FLAPWISE BENDING VIBRATION ANALYSIS OF A ROTATING
DOUBLE TAPERED TIMOSHENKO BEAM
O. Ozdemir Ozgumus, M. O. Kaya*
Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak, Istanbul,
Turkey
Abstract
In this study, free vibration analysis of a rotating, double tapered Timoshenko beam that
undergoes flapwise bending vibration is performed. At the beginning of the study, the
kinetic and the potential energy expressions of this beam model are derived using several
explanatory tables and figures. In the following section, Hamilton’s principle is applied to
the derived energy expressions to obtain the governing differential equations of motion and
the boundary conditions. The parameters for the hub radius, rotational speed, shear
deformation, slenderness ratio and taper ratios are incorporated into the equations of
motion. In the solution part, an efficient mathematical technique, called the Differential
Transform Method (DTM), is used to solve the governing differential equations of motion.
Using the computer package, Mathematica, the effects of the incorporated parameters on
the natural frequencies are investigated and the results are tabulated in several tables and
introduced in several graphics.
Key words: Nonuniform Timoshenko Beam, Tapered Timoshenko Beam, Rotating
Timoshenko Beam, Differential Transform Method, Differential Transformation
*
Corresponding Author: Tel: +90 (212) 2853110, Fax: +90 (212) 2852926
E-mail: kayam@itu.edu.tr (M.O.Kaya)
1
Nomenclature
Us
potential energy due to shear
A
cross-sectional area
Vx ,V y ,Vz
velocity components of point P
b0
beam breadth at the root section
W k  ,  k 
transformed functions
cb
breadth taper ratio
w
flapwise bending displacement
ch
height taper ratio
w
flapwise bending slope
E
Young’s modulus
x
spanwise coordinate
EA
axial rigidity of the beam cross
x
spanwise coordinate parameter
x0 , y 0 , z 0
coordinates of P0
section
EI
bending rigidity of the beam cross
section
G
shear modulus
x1 , y1 , z1
coordinates of P
h0
  
i, j, k
beam height at the root section

hub radius parameter
unit vectors in the x , y and z
W
virtual work of the nonconservative
forces
directions
Iy
second momentof inertia about the

shear angle
0
uniform strain due to the centrifugal
y axis
k
shear correction factor
force
kAG
shear rigidity
 ij
classical strain tensor
L
beam length
 xx
axial strain
P
reference point after deformation
  ,  
transverse normal strains
P0
reference point before deformation

sectional coordinate corresponding
to major principal axis for P0 on the
elastic axis
r

r0
inverse of the slenderness ratio S

natural frequency parameter
position vector of P0

sectional coordinate for P0 normal
to  axis at elastic axis
2

r1
position vector of P

density of the blade material
R
hub radius
A
mass per unit length
S
slenderness ratio

rotation angle due to bending
t
time

circular natural frequency
T
centrifugal force

constant rotational speed
u0
axial displacement due to the

rotational speed parameter
centrifugal force
Ub
potential energy due to bending
1. Introduction
The dynamic characteristics, i.e. natural frequencies and related mode shapes, of rotating
tapered beams are required to determine resonant responses and to perform forced vibration
analysis. Therefore, many investigators have studied rotating tapered beams, which are very
important for the design and performance evaluation in several engineering applications
such as rotating machinery, helicopter blades, robot manipulators, spinning space
structures, etc. Klein [13] used a combination of finite element approach and Rayleigh-Ritz
method to analyse the vibration of tapered beams. Downs [3] applied a dynamic
discretization technique to calculate the natural frequencies of a nonrotating double tapered
beam based on both the Euler-Bernoulli and Timoshenko Beam Theories. Swaminathan
and Rao [15], computed the frequencies of a pretwisted, tapered rotating blade using the
Rayleigh-Ritz method and including the effects of the rotational speed, pretwist angle and
breadth taper. To [5] developed a higher order tapered beam finite element for transverse
vibration of tapered cantilever beam structures. Sato [12] used Ritz method to study a
linearly tapered beam with ends restrained elastically against rotation and subjected to an
axial force. Lau [9] studied the free vibration of tapered beam with end mass by the exact
3
method. Banerjee and Williams [11] derived the exact dynamic stiffness matrices of axial,
torsional and transverse vibrations for a range of tapered beam elements. Williams and
Banerjee [10] studied the free vibration of an axially loaded beam with linear or parabolic
taper, and a stepped approximation is used to model the beam as a rigidly connected set of
uniform members. Storti and Aboelnaga [7], studied the transverse deflections of a straight
tapered symmetric beam attached to a rotating hub as a model for the bending vibration of
blades in turbomachinery. Kim and Dickinson [4] used the Rayleigh-Ritz method to
analyse slender beams subject to various complicated effects. Lee et al. [20] used Green’s
function method in Laplace transform domain to study the vibration of general elastically
restrained tapered beams and obtained the approximate fundamental solution by using a
number of stepped beams to represent the tapered beam. Lee and Kuo [22] used Green’s
function method to study the truncated non-uniform beams on elastic foundation with
polynomial varying bending rigidity and elastically constrained ends, and an exact
fundamental solution is given in power series form. Grossi and Bhat [18] used,
respectively, the Rayleigh-Ritz method and the Rayleigh-Schmidt method to analyse the
truncated tapered beams with rotational constraints at two ends. Naguleswaran [19] used
the Frobenius method to analyse the free vibration of wedge and cone beams and beams
with one constant side and another square-root varying side. Bazoune and Khulief [1]
developed a finite beam element for vibration analysis of a rotating doubly tapered
Timoshenko beam. Khulief and Bazoune [23] extended the work in Bazoune and Khulief
[1] to account for different combinations of the fixed, hinged and free end conditions.
In this study, which is an extension of the authors’ previous works [14, 16, 17], free
vibration analysis of a rotating, double tapered, cantilever Timoshenko beam that
4
undergoes flapwise bending vibration is performed using the Differential Transform
Method, DTM, which is an iterative procedure to obtain analytic Taylor series solutions of
differential equations. The advantage of DTM is its simplicity and accuracy in calculating
the natural frequencies and plotting the mode shapes and also, its wide area of application.
In open literature, there are several studies that used DTM to deal with linear and nonlinear
initial value problems, eigenvalue problems, ordinary and partial differential equations,
aeroelasticity problems, etc. A brief review of these studies is given by Ozdemir Ozgumus
and Kaya [16].
2. Beam Configuration
The governing partial differential equations of motion are derived for the flapwise
bending vibration of a rotating, double tapered, cantilever Timoshenko beam represented
by Fig.1.
Here, a cantilever beam of length L , which is fixed at point O to a rigid hub, is shown.
The hub has the radius, R and rotates in the counter-clockwise direction at a constant
rotational speed,  . The beam tapers linearly from a height h0 at the root to h at the free
end in the xz plane and from a breadth b0 to b in the xy plane. In the right-handed
Cartesian co-ordinate system, the x -axis coincides with the neutral axis of the beam in the
undeflected position, the z -axis is parallel to the axis of rotation (but not coincident) and
the y -axis lies in the plane of rotation.
The following assumptions are made in this study,
a. The flapwise bending displacement is small.
5
b. The planar cross sections that are initially perpendicular to the neutral axis of the beam
remain plane, but no longer perpendicular to the neutral axis during bending.
c. The beam material is homogeneous and isotropic.
3.
Derivation of The Governing Equations of Motion
The cross-sectional and the side views of the flapwise bending displacement of a
rotating Timoshenko beam are introduced in Figs.2(a) and 2(b), respectively.
Here, a reference point is chosen and is represented by P0 before deformation and by P
after deformation.
3.1. Derivation of The Potential Energy Expression
Examining Figs. 2(a) and 2(b), coordinates of the reference point are written as follows
a. Before deformation ( Coordinates of P0 ):
y0   ,
x0  R  x ,
z0  
(1)
After deformation ( Coordinates of P ):
x1  R  x  u 0   ,
y1   ,
z1  w  
(2)
Here, the rotation angle due to bending,  , is small so it is assumed that Sin   .




Knowing that r0 and r1 are the position vectors of P0 and P , respectively, dr0 and dr1
can be given by




dr0  dx0 i  dy 0  j  dz 0 k
and




dr1  dx1 i  dy1  j  dz1 k
(3)


The components of dr0 and dr1 are expressed as follows
dx0  dx ,
dy 0  d ,
dx1  (1  u0   )dx - d ,
dz 0  d
dy1  d ,
(4)
dz1  wdx  d
6
(5)
where
 
denotes differentiation with respect to the spanwise position x .
The classical strain tensor  ij may be obtained using the equilibrium equation below
Eringen [2].
 
 
 
T
dr1  dr1  dr0  dr0  2dx d d   ij dx d d 
(6)
Substituting Eqs. (4) and (5) into Eq. (6), the elements of the strain tensor  ij are
obtained as follows
 xx  u 0    
u 2   2  2  u    w 2 ,
2
2
0
 x  0 ,
2
 x  w        u 0 (7)
In this work;  xx ,  x and  x are used in the calculations because as noted by Hodges
and Dowell [6], for long slender beams, the axial strain  xx is dominant over the transverse
normal strains,   and   . Moreover, the shear strain   is two order smaller than the
other shear strains,  x and  x . Therefore,   ,   and   can be neglected.
In order to obtain simpler expressions for the strain components, higher order terms
should be neglected so an order of magnitude analysis is performed by using the ordering
scheme, taken from Hodges and Dowell [6] and introduced in Table 1.
The Euler-Bernoulli Beam Theory is used by Hodges and Dowell [6]. In the present
work, their formulation is modified for a Timoshenko beam and the following new
expression is added to their ordering scheme as a contribution to the literature.
  w    ( 2 )
(8)
Using Table 1, the strain components in Eq.(7) can be reduced to
7
 xx  u 0    
( w) 2
,
2
 x 0 ,
 x  w  
(9)
Using Eq. (9), the potential energy expressions are derived. The potential energy
contribution due to flapwise bending, U b , is given by
Ub 
L

1 
  E xx2 dd dx


2 0  A

(10)
Substituting the first expression of Eq. (9) into Eq. (10), taking integration over the blade
cross section and referring to the definitions given by Table 2, the following potential
energy expression is obtained for flapwise bending
L
Ub 
L
L
1
1
1
EA(u 0 ) 2 dx   EI y ( ) 2 dx   EAu 0 ( w) 2 dx

20
20
20
(11)
The uniform strain,  o , and the associated axial displacement, u 0 that is a result of the
centrifugal force, T x , are related to each other as follows
u 0 ( x)   0 ( x) 
T ( x)
EA
(12)
where the centrifugal force is given by
L
T ( x)   A 2 ( R  x)dx
(13)
x
L
1 T 2 ( x)
Substituting Eq. (12) into Eq. (11) and noting that the 
dx term is constant and
2 0 EA
will be denoted as C1 , the final form of the bending potential energy is obtained as follows
L
Ub 
1
1 L
EI ( ) 2 dx   T ( w) 2 dx  C1

20
2 0
(14)
8
The potential energy contribution due to shear, U s , is given by
Us 
L

1 
2
 kG x dd dx

20A



(15)
Substituting the third expression of Eq. (9) into Eq. (15) and referring to the definitions
given by Table 2, the following potential energy expression is obtained for shear
L
Us 
1
kAG( w   ) 2 dx
2 0
(16)
Summing Eqs.(14) and (16), the total potential energy expression is obtained
L


1
U   EI ( ) 2  kAG( w   ) 2  T ( w) 2 dx  C1
20
(18)
3.2. Derivation of The Kinetic Energy Expression
The velocity vector of the reference point P due to rotation of the beam is expressed as
follows
 
  r1
V
  k  r1
t
(19)
Substituting the coordinates given by Eq. (2) into Eq.(19), the velocity components are
obtained as follows
V x      ,
V y  ( R  x  u 0   ) ,

Vz  w
(20)
Using Eq. (20), the kinetic energy expression,  , is derived as shown below.



L

1 
   Vx 2  V y2  Vz2 dd dx


2 0  A

(21)
Substituting Eq.(20) into Eq.(21) and referring to the definitions given by Table 3, the
final form of the kinetic energy expression is obtained.
9
L



1
Aw 2  I y 2  I y  2 2 dx  C2

20
(22)
where C2 includes the constant terms AR  x  u0  and I z  2 that appear after
substituting Eq.(20) into Eq.(21).
3.3. Application of The Hamilton’s Principle
The governing equations of motion and the associated boundary conditions can be
derived by means of the Hamilton’s principle, which can be stated in the following form for
an undamped free vibration analysis.
t2
  (U  )dt  0
(23)
t1
Using variational principles, variation of the kinetic and potential energy expressions are
taken and the governing equations of motions of a rotating, nonuniform Timoshenko beam
undergoing flapwise bending vibration are derived as follows
 A
 2 w   w   
 w


kAG
    0
T


2
x  x  x 
t
 x

 2
 
 
 w

 I y 2  I y  2   EI y
   0
  kAG
x 
x 
t
 x

(24a)
(24b)
Additionally, after the application of the Hamilton’s principle, the associated boundary
conditions are obtained as follows

The geometric boundary conditions at the fixed end, x  0 , of the Timoshenko beam,
w0, t    0, t   0

(25a)
The natural boundary conditions at the free end, x  L , of the Timoshenko beam,
10
w
 w

 kAG
   0
x
 x

Shear force:
T
Bending Moment:
EI y

0
x
(25b)
(25c)
The boundary conditions expressed by Eqs. (25b)-(25c) can be simplified by noting
that T  0 at the free end, x  L .
w
  0
x
(26a)

0
x
(26b)
4. Vibration Analysis
4.1. Harmonic Motion Assumption
In order to investigate the free vibration of the beam model considered in this study, a
sinusoidal variation of w( x, t ) and  ( x, t ) with a circular natural frequency,  , is assumed
and the functions are approximated as
wx, t   w x e it
and
 x, t    x e it
(27)
Substituting Eq. (27) into Eqs. (24a) and (24b), the equations of motion are expressed as
follows
A 2 w 
d  dw  d 
 dw
kAG

T


dx  dx  dx 
 dx
I y  2  I y  2 

  0

d 
d 
 dw

 EI y
  kAG
   0
dx 
dx 
 dx

4.2. Tapered Beam Formulation and Dimensionless Parameters
11
(28a)
(28b)
The basic equations for the breadth bx  , the height hx  , the cross-sectional area, Ax
and the second moment ofinertia, I y  x  of a beam that tapers in two planes are as follows
x

b x   b0 1  cb 
L

m
x

hx   h0 1  c h 
L

and
m
x 
x

A x   A0 1  cb  1  c h 
L 
L

n
n
(29a)
m
x 
x

I y  x   I y 0 1  c b  1  c h 
L 
L

and
3n
(29b)
where the breadth taper ratio, cb and the height taper ratio, ch are given by
cb  1 
b
b0
and
ch  1 
h
h0
(30)
The values of the constants, n and m , depend on the type of taper. In this study, n  1
and m  1 values are used to model a beam that tapers linearly in two planes. Since the
Young’s modulus E , the shear modulus G and the material density,  are assumed to be
constant, the mass per unit length A , the flapwise bending rigidity EI y and the shear
rigidity kAG vary according to the Eqs. (29a) and (29b).
In order to make comparisons with the results in open literature, the following
dimensionless parameters can be introduced.
x
x
L

R
L
w
w~ 
L
A0 L4 2
 
EI y 0
2
A0 L4  2
 
EI y 0
2
r2 
I y0
1

S 2 A0 L2
s2 
EI y 0
kA0 GL2
(31)
Substituting the tapered beam formulas and the dimensionless parameters into Eqs.(28a)
and (28b), the following dimensionless equations of motion are obtained for the linear taper
case ( m  1 , n  1 ).
12
 cb c h
1
1
x2




cb  c h  1 



c


c


1

c

c


c
c


x



b
h
b
h
b h
 4
2
3
2

~    2
 dw
x3
x4
(32a)
cb  ch  cb ch   cb ch      1  cb x 1  ch x w~ 
3
4
 dx    
2
~
 1  d 
 dw




1

c
x
1

c
x
    0



b
h

 s   dx 
 dx

d
dx
d 
3 d 
3
2
2
2
1  c b x 1  c h x 
  r    1  c b x 1  c h x   
dx 
dx 
~
1
 dw




1

c
x
1

c
x
   0

b
h
2
s
 dx



(32b)
Additionally, substituting the dimensionless parameter into Eqs.(25a)-(26b), the
dimensionless boundary conditions of a rotating, cantilever Timoshenko beam can be
obtained as follows
At x  0
~   0
w
At x  1
~
dw
  0
dx
(33a)
and
d
0
dx
(33b)
5. The Differential Transform Method
The differential transform method is a transformation technique based on the Taylor
series expansion and is a useful tool to obtain analytical solutions of the differential
equations. In this method, certain transformation rules are applied to both the governing
differential equations of motion and the boundary conditions of the system in order to
transform them into a set of algebraic equations. The solution of these algebraic equations
gives the desired results of the problem. It is different from high-order Taylor series method
because Taylor series method requires symbolic computation of the necessary derivatives
13
of the data functions and is expensive for large orders. Details of the application procedure
of DTM is explained by Ozdemir Ozgumus and Kaya [16] using several explanatory tables.
6.
Formulation with DTM
In the solution step, DTM is applied to Eqs.(32a) and (32b) and the following
expressions are obtained.
2  3  c  c   3  4  c c k  1k  2W k  2 
1
1




b
h
b h
2
2
2
6
12
s 


k  12   

cb  c h
s22

cb c h
1  2 

 cb  c h 




W
k

1

k
k

1





W k  

2
s 2  2 2   2 



 k  1k  1
2 




c

c


c
c

c

c
b
h
b h
b
h

W k  1 
3
2 

(34a)
  2 k  1k  2  
cb  c h
k  1
 2 
 cb c hW k  2  2 2  k  1  2 2 k  1 k  
4
s 
s 


cb c h
k  1 k  1  0
s22
k  1k  2 k  2  cb  3ch k  12  k  1 



 r 2  2   2  k  
s

cb  c h
 2
2
2
2
ch k  1k  13cb  ch   s 2  r    cb  3ch   k  1 

3ch cb  ch k k  1  12




(34b)

cb c h

3
2
2
2
cb ch k  1k  2   s 2 3ch cb  ch r     k  2 

c
3cb  ch r 2  2   2  k  3  cb ch 3 r 2  2   2  k  4 
k  1W k  1  cb  ch k W k   cb ch k  1W k  1  0
2
h
s2
s2
s2
Additionally, DTM is applied to Eqs.(33a)-(33c) at x0  0 and the following
transformed boundary conditions are obtained.
at x  0
W 0   0  0
(35a)
14


 kW k    k   0
at x  1
and
k 0
 k k   0
(35b)
k 0
~x  and   x  ,
In Eqs.(34a)-(35b), W k  and  k  are the differential transforms of w
respectively. Using Eqs. (34a) and (34b), W k  and  k  values can now be evaluated in
terms of cb , c h ,  , , d1 and d 2 for k  2,3... . The results calculated in Mathematica for the
  0 , r  0.02 , k  2 3 and E G  8 3 values are as follows
W 2  3750
W 3 
d 2  cb  ch d1
7500  6  4ch  4cb  3ch cb  2
781250d1  1250.04cb  3750c h d 2  2499.96cb  c h d 2

7500  6  4c h  4cb  3c h cb  2
2499.96c c


cb  ch d1  d 2 cb  ch 
 2.04  2   2 d1
 18750000
2
7500  6  4c h  4cb  3c h cb 
7500  6  4c h  4cb  3c h cb  2
h b


2
 2  312.5d1  0.5cb  3ch d 2
 3  104.17cb  c h d1  781250
cb  ch d1  d 2
7500  6  4c h  4cb  3c h cb  2

0.33cb  ch  625d1  cb  3ch d 2    104.17  ch cb  ch2  0.000067 2   2 d 2
Here the constants d1 and d 2 that appear in W k  ’s and  k  ’s are defined as follows
~
 dw
d1  W 1   
 dx  x 0
7.
and
 d 
d 2   1   
 dx  x 0
(36)
Results and Discussions
The computer package Mathematica is used to write a code for the expressions given by
Eqs.(34a)-(35b). The effects of the rotational speed, hub radius, slenderness ratio and taper
ratios on the natural frequencies are investigated and the related graphics are plotted.
15
Additionally, in order to validate the calculated results, comparisons with the studies in
open literature are made and a very good agreement between the results is observed.
In Fig.3, convergence of the first four natural frequencies with respect to the number of
terms, m , used in the DTM application is introduced. In order to evaluate up to the fourth
natural frequency with five-digit precision, it was necessary to take 39 terms. During the
calculations, it is noticed that when the rotational speed parameter is increased, the number
of the terms has to be increased to achieve the same accuracy. Additionally, here it is seen
that higher modes appear when more terms are taken into account in DTM application.
Thus, depending on the order of the required mode, one must try a few values for the term
number at the beginning of the Mathematica calculations in order to find the adequate
number of terms.
In Fig.3, variation of the first three natural frequencies of a rotating, tapered Timoshenko
beam with respect to the rotational speed parameter,  and the hub radius parameter,  , is
introduced.As expected, the natural frequencies increase with the increasing rotational
speed parameter due to the stiffening effect of the centrifugal force that is directly
proportional to the square of the rotational speed. Moreover, as it is seen in Fig.3,  makes
the rate of increase of the natural frequencies larger because the centrifugal force that is
directly proportional to the hub radius makes the beam stiffer with the increasing  .
In Table 4, variation of the natural frequencies of a uniform beam with respect to the
inverse of the slenderness ratio, r , and the rotational speed parameter,  , is introduced
and the results are compared with the ones given by Banerjee [10]. Besides increasing with
the rotational speed parameter, the natural frequencies decrease as the inverse of the
16
slenderness ratio, r , increases. At   12 , the decrease in the frequencies due to r is 7 %
for the first mode, 23.12 % for the second mode, 37 % for the third mode and 59.7 % for
the fourth mode. Comparing the percentage decrease in the frequencies, it is noticed that
the effect of the slenderness ratio is dominant on the higher modes and this effect
diminishes rapidly as the frequency order decreases. It is something expected because the
Timoshenko Beam Theory is used when the higher mode frequencies are of interest. The
effect of the slenderness ratio can be observed better in Fig.5 where variation of the natural
frequencies with respect to the inverse of the slenderness ratio, r is shown.
In Table5, variation of the natural frequencies of a nonrotating Timoshenko beam with
respect to different combinations of breadth and height taper ratios is given as reference
values for the future studies and the results are compared with the ones calculated by
Downs [3].
Furthermore, in order to observe the effects of the taper ratios, Figs.6(a) and 6(b) can be
considered. As it is seen in Figs.6(a) and 6(b), the breadth taper ratio, cb , has very little,
even no influence on the flapwise bending frequencies while the height taper ratio, ch , has
a linear decreasing effect on the natural frequencies except the fundamental natural
frequency, which increases a little with ch .
8. Conclusion
The main contributions of this study to the literature appear in the derivation of the
governing equations of motion and they can be listed as follows:
17

Derivation of both the potential and the kinetic energy expressions are made in a
very detailed and clear way.

In the study of Hodges and Dowell [6], the Euler-Bernoulli Beam Theory is used
and in the present study, their formulation is modified for the Timoshenko Beam Theory
and a new expression,   w    ( 2 ) , is added to their ordering scheme as a
contribution to the literature.
The effects of the slenderness ratio, hub radius, rotational speed and taper ratios on the
natural frequencies are examined. The following results are obtained:

The natural frequencies increase with the increasing rotational speed and this rate of
increase becomes larger with the increasing hub radius parameter,  .
 The effect of the rotational speed is dominant on the fundamental natural frequency
and this effect diminishes rapidly as the frequency order increases.
 The height taper ratio has a little increasing effect on the fundamental natural
frequency. The other natural frequencies decrease as the height taper ratio increases.
 The breadth taper ratio has very little, even no influence on the bending frequencies.
 Inverse of the slenderness ratio has a decreasing effect on the natural frequencies.
Therefore, natural frequencies of a Timoshenko beam are lower than the natural
frequencies of an Euler-Bernoulli beam.
18
REFERENCES
[1] A.Bazoune, Y. A. Khulief (1992) A finite beam element for vibration analysis of
rotating tapered Timoshenko beams. J. Sound Vib. 156: 141-164
[2] A.C. Eringen (1980) Mechanics of Continua, Robert E. Krieger Publishing Company,
Huntington, New York.
[3] B.Downs (1977) Transverse vibrations of cantilever beam having unequal breadth and
depth tapers. Asme J. Appl. Mech. 44: 737-742.
[4] C.S. Kim, S.M. Dickinson (1988). On the analysis of laterally vibrating slender beams
subject to various complicating effects. J. Sound Vib. 122: 441-455.
[5] C.W.S. To (1979) Higher order tapered beam finite elements for vibration analysis. J.
Sound Vib. 63: 33-50.
[6] D.H. Hodges, E.H. Dowell (1974) Nonlinear equations of motion for the elastic bending
and torsion of twisted nonuniform rotor blades. NASA TN D-7818.
[7] D. Storti, Y.Aboelnaga (1987) Bending vibration of a class of rotating beams with
hypergeometric solutions. Asme J. Appl. Mech. 54: 311-314.
[8] F.W. Williams, J.R. Banerjee (1985) Flexural vibration of axially loaded beams with
linear or parabolic taper. J. Sound Vib. 99: 121-138.
[9] J.H. Lau (1984)Vibration frequencies of tapered bars with end mass. Asme J. Appl.
Mech. 51: 179-181.
[10] J.R. Banerjee (2001), Dynamic stiffness formulation and free vibration analysis of
centrifugally stiffened Timoshenko beams. J. Sound Vib. 247(1): 97-115.
19
[11] J.R. Banerjee, F.W. Williams (1985) Exact Bernoulli-Euler dynamic stiffness matrix
for a range of tapered beams. Int. J. Num. Methods Engrg. 21: 2289-2302.
[12] K.Sato (1980) Transverse vibrations of linearly tapered beams with ends restrained
elastically against rotation subjected to axial force. Int. J. Mech. Sci. 22: 109-115.
[13] L. Klein (1974) Transverse vibrations of non-uniform beam, J. Sound Vibr. 37: 491505.
[14] M.O. Kaya (2006) Free vibration analysis of rotating Timoshenko beams by
differential transform method. Aircr. Eng. Aerosp. Tec. 78 (3): 194-203.
[15] M. Swaminathan, J. S., Rao (1977) Vibrations of rotating, pretwisted and tapered
blades. Mech. Mach. Theory 12: 331-337.
[16] O. Ozdemir Ozgumus, M.O. Kaya (2006) Flexural vibration analysis of double tapered
rotating Euler-Bernoulli beam by using the differential transform method, Meccanica,
41(6), 661-670
[17] Ö. Özdemir, M.O. Kaya (2006) Flapwise bending vibration analysis of a rotating
tapered cantilevered Bernoulli-Euler beam by differential transform method. J. Sound Vib.
289: 413–420.
[18] R.O. Grossi, R.B. Bhat (1991) A note on vibrating tapered beams. J. Sound Vib. 147:
174-178.
[19] S. Naguleswaran (1994) Vibration in the two principal planes of a non-uniform beam
of rectangular cross-section, one side of which varies as the square root of the axial coordinate. J. Sound Vib. 172: 305-319.
20
[20] S.Y. Lee, H.Y. Ke, Y.H. Kuo (1990) Analysis of non-uniform beam vibration. J.
Sound Vib. 142: 15-29.
[21] S.Y. Lee, S.M. Lin (1994) Bending vibrations of rotating non-uniform Timoshenko
beams with an elastically restrained root. J. Appl. Mech. 61: 949-955.
[22] S.Y. Lee, Y.H. Kuo (1992) Exact solution for the analysis of general elastically
restrained non-uniform beams. Asme J. Appl. Mech. 59: 205-212.
[23] Y. A. Khulief, A.Bazoune (1992) Frequencies of rotating tapered Timoshenko beams
with different boundary conditions. Comput. Struct. 42: 781-795.
21

z
y
RR
b
O
h0
h
x
L
b0
Figure 1. Configuration of a rotating, double tapered, cantilever Timoshenko beam
z
z
 


P

Middle
Plane
P

w
z1

P0
w

z0
z1
P0 u 0

y
y 0  y1
R
w
z0
x
x
x0
x1
(a)
(b)
Figure 2. (a) Cross sectional view (b) Side view of a rotating Timoshenko beam before and
after flapwise bending deformation
22
5
Log (  )
4th Mode
4
3rd Mode
2nd Mode
3
1st Mode
2
Log ( m )
1
1.5
2.0
2.5
3.0
3.5
4.0
Figure 3. Convergence of the first four natural frequencies (   0 ,   4 , cb  ch  0.5 ,
r  0.08 ,   0.3 , k  0.85 )
23
60
Natural Frequencies
(3)
40
(2)
20
(1)
0
0
2
4

6
8
10
Figure 4. Variation of the natural frequencies with respect to the hub radius parameter, 
and the rotational speed parameter,  . ( r  0.08 ,   0.3 , k  0.5 ,   1,
  0.5 ,
;   0,
)
24
;
Natural Frequencies
120
80
(4)
40
(3)
(2)
(1)
0
0.00
0.04
r
0.08
0.12
Figure 5. Variation of the natural frequencies of a uniform rotating Timoshenko beam with
respect to the inverse of the slenderness ratio effect, r (   0 ,   4 , k  2 3 ,
E G  8 3 , cb  c h  0 )
25
4th Mode
Natural Frequencies
120
3rd Mode
80
2nd Mode
40
1st Mode
0
(a)
0.0
0.2
0.4
cb
0.6
0.8
1.0
120
Natural Frequencies
4th Mode
80
3rd Mode
40
2nd Mode
1st Mode
0
(b)
0.0
0.2
0.4
ch
0.6
0.8
1.0
Figure 6. Effects of the taper ratios, cb and ch , on the natural frequencies (   0 ,   0 ,
r  0.08 ,   0.3 , k  0.85 , Euler,
,Timoshenko,
26
)
Table 1. Ordering scheme for Timoshenko beam formulation
x
 (1)
L

w
 ( )
L
  ( )

L
L
 ( )
u0
 ( 2 )
L
 ( )
  w    ( 2 )
Table 2. Area integrals for the potential energy expression
 dd  A

A
2
 
dd  I z
A
 
2

2
dd  I y
A
 dd  dd   dd  0
  2 dd  J
A
A
A
A
Table 3 Area integrals for the kinetic energy expression
 dd  m
 
2
dd  I z
A
A
 
A
 dd  dd   dd  0
A
A
A
27
2
dd  I y
Table 4. Variation of the natural frequencies of a uniform Timoshenko beam with respect
to the inverse of the slenderness ratio parameter, r and rotational speed parameter, 
( cb  c h  0 , k  2 3 , E G  8 3 ,   0 )
Natural Frequencies

r
0
4
8
Present Ref.[13] Present Ref.[13] Present Ref.[13]
3.51601 3.5160 5.58500 5.585 9.25684 9.2568
22.0345
24.2733
29.9954
0
61.6971
63.9666
70.2929
120.901
123.261
130.049
3.49980 3.4998 5.56158 5.5616 9.20959 9.2096
21.3547
23.6061
29.3215
0.02
57.4705
59.8117
66.2748
106.926
109.459
116.665
3.45267 3.4527 5.49511 5.4951 9.08544 9.0854
19.6497
21.9557
27.7082
0.04
48.8891
51.4822
58.4507
84.1133
87.1836
95.6423
3.37873 3.3787 5.39542 5.3954 8.92085 8.9208
17.5470
19.9662
25.8362
0.06
40.7447
43.7365
51.4154
66.3623
70.1298
79.9414
3.28370 3.2837 5.27486 5.2749 8.74555 8.7456
15.4883
18.0628
24.0479
0.08
34.3005
37.7317
45.9683
53.6516
57.9491
67.8215
3.17377 3.1738 5.14482 5.1448 8.57351 8.5735
13.6607
16.3946
22.3506
0.1
29.3614
33.1793
41.4632
43.9102
47.8101
53.2833
-
28
12
Present Ref.[13]
13.1702 13.170
37.6031
79.6144
140.534
13.0870 13.087
36.8659
75.6698
127.604
12.8934 12.893
35.1811
68.2339
107.887
12.6724 12.672
33.2672
61.6011
93.0672
12.4581 12.458
31.2846
55.9744
77.1047
12.2467 12.247
28.9100
49.6484
56.6750
-
Table 5. Variation of the natural frequencies of a nonrotating Timoshenko beam with
different combinations of breadth and height taper ratios ( r  0.08 , E kG  3.059 ,   0 )
ch
0
0.2
0.4
0.6
0.8
cb
0
3.32404
3.32405§
16.2889
16.2890§
36.7073
36.7078§
58.2778
58.2788§
3.42466
15.8905
35.4301
56.8910
3.56054
15.3528
33.7876
54.7561
3.76227
3.76228§
14.6448
14.6449§
31.6239
31.6243§
51.6225
51.6225§
4.11768
13.7574
28.6360
46.8288
§
0.1
3.42876
16.4625
36.8384
58.394
3.52998
16.0538
35.5595
57.0023
36.6661
15.5057
33.9159
54.8654
3.86943
14.7872
31.7515
51.7327
4.22692
13.8888
28.7620
46.9419
0.2
3.5485
16.6561
36.9869
58.5281
3.65042
16.2362
35.7054
57.1293
3.78791
15.6769
34.0602
54.9893
3.99193
14.9472
31.8945
51.8566
4.35163
14.0370
28.9031
47.0683
0.3
3.68722
3.68723§
16.8750
16.8752§
37.1583
37.1588§
58.6862
58.6872§
3.78999
16.4429
35.8731
57.2775
3.9285
15.8715
34.2253
55.1324
4.13388
4.13389§
15.1296
15.1297§
32.0578
32.0582§
51.9987
51.9995§
4.49590
14.2068
29.0641
47.2125
0.4
3.85054
17.127
37.3608
58.8777
3.95437
16.6816
36.0704
57.4552
4.09410
16.0970
34.4189
55.3024
4.30106
15.3418
32.2486
52.1660
4.66552
14.4055
29.2519
47.3811
Downs [3]
29
0.5
4.04668
17.4242
37.6076
59.1179
4.15186
16.9642
36.3101
57.6759
4.29314
16.3651
34.6531
55.5115
4.50198
15.5954
32.4786
52.3699
4.86899
14.6445
29.4779
47.5848
0.6
4.28828
4.28829§
17.7869
17.7871§
37.9216
37.9221§
59.4326
59.4336§
4.39529
17.3110
36.6141
57.9630
4.5386
16.6961
34.9495
55.7815
4.74978
4.74979§
15.9106
15.9107§
32.7688
32.7693§
52.6309
52.6316§
5.11941
14.9439
29.7625
47.8431
0.7
4.59598
18.253
38.347
59.8717
4.70561
17.7598
37.0259
58.3626
4.85176
17.1278
35.3502
56.1557
5.06597
16.3251
33.1603
52.9901
5.43815
15.3418
30.1461
48.1956
0.8
5.00621
5.00623§
18.9024
18.9026§
38.9849
38.9854§
60.5467
60.5477§
5.11985
18.3917
37.6454
58.9813
5.27027
17.7423
35.9542
56.7371
5.48870
5.48871§
16.9223
16.9224§
33.7509
33.7513§
53.5473
53.5481§
5.86286
15.9229
30.7255
48.7392
LIST OF FIGURES
Figure 1. Configuration of a rotating, double tapered, cantilever Timoshenko beam
Figure 2. (a) Cross sectional view (b) Longitudinal view of a rotating Timoshenko beam
before and after flapwise bending deformation
Figure 3. Convergence of the first four natural frequencies
Figure 4. Variation of the natural frequencies with respect to the hub radius parameter, 
and the rotational speed parameter,  .
Figure 5. Variation of the natural frequencies of a uniform rotating Timoshenko beam with
respect to the inverse of the slenderness ratio effect, r
Figure 6. Effects of the taper ratios, cb and ch , on the natural frequencies
Figure 7. Variation of the natural frequencies with respect to,  , cb and ch
LIST OF TABLES
Table 1. Ordering scheme for Timoshenko beam formulation
Table 2. Area integrals for the potential energy expression
Table 3 Area integrals for the kinetic energy expression
Table 4. Variation of the natural frequencies of a uniform Timoshenko beam with respect to
the inverse of the slenderness ratio parameter, r and rotational speed parameter, 
Table 5. Variation of the natural frequencies of a nonrotating Timoshenko beam with
different combinations of breadth and height taper ratios
30