FLEXURAL-TORSIONAL COUPLED VIBRATION ANALYSIS OF A THINWALLED CLOSED SECTION COMPOSITE TIMOSHENKO BEAM BY USING
THE DIFFERENTIAL TRANSFORM METHOD
Metin O. Kaya and Özge Özdemir
Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak,
Istanbul, Turkey
Abstract. In this study, a new mathematical technique called the Differential Transform Method
(DTM) is introduced to analyse the free undamped vibration of an axially loaded, thin-walled closed
section composite Timoshenko beam including the material coupling between the bending and
torsional modes of deformation, which is usually present in laminated composite beams due to ply
orientation. The partial differential equations of motion are derived applying the Hamilton’s principle
and solved using DTM. Natural frequencies are calculated, related graphics and the mode shapes are
plotted. The effects of the bending-torsion coupling and the axial force are investigated and the results
are compared with the studies in literature.
1. Introduction
A composite thin-walled beam with length
L , cross sectional dimension B and wall
thickness d is shown in Fig.1. The
geometric dimensions are assumed to be
d  B so the terms related to the warping
stiffness and the warping inertia are small
enough to be neglected.
Figure 1. Configuration of an axially
loaded composite Timoshenko beam
The bending motion in the z direction,
the torsional rotation about the x axis and
the rotation of the cross section due to
bending alone are represented by w( x, t ) ,
 ( x, t ) and  ( x, t ) , respectively. A
constant axial force P acts through the
centroid of the cross section which
coincides with the x axis. P is positive
when it is compressive as in Fig.1.
2. Formulation
The governing undamped partial differential equations of motion are derived for the
free vibration analysis of the beam model represented by Fig.1. After the application of the
Hamilton’s principle, the following equations of motion are obtained as follows
 I  EI   kAGw     K   0
(1)
  Pw  kAGw     0
 w
(2)
 I s  PI s     K   GJ   0
(3)
Here,   A is the mass per unit length; I s is the polar mass moment of inertia; K and
kGA are flexure–torsion coupling rigidity and shear rigidity of the beam, respectively.
The boundary conditions at x  0 and x  L for Eqs. (1)-(3) are as follows
EI   k   0
 Pw  kAGw   w  0
 PI s     K   GJ   0
(4)
(5)
(6)
A sinusoidal variation of w( x, t ) ,  ( 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 ,
 x, t    x e it ,
 x, t    x e it
(7)
The following nondimensional parameters can be used to simplify the equations of motion

x
,
L
W   
W
,
L
r2 
I
,
AL2
 *  d   ,    1  *
d
(8)
L
Substituting Eqs.(7) and (8) into Eqs.(4)-(5), the dimensionless equations of motion are
obtained as follows
A1 **  A2  A3W *  A4 **  0
(9)
B1W **  B2W  B3 *  0
(10)
C1 **  C2  C3 **  0
(11)
where the dimensionless coefficients are
A1  1 ,
B2 
A2 
L2  2

kAG
L4 r 2
EI
2 
B3  1
kAGL2
EI
A3 
C1  1 
PI s
GJ
kAGL2
EI
C2 
A4 
K
EI
I s L2 2

GJ
B1  1 
C3 
P
kAG
(12)
K
GJ
3. The Differential Transform Method
The differential transform method is a transformation technique based on the Taylor series
expansion and is used to obtain analytical solutions of the differential equations. In this
method, certain transformation rules are applied and the governing differential equations and
the boundary conditions of the system are transformed into a set of algebraic equations in
terms of the differential transforms of the original functions and the solution of these
algebraic equations gives the desired solution of the problem.
A function f x  , which is analytic in a domain D, can be represented by a power series
with a center at x 0 , any point in D. The differential transform of the function is given by
F k  
1  d k f ( x) 


k!  dx k  x  x
(13)
0
where f x  is the original function and F k  is the transformed function. The inverse
transformation is defined as

f ( x)   ( x  x0 ) k F k 
(14)
k 0
Combining Eqs. (13) and (14) and expressing f x  by a finite series, we get
( x  x0 ) k
k!
k 0
m
f ( x)  
 d k f ( x) 


k
dx

 x  x0
(15)
Here, the value of m depends on the convergence of the natural frequencies [1]. Theorems
that are frequently used in the transformation procedure are introduced in Table 1 and
theorems that are used for boundary conditions are introduced in Table 2 [2].
Table 1. Basic theorems of DTM
Original
Function
DTM
f x  g x  hx
F k   Gk   H k 
f x   g x 
F k   Gk 
f x  g xhx F k   k Gk  l H l 

l 0
f x  
d n g x 
dx n
f x   x n
F k  
k  n ! Gk  n
k!
0 if k  n
F k    k  n   
1 if k  n
Table 2. DTM theorems for boundary conditions
x0
Boundary Condition
df
(0)  0
dx
df
( 0)  0
dx
x 1
Transformed B.C.
F ( 0)  0
Boundary Condition
f (1)  0
d3 f
(0)  0
dx 3
 F (k )  0
k 0
F (1)  0
df
(1)  0
dx
2
d2 f
(0)  0
dx 2
Transformed B.C.


 kF (k )  0
k 0

F ( 2)  0
d f
(1)  0
dx 2
 k (k  1) F (k )  0
F (3)  0
d3 f
(1)  0
dx3
 (k  1)(k  2)kF(k )  0
k 0

k 0
4. Formulation with DTM
In the solution step, the differential transform method is applied to Eqs.(9)-(11). Here we
quit using the bar symbol on  , W ,  and instead, we use  , W ,  .
A1 k  2k  1 k  2  A2 k   A3 k  1W k  1  A4 k  2k  1 k  2  0
(16)
B1 k  2k  1W k  2  B2W k   B3 k  1 k  1  0
(17)
C1 k  2k  1 k  2  C2 k   C3 k  2k  1 k  2  0
(18)
Applying DTM to Eqs. (4)–(6), the boundary conditions are given as follows
at   0

 0  W 0   0  0
at   1

k  1 k  1  A4 k  1 k  1  0
B1 k  1wk  1   k   0
C1 k  1 k  1  C3 k  1 k  1  0
(19)
(20)
(21)
(22)
5. Results and Discussion
In order to validate the computed results, an illustrative example, taken from Ref [3], is
solved and the results are compared with the ones in the same reference. Additionally, the
mode shapes of the beam are plotted.
Variation of the first five natural frequencies (coupled and uncoupled) of the above
example with respect to the axial force is introduced in Table 3 and compared with the results
of Ref. [3] and [4]. Here, it is noticed that the natural frequencies decrease as the axial force
varies from tension P  7.5 to compression P  7.5 . Additionally,it is seen that the
coupled natural frequencies are lower than the uncoupled ones. However, the fourth natural
frequency becomes less when the bending-torsion coupling is ignored.
Table3. Natural frequencies with respect to the axial force
Natural Frequencies
r 2  0.00002322
P0
P  7.5
P  7.5
Present
Ref. [4]
Present§
Ref.[3]§
Present
Present§
Ref.[3]§
Present
Ref.[4]
Present§
Ref.[3]§
40.975
40.97
37.106
37.1
35.283
30.747
30.75
28.064
28.06
21.987
21.99
224.259
224.25
197.672
197.7
217.341
189.779
189.8
210.162
210.16
181.495
181.5
598.668
598.66
525.665
525.6
592.626
518.791
518.8
586.519
586.51
511.818
511.9
647.595
647.59
648.495
648.6
647.411
648.269
648.3
647.228
647.22
648.047
648
1125.71
1125.71
992.878
-
1119.85
986.199
-
1113.950
1113.95
979.473
-
§ Natural frequencies with coupling
The effects of the axial force P and the Timoshenko effect, r , on the first four natural
frequencies are introduced in Figs. 2(a-d). When Fig. 2 is examined, it is noticed that the natural
frequencies decrease with the increasing rotary inertia parameter, r , because Timoshenko effect
decreases the natural frequencies and this effect is more dominant on higher modes as expected.
Additionally, since the fourth mode is torsion, the Timoshenko effect makes a slight change in
the fourth natural frequency value.
200.00
2nd Natural Frequency (Hz)
1st Natural Frequency (Hz)
40.00
36.00
32.00
28.00
24.00
20.00
192.00
188.00
184.00
180.00
-8.00
-4.00
0.00
4.00
8.00
Force (N)
-8.00
-4.00
-8.00
-4.00
648.50
4th Natural Frequency (Hz)
528.00
3rd Natural Frequency (Hz)
196.00
524.00
520.00
516.00
512.00
508.00
0.00
4.00
0.00
4.00
Force (N)
8.00
648.40
648.30
648.20
648.10
648.00
-8.00
-4.00
0.00
4.00
8.00
8.00
Force (N)
Force (N)
Figure 2. Effect of the Timoshenko effect on the first four natural frequencies (
, Timoshenko ;
, Euler)
Mode shapes of the considered beam under the effect of the compressive axial force
( P  7.5 ) are introduced with bending-torsion coupling in Figs. 5(a-d). When these figures are
considered, it can be noticed that the first three normal modes are bending modes while the
fourth normal mode is the fundamental torsion mode.
1

0.75
0.5
Second Mode Shapes
First Mode Shapes
1
w
0.25
0
0.25

0.5

0.75
0.5
0.25
w
0
0.25
0.5

0.75
1
0.2
0.4
0.6
ξ
0.8
1
0.2
0.4
0.6
ξ
0.8
1
1
Fourth Mode Shapes
Third Mode Shapes
1

0.75
0.5
w
0.25
0
0.25

0.5
0.75

0.75
0.5
0.25

0
w
0.2
0.4
0.6
0.8
1
0.2
ξ
0.4
0.6
0.8
ξ
Figure 5. The first four normal mode shapes of the composite beam with bending-torsion coupling
(
,;
, ;
,)
References
S.H. Ho and C.K. Chen, Analysis of General Elastically End Restrained Non-Uniform Beams
Using Differential Transform, Applied Mathematical Modeling 22 (1998) 219-234
Özdemir Ö, Kaya MO, Flapwise Bending Vibration Analysis of a Rotating Tapered
Cantilevered Bernoulli-Euler Beam by Differential Transform Method, Journal of Sound and
Vibration (In Press).
J.R. Banerjee, Free vibration of axially loaded composite Timoshenko beams using the
dynamic stiffness matrix method, Computers and Structures 69 (1998) 197-208
J. Li, R. Shen, H. Hua and X. Jin, Bending-torsional coupled vibration of axially loaded
composite Timoshenko thin-walled beam with closed cross-section, Composite Structures, 64
(2004) 23-35
1