Torsional Vibration Analysis of Pre-Twisted

Cantilever Beam using FEA

Naveen Kishore.P
Department of Mechanical Engineering
J.B.Institute of Engineering and Technology
Hyderabad, Andhra Pradesh, INDIA.
P.Kiran Prasad
Department of Mechanical Engineering
Kakinada Institute of Science and Technology
Kakinada, Andhra Pradesh, INDIA.
Abstract- Structures having the shape of blades are often found in several practical engineering applications such as
turbines and aircraft rotary wings. For reliable and economic designs of the structures, it is necessary to estimate the
modal characteristics of those structures accurately. Among the dynamic characteristics of these structures, determining
the natural frequencies and associated mode shapes are of fundamental importance in the study of resonant responses. A
single free standing blade can be considered as a pretwisted cantilever beam with a rectangular cross-section. The
torsional vibration of pre twist cantilever beam of rectangular cross section is done so that this resembles to a blade.
The differential equation for the torsional vibration of pre-twisted cantilever beam of rectangular cross section has been
obtained. The beam is considered as Timoshenko beam instead of Euler-Bernoulli Beam or Rayleigh Beam because it will
consider shear correction factor, rotary inertia, warping constant. This is again solved by fem software ANSYS and their
results are compared.
Keywords- Pre-twisted Cantilever beam, Finite Element Analysis (FEA).
I. INTRODUCTION
The torsional vibration of a rotating structure can occur in many engineering applications such as turbo-machinery
blades, slewing robot arms, aircraft propellers, helicopter rotors, and spinning spacecraft. To design these
components, the dynamic characteristic, especially near resonant condition, need to be well examined to assure a
safe operation. Among the dynamic characteristics of these structures, determining the natural frequencies and
associated mode shapes are of fundamental importance in the study of resonant responses. It is very important for
manufacturers of turbo machinery components to know the natural frequencies of the rotor blades, because they
have to make sure that the turbine on which the blade is to be mounted does not have some of the same natural
frequencies as the rotor blade. Otherwise, a resonance may occur in the whole structure of the turbine, leading to
undammed vibrations, which may eventually wreck the whole turbine.
Figure 1. Schematic view of a part of a steamturbine
A single free standing blade can be considered as a pre-twisted cantilever beam with a rectangular cross-section.
Vibration characteristics of such a blade are always coupled between the two bending modes in the flap wise and
chord wise directions and the torsion mode. The problem is also complicated by several second order effects such as
shear deformations, rotary inertia, and fibre bending in torsion, warping of the cross-section, root fixing and Coriolis
accelerations.
International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 73 ISSN: 2278-621X
Figure 2. Pre-Twisted BeamModels
The torsional vibration occurs when the centroid and the shear centre of the cross section of the beam do not
coincide. This lack of coincidence between the centroid and the shear centre occurs when the beam has less than two
axis of symmetry or has anisotropy in the material. This makes the torsional axis different from the elastic axis and
thus causes torsional vibration when flexural vibration occurs. When the beam is isotropic and the cross-section of it
has two axes of symmetry, centroid and shear centre coincides and flexural vibrations and torsional vibration
become independent. The flexural-torsional coupled vibration can be analyzed by combining one of the beam
theories for bending with a torsional theory and a consideration of the various warping effects. The simplest model
for the analysis of coupled bending and torsional vibration is combining the classical Bernoulli- Euler theory for
bending and St. Venant theory for torsion .Inclusion of a warping effect, Bishop and Miao results in a better
approximation, especially for higher modes. Also, for non slender beams, applying the Timoshenko Beam theory
instead of the Bernoulli-Euler theory along with the inclusion of a warping effect can improve the accuracy for
higher modes.
1.1 Types of Vibrations-
Vibration can be defined as regularly repeated movement of a physical object about a fixed point. Vibrations can be
classified based on various factors like
a) Nature of excitation (usually the excitation will be periodic).
b) Nature of displacement.
1.2 Nature of Vibration-
Here the vibration depends on the nature of deformation of the beam, when the external forces act on the system.
These are of two types namely:-
a) Flexural or transverse vibration.
b) Torsional vibration.
1.3 Flexural Vibration or Transverse Vibration-
These vibrations are only due to the bending of the beam. When the centroid and shear centre both coincides and
the load also falls at the same location, then the beam or the physical system is said to have only bending. There are
two components of this bending vibration namely:
a) Transverse vibration or Flexural vibration
b) Longitudinal vibration.
1.4 Torsional Vibration-
In the engineering applications it is always not possible to have the cross section which is symmetric. There can be
components with various cross sections, with different axis of symmetry.
a) Double axis symmetry
b) Single axis symmetry
c) Zero axis symmetry or unsymmetric bodies
1.5 Beam Theory-
The study of the torsional vibration start from the basic beam theories, it is important to review and study the
derivation of the various basic beam theories prior to the study of torsional vibration of beams. Here, basic beam
theories of Euler-Bernoulli, Rayleigh, shear and Timoshenko beam theories are reviewed from their derivation. The
assumptions made by all models are as follows.
1. One dimension (the axial direction) is considerably larger than the other two.
2. The material is linear elastic (Hookean).
International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 74 ISSN: 2278-621X
3. The Poisson effect is neglected.
4. The cross-sectional area is symmetric so that the neutral and centroidal axes coincide.
5. The angle of rotation is small so that the small angle assumption can be used.
Euler beam theory discussed was considered, i.e., shear centre and warping are considered to determine the natural
frequencies of a cantilever beam in this paper.
1.6 The Timoshenko Beam Model-
As mentioned earlier that the Euler Bernoulli beam theory can approximate the natural frequency in case of higher
frequency modes and slender beam, hence Timoshenko beam theory can be implemented in such situations. In this
theory deformation due to transverse shear and kinetic energy due to rotation of the cross-section become important.
Energy expressions include both shear deformation and rotary inertia.
The assumption made in the previous theory that the plane sections which are normal to the undeformed centroidal
axis remain plane after bending, will be retained. However, it will no longer be assumed that these sections remain
normal to the deformed axis

Figure 3. Timoshenko Beam
Assumptions made in Timoshenko Beam theory:-
a) Plane sections such as ab, originally normal to the centreline of the beam in the undeformed geometry,
remain plane but not necessarily normal to the centreline in the deformed state.
b) The cross-sections do not stretch or shorten, i.e., they are assumed to act
Like rigid surfaces.
The Timoshenko beam considers few factors which Euler Bernoulli beam theory does not consider. Hence it can be
predicted that Timoshenko beam can give accurate results while considering higher frequencies and slender beams.
Hence Timoshenko beam is considered in the present project.
II. FINITE ELEMENT SOLUTION
The Finite Element Method (FEM) is a numerical procedure that can be used to obtain solutions to a large class of
engineering problems involving stress analysis, heat transfer, fluid flow and vibration and acoustics.
Figure 4. Description of the finite element
International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 75 ISSN: 2278-621X
Basic ideas of the FEM originated from advances in aircraft structural analysis. The origin of the modern FEM may
be traced back to the early 20
th
century, when some investigators approximated and modelled elastic continua using
discrete equivalent elastic bars. However, Courant has been credited with being the first person to develop the FEM.
He used piecewise polynomial interpolation over triangular sub regions to investigate torsion problems in a paper
published in 1943. The next significant step in the utilization of Finite Element Method was taken by Boeing. In the
1950s Boeing, followed by others, used triangular stress elements to model airplane wings. But the term finite
element was first coined and used by Clough in 1960. And since its inception, the literature on finite element
applications has grown exponentially, and today there are numerous journals that are primarily devoted to the theory
and application of the method.
III. RESULTS AND DISCUSSION
In order to validate the proposed finite element model for the vibration analysis of pre twisted Timoshenko beam,
various numerical results are obtained and compared with available solutions in the published literature.
Mode shape of 0.4 radian twist with b/h =1/4 is shown below



International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 76 ISSN: 2278-621X
In order to validate the proposed finite element model for the vibration analysis of pre twisted Timoshenko beam,
various numerical results are obtained and compared with available solutions in the published literature.
Comparison between ANSYS results and mathematical modelling results
Tab 1. Values of Frequencies for b/h =1/4 and for 0.4 radian twist
Mode
no
Fem solution Mathematical
solution
Difference between the
solutions in %
0 2.4097 E+04 2.43970E+04 1.23E+00
1 2.09927E+05 2.10927E+05 4.74E-01
2 5.70624E+05 5.77624E+05 1.21E+00
3 1.11913E+05 1.14113E+05 1.93E+00
4 1.865250E+06 1.88725E+06 1.17E+00
The various examples are considered to evaluate the present finite element formulation for the effects of related
parameters (e.g. twist angle, length, breadth to depth ratio) on the natural frequencies of the pre-twisted cantilever
Timoshenko beams. The natural frequency ratios for the first five modes of vibration are obtained for different
breadth to height ratio and in each case with different twist angles. Graphs are plotted with frequency and twist and
it can easily be checked out from the Figures that the natural frequencies increase as the twist angle increases.
0.10 0.15 0.20 0.25 0.30 0.35 0.40
0
500000
1000000
1500000
2000000
f
r
e
q
u
e
n
c
y

(
h
e
r
t
z
)
twist (radian)
frequency
frequency
frequency
frequency
frequency
Graph 1. Combined graph for different mode between Frequency Vs twist for b/h ratio =1/4
International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 77 ISSN: 2278-621X
0.10 0.15 0.20 0.25 0.30 0.35 0.40
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
1000000
1100000
1200000
f
r
e
n
q
u
e
n
c
y

(
h
e
r
t
z
)
twist (radian)
frenquency
frenquency
frenquency
frenquency
frenquency
Graph 2. Combined graph for different mode between Frequency Vs twist for b/h ratio =1/5
Graphs between deflection and frequency for different breadth to height ratio and twist angles are shown below

0 50000 100000 150000 200000 250000
0.0042
0.0044
0.0046
0.0048
0.0050
0.0052
0.0054
0.0056
0.0058
d
e
f
l
e
c
t
i
o
n
frequency
deflection
0 500000 1000000 1500000 2000000
0.0038
0.0040
0.0042
0.0044
0.0046
0.0048
0.0050
0.0052
d
e
f
l
e
c
t
i
o
n
frequency
deflection
Graph 3. Deflection Vs frequency for b/h=1/5 and for a twist of
0.4 radians
Graph 4. Deflection Vs frequency for b/h=1/4 and for a twist of
0.4 radians
IV. CONCLUSION
The equations of motion for the torsional vibration analysis of blades, which have a pre-twisted cross-section,
arbitrary orientation are derived.
The equations of motion are transformed into a dimensionless form by employing dimensionless variables and
several dimensionless parameters representing area moment of inertia ratio, the pre-twist angle, are identified.
The Garlekins method used here to solve the equation gives an upper bound of frequencies. The resultant
obtained for various frequencies of torsional vibrations shows that it increases with the amount of pre-twisted
and the thinness of the beam.
Numerical results of frequencies in different cases are validated the applicability of the proposed method for
solving such an engineering problem. The pre-twisted angles influence the natural frequencies of the beams.
(Hertz)
International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 78 ISSN: 2278-621X
The natural frequencies found were compared with the simulated FEM results. The simulation was carried in
software ANSYS
The simulation software provides the investigator with different mode shape and frequency for all the beam
geometry and the resonant data obtained is compared with the mathematical models.
It has been noted form graphs that for all the cases twist with 0.5 radians has maximum natural frequency.
REFERENCES
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nd
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International Journal of Latest Trends in Engineering and Technology (IJLTET)
Vol. 2 Issue 1 January 2013 79 ISSN: 2278-621X