4. ANKARA INTERNATIONAL AEROSPACE CONFERENCE
27-29 September, 2007 - METU, Ankara
AIAC-2007-000
FLEXURAL–TORSIONAL COUPLED VIBRATION ANALYSIS OF A ROTATING CLOSED
SECTION COMPOSITE TIMOSHENKO BEAM BY USING DTM
O. Ozdemir Ozgumus 1, M. O. Kaya2
Istanbul Technical University, Faculty of Aeronautics and Astronautics, Istanbul, Turkey
ABSTRACT
This study introduces the Differential Transform Method (DTM) to analyse the free vibration response of
a rotating, closed section, composite, Timoshenko beam which features material coupling between flapwise
bending and torsional vibrations due to ply orientation. The governing differential equations of motion are
derived using Hamilton’s principle and solved by applying DTM. The natural frequencies are calculated and
the effects of the bending-torsion coupling, the slenderness ratio and several other parameters on the natural
frequencies are investigated using the computer package, Mathematica. Wherever possible, comparisons are
made with the studies in open literature.
COMPOSITE BEAM MODEL
A straight composite beam with length L , height h and breadth b is shown in Fig. 1. In the righthanded Cartesian coordinate system, the x -axis is the centroidal axis of the beam. The flapwise
displacement, w( x, t ) and the torsional rotation,  ( x, t ) occur about the y -axis and the x -axis,
respectively. Here, x and t , respectively denote the spanwise coordinate and the time. The beam rotates
with a constant angular velocity,  . Since the cross-sections of the beam have symmetry in both planes, the
x -axis is also the locus of the geometric shear centers of the beam cross-sections. Therefore, the beam
features material coupling between flapwise bending and torsional vibrations only due to ply orientation.

z
y
R
h
L
x
b
Figure 1. Configuration of a Uniform, Rotating, Composite, Timoshenko Beam
1
2
Research Assistant in Aeronautics Department, E-mail: ozdemirozg@itu.edu.tr
Associate Professor in Aeronautics Department, E-mail: kayam@itu.edu.tr
AIAC-2007-000
Ozdemir Ozgumus, Kaya
FORMULATION
Potential and Kinetic Energy Expressions
The total potential energy expression, U is given by Ref.[1] and derivation of the kinetic energy
expression,  is made in this study. As a result, the following expressions are obtained for a rotating,
composite Timoshenko beam
L




1
2
2
2
2
2
U   EI y    T w  I  A   2K    kAGw     GJ   dx
20
L



 




1
Aw 2  I  2  2 I z     I y  2  2   2 2     2 dx

20

(1a)
(1b)
x and time t , respectively.
 is the material density; A is the cross sectional area; A is the mass per unit length, I y and I z
are the second moments of inertia of the beam cross section about the y and z axes respectively; I is
the polar mass moment of inertia per unit length about the x axis; EI y , GJ , K and kGA are the flapwise
where primes and dots denote differentiation with respect to spanwise coordinate
Here
bending rigidity, torsional rigidity, bending–torsion coupling rigidity and shear rigidity of the composite beam,
respectively.
Governing Equations of Motion
The governing differential equations of motion and the associated boundary conditions are obtained
applying the Hamilton’s principle to the energy expressions given by Eqs. (1a) and (1b).
Equations of motion:
I y  I y  2  EI y   kAG(w   )  K   2I y   0
(2a)
  Tw  kAGw     0
Aw
(2b)

I   I y  2  GJ   I  AT   K   2 I y   0
(2c)
where T is the centrifugal force that is given by
L
T ( x)   A 2 ( R  x)dx
(3)
x
Boundary Conditions:
 The geometric boundary conditions at the cantilever end,
x  0 , of the composite beam,
w0, t    0, t    0, t   0
 The natural boundary conditions at the free end,
(4a)
x  L , of the composite beam,
Bending moment:
EI y   K   0
(4b)
Shear force:
Tw  kAGw     0
(4c)
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Ankara International Aerospace Conference
AIAC-2007-000
Ozdemir Ozgumus, Kaya
TI  A   K   GJ   0
Torque:
(4d)
Exponential Solution and Dimensionless Equations of Motion
A sinusoidal variation of w( x, t ) ,  ( x, t ) and  ( x, t ) with a circular natural frequency
and the functions are approximated as exponential solutions.
wx, t   w x e it
,
 x, t    x e it
,
 x, t    x e it

is assumed
(5)
Additionally, the following dimensionless parameters are used to be able make comparisons with the
studies in open literature
x
x
~  w ,   R , r2  I
, w
L
L
L
AL2
(6)
Substituting Eq.(5) into Eqs.(1a)-(1c) and using the dimensionless parameters, equations of motion can
be rewritten as follows
A1
~
d 2
d 2
dw

A


A

A


A
0
2
3
4
5
dx
dx 2
dx 2
(7a)
~
~

x 2  d 2w
dw
~  B d  0
 B1  x  
 2  B2  x 
 B3 w
4
2  dx
dx
dx

(7b)

x 2  d 2
d
d 2
 C1  x    2  C 2  x 
 C3  C 4
 C5  0
2  dx
dx
dx 2

(7c)
where the dimensionless coefficients are
A1  1 , A2 
AL4 r 2

EI y
2

 2 
kAGL2
EI y
,
A3 
K
EI y
kAGL2
AL4 r 2
A2  2i
, A5 
EI y
EI y
B1   
1
kAG

2 AL2  2
1
GJ
C1    
2 I s L2  2
C5  2i
,
B2  
(8a)
,
 
B3   

2
,
B4 
AL2 r 2
 
, C3    
Is

kAG
AL2  2
2
,
C2  B2
, C4 
(8b)
K
I s L2  2
 AL2 r 2

(8c)
Is
APPLICATION OF THE DIFFERENTIAL TRANSFORM METHOD
The Differential Transform Method is a transformation technique based on the Taylor series expansion
and it is a useful tool 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
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Ankara International Aerospace Conference
AIAC-2007-000
Ozdemir Ozgumus, Kaya
functions and the solution of these algebraic equations gives the desired solution of the problem with great
accuracy. The application procedure of this method can be found in Ref.[2]. Applying the Differential
Transform Method to Eqs. (7a)-(7b), the following transformed equations of motion are obtained.
A1 k  2k  1 k  2  A2 k   A3 k  2k  1 k  2  A4 k   A5 k  1W k  1  0
(9a)
k k  1

B1 k  2k  1W k  2  B2  k k  1W k  1   B3 
W k   B4 k  1 k  1  0
2 

(9b)
k k  1 

C1 k  2 k  1 k  2  C 2  k k  1 k  1  C 3 
 k  
2 

C 4 k  1k  2  k  2  C 5 k   0
(9c)
where
~,
 k  , W k  and  k  are the transformed functions of  , w
, respectively.
Additionally, transformed boundary conditions are obtained as follows by applying DTM to Eqs.(4a)–(4d)
at
x 0

 0  W 0   0  0
(10a)
at
x 1

A4 k  1 k  1  k  1 k  1  0
(10b)
B1 k  1wk  1   k   0
(10c)
C1 k  1 k  1  C3 k  1 k  1  0
(10d)
RESULTS AND DISCUSSIONS
The computer package Mathematica is used to write a program for the expressions given by Eqs.(9a)(10d). In order to validate the computed results, an illustrative example that studies a nonrotating composite
Timoshenko beam is taken from Ref. [3] is solved by using the formulas given above. When the results are
compared with the ones given in Ref. [3], it is seen that there is a very good agreement between the results.
Additionally, effect of the rotation speed on the natural frequencies is examined and the related graphic is
plotted.
The beam model that is studied in Ref.[3] is a uniform, nonrotating, cantilever glass-epoxy composite
beam with a rectangular cross section with width = 12.7 mm and thickness = 3.18 mm. Unidirectional plies
each having fiber angles of  15 are used in the analysis. The data used for the analysis are as follows:
0
EI  0.2865 Nm 2
GJ  0.1891 Nm 2
kAG  6343.3 N
  0.0544 kg / m
I s  0.777  10 6 kgm
K  0.1143 Nm 2
L  0.1905 m
r 2  0.00002322
  0 rad / sec
In Table 1, the calculated results are compared with the ones given by Ref. [2] and a vewry good
agreement is observed.
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Ankara International Aerospace Conference
AIAC-2007-000
Ozdemir Ozgumus, Kaya
Table 1. Validation of the Calculated Results
Natural Frequencies
DTM
Ref.[3]
30.7471
30.747
189.779
189.779
518.791
518.791
648.169
648.169
986.199
986.199
1564.75
1564.751
1
1
0.75
0.75
Second Mode Shape
First Mode Shapes
In Fig.2, mode shapes of a rotating composite Timoshenko beam are plotted.
0.5
0.25
0
0.25
0.5
0.75
0.2
0.4
0.6
0.8
0.25
0
0.25
0.5
0.75
1
1
1
0.75
0.75
Fourth Mode Shape
Third Mode Shape
0
0.5
0.5
0.25
0
0.25
0.5
0.75
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0.5
0.25
0
0.25
0.5
0.75
0
0.2
0.4
0.6
0.8
1
5
Ankara International Aerospace Conference
AIAC-2007-000
Ozdemir Ozgumus, Kaya
In Table 2., effect of the rotational speed on the natural frequencies is given.
Natural Frequencies
Rotatinal Speed  (rad / sec)
0
50
100
150
200
250
300
30.747
32.363
36.808
43.273
51.081
59.827
69.313
189.779
189.733
189.597
189.376
189.081
188.733
188.368
518.791
518.461
517.459
515.743
513.242
509.847
505.408
648.169
648.231
648.116
647.923
647.651
647.298
646.861
986.199
985.756
984.415
982.144
978.886
974.561
969.059
1564.751
1564.237
1562.684
1560.066
1556.335
1551.425
1545.248
1944.309
1944.231
1943.997
1943.604
1943.051
1942.334
1941.447
2241.307
2240.888
2239.557
2237.104
2233.289
2228.004
2221.298
References
[1] M. O. Kaya, O. Ozdemir Ozgumus, Flexural–Torsional Coupled Vibration Analysis of Axially Loaded
Closed Section Composite Timoshenko Beam by Using DTM, Journal of Sound and Vibration,
doi:10.1016/j.jsv.2007.05.049
[2] Ö. Özdemir, M.O. Kaya, Flapwise Bending Vibration Analysis of A Rotating Tapered Cantilevered
Bernoulli-Euler Beam by Differential Transform Method, Journal of Sound and Vibration 289, 413–420, 2006..
[3] 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, 23-35, 2004.
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Ankara International Aerospace Conference