Usually at the design load, the transmission error has downward protrusions as shown
in section 7.3. The new tooth profile will have no abrupt protrusions at the design load.
· When the input load is lighter than the design load, the curves of the transmission error
should vary smoothly up and down in the mesh cycle, so that the vibration level is
reduced.
A few models have been build with the new tooth profiles that were created by applying a
combination of back lash and (original) curve rotation, model New_A01, New_A02 and
New_A055 which were aimed at obtaining smoother T.E. curves at different levels of
design load. FEA solutions for the transmission errors are shown in Figure 7.7.1, 7.7.2 and
7.7.3.

New_A01
0.0014

0.0012

0.001
TE (rad)

0.0008
design load

0.0006

0.0004

0.0002 5Nm 38.1Nm 76.2Nm

114.3Nm 152.4Nm
0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
ANG (degree)

Figure 7.7. 1 Transmission errors of model New_A01.

New_A02
0.0016 design load

0.0014

0.0012

0.001
TE (rad)

0.0008

0.0006

0.0004
5Nm 38.1Nm 60Nm
0.0002 76.2Nm 114.3Nm 152.4Nm

0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
ANG (degree)

Figure 7.7. 2 Transmission errors of model New_A02.

175
 New_A055 design load

0.002

0.0015

TE (rad)
0.001

5Nm 38.1Nm 76.2Nm

0.0005 114.3Nm 152.4Nm

0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

ANG (degree)

Figure 7.7. 3 Transmission errors of model New_A055.

It can be seen that the smoothness of the transmission error curves have been improved
within different levels of the design load. With an input load lighter than the design loads,
the transmission errors vary smoothly up and down in the mesh cycle as required.
However, if the input load is greater than the design loads, the over load occurs quite
easily.

For the purpose of the tooth flank surface protection while the input load is high, new tooth
models New B01_01 and New B01_03 have been designed with smooth T.E. results
gradually increasing up to the full torque load. The results of the transmission errors are
shown in Figure 7.7.4 and 7.7.5. However, as seen from the results the higher the full
torque load, the more T.E. variations appear at the design load.

5Nm 38.1Nm
New B01_01 76.2Nm 114.3Nm
0.0012 152.4Nm 19.05Nm

0.001

0.0008
TE (rad)

0.0006
full torque load

0.0004 design load

0.0002

0
-16 -12 -8 -4 0 4 8 12 16
A N G ( de gre e )

Figure 7.7. 4 Transmission errors of model New B01_01.

176
 5Nm 38.1Nm
New B01_03 76.2Nm 114.3Nm
152.4Nm
0.0014

0.0012
full torque load
0.001

TE (rad)
0.0008
design load
0.0006

0.0004

0.0002

0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.7. 5 Transmission errors of model New B01_03.

Model New C02_01 was designed to have the features midway between that of the
previous two groups. The transmission errors are shown in Figure 7.7.6.

5Nm 38.1Nm
New C02_01 76.2Nm 114.3Nm
0.0014 152.4Nm 57.15Nm

0.0012

0.001
ft ft
TE (rad)

0.0008

0.0006

0.0004
design load
0.0002

0
-16 -12 -8 -4 0 4 8 12 16
ANG (degree)

Figure 7.7. 6 Transmission errors of model New C02_01.

The meshing gears with the new tooth profiles are still under development. To further
develop gears with the new tooth profiles, however, would require extensive experiments,
which is beyond the scope of the current research.

The new gears may well be suitable for situations where the input load is within the design
load. However, if a root crack is present these gears may not be desirable for gear
diagnostic purposes as only a minimal change in the T.E. was observed. Examples can be
seen in Figures 7.7.7 and 7.7.8 where 3.5 mm root cracks were present.

177
 0.0016 New_A02_C35

0.0014

0.0012

0.001

T.E. (rad)
0.0008
d d d

0.0006

0.0004
5Nm 38.1Nm
0.0002 76.2Nm 114.3Nm
152.4Nm
0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.7. 7 T.E. of model New_A02 with a 3.5mm root crack.

New_A055_C35

0.002
d d d

0.0015
T.E. (rad)

0.001

0.0005 5Nm 38.1Nm

76.2Nm 114.3Nm
152.4Nm
0
-16 -12 -8 -4 0 4 8 12 16

ANG (degree)

Figure 7.7. 8 T.E. of model New_A055 with a 3.5mm root crack.

7.8 Transmission Error Control

Transmission error is load dependent and has a major influence on the system vibrations.
In some applications, a smooth transmission error within a wide range of input load is
desired and sometimes it is vitally important. For example, the T.E. may be quiet when a
submarine is in cruise condition, but it may be loud when sudden acceleration is required.
Gears with certain load capacities (not including high contact ratio gears) are nearly
impossible to design with smooth transmission error over the full range of input loads.
However, to some extent, a smooth transmission error may be possible over a certain range
of input loads.

178
There are two methods of treatment of the gear system. The first method is based on the
studies of the meshing gears with the new tooth profile as in the previous section. These
studies have found that the design loads can vary with minor changes of the centre
distance. Examples can be seen in Figure 7.8.1 and Figure 7.8.2.

New_A02
0.0012

0.001

0.0008 Original design load T.E.

The design load T.E. with
0.0006 modified center distance.

0.0004

0.0002
60Nm 76.2Nm
0
-14 -10 -6 -2 2 6 10 14
ANG (degree)

Figure 7.8. 1 Design load T.E. of model New_A02.

New_A055

0.002

0.0016
Original design load T.E.
The design load T.E. with
modified center distance.
0.0012

0.0008

0.0004

135Nm 114.3Nm
0
-14 -10 -6 -2 2 6 10 14
ANG (degree)

Figure 7.8. 2 Design load T.E. of model New_A055.

With a 0.13 mm change in centre distance, the design load T.E. of model New_A02 was
found to vary smoothly, and more significantly the input design load was found to change
from 60 Nm to 76.2 Nm, an increase of 27% over the original design load. The case for
model New_A055 with a 0.24 mm centre distance change resulted in an increase of 18% of
the design load. Model New_A055 is designed for a higher input load.
179
It has been shown therefore that the design load T.E., to some extent, can be controlled by
modifying the gear centre distance. However, it may not be easy to control such small
movements in the gear centre distance. Reliable measurements would also be required for
applying such control. The reliable signals, however, may not be vibration signals, but
may be higher order components of the actual meshing gears such as stress or strain. This
will be discussed further in the following section.

The second conceptual method for controlling T.E. involves the application of force(s)
along the gear centre distance through the shafts in each mesh cycle, since the centre
distance is sensitive to the transmission error (see 7.4). The control of the magnitude of the
pulse(s) and the phase frequency modulations can be complicated, especially for high-
speed applications.

7.9 FEA of High Order Components – Meshing Gears With Shaft

Rather than considering the use of vibration signals for gear fault detection, an analysis has
attempted to concentrate on the use of high order components of the actual mating gears
with their shafts. Elastic strains (same order of stress) of the mating gears and their shafts
would provide a direct indication of the behaviour of the mating gears. The research on
gear stiffness properties developed in Chapter 4 has found that the distortion field in the
area near the hubs between the contacts varies depending on the number of teeth in contact.
With two teeth in contact, it was found that the reaction forces in the hub tend to spread
symmetrically about the gear body. With a single tooth contact, the symmetrical nature of
the stress field becomes distorted. If a single keyway is used, it has been found that the
variations of the distortion field in the area near the hub become difficult to observe.

In order to find the resulting elastic strain variations on the shaft, a series of 3D gear-shaft
models were established. Figure 7.9.1 shows the basic 3D model that was developed in
this research. A pair of standard involute gears with a 120mm´ 30mm shaft is shown along
with the mapped mesh using 20 node brick elements.

180
 Figure 7.9. 1 The basic 3D model.

The model as shown contained 23673 nodes. If the nodes in the contact area between the
shaft and the gear hub were coupled in rotation, this would simulate a fine spline with an
interference fit between the gear and the shaft. Models containing spline fits can easily be
extended to very large models. An example of an eight-spline shaft with gear hub
interference fit would result in a 50,000-node model. The whole model can be further
extended to 70,000 nodes if a rotor is included, as seen in Figure 7.9.2. The solution of
such a model may require more than three days for one solution using a standard PC (PIII
800 Mhz, 1 GB SDRAM).

Figure 7.9. 2 Splines with interference fit.

181
In the actual FEA solution the model has to be simplified. As long as there is more than a
single keyway between the shaft and the gear, the elastic strains on the shaft may vary
throughout the mesh cycle. The particular measurements of interest are those in the section
of the shaft that is near the gear hub. The section length of interest should cover the
diameter of the shaft (Saint-Venant’s Principle), and for this model the length of 30 mm
was chosen. Figure 7.9.3 shows the locations where the elastic strains EPEL Y were
measured within the global cylindrical coordinate system.

Figure 7.9. 3 The locations where the elastic strain was measured.

FEA solutions for the elastic strains at the specified locations along with the transmission
error were calculated. Instead of using a spline fit, the nodes in the common area between
the shaft and the gear were coupled in rotation. With the input load of 76.2 Nm, the FEA
model was solved with 15 sub steps in each position of the mesh cycle, and the results are
shown in Figure 7.9.4.

182
 E1 (Node17459) E2 (Node17461) E3 (Node17463)
E4 (Node17465) E4a (Node17450) E4b (Node18417) E4c (Node17985)
E5 (Node17467) E5a (Node17452) E5b (Node18419) E5c (Node17987)

1.8E-07

1.3E-07 E5 Series
EPEL Y

8.0E-08

3.0E-08

-5 0 5 10 15 20

-2.0E-08 E4 Series

1.5E-03 Double Zone Single Zone

T.E. (rad)

1.4E-03

1.3E-03
-5 0 5 10 15 20

Roll Angle (degree)

Figure 7.9. 4 Elastic strains over a complete mesh cycle.

It can be seen, over the mesh cycle, that the elastic strain variations at E1, E2 and E3
generally obey Saint-Venant’s Principle. The abrupt protrusions may be due to the use of
the mapped mesh (adaptive mesh may reduce the effect). The elastic strains at the E4 and
E5 positions appear to vary with the transmission error as measured at the input end of the
shaft. It was also noted that the strains were symmetrical to the axle of the shaft since the
elastic strain EPEL Y in each location of the E4 series (E4, E4a, E4b and E4c) or the E5

183
series (E5, E5a, E5b and E5c) were exactly same. Figure 7.9.5 shows the FEA results of
the elastic strain EPEL Y in the E4 and E5 locations with a 4.7 mm root crack.

N19661 N19646 N20613 N20181

2.5E-07 N19663 N19648 N20615 N20183

2.0E-07

E5 Series
1.5E-07
EPEL Y

1.0E-07

5.0E-08
E4 Series

0.0E+00

-5 0 5 10 15 20

-5.0E-08
Double Zone Single Zone
1.46E-03
T.E. (rad)

1.40E-03

1.34E-03

1.28E-03

-5 0 5 10 15 20
Roll Angle (degree)

Figure 7.9. 5 Elastic strains of the model with a 4.7 mm root crack
(FEA solutions were in 20 sub steps).

It should be noted from the results that the root crack could also be found by the elastic
strains, especially by the elastic strains of the E5 series. Once again, the elastic strains of
the E4 series or E5 series are perfectly symmetrical to the axle of the shaft due to the
boundary conditions between the shaft and the gear. If keyways or a splined shaft were
used, combined with manufacturing or other errors, the stress or strain states of the shaft
near the hub will become more complex. Further analysis was conducted with a refined
mesh at the shaft near the hub, where a 4 spline shaft was used. The shaft splines and the
gear were interference fitted. The manufacturing and other geometrical errors were
considered by making the boundary conditions between each of the splines slightly
different. The numbers of the locations for the elastic strain measurements were increased
in the region within 30 mm of the gear hub, as seen in Figure 7.9.6.
184
 Figure 7.9. 6 The relocated series of measurements.

A series of locations were shifted from the E1 series to the E5 series positions, and each of
them contained 4 locations coincident with the nodes in the circle of the section evenly
arranged at 900 intervals. FEA solutions for the elastic strains at the specified locations
along with the transmission error calculations have been carried out, and the results are
shown in Figure 7.9.7.

It can be seen that the elastic strains in each location of the series are no longer symmetrical
about the axis of the shaft, however, the amount of variations from the single zone to the
double zone remain stable. This shows that the strain measurements should be
concentrated on the relative changes of the elastic strain over the mesh cycle. Elastic strain
measurements can also find a small root crack, such as 1.5 mm root crack in the same
model, as shown in Figure 7.9.8.

185
 Node 227 Node1349 Node2261 Node180 Node178 Node1347 Node2259
Node 225 Node 223 Node1345 Node2257 Node176 Node 221 Node1343
Node2255 Node174 Node 219 Node1341 Node2253 Node172

1.8E-07

1.6E-07
E5 Series

1.4E-07

1.2E-07

1.0E-07
EPEL Y

8.0E-08

6.0E-08
E4 Series

4.0E-08

2.0E-08

0.0E+00

4.8E-04

3.6E-04
T.E. (rad)

2.4E-04
Double Zone Single Zone
1.2E-04

0.0E+00
-5 0 5 10 15 20
Roll Angle (degree)

Figure 7.9. 7 Elastic strains over a complete mesh cycle (FEA solutions were in 8 sub steps).
186
 E3 series Node1345 Node2257 Node223 Node176
E4 series Node1347 Node2259 Node225 Node178
E5 series Node1349 Node2261 Node227 Node180

2.0E-06

1.5E-06

1.0E-06

5.0E-07
EPEL Y

0.0E+00

-5 0 5 10 15 20
-5.0E-07

-1.0E-06

-1.5E-06

-2.0E-06

5.0E-04
T.E. (rad)

4.0E-04

3.0E-04

2.0E-04

-5 0 5 10 15 20
Roll Angle (degree)

Figure 7.9. 8 Elastic strains of the model with a 1.5 mm root crack.

For such a case, it would be difficult to find the root crack through the transmission error or
vibration measurements of the gear case.

Perfect involute gears in mesh along with an input shaft have been analysed. The elastic
strains on the surface of the shaft near the gear hub have been found to vary in form with
the transmission error over the mesh cycle. It has been recognized that the elastic strains
have more capability to reveal the details of the gear-shaft system if the measurements
were obtained from both input and output shafts. This investigation has shown that the
presence of a root crack changes the strain measurements on the shaft. The changes in the
shaft strain appear to be larger than the curves providing changes to the T.E. due to the
presence of the crack. Further studies are expected to improve the capabilities of condition
monitoring and controlling transmission system noise and vibrations.
187
8.1 General Introduction

Applications for non-metallic gears are growing rapidly, specially within office equipment
where the demand is for faster and quieter mechanisms. Non-metallic gears are also
getting more involved with the power transmitted systems such as vehicle parts. In
general, thermoplastic poly-condensates are used to manufacture plastics, industrially
known as polyamides (PA). However, reinforced composites with fibers have also become
more realistic in application where non-standard involute tooth forms are used to
effectively reduce wear and extend the gears service life.

The advantages and useful features in using non – metallic gears comprise their
· Low weight,

188
 · High wear resistance, good slip and dry – running properties,
· Good resistance to solvents, fuels and lubricants,
· Low noise and high resilience,
· Non – toxicity.
However, the limitations are also significant including the fact that standard polyamides
have limited resistance to high temperature, have lower load carrying capacity than their
metal counterparts, have high coefficients of thermal expansion and moisture content
which impairs mechanical properties and affects the structure dimensions. They are much
more complex materials than metals and are therefore, more difficult to analyse. As a
result there has been less work done on polymer and composite engineering components
and the engineer is often forced to experiment when developing new products (Walton
1995).

8.2 Previous Research

In recent years, a few different procedures have been developed to model the behavior of
non – metallic gears including material composites. Examples of this can be seen in
references (Yelle 1981; Terashima 1986; Tsukamoto 1986; Janover 1989; Walton 1989;
Enzmann 1990; Tsukamoto 1990; Breeds 1991; Baumgart 1992; Kudinov 1992; Walton
1992; Zhang 1992; Mao 1993; Nabi 1993; Solaro 1993; Tessema 1993; Tessema 1994;
Walton 1994; Koffi 1995; Tessema 1995; Walton 1995; Williams 1995; Du 1997; Nitu
1997; Smith 1997; White 1997; Kleiss 2000; Kurokawa 2000; Luscher 2000; Panhuizen
2000; Bushimata 2001; Wright 2001; Alagoz 2002; Andrei 2002). However, little has been
published to date on the mechanism properties of non-metallic gears in mesh. One of the
rare examples as shown in Figure 8.2.1 (Du 1997) presents results of the gear mesh
stiffness obtained by FEA, where the results for both steel and nylon gears were provided
over a complete mesh cycle.

Despite the lack of detail on the gear parameters (Du 1997), the results for Du’s nylon
gears can be improved either by Walton’s analysis on the flexibility of non – metallic gears
or by using the results on the hand over region in Chapter 6. This is due to the linear
material assumption as used by Du. Tessema and Walton (Tessema 1994) have further

189
confirmed there should be tip-relief different from that used in metal gears. To obtain the
correct transmission error of involute non-metallic gears in mesh is fundamentally
important. The fundamental analysis will give indications for the further analysis including
fiber reinforced materials and tooth profile modifications. The fundamental analysis
including experiments will be presented in this chapter.

Figure 8.2. 1 Gear mesh stiffness (Du 1997).

8.3 Modelling Material Nonlinearities

A number of material-related factors can cause a structures stiffness to change during the
course of an analysis. Non-linear stress-strain relationships of plastic, multilinear elastic,
and hyperelastic materials will cause a structures stiffness to change at different load levels
(and, typically, at different temperatures). Creep, viscoplasticity, and viscoelasticity will
give rise to nonlinearities that can be time, rate, temperature, and stress-related. Swelling
will induce strains that can be a function of temperature, time, neutron flux level (or some
analogous quantity), and stress. Any of these kinds of material properties can be
incorporated into FEA if appropriate element types are used.

The parameters of the gear (sets) that were used in the FE models and the simulation were
shown in Table 4.1. The non-metallic material used here was one of the standard
polyamides – cast nylon PA 6 - G and the Young’s modulus is given in Figure 8.3.1 by H.
Domininghaus (Domininghaus 1993). Poisson’s ratio was 0.4 as given by the material
190
supplier. According to the data that can be obtained, the Multilinear Elastic (MELAS)
material behaviour option was used in the analysis. The ANSYS MELAS material
behaviour option describes a conservative (path-independent) response in which unloading
follows the same stress-strain path as loading. Thus, relatively large load steps might be
appropriate for models that incorporate this type of material nonlinearity. In case the
material displays non-linear or rate-dependent stress-strain behaviour during the analysis, it
must be defined in the non-linear material property relationships in terms of a data table, as
seen in Figure 8.3.2.

Figure 8.3. 1 Stress/strain diagram for PA6 (dry) at various

temperature (Domininghaus 1993).

Figure 8.3. 2 MELA table preview.

191
It has to be stressed that the availability of the material properties is often one of the
limitations that will restrict the analysis. The desired analysis option is Anisotropic
(ANISO), which allows for different stress – strain behaviour in the material x, y and z
directions and more importantly allows for different behaviour in tension and compression.
The research in this current stage used the stress/strain diagram (Figure 8.3.1), which was
produced by the accelerated tensile test (Domininghaus 1993). The analysis results will
likely present some considerable overestimate of the T.E. (displacement) and an
underestimate of the torsional mesh stiffness (polyamide is considered to be some what
stiffer in compression than in tension).

The friction coefficient can also be an important factor in the modelling. Figure 8.3.3
shows that the friction coefficient is dependent on both load and running time
(temperatures).
Polyamide a
Coefficient of sliding friction

0.5 0.8
Friction factor

b c
0.4 PA 0.6

0.3 0.4

d
0.2 0.2

0.1 0
0 0.05 0.1 0.26 0.5 1 2.5 5 10 15 1 2.5 5 10 25 50 100 250 h 500
2
Mean bearing pressure N/mm Running time

(a) Dependence of the coefficient of sliding friction on (b) Coefficient of sliding friction for PA 66 against various
mean bearing pressure. metals as a function of running time (axial) bearing test
Roughness height for PA, polyamide rig, dry, pv=0.015 N/mm2 * m/s; 7.0 psi * ft/min
a Cu/Zn alloy to DIN 17660
R Z =1.5 to 3 μm/0.059 to 0.118 mil . b AI alloy to DIN 1725
c GG 22 to DIN 1691
d Steel 16 MnCr 5 to DIN 17210; 52 HRC

Figure 8.3.3 The friction behaviours of polyamide (Domininghaus 1993).

FE modelling of non-metallic gears in mesh including the variation of friction coefficient is

important since, (for dry unlubricated gears in mesh) wear is dominant in the gear service
life (Walton 1995). While, this will add more difficulties to the modelling task, the contact
ratio can also change dramatically with load and temperature. One of the strategies is to
model the meshing gears with a lower constant friction coefficient (e.g. 0.1). The friction
coefficient of Figure 8.3.3 can then be inserted for each relative meshing tooth pair
according to the contact stress at each mesh position over a mesh cycle, then the solution

192
re-evaluated. Such procedure can be repeated until the results are satisfactory. However,
there is reluctance to do such tremendous FE calculations without considering the use of
the realistic compression stress/strain diagram.

8.4 Element and Mesh

The available elements for the research using the ANSYS MELAS material behaviour
option are plane 42 for 2D modelling and solid 45 for 3D modelling. Both quad and brick
elements can be used without the mid-side nodes. The procedures for applying mesh
adaptation with contact are no different with that of metallic gears. However, when using
solid 45 to build the 3D model, the number of element divisions along the rotation axis will
be limited as the low order element can only handle limited distortions, especially in the
aspect ratio. As the adaptive mesh produced many small elements near the contact points,
it required more divisions along the tooth width, consequently there will be a large number
of distorted elements in the low stress area, as seen in Figure 8.4.1.

Figure 8.4. 1 The element check has shown the distorted elements.
193
The easiest way for applying adaptive mesh without distorted elements is to increase the
element number in the low stress area, but the computational efficiency will be poor.
Alternatively the transition of the division between low and high stress areas can be made,
but that is extremely time consuming for building the model. For the current stage, 2D
plane stress models were used. The model and its detailed adaptive mesh are shown as in
Figure 8.4.2.

Driving

Figure 8.4. 2 The 2D model and the adaptive mesh.

8.5 Transmission Error

The application of polyamide material to sliding contact is highly non-linear. To obtain

accurate solution results for the subsequent calculation of the transmission error, the choice
of the proper number of sub steps is important. The analysis approach here chose 10 sub
steps in an altering range of minimum 5 to maximum 50 sub steps for the solutions. It was
noted that if the major sub steps were set greater than 15, better solutions were achieved.
However, the computing time was greatly affected. For each temperature, the quantity of
FEA calculations was equivalent to that of metallic gears.
194
8.5.1 T.E. as Function of Input Load at Each Temperature

The calculations at each temperature were initially calculated with an input load of 1 Nm.
This load was then gradually increased until the triple contact zone was obvious. Unlike
the nature of HCRG (high contact ratio gears), the triple contacts of LCRG (low contact
ratio gears) always leads to premature contacts, as shown in Figure 8.5.1.

Figure 8.5. 1 The typical triple contacts of the nylon (PA6) gears.

The load and the temperature can easily increase the contact stress at the tooth tips,
therefore high friction forces will be at the contacts where naturally the highest relative
movement (rapid wear) occurs in such condition. The triple contact zones at each
temperature are shown in Figure 8.5.2 to 8.5.15.
0.016 Triple Contact Zone

0.014 120 Nm

0.012
100 Nm

0.01
80 Nm

0.008
60 Nm
0.006
Pitch Point

Pitch Point

Pitch Point

40 Nm
0.004

20 Nm
0.002
10 Nm
5 Nm
0 1 Nm
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Distance (degree)
Figure 8.5. 2 T.E. under various input loads when the temperature is at -40o C .
195
 Pitch Pitch
0.018 Point Point
Triple Contact Zone

0.016

100 Nm
0.014

0.012 80 Nm

0.01
60 Nm
0.008

0.006 40 Nm

0.004
20 Nm
0.002 10 Nm
5 Nm
0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 3 T.E. under various input loads when the temperature is at -20o C .
0.016 Triple Contact Zone

0.014

80 Nm
0.012
70 Nm
0.01
60 Nm

0.008
Pitch Point

Pitch Point

Pitch Point
40 Nm
0.006

0.004
20 Nm

0.002
10 Nm
5 Nm
0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

o
Figure 8.5. 4 T.E. under various input loads when the temperature is at 0 C .
Triple Contact Zone Pitch Pitch
Point Point
0.016

0.014 70 Nm

0.012 60 Nm

0.01 50 Nm

0.008 40 Nm

0.006 30 Nm

0.004 20 Nm

0.002 10 Nm
5 Nm
0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 5 T.E. under various input loads when the temperature is at 23o C .
196
0.021
Triple Contact Zone

0.018 60 Nm

0.015 50 Nm

0.012 40 Nm

0.009 30 Nm

0.006 20 Nm

0.003 10 Nm
5 Nm
1 Nm
0
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 6 T.E. under various input loads when the temperature is at 40o C .

Triple Contact Zone

0.025
50 Nm

0.02
40 Nm

0.015
30 Nm

0.01 20 Nm

0.005 10 Nm

5 Nm
0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 7 T.E. under various input loads when the temperature is at 45o C .

Triple Contact Zone

0.035
50 Nm
0.03

40 Nm
0.025

0.02 30 Nm

0.015
20 Nm

0.01
10 Nm
0.005
5 Nm

0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 8 T.E. under various input loads when the temperature is at 50o C .

197
 Triple Contact Zone
0.05

0.045 50 Nm
0.04

0.035
40 Nm

0.03
30 Nm
0.025

0.02
20 Nm
0.015

0.01 10 Nm
0.005 5 Nm
0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 9 T.E. under various input loads when the temperature is at 60o C .
Triple Contact Zone
0.04
40 Nm
0.035

0.03
30 Nm
0.025

0.02 20 Nm
0.015

0.01 10 Nm
0.005 5 Nm
1 Nm
0
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 10 T.E. under various input loads when the temperature is at 80o C .
Triple Contact Zone
0.035
30 Nm
0.03

0.025
20 Nm
0.02

0.015

0.01 10 Nm

0.005 5 Nm

0 1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 11 T.E. under various input loads when the temperature is at 100o C .
Triple Contact Zone
0.04
0.035 30 Nm
0.03
0.025 20 Nm
0.02
0.015
10 Nm
0.01
0.005
5 Nm
0
1 Nm
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 12 T.E. under various input loads when the temperature is at 120o C .
198
 Triple Contact Zone
0.03
20 Nm
0.025

0.02

0.015 10 Nm
0.01
5 Nm
0.005
1 Nm
0
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 13 T.E. under various input loads when the temperature is at 140o C .
Triple Contact Zone

0.035 20 Nm
0.03
0.025
0.02
10 Nm
0.015
0.01 5 Nm
0.005
1 Nm
0
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 14 T.E. under various input loads when the temperature is at 160o C .

0.06 Triple Contact Zone

20 Nm
0.05

0.04

0.03 10 Nm
0.02
5 Nm
0.01
1 Nm
0
-22 -18 -14 -10 -6 -2 2 6 10 14 18 22

Roll Distance (degree)

Figure 8.5. 15 T.E. under various input loads when the temperature is at 180o C .

Results above have shown the distinguishing characteristics of the T.E. (contrast to that of
metallic gears) due to the flexibility of the material under load and temperature. It should
be noted that,
1. Triple contact does occur, especially when the temperature is not too low. In most
cases, T.E. of triple contact zones have shown higher values than that of the double
contact zones. Such high contact ratio character is a significant difference compared
to that of conventional high contact ratio (metallic) gears (see Appendix C)
2. The triple contact zones do not appear symmetrical over the tooth mesh cycle.
Generally the T.E. of the approach cases are greater than that of recess cases, and the
triple contact zones do not appear symmetrical about the pitch point. This is because,

199
 3. The handover regions (of the double zone) are significantly expanded and the
expansion rates for the approach and the recess cases are significantly different.
Figure 8.5.16 shows the detailed variations of the contact ratio and the width of the
handover regions against load at temperatures of -40 o C , 23o C and 160 o C .

4.5

Width of Hand Over Regions (degree)

4 Recess Case
3.5
3
2.3 o
160 C 2.5
2
2.2 Approach Case
1.5
2.1 o o
1
23 C -40 C 0.5
2 0
0 20 40 60 80
1.9
Load (Nm)
1.8 (b) The handover regions vs. input load
o
at -40 C (when contact ratio < 2).
1.7

1.6 Width of Hand Over Regions (degree) 4.5

4 Recess Case
1.5 3.5
0 20 40 60 80 100 120 3
Load (Nm) 2.5
o 2
(a) Contact ratio vs. input load at -40 C , Approach Case
1.5
o o
23 C and 160 C . 1
0.5
0
0 10 20 30 40
Load (Nm)
(c) The handover regions vs. input load
o
at 23 C (when contact ratio < 2).

Figure 8.5. 16 Detailed variations of the contact ratio and the width of handover
regions against load at temperature of -40 o C , 23o C and 160 o C (the
o
handover regions vs. input load at 160 C is not shown, as more calculations
with light loads were required).

8.5.2 T.E. as Function of Temperature at Certain Input Load

If it was assumed that the input load remains steady at each temperature and the
temperatures are loaded without considering the time, then the T.E. (over a complete mesh

200
cycle) against the temperature can be plotted in a 3D figure. Under various input load, 3D
figures are produced as shown from Figure 8.5.17 to 8.5.25.

Figure 8.5. 17 Under 1 Nm input load, T.E. (over a complete mesh cycle) as
a function of full range temperature.

Figure 8.5. 18 Under 5 Nm input load, T.E. (over a complete mesh cycle) as a
function of full range temperature.
201
Figure 8.5. 19 Under 10 Nm input load, T.E. (over a complete mesh cycle) as
a function of full range temperature.

Figure 8.5. 20 Under 20 Nm input load, T.E. (over a complete mesh cycle) as
a function of full range temperature.

202
Figure 8.5. 21 Under 30 Nm input load, T.E. (over a complete mesh cycle) as
a function of temperature (from -40o C to 120 o C ).

Figure 8.5. 22 Under 40 Nm input load, T.E. (over a complete mesh cycle) as
a function of temperature (from -40o C to 80o C ).

203
Figure 8.5. 23 Under 50 Nm input load, T.E. (over a complete mesh cycle) as
a function of temperature (from -40o C to 60o C ).

Figure 8.5. 24 Under 60 Nm input load, T.E. (over a complete mesh cycle) as
a function of temperature (from -40o C to 40o C ).

204
 Figure 8.5. 25 Under 80 Nm input load, T.E. (over a complete mesh cycle) as
a function of temperature (from -40o C to 0 o C ).

The results show two rapid changes (in most cases) in the region of 45o C to 50o C and the
region over 150o C . This characteristic shows the dependence on the material properties
(Figure 8.3.2).

8.5.3 The Comparison with Metallic Gears

With the different materials (aluminium and nylon), the analysis has produced two groups
of T.E. curves under the same boundary and load conditions as shown in Figure 8.5.26.

The T.E. results presented in Figure 8.5.26 (a) and (b) are significantly different in both the
amplitude (absolute value) and the curve shape (relative value). The main interesting
features of the comparison is the curve shape (relative value), so Figure 8.5.26 (a) and (b)
are compared together to produce Figure 8.5.27.

205
T.E. (Aluminum) rad x 10-4 120 Nm

T.E. (Nylon – PA6) rad x 10-3

100 Nm

80 Nm

60 Nm

50 Nm
40 Nm

30 Nm

20 Nm
10 Nm
5 Nm
1 Nm

Roll Distance (degree) Roll Distance (degree)

(a) Aluminum gears in mesh. (b) Nylon (PA6) gears in mesh (temperature is at
o
-40 C ).

Figure 8.5. 26 T.E. of aluminium gears and nylon gears in mesh (each pair
with the same boundary and load conditions.

120 Nm
-4

100 Nm
-3

80 Nm

60 Nm

50 Nm

40 Nm

30 Nm

20 Nm

10 Nm
5 Nm
1 Nm

Roll Distance (degree)

Figure 8.5. 27 The comparisons when the nylon gear pair is at temperature -40 o C .

206
Similarly, results can be shown for 23o C and 160o C as illustrated in Figure 8.5.28 and 8.5.29.

-4

60 Nm
T.E. (Nylon – PA6) rad x 10-3
50 Nm

40 Nm

30 Nm

20 Nm

10 Nm

5 Nm
1 Nm

Roll Distance (degree)

Figure 8.5. 28 The comparisons when the nylon gear pair is at temperature 23o C .
-4

20 Nm
-3

10 Nm

5 Nm

1 Nm

Roll Distance (degree)

Figure 8.5. 29 The comparisons when the nylon gear pair is at temperature 160o C .

207
By the observation of the comparison figures above, the following conclusions can be
given,
1. When the input load tends to be very light, the T.E. curves of metallic and non-
metallic gears tend to have the same shape (relative values).
2. The curve shape mainly changes (by varying material property) in the low stiffness
region (single contact zone), as observed by the expansion of the handover regions
(when contact ratio < 2) or the handover regions plus the triple contact zone (when
contact ratio > 2).
3. Double zone curve shape changes only when the triple contact zone is significantly
expanded (heavy load, high temperature).

8.6 The Combined Torsional Mesh Stiffness

The combined torsional mesh stiffness under various input loads will present obvious
different behaviours over the mesh cycle. The most complicated changes will occur in the
lower stiffness regions including the single contact zone, the handover regions and the
triple contact zone – which in most cases is still with lower stiffness, as shown in Figure
8.6.1 and 8.6.2.

It can be seen, that before the single zone disappears (contact ratio < 2) the major
variations of the stiffness are in the higher amplitude and the expansion of the handover
regions that cause a minor stiffness curve phase change over the periodical mesh cycles.
After the triple contact takes place, the double zone stiffness rises slowly, while the
stiffness of the triple zone and handover regions are still increasing rapidly with input load
and this fast change causes a major phase decrease over the periodical mesh cycle. Under
extreme conditions such as the temperature reaching 180o C , the flexibility of the gears
allows the triple contact to occur very early on. The triple zone stiffness and its width
increases with the input load, meanwhile the increase in the double zone stiffness continues
after the triple contact occurs. The stiffness curve tends to be smoothed quickly with the
increasing load as shown in Figure 8.6.3.

208
 1 Nm 5 Nm 10 Nm 20 Nm 30 Nm 40 Nm
50 Nm 60 Nm 80 Nm 100 Nm 120 Nm Pitch Point

Combined Torsional Mesh Stiffness (Nm/rad)

Pitch Point
Triple contact zone Phase decrease

8750

8250

7750

7250

6750

6250

5750
Pitch Point

Pitch Point
5250 Single contact zone
-20 -16 -12 -8 -4 0 4 8 12 16 20

Roll Distance (degree)

Figure 8.6. 1 The combined torsional mesh stiffness under various input loads,
temperature -40 o C , Nylon (PA 6).

1 Nm 5Nm 10 Nm 20 Nm 30 Nm
40 Nm 50 Nm 60 Nm 70 Nm pitch point
Combined Torsional Mesh Stiffness (Nm/rad)

5000

4500

4000

3500

3000
-24 -18 -12 -6 0 6 12 18 24

Roll Distance (degree)

Figure 8.6. 2 The combined torsional mesh stiffness under various input loads,
temperature 23o C , Nylon (PA 6).

209
 Triple contact zone 0.1Nm 1Nm 5 Nm
10 Nm 20 Nm pit ch point

Combined Torsional Mesh Stiffness (Nm/rad)

380

330

280

230

180
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Distance (degree)

Figure 8.6. 3 The combined torsional mesh stiffness under various input loads, temperature 180 C ,
o

Nylon (PA 6).

Discussions above have mainly shown the variations of the combined torsional mesh
stiffness against the input loads at different temperatures. For the variations of the
combined torsional mesh stiffness against the temperatures at different input loads, the
details are given as shown in Figure 8.6.4 to 8.6.11.

Figure 8.6. 4 Combined torsional mesh stiffness under 1 Nm input load

against the full range temperature, Nylon (PA 6).
210
Figure 8.6. 5 Combined torsional mesh stiffness under 5 Nm input load against
the full range temperature, Nylon (PA 6).

Figure 8.6. 6 Combined torsional mesh stiffness under 10 Nm input load

against the full range temperature, Nylon (PA 6).

211
Figure 8.6. 7 Combined torsional mesh stiffness under 20 Nm input load against the full
range temperature, Nylon (PA 6).

Figure 8.6. 8 Combined torsional mesh stiffness under 30 Nm input load against temperature
from -40o C to 120o C , Nylon (PA 6).
212
Figure 8.6. 9 Combined torsional mesh stiffness under 40 Nm input load against temperature
from -40o C to 80o C , Nylon (PA 6).

Figure 8.6. 10 Combined torsional mesh stiffness under 50 Nm input load against temperature
from -40o C to 60o C , Nylon (PA 6).
213
Figure 8.6. 11 Combined torsional mesh stiffness under 60 Nm load against temperature from -40o C to
40 C , Nylon (PA 6).
o

It can be seen that the combined torsional mesh stiffness drops with the temperature
increases from -40o C to 40o C . When the temperature is higher than 40o C , the combined
torsional mesh stiffness becomes relatively stable (drops much slower with temperature
increase). The results indicate that this characteristic does not rely on whether the triple
contact has occurred. For example, when the input load is 1 Nm, no triple contact occurs
through the temperature range, while when the input load is 60 Nm, the triple contact
occurs before the temperature reaches 0o C .

The behaviours of the combined torsional mesh stiffness have shown (one of) the reason
why non-metallic gears run smoother than metallic gears. However, when non-metallic
gears run quiet and smooth this doesn’t mean the gears are under good running condition.
It is likely that triple contact has taken place, and consequently this will result in excessive
wear – the most significant factor to the service life.

8.7 Load Sharing Ratio

In low contact ratio (it normally is between 1.3 to 1.6) non-metallic spur gears, the load is
transmitted by one pair and two pairs of teeth alternately when the combination of input
214
load and the temperature is within a limited range, otherwise, two pairs and three pairs of
teeth will transmit the load alternatively.

Due to the material flexibility, the deformed teeth will be rotating (about the tooth roots),
bending, shearing and the tooth flanks becoming flattened at the points of contact. These
factors can easily become significant, altering the load variation outside the normal path of
contact. Triple contact with low contact ratio gears is caused by a significant amount of
contact outside the normal path. Theoretically any contact outside the normal path on
involute spur gears will present premature contact (or corner contact). For the special case
of the triple contact, increasing either the input load or the temperature or both can quickly
increase the percentage of total load on the teeth with premature contact, and the triple zone
can rapidly expand. The curves of tooth load sharing ratio can clearly show these complex
variations from the tooth running in, to running out of contact.

8.7.1 Load Sharing Ratio as Function of Input Load at Each Temperature

It has been shown that tooth load sharing ratio is dependent on the input load (Chapter 7)
for metallic gears in mesh, but for non-metallic gears, the variations can be dramatic and
there will be a very limited common curve of the load sharing ratio (only when the single
contact zone exists). At each of the fourteen different temperatures, the load share against
various input loads is given as shown from Figure 8.7.1 to 8.7.14.

One base pitch 1Nm 5Nm

1 10Nm 20Nm
30Nm 40Nm
50Nm 60Nm
80Nm 100Nm
0.8 120Nm pitch point

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Roll Distance (degree)
Figure 8.7. 1 The load-sharing ratio against input loads at temperature -40o C , Nylon (PA 6).
215
 1Nm 5Nm
One base pitch
10Nm 20Nm
1 30Nm 40Nm
50Nm 60Nm
80Nm 100Nm
pitch point
0.8

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Roll Distance (degree)
Figure 8.7. 2 The load-sharing ratio against input loads at temperature -20o C , Nylon (PA 6).
One base pitch 1Nm 5Nm
1 10Nm 20Nm
30Nm 40Nm
50Nm 60Nm
0.8 70Nm 80Nm
pitch point

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

Roll Distance (degree)

Figure 8.7. 3 The load-sharing ratio against input loads at temperature 0o C , Nylon (PA 6).

One base pitch 1Nm 5Nm

1 10Nm 20Nm
30Nm 40Nm
50Nm 60Nm
0.8 70Nm pitch point

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Roll Distance (degree)
Figure 8.7. 4 The load-sharing ratio against input loads at temperature 23o C , Nylon (PA 6).
216
 One base pitch 1 Nm 5 Nm
1 10 Nm 20 Nm
30 Nm 40 Nm
50 Nm 60 Nm
pitch point
0.8

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

Roll Distance (degree)

Figure 8.7. 5 The load-sharing ratio against input loads at temperature 40 o C , Nylon (PA 6).

One base pitch 1Nm 5Nm

1 10Nm 20Nm
30Nm 40Nm
50Nm pitch point
0.8

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

Roll Distance (degree)

Figure 8.7. 6 The load-sharing ratio against input loads at temperature 45o C , Nylon (PA 6).

One base pitch 1Nm 5Nm

1 10Nm 20Nm
30Nm 40Nm
50Nm pitch point
0.8

0.6

0.4

0.2

0
-19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5 7 9 11 13 15 17

Roll Distance (degree)

Figure 8.7. 7 The load-sharing ratio against input loads at temperature 50 o C , Nylon (PA 6).
217
 One base pitch 1Nm 5Nm
1 10Nm 20Nm
30Nm 40Nm
50Nm pitch point
0.8

0.6

0.4

0.2

0
-20 -16 -12 -8 -4 0 4 8 12 16

Roll Distance (degree)

Figure 8.7. 8 The load-sharing ratio against input loads at temperature 60o C , Nylon (PA 6).

One base pitch 1Nm 5 Nm

1 10 Nm 20 Nm
30 Nm 40 Nm
pitch po int
0.8

0.6

0.4

0.2

0
-20 -16 -12 -8 -4 0 4 8 12 16

Roll Distance (degree)

Figure 8.7. 9 The load-sharing ratio against input loads at temperature 80o C , Nylon (PA 6).

One base pitch

1Nm 5Nm
1
10Nm 20Nm
30Nm pitch po int
0.8

0.6

0.4

0.2

0
-20 -16 -12 -8 -4 0 4 8 12 16

Roll Distance (degree)

Figure 8.7. 10 The load-sharing ratio against input loads at temperature 100o C , Nylon (PA 6).
218
 One base pitch 1Nm 5Nm
1 10Nm 20Nm
30Nm pitch po int
0.8

0.6

0.4

0.2

0
-20 -16 -12 -8 -4 0 4 8 12 16

Roll Distance (degree)

Figure 8.7. 11 The load-sharing ratio against input loads at temperature 120o C , Nylon (PA 6).

One base pitch

1Nm 5Nm
1 10Nm 20Nm
pitch po int

0.8

0.6

0.4

0.2

0
-20 -16 -12 -8 -4 0 4 8 12 16

Roll Distance (degree)

Figure 8.7. 12 The load-sharing ratio against input loads at temperature 140o C , Nylon (PA 6).

One base pitch 1Nm 5Nm

1 10Nm 20Nm
pitch po int

0.8

0.6

0.4

0.2

0
-20 -16 -12 -8 -4 0 4 8 12 16
Roll Distance (degree)
Figure 8.7. 13 The load-sharing ratio against input loads at temperature 160o C , Nylon (PA 6).

219
 One base pitch 1Nm 5Nm
1 10Nm 20Nm
pitch point

0.8

0.6

0.4

0.2

0
-21 -18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

Roll Distance (degree)

Figure 8.7. 14 The load-sharing ratio against input loads at temperature 180o C , Nylon (PA 6).

Results above have shown how quickly the tooth load share can change with the input
loads at different temperatures. It is noted that when the curve of the load sharing ratio
expands over one base pitch (it always happens at the approach case first), triple contact
occurs and consequently the maximum tooth load share will be smaller than 1.

8.7.2 Load Sharing Ratio as Function of Temperature

It will be anticipated with the T.E. and the combined torsional mesh stiffness results, that
the tooth load share will also be a function of temperature when the input load is stable.
However, for different input loads, the temperature varying ranges are normally different.
Those variations are shown in Figure 8.7.15 to 8.7.22.

Figure 8.7. 15 Load-sharing ratio under 1 Nm input load against the full range
temperature, Nylon (PA 6).
220
Figure 8.7. 16 Load-sharing ratio under 5 Nm input load against the full range temperature,
Nylon (PA 6).

Figure 8.7. 17 Load-sharing ratio under 10 Nm input load against the full range temperature,
Nylon (PA 6).
221
Figure 8.7. 18 Load-sharing ratio under 20 Nm input load against the full range temperature, Nylon (PA 6).

o o
Figure 8.7. 19 Load-sharing ratio under 30 Nm input load against temperature from -40 C to 120 C , Nylon (PA 6).

222
Figure 8.7. 20 Load-sharing ratio under 40 Nm input load against temperature from -40o C to
o
80 C , Nylon (PA 6).

Figure 8.7. 21 Load-sharing ratio under 50 Nm input load against temperature from -40o C to
o
60 C , Nylon (PA 6).

223
Figure 8.7. 22 Load-sharing ratio under 60 Nm input load against temperature from -40o C to
o
40 C , Nylon (PA 6).

Results above have shown that increasing temperature will result in tooth load sharing ratio
variation in a similar way as the input torque did under a stable temperature. It is observed
that the most dramatic changes occur about 50o C in most cases, and that is due to the
nature of the material properties (Figure 8.3.1). The results also confirmed that within the
ordinary range of temperatures, unlike the analysis for metallic gears, the behaviour of
Nylon is dependent on the effects of both temperature and input torque.

8.7.3 About the Theoretical Tooth Load Share

If the gears are rigid, the tooth load-sharing ratio will not be influenced by the input load.
In other words, there is only one curve of load sharing ratio for the meshing gears if they
are totally rigid. For elastic gears in mesh, if the input load tends to be very light, the effect
of elastic deformation can be ignored and the curve of the load sharing ratio will tend to
that of the rigid gears in mesh. Theoretically, when the input load tends to zero, the curve
of the load-sharing ratio will be independent of material properties, so that the metallic,
non-metallic and rigid gears will all operate with theoretical tooth load share.
224
The research on the load-sharing ratio of involute spur gears can be found as early as the
1930’s by Walker (Walker 1938). He presented the load distribution on rigid teeth as
shown in Figure 8.7.23.
d e
W

b g
W/2 c f

0 a h
Roll Distance
Figure 8.7. 23 The theoretical load sharing ratio (on rigid teeth).

The tooth load in the double zone bc or fg was exactly half of that in the single zone de.
This figure is still predominant today, as it appears in many publications (AYEL 1984;
Kuang 1992; Castro 1994; Walton 1994; Walton 1995; Sirichai 1996; Sirichai 1999;
Yildirim 1999) and some important gearing handbooks (Maitra 1989; MAAG 1990).
However, the proof for the load share ratio of rigid gears in mesh as shown in Figure 8.7.23
has not been found in the literature. For rigid gears in mesh, neglecting the friction at the
contacts (in the double contact zone), it can be illustrated in Figure 8.7.24. The driving
gear force balance equation can be given, assuming steady state conditions,
T – Fa . rb – Fc . rb = 0, (8.1)
Which can be rearranged to give,
Fa + Fc = T / rb. (8.2)
Equation (8.2) shows that the sum of Fa and Fc remain constant in the double contact zone
if the input load is stable. However, it seems no further information can be provided for the
relation between Fa and Fc. So, Figure 8.7.23 seems to depend on the assumption of Fa = Fc
in the double contact zone.

The options for using FEA to prove the tooth load share of rigid gears in mesh are: (i)
applying very light loads to find out the convergence curve as described in the beginning of
this section and (ii) changing the modulus of elasticity to a very high value.

225
 O2

Fc
C
P
Fa
A
B

qa qc

ra
rc

20o
rb

O1

Figure 8.7. 24 Rigid gears (ratio 1:1) in mesh of double contact zone.

Firstly, the results were calculated for aluminium and nylon gears (PA 6 at 23o), both with
1 Nm input load and compared with the curve of Figure 8.7.23, as shown in Figure 8.7.25.

It can be seen, the major difference between the aluminium gears and the nylon gears are
the curves in their handover regions, but they have very similar (common) double contact
zones. Secondly, the input loads for the nylon gears were further decreased from 1 Nm to
0.01 Nm while the modulus changes according to Figure 8.3.1 from 23o C to 180o C were
used. Results are shown in Figure 8.7.26.

226
 Load Sharing Ratio

O
Nylon (PA 6) gears @ 23 C

Rigid teeth (Walker)

Metallic gears (aluminum)

Roll Distance

Figure 8.7. 25 With light load (1 Nm), the tooth load share of metallic and non-metallic
gears are compared with that of conventional rigid teeth.
Load Sharing Ratio

Metallic gears (aluminum) @ 1 Nm input load.

Rigid teeth (Walker).
O
Nylon (PA 6) gears @ 180 C , 1 Nm input load.
O
Nylon (PA 6) gears @ 180 C , 0.1 Nm input load.
O
Nylon (PA 6) gears @ 180 C , 0.01 Nm input load.

Roll Distance

Figure 8.7. 26 The variations of the nylon gears.

These results show the difference with the curve of conventional rigid teeth in the common
double contact zone compared to nylon gears with the lower modulus, but when the input
load tends to the minimum, the additional difference will also tend to disappear. Finally,
the result of the gears with higher modulus of elasticity (106 Gpa) is given and compared
with that of aluminium gears (with 1 Nm input load) as shown in Figure 8.7.27.

227
 Load Sharing Ratio
· Aluminum gears @ 1 Nm load.
· Modulus of elasticity = 69 Gpa.
· Tooth tip fillet radius = 0.5 mm.

· Metallic gears @ 0.01 Nm load.

· Modulus of elasticity = 106 Gpa.
· Tooth tip fillet radius = 0.25 mm.

Roll Distance
Figure 8.7. 27 The result of gears with higher modulus of elasticity.

The results above have shown that there is no possibility for the load-sharing ratio to
become level over the double contact zone, so that the contact forces Fa and Fc will not tend
to be equal. The magnitude of contact force seems to depend on the distance from the
contact point to the hub centre, and the distance varies over the double zone almost
linearly. The difference that can be observed in Figure 8.7.27 is in the contact ratio which
increased from 1.536 to 1.564 due to the tooth tip fillet radius decreasing from 0.5 mm to
0.25 mm (the theoretical contact ratio 1.59 is based on the involute tooth profile without tip
modification).

In conclusion, the theoretical tooth load share ratio should be produced by using FEA, if
the input load were very light or the modulus of elasticity were sufficiently high and the
tooth profile was of involute form. However, it can be very difficult to obtain the FEA
solutions (in or near the handover regions) if, for example, there is no tooth tip fillet. More
about the effects on tooth tip fillet radius can be referred to Appendix C where results of
gears without tooth tip fillet are given, but the mesh density was higher than 100 thousand
nodes.

Finally, a simplified theoretical tooth load share ratio can be produced, which is based on
the theoretical contact ratio Cr (1.59), base pitch Pb (15.652 degree) and the assumption of
the linear variations in the double zones, based on the FEA results, as shown in Figure
8.7.28.
228
 Cr*Pb

Load Sharing Ratio

(2-Cr)*Pb
0.5

The closest FEA simulation.

Simplified theoretical tooth load share.
Conventional tooth load share of rigid teeth.

Pitch Point

Roll Distance

Figure 8.7. 28 The simplified theoretical tooth load share.

Actual applications may required minor FEA calculations to produce the curve of
theoretical tooth load share ratio compared to the above simplified curve.

8.8 Optimal Tip Relief

The research has found that applying long tip relief can achieve a (very) low design load,
and premature contact occurs when the input load is higher than the design load. In such a
case, the design load will not increase significantly with a large amount of relief Ca,
because the tipping has also weakened the tooth. This means over tipping (with long
modification) can degrade the performance of the tooth. Nylon gears have naturally wider
handover regions, higher damping and lower inertia, so long tip relief will not be
considered necessary for general applications. On the other hand short type relief is very
much needed to minimise the high stress that occurs at the premature contact (tooth tip) to
prevent the friction coefficient increasing (see Figure 8.3.3), and consequently wear will be
reduced as wear is the predominant mode of failure in the applications (Walton 1995).

The desired tipping should have the effect of model R5out12 of aluminium gears as shown
in section 7.4.9. Model R5out12 has presented the longest short type relief; there is no
rigid body motion when the rotation is over a complete mesh cycle. The effect of the

229
premature contact at the relief starting point S1 (depend on the amount of Ca) is a minimum
compared to other types of short relief. The more important fact is that the position S1 only
relies on the gear geometric parameters, so the S1 of the aluminium gears is valid for the
gears with different materials including nylon (PA 6). The analysis on such nylon gears
has been carried out with three models with the third model being for the simulation of the
commercially purchased nylon (PA 6) test gears. Tooth relief parameters of each model
are listed in Table 8.1.

Parameters Ca Ln
a (degree)
Model name Absolute value (mm) Normalized value Absolute value (mm) Normalized value

Model 1 0.0326 0.27 2.44 0.68 0.6

Model 2 0.0545 0.45 2.44 0.68 1.0
Model 3 0.2 1.67 2.49 0.69
Table 8.1. Tooth relief parameters.

The form of the modified profile is the original involute curve rotated by 0.6 and 1.0 degree
for model 1 and 2 respectively, and a straight line is used for model 3. More details about
the test gears will be presented in the next section, and the further optimum form of the
modified profile (using circular form) can be referred to Appendix E. FEA results are
presented in Figure 8.8.1 to 8.8.8, with a temperature of 23o C.
Single zone protrusions
0.0014

0.0012

0.001
T.E. (rad)

0.0008

0.1Nm 1Nm

5Nm pitch po int

Pitch Point

Pitch Point

0.0006
Pitch Point

0.0004

0.0002

0
-24 -18 -12 -6 0 6 12 18 24
Roll Angle (degree)

Figure 8.8. 1. The T.E. of model 1 with lighter input loads, Nylon (PA 6).
230
 Triple Zone 10 Nm 20 Nm 30 Nm 40 Nm
60 Nm 80 Nm pitch point
0.018

0.016

0.014

0.012

T.E. (rad)
0.01

0.008

0.006

0.004

0.002

0
-24 -18 -12 -6 0 6 12 18 24

Roll Angle (degree)

Figure 8.8. 2. The T.E. of model 1 with heavier input loads, Nylon (PA 6).

Triple Zone
Combined Torsional Mesh Stiffness (Nm/rad)

5000

4500

4000

3500

Single zone
3000 with premature
contacts.
Amount protection
for load 0.1 Nm.
2500
0.1 Nm 1 Nm 5Nm 10 Nm 20 Nm
30 Nm 40 Nm 60 Nm 80 Nm pitch point
2000
-24 -18 -12 -6 0 6 12 18 24
Roll Distance (degree)

Figure 8.8. 3. The combined torsional mesh stiffness of model 1, Nylon (PA 6).

0.1Nm 1Nm 5Nm 10 Nm pitch po int

Single zone protrusions
0.0025

0.002
T.E. (rad)

0.0015

0.001

0.0005

0
-24 -18 -12 -6 0 6 12 18 24
Roll Angle (degree)

Figure 8.8. 4. The T.E. of model 2 with lighter input loads, Nylon (PA 6).
231
 20 Nm 30 Nm 40 Nm 60 Nm 80 Nm pitch point
0.02
Triple Zone
0.018

0.016

0.014

0.012

0.01
T.E.

0.008

0.006

0.004

0.002

0
-24 -18 -12 -6 0 6 12 18 24
Roll Angle (degree)

Figure 8.8. 5. The T.E. of model 2 with heavier input loads, Nylon (PA 6).

5000
(Nm/rad)

4500

4000
Combined Torsional Mesh Stiffness

3500

3000

2500

2000

0.1 Nm 1 Nm
1500
5Nm 10 Nm
20 Nm 30 Nm
1000 40 Nm 60 Nm
80 Nm pitch point
500

0
-24 -18 -12 -6 0 6 12 18 24
Roll Distance (degree)

Figure 8.8. 6. The combined torsional mesh stiffness of model 2, Nylon (PA 6).

Results above have shown that the premature contacts have already occurred with the load
of 5 Nm for model 1 and 10 Nm for model 2, by observing the protrusions of the T.E.
curves. Even for model 2, the amount of relief is far from sufficient considering that the
temperature could be higher. The amount of relief of model 3 was more than three times
that of model 2 and 1.67 times the maximum allowance of current standards. With the
simplest relief (straight line), model 3 presents the results as shown in Figure 8.8.7 and
8.8.8.
232
 0.009
4.927 Nm 12.91Nm 20.09 Nm 30Nm 40Nm pitch po int

T.E. protrusion
0.008

0.007

0.006
T.E. (rad)

0.005

0.004

0.003

0.002

0.001

0
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

Figure 8.8. 7. The T.E. of model 3, Nylon (PA 6).

6000
Combined Torsional Mesh Stiffness

5500

5000
(Nm/rad)

4500

4000

4.927 Nm 12.91Nm
20.09 Nm 30Nm
3500 40Nm pitch po int

3000
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

Figure 8.8. 8. The combined torsional mesh stiffness of model 3, Nylon (PA 6).

It can be seen that Model 3 has gained a wider load range, which is limited to 30 Nm at 23o
C. However, the maximum allowable load (30 Nm) can significantly decrease when the
temperature is above 50o C due to the nature of the material that has been presented in this
chapter.

Model 3 was considered nearly to the limit of the amount of relief, as further increasing the
amount of relief will lead to premature contact at the relief starting point becoming
significant. Even if a circular form of tipping may improve the contact, the top contact
(Munro 1999) may occur especially when temperature is high.
233
The benefit from the tip relief applied on the standard 20o pressure angle involute Nylon
spur gears has been found to be limited by the load compared to that of metallic gears. The
benefit will be further diminished considering the operating temperature influence.

8.9 Experimental Work

8.9.1 Introduction

This section presents data including previous results measured from a static test rig on
aluminium gears (Sirichai 1999) and tooth profile inspections through a Higler – Optical
projector (Barker 2002). All data produced for non-metallic gears were based on the
commercial purchased nylon (PA 6) gears supplied by Hoffman Engineering, Perth. The
nylon gears, however, were manufactured with “commercial standard” tip relief, the details
and its inspections are presented in the next section.

8.9.2 Tooth Profile Inspection

Measurement of tooth profile error requires the highest accuracy possible. The traditional
methods with a Hilger – Optical Projector proved to provide the highest accuracy out of the
existing equipment at Curtin University of Technology, as shown in Figure 8.9.1. The
optical projector works by producing a magnified silhouette of the object that can be
measured or compared to a translucence master copy for a rapid gear examination, which is
with the capabilities for coordinate measurements to an accuracy of 0.0001 in and angular
measurements to one minute. The twenty-one data point inspection, divided the working
depth into an equal number of increments, providing a sufficiently detailed analysis.

Some of the inspection results of the aluminium gears (manufactured at Curtin University
and specified as in Table 4.1) are presented in Figure 8.9.2.

234
 Figure 8.9 1 The Hilger – Optical Projector.

Tooth 1 Tooth 2 Tooth 3

7 7 One of the data 7 One of the increments

inspection points
Y-axis position (mm)

5 5 5

3 3 3
Involute
Involute Involute
Error x 10
1 Error x 10 1 1 Error x 10

-1 0 2 4 6 8 10 12 -1 0 2 4 6 8 10 12 -1 0 2 4 6 8 10 12
Working depth (mm)
-3 -3 -3

-5 -5 -5

-7 -7 -7

Figure 8.9 2 Comparisons of three aluminium gear teeth inspected using twenty-one data
points to represent the measured profile (Error is the difference between the
involute and the measured tooth profile observed in Y-axis direction).

Three typical tooth profiles have been presented, where the tooth tips are predominantly
negative. The tooth tip results were partly due to the small fillet radius that may be
reasonable. However, the profile error was indicated with a high level of waviness on both
sides of the tooth flanks.

235
Except for the material property differences, the nylon gears are also specified as in Table
4.1, and a standard drawing was also provided to the manufacturer, however, the gears
were obtained with a ‘commercial standard’ tip relief. The ‘commercial standard’ tip relief
was given afterward by the manufacturer, as shown in Figure 8.9.3.

Mn p 0.005Mn = Ca
TIP RELIEF
Mn p /2 Mn p /2

o
Mn 20
2.25 Mn

0.3 Mn = Dla
1.25 Mn

RACK RACK
DATUM UNE Pf = 0.38Mn

RACK PROFILE AS PER DIN 867(FEB 86)

Normal Module Mn - 25.4/DP
Pressure angle a - 20o
Fillet radius pf - 0.38 Mn
Tooth depth hp - 2.25 Mn
Gear addendum hap - Mn
Gear dedendum hfp - 1.25 Mn
Pitch P - p Mn

TIP RELIEF
Height d la = 0.3 Mn
Relief Ca = 0.005 Mn.

NOTE: THIS PROFILE IS ALSO AS PER ISO 53 (1974)

Figure 8.9 3 The manufacturer specified tip relief of the nylon gears.

The tooth profile inspections were focused on the comparisons between the true involute
curve, the curve of the manufacturer specified tooth profile and the curve of the measured
inspection. At each data point, errors were defined as the Y – axis position differences
from the true involute curve and multiplied 10 times for clear display. Four typical
inspection results of the nylon gear teeth are presented in Figure 8.9.4.

236
 7 Tooth No.1 7 Tooth No.2
Ambient Temperature: 24.8o C Ambient Temperature: 24.8o C

5 5

3 3
Y-axis position (mm)

True involute True involute

Specified (Error x 10) Specified (Error x 10)
1 1
M easured (Error x 10) M easured (Error x 10)

0 2 4 6 8 10 0 2 4 6 8 10
-1 Working depth (mm) -1

-3 -3

-5 -5

o ne o f the da ta
ins pe c tio n po int
-7 -7

7 Tooth No.3 7 Tooth No.4

Ambient Temperature: 23.5o C Ambient Temperature: 23.5o C

5 5

3 3
Y-axis position (mm)

True involute True involute

1 Specified (Error x 10) 1 Specified (Error x 10)
M easured (Error x 10) M easured (Error x 10)

0 2 4 6 8 10 0 2 4 6 8 10
-1 Working depth (mm) -1
Pitch Circle

-3 -3

-5 -5

One of the increments

-7 -7

Figure 8.9 4 Comparisons of four nylon (PA 6) gear teeth inspected using twenty-one
data points, the ‘Error’ of manufacturer and inspection results compared
to the true involute.

237
As can be seen in Figure 8.9.4, the tooth errors are much greater than that of the aluminium
gears. It is important to observe that the tip negative errors are more than twice the amount
of relief Ca, which was specified by the manufacturer, and due to the actual tooth thickness
at the pitch circle, the equivalent relief length Ln (same as the specified DLa in Figure 8.8.3)
will be greater than 0.3 Mn. These important features will be taken into account in the FEA
simulations, in order to compare with the measurement of the static T.E.

8.9.3 The Measurement of Static T.E.

The measurement of static T.E. traditionally doesn’t provide much detail over the mesh
cycle, as for rigid structure(s) any errors will be incorporated into the total displacement, in
which the elastic deformations are small. The detailed discussions have been presented in
Chapter 7.

While steps have been taken to continue to improve the qualities of the test gears and the
test rig assembly, the requirements for the rigidity of the shafts, bearings and the bearings
supports (such as pedestals) may not have been fully realized. It has been found that the
more rigid test rig will produce more accurate results, such as the test rig of the University
of Huddersfield, UK, which is a few tonnes heavier than the one used in this research (as
shown in Figure 8.9.5), and which has produced the classical results since Harris (Harris
1958).

The rigidity of the test rig is significant only when the test gear hub centre movements can
be limited during the test. The analysis results in section 7.5 have shown that the centre
distance movement of 0.005 mm will cause the measured T.E. to have a considerable error.

When gears mate with a single pair in contact, the local elastic deformation tends to occur
predominantly near the contact teeth, while in the double contact zone, it will occur over
the whole structure. Such a phenomenon plus the effects of other errors can cause slight
variations in the centre distance especially when the support bearings are self-cantering.

238
 TE = q1 -q2
q1

Drive shaft
Input Torque
q2
Adjustable
pedestal
Nikon autocollimator

Fixed rotary table

Reflectors
Fixed shaft

Fixed pedestal

Watts autocollimator

Torque arm

Figure 8.9 5 Illustration of the test rig components and static T.E. measurement.

The tests on the aluminium gears with the tooth profile errors as shown in Figure 8.9.2
have produced the following results: (i) the torsional stiffness measurement, which was
carried out previously in Curtin University. It is noted that the driven gear was much stiffer
than the driving gear and its hub was rigidly fixed (Sirichai 1999) as shown in Figure 8.9.6,
so that the test result (as shown in Figure 8.9.6) was similar to the FEA simulated
individual torsional mesh stiffness that was presented in Chapter 6.

239
 Rotary Table
Fixed Gear
F
Driving Gear

F Figure 8.9.6 The torsional stiffness test rig lay

Figure 8.8Figure 8.9 6 out (Sirichai, 1999).

35000
Observed Observed
Single Zone Double Zone
Torsional stiffness (N.m/rad)

32500

30000

Theoretical Theoretical Theoretical

27500 Double Zone Single Zone Double Zone

25000
0 5 10 15 20 25 30
Meshing position (deg.)

Figure 8.9 7 The torsional stiffness measurement of the aluminium gears (Sirichai, 1999).

(ii) The static T.E. measurement, which was carried with two identical aluminum gears on
the test rig, as shown in Figure 8.9.5. The test results produced with the load 78.5 Nm are
shown in Figure 8.9.7.

It can be seen that the measured T.E. did not present quality data for the research, due to
the effects of the profile and testing errors which largely over ride the deformations of the
gears themselves. The T.E. measured at point A, as shown in Figure 8.9.8, was due to the
negative tooth (tip) profile errors and shows a similar effect of a short tip relief. It has been
proven that increasing the load did not achieve an improvement in the results. Comparison

240
between the test (i) and (ii) will show how the rigidity changes on the gear hub can affect
the tests result and it has been explained in the beginning of this section. However, the
developed test rig (Figure 8.9.5) is well suitable for non-metallic gear tests.

A
0.003

T.E. (rad)
0.002

Single Zone
0.001

0
0 5 10 15

Meshing position (deg.)

Figure 8.9 8 The static T.E. measurement of the aluminium gears (Barker 2002).

The tests on the nylon gears with the tooth profile errors as shown in Figure 8.9.4 have
produced a better result compared to that of the aluminium gears, even though the profile
errors are larger. This is due to the increased material flexibility; where the chaotic
conditions at contact(s) have been reduced (larger contact area) and the major elastic
deformations will predominantly take up the total displacement under a considerable
heavier load. In order to conduct FEA modelling of the gears with the inspection measured
tooth profile (Figure 8.9.4), the relief parameters have been carefully chosen, as shown in
Figure 8.9.9.
0.2 mm
2.49 mm

Modified profile
(straight line)

Figure 8.9 9 The tooth form to simulate the measured tooth profile.

The experimental results of static T.E. measurement under a series of input loads at
temperature 20o C are shown in Figure 8.9.10, and the comparison with FEA simulation is
also made, but the simulation can only be carried out with the temperature 23o C, Figure
8.3.1.
241
 Experimental_4.927 Nm Experimental_12.91 Nm Experimental_20.09 Nm
Experimental_30 Nm Experimental_40 Nm FEA_4.927 Nm
FEA_12.91 Nm FEA_20.09 Nm FEA_30Nm
FEA_40Nm

0.008

0.007

0.006

Pitch
Point
0.005

0.004

0.003

0.002

0.001

0
-10.5 -7 -3.5 0 3.5 7 10.5

Roll Angle (degree)

Figure 8.9 10 Experimental and numerical results.

In observation of Figure 8.9.10, the experimental data shows a stiffer material than the
actual gears by their magnitude and curve shape. The error analysis at each data point was
calculated by,
T .E .FEA - T .E .Exp .
Relative Error = ´ 100. (8.3)
T .E .Exp .

Figure 8.8.11 gives an overview of the relative errors.

4.927 Nm 12.91Nm 20.09 Nm 30 Nm 40 Nm
160

140
Relative Error (%)

120

100

80

60
Point
Pitch

40

20

0
-12 -9 -6 -3 0 3 6 9 12

Roll Angle (degree)

Figure 8.9 11 Relative errors between the experimental and numerical results.

242
When the input load is 40 Nm the relative errors are between 5.2 % and 19.75 %, but in
most load cases the relative errors are considerable larger, between 50 % and 80 %.
However, the largest errors occur when the load is 12.91 Nm, between 110 % and 140 %.
In spite of the tooth profile errors, this shows the non-linear FEA modelling has not
matched well to the actual material behaviours. As discussed in section 8.3, this is often
the case for non-metallic gears.

The comparisons of the load – deflection behaviours of the structure produced the
combined torsional deflections (T.E.) as a function of the input loads at mesh position 0.5
degree (single contact zone) and 7.5 degree (double contact zone) as shown in Figure
8.9.12, and shows the major reason for the large errors.

M esh po sitio n: 0.5 degree M esh po sitio n: 7.5 degree

CombinedTorsionalDeflection (rad)

CombinedTorsionalDeflection (rad)
0.008 0.006

0.006
0.004

0.004

0.002
0.002
Experiment Experiment
FEA FEA
0 0
0 10 20 30 40 0 10 20 30 40
Input Load (Nm) Input Load (Nm)

Figure 8.9 12 The deflection vs. load relationships at mesh position 0.5 and 7.5 degree.

8.10 Conclusions

This chapter has presented general finite element formulations of the change over process
of the 20o pressure angle standard non-metallic (PA 6) involute spur gears in mesh. The
results of the change over process have taken into account static T.E., combined torsional
mesh stiffness and load sharing ratio, as solved by using the ANSYS® non-linear material
MELAS option. Those formulations, presented as functions of both input load and
temperature, demonstrated the major characteristics of large tooth deflection, the triple
contact caused by the large elastic deformation and the significant difference between tooth
mating approach and recess cases. As a consequence of heavy load (torque, temperature or
the combination), tooth load sharing can be widely expanded covering more than a
243
complete mesh cycle with more teeth come into the contact zone (triple contact) so that the
gears will be running smoother (with smoother T.E. and stiffness), however, the excessive
wear will then take place and subsequently reduce the gear service life.

The detailed variations of the contact ratio and the handover regions have been obtained
and comparisons between the results of the metallic and the non-metallic gears have been
provided. Experimental results have further confirmed the major variation trend of the
static T.E.

One of the important conclusions, involving previous tests on the aluminium gears, is that
the rigidity of the lateral movement constraint on the test gear hubs is most critical for a
metallic gear test rig. Within normal industry manufacturing standards, there will be a
considerable pitch and tooth profile error, as shown by the tooth inspections, but as long as
the lateral movements can be limited, the major trend of test results will be obtained.
Otherwise, there is no point to purchase expensive precision test gears.

The related error analysis also indicates several possibilities for improving the accuracy of
both numerical and experimental results. In particular, it is critical for further numerical
research that more realistic material properties are obtained for non-metallic materials.

Finally, the investigations of the tip relieved gears in mesh have shown that standard 20o
pressure angle nylon gears have a limited range of load and temperature to prevent
excessive wear. It can be expected that the power-weight ratio will not be significantly
improved unless an improvement can be gained on the material properties (such as using
fiber composite materials). This indicates that the application of non-metallic gears should
not follow that of conventional metallic gears, and requires more efforts on the design
according to its own characters. A high-pressure angle involute spur gear has been
recommended by Seager (Seager 1975) because it is consistent also with low sliding speed
and high tooth strength. Recent experiment work on high-pressure-angle nylon gears has
achieved considerable success by Prof. D Walton, (Walton, 1995). A related FEA of high-
pressure angle nylon gears can be referred in Appendix E.

244
9.1 General Conclusions

This thesis has presented the development of a numerical approach for predicting the
characteristics of involute spur gears in mesh for various combinations of load and
temperature for metallic and non-metallic materials. The numerical models were formed
over the complete mesh cycle providing detailed information of the change over process as
teeth enter and leave the contact region. Various items of interest for involute spur gears
were investigated including profile modifications, localised tooth cracks, and non-linear
material behaviour for non-metallic gears.

The major thrust of the study has been the development of efficient and reliable numerical
solution techniques using automatic mesh adaptation with contact and the element birth and
death option. This included a fundamental study on the effect of plane strain and plane
stress elements. The numerical approach enabled the development of a software program
based on the software package ANSYS® to predict the effect of the gears on torsional mesh
stiffness, transmission error and load sharing ratio, and provided basic information for
condition monitoring and optimal gear design.

(i) Plane strain vs. plane stress

A wide range of FEA problems are often solved with 2D assumptions because of the
greatly increased computational efficiency and reduced cost. However, when the
245
numerical analysis involves non-linear factors such as contact, fracture or other extreme
load cases, the valid 2D assumptions can be restricted to a very narrow range. A large
number of FEA models and calculations were made for the investigations. The
comparisons concentrated on the torsional stiffness, first maximum principal stress and
stress intensity factors that were obtained under assumptions of plane stress, plane strain
and 3D analysis. With the conclusion of this research, errors were found in the literature of
previous research studies, especially when 2D assumptions were used with solid gears.
The understanding of these results has helped to avoid large errors in the analysis.

(ii) Mesh adaptation with contact

The accuracy and the efficiency of the developed numerical models have provided the
numerical results with more details of the change over process than previously obtainable,
and this has proven to be an important basic tool in the numerical analysis of gears in mesh.
The extension onto 3D models (using brick elements) shows great potential in analysis of
all types of gears.

(iii) Element birth and death option

When the involute tooth profile modification presents a negative (or partly negative) tooth
profile, the numerical solution will have instabilities. The element birth and death option
has been proven to be one of the practical solution methods, which ensure the numerical
results are accurately incorporated into the rigid body motion and the elastic deformation.

(iv) Handover Regions

The handover region of involute spur gears can include both the approach and recess cases,
and it generally exists when gears are considered elastic. It is a key component of gear
behaviour that can be found in the combined torsional mesh stiffness, transmission error,
load-sharing ratio, ratio of local deformation and even in the individual torsional mesh
stiffness characteristics. In particular, the handover region of the static T.E. represents the
TE o.p.c. In previous research, the theory and evaluation methods of TE o.p.c. were developed
mathematically (geometrical analysis), and they were found to be suitable for gears with
geometrical errors. However, some confusion may arise when one attempts to apply the

246
theory and the methods to elastically deformed gears. The numerical evaluations presented
in this thesis may be one of the alternative solutions.

(v) Tooth profile modifications

The research on tooth profile modifications has presented the effects on the transmission
error of various types of profile modifications including short, intermediate, long and
excessively long profile modifications with the results illustrated using the Harris Map.
The critical relief starting points S1 and S2 were defined as major aspects of the profile
modifications. S1 denotes the relief midway between the short and long tooth profile
modifications, producing a T.E. curve (in a mesh cycle) consisting of a single zone,
handover region and a point of double contact so that when the load tends to zero, the
handover region disappears and the contact ratio decreases to 1. Significantly with this
relief, the contact stress near the tooth tip has been minimised (Appendix D), to yield the
important optimal tooth tip-relief providing improved tooth flank surface protection.

The critical relief starting point S2 was defined for long tooth profile modifications. Relief
with the starting point S2 will produce a T.E. curve at the design load smoother than others
(no protrusions), but when the input load exceeds the design load premature contact will
immediately occur. On the other hand, if the relief starting point extends beyond S2 (relief
is too long) there will be constant T.E. protrusions due to rigid body motion occurring in
both the single and double contact zone, and premature contact can easily occur. With the
critical relief starting points S1 and S2, a range of long tooth profile modifications can be
developed.

Further development on the tooth profile modifications included the tip-relief (short) on
standard 20o pressure angle nylon gears (chapter 8) and 30o pressure angle nylon gears
(Appendix). Although the optimal tip-relief was proven to be valid for non-metallic gears,
the effects on 20o pressure angle nylon gears have further confirmed its disadvantages due
to the excessive large deflection under high torque and temperature conditions. Applying
the optimal tip-relief on 30o pressure angle nylon gears has effectively reduced the contact
stress near the tooth tip so that the reduction in surface wear can be achieved, especially
when a circular form was used as the modified tooth profile.

247
Finally, the development on tooth profile modifications involving high contact ratio gears
in mesh was presented (Appendix C). The results have shown that the current standards
are not well suitable for the applications of HCRG. The optimal (short) tip-relief of HCRG
was found to be critical, where any other relief length above the optimal tip-relief length
will result in the contact ratio decreasing below 2, hence losing (part or full of) the benefit
of HCRG.

(vi) Analysis of non-metallic gears

General finite element models of involute spur gears using non-metallic (PA 6) materials
have been formulated. The models investigated the details of the standard 20o pressure
angle change over regions for the static T.E., combined torsional mesh stiffness and load
sharing ratio, produced by using ANSYS® non-linear material MELAS option. Those
formulations, presented as functions of both input load and temperature, demonstrated the
phase change of the major characteristics and the triple contact that can be caused by the
large elastic deformation and the significant difference between tooth mating for the
approach and recess cases. As a consequence of heavy load (torque, temperature or the
combination), tooth load sharing can be widely expanded covering more than a complete
mesh cycle with more teeth come into the load sharing (triple contact) so that the gears will
run smoother (with smoother T.E. and stiffness). However, excessive wear may then take
place which is one of the major concerns for the gear service life. Tip-relief has been
found to have little effect on such a case, and the requirement for adapted gear design to
such characteristics was outlined.

The related error analysis also indicated several possibilities for improving the accuracy of
both numerical and experimental results. In particular, it was noted as being critical for
further numerical research that more realistic material properties are obtained for non-
metallic materials.

(vii) Condition monitoring gear system using high order strain measurement

Perfect involute gears in mesh along with an input shaft have been analysed. The elastic
strains on the surface of the shaft near the gear hub have been found to vary in form with
the transmission error over the mesh cycle. It has been recognized that the elastic strains
248
have more capability to reveal the details of the gear-shaft system if the measurements
were obtained from both input and output shafts. This investigation has shown that the
presence of a root crack changes the strain measurements on the shaft. The changes in the
shaft strain appear to be larger than the curves providing changes to the T.E. due to the
presence of the crack. Further studies are expected to improve the capabilities of condition
monitoring and controlling transmission system noise and vibrations.

9.2 Future Work

The following areas have been noted as being worthy of further research, in the light of this
thesis.

· Further numerical method investigation and study should be conducted on gears in

mesh with general stress formulations over a complete mesh cycle.

· Further numerical method investigation and study should be conducted on gears in

mesh under dynamic situations with and without cracked teeth, surface pitting and wear
using the finite element method.

· Further numerical method investigation and study should be conducted for all types of
gear mesh with and without tooth damage. This could include helical, spiral bevel and
other gear tooth forms.

· Further numerical method investigation and study should be conducted on a whole

gearbox with all elements in the system including the gear casing with and without
tooth damage.

· Further numerical method investigation and study on non-metallic gears should be

conducted including the effect of thermal expansions. This would involve the use of a
coupled field analysis.

· Further experimental testing should be carried out with the aim of eliminating the
lateral movement of test gear hub centres.

249
Aida, T. (1968). “Fundamental Research on Gear Noise and Vibration I.” Transaction of the
Japanese Society of Mechanical Engineers 34: 2226-2264.
Aida, T. (1969). “Fundamental Research on Gear Noise and Vibration II.” Transaction of the
Japanese Society of Mechanical Engineers 35: 2113-2119.
Aida, T., Sato, S. and Fukuma, H. (1967). Properties of Gear Noise and Its Generating Mechanism.
Proceedings of the Japanese Society of Mechanical Engineers Semi-International Gearing Symposium.
Alagoz, Q., Arikan, M. A. S., Bilir, O. G. and Parnas, L. (2002). “3-D finite element analysis of
long-fiber reinforced composite spur gears.” Gear Technology 19(2): 12-19.
Alattass, M. (1994). Experimental Study of Fault Influence on Vibration and Noise Measurements
in a Gear Transmission Mechanism. International Gearing Conference, UK.
Anderson, T. L. (1995). Fracture mechanics: fundamentals and applications, Boca Raton : CRC Press.
Andrei, L; Andrei, G; Epureanu, A; Oancea, N; Walton, D. (2002). “Numerical simulation and
generation of curved face width gears.” International Journal of Machine Tools & Manufacture
(UK) 42(1): 1-6.
ANSYS (2000). User's Manual, ANSYS, Inc.
Arafa, M. H. and Megahed, M. M. (1999). “Evaluation of spur gear mesh compliance using the
finite element method.” Proceedings of the Institution of Mechanical Engineers. Part C, Journal
of Mechanical Engineering Science v. 213 noC6: 569-79.
Arikan, M. A. S. and Kaftanoglu, B. (1989). “Dynamic Load and Root Stress Analysis of Spur
Gears.” Annal of the CIRP 38: 171-174.
Arikan, M. A. S. and Kaftanoglu, B. (1991). Dynamic Load and Contact Stress Analysis of Spur
Gears. Advance in Design Automation, USA.
Arikan, M. A. S. and Tamar, M. (1992). Tooth Contact and 3-D Stress Analysis of Involute
Helical Gears. 6th International Power Transmission and Gearing Conference.
Arthur, P. and Chong, K. P. (1987). Elasticity in Engineering Mechanics, New York: Elsevier.
AUTODESK. (1997)., AUTODESK, USA.
AYEL, J. (1984). “Engrenages. Introduction a la Lubrification des Engrenages. La Lubrification
Industrielle. Transmissions, Compressures, Turbines.” Collection Coloques et Seminaires 39,
Paris: Ed. Technip. cap. 4: 183-296, tome 1.
Babuska, I. (1988). "The p- and hp- Versions Of The Finite Element Method The State Of The Art," in
Finite Elements: Theory And Application. Springer-Verlag, Berlin, D.L. Doyer et al. (eds).
Bahgat, B. M., Osman, M. O. M. and Sankar, T. S. (1981). “On the Dynamic Gear Tooth Loading as
Effected by Bearing Clearances in High Speed Machinery.” Journal of Mechanical Design: 81-DET-112.
Bahgat, B. M., Osman, M. O. M. and Sankar, T. S. (1983). “On the Spur-Gear Dynamic Tooth-
Load under Consideration of System Elasticity and Tooth Involute Profile.” Journal of
Mechanisms, Transmissions, and Automation in Design 105(September): 302-309.
Bard, C., Remond, D. and Play, D. (1994). New Transmission Error Measurement for Heavy
Load Gears. International Gearing Conference, UK.
Baret, C., Coccolo, G. and Raffa, F. A. (1994). 3d Stress Analysis of Spur Gears with Profile
Errors and Modifications Using P-Fem Models. International Gearing Conference, UK.
Barker., J (2002). "Gear tooth inspection and static transmission error test rig improvement," B.
Eng. Project Report. Perth, Curtin University of Technology.
Barnett, D., Agarwal, A. and Braun, E. (1988). “Load sharing in high contact ratio truck
transmission gearing.” SAE technical paper 885125 2: 252-261.
Barnett, O. W. and Yildrim, N. (1994). Loaded Transmission Error Predictions Using a Computer
Model and Its Verification. International Gearing Conference, UK.
Baron, E., Favre, B. and Mairesse, P. (1988). Analysis of Relation between Gear Noise and
Transmission Error. Proc. Internoise, '88.

250
Baumgart, J. (1992). “Investigation on the Optimization of Plastics/Steel Toothed Wheel Pairs.”
Gov. Res. Announc. Index (USA): 260.
Blazakis, A. and Houser, D. R. (1994). Finite Element and Experimental Analysis of the Effect of Thin-
Rimmed Gear Geometry on Spur Gear Fillet Stresses. International Gearing Conference, UK.
Breeds, A. R. (1991). Non-metallic gears for power transmission, The University of Birmingham, U.K.
Brousseau, J., Gosselin, C. and Cloutier, L. (1994). Reference Point, Mesh Stiffness, and Gear
Dynamic Models. International Gearing Conference, University of Newcastle, UK.
BS (1970). 436: Part 2: 1970 Specification for spur and helical gears. Part 2: basic rack form,
modules and accuracy. (British Standards Institution, London).
Bushimata, T; Sakuta, H; Asai, T. (2001). “Wear of Plastic Spur Gear Made by Injection
Molding.” Journal of Japanese Society of Tribologists 46(11): 889-896.
Castro, J., Seabra, J. and Almacinha, J. (1994). Recent Advances in Experimental Mechanics.
Experimental and analytical evaluation of the teeth load distribution in a spur gear and rack
pair. Balkema, Rotterdam.
Chabert, G., Tran, T.D. and Mathis, R. (1974). “An Evaluation of Stresses and Deflection of
Spur Gear Teeth under Strain.” Journal of Engineering for Industry: 85-93.
Chen, C.-H., Wang, Y. and Colbourne, J. R. (1992). A General formula for Determining the
Normal to the Path of Contact in Parallel-Axis Gearing. 6th International Power Transmission
and Gearing Conference.
Chen, J. S., Litvin, F. L. and Shabana, A. A. (1994). Computerized Simulation of Meshing and
Contact of Loaded Gear Drives. International Gearing Conference, UK.
Cheng, Y. and Lim, T. C. (1998). Dynamic Analysis of High Speed Hypoid Gears with Emphasis on
Automotive Axle Noise Problem. ASME Design Engineering Technical Conference, Atlanta, GA, USA.
Cheung, Y. K. and Zienkiewicz, O. C. (1965). “Finite Elements in the Solution of Field
Problems.” Engineer 220: 507-510.
Childs, D. B. (1980). Estimation of Seal Bearing Stiffness and Damping Parameters from
Experimental Data. Second International Conference: Vibration in Rotating Machinery, IME.
Choy, F. K. and Polyshchuk, V. (1995). “Dynamic Analysis and Experimental Correlation of a
Gear Transmission System.” International Journal of Turbo and Jet Engines 12: 269-281.
Choy, F. K., Tu, Y. K., Zakrajsek, J. J. and Townsend, D. P. (1991). “Effect of Gear Box
Vibration and Mass Imbalance on the Dynamics of Multistage Gear Transmission.” Journal of
Vibration and Acoustics 113: 333-344.
Choy, F. K., Ruan, Y. F., Tu, Y. K., Zakrajsek, J. J. and Townsend, D. P. (1992). “Modal
Analysis of Multistage Gear Systems Coupled with Gearbox Vibrations.” Journal of
Mechanical Design 114: 486-496.
Choy, F. K., Ruan, Y. F., Zakrajsek, J. J. and Oswald, F. B. (1993). “Modal Simulation of Gear
Box Vibration with Experimental Correlation.” Journal of Propulsion and Power 9(2): 301-306.
Choy, F. K., Polyshchuk, V., Zakrajsek, J. J., Handschuh, R. F. and Townsend, D. P. (1996).
“Analysis of the Effects of Surface Pitting and Wear on the Vibrations of a Gear Transmission
System.” Tribology International 29(1): 77-83.
Choy, F. K., Huang, S., Zakrajsek, J. J., Handschuh, R. F. and Townsend, D. P. (1996).
“Vibration Signature Analysis of a Faulted Gear Transmission System.” Journal of Propulsion
and Power 12(2): 289-295.
Clough, R. W. (1960). The finite element method in plane stress analysis. ASCE, Proceedings of
the Second Conference on Electronic Computation.
Cook, R. D. (1989). Concepts and Applications of Finite Element Analysis, John Wiley & Sons, N.Y: 578.
Cornell, R. W. (1981). “Compliance and Stress Sensitivity of Spur Gear Teeth.” Journal of
Mechanical Design 103: 447-459.
Cornell, R. W. and Westervelt, W. W. (1978). “Dynamic tooth loads and stressing for high
contact ratio spur gears.” ASME Journal of Mechanical Design 100(1): 69-76.
Courant, R. (1943). “Variational method for the solution of problems of equilibrium and
vibrations.” Bulletin of the American Mathematical Society 49: 1-23.

251
Coy, J. J. and Chao, C. H. (1982). “A Method of Selecting Grid Size to Account for Hertz Deformation
in Finite Element Analysis of Spur Gears.” ASME Journal of Mechanical Design 103: 759-459.
Daly, K. J. and Smith, J. D. (1979). Using Gratings in Driveline Noise Problems. Proc., Noise &
Vib. of Engineering and Transmissions, Cranfield.
Daniewicz, S. R., Collins, J. A. and Houser, D. R. (1994). “The Stress Intensity Factor and
Stiffness for a Cracked Spur Gear Tooth.” Journal of Mechanical Design 116: 697-700.
DIN (1986). Rack profile as PER DIN 876 (FEB 86), Supplied by Hofmann Engineering, Perth.
Domininghaus, H. (1993). Plastics for Engineers, Bondway Publishing.
Donno, M. D. and Litvin, F. L. (1998). Computerized Design, Generation and Simulation of
Meshing of a Spiroid Worm-Gear Drive with Double-Crowned Worm. 1998 ASME Design
Engineering Technical Conference, Atlanta, GA, USA.
Dooner, D. B. and Seireg, A. A. (1998). Concurrent Engineering of Toothed Bodies for
Generalized Function Transmission. 1998 ASME Design Engineering Technical Conference,
Atlanta, GA, USA.
Drago, R. (1974). “Heavy lift helicopter brings up drive ideas.” Power Transmission Design: 49-63.
Drago, R. J., Lenski, J. W. and Royal, A. (1979). An Analytical Approach to the Source
Reduction of Noise and Vibration in Highly Loaded Mechanical Power Transmission Systems.
Proceeding of the Fifth World Congress on Theory of Machines and Mechanisms.
Du, S. (1997). Dynamic Modelling and Simulation of Gear Transmission Error for Gearbox
Vibration Analysis, Ph.D Thesis, University of New South Wales, Sydney, Australia.
Du, S.; Randall, R. B.; Kelly, D. W. (1998). “Modelling of spur gear mesh stiffness and static
transmission error.” Proceedings of the Institution of Mechanical Engineers. Journal of
Mechanical Engineering Science. 212(C4): 287-297.
Dvorak, P. (1999). “Meshless analysis breaks with FEA traditions.” Machine Design; Cleveland
71(23): 82-.
Elkholdy, A. H. (1985). “Tooth load sharing in high contact ratio spur gears.” Trans. ASME.J. of
Mechanisms, Transmission and Automn in Des. 107: 11-16.
Elkholdy, A. H., Elsharkawy, A. A. and S., Y. A. (1998). “Effect of Meshing Tooth Stiffness and
Manufacturing Error on the Analysis of Straight Bevel Gears.” Mechanics of Structures and
Machines. 21(1): pp 41-.
El-Saeidy Fawzi, M. A. (1991). “Effect of Tooth Backlash and Ball Bearing Deadband Clearance on
Vibration Spectrum in Spur Gear Boxes.” Journal of Acoustical Society of America 89(6): 2766-2773.
Enzmann, B. (1990). “Special Gearings of Plastic for Quiet Gears.” Antriebstechnik 5: 42-44.
Feng, P.-H. (1998). Determination of Principal Curvatures and Contact Ellipse for Profile
Crowned. 1998 ASME Design Engineering Technical Conference, Atlanta, GA, USA.
Filiz, H. I. and Eyercioglu, O. (1995). “Evaluation of Gear Tooth Stresses by Finite Element
Method.” Journal of Engineering for Industry 117: 232-239.
Forrester, B. D. (1996). Advance Vibration Analysis Techniques for Fault Detection and Diagnosis in
Geared Transmission System, Ph.D Thesis, Swinburne University of Technology, Melbourne, Australia.
Fujio, H. (1992). Laser Holographic Measurement of Tooth Flank form of Cylindrical Involute Gear: Part
2- for Helical Gear Tooth. 6th International Power Transmission and Gearing Conference.
Gabbert, U. and Graeff-Weinberg, K. (1994). “Eine pNh-Elementformulirung für die
Kontaktanalyse.” ZAMM74 4: 195-197.
Gargiulo, E. P. J. (1980). “A Simple Way to Estimate Bearing Stiffness.” Machine Design v. 70
no12 (July 9 '98) p. 132+: 107-110.
Gosselin, C., Gagnon, P. and Cloutier, L. (1998). “Accurate Tooth Stiffness of Spiral Bevel Gear
Teeth by the Finite Strip Method.” Journal of Mechanical Design v. 119 (Dec. '97) p. 518-21
120(4): 599.
Gosselin, C. (1994). A Review of the Current Contact Stress and Deformation formulations
Compared to Finite Element Analysis. International Gearing Conference, UK.
Gregory, R. W., Harris, S. L. and Munro, R. G. (1963). Dynamic Behaviour of Spur Gears.
Proceedings of the Institute of Mechanical Engineers.

252
Gregory, R. W., Harris, S. L. and Munro, R. G. (1963). “A Method of Measuring Transmission
Error in Spur Gear of 1:1 Ratio.” J. SCI. INSTRUM. 40: 5-9.
Gregory, R. W., Harris, S. L. and Munro, R. G. (1963). Torsional Motions of a Pair of Spur
Gears. Proc. I. Mech. E.
Harris, S. L. (1958). Dynamic Loads on the Teeth of Spur Gears. Proc. I. Mech. E.
Hayashi, I. and Hayashi, T. (1979). New measuring Method of a Single Flank Transmission Error
of a pair of Gears. Proc., Fifth World Congress on Theory of Machines and Mechanisms.
Hayashi, I. and Hayashi, T. (1981). Development of the Dynamic Measurement Method of
Transmission Error of a Gear Pair. International Symposium on Gearing & Power Transmissions,
Tokyo, JAPAN.
Hoff, N. H. (1956). Analysis of Structures. New York, Wiley, New York.
Houser, D. R. (1990). Research of the Ohio State University Gear Dynamics and Gear Noise
Research Laboratory. First International Conference, Gearbox Noise and Vibration.
Houser, D. R., Bolze, V. M. and Graber, J. M. (1996). A Comparison of Predicted and Measured
Dynamic and Static Transmission Error for Spur and Helical Gear Sets. 14th Intl. Modal Anal.
Conf., USA.
Houser, D. R. and Blankenship, G. W. (1989). Methods for Measuring Gear Transmission Error
under Load and At Operating Speeds., SAE Technical Paper No. 891869: 1-8.
Howard, I., Jia, S. and Wang, J. (2001). “The dynamic modelling of a spur gear in mesh including
friction and a crack.” Mechanical Systems and Signal Processing 15(5): 831-853.
Iida, H., Tamura, A. and Yamamoto, H. (1986). “Dynamic Characteristics of a Gear Train
System with Softly Supported Shafts.” Bulletin of the JSME 29(252): 1811-1817.
Iida, H. and Tamura, A. (1984). Couple Torsional-Flexural Vibration of a Shaft in a Geared
System. Proceeding of the Third international Conference on Vibrations in Rotating Machinery.
Iida, H. (1980). “Coupled Torsional-Flexural Vibration of a Shaft in a Geared System of Rotors.”
Bulletin of the JSME 23(186): 2111-2117.
Inoue, K., Townsend, D. P. and Coy, J. J. (1992). Optimum Design of Gearbox for Low
Vibration. 6th International Power Transmission and Gearing Conference, USA.
ISO/DIS (1983). 6336: 1983, Parts 1-4. (Organization for International Standardizations, Belgium).
Iwatsubo, T., Arii, S. and Kawai, R. (1984). “Couple Lateral-Torsional Vibration of Rotor System
Trained by Gears.” Bulletin of the JSME 27(224): 271-277.
Iwatsubo, T., Arii, S. and Kawai, R. (1984). The Coupled Lateral Torsional Vibration of a Geared
Rotor System. Proceeding of the Third international Conference on Vibrations in Rotating Machinery.
Janover, J. S. and Sehring, J. F. (1989). “Finite-Element Analysis of Plastic Gear Transmission.”
Plast. Des. Forum 14(6): 69, 72, 74-75, 81.
Jia, S., Wang, J. and Howard, I. (2000). The Modelling of the Vibration Response of Spur Gears
with Localised Teeth Crack. 13th International Congress on Condition Monitoring and
Diagnostic Engineering Management, Houston, Texas, USA.
Jia, S. Howard, I. and Wang, J. (2001). A common formula for spur gear mesh stiffness. The
JSME International Conference, Fukuoka, JAPAN.
Johnson, D. C. (1962). “Modes and Frequencies of Shafts Coupled by Straight Spur Gears.”
Journal of Mechanical Engineering Science 4(3): 241-250.
Kahraman, A. (1992). “Dynamic Analysis of Geared Rotors by Finite Elements.” Journal of
Mechanical Design 114(September): 507-514.
Kahraman, A. (1993). “Effect of Axial Vibrations on the Dynamics of a Helical Gear Pair.”
Journal of Vibration and Acoustics 115(1): 33-39.
Kahraman, A. and Singh, R. (1991). “Interaction between Time-Varying Mesh Stiffness and
Clearance Non-Linearities in a Geared System.” Journal of Sound and Vibration 146(1): 135-156.
Kahraman, A. and Singh, R. (1991). “Non-Linear Dynamics of a Geared Rotor-Bearing System
with Multiple Clearances.” Journal of Sound and Vibration 144(3): 469-506.
Kahraman, A. and Blankenship, G. W. (1999). “Effect of involute contact ratio on spur gear
dynamics.” Journal of Mechanical Design, Transactions Of the ASME [J Mech Des, Trans
ASME] 121(1): 112-118.
253
Kalluri, R. and Houser, D. R. (1998). A Possible Strategy for Incorporating Edge Effects in Root
Stress Estimation of Helical Gears. 1998 ASME Design Engineering Technical Conference,
Atlanta, GA, USA.
Kasuba, R. (1961). An Analytical and experimental Study of Dynamic Loads on Spur Gear Teeth,,
University of Illinois.
Kasuba, R. (1971). “Dynamic Loads on Spur Gear Teeth by Analog Computation.” American
Society of Mechanical Engineers: 71-DE-26.
Kato, M. (1994). Evaluation of Sound Power Radiated by a Gearbox. 1994 International Gearing
Conference, University of Newcastle, UK.
Kim, B. S. (1998). “Crack face friction effects on Mode II stress intensities for a surface-cracked coating in
two-dimensional rolling contact.” Tribology Transactions; Park Ridge. 41(1): 35-.
Kin, V. (1992). Computerized Analysis of Gear Meshing Based on Coordinate Measurement Data.
6th International Power Transmission and Gearing Conference, USA.
Kishor, B. (1979). Effect of Gear Errors on Non-linear Vibrations of a Gear-train System.
Proceedings of the Fifth World Congress on Theory of Machines and Mechanisms.
Kissling, U. L. (1994). Improving Gearbox Design by Highly Integrated Calculation Programs.
International Gearing Conference, University of Newcastle, UK.
Kiyono, S. and Fujii, Y. (1981). “Analysis of Vibration of Bevel Gears.” Bulletin of the JSME
24(188): 441 -446.
Kleiss, N. J. J.; Kleiss, R. E. (2000). “A practical guide for molding better plastic geared
transmissions.” Gear Technology (USA) 17(3): 20-25.
Kobler, H. K., Pratt, A. and Thomson, A. M. (1970). Dynamics and Noise of Parallel Axis
Gearing. Proceedings of the Institute of Mechanical Engineers.
Koffi, D. and Cuilliere, J. C. (1995). “Study of regimes and lubrication parameters of gears of
thermoplastic materials.” Materiaux et Techniques (Paris) 83(3-4): 33-39.
Kowalczyk, P. (1994). “Finite-Deformation Interface formulation for Frictionless Contact
Problems.” Communications in Numerical Methods in Engineering 10: 879-893.
Kral, S. (1991). “The Effect of Deformations of Teeth on Tribological Properties of Metal--Plastic
Gearing.” Ropa a Uhlie (Czechoslovakia) 33(10): 614-618.
Kuang, J. H. and Yang, Y. T. (1992). An Estimate of Mesh Stiffness and Load Sharing Ratio of a
Spur Gear Pair. 6th International Power Transmission and Gearing Conference, USA.
Kubo, A. and Kiyono, S. (1980). “Vibration excitation of cylindrical involute gears.” JSME
Bulletin 23(183): 1536-1543.
Kubo, A. (1992). Laser Holographic Measurement of Tooth Flank form of Cylindrical Involute
Gear: Part 1- Principle and Measurement of Spur Gear. 6th International Power Transmission
and Gearing Conference.
Kudinov, A. T. Starzhinsky, V. E.; Osipenko, S. A. (1992). “Calculation of Technological
Shrinkage in Cast Toothed Gears From Thermoplastics.” Plasticheskie Massy (Russia) 4: 53-55.
Kurokawa, M.; Uchiyama, Y.; Nagai, S. (2000). “Tribological properties and gear performance
of polyoxymethylene composites.” Journal of Tribology (Transactions of the ASME) (USA)
122(4): 809-814.
Kurowski, P. (2000). “Say good-bye to defeaturing and meshing.” Machine Design; Cleveland;
Aug 17, 2000 72(16): 71-78.
Laschet, A. and Troeder, C. (1984). “Torsional and Flexural Vibrations in Drive Systems: A
Computer Simulation.” Computers in Mechanical Engineering 3: 32-43.
Lee, C., Lin, H. H., Oswald, F. B. and Townsend, D. P. (1991). “Influence of linear profile
modification and loading conditions on the dynamic load and stress of high contact ratio spur
gears.” Trans. ASME.J. of Mechanical Design 113: 473-480.
Lee, C. Y. and Oden, J. T. (1993). “Theory and approximation of quasistatic frictional contact
problems.” Comput. Methods. Appl. Mech. Engng. 106: 407-429.
Lepi, S. M. (1998). Practical guide to finite elements : a solid mechanics approach, New York :
Marcel Dekker.
Levy, S. (1953). “Structural analysis and influence coefficients for Delta wings.” J. Aero. Sci 20(7): 449-54.
254
Lewicki, D. (1994). Improvements in Spiral-Bevel Gears to Reduce Noise and Increase Strength.
International Gearing Conference, University of Newcastle, UK.
Lewicki, D. G. and Ballarini, R. (1997). “Effect of Rim Thickness on Gear Crack Propagation
Path.” Journal of Mechanical Design 119: 88-95.
Li, R. and Tang, Q. (1994). Deformation numerical analysis of meshing gears and tooth profile
modification. 7994 International Gearing Conference, University of Newcastle, UK.
Lim, T. C. and Singh, R. (1990). “Vibration Transmission through Rolling Element Bearings, Part
I: Bearing Stiffness formulation.” Journal of Sound and Vibration 139(2): 179-199.
Lim, T. C. and Singh, R. (1990). “Vibration Transmission through Rolling Element Bearings, Part
II: System Studies.” Journal of Sound and Vibration 139(2): 201-225.
Lim, T. C. and Singh, R. (1991). “Vibration Transmission through Rolling Element Bearings, Part
III: Geared Rotor System Studies.” Journal of Sound and Vibration 151(1): 31-54.
Lim, T. C. and Singh, R. (1992). “Vibration Transmission through Rolling Element Bearings, Part
IV: Statistical Energy Analysis.” Journal of Sound and Vibration 153(1): 37-50.
Lin, H. H., Huston, R. L. and Coy, J. J. (1988). “On Dynamic Loads in Parallel Shaft
Transmissions: Part 1.” Journal of Mechanisms, Transmissions, and Automation in Design
110(June): 221-225.
Lin, H. H., Lee, C., Oswald, F. B. and Townsend, D. P. (1993). “Computer aided design of high
contact ratio gears for minimum dynamic load and stress.” Trans. ASME.J. of Mechanical
Design 115: 171-178.
Lin, H. H., Oswald, F. B. and Townsend, D. P. (1989a). Dynamic Loading of Spur Gears with
Linear or Parabolic Tooth Profile Modifications. Proceedings of the 1989 International Power
Transmission and Gearing Conference, Chicago, USA.
Lin, H. H., Oswald, F. B. and Townsend, D. P. (1994). “Dynamic Loading of Spur Gears with
Linear or Parabolic Tooth Profile Modifications.” Mech. Mach. Theory 29(8): 1115-1129.
Lin, H. H., Wang, J., Oswald, F. B. and Coy, J. J. (1994). “Effect of extended tooth contact on
the modeling of spur gear transmissions.” Gear Technology: 18-25.
Liou, C.-H. (1992). Effect of Contact Ratio on Spur Gear Dynamic Load. 6th International Power
Transmission and Gearing Conference, USA.
Liou, C.-H. (1996). “Effect of Contact Ratio on Spur Gear Dynamic Load with No Tooth Profile
Modifications.” Journal of Mechanical Design 118(September): 439-443.
Litvin, F. L. (1989). Theory of Gearing, NASA Reference Publication, NASA.
Litvin, F. L., Egelja, A. M. and De Donno, M. (1998). “Computerized determination of
singularities and envelopes to families of contact lines on gear tooth surfaces.” Computer
Methods in Applied Mechanics and Engineering 158(1/2): 23.
Litvin, F. L., Lu, J. and Howkins, M. (1998). “Computerized simulation of meshing of
conventional helical involute gears and modification of geometry.” Mechanism and machine
theory. 34(1): 123.
Litvin, F. L., Wang, A. G. and Handschuh, R. F. (1998). “Computerized generation and
simulation of meshing and contact of spiral bevel gears with improved geometry.” Computer
Methods in Applied Mechanics and Engineering 158(1/2): 35.
Litvin, F. L. and Feng, P.-H. (1996). “Computerized Design and Generation of Cycloidal
Gearings.” Mech. Mach. Theory 31(7): 891-911.
Litvin, F. L. and Hsiao, C. L. (1993). “Computerized Simulation of Meshing and Contact of Enveloping
Gear Tooth Surfaces.” Computer Methods in Applied Mechanics and Engineering 102: 337-336.
Litvin, F. L. and Lu, J. (1993). “Computerized Simulation of Generation, Meshing and Contact of
Double Circular-Arc Helical Gears.” Mathl. Comput. Modelling 18(5): 31-47.
Litvin, F. L. and Lu, J. (1995). “Computerized Design and Generation of Double Circular-Arc
Helical Gears with Low Transmission Errors.” Computer Methods in Applied Mechanics and
Engineering 127: 57-86.
Litvin, F. L. and Seol, I. H. (1996). “Computerized Determination of Gear Tooth Surface as
Envelope to Two Parameter Family of Surfaces.” Computer Methods in Applied Mechanics
and Engineering 138: 213-225.
255
Litvin, F. L. (1995). “Computerized Design and Generation of Low-Noise Helical Gears with
Modified Surface Topology.” Journal of Mechanical Design 117(June): 254-261.
Litvin, F. L. (1996). “Application of Finite Element Analysis for Determination for Load Share,
Real Contact Ratio, Precision of Motion, and Stress Analysis.” Journal of Mechanical Design
118(December): 556-567.
Litvin, F. L. and Kim, D. H. (1997). “Computerized design, generation and simulation of meshing
of modified involute spur gears with localized bearing contact and reduced level of
transmission errors.” Journal of Mechanical Design 119: 96-100.
Lu, J., Litvin, F. L. and Chen, J. S. (1995). “Load Share and Finite Element Stress Analysis for
Double Circular-Arc Helical Gears.” Mathl. Comput. Modelling 21(10),: 13-30.
Luscher, A. Houser, D; Snow, C. (2000). “An investigation of the geometry and transmission
error of injection molded gears.” Journal of Injection Molding Technology (USA) 4(4): 177-190.
MAAG (1990). MAAG Gear Book-Calculation and practice of gears, gear drives, toothed
couplings and synchronous clutch couplings, MAAG Gear Company LTD., Zurich.
Mahalingam, S. (1968). “Couple Vibration of Gear Systems.” Journal of the Royal Aeronautical
Society 72: 522-526.
Maitra, M. G. (1989). Handbook of Gear Design. New Delhi, Tata McGraw-Hill Publishing
Company Limited.
Mao, K. (1993). The performance of dry running non-metallic gears, The University of
Birmingham, UK.
Mark, W. D. (1992). "Elements of Gear Noise Prediction." Noise and Vibration Control Engineering:
Principles and Applications, L.L. Beranek and I.L. Ver, eds., John Wiley & Sons, Inc.: 735-770.
Maruyama, N., Morikawa, K. and Hitomi, N. (1992). Gear Case Shape and Rib Distribution for
Reducing Automobile Transmission Gear Noise. 6th International Power Transmission and
Gearing Conference, Scottdale, Arizona, USA.
McFadden, P. D. (1985). Analysis of the Vibration of the Input Bevel Pinion in RAN Wessex
Helicopter Main Rotor Gearbox Wak143 Prior to Failure., Department of Defence,
Aeronautical Research Laboratory.
Moriwaki, I. (1993). “Global Local Finite Element Method (Glfem) in Gear Tooth Stress
Analysis.” Journal of Mechanical Design 115(December): 1008-1012.
Munro, R. G. (1962). The dynamic behavior of spur gears, Cambridge University.
Munro, R. G. (1997). The Use of Optical Gratings in Gear Metrology. Third International
Conference on Laser Metrology and Machine Performance, UK.
Munro, R. G., Yildirim, N. and Hall, D. M. (1990). “Optimum profile relief and transmission
error in spur gears.” In Proceedings of IMechE Conference on Gearbox Noise and Vibration
paper C404/032: 35-41.
Munro, R. G. and Yildirim, N. (1994). Some Measurements of Static and Dynamic Transmission
Error of Spur Gears. International Gearing Conference, UK.
Munro, R. G.; Morrish., L.; Palmer, D. (1999). “Gear transmission error outside the normal path
of contact due to corner and top contact.” Proceedings of the Institution of Mechanical
Engineers. Part C, Journal of Mechanical Engineering Science 213(C4): 389-400.
Muthukumar, R. and Raghavan, M. R. (1987). “Estimation of Gear Tooth Deflection by the
Finite Element Method.” Mech. Mach. Theory 22: 177-181.
Nabi, S. M., Ganesan, N. (1993). “Static Stress Analysis of Composite Spur Gears Using 3D-
Finite Element and Cyclic Symmetric Approach.” Composite Structures (UK) 25(1-4): 541-546.
Nadolski, W. and Pielorz, A. (1998). “The influence of variable stiffness of teeth on dynamic loads
in single-gear transmission.” Archive of applied mechanics = ingenieur-archiv 68(3/4): 185.
Nakada, T. and Utagawa, M. (1956). The Dynamic Loads on Gears Caused by the Varying Elasticity of
the Mating Teeth of Spur Gears. The Sixth Japanese National Congress on Applied Mechanics.
Narasimhan, R. and Rosakis, A. J. (1988). Three Dimensional Effects Near a Crack Tip in a
Ductile Three Point Bend Specimen - Part I: A Numerical Investigation. Pasadena, CA,
California Institute of Technology, Division of Engineering and Applied Science.

256
Neriya, S. V., Bhat, R. B. and Sankar, T. S. (1985). “Coupled Torsional-Flexural Vibration of a
Geared Shaft System Using Finite Element Analysis.” Shock and Vibration Bulletin 55(3): 13-25.
Niazy, A.-S. M. (1997). “What you should know about FEA.” Machine Design; Cleveland 69(21): 55-58.
Niemann, G. and Baethge, J. (1970). “Transmission error, tooth stiffness, and noise of parallel
axis gears.” VDI-Z Vol 2, No 4 and No 8.
Nitu, C. and Grecu, E. (1997). “Plastic gears.” Constructia de Masini (Romania) 49(11-12): 76-80.
Okamoto, N. (1994). Creep in Connections Between Gears and Shafts. 1994 International Gearing
Conference, University of Newcastle, UK.
Ong, J. H. (1992). “Finite Elements for Vibration of Vehicle Transmission System.” Computers in
Industry 18: 257-263.
Oswald, F. B. (1992). Comparison of Analysis and Experiment for Gearbox Noise. 6th
International Power Transmission and Gearing Conference, USA.
Oswald, F. B. (1992). Effect of Operations Condition on Gearbox Noise. 6th International Power
Transmission and Gearing Conference, Scottdale, Arizona, USA.
Oswald, F. B. (1994). Influence of Gear Design Parameters on Gearbox Radiated Noise. 1994
International Gearing Conference, University of Newcastle, UK.
Ozguven, H. N. and Houser, D. R. (1988). “Dynamic Analysis of High Speed Gears by Using
Loaded Static Transmission Error.” Journal of Sound and Vibration 125: 71-83.
Ozguven, H. N. and Houser, D. R. (1988). “Mathematical Models Used in Gear Dynamics- A
Review.” Journal of Sound and Vibration 121(3): 383-411.
Paczelt, I.; Szabo., B.A.; Szabo, T. (1999). “Solution of Contact Problem Using the hp-Version of
the Finite Element Method.” Computers and Mathematics with Applications 38: 49-69.
Palmer, D. and Munro, R. G. (1995). Measurements of transmission error, vibration and noise in
spur gears. British Gear Association Congress, Suite45, IMEX Park, Shobnall Rd., Burton on Trent.
Panhuizen, F. (2000). “New approaches to noise reduction through the use of thermoplastics.”
Motor Industry Research Association (MIRA), New Technology (UK): 57.
Perret-Liaudet, J. and Sabot, J. (1992). Dynamics of Truck Gearbox. 6th International Power
Transmission and Gearing Conference, Scottdale, Arizona, USA.
Prasad, V. S., Dukkipati, R. V. and Osman, M. O. M. (1992). A Group Theory for Composite
Gear Trains Employing Three-Shafts. 6th International Power Transmission and Gearing Conference.
Ramamurti, V. and Rao, M. A. (1988). “Dynamic Analysis of Spur Gear Teeth.” Computers and
Structures 29(5): 831-843.
Randall, R. B. (1980). Advances of the Application of Cepstrum Analysis to Gearbox Diagnosis.
Second International Conference: Vibration in Rotating Machinery.
Randall, R. B. (1984). Seperating Excitation and Structural Response Effects in Gearboxes.
IMECHE, Third International Conference on Vibration in Rotating Machinery.
Randall, R. B. and Deyang, L. (1990). Hibert Transform Techniques for Torsional Vibration
Analysis. The Institution of Engineers Australia Vibration and Noise Conference, Australia.
Randall, R. B. and Kelly, D. W. (1990). Analytical and Experimental Vibration Analysis of a Gearbox
Casing. The Institution of Engineers Australia Vibration and Noise Conference, Melbourn, Australia.
Rao, C. R. M. and Muthuveerappan, G. (1993). “Finite Element Modelling and Stress Analysis
of Helical Gear Teeth.” Computers and Structures 49(6): 1095-1106.
Rebbechi, B., Oswald, F. B. and Townsend, D. P. (1996). Measurement of Gear Tooth Dynamic
Friction. Transmission and Gearing Conference, USA.
Rebbechi, B. and Crisp, J. D. C. (1981). A New formulation of Spur-Gear Vibration. International
Symposium on Gearing & Power Transmissions., Tokyo.
Rebbechi, B. and Crisp, J. D. C. (1983). The Kinetics of the Contact Point, and Oscillatory
Mechanisms in Resilient. The sixth world congress on theory of machines and mechanisms.
Rebbechi, B. (1992). A Comparison Between Theoretical Prediction and Experimental
Measurement of the Dynamic Behaviour of Spur Gears. 6th International Power Transmission
and Gearing Conference, USA.
Refaat, M. H. (1995). Nonlinear Finite Element Analysis of Frictional Contact Problem Using
Variational Inequalities,, University of Toronto, Toronto.
257
Refaat, M. H. and Meguid, S. A. (1994a). A Novel Finite Element Analysis of the General Contact
Problem Using Variational Inequalities. Recent Advances in Structural Mechanics ASME.
Refaat, M. H. and Meguid, S. A. (1995). “On the Contact Stress Analysis of Spur Gears Using
Variational Inequalities.” Computers and Structures 57(5): 871-882.
Regalado, I. and Houser, D. R. (1998). Profile Modifications for Minimum Static Transmission Error in
Cylindrical Gears. 1998 ASME Design Engineering Technical Conference, Atlanta, GA, USA.
Remmers, E. P. (1978). “Gear Mesh Excitation Spectra for Arbitrary Tooth Spacing Errors, Load
and Design Contact Ratio.” Journal of Mechanical Design, 100(October): 715-722.
Richard, E. D., J., Echempati, R. and Ellis, J. (1998). Design and Stress Analysis of Gears Using
the Boundary Element Method. 1998 ASME Design Engineering Technical Conference,
Atlanta, GA, USA.
Richards, R. R. (2001). Principles of solid mechanics, Boca Raton, Fla. : CRC Press, c2001.
Richardson, H. H. (1958). Static and dynamic load, stress, and deflection cycles in spur gear
systems, Massachusetts Institute of Technology.
Rollinger, J. G. and Harker, R. J. (1967). “Instability Potential of High Speed Gearing.” The
Journal of the Industrial Mathematics Society 17: 39-55.
Roosmalen, A. N. J. V. (1995). Noise Generation Mechanism of Gear Transmissions. IMechE 1995.
Rosen, M. K. and Frint, H. K. (1982). “Design of high contact ratio gears.” Journal of the
American Helicopter Society: 65-73.
Rosinski, J. (1992). Dynamic Loads in Geared Transmission Systems in Mining Machinery.
Vibrations in Rotating Machinery, Proceedings of the Institute of Mechanical Engineers.
Rosinski, J., Hofmann, D. A. and Pennell, J. A. (1994). Dynamic transmission error
measurements in the time domain in high speed gears. International Gearing Conference,
University of Newcastle, UK.
Sabot, J. and Perret-Liaudet, J. (1994). Computation of the Noise Radiated by a Simplified
Gearbox. 1994 International Gearing Conference, University of Newcastle, UK.
Seager, D. L. (1975). Separation of gear teeth in approach and recess, and the likelihood of corner
contact. 30th Annual Meeting of American Society of Lubrication Engineers.
Seol, I. H. and Litvin, F. L. (1996). “Computerized Design, Generation and Simulation of Meshing
and Contact of Modified Involute, Klingelnberg and Flender Type Worm-Gear Drives.” Journal
of Mechanical Design 118(December): 551-555.
Seol, I. H. and Litvin, F. L. (1996). “Computerized Design, Generation and Simulation of Meshing
and Contact of Worm-Gear Drives with Improved Geometry.” Computer Methods in Applied
Mechanics and Engineering 138: 73-103.
Sirichai, S. (1999). Torsional properties of spur gears in mesh using nonlinear finite element
analysis, Ph.D Thesis, Curtin University of Technology, Australia.
Sirichai, y., Howard, I., Morgan, L. and Teh, K. (1998). A finite element model of spur gears in
mesh with single tooth crack. ISASTI '98, -.
Sirichai, S. (1996). The Measurement of Static Torsional Stiffness of Gears in Mesh. Third
International Symposium on Measurement Technology and Intelligent Instruments, Japan.
Sirichai, S. (1997). Finite Element Analysis of Gears in Mesh. Fifth International Congress on
Sound and Vibration, Australia.
Smith, J. D. (1983). Gears and Their Vibration. USA, Marcel Dekker.
Smith, J. D. (1987). “Gear Transmission Error Accuracy with Small Rotary Encoders.” Journal of
Mechanical Engineering Science 201(c2): 133-5.
Smith, J. D. (1999). Gear noise and vibration, New York : Marcel Dekker, 1999.
Smith, R. E. (1987). The Relationship of Measured Gear Noise to Measured Gear Transmission
Errors. American Gear Manufacturers Association.
Smith, Z.; Fletcher, M. (1997). “Shifting gears--exploiting the potential of plastics.” Gear
Technology (USA) 14(1): 35-39.
Solaro, M. (1993). “Criteria for the design of gears using thermoplastic materials.” Rev. Plast.
Mod. (Spain) 66(446): 153-157.

258
Stadtfeld, H. J. (1998). Single Flank Test, Structure-Borne Noise Analysis and Digital Imaging of
Tooth Contact. 1998 ASME Design Engineering Technical Conference, Atlanta, GA, USA.
Strang, G., Fix, G. J. (1973). An Analysis of the Finite Element Method. Englewood Cliffs, NJ,
Prentice-Hall.
Su, X. and Houser, D. R. (1998). Calculation of Wobble From Lead Inspection Data Gear Teeth
with Modifications. 1998 ASME Design Engineering Technical Conference, Atlanta, GA, USA.
Sundarajan, S. and Amin, S. (1991). “Finite Element Analysis of Ring Gear/Casing Spline
Contact.” Journal of Propulsion and Power 7(4): 602-604.
Sundarajan, S. and Young, B. G. (1990). “Finite Element Analysis of Large Spur and Helical
Gear Systems.” Journal of Propulsion and Power 6(4): 451-454.
Svicarevich, P., Howard, I. and Wang, J. (1999). The Dynamic Modelling of a Spur Gear in
Mesh Including Friction. Internation conference on applications of modal analysis'99, Gold
Coast, Queensland, Australia.
Sweeney, P. J. and Randall, R. B. (1996). “Gear Transmission Error Measurement Using Phase
Demodulation.” Journal of Mechanical Engineering Science 210(3): 201-213.
SzabÒ, B., Babuska, I. (1991). Finite Element Analysis. N.Y., John Wiley & Sons, Inc.
Tavares, H. F. and Prodonoff, V. (1986). “A New Approach for Gearbox Modeling in Finite
Element Analysis of Torsional Vibration of Gear-Branched Propulsion Systems.” Shock and
Vibration Bulletin 56(2): 117-125.
Terashima, K., Tukamoto, N. and Nishida, N. (1986). “Development of plastic gears for power
transmission (Design on load carrying capacity).” Bulletin of the JSME 29(250): 1326-1329.
Terauchi, Y. and Nagamura, K. (1981). On tooth deflection calculation and profile modification
of spur gear teeth. Int. Symp. on Gearing and Power Trans., Tokyo, Japan.
Terauchi, Y. and Nagamura, K. (1981). “Study on Deflection of Spur Gear Teeth.” Bulletin ofthe
JSME 24(188): 447-452.
Tessema, A. A., Walton, D and Shippen, J (1993). “Load sharing in Non-metallic gears.” KOLA
ZEBATE KZ'93 Wyiwarzanie, Pomiary, eksploatacja, Poznan: 221-227.
Tessema, A. A., Walton, D. (1994). Flexibility effects in non-metallic gears. Proc. of the 1994 Int.
Gearing Conf., Newcastle upon Tyne.
Tessema, A. W., D; Weale, D. (1995). “The effect of web and flange thickness on non-metallic
gear performance.” Gear Technology (USA) 12(6): 30-35.
Tobe, T., Sato, K. and Takatsu, N. (1976). “Statistical Analysis of Dynamic Loads on Spur Gear
Teeth.” Bulletin of the JSME 19(133): 808-813.
Tordian, G. V. and Geraldin, H. (1967). Dynamic Measurement of the Transmission Error in
Gears. Proc. JSME Semi-International Symposium.
Townsend, D. P. (1991). Dudley's Gear Handbook.
Townsend, D. P., Baber, B. B. and Nagy, A. (1979). “Evaluation of high contact ratio spur gears
with profile modification.” NASA technical paper 1458.
Tsai, Y. C. and Chang, H. L. (1992). Reconsideration of the Logic Gear Using a General
Mathematical Model of Composite Gear Profiles. 6th International Power Transmission and
Gearing Conference, Scotts dale, Arizona.
Tsukamoto, N. and Terashima, K (1986). “Development of plastic gears for power transmission
(various methods of lengthening the life of plastic gears and their effect).” Bulletin of the JSME
29(247): 249-255.
Tsukamoto, N.; Marayama, H.; Terashima, K. (1990). “Fundamental Characteristics of Plastic
Gears Made of Thermoplastic Resin Filled With Various Fibers.” JSME International Journal,
Series III 33(4): 590-596.
Tuma, J., Kubena, R. and Nykl, V. (1994). Assessment of Gear Quality Considering the Time
Domain Analysis of Noise and Vibration Signals. International Gearing Conference, UK.
Umezawa, K., Ajima, T. and Houjoh, H. (1986). “Vibration of Three Axes Gear System.”
Bulletin of the JSME 29(249): 950-957.
Underhill, W. R. C., Dokainish, M. A. and Oravas, G. E. (1997). “A Method for Contact Problem
Using Virtual Elements.” Computer Methods in Applied Mechanics and Engineering 143: 229-247.
259
Velex, P., Maatar, M. and Octrue, M. (1995). Loaded Transmission Error Predictions and
Measurements on Spur Gears with Profile Reliefs. IMechE.
Velex, P. and Berthe, D. (1989). Dynamic tooth loads on geared tran. Proc. of ASME 5th Int.
Power Trans. and Gearing conf., Chicago, IL.
Velinsky, S. A. (1996). “Torsional Stiffness of a Machine Drive Element with Tensile Web
Structure.” Mechanism and machine theory. 31(8): 1043.
Vijayakar, S. M., Busby, H. R. and Houser, D. R. (1988). “Linearization of Multibody Frictional
Contact Problems.” Computers and Structures 29(4): 569-576.
Vijayakar, S. M. and Houser, D. R. (1993). “Contact Analysis of Gears Using a Combined Finite
Element and Surface Integral Method.” Gear Technology: 26-33.
Vijayarangan, S. and Ganesan, N. (1994). “Static Contact Stress Analysis of a Spur Gear Tooth Using
the Finite Element Method, Including Frictional Effects.” Computers and Structures 51(6): 765-770.
Vinayak, H. and Houser, D. R. (1992). A Comparison of Analytical Predictions with Experiment
Measurements of Transmission Error of Misaligned Loaded Gears. 6th International Power
Transmission and Gearing Conference, USA.
Walford, T. L. H. and Stone, B. J. (1980). Some Damping and Stiffness Characteristics of
Angular Contact Bearings under Oscillating Radial Load. Second International Conference:
Vibration in Rotating Machinery.
Walker, H. (1938). “Gear tooth deflection and profile modification.” The Engineer: 319-412 and 434-436.
Wallace, D. B. and Seireg, A. (1973). “Computer Simulation of Dynamic Stress, Deformation, and
Fracture of Gear Teeth.” Journal of Engineering for Industry: 1108-1114.
Walton, D. (1992). A new look at testing and rating non-metallic gears. 3rd World Congress on
Gearing and Power Transmissions, Paris, France.
Walton, D., White, J. and Weale, D. J. (1998). Developments and Advancements in Plastic
Gearing Technology. Society of Plastics Engineers Annual Technical Conference and Exhibition.
Walton, D. (1994). “Load Sharing in Metallic and Non-metallic Gears.” Proc Instn Mech Engrs,
Part C: Journal of Mechanical Engineering Science 208: 81-87.
Walton, D. (1995). “A note on tip relief and backlash allowances in non-metallic gears.” Journal of
Mechanical Engineering Science 209: 383-388.
Walton, D., Shi, Y. W. (1989). “A Comparison of Ratings for Plastic Gears.” Proc. Inst. Mech.
Eng. C, J. Mech. Eng. Sci. 203: 31-38.
Wang, C. C. (1978). “Rotational Vibration with Backlash: Part 1.” Journal of Mechanical Design
100: 363-373.
Wang, C. C. (1985). “On analytical Evaluation of Gear Dynamic Factors Based on Rigid Body
Dynamics.” Journal of Mechanisms, Transmissions, and Automation in Design 107: 301-311.
Wang, J., Howard, I. and Jia, S. (2000). FEA Analysis of Spur Gears in Mesh Using a Torsional
Stiffness Approach. COMADEM, Houston, Texas, USA.
Wang, S. M. (1974). “Analysis of Non-linear Transient Motion of a Geared Torsional.” Journal of
Engineering for Industry 96: 51-59.
Wang, S. M. and Morse, I. E. (1972). “Torsional Response of a Gear Train System.” Journal of
Engineering for Industry, Transactions of the ASME 94: 583-594.
Wang, Y.-F. and Tong, Z.-F. (1996). “Influence of Gear Errors on Acceleration Noise From Spur
Gears.” J. Accoust. Soc. Am. 99(5): 2922-2929.
Weber, C. (1949). The deformation of loaded gears, and the effect on their load - carrying capacity,
D.S.I.R. Sponsored Research (Germany), Report No. 3.
Welbourn, D. B. (1972). Forcing Frequencies Due to Gears. Vibr., in Rotating Syst. Conf.
Welbourn, D. B. (1979). Fundamental Knowledge of Gear Noise - A Survey,. Noise & Vib. of
Engineering and Transmissions, Cranfield, UK.
White, J. Walton, D; Wright, NA; Kukureka, SN; Du Gard, FJ; Driver, CL; Duke, BR; Nicoll, KA
(1997). An evaluation of polymer composite materials for use in mechanical power
transmission drives. EUROMAT 97: 5th European Conference on Advanced Materials and
Processes and Applications, Maastricht, Netherlands.

260
Wilcox, L. and Coleman, W. (1973). “Application of Finite Element to the Analysis of Gear Tooth
Stresses.” Journal of Engineering for Industry, Transactions of the ASME: 1139-1148.
Williams, EH; Quinn, KR (1995). “Materials for plastic gears.” Mach. Des. 67(2): 50-52.
Wright, N. A. and Kukureka, S. N. (2001). Wear testing and measurement techniques for polymer
composite gears. 13th International Conference on Wear of Materials, Part 2, Vancouver, BC, Canada.
Yang, D. C. H. and Sun, Z. S. (1985). “A rotary model for spur gear dynamics.” ASME Trans.,
Journal of Mech., Trans., and Automation in Design 107: 529-535.
Yau, E., Busby, H. R. and Houser, D. R. (1994). “A Rayleigh-Ritz Approach to Modeling
Bending and Shear Deflections of Gear Teeth.” Computers and Structures 50(5): 705-713.
Yelle, H. and Burns, D. J. (1981). “Calculation of contact ratios for plastic/plastic or plastic/steel
spur gear pairs.” Trans. Of tlie ASME.J. of Mechanical Design 103: 528-542.
Yildirim, N. (1994). Theoretical and experimental research in high contact ratio spur gearing, Ph.D
Thesis, Huddersfield University, UK.
Yildirim, N. and. Munro, R. G. (1999). “A new type of profile relief for high contact ratio spur
gears.” Proceedings of Inst. Mech Engrs, Part C 213: 563-568.
Yildirim, N. and. Munro, R. G. (1999). “A systematic approach to profile relief design of low and
high contact ratio spur gears.” Proceedings of the Institution of Mechanical Engineers. Part C,
Journal of Mechanical Engineering Science 213(C6): 551-62.
Zakrajsek, J. J. (1990). Gear Noise, Vibration, and Diagnostic Studies at NASA Lewis Research
Centre. First International Conference, Gearbox Noise and Vibration.
Zeman, J. (1957). Dynamische Zusatzkrafter in Zahnbradgetrieben. Zeitschrift des Vereines
Deutscher Ingiere(99).
Zhang, H. and Li, J. (1992). Thermo-viscoelastic analysis of fiber reinforced plastic gears. The
Second International Symposium on Composite Materials and Structures, Beijing, China,
Peking University Press (China).
Zhang, T., Kohler, H. K. and lack, G. K. (1994). Noise Optimisation of a Double Helical Parallel
Shaft Gearbox. International Gearing Conference, University of Newcastle, UK.
Zhang, T. and Kohler, H. K. (1994). A Gearbox Structural Optimisation Procedure for Minimising
Noise Radiation. international gearing conference, Uk.
Zhang, Y. and Fang, Z. (1998). “Analysis of tooth contact and load distribution of helical gears
with crossed axes.” Mechanism and machine theory. 34(1): 41.
Zhang, Y. (1994). “Computerized Analysis of Meshing and Contact of Gear Real Tooth Surfaces.”
Journal of Mechanical Design 116(September): 667-682.
Zhong, Z. H. and Mackerle, J. (1992). “Static contact problems - A rewiew.” Eng. Comput. 9: 3-37.
Zhong-Sheng, L., Su-Huan, C. and Tao, X. (1993). “Derivatives of Eigenvalues for Torsional
Vibration of Geared Shaft Systems.” Journal of Vibration and Acoustics 115(July): 277-276.

261
/GST,ON ! Turns graphical solution tracking on
!*
*DIM,ANG,,30 ! Defines, ANG( ), an array parameter and its dimensions of
ANG(1)=-7.5,-7,-6.5,-6,-5.5 meshing position. The mesh positions from –7.5 to 7.5 degree
ANG(6)=-5,-4.5,-4,-3.5,-3 cover (nearly) one base pitch – via (nearly) mid-double zone –
ANG(11)=-2.5,-2,-1.5,-1,-0.5 single zone - (nearly) mid-double zone, and which is according to
ANG(16)=0.5,1,1.5,2,2.5 the 2D model of ‘2dMeshedatPitch.db’
ANG(21)=3,3.5,4,4.5,5
ANG(26)=5.5,6,6.5,7,7.5
FNAM=1 ! Set initial meshing position.
PARSAV,ALL,ANGLE_A,, ! Write all parameter to file.
:DOLOOP ! :DOLOOP of meshing position in a cyclic.
RESUME,2dMeshedatPitch,db,,0,0 ! Resumes the 2D model (in the working directory) which has pre-
!* defined the material properties, element type and initial meshed,
!* contact at the pitch point.
PARRES,,ANGLE_A,, ,, ! Replace current parameter set with all array parameter from file.
/PREP7 ! Enter pre-processor
!*
TOQ=152.4 ! TOQ is the input torque load (currently set to 152.4 Nm).
!*
CSYS,1
FLST,3,48,5,ORDE,2
FITEM,3,1
FITEM,3,-48
AGEN, ,P51X, , , ,ANG(%FNAM%), , , ,1
WPSTYLE,,,,,,,,1
wpstyle,0.05,0.1,-1,1,0.003,1,2,,5 Turn the driving and driven gear with angle ANG(%FNAM%)
CSYS,4 and -ANG(%FNAM%) respectively.
FLST,3,48,5,ORDE,2
FITEM,3,49
FITEM,3,-96
AGEN, ,P51X, , , ,-ANG(%FNAM%), , , ,1
!*
SMRT,1 ! Start the mesh adaptation on current model.
ESIZE,0.5,0,
FLST,5,6,5,ORDE,4
FITEM,5,24
FITEM,5,-26
FITEM,5,72
FITEM,5,-74
CM,_Y,AREA
ASEL, , , ,P51X
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
!*
ACLEAR,_Y1
AMESH,_Y1
!*
CMDELE,_Y
CMDELE,_Y1

262
CMDELE,_Y2
!*
/COM, CONTACT PAIR CREATION - START
CM,_NODECM,NODE
CM,_ELEMCM,ELEM
CM,_LINECM,LINE
CM,_AREACM,AREA
/GSAV,cwz,gsav,,temp
MP,MU,1,0.06
MAT,1
R,3
REAL,3
ET,2,169
ET,3,172
KEYOPT,3,9,0
! Generate the target surface
LSEL,S,,,258
LSEL,A,,,261
LSEL,A,,,265
CM,_TARGET,LINE
TYPE,2
NSLL,S,1
ESLN,S,0
ESURF,ALL
CMSEL,S,_ELEMCM
! Generate the contact surface
LSEL,S,,,50
LSEL,A,,,53
LSEL,A,,,56
CM,_CONTACT,LINE Generate the contact and the target elements.
TYPE,3
NSLL,S,1
ESLN,S,0
ESURF,ALL
ALLSEL
ESEL,ALL
ESEL,S,TYPE,,2
ESEL,A,TYPE,,3
ESEL,R,REAL,,3
/PSYMB,ESYS,1
/PNUM,TYPE,1
/NUM,1
EPLOT
ESEL,ALL
ESEL,S,TYPE,,2
ESEL,A,TYPE,,3
ESEL,R,REAL,,3
CMSEL,A,_NODECM
CMDEL,_NODECM
CMSEL,A,_ELEMCM
CMDEL,_ELEMCM
CMSEL,S,_LINECM
CMDEL,_LINECM
CMSEL,S,_AREACM
CMDEL,_AREACM
/GRES,cwz,gsav
CMDEL,_TARGET
CMDEL,_CONTACT
/COM, CONTACT PAIR CREATION - END
!*
FLST,2,76,1,ORDE,13
FITEM,2,4858
FITEM,2,4862
FITEM,2,-4893
FITEM,2,8219
FITEM,2,-8223
FITEM,2,8346
FITEM,2,8350 Rotate the nodal coordinate of the driven gear hub nodes
FITEM,2,-8380 into the polar coordinate system which is located at the
FITEM,2,8863 driven gear hub centre.
FITEM,2,8865
FITEM,2,8869
FITEM,2,-8871
263
FITEM,2,8877
NROTAT,P51X
CSYS,1
FLST,2,76,1,ORDE,13
FITEM,2,2938
FITEM,2,2942
FITEM,2,-2973
FITEM,2,4518
FITEM,2,4522
FITEM,2,-4553 Rotate the nodal coordinate of the driving gear hub nodes
FITEM,2,4777 into the global cylindrical coordinate system which is
FITEM,2,-4780 located at the driving gear hub centre.
FITEM,2,4790
FITEM,2,4795
FITEM,2,4799
FITEM,2,-4801
FITEM,2,4807
NROTAT,P51X
!*
!*
FLST,2,76,1,ORDE,13
FITEM,2,2938
FITEM,2,2942
FITEM,2,-2973
FITEM,2,4518
FITEM,2,4522
FITEM,2,-4553
FITEM,2,4777
FITEM,2,-4780
FITEM,2,4790
FITEM,2,4795
FITEM,2,4799
FITEM,2,-4801
FITEM,2,4807
/GO
D,P51X, , , , , ,UX, , , , ,
!*
!* Apply boundary conditions on driving and driven gear
FLST,2,76,1,ORDE,13 hub respectively.
FITEM,2,4858
FITEM,2,4862
FITEM,2,-4893
FITEM,2,8219
FITEM,2,-8223
FITEM,2,8346
FITEM,2,8350
FITEM,2,-8380
FITEM,2,8863
FITEM,2,8865
FITEM,2,8869
FITEM,2,-8871
FITEM,2,8877
!*
/GO
D,P51X, , , , , ,ALL, , , , ,
!*
!*
FLST,2,76,1,ORDE,13
FITEM,2,2938
FITEM,2,2942
FITEM,2,-2973
FITEM,2,4518
FITEM,2,4522
FITEM,2,-4553
FITEM,2,4777 Apply nodal forces on driving gear hub.
FITEM,2,-4780
FITEM,2,4790
FITEM,2,4795
FITEM,2,4799
FITEM,2,-4801
FITEM,2,4807
/GO
F,P51X,FY,-TOQ/1.14
264
!*
!*
FLST,4,76,1,ORDE,13
FITEM,4,2938
FITEM,4,2942
FITEM,4,-2973
FITEM,4,4518
FITEM,4,4522
FITEM,4,-4553
FITEM,4,4777 Apply nodal coupling on driving gear hub.
FITEM,4,-4780
FITEM,4,4790
FITEM,4,4795
FITEM,4,4799
FITEM,4,-4801
FITEM,4,4807
CP,1,UY,P51X
FINISH ! The pre-processor finished.
!*
/SOLU ! Enter the solution.
!*
NLGEOM,1 ! Turn the Large-deflection key on.
NROPT,AUTO, , ! Let the program choose the Nonlinear Options (default).
LUMPM,0
EQSLV,ICCG,1.0e-6,0, ! Equation solver is ICCG with its tolerance 1.0e-6.
PRECISION,0
MSAVE,0
PIVCHECK,1
SSTIF
PSTRES
TOFFST,0,
!*
TIME,1
AUTOTS,1
NSUBST,6,10,5,0 ! The substep is 6 within the minimum 5 and the maximum 10
KBC,0 substeps.
!*
TSRES,ERASE
FINISH
/SOLU
/STATUS,SOLU
SOLVE ! Starts solving the problem.
/POST1 ! Enter the postprocessor.
RSYS,1 ! Turn the active coordinate system into the global cylindrical.
PRNSOL,DOF,U,Y
*CFOPEN,2D%TOQ%Nm,lis,,APPEND ! Open the text document ‘2D154.2Nm.lis’ for appending.
!*CFOPEN,2Dstandard%TOQ%Nm,lis,L:\FEAResults\,APPEND ! ‘L’ is network drive (optional)
*CFWRITE,,%ANG(FNAM)%,UY%FNAM%,UY('No.') ! Write the driving gear hub master node DOF solution UY to the
*CFCLOSE text file above.
SAVE,2Dstandard%TOQ%Nm%ANG(FNAM)%,DB, ! Save all solution in the data base (in the working directory).
!SAVE,2Dstandard%TOQ%Nm%ANG(FNAM)%,DB,L:\FEAResults\,0,0 ! ! ‘L’ is network drive (optional)
FINISH
FNAM=FNAM+1 ! Set next meshing position.
*IF,FNAM,GT,30,:ENDLOOP
PARSAV,ALL,ANGLE_A,, ! Write all parameter to file.
*GO,:DOLOOP ! Go back to :DOLOOP line action.
:ENDLOOP ! End a do-loop when completed a cyclic mesh.
FINISH
!/CLEAR,START ! Start next loop with different torque load (optional).
!/INPUT,2D_2nd,log,,, 0 ! ‘2D_2nd.log’ is in the working directory, otherwise the program
will stop.

265
/GST,ON ! Turns graphical solution tracking on
*DIM,ANG,,30 ! Defines, ANG( ), an array parameter and its dimensions of
ANG(1)=-7.5,-7,-6.5,-6,-5.5 meshing position. The mesh positions from –7.5 to 7.5 degree
ANG(6)=-5,-4.5,-4,-3.5,-3 cover (nearly) one base pitch – via (nearly) mid-double zone –
ANG(11)=-2.5,-2,-1.5,-1,-0.5 single zone - (nearly) mid-double zone, and which is according to
ANG(16)=0.5,1,1.5,2,2.5 the 2D model of ‘PA6_2d.db’
ANG(21)=3,3.5,4,4.5,5
ANG(26)=5.5,6,6.5,7,7.5
FNAM=1 ! Set initial meshing position.
PARSAV,ALL,ANGLE_A,, ! Write all parameter to file.
:DOLOOP ! :DOLOOP of meshing position in a cyclic.
RESUME,PA6_2d,db,,0,0 ! Resumes the 2D model (in the working directory) which has pre-
!* defined the material properties, element type and initial meshed,
!* contact at the pitch point.
PARRES,,ANGLE_A,, ,, ! Replace current parameter set with all array parameter from file.
/PREP7 ! Enter pre-processor
!*
TOQ=20 ! TOQ is the input torque load (currently set to 20 Nm).
!*
MPDATA,EX,2,,1000 ! Using MPDATA to define the linear materials properties.
MPDATA,PRXY,2,,0.4 Note: the linear property EX2 can not exceed the value that
!* is defined below.
TB,MELA,2,1,30 ! Using TB to define the MELA option of 30 points modulus data
TBTEMP,60 ! at the temperature of 60 degree.
TBPT,,0.005,1.4
TBPT,,0.01,3.2
TBPT,,0.015,4.8
TBPT,,0.02,6.76
TBPT,,0.025,8.2
TBPT,,0.03,10
TBPT,,0.035,11.33
TBPT,,0.04,12.86
TBPT,,0.045,14.4
TBPT,,0.05,16
TBPT,,0.055,17.4
TBPT,,0.06,18.66
TBPT,,0.065,20
TBPT,,0.07,21.25
TBPT,,0.075,22.31
TBPT,,0.08,23.41
TBPT,,0.085,24.7
TBPT,,0.09,25.56
TBPT,,0.095,26.4
TBPT,,0.1,27.33
TBPT,,0.105,28
TBPT,,0.11,28.73
TBPT,,0.115,29.33
TBPT,,0.12,29.86
TBPT,,0.125,30.13
TBPT,,0.13,30.8
TBPT,,0.135,31.06
TBPT,,0.14,31.26
TBPT,,0.145,31.33
TBPT,,0.15,31.26
!*
FLST,3,47,5,ORDE,2
FITEM,3,48
FITEM,3,-94
266
AGEN, ,P51X, , , ,ANG(%FNAM%), , , ,1 ! Turn the driving gear with angle ANG(%FNAM%).
CSYS,1 ! Turn the active coordinate system into the global cylindrical.
FLST,3,47,5,ORDE,2
FITEM,3,1
FITEM,3,-47
AGEN, ,P51X, , , ,-ANG(%FNAM%), , , ,1 ! Turn the driven gear with angle - ANG(%FNAM%).
!*
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
……………………………………………………….. The rest of the preprocessor is similar as in Appendix A.
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
………………………………………………………..
!*
/SOLU ! Enter the solution.
!*
NLGEOM,1 ! Turn the Large-deflection key on.
NROPT,AUTO, , ! Let the program choose the Nonlinear Options (default).
LUMPM,0
EQSLV,ICCG,1.0e-5,0, ! Equation solver is ICCG with its tolerance 1.0e-5.
PRECISION,0
MSAVE,0
PIVCHECK,1
SSTIF
PSTRES
TOFFST,0,
!*
TIME,1
AUTOTS,1
NSUBST,10,50,5,0 ! The substep is 10 within the minimum 5 and the maximum 50
KBC,0
!*
TSRES,ERASE
AUTOTS,-1.0
NEQIT,80 ! The number of equilibriums sets 80 for each substep.
/STATUS,SOLU
SOLVE
!*
/POST1
RSYS,1
PRNSOL,DOF,U,Y
*CFOPEN,PA6-%TOQ%Nm,lis,,APPEND
*CFWRITE,,%ANG(FNAM)%,UY%FNAM%,UY(671)
*CFCLOSE
SAVE,Pa6-2Dstandard%TOQ%Nm%ANG(FNAM)%,DB,
FINISH
FNAM=FNAM+1
*IF,FNAM,GT,30,:ENDLOOP
PARSAV,ALL,ANGLE_A,,
*GO,:DOLOOP
:ENDLOOP
FINISH
!/CLEAR,START ! Start next loop with different torque load or temperature (optional).
!/INPUT,2D_2nd,log,,, 0 ! ‘2D_2nd.log’ is in the working directory, otherwise the program
will stop.

267
C-1 Abstract: High contact ratio gears have been demonstrated to provide significant
advantages in decreasing tooth root and contact stresses with potential flow-on benefits for
increased load carrying capacity. Previous investigations with high contact ratio gears have
involved analytical, numerical and experimental aspects. Much of the earlier numerical
work using FEA was limited in its usefulness due to several factors; (i) the difficulty in
predicting load sharing over roll angles covering two or three teeth simultaneously in mesh,
(ii) the difficulty for the analysis to obtain quality results when modeling Hertzian contact
deflection simultaneously with the bending, shear and angular deflections, and (iii) the
problem of primary unconstrained body motion when profile modifications were applied.
This paper presents methods and results for overcoming these difficulties with recent
computer hardware and software improvements. Particular developments discussed include
the use of FE analysis of High Contact Ratio Gears in mesh and the results obtained when
adaptive meshing and element birth and death options are used. The details of transmission
error, combined torsional mesh stiffness and load sharing ratio against various input loads
over a complete mesh cycle are also given. Results with various tooth profile
modifications will also be presented.

C-2 Introduction. Previous research, (Townsend 1979; Rosen 1982; Elkholdy 1985;
Barnett 1988; Lee 1991; Lin 1993; Yildirim 1999), has shown that high contact ratio gears
can be designed to decrease bending stress by 20 percent and decrease contact stress by 30
percent (Drago 1974), thereby potentially enabling a higher power to weight ratio, longer
service life and greater reliability.

Historically, the most significant step forward in the study of spur gears in mesh was made
in 1958 by Harris (Harris 1958), with the fundamental work known as the Harris Map
which has helped to provide a foundational understanding on conventional gearing (with
gear contact ratios between 1.2 to 1.6). However, the Harris Map doesn’t seem to
automatically cover the characteristics of high contact ratio gears in mesh. Recent research
(Townsend 1979; Rosen 1982; Elkholdy 1985; Barnett 1988; Lee 1991; Lin 1993; Yildirim
1999) on high contact ratio gears was based on a limited range of input loads (centered
mainly on the design load). When tooth profile modifications are applied it is important to
consider the effect of large and small loads on the gear behaviour. Detailed analysis on
high contact ratio gears in mesh with wide range of input loads has been more difficult than
the analysis on conventional low contact ratio gears for a number of reasons; (i)
experimental difficulties where high contact ratio gears appear more sensitive to tooth
spacing variations and other geometrical errors, (ii) mathematical models were either over
complicated or used too many assumptions. With recent computing advances, numerical
methods applied in the research of high contact ratio gears have become more realistic.

268
This paper presents results obtained by using customized ANSYS® APDL looping
programs. Particular developments discussed include the use of FE analysis of high contact
ratio gears in mesh and the results obtained when adaptive meshing and element birth and
death options are used. The details of transmission error, combined torsional mesh
stiffness and load sharing ratio against various input loads over a complete mesh cycle are
given. Results with various tooth profile modifications will also be presented.

C-3 Analysis of Standard Involute Gears in Mesh. The major components of tooth
deformation in a loaded involute elastic gear are: (i) tooth rotation about its root, (ii) local
Hertzian contact deformation, (iii) bending displacement and (iv) shearing displacement. In
particular the tooth rotation can be up to 5 to 10 times larger than the other deformation
components. The complete analysis then requires FE modeling of the entire gear in mesh.

The involute tooth form of high contact ratio gears has been presented by Cornell (Cornell
1978; Cornell 1981) and it has also been used by other researchers (Lee 1991; Lin 1993).
The tooth parameters used in this paper are shown in Table C-1. The FE model and its auto
– mesh adaptation with contact is shown in Figure C1.

Gear tooth ------------------------------------------------------------------------------- Standard involute tooth

Material -------------------------------------------------------------------------------------------------------- Steel
Friction coefficient ------------------------------------------------------------------------------------------------ 0.1
Number of teeth ------------------------------------------------------------------------------------------------- 32
Module M, mm (diametral pitch P, 1/in.) --------------------------------------------------------------- 3.18 (8)
Pressure angle, deg---------------------------------------------------------------------------------------------- 20
Addendum, mm--------------------------------------------------------------------------------------------- 1.53 * M
Face width, mm (in.)-------------------------------------------------------------------------------------- 25.4 (1.0)
Theoretical contact ratio----------------------------------------------------------------------------------------- 2.40
Table C-1. Tooth parameters.

O2

O1

Figure C. 1 FE model of the gears in mesh (ratio 1:1) and its auto – mesh
adaptation with contacts.
For a particular input load, the FEA solution for the displacement of the driving gear hub,
with the driven gear restrained at its hub, was obtained for different mesh positions (about
69) over a complete mesh cycle. Results for a series of input loads can produce a map that
shows the static Transmission Error (T.E.) over a complete mesh cycle as a function of
input load. The same calculations can also produce the map of combined torsional mesh
stiffness and load shearing ratio, as shown in Figure C2.
269
 Triple Zone max
Triple Zone min 550 Nm
8

7 450 Nm

–4 6
350 Nm
5
x

4 250 Nm

3
150 Nm
2

1 50 Nm

0
-18 -12 -6 0 6 12 18

76
Combined Torsional Mesh
Stiffness (Nm/rad x 10 4)

74

72
Pitch point

Pitch point

Pitch point
70
68

66

64
-18 -12 -6 0 6 12 18

I II III IV V VI
Load Sharing Ratio

0.6 1Nm
Pitch point

Pitch point
0.5 50 Nm
150 Nm
0.4 250 Nm
350 Nm
0.3
Pitch point

450 Nm
0.2 550 Nm

0.1
0
-18 -12 -6 0 6 12 18
Roll Distance (degree)

Figure C. 2 The change over process under various input loadsof the
involute gears in mesh (model 0).

Automatic mesh adaptation with contact has been known to be able to cope with the non-
linearities present between contacts while avoiding the use of very large models. When the
size of the elements (near the contacts) is between 0.3 to 0.5 mm, high quality results can
be obtained, especially for the detailed change over region between the triple and double
contact zones. As shown in Figure 2, the change over regions are denoted as regions I, III
and V for the triple pair mating approach case, and regions II, IV and VI for the recess
case. These regions, defined as Handover Regions, exist between the actual (loaded) and
the theoretical (unloaded) higher stiffness zone. The higher stiffness zone is the double
contact zone (when the contact ratio < 2) or the triple contact zone (when the contact ratio
> 2). Generally, under a certain input load, the handover regions of approach and recess
are different in their width and they vary non–linearly with input load. The expansion of
handover regions will subsequently increase the contact ratio, and this can be seen in
Figure C3.
270
 Approach case 2.54
1
2.52

0.8 2.5
2.48
0.6 Recess 2.46
case 2.44
0.4
2.42

0.2 2.4
2.38
0 2.36
0 100 200 300 400 500 600 0 100 200 300 400 500 600
Input Load (Nm) Input Load (Nm)

Figure C. 3 The variation of handover regions and contact ratio with a series input loads.
Previous investigation and discussion of the handover regions can be found in the work by
Seager (Seager 1975), by whom the term tooth “separation” was used. Since then,
researchers (Lin 1994; Munro 1999) have attempted to describe the handover region with
analytical methods (geometrically), but the (T.E.) curves they produced in the handover
regions were independent of the material properties.

Detailed handover region variations can be used to evaluate the amount of contact(s)
outside the normal (theoretical) path that is due to the material flexibility. For the
evaluation of the alignment, or geometrical errors such as manufacturing errors and wear,
Munro (Munro 1999) has provided a complete theory with the calculation equations. For
the general case, the evaluation of the total amount of contact(s) outside the normal path
should be separated in two individual portions (approach and recess) as described above.

Because the handover regions exist when gears are loaded, tip relief is required to avoid the
consequence caused by contact outside the normal path – the premature contact(s). In the
past, however, researchers were required to minimise the dynamic response on the design
load with long tip relief or profile modifications and the relevant research was based on the
use of a limited range of input loads. Due to the meshing sensitivity of high contact ratio
gears, profile modifications may also change the nature of the meshing significantly when
the actual load is other than the design load, and the relevant investigations are as shown in
the next section.

C-4 Analysis with Tooth Profile Modifications. The conventional amount of tip relief
is as given in the existing standards such as British Standard (BS 1970) and ISO (ISO/DIS
1983), where the maximum amount of tip and flank modifications are defined as shown in
Figure 4, including the parameters Ca max = 0.02M and DLa max = 0.6M to prevent the
possibility of excess relief. Tooth profile modification of high contact ratio gears will have
more effect on changing the gear transmission error (T.E.), combined torsional mesh
stiffness and the shared tooth load. The existing standards applicable for high contact ratio
gears should be followed with care to ensure rigid body motion leading to premature
contact and large dynamic loads do not occur. The standard tip relief limitations have been
chosen as reference values to normalize the amount of profile modification, for example,
the amount of tip relief Ca = 1.0 means (actual amount relief)/Ca max = 1.

There are different forms that can be chosen for the modified tip profile including linear
and parabolic variations. The parabolic form presented by Walker (Walker 1938)
attempted to remove material from the tooth tip with consideration of the effects of the
271
deformed tooth, much like that of a bent cantilever beam applied on top of the involute
curve.
Ca (max. 0.02*M)
The modified profile form used in this T2
T1

research involves the original involute and a Δ La

Modified profile (max. 0.6*M)
relief was achieved by rotating the original S

curve through an angle a about the relief

starting point S, as seen in Figure C4.
Such relief is worthy of consideration for
P (pitch point)
various reasons;
(i) the major component of the tooth
displacement is the tooth rotation about its
root, as described previously and (ii) it
may be easier to manufacture.
Figure C. 4 Gear tooth with modified tooth profile.
The investigation involved a series of
tooth-relieved models, with each model, Figure C. 4 Gear tooth with modified tooth profile.
having a particular Ca and DLa combination.
In this paper, four of the models and their results will be presented. The selected models
with their relief parameters are listed in Table C-2. The effects of different relief length DLa
on the TE, load sharing ratio and mesh stiffness are further investigated.

Parameters Ca DLa
a (degree)
Model name Absolute value (mm) Normalized value Absolute value (mm) Normalized value

Model 1 0.0156 0.245 0.661 0.35 1.0

Model 2 0.0108 0.17 1.558 0.82 0.3
Model 3 0.0162 0.255 2.359 1.24 0.3
Model 4 0.0174 0.274 0.928 0.49 0.8

Table C-2. Tooth relief parameters.

Model 1 is typical of a short type relief, where the relief length DLa is only about one third
of the standard limit. As the FEA results show in Figure C5, the relief generally results in a
theoretical reduction of contact ratio with wider handover regions compared to the results
illustrated in Figure C2. This model was designed to take a maximum input load of 350
Nm. Overload will cause corner contact, while the double T.E. zone will protrude
upwards. On the other hand, the triple zone T.E. results also show a downward protrusion
in most cases indicating that the amount of relief was too large, with premature contact
occurring at the relief starting point S. So, this type of relief may only be suitable for light
load designs (with limited Ca).

Model 2. The length of the modification here was 0.82 of the maximum allowable with the
existing standard. On the original involute profile, segment ST1 was one-third the length of
PT1 as shown in Figure C4. So this model was close to the one carried out by Lee and Lin
(Lee 1991; Lin 1993), though the design load was 100 Nm lower due to the smaller amount
of relief Ca. The most significant influence on solving this model was the length of the
modification DLa, which was long enough to cause rigid body (rotational) motion, which
will be incorporated into the total displacement, including the T.E. in the triple contact
zone. Under such conditions, the FEA solution will suffer from difficulties in attempting to

272
solve the primary unconstrained body motion of the driving gear, and the system matrix
will become singular. One of the methods that can be used in solving such problems
involves using a temporary link (or element connection) between a solid base and the
driving gear to avoid the ill – conditioned system matrix. When the solution commences,
(the pair of teeth are already in contact), the ANSYS® element birth and death option can
be used to deactivate the link. Figure C6 shows the results for model 2.
The relief starting point S of model 2 remained inside the triple contact zone under both
unloaded and loaded conditions, so there was a small variable (load-dependent) distance
away from the HP2DTC (the highest point of second double tooth contact) (Lee 1991; Lin
1993).
Triple Zone max One base pitch

8 550 Nm

Theoretical Triple T.E. protrusions.

Zone (min)
7
450 Nm

6
–4

350 Nm
5
x

Pitch point
Pitch point

4
250 Nm

150 Nm
2

1
50 Nm

0 1 Nm
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Combined Torsional Mesh
Stiffness (Nm/rad x 10 4)

76

72

68

64

60
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

0.7 1 Nm
Load Sharing Ratio

Pitch point

Pitch point

50 Nm
0.6 150 Nm
250 Nm
0.5
350 Nm
0.4 450 Nm
550 Nm
Pitch point

0.3 pitch point

0.2

0.1

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Roll Distance (degree)
Figure C. 5 FEA results of Model 1.
273
 Triple Zone
7
450 Nm

Pitch point

Pitch point

Pitch point
6

350 Nm
5
–4

Design load
x

4
250 Nm

150 Nm
2

100 Nm

1
60 Nm

25 Nm
0 1 Nm
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Single Zone
Combined Torsional Mesh
Stiffness (Nm/rad x 10 4)

70

65

60

55

50

45
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

1 Nm
25 Nm
Pitch point

Pitch point
1
60 Nm
Load Sharing Ratio

100 Nm
0.8
150 Nm
250 Nm
0.6
350 Nm
450 Nm
0.4
Pitch point

pitch point

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Roll Distance (degree)

Figure C. 6 FEA results of Model 2.

The most notable variations in the results of model 2 are; 0.6

Load Sharing Ratio

0.4
(i) Large T.E. fluctuations are present below the design 0.2
load, and increasing the load will expand the triple 0
zone with further large T.E. fluctuations. -14 -10 -6 -2 2
Input Load (Nm)
6 10 14

Figure C. 7 Load sharing ratio is at the design load (250 Nm).

Figure C. 7 Load sharing ratio at the
(ii) During the loading phase (gears are expected to reach design load (250 Nm).
2.5
the design load) or in and out of cruise condition, 2.3
Contact Ratio

gears will experience a dramatic change from LCRG 2.1

to HCRG. The load sharing ratio and contact ratio 1.9

variations against input load will be as shown in 1.7

Figure C7 and C8 respectively. 0 100 200 300

Input Load (Nm)
400

Figure C. 8 Contact ratio variations of Model 2. Figure C. 8 Contact ratio variations of

Model 2.
274
(iii)Corner contact can still happen when the load is very light (gears operating under the
cruise condition).

Model 3. The relief length here has exceeded the limit, which is 1.24 times of the
maximum allowable by the current standards. Because the relief starting point S is located
inside the double contact zone, (rotational) rigid body motion will be further incorporated
into the part of the double zone displacement (near the double zone boundary area).
Consequently no smooth T.E. curve will be found for the design load, as seen in Figure C9.
In principle, this will be the same as for the tip relief situation for low contact ratio gears
where the relief starting point should not be too close to the pitch point (near or inside the
single zone).

Triple Zone

10
650 Nm

8
–4

450 Nm
x

350 Nm

4
250 Nm

2 125 Nm

50 Nm
25 Nm
0 1 Nm
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Single Zone
Combined Torsional Mesh
Stiffness (Nm/rad x 10 4)

70

65

60

55

50

45
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

1Nm
25 Nm
1
Load Sharing Ratio

50 Nm
Pitch point

Pitch point

125 Nm
0.8
250 Nm
350 Nm
0.6
450 Nm
650 Nm
Pitch point

0.4
pit ch point

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18
Roll Distance (degree)

Figure C. 9 FEA results of Model 3.

275
On the other hand, the contact ratio will be further decreased compared to Model 2, as
shown in Figure C10 and corner contact can occur when the load is very light.
Figure C. 10 Contact ratio variations of Model 3.
Model 4. Model 4 is classified as a short type relief, 2.4

because there is no rigid body (rotational) motion when the

Ratio
2.1

gears rotate throughout the mesh cycle. This type of relief

Contact Ratio
is the longest of the short relief classifications. The critical

Contact
1.8

relief parameter DLa was found to be (the normalized) 0.49

for this gear model. For gears with different geometrical 1.5

parameters this critical relief parameter will change. 1.2

0 100 200 300 400 500 600

The FEA results are shown in Figure C11. Load (Nm)

Figure C 10 Contact ratio variations
of Model 3.
One base pitch
550 Nm
8

Double Zone Protrusions

7 450 Nm

6
–4

Design (or max) load 350 Nm

5
x

4 250 Nm

3
150 Nm
2
100 Nm

1 50 Nm

1 Nm
0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

78
Combined Torsional Mesh
Stiffness (Nm/rad x 10 4)

B A C
75

72

69

66

63
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

0.6
Pitch point

Pitch point
Load Sharing Ratio

0.5

0.4 A 1 Nm
50 Nm
0.3 150 Nm
250 Nm C
B 350 Nm
0.2
Pitch point

450 Nm
0.1 550 Nm
pitch point
0
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14

Roll Distance (degree)

Figure C. 11 FEA results of Model 4.
276
The results of model 4 have shown that the region of triple contact can diminish to one
point at the pitch point A (or B and C) when the input load tends to be very light. At the
pitch point A (or B and C), the T.E. remains the value as if the gears were without tooth
profile modification (pure involute) under the same load and boundary conditions, so too
the combined torsional mesh stiffness and the load-sharing ratio. However, the FEA results
show the minor difference in the amplitude of model 4 and model 0 (involute gears in
mesh), especially in the combined torsional mesh stiffness (Figure C12). This difference
was due to the different mesh density, in the numerical models as a higher mesh density
was used in model 4, resulting in smaller elements which are principally stiffer.

Model 4
350 Nm
0.0005
Model 0

0.0004
250 Nm

0.0003

150 Nm
0.0002

0.0001
50 Nm

800000 1Nm 50 Nm 150 Nm 250 Nm 350 Nm 450 Nm 550 Nm

1Nm 50 Nm 150 Nm 250 Nm 350 Nm 450 Nm 550 Nm
Combined Torsional Mesh

B A C
Stiffness (Nm/rad)

750000

Model 4

700000

650000
Model 0 Here presents a m
nor difference between the
approach and recess cases.
600000

0.6
Load Sharing Ratio

0.5

0.4 A
1Nm 1Nm
0.3 50 Nm 50 Nm
B 150 Nm 150 Nm C
0.2 Model 4 250 Nm 250 Nm Model 0
350 Nm 350 Nm
0.1 450 Nm 450 Nm
550 Nm 550 Nm
0
-15 -10 -5 0 5 10 15
Roll Distance (degree)
Figure C. 12 . The comparisons between model 4 and model 0.

The results also show that if the gears of Model 4 are rigid enough or the input load tends
to zero, the tooth load-sharing ratio will show an exact space of a complete mesh cycle, and
the contact ratio will tend towards 2. With such meshing characteristics, the consequence

277
of the (possible) premature contact at the relief starting point is minimised and the designed
maximum load can be set higher, compared to Model 1. The other important characteristic
of this model was that the contact ratio was always above
2, as shown in Figure C13. 2.5

Figure C. 13 Contact ratio variations of Model 4. 2.4

Contact Ratio
2.3
C-5 Conclusions This paper has outlined methods for
2.2
using FEA of high contact ratio gears in mesh, including
2.1
adaptive meshing and element birth and death options.
2
These methods have been shown to be proficient over a 0 100 200 300 400 500 600

range of loads and teeth profile modifications, including Lo ad (Nm)

those instances when rigid body motion is required. Figure C. 13 Contact ratio variations of Model 4.

1. With detailed results in the changeover process of the involute HCRG in mesh, the
definition of handover region is more clearly defined. The existence of the handover
region(s) presents the contact(s) outside the normal path of contact due to the material
flexibility, and is one of the primary demands for tooth profile modification. Handover
regions exist in the static T.E., combined torsional mesh stiffness and load-sharing ratio. In
particular, the handover region of the static T.E. represents the TE o.p.c. In previous
research, the theory and evaluation methods of TE o.p.c. were developed mathematically
(geometric analysis), and they are very suitable for gears with geometrical errors.
However, some confusion may arise when one attempts to apply the theory and the
methods to elastically deformed gears. The evaluations for the different cases should be
separated. The numerical evaluations presented in this paper may be one of the alternative
solutions.
2. Four types of tooth profile modification (typical short, long, longer and optimal
short) and their analysis results have been presented. The length of the tooth profile
modifications was found to be the most critical relief parameter over a wide range of load.
For long tooth profile modifications, if the relief length exceeds the maximum allowable by
current standards, it was shown to produce a large dynamic response below the design load,
and premature contact(s) may occur as the contact ratio drops below 2. Moreover, if the
relief length has reached to the double contact zone, the smooth design load (T.E.) will not
be achieved. Further increase in the relief length and the amount of relief, will increase the
potential for flank surface scoring when the gears are operated at less than the design load.
It is suggested that the relief, at least, remains within the maximum allowable by current
standards (for this particular gear). However, there will be minor variations on the relief
length limitation if gears take different geometrical parameters.
3. Even if the optimal design load for minimum dynamic response can be found within
the maximum allowable of current standards, one still has to be concerned for the response
when the input load is lighter.
4. The optimal relief for normal applications has been found as that which keeps the
contact ratio above 2. The premature contact(s) at the tooth tip can be prevented within the
designed maximum load, and at the relief starting point the possibility of premature contact
(depending on the amount of relief) can be minimised. There is also a minor variation on
the optimal relief for gears with different geometrical parameters. Generally, the higher the
contact ratio, the shorter the optimal relief will be. The gear material properties have been
found to be of no influence on this.
278
D-1 Introduction

High-speed gears can be excited by static T.E. resulting in high vibration levels (Walker
1938; Harris 1958; Gregory 1963; Welbourn 1979; Lin 1994). Understanding the
generation of static T.E., combined torsional mesh stiffness, tooth load share and other
mechanism properties over the mesh cycle with various types of tooth profile irregularities
will enable further dynamic modelling to be more realistic.

HCRG (High Contact Ratio Gears) are expected to be more sensitive to irregularities in the
tooth profile than LCRG (Low Contact Ratio Gears) and previous research results have
proved that mainly in the dynamic response (Cornell 1978; Rosen 1982; Barnett 1988; Lee
1991; Lin 1993). However, detailed analysis of the static meshing process with HCRG has
been far from sufficient and has result in many assumptions being incorporated into the
dynamic models. Dynamic modelling of gear systems is often aimed at understanding the
system diagnostics and condition monitoring (Du 1997; Howard 1998; Svicarevich 1999;
Howard 2001), in which one of the critical stages is to have the capability of determining
the difference between surface pitting and a tooth crack. Accurate knowledge of the effect
and influence of tooth profile irregularities will assist this overall goal.

As part of an ongoing effort, this paper will present the analysis of FEA adaptive mesh
using ANSYS® showing results of the meshing process of HCRG with and without surface
pitting, tooth root crack and tooth tip fillet, in particular, the static T.E., combined torsional
mesh stiffness, load sharing ratio and contact stress are presented at the design load and
over a complete mesh cycle. Some results of a related LCRG are also presented so that the
characteristics of HCRG are further highlighted.

D-2 Optimal Tip-relief in General Applications

Applying tip-relief to reduce the potential for gear flank surface damage and fluctuations in
dynamic load and noise has been an accepted practice for many years. For general
applications, short type relieves have been widely used. In order for these results to be as
realistic as possible, the analysis on the effects of tooth profile pitting and root crack were
carried out with a short type tooth tip-relieved gears. The analysis of short tip-relief has
been presented in chapter 7, where the resulting characteristics have been shown to mainly
depend on the relief length as shown by other recent publications (Yildirim 1999).
However, further analysis with short type relief is still needed to determine the optimal
short tip-relief for particular applications.

279
The involute tooth form of high contact ratio gears has been presented by Cornell (Cornell
1978; Cornell 1981) and it has also been used by other researchers (Lee 1991; Lin 1993).
The tooth parameters used in this paper

La (max. 0.6*M)
are shown in Table D-1. The tooth Ca (max. 0.02*M)

profile of the HCRG and its related

LCRG are shown in Figure D1. It can
T T
be seen, the only difference with the 2 1

1.53 * M
HCRG is the addendum of the LCRG,
which is the standard 1.0 M, thus the Modified tooth profile

1*M
theoretical contact ratio decreases from
2.4 to 1.71. For the comparison, there pitch point
are no changes made on tooth root
fillet of the LCRG (just for this
section). To find out the optimal tip-
relief, the analysis was carried out
beginning with the LCRG, where a
series of FE models were built with
different Ca and DLa combinations as
Figure D1 The tooth profile of the HCRG and its
shown in Table D-2. related LCRG with details of short modification.
Figure D 1 The tooth profile of the CRG
and its related LCRG with details of short modification.
Gear tooth -------------------------------------------------------------------------------------------- Standard involute tooth
Material --------------------------------------------------------------------------------------------------------------------- Steel
Friction coefficient -----------------------------------------------------------------------------------------------------------0.1
Number of teeth--------------------------------------------------------------------------------------------------------------- 32
Module M, mm (diametral pitch P, 1/in.) --------------------------------------------------------------------------- 3.18 (8)
Pressure angle, deg ---------------------------------------------------------------------------------------------------------- 20
Addendum, mm ---------------------------------------------------------------------------------------------------------1.53 * M
Face width, mm (in.) --------------------------------------------------------------------------------------------------25.4 (1.0)
Theoretical contact ratio---------------------------------------------------------------------------------------------------- 2.40
Design load --------------------------------------------------------------------------------------------------------------350 Nm

Table D-1 Tooth parameters of the HCRG.

Parameters Ca DLa
Model name Absolute value Normalized value
Normalized value Absolute value (mm)
(mm) (Ln)
Model 1 0.021 0.325 0.79 0.41
Model 2 0.023 0.360 1.05 0.55
Model 3 0.028 0.445 1.45 0.76
Model 4 0.031 0.49 1.59 0.83

Table D-2 Tooth relief parameters of the LCRG.

The standard tip-relief limitations, such as BS (1970) and ISO (ISO/DIS 1983) use Ca max. =
0.02M and DLa max. = 0.6M, and these have been chosen as the reference value here to
normalise the modification parameters, for example the normalised relief length
actual , relief ,length , DLa
Ln = max .allowable ,relief ,length ,0.6 M . FEA results of each model for the static T.E. and tooth
contact stresses were obtained at the design load 350 Nm, as shown in Figure D2, where
the results of involute (unmodified) gears are also presented. It can be seen that the T.E. of
involute gears have double contact zones, handover region(s) and single contact zones. For
tooth tip-relieved gears (model 1 to model 4), the single zone and the handover regions
expand with the increase of the relief length DLa. Meanwhile, the T.E. of double contact
280
zone will decrease; when the relief length reached to 1.59 mm (model 4), there are only
single points remaining at each original double contact T.E. zone. Further increase of the
relief length changes the T.E. at (or near) the middle of the double contact zone that starts
to increase meaning that rigid body motion will be incorporated into the T.E. and the tip-
relief will be subject to long modifications, but this will not be discussed in this paper.
Single contact T.E.
0.0008
Static Transmission Error (rad)

0.0007

0.0006

0.0005
Double contact T.E.
0.0004 Double contact T.E. of the handover
region (approach).

0.0003 Involute
M odel 1
0.0002 M odel 2
M odel 3

0.0001 M odel 4
pitch point

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

a

3000

d
2000
c

f e
1000 b

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

-1000

-2000

-3000

-4000 S3_Involute SEQV_Involute S3_M odel 1 SEQV_M odel 1

S3_M odel 2 SEQV_M odel 2 S3_M odel 3 SEQV_M odel 3
S3_M odel 4 SEQV_M odel 4 pitch point

Figure D 2 FEA results of each model for the static T.E. and tooth contact stresses.

Tooth contact stresses over the mesh cycle for each model have shown dramatic changes,
in particular, the peak stress values decrease when the relief length was gradually increased

281
from model 1 to model 4. The peak stress values of model 1, for example, show that the
von Mises stress at c would have a value greater than that of model 2 at d if the amount of
relief Ca were sufficient to avoid the premature contact at the tooth tip b (9.75 degrees).
Figure D3 shows the von Mises stress of model 1 at mesh position 9.75 degrees where the
tooth contact stress of MT has shown the peak value in Figure D2 at b. The other peak
value at c occurred when contact was at the relief starting point.

AT
MT

RT

Driving

Figure D 3 The von Mises stress of model 1 at the mesh position 9.75 degrees.
MT: mid-tooth which presents the tooth contact stresses in Figure C 15.
AT: approach tooth.
RT: recess tooth.
Roll angle = 0: when MT contacts at the (un-loaded) pitch point.

The other peak values at d, e and f of model 2, 3 and 4 occurred when contact was at the
starting points of their relief. The maximum stress reductions have been found by model 4
due to its relief (Ca = 0.031 mm), being larger than that of the other models. However,
confirmation of the optimal short tip-relief needs further analysis carried out with Ca of
model 4 increasing to the maximum allowable 0.02M, which is 0.064 mm. This is shown
in Figure D4.

With the maximum allowable Ca, the tooth contact stresses did not show higher values than
normal particularly at the relief starting point. So, for this particular (LCRG) gear, relief
length of 1.59 mm or the normalised value 0.83 will achieve an optimal performance. The
possible premature contact at the relief starting point (when Ca is up to the maximum
allowable) will be at a minimum compared to a relief length shorter than the normalised
0.83.

One of the characteristics of the optimal tip-relief is that the contact ratio (with gears ratio
1:1) will decrease to 1 when the input load tends to zero, as shown in Figure D5.

282
 0.0008

Static Transmission Error

0.0007

0.0006

0.0005
(rad)
0.0004

0.0003

0.0002 M odel 4
pitch point
0.0001
M odel 4 (Ca max.)
0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

2000 S3_M odel 4

SEQV_M odel 4
1500 pitch point
S3_M odel 4 (Ca max.)
1000 SEQV_M odel 4 (Ca max.)

500

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88
-500

-1000

-1500

-2000

Figure D 4 T.E. and contact stresses of model 4 with the maximum relief allowable at the
design load (350 Nm).
Static Transmission Error (rad)

0.0000025

0.000002

0.0000015

0.000001

0.0000005

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

Figure D 5 T.E. of model 4 with the maximum relief allowable under 1 Nm input load.

With the same procedure, the critical relief length DLa for the HCRG (with the addendum
1.53M) was found to be 0.928 mm (0.49 in normalized value). One of the characteristics
of the optimal tip-relief of the HCRG is that the contact ratio (with gear ratio 1:1) will
decrease to 2 when the input load tends to zero.
283
D-3 The Analysis of Tooth Tip-relieved Gears with Crack and Pitting

The most notable tooth profile irregularities are the tooth root crack and surface pitting.
The analysis is based on the Model 4 that was presented in this appendix part I, and here it
is named Rfif08. Figure D6 shows the detailed features of the model, and Figure D7 gives
the FEA model overview and mesh.
The crack mouth as shown was
T
located at the middle of the tooth
root fillet curve BC, and the crack

3 mm
length of 1mm and 2 mm was

4.2 mm
considered. Three instances of
(0.1mm radius) circular pitting Pitting No.3

6.8 mm
were simulated. The dedendum Pitting No.2
section of the drive gear is often the Pitch point

first to experience serious pitting

damage which was represented by
the pitting No. 1, however as
Pitting No.1
operation continues, pitting usually
B
progresses to the point where a
considerable portion of the tooth
surfaces have developed pitting, 55 O C

and those were represented by the

pitting No. 2 and No. 3. In order to
obtain detailed quality FEA Figure D6 The detailed tooth irregularities on the
solutions over the mesh cycle (s), tip-relieved tooth for 2D FE modeling.
adaptive mesh was used that was incorporated with the customised ANSYS® APDL
looping programs, and 206 data points were produced in a complete looping to cover roll
angles of three teeth. The program also output the contact stresses at each mesh position to
determine the maximum vonMises (SEQV) and the minimum principal stress (S3) in the
flank surface TB as shown in Figure D 6.
Figure D 6
AT

MT

RT

Driving

Figure D 7 Model over view and detailed adaptive mesh.

MT: mid-tooth with profile disfeaturing
AT: approach tooth
RT: recess tooth
Roll angle = 0: when MT is contacted at the pitch point (un-loaded).

284
D-3-1 The Effects of a Single Tooth Root Crack

The FEA results of static T.E. and combined torsional mesh stiffness are presented in
Figure D8, where the results of a single tooth root crack of 1mm and 2mm are compared
with that of undamaged gears. For LCRG, a tooth root crack can only cause changes to the
static T.E. or combined torsional mesh stiffness in just one complete mesh cycle (Sirichai
1999; Wang 2000; Wang 2002), but for HCRG that covers the entire region from a to b
(see Figure D8), which is significantly extended causing more global effect. The effect of a
1mm root crack is also significant compared to that for such a small root crack with LCRG
that is usually insensitive. The maximum effects of a root crack in both cases are observed
at the location c that is between the high stiffness (triple contact) and the low stiffness
(double contact) regions in the recess case.

It is also noted that when the crack length was 2mm, small protrusions were found at mesh
positions c and d in the curve of both static T.E. and combined torsional mesh stiffness.
This shows that the tip-relief becomes insufficient in this case. Further study with this case
has produced tooth load sharing ratio of AT, MT and RT with and without the root crack,
and the contact stresses for those three teeth for the cracked gears only are shown in Figure
D9.

0.0006 c d
b
0.0005
Transmission Error (rad)

a
0.0004

0.0003

0.0002

Rf if 08_2 mm root crack

Rf if 08_1mm root crack
0.0001
Rf if 08(undamaged)
pit ch point

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

760000
a
Combined Torsional Mesh
Stiffness (Nm/rad)

720000

680000

640000
b
c d
600000
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

Figure D 8 Static T.E. and combined torsional mesh stiffness of tooth with a single root
crack 1mm and 2mm compared with that of undamaged gears.

285
 a c d b
0.6

Load Sharing Ratio

0.5

0.4 RT
0.3 M T with 2 mm AT
root crack
0.2

0.1 undamaged
pit ch point
0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88
Roll Angle (degree)

2000

1500

1000

500

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88
-500

-1000

-1500

-2000
S3_AT SEQV_AT S3_MT SEQV_MT S3_RT SEQV_RT pit ch point

Figure D 9 Tooth load sharing and the contact stress when the tooth (MT: mid-tooth)
contains a 2mm root crack.

Figure D9 has shown that the damaged tooth MT not only changes its own load-sharing
ratio but also that for the neighbouring teeth RT and AT. In the affect range that is
observed from a to b, teeth RT and AT tend to support the damaged tooth MT, in
particular, AT takes more load from MT in the cross section (double contact zone) cd, and
this leads to the stresses changes as shown with MT decreasing and AT increasing.

It is also noted that there are two abruptions in MT stress curves SEQV or S3 due to the
insufficient tip-relief causing the premature contacts, but the stresses abruptions do not
show any obvious signs that their effects have been transmitted to the neighbouring teeth.
One of the abruptions, however, was not positioned at d, where it was expected to be.

Consider that the 2mm root crack is small, it may still enable the gears to keep running foe
a long period without crack extension. Gears running under such condition will cause flank
surface scoring damage, but initial pitting is most likely to occur near c and d once enough
stress cycles have been built up. The position c is somewhere above the pitch point
(depending on the input load) which is significantly different with where the conventional
initial pitting that appears in the dedendum section, but contact at d is near the tooth tip of
MT and below the pitch point of AT.

D-3-2 The Effects of Tooth Surface Pitting

Further FEA simulations were carried out when the tooth MT contains multiple pitting as
shown in Figure D6 (without the tooth root crack). In the proposed rolling angle from –
16.88 degree to 16.88 degree (where the rolling angle is 0 degree when the tooth MT is
contacted at its pitch point), the results of static T.E., combined torsional mesh stiffness and

286
load sharing ratio, with and without the pitting, and the results of the contact stresses for
the teeth RT, MT and AT were produced when the tooth MT contained the pitting.

All those results were aligned as shown in Figure D10. Pitting No. 1 is located in the
dedendum section of the driving gear tooth MT where the initial pitting is likely to appear.
This pitting has shown its wider impact on the T.E. and the combined torsional mesh
stiffness than the other pitting locations. While, pitting No. 2 has shown its impact by the
largest amplitude, particularly in the contact stresses. Pitting No. 2 is located where the
premature contact occurs if a (2mm) root crack has occurred. Pitting No. 3 has shown
more stable characteristics, and the location (addendum section) is where there is usually
less chance to have pitting damage.

All the pitting locations have shown the major characteristics of localized changes to the
T.E. and to the combined torsional mesh stiffness. The tooth load-sharing ratio of the tooth
MT (with pitting) does affect that of its neighbouring teeth (also locally), but its stresses
does not appear to affect the others. Such characteristics are significant different with that
of the tooth with crack damage. Surface pitting (initial) appears to induce high frequency
changes compared to a cracked tooth. Both pitting and a root crack should affect the
system dynamic behaviour to excite further vibration. In particular, the large stress
amplitude of the tooth with pitting will not transmit to the other teeth, but when the tooth
has a root crack, the stress pattern of the damaged tooth and its neighbouring teeth are
widely changed. The stress behaviours have provided the possibilities of condition
monitoring gear systems using high order strain measurement (Wang 2002), because elastic
strain is of the same order as stress.

287
 Pitting No. 1 Pitting No. 2 Pitting No. 3
0.00055

0.00053

0.00051

0.00049

0.00047

0.00045
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88
Combined Torsional Mesh Stiffness

760000

720000
(Nm/rad)

680000

640000

600000
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

0.7 without pitting

Load Sharing Ratio

0.6
0.5
0.4 RT
0.3 AT
M T with
0.2 multi-pitting
0.1
0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

8000
S3_M T SEQV_M T
SEQV (Mpa)

S3_AT SEQV_AT
6000 S3_RT SEQV_RT
pitch point
4000

2000

-2000
S3 (Mpa)

-4000

-6000

-8000
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

Figure D 10 The meshing process when the tooth MT presents multi-pitting.

D-3-3 3D Modelling of Surface Pitting of the LCRG and HCRG

More realistic modelling of tooth surface pitting will involve 3D FEA, in which mesh
adaptation with contact(s) using brick element and symmetrical mid-plane modelling are
very important for quality solutions without using huge models (mesh density was under
40,000 nodes). The detailed FE model of the HCRG with a single surface pit is shown in
Figure D11.
288
 MT RT

AT

Driving

This edge is on the pitch line

h
l

MT

Figure D 11 3D FE model and the adaptive mesh with contact also with single pit on MT
of driving gear (the maximum depth in the pit centre is 0.1 mm).

It can be seen that a rectangular pit (h x l) was positioned just under the pitch point (or the
pitch line) because the maximum tooth contact stress will be in this area for gears with
optimal tip-relief.

According to Vizintin (Vizintin 2002), a 2mm x 4mm pit will cause WPC (wear – particle
concentration) exceeding the critical limit, and this pit could develop fast to destructive
pitting. So the FE modelling concentrated on this critical pitting and one of the analysis
results is shown in Figure D12.

289
 MT

AT RT

Figure D 12 The von Mises stress of the (driving) gear appeared with a critical pit (2mm
x 4mm).

The static T.E. were produced by the looping programs for different mesh positions over
the mesh cycle, where the pit was located in the same position (in the middle of the tooth
face width and just below the pitch point) for both HCRG and LCRG. The results were also
compared with that of the 2D models, as shown in Figure D13.

290
 HCRG(2D undamaged) HCRG(3D undamaged) HCRG(pitting 2x4) HCRG(pitting 0.5x1.4)
LCRG(3D undamaged) LCRG(2D undamaged) LCRG(pitting 2x4) pitch point

0.00073

0.00068
Transmission Error (rad)

0.00063

0.00058

0.00053

0.00048

0.00043
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

Figure D 13 Static T.E. of the HCRG and the LCRG.

It can be seen that the static T.E. has been overstated by 2D modelling compared to that of
3D modelling, however, the major difference between the 2D and the 3D FE modelling, in
particular for the HCRG modelling, is the different triple contact regions and that is due to
the 3D modelling having a larger contact area than the 2D’s (see Figure D3 and D7). It
should be noted that the HCRG with the small pitting, (0.5mm x 1.4mm) close to pitting
initialisation (Vizintin 2002), hardly makes any changes to the static T.E. However the
results for critical pitting size (2mm x 4mm) can cause significant change to the T.E.
magnitude for the LCRG (in the half of the single contact zone) and over a wider range for
the HCRG (in the complete double contact zone).

Compared to results of the single root crack damage, the following conclusions can be
given,
· For the LCRG, the T.E. (and other mechanism properties) can have an abrupt
change in the half of the single zone for a single critical pit, but for a root crack the
change will be over the complete mesh cycle.
· For the HCRG, the single critical pit affects the T.E. over the entire double contact
zone whereas the single root crack affects the T.E. over the two complete mesh
cycle.

291
D-4 The Effect of Varying Tooth Tip Fillet Size

Considering the variations to tooth tip fillet

radius of 0.5 mm, 0.25 mm and without the
tip fillet, two FEA models, Model L and
Model H were studied. Model L, with the

1.53 * M
tooth addendum 1*M, has an ideal contact

1*M
ratio 1.71 (for 1:1 gear ratio) and is a Tip fillet

conventional LCRG. Model H with the pitch

addendum increasing to 1.53*M becomes a

HCRG (ideal contact ratio 2.4) that is very
much the same as the model used in
previous sections, where tip-relief was not
applied (see Figure D14), where M is the
module. The FEA solution for the
transmission error has produced six results
for those two models, as shown in Figure D14 The tooth of Model L (addendum 1*M)
Figure D15. and Model H (addendum 1.53*M).
Figure D 14 The tooth of Model L (addendum 1*M) and Model H (addendum 1.53*M).

0.00080 0.25 mm fillet radius 0.5 mm fillet radius

without tip fillet pitch point

0.00075
T.E. (Model L) (rad)

0.00070

0.00065

0.00060

0.00055

0.00050

0.00045
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

0.00055
T.E. (Model H) (rad)

0.00050

0.00045

0.00040
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88

Roll Angle (degree)

Figure D 15 T.E. of Model L and Model H.

292
It can be seen that even a small tip fillet can change the T.E. significantly over the mesh
cycle, in particular, the contact ratio decreases when the tip fillet radius increases, as shown
in Table D-3.

Theoretical Contact Ratio Loaded Contact Ratio Loaded Contact Ratio Loaded Contact Ratio
(un-loaded & no tip fillet) (no tip fillet) (0.25 mm tip fillet (0.5 mm tip fillet
radius) radius)
Model L 1.69 1.88 1.82 1.78
Model H 2.4 2.49 2.45 2.22
* Loaded case for all the models was the design load of 350 Nm.

Table D-3 The contact Ratio under various conditions.

The results also show that the T.E. (relative amplitude value) decreases 70 % from LCRG
to HCRG. Tooth contact stresses have also been produced over the mesh cycle under
various tip fillet radius, as shown in Figure D16.

12000
S3(no tip fillet) SEQV(no tip fillet)
S3(0.25mm tip fillet radius) SEQV(0.25mm tip fillet radius)
10000 S3(0.5mm tip fillet radius) SEQV(0.5mm tip fillet radius)

8000
SEQV (Mpa)

6000

4000

2000

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88
-2000

-4000 Approach case Recess case

S3 (Mpa)

-6000

-8000

-10000

-12000
Model
6000
S3(no tip fillet) SEQV(no tip fillet)
SEQV (Mpa)

S3(0.25mm tip fillet radius) SEQV(0.25mm tip fillet radius)

4000 S3(0.5mm tip fillet radius) SEQV(0.5mm tip fillet radius)

2000

0
-16.88 -12.66 -8.44 -4.22 0 4.22 8.44 12.66 16.88
S3 (Mpa)

-2000
Approach
case Recess
-4000
case

-6000
Model H
Roll Angle (degree)

Figure D 16 Tooth contact stress variations under various tip fillet radius.(input load 350 Nm).

293
In the tooth contact region, away from the tooth tip, the stresses were decreased nearly 30%
from LCRG to HCRG because of the tooth load reduction due to the tooth load sharing
which is close to previous research results (Drago 1974). The tooth tip fillet can heavily
influence the tooth contact stresses in the tooth contact approach and recess cases when tip-
relief was not applied. The following points of significance were noted.
· Increasing the tooth tip fillet radius will decrease the contact stresses, but the stress
decrease in the tooth approach and recess cases are not the same, as can be seen in
Table D-4 and Table D-5. The fastest stress decrease is always in the tooth contact
recess case (the point of highest contact stress). So, it is generally more important
for the driving gear teeth to have tip-relief, but for the case of LCRG, it is important
to have tip-relief on both driving and driven gear teeth due to the high contact
stresses in the tooth approach case. However, the contact stresses of HCRG (Model
H) in the approach case are relatively low in magnitude, and remain relatively
stable when the tooth tip fillet radius change value. So, tip-relief of HCRG is not
necessarily applied to the same degree on driving and driven gears. It is possible
for the driven gear to have tip-relief with smaller scale than that of the driving
gears, and this may help to prevent HCRG lose their contact ratio dramatically
when the input load is light (below the design load) (see Part I of this appendix).
· Generally, there is about 30% difference in contact stress between LCRG and
HCRG. This difference can rise up to 80% when the contact is at the tooth tip. The
different matting angle between the tooth tip and flank surface should be one of the
reasons along with the different tooth load share ratio. LCRG have a greater
matting angle than that of HCRG and its tooth is also stronger, which makes it more
important for LCRG to have tip-relief from the stress reduction point of view, and
tip-relief on LCRG is also more effective than that on HCRG, as Table D-4 and D-5
have shown. Also evident is the fact that LCRG decrease the stresses faster than
that of HCRG with increasing tooth tip fillet radius. Increasing gear pressure angle
will have a similar influence to that of decreasing contact ratio, where the higher the
pressure angle, the higher the contact stress will be when the contact is at the tooth
tip, and it will be more important to have tip-relief, and the tip-relief will be more
effective.

Stress difference
SEQVmax of SEQVmax of Model H
between the
Model L (Mpa) (Mpa)
models
Approach Recess Approach Recess Approach Recess
Fillet radius = 0 mm 8851.2 10940 1568.6 3593.8 82.28% 67.15%
Fillet radius = 0.25 mm 3651.5 3217.3 1706.1 2021 53.28% 37.18%
Fillet radius = 0.5 mm 2314.7 2131.7 1248 1707.7 46.08% 19.89%
Stress reduction by tip
fillet (radius=0.25mm)
58.75% 70.59% -8.77% 43.76% ----- -----
Stress reduction by tip
fillet (radius=0.5mm)
73.85% 80.51% 20.44% 52.48% ----- -----

Table D-4 The variations with tooth tip fillet radius of maximum von Mises stress
SEQVmax in approach and recess cases and the comparison between
Model L and Model H.

294
 S3min of Model L S3min of Model H Stress difference
(Mpa) (Mpa) between the models
Approach Recess Approach Recess Approach Recess
Fillet radius = 0 mm -10198 -11707 -1810.7 -3720.8 82.24% 68.22%
Fillet radius = 0.25 mm -3978.3 -3672 -1943.9 -2330.8 51.14% 36.53%
Fillet radius = 0.5 mm -2659.2 -2425.6 -1681 -1966.4 36.79% 18.93%
Stress reduction by tip
fillet (radius=0.25mm)
60.99% 68.63% -7.36% 37.36% ----- -----
Stress reduction by tip
fillet (radius=0.5mm)
73.92% 79.28% 7.16% 47.15% ----- -----

Table D-5 The variations with tooth tip fillet radius of the minimum principal stress
S3min in approach and recess cases and the comparison between Model L
and Model H.

D-5 Conclusions

Finite element analysis of spur gears in mesh was carried out. It has presented that the
different characteristics of the LCRG and the HCRG when the tooth addendum was taken
the different values. To achieve high quality solutions, the adaptive mesh was used in the
customized loop programs that produced the static T.E., combined torsional mesh stiffness,
load sharing ratio and contact stress over a complete mesh cycle. So that the detailed
analysis can be done that including,
1. The optimal tip-relief for general applications (surface protections) was achieved by
comparing the results of the gears with different tooth relief lengths particularly is the
results of the contact stress over the mesh cycle. The meshing characteristics of the
optimal tip-relief were presented by the variations of the static T.E. over the mesh cycle
and it has clearly shown that the gears contact ratio will decrease to 1 when the load
tends to zero.
2. Analysis on the tooth irregularities (surface pitting and root crack) was concentrated on
the different effects between the LCRG and the HCRG. In particular the modelling for
the surface pitting were carried with 2D and 3D cases. It was found that for the given
tooth irregularities HCRG is less sensitive in the magnitudes than that of LCRG in most
mechanism properties (such as the static T.E.), but it has a wider affected range (mesh
angle) than that of LCRG over the mesh cycle.
3. In the cases of true involute tooth profile with or without tooth tip fillet, it has been
found that the LCRG is more sensitive to its mechanism properties response to the load
over the mesh cycle than that of the HCRG, there is 70% more response in static T.E.
and 30% to 80% more response in contact stress of the LCRG than that of the HCRG.
The results are also shown that the (theoretical) contact ratio decreased by any applied
tip fillet and the decrease ratios are almost the same for both the LCRG and the HCRG.
It should be noted that the contact ratio of the HCRG presented in this research is sufficient
high (2.4), the gear hub centre is rigidly fixed (no lateral movement) and the length of the
profile modification was limited, so that the HCRG has shown its stabilities and advantages
override the LCRG. Otherwise, when the contact ratio is too close to 2, the gears may
become very sensitive as any errors (including wear, the lateral movement) and profile
modifications (including tooth tip fillet) could cause the meshing characteristics abruptly
change between that of HCRG and LCRG.

295
E-1 Introduction

The previous results have shown (Chapter 8) that for the standard 20o pressure angle nylon
gears there was little that could be done on the tooth profile modifications to avoid the
premature contact under normal operating load conditions due to the large deformations.
The premature contact(s) near the tooth tip can raise the friction coefficient and with
relative high-speed movement the temperature can also be raised rapidly which generally
decreases the meshing stiffness and also further modifies the friction coefficient
(Domininghaus 1993). Consequently excessive wear is often a problem in the applications
of standard 20o pressure angle nylon gears.

High-pressure angle involute spur gears were recommended (Seager 1975) because it is
consistent also with low sliding speed and high tooth strength. Increasing the tooth
strength can potentially increase the effects of the tooth profile modifications. This
research will present FEA results on the standard 30o pressure angle involute spur gears
(Nylon PA6) in mesh. The results of the static transmission error, combined torsional mesh
stiffness, load sharing ratio and the contact stresses are presented over a complete mesh
cycle for both the pure involute gear model and the (optimal) tip-relieved gear model so
that the effects of an optimal tooth profile modification can be observed.

E-2 Analysis of Standard High Pressure Angle Involute Gears in Mesh.

A standard 30o pressure angle involute spur gear (nylon PA 6 at temperature 23o C) was
investigated with gear parameters as given in Table E-1.
Gear tooth ---------------------------------------------------------------------------------------- Standard involute tooth
Material------------------------------------------------------------------------------------------------------ Nylon (PA 6G)
Friction coefficient -------------------------------------------------------------------------------------------------------0.1
Number of teeth ----------------------------------------------------------------------------------------------------------- 32
Module M ------------------------------------------------------------------------------------------------------------- 6 mm
Pressure angle, deg---------------------------------------------------------------------------------------------------------30
Addendum------------------------------------------------------------------------------------------------------------- 1 * M
Dedendum---------------------------------------------------------------------------------------------------------- 1.25 * M
Face width------------------------------------------------------------------------------------------------------------ 15 mm
Hub radius------------------------------------------------------------------------------------------------------------ 15 mm
Theoretical contact ratio-------------------------------------------------------------------------------------------------1.3

Table E-1. Gear parameters.

The FE model with the gear ratio 1:1 is shown in Figure E1, where the automatic mesh
adaptation with contact was incorporated into the looping program for each mesh position
over the mesh cycle.

296
Figure E 1 FE model of the gears in mesh (ratio 1:1) and its auto – mesh adaptation with contact.

The FEA results of the T.E., combined torsional mesh stiffness and load sharing ratio under
various input loads are given in Figure E2, E3 and E4 respectively.

0.1 Nm 1 Nm 5Nm 10 Nm 20 Nm
40 Nm 80 Nm 120 Nm 160 Nm pitch point
0.035

0.03

0.025
Transmission Error (rad)

0.02

0.015

0.01

0.005

0
-24 -18 -12 -6 0 6 12 18 24

Roll Angle (degree)

Figure E 2 T.E. of the 30o pressure angle involute spur gears in mesh under various input loads.

297
 0.1 Nm 1 Nm 5Nm 10 Nm 20 Nm
40 Nm 80 Nm 120 Nm 160 Nm pitch point

Phase decrease
5300
Triple
Combined Torsional Mesh Stiffness (Nm/rad) Zone

4800

4300

3800

Single Zone

3300
-24 -18 -12 -6 0 6 12 18 24
Roll Angle (degree)

Figure E 3 Combined torsional mesh stiffness of the 30o pressure angle involute spur
gears in mesh under various in put loads.

One base pitch

0.1 Nm
1 1 Nm
5Nm
10 Nm
20 Nm
Load Sharing Ratio

0.8 40 Nm
60 Nm
80 Nm
100 Nm
0.6
120 Nm
160 Nm
pitch point
0.4

0.2

0
-18 -12 -6 0 6 12

Roll Angle (degree)

Figure E 4 Load sharing ratio of the 30o pressure angle involute spur gears in mesh under
various input loads.

The results have shown that the tooth load capability is significantly increased where the
triple contact can be observed when the input load is up to 160 Nm (at the temperature 23o
C). A special feature is also observed particularly showing in the load-sharing ratio (Figure
E4), where the curve of 120 Nm crosses over one base pitch without causing triple contact.
It is known from previous analysis that the triple contact would occur here for gears with
20o pressure angle.
298
More detailed comparisons of these gears with the standard 20o pressure angle gears are
shown in Figure E5, E6 and E7. It should be noted here that the relative T.E. of 20o
pressure angle gears decrease as seen in Figure 5 at 60 Nm and 80 Nm due to the triple
contact. From the load-sharing ratio (Figure E7), it can be observed that the triple contact
occurred just before the load reached 50 Nm for the 20o pressure angle gears and just over
120 Nm for 30o pressure angle gears. The combined torsional mesh stiffness has shown
about a 7% increase by using the high-pressure angle gears.

One base pitch

0.018
1Nm

5Nm
0.016
10 Nm

30o pressure angle

20 Nm
0.014
40 Nm
Transmission Error (rad)

60 Nm
0.012
80 Nm

pit ch point
0.01
1Nm

0.008 5Nm

10 Nm

20o pressure angle

0.006 20 Nm

40 Nm

0.004 60 Nm

80 Nm
0.002

0
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

20o pressure angle

0.000029
gears under load of
Transmission Error (rad)

0.1 Nm.
0.000027
30o pressure angle
0.000025
gears under load of
0.1 Nm.

0.000023

0.000021

0.000019
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

Roll Angle (degree)

Figure E 5 T.E. Comparisons between the 30o pressure angle nylon gears and the 20o
pressure angle nylon gears in mesh.

299
 5500
0.1Nm

1Nm

30o pressure angle

Combined Torsional Mesh Stiffness (Nm/rad)
5Nm

5000 10 Nm

20 Nm

40 Nm

60 Nm
4500 80 Nm

pit ch point

0.1Nm

1Nm
4000

20o pressure angle

5Nm

10 Nm

20 Nm

40 Nm
3500
60 Nm

80 Nm
One base pitch

3000
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

Figure E 6 The comparisons of stiffness as the pressure angle changes from 20o to 30o.

1Nm_20p 5Nm_20p 10Nm_20p 20Nm_20p 30Nm_20p

40Nm_20p 50Nm_20p 60Nm_20p 70Nm_20p pitch point
1 Nm_30p 5Nm_30p 10 Nm_30p 20 Nm_30p 40 Nm_30p
60 Nm_30p 80 Nm_30p 100 Nm_30p 120 Nm _30p 160 Nm_30p

One base pitch

1

0.8
Load Sharing Ratio

0.6

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15
Roll Angle (degree)

Figure E 7 Load sharing ratio under various input loads for 20o and 30o pressure angle gears.

300
During the mesh cycle, the stresses on the tooth flank surface, especially the maximum von
Mises stress (SEQV) and the minimum principal stress (S3) can be determined where the
contact occurs. By using FE post processing, the stress plot can be obtained as shown in
Figure E8.

SEQV_0.1Nm

SEQV_1Nm

60 SEQV_5 Nm

SEQV_10 Nm
SEQV (Mpa)

SEQV_20 Nm

40 SEQV_40 Nm

SEQV_60 Nm

SEQV_80 Nm

20 SEQV_100 Nm

SEQV_120 Nm

SEQV_160 Nm
0 pit ch point
-24 -18 -12 -6 0 6 12 18 24
S3_0.1Nm

S3_1Nm
-20 S3_5 Nm
S3 (Mpa)

S3_10 Nm

S3_20 Nm
-40 S3_40 Nm

S3_60 Nm

S3_80 Nm
-60
S3_100 Nm

S3_120 Nm

S3_160 Nm
-80
Roll Angle (degree)

Figure E 8 The tooth (contact) stresses of the 30o pressure angle nylon gears (gear ratio
1:1) over the mesh cycle under various input loads.

It should be noted that the stresses near the tooth tip could be very high (with the load).
This indicates that tip-relief may be more essential for gears with stronger teeth.

E-3 Analysis with the Optimal Tooth Profile Modifications.

The motivation for the research on the tooth profile modification of high pressure angle
gears is to reduce the peak stresses as shown in Figure E8, especially when the input load is
in operating range.

The principle of optimal tip-relief (in minimising the contact stress) has been presented in
Appendix D. Because the tip-relief is independent of the material properties it will be valid
for the application of non-metallic gears. However, increasing the gear pressure angle will
change the profile of the optimal tip-relief, in which the most critical relief component DLa
(relief length) has to be re-evaluated. The other component Ca (the amount of relief) will
also need to be significantly increased due to the material flexibility. The normalization
actual , amount ,of , relief ,C
can be expressed as allowable,C ,0.02 M ,by ,currenta , standards. and will usually have a value greater
a max

301
than 2.0. Through a series of FE calculations with different relief lengths on the 30o
pressure angle gears in mesh, the critical relief length was determined to be 1.48 mm (0.41
normalized). The tooth profile with the optimal tip-relief is illustrated in Figure E9.

Ca (Normalized value > 2).

T1
T2 DLa = 1.48 mm
(Normalized = 0.41)
S

Modified tooth
profile (circular
curve with radius
of 5.3 mm) P (pitch point).

Figure E 9 Illustration of the optimal tooth tip-relief.

The use of the circular modified tooth profile can be referred to Walton (Walton 1995).
The modified tooth profile ST2 (as shown in Figure 9) is in a circular form. Once the
amount of the modification Ca (or T1T2) is chosen (here T1T2 is ¼ of the tooth top land
width), the minimum radius of the curve ST2 can be found (5.3 mm) which ensures that
modified tooth profile ST2 is inside the original tooth profile (no positive materials). Thus
the features of easy manufacturing and the maximum smoothness near the relief starting
point (S) were achieved.

The following FEA results were considered with Ca = 0.47 mm, which is 3.92 in
normalized value. The results of the static T.E., combined torsional mesh stiffness, tooth
load sharing ratio and contact stresses were obtained for the temperature of 23o C, as shown
in Figure E10 and the static T.E. under lighter loads is shown in Figure E11.

302
 0.035

160 Nm
0.03
Transmission Error (rad)
Transmission Error (rad)
0.025
120 Nm

0.02 100 Nm

80 Nm
0.015

60 Nm
0.01
40 Nm

0.005 20 Nm
10 Nm
5 Nm
0
-24 -18 -12 -6 0 6 12 18 24

5300
Combined Torsional Mesh
Stiffness (Nm/rad)

4800

4300

3800

3300
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

0.1Nm
1 1Nm
Load Sharing Ratio

5Nm
0.8 10 Nm
20 Nm
0.6 40 Nm
60 Nm
80 Nm
0.4
100 Nm
120 Nm
0.2 160 Nm
pit ch point
0
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

50
S3 (Mpa) SEQV (Mpa)

40
30
20
10
0
-10 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

-20
-30
-40
-50

Figure E 10 The change over process of the tip-relieved gears.

303
 0.00028

T.E. (rad)
0.00026

0.00024

0.00022
Load = 1 Nm

0.0002
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

0.00003

0.000028
T.E. (rad)

0.000026

0.000024

0.000022
Load = 0.1 Nm

0.00002
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24
Roll Angle (degree)

Figure E 11 The T.E. of the tip-relieved gears with lighter loads.

The comparisons between the tip-relieved and the unmodified (involute) gears have been
made with the static T.E. shown in Figure E12, the combined torsional mesh stiffness
shown in Figure E13 and the load-sharing ratio shown in Figure E14.

Figure E15 shows the stress comparisons at each input load. It should be noted that the
stresses near the pitch point have remained constant in all cases. The optimal tip-relief (in
current stage) is aimed at decreasing the peak stresses near the tooth tip to be close or
below the constant stresses at the pitch point. These results have shown, except for the
very light loads (0.1 and 1 Nm), that this primary goal has been achieved.

The optimal tip-relief not only decreases the peak stress but also shifts where the maximum
stresses occur from being near the tooth tip to the relief starting point as shown in Figure
E16, and this has indicated that further improvement on the optimal tip-relief may be
achieved by smoothing the modified tooth profile near the relief starting point. Care must
be taken here as any change made to the critical relief length could cause a big difference in
the meshing characteristics, for example, if the critical relief length is shortened then the
stresses at the relief starting point could rise rapidly, and if the critical relief length is
extended then it could cause dynamic impact over the mesh cycle when the load is light.
However, using circular profile modifications (as shown in the current results) could be one
of the easiest manufacturing methods.

304
 Tip-relieved
0.035 Involute
Pitch point

160 Nm
0.03

0.025
Static Transmission Error (rad)

120 Nm

0.02
100 Nm

80 Nm
0.015

60 Nm

0.01

40 Nm

0.005
20 Nm

10 Nm

5 Nm
0 1 Nm

-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

Roll Angle (degree)

0.00003

0.000028
0.1 Nm
0.000026

0.000024

0.000022

0.00002

-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

Figure E 12 Comparisons between the T.E. of the tip-relieved and the unmodified
(involute) gears.

305
 0.1 Nm_T 5Nm_T 20 Nm_T 40 Nm_T 80 Nm_T
160 Nm_T 0.1 Nm_INV 5Nm_INV 20 Nm_INV 40 Nm_INV
80 Nm_INV 160 Nm_INV pitch point

5200
Combined Torsional Mesh Stiffness (Nm/rad)

4700

4200

3700

3200
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24

Roll Angle (degree)

Figure E 13 Comparisons between the combined torsional mesh stiffness of the tip-
relieved and the unmodified (involute) gears.

0.1 Nm_Involute
1 5Nm_Involute
10 Nm_Involute
40 Nm_Involute
80 Nm_Involute
160 Nm_Involute
0.8 pitch point
Load Sharing Ratio

0.1 Nm_Tip-relieved
5Nm_Tip-relieved
10 Nm_Tip-relieved
0.6 40 Nm_Tip-relieved
80 Nm_Tip-relieved
160 Nm_Tip-relieved

0.4

0.2

0
-18 -15 -12 -9 -6 -3 0 3 6 9 12 15 18

Roll Angle (degree)

Figure E 14 Comparisons between the load-sharing ratios of the tip-relieved and the
unmodified (involute) gears.

306
E-4 Conclusions.

Finite element analysis of 30o pressure angle nylon (PA 6) gears in mesh has been
presented in this paper. The advantages of using the high-pressure angle for non-metallic
gears were outlined by comparing with the conventional 20o pressure angle gears,
especially in their static T.E., combined torsional mesh stiffness and load sharing ratio. It
has been shown that high-pressure angle gears can develop high contact stresses when
contact occurs at the tooth tip. In order to prevent excessive wear (relief is too short), as
well as dynamic impact when load is light (relief is too long), an optimal tip-relief with a
circular form was investigated. The successful achievement has been shown by the
detailed comparisons between the results of the tip-relieved and the unmodified (involute)
gears in mesh, especially in that the peak stresses have been successfully reduced over a
wide range of the input load.

5. 00E -01 S3_0.1Nm 4. 00E +00

S3_1Nm
(involut e) (involut e)
4. 00E -01
SEQV_0.1Nm 3. 00E +00
SEQV_1Nm
3. 00E -01
(involut e) (involut e)
S3_0.1Nm S3_1Nm (t ip-
2. 00E +00
2. 00E -01 (t ip-relieved) relieved)
SEQV_0.1Nm SEQV_1Nm
1. 00E -01 1. 00E +00
(t ip-relieved) (t ip-relieved)
0. 00E +00 0. 00E +00
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28
-1. 00E -01
-1. 00E +00

-2. 00E -01

-2. 00E +00
-3. 00E -01

-3. 00E +00

-4. 00E -01

-5. 00E -01 -4. 00E +00

10 S3_5 Nm 15 S3_10 Nm
(involut e) (involut e)
8
SEQV_5 Nm SEQV_10 Nm
10
6
(involut e) (involut e)
S3_5 Nm (t ip- S3_10 Nm
4 relieved) (t ip-relieved)
5
SEQV_5 Nm SEQV_10 Nm
2
(t ip-relieved) (t ip-relieved)
0 0

-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28

-2

-5
-4

-6
-10

-8

-10 -15

307
 25 S3_20 Nm 40 S3_40 Nm
(involut e) (involut e)
20
SEQV_20 Nm 30 SEQV_40 Nm
15
(involut e) (involut e)
S3_20 Nm 20 S3_40 Nm
10 (t ip-relieved) (t ip-relieved)
SEQV_20 Nm 10 SEQV_40 Nm
5
(t ip-relieved) (t ip-relieved)
0 0
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28
-5
-10

-10
-20
-15

-30
-20

-25 -40

50 S3_60 Nm 60 S3_80 Nm
(involut e) (involut e)
40
SEQV_60 Nm SEQV_80 Nm
30
(involut e) 40 (involut e)
S3_60 Nm S3_80 Nm
20 (t ip-relieved) (t ip-relieved)
20
SEQV_60 Nm SEQV_80 Nm
10
(t ip-relieved) (t ip-relieved)
0 0

-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28 -24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28

-10

-20
-20

-30
-40

-40

-50 -60

65 S3_100 Nm S3_120 Nm
62
(involut e) (involut e)
52 SEQV_100 Nm 49 SEQV_120 Nm
(involut e) (involut e)
39 36
S3_100 Nm S3_120 Nm
26 (t ip-relieved) 23 (t ip-relieved)
SEQV_100 Nm SEQV_120 Nm
13 10 (t ip-relieved)
(t ip-relieved)
0 -3
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28
-24 -20 -16 -12 -8 -4 0 4 8 12 16 20 24 28
-13 -16

-26 -29

-39 -42

-52 -55

-65 -68

S3_160 Nm
71
(involut e)
58 SEQV_160 Nm
45 (involut e)
S3_160 Nm
32
(t ip-relieved)
19
SEQV_160 Nm
6 (t ip-relieved)

-24 -20 -16 -12 -8 -4 -7 0 4 8 12 16 20 24 28

-20

-33

-46

-59

-72

-85

Figure E 15 The stress comparisons between the tip-relieved and the unmodified gears at
each load.

308
 SEQVmax of the SEQVmax of the tip-
involute gears. relieved gears.
Input load = 0.1 Nm Input load = 0.1 Nm

SEQVmax of the SEQVmax of the tip-

involute gears. relieved gears.
Input load = 40 Nm Input load = 40 Nm

SEQVmax of the SEQVmax of the tip-

involute gears. relieved gears.
Input load = 80 Nm Input load = 80 Nm

SEQVmax of the SEQVmax of the tip-

involute gears. relieved gears.
Input load = 160 Nm Input load = 160 Nm

Figure E 16 The optimal tip-relief not only decreases the peak stress but also shifts the
maximum stress position from near the tooth tip to the relief starting point.

309