Free vibration analysis of pre-twisted rotating FGM beams

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


wðx; tÞ ¼Xl3

j¼1


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


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


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


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


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


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


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


� �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


0

3jdx

0

@

1

A

8<

:

þZ1

0

Jq11

2L2 � x2� �

/0

3i/0

3jdx

0

@

1

A

9=

;q3j ¼ 0

ð56Þ


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


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


� �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


123

KGB2ij ,

1

2

Z1

0

1� n2� �

w0

aiw0

ajdn ð76Þ

KB2ij ,

Z1

0

1

2k þ 1ð Þ þ 1


� �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


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


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


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


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


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


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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