Free vibration analysis of pre-twisted rotating FGM beams
M. N. V. Ramesh • N. Mohan Rao
Received: 10 April 2013 / Accepted: 24 September 2013 / Published online: 11 October 2013
� Springer Science+Business Media Dordrecht 2013
Abstract The natural frequencies of vibration of a
rotating pre-twisted functionally graded cantilever
beam are investigated. Rotating cantilever beam with
pre-twist made of a functionally gradient material
(FGM) consisting of metal and ceramic is considered
for the study. The material properties of the FGM
beam symmetrically vary continuously in thickness
direction from core at mid section to the outer surfaces
according to a power-law form. Equations of motion
for free vibration are derived using Lagrange’s
equation and the natural frequencies are determined
using Rayleigh–Ritz method. The effect of parameters
such as the pre-twist angle, power law index, hub
radius and rotational speed on the natural frequencies
of rotating functionally graded pre-twisted cantilever
beams are examined through numerical studies and
comparison is made with the numerical results
obtained using other methods reported in literature.
The effect of coupling between chordwise and flap-
wise bending modes on the natural frequencies has
also been investigated.
Keywords Functionally graded beam �Rotating pre-twisted beam �Natural frequencies �Flapwise vibrations � Chordwise vibrations
1 Introduction
The concept of functionally graded materials (FGM)
was originally introduced in the year 1984 in Japan by
a group of material science scientists as ultra high
temperature resistant materials. Various rotating
structures, like turbo machinery, wind blades and
aircraft rotary wings are pre-twisted beams that are
often subjected to vibration with larger pressure
loadings, temperature gradients and thermo elastically
induced loadings. Due to the high potentiality of these
materials, study of natural frequencies of these
structures is important during design and analysis
phase for estimating their dynamic behaviour.
During the past decade, many researchers have
focused on the vibration characteristics of stationary
and rotating functionally graded beams employing
various methods of analysis. Among them the salient
works are presented here. Sankar (2001) found an
elasticity solution for transversely loaded functionally
graded beam and a simple Euler–Bernoulli beam
theory was also developed with the assumption that
plane sections remain plane and normal to the beam
axis. Aydogdu and Taskin (2007) developed an
equation of motion for functionally graded beam by
M. N. V. Ramesh (&)
Department of Mechanical Engineering, Nalla Malla
Reddy Engineering College, Hyderabad 500088, India
e-mail: [email protected]
N. Mohan Rao
Department of Mechanical Engineering, JNTUK College
of Engineering Vizianagaram, Vizianagaram 535003,
India
e-mail: [email protected]
123
Int J Mech Mater Des (2013) 9:367–383
DOI 10.1007/s10999-013-9226-x
using Hamilton principle. In these the Young’s
modulus is assumed to vary along the thickness.
Higher order shear deformation theories and classical
beam theories were used for the analysis of the FG
beam. Benatta et al. (2008) presented higher order
flexural theories for short functionally graded sym-
metric beams with three point bending. The governing
equations were derived using the principle of virtual
work. Kadoli et al. (2008) studied the static behavior
of metal ceramic FG beam with the composition
variation based on power law exponent at ambient
temperatures. Higher order beam theory was used to
study the static displacement field components, axial
stress and shear stress distribution in various FGM
beams. Kapuria et al. (2008) validated the static and
free vibration response using modified rule of mixtures
for a third ordered zigzag theory based model on
functionally graded layered beams. Li (2008) pre-
sented new unified approach for analyzing the static
and dynamic behavior of FG beams including the
rotary and shear deformation. A single fourth order
governing partial differential equation was derived
and all physical quantities were obtained from the
solution of the equation. Yang and Chen (2008)
presented a free vibration and elastic buckling analysis
of functionally graded beams with open edge cracks
using Bernoulli beam theory and the rotational spring
model for different end supports.
A solution for free vibration of functionally graded
beams resting on a Winkler–Pasternak elastic founda-
tion based on two dimensional theory of elasticity was
presented by Ying et al. (2008). Mechanical behavior of
a non linear functionally graded cantilever beam
subjected to an end force was investigated by Kang
and Li (2009) using large and small deformation
theories. The young’s modulus of the material is
assumed to vary along the thickness according to a
power law and its effect on the deflections of the beam
was analyzed. Li and Shi (2009) studied the free
vibration of functionally graded piezoelectric material
beam under different boundary conditions using the
state-space based differential quadrature method.
Simsek (2009) adopted Ritz method for static analysis
of a functionally graded beam subjected to uniformly
distributed load. Simsek (2010) also investigated
the vibration analysis of simply supported beam
with moving mass by employing Euler-Ber-
noulli, Timoshenko and shear deformation theory.
Simsek and Kocaturk (2009) investigated free vibration
characteristics and the dynamic behavior of a function-
ally graded simply supported beam with a concentrated
moving harmonic load under the assumption of Euler-
Bernoulli beam theory. Lagrange’s equations were used
to derive system of equations of motion. Sina et al.
(2009) analyzed the free vibration of functionally graded
beams using a new beam theory based on Hamilton’s
principle, that is different from traditional first order
shear deformation beam theory to derive the governing
equations of motion. Alshorbagy et al. (2011) presented
dynamic characteristics of functionally graded beam
whose properties vary axially or transversely through the
thickness based on a power law form. Euler–Bernoulli
beam theory was assumed, to derive system of equations
of motion based on the principle of virtual work.
Fazelzadeh et al. (2007) presented vibration analysis
of a thin walled-blade made of FGMs operating under
high temperature supersonic gas flow. First order shear
deformation theory was used to derive the governing
equations by considering the effects of rotary inertias
and presetting angle. Thin walled rotating blades made
of FGMs used in turbo-machines subjected to aerother-
moelastic loads were analysed as beams using first order
shear deformation theory by Fazelzadeh and Hosseini
(2007). Yoo et al. (1995) and Yoo and Shin (1998)
presented free vibration analysis of a homogeneous
rotating beam. Piovan and Sampaio (2009) developed a
nonlinear beam model to study the influence of graded
properties on the damping effect and geometric stiff-
ening of a rotating beam. Naguleswaran (1994) eval-
uated lateral vibrations of doubly symmetric Euler–
Bernoulli beam for different end conditions.
The constant requirements of the industry force the
engineering community in quest of new concepts and
new strategies in order to improve the structural
response of structures as well as to enhance the
strength of materials. This is particularly essential in
the case of rotating beams that are subjected to severe
vibrations with large pressure loadings, high rotating
accelerations, centrifugal forces, geometric stiffening,
among others. The earlier works are mainly confined
to stationary beams and on homogeneous and FGMs.
However studies on rotating beams made of FGMs are
scarcely reported. To the authors’ knowledge, the
effect of rotating speed, centrifugal force, hub radius,
material morphology on natural frequencies of func-
tionally graded rotating beam with pre-twist is not
reported in the literature. In this paper, the dynamic
modeling method which was proposed by Yoo et al.
368 M. N. V. Ramesh, N. Mohan Rao
123
(2001) for homogeneous rotating cantilever using the
hybrid deformation variables is extended for estimat-
ing natural frequencies of a functionally graded
rotating pre-twisted cantilever beam. The equations
of motion are derived for both stretching and bending
configurations and are simplified for bending config-
uration by neglecting the coupling effect between
stretching and the bending motion as the coupling
effect becomes negligible for slender beams. The
effect of pre-twist angle, power law index, angular
speed and hub radius on the natural frequencies of a
functionally graded rotating beam is investigated for
bending configuration in chordwise and flapwise
modes through numerical studies and comparison is
made with the results reported in available literature.
An attempt is also made to study the coupling effect
between the chordwise and flapwise bending motions.
2 Functionally graded beam
Consider a functionally graded beam with length L,
width b and total thickness h and composed of a
metallic core and ceramic surfaces as shown in Fig. 1.
The graded material properties vary symmetrically
along the thickness direction from core towards
surface according to a power law given below:
P zð Þ ¼ P mð Þ þ P cð Þ �P mð Þ� � 2� z
h
����
����
n
ð1Þ
where PðzÞ represents an effective material property
(i.e., density, q or Young’s modulus, E), PðmÞ and
PðcÞ are metallic and ceramic properties respectively.
The volume fraction exponent or power law index, n is
a variable whose value is C0 and the variation in
properties of the beam depends on its magnitude. The
functionally graded beam is made of temperature
resistant ceramic at the surface (at z = ?h/2 and -h/2)
and tough metallic core (at z = 0), in which the
proportion of the metal to ceramic symmetrically
varies as per power law given above.
3 Equation of motion
For the problem considered in this study, the equations
of motion are obtained under the assumptions namely
the material properties vary only along the thickness
direction according to a power law, the neutral and
centroidal axes in the cross section of the rotating
beam coincide so that effects due to eccentricity and
torsion are not considered. Shear and rotary inertia
effects of the beam are neglected as beam has slender
shape and the pre-twist angle varies continuously from
the hub to the free end of the beam.
Figure 2 shows the deformation of the neutral axis
of a beam fixed to a rigid hub rotating about the axis k:
The angle of pre-twist, a varies continuously from hub
to tip (i.e., at rigid hub), where i; j and k are attached,
towards free end of the F G beam while l; m and n are
attached at free end. No external force acts on the FG
beam and the beam is attached to a rigid hub which
rotates with constant angular speed. The rotation of the
beam is characterized by means of an assumed rotation
Fig. 1 Geometry of functionally graded beam
Free vibration analysis 369
123
X (t) around the k-axis. The position of a generic point
on the neutral axis of the FG beam located at P0
(before deformation) changes to P (after deformation)
and its elastic deformation is denoted as d that has
three components in three dimensional space. Con-
ventionally the differential equations of motion are
derived by approximating the three Cartesian vari-
ables, u, v and w. In the present work, a hybrid set of
Cartesian variables v and w and a non Cartesian
variable s are approximated by spatial functions and
corresponding coordinates are employed to derive the
equations of motion.
3.1 Approximation of deformation variables
By employing the Rayleigh–Ritz method, the defor-
mation variables are approximated as follows:
sðx; tÞ ¼Xl1
j¼1
/1jðxÞq1jðtÞ ð2Þ
vðx; tÞ ¼Xl2
j¼1
/2jðxÞq2jðtÞ ð3Þ
wðx; tÞ ¼Xl3
j¼1
/3jðxÞq3jðtÞ ð4Þ
In the above equations, /1j,/2j and /3j are the
assumed modal functions for s, v and w respectively.
Any compact set of functions which satisfy the
essential boundary conditions of the cantilever beam
can be used as the test functions. The qij’s are the
generalized coordinates and l1, l2 and l3 are the
number of assumed modes used for s, v and w respec-
tively. The total number of modes, l, equal to the sum
of individual modes i.e., l = l1 ? l2 ? l3
The geometric relation between the arc length
stretch s and Cartesian variables u, v and w is given
(Yoo et al. 1995) as
s ¼ uþ 1
2
Zx
0
v0
� �2
þ w0
� �2� �
dr ð5Þ
or
u ¼ s� 1
2
Zx
0
v0
� �2
þ w0
� �2� �
dr ð6Þ
where a symbol with a prime (0) represents the partial
derivative of the symbol with respect to the integral
domain variable (i.e., a dummy variable) r.
3.2 The kinetic energy of the system
The velocity of a generic point P can be obtained as
v~P ¼ v~O þAdp~
dtþ x~A � p~ ð7Þ
where v~o is the velocity of point O that is a reference
point identifying a point fixed in the rigid frame A; x~o
angular velocity of the frame A; and P~ is the vector
Fig. 2 Configuration of the
functionally graded rotating
beam
370 M. N. V. Ramesh, N. Mohan Rao
123
from point O to Po the termAdp~dt
is the time derivative of
vector P~ in the reference frame A and the terms P~; v~o
and x~A can be expressed as follows
P~ ¼ xþ uð Þiþ vjþ wk ð8Þ
v~O ¼ rXj ð9Þ
x~A ¼ Xk ð10Þ
v~p ¼ ð _u� XvÞiþ ½ _vþ Xðr þ xþ uÞ�jþ _wk ð11Þ
where i; j and k are orthogonal unit vectors fixed in A
and r is the distance from the axis of rotation to point O
(i.e., radius of the rigid frame) and X is the angular
speed of the rigid frame.
Using the Eq. (7), the kinetic energy of the rotating
beam is derived as
T ¼ 1
2
Z
v
Jq11
Av~:v~dv ð12Þ
where
Jq11 ¼
Z
A
q zð ÞdA ð13Þ
In which, A is the cross section, qðzÞ is the mass
density per unit volume of the functionally graded
beam, V is the volume.
Substituting the Eqs. (2) to (4) into Eq. (12) and
taking partial derivatives of T with respect to q1j and _qij
and neglecting the higher order non-linear terms, the
following equations are obtained.
oT
oq1i
¼ XXl2
j¼1
ZL
0
Jq11/1i/2j _q2jdx
þ X2Xl1
j¼1
ZL
0
Jq11/1i/1jq1jdx
þ X2
ZL
0
Jq11 r þ xð Þ/1idx ð14Þ
oT
oq2i
¼ X2Xl2
j¼1
ZL
0
Jq11/2i/2jq2jdx
� XXl1
j¼1
ZL
0
Jq11/2i/1j _q1jdx
� X2Xl2
j¼1
ZL
0
Jq11r L� xð Þ/02i/
0
2jq2jdx
2
4
þ 1
2
Xl2
j¼1
ZL
0
Jq11 L2 � x2� �
/0
2i/0
2jq2jdx
3
5
ð15Þ
oT
oq3i
¼ �X2Xl3
j¼1
ZL
0
Jq11r L� xð Þ/03i/
0
3jq3jdx
2
4
þ 1
2
Xl3
j¼1
ZL
0
Jq11 L2 � x2� �
/0
3i/0
3jq3jdx
3
5
ð16Þ
d
dt
oT
o _q1i
¼Xl1
j¼1
ZL
0
Jq11/1i/1j €q1jdx
� XXl2
j¼1
ZL
0
Jq11/1i/2j _q2jdx ð17Þ
d
dt
oT
o _q2i
¼Xl2
j¼1
ZL
0
Jq11/2i/2j €q1jdx
þ XXl1
j¼1
ZL
0
Jq11/2i/1j _q1jdx ð18Þ
d
dt
oT
o _q3i
¼Xl3
j¼1
ZL
0
Jq11/3i/3j €q3jdx ð19Þ
3.3 Strain energy of the system
Based on the assumptions, the total elastic strain
energy of a functionally graded beam can be written as
U ¼ 1
2EðZÞA
Z
L
ds
dx
2
dxþ 1
2EðzÞIð3Þ
Z
L
d2v
dx2
2
dx
þ 1
2EðzÞIð2Þ
Z
L
d2w
dx2
2
dx
þ EðzÞIð23Þ
Z
L
d2v
dx2
d2w
dx2
dx
ð20Þ
where E zð Þ is the Young’s modulus, A is the cross
sectional area of the beam, Ið2Þ; Ið3Þ and Ið23Þ are second
Free vibration analysis 371
123
area moments of inertia and second area product of
inertia of the cross section, L is length of the beam.
Using principal moments of cross sections, JE22;yy
and JE22;zz; the flexural rigidities of the functionally
graded beam can be expressed as
EðzÞIð2Þ ¼JE
22;yy þ JE22;zz
2þ
JE22;yy � JE
22;zz
2cosð2hÞ
ð21Þ
EðzÞIð3Þ ¼JE
22;yy þ JE22;zz
2�
JE22;yy � JE
22;zz
2cosð2hÞ
ð22Þ
EðzÞIð23Þ ¼JE
22;yy � JE22;zz
2sinð2hÞ ð23Þ
where h is pre-twist angle of a cross section with
reference to the fixed end of the functionally graded
beam given by h ¼ ax
L; and ð24Þ
JE11 ¼
Z
A
E zð ÞdA; ð25Þ
JE22;yy ¼
Z
A
E zð Þz2dA ð26Þ
JE22;zz ¼
Z
A
E zð Þy2dA ð27Þ
Substituting Eqs. (2–4) in strain energy expression
of Eq. (20), results in
U ¼ 1
2EðZÞA
Z
L
Xl1
j¼1
/0
1jq1j
!2
dx
þ 1
2EðzÞIð3Þ
Z
L
Xl2
j¼1
/00
2jq2j
!2
dx
þ 1
2EðzÞIð2Þ
Z
L
Xl3
j¼1
/00
3jq3j
!2
dx
þ EðzÞIð23Þ
Z
L
Xl2
j¼1
/00
2jq2j
!Xl3
j¼1
/00
3jq3j
!
dx
ð28Þ
The partial derivatives of strain energy with respect
to q1i, q2i and q3i are
oU
oq1i
¼ EðzÞAXl1
j¼1
ZL
0
/0
1i/0
1jq1jdx ð29Þ
oU
oq2i
¼ EðzÞIð3ÞXl2
j¼1
ZL
0
/00
2i/00
2jq2jdx
þ EðzÞIð23ÞXl3
j¼1
ZL
0
/00
2i/00
3jq3jdx ð30Þ
oU
oq3i
¼ EðzÞIð2ÞXl3
J¼1
ZL
0
/00
3i/00
3jq3jdx
þ EðzÞIð23ÞXl2
J¼1
ZL
0
/00
3i/00
2jq2jdx ð31Þ
The Lagrange’s equations of motion for free
vibration of distributed parameter system can be
obtained as
d
dt
oT
o _qi
� oT
oqi
þ oU
oqi
¼ 0 i ¼ 1; 2; 3. . .l ð32Þ
The linearized equations of motion can be obtained
as follows
Xl1
j¼1
ZL
0
Jq11/1i/1jdx
0
@
1
A€qj
� X2Xl1
j¼1
ZL
0
Jq11/1i/1jdx
0
@
1
Aq1j
þ EðzÞAXl1
j¼1
ZL
0
/0
1i/0
1jdx
0
@
1
Aq1j
� 2XXl2
j¼1
ZL
0
Jq11/1i/2jdx
0
@
1
A _q2j
¼ X2
ZL
0
Jq11x/1idxþ rX2
ZL
0
Jq11/1idx ð33Þ
372 M. N. V. Ramesh, N. Mohan Rao
123
Xl2
j¼1
ZL
0
qðzÞ/2i/2jdx
0
@
1
A€q2j
þ 2XXl1
j¼1
ZL
0
Jq11/2i/1jdx
0
@
1
A _q1j
þEðzÞIð3ÞXl2
j¼1
ZL
0
/00
2i/00
2jdx
0
@
1
Aq2j
þX2Xl2
j¼1
r
ZL
0
Jq11 L� xð Þ/02i/
0
2jdx
0
@
1
A
8<
:
þZL
0
Jq11
2L2 � x2� �
/0
2i/0
2jdx
0
@
1
A�ZL
0
Jq11/2i/2jdx
0
@
1
A
9=
;q2j
þEðzÞIð23ÞXl3
j¼1
ZL
0
/00
2i/00
3jdx
0
@
1
Aq3j ¼ 0
ð34Þ
Xl3
j¼1
Z1
0
Jq11/3i/3jdx
0
@
1
A€q3j þ EðzÞIð2ÞXl3
j¼1
Z1
0
/00
3i/00
3jdx
0
@
1
Aq3j
þEðzÞIð23ÞXl2
j¼1
Z1
0
/00
3i/00
2jdx
0
@
1
Aq2j
þX2Xl3
j¼1
r
ZL
0
Jq11 L� xð Þ/03i/
0
3jdx
0
@
1
A
8<
:
þZL
0
L2 � x2� �
/0
3i/0
3jdx
0
@
1
A
9=
;q3j ¼ 0
ð35Þ
where a symbol with double prime (00) represents the
second derivative of the symbol with respect to the
integral domain variable.
3.4 Dimensionless transformation
For the analysis, the equations in dimensionless form
may be obtained by introducing following dimension-
less variables in the equation.
s,t
Tð36Þ
n,x
Lð37Þ
hj,qj
Lð38Þ
d,r
Lð39Þ
c, TX ð40Þ
k,JE
22;yy
JE22;zz
ð41Þ
where s, d, c and k are dimensionless time, hub radius
ratio, dimensionless angular speed and flexural rigid-
ity modulus ratio respectively.
For calculating rigidity modulus, using Eq. (41), the
Eqs. (21–23) can be written as
EðzÞIð2ÞJE
22;zz
¼ 1
2k þ 1ð Þ � 1
2k � 1ð Þ cosð2anÞ ð42Þ
EðzÞIð3ÞJE
22;zz
¼ 1
2k þ 1ð Þ þ 1
2k � 1ð Þ cosð2anÞ ð43Þ
EðzÞIð23ÞJE
22;zz
¼ 1
2k � 1ð Þ sinð2anÞ ð44Þ
4 Free vibration analysis
The Eq. (34) governs the chordwise bending vibration
of the functionally graded rotating beam which is
coupled with the Eqs. (33) and (35). With the
assumption that the natural frequencies of stretching
motion are far greater than the bending motion and the
coupling effect become negligible for the slender
beams (Yoo and Shin 1998), the stretching equation of
motion and coupling effect between the stretching and
bending motions are ignored.
4.1 Determination of natural frequencies
neglecting coupling terms in bending motions
Coupling terms involved between chordwise and
flapwise bending equations in Eqs. (34) and (35) are
neglected and are used to evaluate chordwise and
flapwise bending natural frequencies.
(a) Chordwise bending natural frequencies
Free vibration analysis 373
123
The Eq. (34) can be modified as
Xl2
j¼1
Z1
0
Jq11/2i/2jdx
0
@
1
A€q2j�X2
Z1
0
Jq11/2i/2jdx
0
@
1
Aq2j
2
4
þ EðzÞIð3Þ
Z1
0
/00
2i/00
2jdx
0
@
1
Aq2j
þX2 r
Z1
0
Jq11 L� xð Þ/02i/
0
2jdx
0
@
1
A
8<
:q2j
þZ1
0
Jq11
2L2� x2� �
/0
2i/0
2jdx
0
@
1
Aq2j
9=
;¼ 0
ð45ÞThe Eq. (45) involves the parameters L, X, k, a, x
and E(z), q(z), which are the properties may vary along
the transverse direction of the beam.
After introducing the dimensionless variables from
Eq. (36) to Eq. (41) in Eq. (45), the equation modifies
to
Xl2
j¼i
Z 1
0
waiwbjdf
€h2j þ
Xl2
j¼1
Z 1
0
1
2k þ 1ð Þ
��
� 1
2k � 1ð Þ cosð2anÞ
�w00
aiw00
bjdf
�h2j
þc2Xl2
j¼1
dZ1
0
1� fð Þw0aiw0
bjdf
0
@
8<
:
þ 1
2
Z1
0
1� f2� �
w0
aiw0
bjdf
1
Ah2j
�Z1
0
waiwbjdf
0
@
1
Ah2j
9=
;
3
5 ¼ 0
ð46Þ
where
T ,J
q11L4
JE22;ZZ
!12
ð47Þ
Eq. (46) can be written as
Xl2
j¼1
M22ij
h€hj þ KB2
ij hj þ c2 KG2ij �M22
ij
� �hj
i¼ 0 ð48Þ
where
Mabij ,
Z1
0
waiwbjdf ð49Þ
KBaij ,
Z1
0
1
2k þ 1ð Þ � 1
2k � 1ð Þ cosð2anÞ
� �w00
aiw00
ajdn
ð50Þ
KGaij ,
Z1
0
1� nð Þw0aiw0
ajdnþ 1
2
Z1
0
1� n2� �
w0
aiw0
ajdn
ð51Þ
where wai is a function of n has the same functional
value of x.
From Eq. (48), an eigenvalue problem can be
derived by assuming that h’s are harmonic functions of
s expressed as
h ¼ ejxsH ð52Þ
where j is the imaginary number, x is the ratio of the
chordwise bending natural frequency to the reference
frequency, and H is a constant column matrix
characterizing the deflection shape for synchronous
motion and this yields
x2MH ¼ KCH ð53Þ
where M is Mass matrix and KC stiffness matrix which
consists of elements that are defined as
Mij,M22ij ð54Þ
KCij ,KB2
ij þ c2 KG2ij �M22
ij
� �ð55Þ
(b) Flapwise bending natural frequencies
The Eq. (35) can be modified as
Xl3
j¼1
Z1
0
Jq11/3i/3jdx
0
@
1
A€q3j þ EðzÞI 2ð ÞXl3
j¼1
Z1
0
/00
3i/00
3jdx
0
@
1
Aq3j
þ X2Xl3
j¼1
r
Z1
0
Jq11 L� xð Þ/03i/
0
3jdx
0
@
1
A
8<
:
þZ1
0
Jq11
2L2 � x2� �
/0
3i/0
3jdx
0
@
1
A
9=
;q3j ¼ 0
ð56Þ
374 M. N. V. Ramesh, N. Mohan Rao
123
The Eq. (56) involves the parameters L, X, k, a, x
and E(z), q(z), which are the properties may vary along
the transverse direction of the beam.
The equations in dimensionless form may be
obtained by introducing the dimensionless variable
from Eq. (36) to (41) in Eq. (56) and the equation
results
Xl3
j¼i
Z1
0
waiwbjdf
0
@
1
A€h3j
þXl3
j¼1
Z1
0
1
2k þ 1ð Þ þ 1
2k � 1ð Þ cosð2anÞ
w00
aiw00
bjdf
� �h3j
c2Xl3
j¼1
dZ1
0
1� fð Þw0aiw0
bjdf
0
@
1
A
8<
:
þ 1
2
Z1
0
w0
aiw0
bjdf
0
@
1
A
9=
;h3j ¼ 0
ð57ÞEquation (57) can be written as
Xl3
j¼1
M33ij
h€h3j þ KB3
ij h3j þ c2KG3ij h3j
i¼ 0 ð58Þ
where
T,J
q11L4
JE22;zz
!12
ð59Þ
Mabij ,
Z1
0
waiwbjdn ð60Þ
KBaij ,
Z1
0
1
2k þ 1ð Þ þ 1
2k � 1ð Þ cosð2anÞ
� �w00
aiw00
ajdn
ð61Þ
KGaij ,
Z1
0
1� nð Þw0aiw0
ajdnþ 1
2
Z1
0
1� n2� �
w0
aiw0
ajdn
ð62ÞFrom Eq. (58), the eigen value problem for the
flapwise bending vibration of functionally graded
rotating cantilever beam can be formulated as
x2MH ¼ KFH ð63Þ
where M and KF are defined as
Mij,M33ij ð64Þ
KFij ,KB3
ij þ c2KG3ij ð65Þ
4.2 Determination of natural frequencies including
the coupling terms in bending motions
Equations (34) and (35), are used to estimate the
natural frequencies of rotating functionally graded
pre-twisted beam. Coupling terms involved between
chordwise and flapwise bending equations are con-
sidered to evaluate natural frequencies called coupled
natural frequencies in this section.
These equations are expressed in matrix form as
M€hþ Kh ¼ 0 ð66Þ
where
M � M22 0
0 M33
� �ð67Þ
K � K22 K23
K32 K33
� �ð68Þ
h � h2
h3
� �ð69Þ
In Eqs. (67) and (69), M22 and h2 are matrices
which are composed of M22ij and h2j respectively. The
sub matrices in matrix K are defined as
K22ij , c2 KGB2
ij þ dKGA2ij � M22
ij
� �þ KB2
ij
h ið70Þ
K23ij ,KB23
ij ð71Þ
K32ij ,KB32
ij ð72Þ
K33ij , c2 KGB3
ij þ dKGA3ij
� �þ KB3
ij
h ið73Þ
where
Mabij ,
Z1
0
waiwbjdn ð74Þ
KGAaij ,
Z1
0
1� nð Þw0aiw0
ajdn ð75Þ
Free vibration analysis 375
123
KGB2ij ,
1
2
Z1
0
1� n2� �
w0
aiw0
ajdn ð76Þ
KB2ij ,
Z1
0
1
2k þ 1ð Þ þ 1
2k � 1ð Þ cosð2anÞ
� �w00
aiw00
ajdn
ð77Þ
KBabij ,
Z1
0
1
2k � 1ð Þ sinð2anÞw00aiw
00
bjdn ð78Þ
The natural frequencies of a rotating functionally
graded pre twisted beam can be obtained from solution
of Eq. (66).
5 Numerical results and discussion
The procedures presented above are demonstrated
through numerical examples for rotating functionally
graded cantilever pre-twisted beam with the properties
given in the Table 1.
5.1 Numerical example 1
In this example the chordwise bending natural fre-
quencies are determined for a homogeneous metallic
beam, ifPðmÞ = PðcÞ in Eq. 1, having geometrical
dimensions breadth = 22.12 mm, height = 2.66 mm
and length = 152.40 mm. At zero rotational speed,
with clamped-free (clamped at x = 0 and free at
x = L) boundary conditions, the chordwise bending
frequencies are calculated with ten assumed modes to
obtain the three lowest natural frequencies. The
procedure presented in this paper does not consider
rotary inertia effects and the results obtained closely
match with those based on finite element method that
includes rotary inertia effects and the variation is
within 0.2 %. It is also observed that the present
results are in agreement with a variation less than
Table 1 Properties of metallic (steel) and ceramic (alumina)
materials
Properties of materials Steel Alumina (Al2O3)
Young’s modulus, E (GPa) 214.00 390.00
Material density, q (kg/m3) 7800.00 3200.00
Table 2 Comparison of
natural frequencies of a
metallic (steel) cantilever
beam (Hz)
Present
approach
Ref. (Piovan and
Sampaio 2009) (FEM)
Ref. (Piovan and
Sampaio 2009)
(experimental)
% variation (with
experimental)
96.9 97.0 97.0 0.10
607.3 607.5 610.0 0.44
1700.4 1697.0 1693.0 0.44
Table 3 Comparison of the first chordwise bending natural frequencies and the first flapwise bending natural frequencies at n = 0,
4, 8, a = 0.0, d = 0.0, 0.5, 2.0 and k = 1.0
N (rps) d Chordwise Flapwise from Eq. (36)
n 0 4 8 0 4 8
2 0.0 35.67792 19.46406 18.30947 35.73393 19.56654 18.41838
0.5 35.72192 19.54458 18.39505 35.77786 19.64665 18.50345
2.0 35.85359 19.78417 18.64939 35.90933 19.885 18.75633
25 0.0 37.26551 22.05061 21.00614 44.87447 33.33511 32.65361
0.5 43.3133 31.10963 30.36529 50.01042 39.91001 39.33257
2.0 57.72327 49.00565 48.5085 62.9045 55.01413 54.57174
50 0.0 41.1941 26.48727 25.87316 64.7839 56.77268 56.2976
0.5 60.15645 51.10753 50.55035 78.22275 71.49811 71.1009
2.0 96.63529 90.70758 90.34597 108.8043 103.5754 103.2589
376 M. N. V. Ramesh, N. Mohan Rao
123
0.5 % with the experimental results reported by
Piovan and Sampaio (2009) presented in Table 2.
5.2 Numerical example 2
In this example, a functionally graded non rotating
beam without pre-twist with dimensions: length =
1000 mm, breadth = 20 mm and height = 10 mm is
considered for the analysis. Steel is considered as
metallic constituent and Alumina as ceramic constit-
uent whose mechanical properties are given in
Table 1. The variation of the lowest three chordwise
bending natural frequencies of a functionally graded
beam with respect to variation in power law index, n is
presented in Fig. 3 and compared with those reported
by Piovan and Sampaio (2009). It has been observed
that the variation of the three natural frequencies with
respect to power law index, n follows the same trend as
that of the results obtained by finite element method
(Piovan and Sampaio 2009). The three natural fre-
quencies decreased rapidly with an increase in power
law index up to a certain value and there after the
frequencies are relatively un-effected by change in
n value.
From the above examples, it is inferred that the
present analysis is appropriate and hence the same
approach has been adopted for detailed study on the
beam having same physical dimensions and properties
considered in example 2.
The influence of pre-twist angle on first three
chordwise bending natural frequencies at various
angular speeds has been examined and presented in
Figs. 4, 5, and 6. It has been observed that the
chordwise bending natural frequencies increase with
an increase in angular speed. It is to be noted that as the
pre-twist angle increases the chordwise bending
natural frequencies decrease and the extent of reduc-
tion is pronounced at higher pre-twist angles and also
at higher modes (At zero angular speed, the percentage
reduction in first mode is 6.38, in second mode is 20.58
and in third mode is 24.69). However, this feature is
less pronounced at higher angular speed as the
Fig. 4 Variation of first chordwise bending natural frequency
of rotating functionally graded pre-twisted beam
Fig. 3 Chordwise bending natural frequency variation of non
rotating functionally graded beam without pre-twist
Fig. 5 Variation of second chordwise bending natural fre-
quency of rotating functionally graded pre-twisted beam
Free vibration analysis 377
123
frequencies converge at higher angular speeds. This
phenomenon could be attributed to increase in
centrifugal inertia force at higher angular speeds.
In Fig. 7, the relation between chordwise bending
natural frequencies and power law index, n has been
examined at constant angular speed of the beam and at
k = 0.25 and d = 1.0, by varying the pre-twist angle
between 0� and 45� at an interval of 15�. It has been
observed that the pre-twist angle has negligible
influence on first frequency while there is marginal
influence on second frequency and noticeable influ-
ence on third frequency. As the pre-twist angle
increases, there is an increase in frequency at all
values of power law index. At lower n values the
frequency decrease drastically and they become
asymptotic to the power law index axis. As the beam
approaches metallic composition marginal change
occurs in frequencies. It is also to be noted that with an
increase in pre-twist angle the difference in the
frequencies increases (notice the gap between the
different lines of 1st, 2nd and 3rd frequencies).
In Table 3, the chordwise bending natural frequen-
cies are compared with the flapwise bending natural
frequencies. The chordwise natural frequencies
obtained from Eq. 48 match well with those obtained
Fig. 6 Variation of third chordwise bending natural frequency
of rotating functionally graded pre-twisted beam
Fig. 7 Variation of chordwise bending natural frequencies with
power law index
Fig. 8 Variation in flapwise and chordwise bending natural
frequencies for k\1
Fig. 9 Variation in chordwise and flapwise bending natural
frequencies for k[1
378 M. N. V. Ramesh, N. Mohan Rao
123
from the relation between chordwise and flapwise
natural frequencies that originally proposed by Na-
guleswaran (1994).
Figures 8 and 9 show the variation of chaordwise
and flapwise bending natural frequencies with an
increase in angular speed of the functionally graded
pre-twisted rotating beam. For a beam having
modulus ratio (k) \ 1, at lower speeds the chordwise
bending frequencies are higher as compared to
flapwise bending natural frequencies. However, as
the speed increases, there is a tendency for the
flapwise bending frequencies to overtake chordwise
bending frequencies. The rate of increase in fre-
quency, i.e., change in frequency for a change in
angular speed, is lower at lower speeds as compared
that at higher speeds. In addition, at higher speeds
the rate of increase in frequencies tends to remain
constant. For the beam having, modulus ratio
(k) [ 1, flapwise natural bending frequencies are
higher than chordwise bending natural frequencies at
all speeds. However, it may be noted that the
differences in the frequencies gets reduced at higher
angular speeds.
Figure 10 shows the relation between flapwise
bending natural frequencies and power law index, n at
constant angular speed of the beam, at k = 4.0 and
d = 1.0. Variation of pre-twist angle between 0� and
45� at an interval of 15� has negligible influence on
first frequency while there is marginal influence on
second frequency and noticeable influence on third
frequency, in that as the pre-twist angle increases,
frequency decreases at all values of power law index.
At lower n values the frequencies decrease drastically
and they become asymptotic to the power law index
axis. It is also to be noted that with an increase in pre-
twist angle there is a rapid decrease in the magnitude
of the natural frequency, which is evident from the
widening of the interval between the lines of third
natural frequency of Fig. 10.
Figure 11 shows the influence of d (hub radius
ratio) on the relation between frequency and angular
speed. It is observed that as the value of d increases
(increase in hub radius), the frequency increases. InFig. 10 Flapwise natural frequency variation of rotating
functionally graded pre-twisted beam
Table 4 Comparison of the first chordwise bending natural frequencies with and without coupling terms at n ¼ 0; 4;
8; a¼ 300; d¼ 0:0; 0:5; 2:0; k ¼ 0:25
N (rps) d Chordwise (without coupling terms) Chordwise (with coupling terms) % variation for different n values*
n 0 1 2 0 1 2 0 1 2
2 0.0 35.45 23.80 21.19 35.31 23.71 21.11 0.40 0.38 0.38
0.5 35.49 23.87 21.27 35.35 23.78 21.18 0.40 0.38 0.42
2.0 35.63 24.07 21.49 35.49 23.97 21.41 0.40 0.42 0.37
25 0.0 37.07 26.07 23.66 37.16 25.92 23.58 -0.22 0.58 0.34
0.5 43.16 34.12 32.30 43.23 34.02 32.25 -0.16 0.29 0.15
2.0 57.61 51.06 49.81 57.67 51.00 49.78 -0.10 0.12 0.06
50 0.0 41.07 30.78 28.50 40.99 30.76 28.48 0.19 0.06 0.07
0.5 60.08 53.36 52.01 60.04 53.35 52.00 0.07 0.02 0.02
2.0 96.59 92.17 91.30 96.57 92.17 91.30 0.02 0.00 0.00
� fwoc�fwc
fwoc100
� �; fwoc—frequency without coupling terms and fwc—frequency with coupling terms
Free vibration analysis 379
123
Table 5 Comparison of the first flapwise bending natural frequencies with and without coupling terms at
n ¼ 0; 4; 8;a¼ 300; d¼ 0:0; 0:5; 2:0; k ¼ 0:25
N (rps) d Flapwise (without coupling terms) Flapwise (with coupling terms) % variation for different n values*
n 0 1 2 0 1 2 0 1 2
2 0.0 18.36 12.43 11.11 18.01 12.20 10.90 1.94 1.88 1.93
0.5 18.45 12.56 11.25 18.10 12.32 11.04 1.93 1.95 1.90
2.0 18.70 12.92 11.66 18.35 12.70 11.46 1.90 1.73 1.74
25 0.0 32.53 29.42 28.82 32.23 29.41 28.81 0.93 0.03 0.03
0.5 39.22 36.61 36.11 39.00 36.66 36.12 0.56 -0.14 -0.03
2.0 54.48 52.47 52.07 54.35 52.50 52.08 0.24 -0.06 -0.02
50 0.0 56.15 53.91 53.44 56.17 53.91 53.44 -0.03 0.00 0.00
0.5 70.98 69.11 68.73 70.99 69.11 68.73 -0.01 0.00 0.00
2.0 103.1 101.6 101.3 103.17 101.67 101.3 0.00 0.00 0.00
Fig. 11 Variation in chordwise and flapwise bending natural frequencies at different hub radius ratios for k\1
380 M. N. V. Ramesh, N. Mohan Rao
123
addition, with an increase in angular speed, there is a
tendency for convergence of frequencies.
The relation between power law index, n and hub
radius ratio,d at various speeds for chordwise fre-
quencies with and without coupling effect is presented
in Table 4. It is observed from the table that the
variation in frequencies with variation in ‘n’ reduces
with an increase in rotational speed. It has also been
observed that coupling effect gets reduced with an
increase in rotational speed of the beam. Similar trends
have been observed in respect of flapwise bending
frequency (Table 5).
Figure 12 shows the influence of pre-twist angle on
the relation between frequency and angular speed.
d = 1.0 and n = 2.0. It is observed that, as the speed
increases, natural frequency of vibration increases. ItFig. 12 Variation of coupled natural frequencies with angular
speed
Fig. 13 Veering between first and second coupled natural frequencies when power law index, n = 0.0, 2.0, 4.0 and 8.0
Free vibration analysis 381
123
is interesting to note that gradual veering occurs
between the first and second natural frequencies. The
veer occurs for the beam with smaller pre-twit angle at
lower speeds than the beam with higher pre twist
angle. It is also to be noted that at the veer region the
first and second frequencies are widely separated in
respect of higher pre-twist angle than in the case of
lower pre-twist angle.
Figure 13 shows the variation of natural frequen-
cies of rotating functionally graded pre-twisted beam
with angular speed. The speed at which veering occurs
varies with a change in power law index in that at
higher ‘n’ values veering occurs at lower angular
speeds. It is also observed that the gap between the loci
decreases at the location of veering as power law index
increases.
6 Conclusion
In this work, the equations of motion for rotating
functionally graded cantilever pre-twisted beam
attached to a rigid hub are derived. The effects of
pre-twist angle, power law index, angular speed and
hub radius on the natural frequencies of a functionally
graded rotating pre-twisted beam are investigated for
bending configuration in chordwise and flapwise
modes through numerical study. The results are
compared with those reported in the literature based
on experimental and finite element methods. The
effect of coupling between chordwise and flapwise
bending modes on the natural frequencies has also
been investigated.
The salient observations from the study are: (i) the
chordwise natural frequency increases with an
increase in angular speed due to increase in centrifugal
inertia force. (ii) chordwise natural frequencies of
functionally graded rotating beam decrease rapidly
when the beam composition changes from ceramic to
ceramic-metal structure. As the beam composition
approaches that of metal, the frequency change is
marginal. The similar observations have been found in
respect of flapwise bending frequencies. (iii) when
k \ 1, the chordwise bending frequencies are higher
as compared to flapwise bending natural frequencies,
while an opposite trend has been observed when
k [ 1. (iv) the influence of pre-twist angle is signif-
icant on the higher order frequencies as compared to
that of lower order in both chordwise and flapwise
bending natural frequencies. In addition, with an
increase in pre-twist angle the difference in frequen-
cies increase. (v) It has been observed that veering
occurs in first and second natural frequencies when
angular speed increases. At lower power law index
veering occurs at higher speeds and pre-twist angle has
an influence on the loci veering. The procedure
presented in this paper not considered the shear effect,
the accuracy of the results may improve by consider-
ing the shear effects. The modeling method proposed
in the present study can usefully be employed for the
design and analysis of rotating functionally graded
beam structures, as an alternative to elaborate exper-
imental work as well as other methods like finite
element method.
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