The International Journal of Advanced Manufacturing Technology (2023) 129:1999–2010

https://doi.org/10.1007/s00170-023-12340-x

ORIGINAL ARTICLE

Design for manufacturing of a spur gears profile modification based

on the static transmission error for improving the dynamic behavior
Michele Abruzzo1 · Marco Beghini1 · Luca Romoli1 · Ciro Santus1

Received: 8 March 2023 / Accepted: 17 September 2023 / Published online: 11 October 2023
© The Author(s) 2023

Abstract
This study presents a novel iterative algorithm for obtaining an improved profile modification for spur gears based on
minimizing the peak-to-peak static transmission error. Two improved profile modifications, obtained with different initial
constraints applied to the tooth profile, were calculated and characterized. The static behavior of the gears was analyzed using
a finite element model, by which the modifications were compared considering the maximum contact pressure and peak-to-
peak static transmission error. The dynamic behavior of the gears was analyzed using a lumped parameter model. The dynamic
overload and the dynamic transmission error were evaluated under several working regimes. The grinding wheel profiles used
to realize the proposed profile modifications and the total volume of material removed were calculated using a CAD model.
The results showed that the proposed modifications lead to a significant improvement in the static and dynamic behavior of
the transmission by reducing both the maximum dynamic overload and meshing noise. The profiles of the grinding wheel
were found to be accurately represented by polynomial functions, thus a feasible manufacturing process can be obtained. The
total volume of material removed was lower than that produced by commonly adopted profile modifications, thus reducing
material waste and increasing the productivity of the manufacturing process compared to standard profile modifications.

Keywords Spur gears · Profile modifications · Tip relief · Static and dynamic behavior · Gear manufacturing

Nomenclature
MT , MN Static torque and Nominal torque
a, b Center distance and backlash q Total relative mesh displacement
fm Meshing frequency q0 , qu Static and zero load transmission error
h Profile modification qrms Root mean square of the mesh displacement
ha , hf Addendum and Dedendum Rb , Rhub Base radius and Hub radius
h max Maximum profile modification Vn Normalized volume of removed material
I Rotational inertia of the gears Z, α Number of teeth and pressure angle
Kv Dynamic overload factor φ, φ0 Roll angle and gear angular step
L, m n Face width and module φin Profile modification starting point
ψ Normalized roll angle
Marco Beghini, Luca Romoli, and Ciro Santus contributed equally to
this work.

B Michele Abruzzo 1 Introduction

michele.abruzzo@ing.unipi.it
Marco Beghini In high-performance geared transmissions, accurate static
marco.beghini@unipi.it
and dynamic models are fundamental for obtaining a real-
Luca Romoli istic characterization of the system’s behavior by which the
luca.romoli@unipi.it
transmission’s performances can be improved. Static models
Ciro Santus study the transmission at low angular speeds (typically a few
ciro.santus@unipi.it
hundred revolutions per minute), neglecting the influence of
1 Dipartimento di Ingegneria Civile e Industriale, Universitá any gear-induced vibration on the meshing conditions. On
di Pisa, Largo Lucio Lazzarino 2, Pisa 56122, Tuscany, Italy the contrary, dynamic models include several high-frequency

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2000 The International Journal of Advanced Manufacturing Technology (2023) 129:1999–2010

vibrating motions superimposed on the nominal gear con- with some exceptions [25, 30], only the static behavior of the
stant angular speed, which may be different depending on transmission is considered.
the working regime considered. Moreover, as the working From a manufacturing point of view, introducing a profile
regimes are often characterized by intersections with the modification increases the complexity of the production pro-
system’s resonance frequencies (amplifying gear-induced cess, the overall cost, the production time, and the material
vibrations), it is necessary to consider either the static or waste. The commonly adopted profile modification is usu-
dynamic behavior of the transmission. ally linear or parabolic, starts from the pitch point, and is
Öezgüven and Houser [1] and Wang et al. [2] reviewed defined by the maximum entity of the modification at the
the mathematical models used to model gear transmissions tooth tip. This produces a significant quantity of removed
up to the 1990s and indicated the parameters of paramount material, especially for high-profile modifications, and deter-
importance when analyzing the system dynamics. Of all these mines contact conditions that involve the nominal involute
parameters, profile modifications are essential for any geared profile only up to the pitch point, with an increase in the
transmission system. Profile modifications are defined as the maximum contact pressures. The design for manufacturing of
amount of material removed in the direction normal to the the profile modification can reduce the volumes of removed
nominal profile of the tooth, in general a circular involute, material and manufacturing time compared with the com-
to obtain the desired tooth profile. The profile modifications monly adopted profile modifications, and also improve both
can be uniform along the gear width (i.e., tip relief), a local the static and dynamic meshing conditions.
modification of the tooth’s external edges (i.e., tip crowning), To respond to the critical issues described so far, the
or a combination of the two. A suitable profile modification, present work proposes a method to improve spur gear profiles
usually optimized for specific working regimes with static based on the transmission error. The transmission error is the
analyses, prevents collisions when a new teeth pair engages, most used index to represent the non-ideal behavior of the
increases the overall efficiency of the transmission, reduces meshing gears due to the gear compliance and tooth profile
noise and vibrations, and so on. modifications (compared to the ideally rigid nominal involute
Kahraman studied the effect of the involute contact ratio profile). Modifying the transmission error in static conditions
on the torsional dynamics of spur gears using gears with through a profile modification can reduce the meshing stiff-
the same geometry but different outer diameters [3]. Other ness fluctuations in the working regimes of interest, thereby
researchers studied standard profile modifications (i.e., lin- achieving better dynamic performances. Following this logic,
ear or parabolic tip relief) [4, 5] and analyzed their influence the static transmission error function was calculated for the
on the dynamic tooth root strain. Eritenel and Parker [6] nominal involute profile. After selecting the nominal torque
derived an analytical solution for the nonlinear vibrations of the transmission, two profile modification functions with
of spur gears with tooth profile modifications and analyzed improved dynamic performances were obtained using an iter-
their influence on total and partial contact loss. Moreover, ative algorithm. The proposed profile modifications were
tooth surface microscopic features can influence the mesh- compared in terms of peak-to-peak static transmission error
ing stiffness and the overall efficiency of the transmission [7, and maximum contact pressure in static conditions. A lumped
8]. Narrow-faced spur and helical gears [9], multimesh gear parameter model was introduced to compare the dynamic
vibration with profile modifications [10], and gear manufac- performances of the modified profiles in terms of dynamic
turing errors [11] have also been studied. overload factor and dynamic transmission error. The grinding
In this context, several studies demonstrated that the pro- wheel profiles used to machine the proposed profile modifi-
file modifications play a fundamental role both in the static cations were defined. Lastly, the volume of material removed
and dynamic behavior of geared transmission systems [12– was evaluated for several profile modifications.
14]. The profile modifications influence meshing stiffness,
load distribution, contact ratio, corner contacts, and trans-
mission error. Consequently, optimizing the shape and entity 2 Profile modification definition
of the profile modification can improve the behavior of the
transmission. Several authors have studied methods to opti- As the main purpose is to reduce the vibrations induced
mize the gear profile and the static performance of gear trains by a non-constant meshing stiffness, the definition of the
[15–20]. Sankar et al. [21, 22] optimized the gear tooth improved profile modification is based on the minimization
profile design using finite element models to increase the of the peak-to-peak static transmission error qPTP , which is
overall tooth strength. Other studies [23–27] have explored the difference between the maximum and the minimum static
the optimal profile modification to minimize the maximum transmission error in a single meshing cycle (Fig. 1). Ideally,
contact pressure. Genetic algorithms have been successfully if the peak-to-peak static transmission error was identically
employed to optimize the macro and micro-geometry of spur zero in a complete meshing cycle at a given constant torque,
gears [28, 29]. The main limitation of these studies is that, the meshing stiffness would be constant, and the self-induced

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The gears feature a contact ratio equal to 1.37, meaning

that the meshing teeth pairs vary between 1, in the Single
Teeth Contact (STC) zone, and 2, in the Double Teeth Contact
(DTC) zone. As can be observed in Fig. 1, the transition from
the STC to the DTC zone is gradual and the peak-to-peak
static transmission error of the nominal involute profile is
equal to 3.72 μm.
Considering the nominal involute profile, the variation of
the static transmission error is generated by the compliance
of the spur gears, which cannot be eliminated or significantly
modified. Consequently, the peak-to-peak static transmission
Fig. 1 Static transmission error curve at MT = MN = 200 Nm error can be reduced only through a profile modification,
which increases the static transmission error in the DTC zone
due to a rigid motion of the gears. Moreover, the peak-to-
vibrations would be absent. As a consequence, minimiz- peak static transmission error cannot be reduced identically
ing the peak-to-peak static transmission error is expected to to zero, as a lower limit is given by the tooth compliance in
improve the static and dynamic behavior of the transmission. the STC zone (Fig. 1). For this purpose, the next step of the
In the proposed method, the static transmission error is evalu- algorithm consists in identifying the value of the static trans-
ated and assigned to the tooth profile as a material subtraction mission error to which all points along the meshing arc must
until a minimum peak-to-peak transmission error is reached, tend, which is given by the minimum value of the static trans-
allowing to uniquely determine the tooth profile points to mission error when only two teeth are in contact qmin (Fig. 2).
be modified and to intervene on any gear geometry without The intersection of the horizontal line passing through qmin
losing generality. and the static transmission error curve defines the starting
For a gear pair, the first step to define the improved profile point of the profile modification φin . The ending point of
modification is the calculation of the static transmission error the profile modification is at the tooth tip. The difference
curve. The nominal torque MN is chosen based on the actual between qmin and the static transmission error defines the
working regimes of the transmission. A static finite element total profile modification assigned at each meshing point in
model [31] is used to calculate the static transmission error q0 the direction normal to the nominal involute profile h(φ), as
under a static torque MT equal to the nominal one MT = MN , shown in Fig. 2 and Eq. (1).
thus obtaining the starting point of the iterative algorithm. In 
the present work, the gear parameters (reported in Table 1) qmin − q0 (φ) if qmin − q0 (φ) ≥ 0
are taken from [32], and the static transmission error obtained h(φ) = (1)
0 else
for MT = MN = 200 Nm as a function of the normalized
roll angle ψ (the ratio between the roll angle φ and the gear The static transmission for MT = MN is calculated with
angular step φ0 ) is shown in Fig. 1. the updated geometry (Fig. 3) and used as the new pro-
file to be modified. The profile modification increases the
Table 1 Parameters of the transmission static transmission error in the transition region between the
Parameter Symbol Value STC and the DTC zone (tooth tip modification), causing an

Module mn 3 mm
Number of teeth Z 50
Pressure angle α 20◦
Base radius Rb 70.48 mm
Addendum ha 1.87 mm
Dedendum hf 3.75 mm
Face width L 20 mm
Center distance a 150 mm
Hub radius Rhub 25 mm
Backlash b 0.5 mm
Rotational inertia I 12846 kgmm2
Fig. 2 Definition of the profile modification

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• The first profile modification is obtained by applying

the iterative algorithm to the nominal involute pro-
file without any constraint on the modification shape
(hereafter defined as “Type 1”). The maximum profile
modification at the tooth tip is limited by strength require-
ments (not exceeding the scuffing limit) and is oriented
to improve the dynamic behavior of the transmission.
• The second profile modification is obtained by applying
the iterative algorithm to a profile calculated in static con-
ditions to reduce the maximum contact pressures in one
meshing cycle and derived from [24] (hereafter defined as
Fig. 3 Updated gear tooth profile “Type 2”). The starting modification shape is a parabolic
and linear spline, which produces an initial static trans-
increase in the lower limit of the peak-to-peak transmission mission error different from the nominal involute profile.
error and the need to avoid excessive profile modifications. In this case, the purpose of the algorithm is to modify the
Consequently, the procedure is applied iteratively until the material removal at each point, improving the dynamic
difference between qPTP and its lower limit (Fig. 1) is lower performances of the starting tooth profile while main-
than 10%. The proposed algorithm (reported in Fig. 4) can taining the same static performances. As in the previous
be applied to improve the material removal of any tooth pro- case, the maximum profile modification at the tooth tip
file, regardless of whether it is the nominal involute profile is limited by strength requirements.
or an existing profile modification. There are infinite solu-
tions with equal peak-to-peak transmission error, allowing The spur gears from [32] were previously characterized
the application of other constraints or starting points to sat- using a dynamic model based on a novel meshing stiffness
isfy different requirements (for example, static or endurance definition, featuring high precision and the capability to repli-
requirements). cate the experimental behavior of the transmission [31]. As
It is worth noting that the algorithm can be applied to the model can simulate any profile modification while main-
any gear type. However, most advantages are obtained for taining the same accuracy, the theoretical results presented
spur gears, which feature higher contact irregularity, lower in the following may be considered accurately representative
contact ratio, and a higher margin of improvement for the of the actual dynamic behavior of the transmission.
transmission’s dynamics. For example, the intrinsic meshing
regularity of hypoid or helical gears makes profile modifi-
2.1 Type 1
cations less relevant to reduce vibrations, thus the profile
modifications are mainly employed to avoid edge contact or
The type 1 profile modification was obtained by applying
modify the meshing contact pattern for those gears.
the iterative algorithm starting from the nominal involute
In the present work, the method was applied to obtain two
profile. In Fig. 5, the profile modification h is represented
different profile modifications calculated for the gears from
as a function of the normalized roll angle ψ. It can be
[32]:
noticed that convergence needs only a few steps, and a

Fig. 4 Optimization algorithm Fig. 5 Type 1 profile modification as a function of ψ

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Fig. 7 Response surface of the gear pair with the type 1 profile modi-
Fig. 6 Static transmission error at MT = MN = 200 Nm with the type fication
1 profile modification

2.2 Type 2
profile modification different from commonly used shapes
The type 2 profile modification was obtained by applying the
is obtained. The differences between the curves at each iter-
iterative algorithm to a profile modified using a spline, which
ation are the starting point of the profile modification and the
is made of a parabolic and a linear function of φ [24].
maximum profile modification at the tooth tip h max , which is
The tip relief h is represented as a function of the roll angle
adjusted to avoid corner contacts (i.e., based on the maximum
φ in Fig. 8. Three steps are needed to reach convergence,
contact pressure).
obtaining the improved spline. Compared to the initial curve,
The static transmission error at MT = MN = 200 Nm
the entity of the tip relief at each value of φ is increased. The
for the type 1 profile modification is represented in Fig. 6
position of the spline node moves towards the starting point
for all the iterations required to reach convergence. All the
of the profile modification, extending the linear part of the
static transmission error curves show the same trend in the
spline. The maximum profile modification is unaffected by
STC zone, which is unaltered by the profile modification.
the iterative algorithm.
The minimum value of the static transmission error in the
The static transmission error at MT = MN = 200 Nm
DTC zone reaches the value of qmin for all the iterations con-
for the type 2 profile modification is represented in Fig. 9
sidered. The static transmission error of the modified profiles
for all the iterations considered. The final curve shows the
in the transition from the STC to the DTC zone is slightly
same features of the type 1 profile modification, with a slight
increased due to the presence of a profile modification, caus-
difference in the static transmission error trend in the DTC
ing an increase of qPTP and a negligible angular extension of
zone. At the final step, qPTP is reduced from 3.72 to 0.62 μm,
the STC zone. At convergence, qPTP is reduced from 3.72 to
obtaining an 83% reduction.
0.70 μm, thus obtaining an 81% reduction compared to the
The type 2 profile modification was included in a static
unmodified profile.
finite element model to obtain the related response surface
The obtained profile modification was included in a static
finite element model to calculate the response surface of
the modified gear pair. The results obtained for any loading
regime and angular configuration are reported in Fig. 7, in
which q0 is represented as a function of ψ for several torque
regimes. The variability of the gear pair compliance with the
contact position along the tooth flank, the number of teeth in
contact, and the transmitted torque can be observed. The STC
zone, the DTC zone, and the static transmission error at zero
load qu are highlighted. Compared to the nominal involute
profile, the static transmission error at zero load is non-null
and constitutes the main component of the static transmis-
sion error at low nominal torques (idle regimes). Moreover,
the static transmission error at zero load is uniform in the
DTC zone. Fig. 8 Type 2 profile modification as a function of φ

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Fig. 9 Static transmission error at MT = MN = 200 Nm with the type Fig. 11 Maximum contact pressure pmax for the two profile modifica-
2 modification tion as a function of ψ for MT = MN = 200 Nm

The peak-to-peak static transmission error qPTP is repre-

(Fig. 10). Compared to the type 1 profile modification, the
sented as a function of MT in Fig. 12. For the unmodified
response surface obtained presents the same features, but the
profile, qPTP monotonically increases with MT . For the mod-
maximum unloaded transmission error is not uniform and
ified gears, the peak-to-peak static transmission error shows
shows a peak at the center of the DTC zone.
a minimum for the nominal torque and then increases as the
static torque is either increased or decreased, depending on
which component prevails in the determination of the total
relative mesh displacement (i.e., the static transmission error
3 Static behavior
at zero load at lower static torques and the compliance of the
gear elements at the highest static torques). Regardless of
The profile modifications obtained were compared consid-
the adopted profile modification, the minimum value of qPTP
ering the maximum contact pressure in a meshing cycle and
is obtained at the nominal torque MT = MN = 200 Nm.
peak-to-peak transmission error as a function of the applied
For MT > MN , the peak-to-peak static transmission error of
torque.
the modified profiles increases with MT and is always lower
The maximum contact pressure pmax is represented as a
than the value obtained for the nominal involute profile. For
function of ψ in Fig. 11. As expected, pmax is uniform along
MT ≤ MN , the peak-to-peak static transmission error of the
the STC zone. The maximum value of pmax is experienced in
modified profiles increases as the nominal torque is reduced
the transition from the STC to the DTC zone and is higher for
and becomes higher than the value obtained for the nominal
the type 1 profile modification, reaching a maximum value
involute profile when MT < 90 Nm.
of 1630 MPa. Considering a maximum contact pressure of 2
The static results show that the two profile modifica-
GPa to avoid scuffing failure (typical of case-hardened steel
tions are equivalent from the point of view of peak-to-peak
gears), the safety factors obtained for the type 1 and 2 profile
static transmission error. However, the type 2 profile mod-
modifications are equal to 1.25 and 1.74, respectively. The
ification features a more regular profile (produced by the
relative difference between the two profile modifications in
the transition zone is about 40%.

Fig. 10 Response surface of the gear pair with the type 2 profile mod- Fig. 12 Peak-to-peak static transmission error represented as a function
ification of MT

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initial constraints), leading to lower maximum pressures can be identified (points 1, 2, and 3 in Fig. 13). A bifurcation
experienced in a single meshing cycle during the transition in the dynamic response of the system is experienced dur-
from the STC to the DTC zone. ing the Ramp-Down simulation at all the resonance peaks
of the system when no profile modifications are present.
Moreover, the maximum K v is about 4.25 and is experi-
4 Dynamic behavior enced during the Ramp-Down phase at 4.5 kHz (point 3).
The implementation of the type 1 profile modification pro-
The dynamic behavior of the geared transmission system was duces a significant improvement in the dynamic response.
studied using a lumped parameter model [31]. In particular, Outside any near-resonant condition K v = 1, meaning that
a meshing frequency f m up to 5 kHz was considered, allow- no dynamic overload is experienced by the transmission sys-
ing the investigation of the resonance peaks in the dynamic tem. In resonant conditions, the bifurcation phenomena are
response of the system in this range. Three static torques reduced (points 2 and 3) or eliminated (point 1). Any reso-
MT = [100, 200, 400] Nm were considered to investigate nance peak of the system is narrower and less intense. The
the usefulness of the proposed profile modification outside maximum K v is still experienced during the Ramp-Down
the working regime at the nominal torque. At each static phase at 4.5 kHz (point 3) but is equal to 2.75. A 35% reduc-
torque, the dynamic response was simulated by increasing tion compared to the gears without profile modifications was
(Ramp-Up or RU) and decreasing (Ramp-Down or RD) the consequently expected. The dynamic response of the type
meshing frequency to highlight the effect of any bifurcation 2 profile modification shows the same features as the type
phenomena, which may lead to a different dynamic response 1 profile modification, but the improvement is limited: the
depending on the sign of the gear’s angular acceleration [31]. resonance peaks are wider and more intense (points 2 and
In all the simulations, a reasonable damping value was esti- 3). Overall, the maximum K v is still experienced at 4.5 kHz
mated using the results of previous studies [33]. At each (point 3) and is equal to 3, with a 30% reduction compared
static torque and meshing frequency, the dynamic overload to the gears without profile modifications.
K v (defined as the ratio between the maximum instanta- The same improvements in the dynamic response at MT =
neously transmitted torque and the static one) and the root MN = 200 Nm can be observed considering qrms (Fig. 14).
mean square of the total relative displacement qrms were used For the modified gears, qrms is null outside any resonant
to compare the unmodified gears against the two proposed condition, thus indicating no dynamic amplification of the
profile modifications. nominal load (as observed in Fig. 13). All the resonance
peaks of the system of the modified gears are narrower and
4.1 Nominal torque less intense, denoting a significant reduction in the transmis-
sion’s noise. For the gears without profile modifications, the
The comparison between the dynamic overload at MT = maximum value of qrms is about 22.5 μm and is experienced
MN = 200 Nm for the unmodified gears and the proposed during the Ramp-Down phase at 4.5 kHz (point 3). Compared
profile modifications is represented in Fig. 13. to the unmodified gears, the maximum value of qrms for the
At least three resonance peaks, attributed to the main har- type 1 profile modification is obtained at the same frequency,
monic of the meshing frequency and its super-harmonics,

Fig. 14 Root mean square of the total relative mesh displacement for
Fig. 13 Dynamic overload for the unmodified gears and the proposed the gears without profile modifications and for the proposed profile
profile modifications at MT = MN = 200 Nm modifications at MT = MN = 200 Nm

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but it is 44.5 % lower. The maximum qrms for the type 2 pro-
file modification is obtained during the Ramp-Down phase at
2.25 kHz (point 2), obtaining a 33.3 % reduction compared
to the gears without profile modification.

4.2 Half the nominal torque

The comparison between the dynamic overload at MT =

100 Nm for the gears without profile modifications and the
proposed profile modifications is represented in Fig. 15.
Compared to the nominal torque, the three main reso-
Fig. 16 Root mean square of the total relative mesh displacement for
nance peaks of the system are shifted at lower frequencies the gears without profile modifications and for the proposed profile
due to the lower instantaneous meshing stiffness (points 1, modifications at MT = 100 Nm
2, and 3). The unmodified gears show a bifurcation in the
dynamic response at each resonance peak. The maximum
2 kHz and is 22 % (type 1) and 33 % (type 2) lower than the
K v is experienced at 4.25 kHz (point 3) and is equal to 4.5.
unmodified gears.
Even outside the nominal torque regime, the proposed modi-
fications improve the dynamic behavior of the system. K v is
reduced by at least 25% outside resonant conditions for both 4.3 Double the nominal torque
the proposed profile modifications. The type 1 profile modifi-
cation shows bifurcation phenomena at each resonance peak. The comparison between the dynamic overload at MT =
The maximum K v is experienced at 2 kHz and is 28% lower 400 Nm for the gears without profile modifications and the
than the unmodified gears. At 4.25 kHz (point 3), the type 1 proposed profile modifications is represented in Fig. 17.
modification achieves a 53% reduction of K v . The type 2 tip Compared to the nominal torque, the three main resonance
relief eliminates some of the bifurcation phenomena present peaks of the system are shifted at higher frequencies due to
(points 1 and 3). The maximum K v is experienced at 2 kHz the higher instantaneous meshing stiffness (points 1, 2, and
and is 34% lower than the unmodified gears. At 4.25 kHz 3). The unmodified gears show a bifurcation in the dynamic
(point 3), the type 2 modification achieves a 60% reduction response only at 1.2 kHz and 4.5 kHz (points 1 and 3). The
of the dynamic overload. maximum K v is experienced at 4 kHz (point 3) and is equal to
The same considerations about the dynamic response of 3.5. Moreover, the resonance peak at 4.5 kHz shows a sudden
the system can be made considering qrms (Fig. 16). The pro- change in shape compared to the lower torques. The pro-
posed modifications produce a significant reduction of qrms posed modifications produce an improvement in the dynamic
outside the resonant conditions. For the unmodified gears, response of the system outside the resonant conditions. The
the maximum value of qrms is experienced at 1 kHz (point 1) K v values obtained with the type 1 profile modification are
and is equal to 13.5 μm. At 1 kHz (point 1), the reduction of lower or at most equal to the values obtained with the unmod-
qrms obtained with the proposed modifications is about 60%. ified gears, so the usage of such a modification can still
For the modified gears, the maximum qrms is experienced at improve the dynamic behavior of the transmission. The K v

Fig. 15 Dynamic overload for the gears without profile modifications Fig. 17 Dynamic overload for the gears without profile modifications
and for the proposed profile modifications at MT = 100 Nm and for the proposed profile modifications at MT = 400 Nm

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As is typical in high-performance applications with high

production volumes, it was assumed that the initial rough-
ing phase is performed via gear hobbing, which is a process
with a higher production rate than other gear generation pro-
cesses. The gear hob shapes the nominal involute profile
starting from the raw disk combining a cutting motion of
the tool with a relative piece-tool movement that simulates
the gear meshing. The typical output obtained at the end of
the hobbing phase is a nominal involute profile character-
ized by a roughness average in the range of [1.6 ÷ 3.2] μm.
Hobbing is followed by a finishing process that produces
Fig. 18 Root mean square of the total relative mesh displacement for the profile modifications and the required tooth surface tol-
the gears without profile modifications and for the proposed profile erances by removing the machining allowance. Considering
modifications at MT = 400 Nm that high-performance spur gears are often characterized by
heat treatments which increase hardness and make the fin-
ishing process difficult to be performed with standard tools,
values obtained with the type 2 profile modification are higher it was assumed that the finishing is performed with a prop-
than the ones obtained for the unmodified gears at 1.2 kHz erly shaped grinding wheel, whose active surface replicates
and 2.4 kHz (points 1 and 2), so the use of such a modification the final shape of the tooth flank. As schematically shown
at the considered torque regime becomes questionable. in Fig. 19, the grinding wheel is first brought into position
Despite the results obtained for K v , the proposed profile outside the gear at a radial position equal to the total cutting
modifications reduce qrms at all the working regimes even depth. The feeding motion is parallel to the gear rotational
at MT = 400 Nm (Fig. 18). In particular, a 50 % reduction axis, and the opposite tooth flanks of each tooth vane are
of qrms is achieved outside the resonant conditions. For the machined simultaneously, achieving the required surface fin-
unmodified gears, the maximum value of qrms is experienced ish and final tooth shape. After machining each tooth vane,
at 2 kHz (point 2) and is equal to 32.5 μm, for which a 14% the gear rotates by an angular step around its axis, preparing
reduction is obtained with the proposed profile modifications. the next tooth vane.
The modified gears significantly improve the dynamic Table 2 shows the characteristic points of the grinding
behavior of the transmission. Bifurcation phenomena are wheel surface, measured in a reference system having its
reduced or eliminated, the maximum overload is reduced by origin at the intersection between a plane perpendicular to
at least 25%, and the gearbox is less noisy at all the working the grinding wheel axis and its external diameter (Fig. 19).
regimes considered. All the points were calculated considering a high-precision
The type 1 profile modification, obtained without any ini- execution of the grinding wheel profile (resolution lower
tial constraint on the profile shape, produces better dynamic than 1 μm). Neglecting the tooth root profile, typically
results. As the two profile modifications feature the same obtained through other machining processes, the grind-
peak-to-peak static transmission error, the different dynamic ing wheel profiles can be approximated by a single-node
behavior can be attributed to the static transmission error
trend in a single meshing cycle. The type 1 profile modifi-
cation shows a constant static transmission error in the DTC
zone (Fig. 6), producing constant meshing stiffness and lower
self-induced dynamic overloads. On the contrary, the static
transmission error in the DTC zone obtained with the type 2
profile modification features the superimposition of a higher
frequency wave (Fig. 9), producing higher fluctuations of the
meshing stiffness and slightly worse dynamic behavior.

5 Design for manufacturing

For the sake of simplicity, the manufacturing was charac-

terized by considering the mass production of spur gears
designed to operate with the same nominal torque. Fig. 19 Tooth vane and profile modification finishing

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Table 2 Grinding wheel active surface profiles

Type 1 Type 2
x (mm) y (mm) x (mm) y (mm)

0.8323 0.0089 0.8323 0.0089

1.1504 0.1251 1.1504 0.1251
1.3214 0.4174 1.3214 0.4174
1.3914 0.7636 1.3914 0.7636
1.5571 1.4674 1.5571 1.4674
1.7532 2.1674 1.7532 2.1674
1.9765 2.8674 1.9765 2.8674
2.2249 3.5674 2.2249 3.5674 Fig. 21 Vertical difference between the calculated grinding wheel pro-
2.4971 4.2674 2.4971 4.2674 files and their quadratic interpolation
2.7356 4.8374 2.7675 4.9111
2.7716 4.9204 2.8306 5.0552
2.8081 5.0039 2.8451 5.0880 The manufacturing of the proposed profile modifications
2.8451 5.0880 2.8638 5.1302 was also compared in terms of productivity with two standard
2.8827 5.1725 2.8932 5.1961
profile modifications for industrial applications: the linear
2.9208 5.2576 2.9208 5.2576
and the parabolic tip reliefs applied starting from the pitch
point. The initial manufacturing process used to obtain the
2.9595 5.3431 2.9573 5.3384
nominal involute profile was assumed to be the same for all
2.9987 5.4292 3.0009 5.4340
the modifications considered, as it depends only on the gear
3.0385 5.5159 3.0385 5.5159
macro-geometry. The comparison was thus made consider-
3.0803 5.6061 3.0803 5.6010
ing only the finishing phase, which can be obtained using
differently shaped grinding wheels depending on the tooth
profile to be obtained. The profile modifications were com-
spline, which is made of the nominal involute circle and a pared under the same maximum profile modification at the
second-order polynomial function (Fig. 20). The two profiles tooth tip (6 μm) using the total volume of removed material
share the initial curve, a perfect involute circle, and diverge per gear during the finishing phase normalized with respect
from the modification starting point, which is moved forward to the type 2 profile modification Vn , which is directly related
for the type 2 profile modification. Macroscopically, the tooth to the productivity of the manufacturing process.
profiles appear to be identical. The difference between the Compared to the nominal involute profile (Fig. 22), the
calculated profiles and their quadratic interpolation is lower total volume of removed material for each gear (Table 1) is
than a micron (Fig. 21). The profile modifications proposed, about 5 mm3 for either of the proposed profile modifications,
despite their apparent complex shape, are obtainable through with a negligible difference between type 1 and type 2 pro-
low-degree polynomials, justifying their ease of execution in files. Vn is higher for the standard profile modifications, with
industrial practice. a 150% and a 73% increase for the linear and the parabolic tip
relief modifications, respectively. Our profile modifications
can thus be obtained faster and with less material waste than

Fig. 20 Grinding wheel active surface obtained for the type 1 and the
type 2 profile modifications. The interpolation of the profile is repre- Fig. 22 Normalized volume of material removed for the tip relief func-
sented using a dashed line tions considered

123
The International Journal of Advanced Manufacturing Technology (2023) 129:1999–2010 2009

standard profile modifications obtained using the same manu- static and dynamic performances of the whole transmission
facturing process. Furthermore, the lower volume of material system.
removed reduces the grinding wheel wear experienced dur- Our iterative approach could be applied to define sev-
ing the finishing phase, increasing process productivity and eral profile modifications for different industrial applications.
reducing production downtime, especially in the mass pro- In particular, the constraints and requirements could be
duction of spur gears. changed, thus obtaining a tooth profile that could improve
several aspects of gear meshing and manufacturing.

Author contribution All the authors contributed to the study concep-

6 Conclusions tualization, validation, writing, reviewing and editing. Methodology,
material and software preparation, investigation, data collection and
An iterative algorithm based on the static transmission error analysis were performed by Michele Abruzzo. The first draft of the
was used to define a profile modification capable of improv- manuscript was written by Michele Abruzzo and all authors commented
on previous versions of the manuscript. All authors read and approved
ing the dynamic behavior of spur gears. The proposed the final manuscript.
approach was applied to a nominal involute profile (type 1)
and to a profile calculated to reduce the maximum contact Funding Open access funding provided by Università di Pisa within the
pressure in one meshing cycle (type 2). In both cases, the cal- CRUI-CARE Agreement. Open access funding provided by Università
di Pisa within the CRUI-CARE Agreement. The authors declare that no
culation was based on the minimization of the peak-to-peak funds, grants, or other support were received during the preparation of
static transmission error, and only three steps were needed to this manuscript.
reach convergence.
The static results show that the proposed modifications Declarations
radically modify the peak-to-peak static transmission error
curves, which present a minimum for the nominal torque
considered. In addition, the maximum contact pressure expe- Competing of interests The authors declare no competing interests.
rienced during a single meshing cycle was reduced with the
Open Access This article is licensed under a Creative Commons
proposed type 2 profile modification. Attribution 4.0 International License, which permits use, sharing, adap-
The proposed modifications can be adopted to reduce the tation, distribution and reproduction in any medium or format, as
dynamic overload and the transmission noise, even outside long as you give appropriate credit to the original author(s) and the
the nominal torque regime. The dynamic overload obtained source, provide a link to the Creative Commons licence, and indi-
cate if changes were made. The images or other third party material
with the modified gear profiles was significantly reduced out- in this article are included in the article’s Creative Commons licence,
side any resonant condition and all the resonance peaks were unless indicated otherwise in a credit line to the material. If material
narrower and less intense compared to the unmodified gears. is not included in the article’s Creative Commons licence and your
In addition, some of the bifurcation phenomena experienced intended use is not permitted by statutory regulation or exceeds the
permitted use, you will need to obtain permission directly from the copy-
by the unmodified gears during the Ramp-Down were elimi- right holder. To view a copy of this licence, visit http://creativecomm
nated. The type 1 profile modification showed a very different ons.org/licenses/by/4.0/.
shape from any industrial standard but obtained a significant
reduction in the dynamic overload and the transmission noise
at any static torque considered, outperforming the typical
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