16th. Annual (International) Conference on Mechanical Engineering-ISME2008
May 14-16, 2008, Shahid Bahonar University of Kerman, Iran
ISME2008_2332
Rotors Frequency and Time Response of Torsional Vibration Using Hybrid Modeling
M. Tahani
Assistant professor, Ferdowsi
University of Mashhad, Iran
mtahani@ ferdowsi.um.ac.ir
S. Soheili
PhD student, Ferdowsi
University of Mashhad, Iran
soheili78@yahoo.com
M. Abachizadeh
MSc of Mechanical Engineering,
Ferdowsi University of Mashhad, Iran
m_abachizadeh@yahoo.com
A. Farshidianfar
Associate professor, Ferdowsi
University of Mashhad, Iran
farshid@ferdowsi.um.ac.ir
Abstract
In this paper, the application of distributed-lumped
(hybrid) modeling technique (DLMT) in modeling of
the forced torsional vibration of systems is investigated.
To illustrate the simplicity and efficiency of the method,
an industrial example of rotating shaft with a lumped
element subjected to various torques is analyzed.
Natural frequencies obtained by this method are
compared with those obtained by using finite element
and Holzer's method. In addition, time responses of the
system subjected to three different types of torsional
forces are computed by this method.
Keywords: Distributed, Lumped, Modeling, Frequency
response, Time response.
Introduction
The question of vibration model of industrial systems,
especially rotating shafts, is the basic consideration in
engineering design of dynamic systems. Not only the
avoidance from natural frequencies of such systems
have been observed since long before, but the condition
monitoring (CM) of highly sensitive and precious plants
such as turbines and jet rotors are applied widely using
accurate vibration model of systems. Condition
Monitoring for rotating machinery incorporates a wide
range of techniques, such as oil analysis, wear-debris
analysis, ultrasonic, corrosion, and vibration analysis.
Vibration condition monitoring is, arguably, the oldest
type of machinery condition monitoring. Measured
vibration signals can reveal important and detailed
information about possible fault which may exist in a
machine [1]. Fault identification in rotating machinery
using vibration analysis is a constantly expanding field.
Developments are continually made with the use of new
analysis methods, increased computing power,
measurement techniques, and so on.
Among different methods of modeling systems such as
lumped-lumped modeling technique (LLMT) and
distributed-lumped modeling technique (DLMT), or
numerical and approximate methods such as transfer
matrix method (TMM) and finite element method
(FEM), it is clear that the model combined with both the
distributed and lumped elements, is the best
representative of complex and accurate systems [2].
Many industrial systems can be modeled as a rotating
shaft with disks on it, such as gear systems, propellers,
pumps, turbines, compressors, jet engines, etc. In such
systems, the disk, which is the representative of blade,
gear, etc., is impressed by different loads, which affect
the vibration of system and lead to different frequency
and time responses. Comparison between safe and
defected system responses brings us an effective and
advantageous method to the condition monitoring of
expensive and important systems such as jet engines.
In this study, for a simple example, the torsional
vibration of a general three stage distributed-lumpeddistributed system is considered. The system is modeled
by distributed-lumped technique, and the natural
frequencies are investigated for two sets of boundary
conditions (BCs): clamped-free and clamped-clamped
shaft, which are more common in real systems. To
check the correctness and accuracy of the present
method, the natural frequencies and mode shapes of an
industrial example of the system achieved from this
method are compared with those obtained by utilizing
the commercial finite element package of ANSYS,
revision 9 and also Holzer's method. The frequency
responses are computed in response to the limited step,
limited ramp, and delta force functions, using hybrid
model of the system for clamped-free shaft. Time
responses are also calculated employing both the
inverse Fourier transform (IFT) and convolution integral
together.
The General Distributed-Lumped Model
Generally speaking, hybrid modeling technique deals
with systems by dividing them into two element types.
1) The distributed element, which is the main part
of shafts, rotors or any other continuous part of
the systems with distributed mass or inertia.
2) The lumped element which is the
supplementary part of shafts, rotors, etc. with
concentrated mass or inertia such as disks,
gears, propellers, blades, and so on.
In this way, a system is considered as a combined set of
distributed and lumped elements, in which the vibration
of final model is obtained by setting the distributed and
lumped matrices of different parts and combining them
together (see Figure 1). Distributed and lumped matrices
are formed according to the analytical equations of
motion, so this is an exact method in contrast with other
approximate methods such as transfer matrix method,
finite element method, and so on. Another advantage of
this method compared with analytical method is that the
continuity conditions between distributed and lumped
elements are identically satisfied, and it remains only to
satisfy the boundary conditions of the system. To this
end, there is no difficulty in using this approach for
analyzing systems with mixed series of distributed and
lumped elements as it is shown by Bartlett et al. [3].
 2k
 2k  0
x
in which, k is the main function as:
2
k   ( x, s ) or T ( x, s)
(9)
and
s

E
The general solution of equation (8) is given by:
k  r1ex  r2e x
where
(10)
(11)
e x  cosh x  sinh x
Figure 1 - General series representation of a distributed
lumped parameter system (Hybrid Model)
(8)
e  x  cosh x  sinh x
Therefore, the solution for k can be rewritten as:
(12)
k  (r1  r2 ) coshx  (r1  r2 ) sinh x
Deriving Transfer Matrix for the Distributed
Element
The equations of motion for torsional vibration of a rod
with the density  , the shear modulus of elasticity G
and the polar moment of inertia J can be expressed by
the following equations (e.g., see [4,5]):
T ( x, t )
 2 ( x, t )
 J
(1)
x
t 2
 ( x, t )
1
(2)

T ( x, t )
x
GJ
where  ( x, t ) and T(x,t) are the torsional angle and torque
functions, respectively; x is the distance along a section
and t is time.
Differentiating equation (2) with respect to x and
substituting for T x in equation (1) yields:
 2 ( x, t )
  2 ( x, t )
(13)
Hence, the solution of equations (7) will be in the
following form:
T ( x, s)  A coshx  B sinh x
(14)
 ( x, s)  C sinh x  D coshx
The unknown constants of integration, A and D, are
obtained by imposing the boundary conditions at x=0.
That is,
A  T (0, s)
(15)
D   (0, s)
Next, it remains to find B and C in equations (14).
Differentiating equations (14) with respect to x and
substituting for T x and  x from Laplace
transformation of equations (1) and (2), respectively,
gives:
Js 2 ( x, s)  A sinh x  B cosh x
G t 2
x 2
Also differentiating equation (2) twice with respect to t
results in:
(16)
1
T ( x, s)  C cosh x  D sinh x
GJ
Upon substitution of x=0 in equations (16), the following
results would be obtained:
 3 ( x, t )
B  sJ G  (0, s )    (0, s )

(3)
1  2T ( x, t )
(4)
GJ
xt
t 2
Moreover, differentiating equation (1) with respect to x
gives:
2
 2T ( x, t )

 J
 3 ( x, t )
(5)
x 2
xt 2
and substituting equation (4) into (5) results in:
 2T ( x, t )

  2T ( x, t )
(6)
G t 2
x 2
Equations (3) and (6) are the main equations of
torsional vibration of the shaft. Assuming zero initial
conditions, Laplace transformation of equations (6) and
(3) gives:
 2T ( s, t )
x
2
 2 ( s, t )


G

s 2T ( s, t )  0
(7)
 s  ( s, t )  0
G
x 2
where s is the Laplace transform variable. Equations (7)
can be written in the compact form as follows:
2
C
1
T (0, s ) 
1
T (0, s )
(17)

sJ E
Hence, the solution of equations (7) for the jth element
can be expressed in the matrix form as follows:
 cosh j x  j sinh  j x  T (0, s)


T j ( x, s) 
 1
 j


(18)


  sinh  x cosh x  


(
x
,
s
)

(
0
,
s
)
j
j




j
 j
  j




where
 j  sJ j  j G j
(19)
According to Figure 2, for the jth element at x=0 there
would be:
T j (0, s )  T j 1 ( s)
(20)
 j (0, s)   j 1 ( s)
Therefore, the main matrix for the distributed element
representative of torsional vibration is as follows:
 cosh  j l j
T j ( s) 

 1

   sinh  l
j j

 j ( s)
  j

 j sinh  j l j 
T j 1 ( s) 



cosh  j l j  

(
s
)


  j 1 
(21)
Figure 2– General model of the rotating rotor system
Deriving Transfer Matrix for the Lumped Element
The equation of motion and continuity conditions in
Laplace domain of the jth lumped element, which is
exposed to the applied force f in the radius r are written
as:
T j ( s)  T j 1 ( s )  f j ( s )r j  J dj s 2 j ( s )
 j ( s)   j 1 ( s )
(22)
Figure 3– Hybrid model of the rotating rotor system
where

J 2d s 2 

1 
0
TL 2  1
(27)
Substituting equations (24) into (26) yields [6,7]:
1


cosh 2l  J 2d  1s 2 sinh 2l  sinh 2l  J 2d s 2 cosh2 l
 T0 
T3  
2


 
 
 3   1 sinh 2l  J d   2 s 2 sinh 2 l cosh 2l  1 J d  1s 2 sinh 2l   0 
2
2
2


where J dj is the mass polar moment of inertia for disk
 cosh l  sinh l   fr 
  1


 sinh l cosh l   0 
and r is the radius which the force is applied. Therefore,
equation (22) can be expressed in the matrix form as:

T j (s) 
 1 J dj s 2  
T j 1 (s) 
  f j r 
(23)





1  

 j (s)
 0
 j 1(s)
  0 
(28)
Equation (23) may be shown in the simple form as:
T3 
T0 
 fr 
(29)
   C    D



 0 
 3
 0
where
Illustrative Example
In this section, the methodology outlined previously is
applied to a shaft with a disk on its middle (see Figure
2), which is a simplified model for useful and common
industrial systems. The properties of the system
considered here are shown in Table 1. As mentioned
already, the present method can be used for analyzing
systems with any number of distributed and lumped
elements without any increasing in difficulty.
Table 1– Properties of the system
Shaft Length 2l
4m
Shaft Diameter d shaft
0.15 m
Mass of Shaft per Unit Length m
Density of Shaft Material ρ
Modulus of Elasticity of Shaft E
Shear Modulus of Shaft G
Mass of Disk
Radius of Disk
Mass Moment of Inertia for Disk
Disk Thickness
137.837 kg
7800 kg/m3
200 GPa
80 GPa
100 kg
1m
50 kgm2
0.08 m
DLMT Solution
To represent the main hybrid model of the system, one
should notice that the system is combined of two
distributed and one lumped elements (Figure 3). For the
distributed elements 1 and 3 the transfer matrices can be
written according to equation (21) as follows:
T0  T3 
T1 
T2 
   TD 1   ,    TD 3  
1 
 2 
 0   3 
(24)
 coshl


 sinh l 

coshl 

 cosh l  sinh l 
(30b)
[ D]   1

 sinh l cosh l 
Equation (29) is the transfer matrix of the overall
system relating torques and torsional angles of the left
and right ends of the system.
The Laplace transform variable ‘s’, in general, is the
representative of equation s    i ; in which the real
part ( ) shows damping, and the imaginary part ( )
shows the vibrating frequency. It is assumed in the
present example that   0 and, therefore, equation
(29) will be altered from Laplace domain into frequency
domain by putting s  i [6].
For each sets of boundary conditions, one characteristic
equation can be obtained that its solution will give the
natural frequencies of the system. In the following, two
numerical examples are presented for a rotating shaft
with a lumped mass on it with two different boundary
conditions (i.e., clamped-free and clamped-clamped) to
check the accuracy of the present method.
Clamped-Free System
Assuming that the shaft is clamped at the position zero,
and free at the position 3, the boundary conditions will
be (see Figure 2):
 0  0 (at clamped end )
(31)
(25)
According to the above relations, equation (29) can be
rearranged as follows:
(26)
 1
T0   C11
   C
 3   21
 C11
Also the transfer matrix for the lumped element is:
T2 
T1   fr 
   TL 2    

 2 
1   0 
(30a)
T3  0 (at free end )
where
TD    1 sinh l
1 d 1 2


d 2
2
 cosh 2l  2 J 2  s sinh 2l  sinh 2l  J 2 s cosh l 
[C ]  

 1 sinh 2l  J d   2 s 2 sinh 2 l cosh 2l  1 J d  1s 2 sinh 2l 
2
2
2


C12
D



 11
0
 T3  
 fr 
C11
D11
  


C 21C12
C 21




0



 C 22

D  D21 0  0 

 C11 11

C11

(32)
where  0 , T3 , and f are the inputs and  3 and T0 are
the outputs of the system.
In this case, the natural frequencies are obtained by
plotting  3  0 , for instance, as shown in Figure 4. In
this view, the natural frequencies occur at the peaks of
the spectrum. From equation (32), it is clear that the
peaks are the result of denominator approaching zero.
Since in all relations C11 is the denominator, so the
natural frequencies are the roots of C11 .
0.7
Hence, equation (29) can be rearranged as:
 C11
T3   C 21
  1
T0  
 C 21

C C 
C

C12  11 22 
D11  D21 11
C 21   3  
C 21
 
C
D
  0  
 22
 21


C 21
C
21



0
 fr 


0 
0 


(35)
where  0 ,  3 , and f are the inputs and T3 and T0 are
the outputs of system. The natural frequencies are
obtained by plotting 1 fr , for instance, as shown in
Figure 5. Similar to the discussion presented in the
previous part, the roots of C 21 should be computed in
this case. The results for this case are listed in Table 3.
0.6
1.2
x 10
-10
1
0.4
0.8
0.3
theta1/fr
theta3/theta0
0.5
0.2
0.1
0
0.4
0.6
0.4
0.6
0.8
1
1.2
1.4
w(rad/s)
1.6
1.8
2
x 10
0.2
4
Figure 4– Frequency spectrum for clamped-free BCs
( 3  0 vs.  (rad s) )
Other than that, substituting the relations (31) in
equation (29) and neglecting the term coincides with f
(because the natural frequencies are independent of
applied force) gives:
C11T0  0
(33)
C 21T0   3
The first equation is satisfied when C11  0 , which is
another reason for computing the roots of C11 to find
the natural frequencies as well. The results for this case
are listed in Table 2.
Table 2- Natural frequencies of clamped-free rotating
system.
Error
Error
Percent
Percent
Frequency DLMT FEM Holzer's
(FEM in (Holzer in
(Hz)
Method Method Method
respect to respect to
DLMT)
DLMT)
1
31.4
31.7
31.4
0.37
1.50
2
397.9 410.0
383.4
3.06
3.64
3
795.8 816.6
798.6
2.60
0.35
4
1193.7 1224.0 1134.8
2.50
4.93
5
1591.5 1632.0 1576.6
2.54
1.00
6
1989.4 2040.0 1862.1
2.54
6.40
7
2403.2 2449.0 2315.7
1.91
3.64
8
2875.2 2859.0 2548.8
2.65
8.48
9
3199.0 3270.0 2996.0
2.22
6.34
Clamped-Clamped System
In this case, the boundary conditions are expressed as:
0  0
(34)
3  0
0
1.4
1.6
1.8
2
2.2
2.4
w(rad/s)
2.6
2.8
3
3.2
x 10
4
Figure 5– Frequency spectrum for clamped-clamped
BCs ( 1 fr vs.  (rad s) )
Table 3- Natural frequencies of clamped- clamped
rotating system.
Frequency DLMT FEM Holzer's
(Hz)
Method Method Method
1
2
3
4
5
6
7
43.9
802.0
1602.0
2402.0
3203.0
4003.5
4804.0
45.2
815.3
1631.0
2449.0
3269.0
4093.0
4921.0
44.4
802.1
1582.0
2320.5
2998.5
Error
Error
Percent
Percent
(FEM in (Holzer in
respect to respect to
DLMT)
DLMT)
2.81
1.09
1.72
0.08
1.86
1.28
1.96
3.24
2.08
6.37
2.24
2.44
FEM Solution
To contrast and confirm the results with another
method, the finite element method is used to investigate
the natural frequencies. The system is modeled by
ANSYS (9) software, and meshed using brick 45 (8
nodes 3D) elements. Block Lanczos solver of ANSYS is
used in the analysis,and each distributed part is divided
into 100 elements. The natural frequencies are listed in
Tables 2 and 3. The first two mode shapes for clampedfree and clamped-clamped boundary conditions are
shown in Figures 6-9.
Holzer's Method
In order to verify the results of the two techniques,
Holzer's method is also applied to obtain the natural
frequencies of the mentioned system.
Figure 6– 1st mode shape for clamped-free BCs
Figure 8–1st mode shape for clamped-clamped BCs
Figure 7– 2nd mode shape for clamped-free BCs
Figure 9–2nd mode shape for clamped-clamped BCs
The employed equations are discussed extensively in
[4]. Each distributed part (shaft) is divided into 10
stations (rigid disk elements) and 10 fields (massless
shaft elements) and the frequency equation is solved by
using MATLAB software. The results are listed in
Tables 1 and 2 for the clamped-free and clampedclamped boundary conditions, respectively.
In equation (36), x( ) is  0 fr ,  3 fr or 1 fr
which are obtained from equations (32) or (26). Since
the referred functions are complex, the integral relation
can be expressed numerically as:
Frequency and Time Responses
Since the rotor systems are usually subjected to different
external torques, the effect of three important types of
torques on a clamped-free shaft is investigated.
To acquire the frequency response, three types of
applied torques (limited step, limited ramp and impulse
functions) in Laplace form are substituted into equation
(32). These three types of torques are shown in Figures
10-12. The frequency responses for  3 and T0 show
that these parameters tend to infinity at the natural
frequencies. Also the disk displacement can be obtained
through equation (26). However, for the sake of brevity,
theses parameters are not presented here.
To compute the time response, both the inverse Fourier
transform and convolution integral are used together.
Firstly, the inverse Fourier transform (IFT) which is
expressed as [8]:
1
x(t ) 
2

 x( ) e
i t
d
(36)

is used to find the time response of the system to the
unit impulse torque (without delay).
x(t ) 
k
A(r ) cos(r t )  B(r ) sin(r t )

1
(37)
r 1
where
r  r
(38)
In equations (38), A( r ) and B( r ) are the real and
imaginary parts of the function x( ) , respectively.
Secondly, the convolution integral, which is expressed
in the following form:
t
y (t ) 
 f ( ) x(t   )d
(39)
0
is used to compute time response of the system to the
torques mentioned before. In equation (39), f ( ) is the
torque defined in the matrix form, and the whole
relation can be computed numerically (for example, by
using function ‘conv’ in MATLAB software [9]).
Figures 13-15 show the dynamic responses of 1 ,  3 ,
and T0 , respectively. It can be seen that the first part of
spectrum has zero quantity (no force is applied). The
second part shows vibration, where the amplitude is two
times greater in response to limited step comparing with
limited ramp, as it is expected. The third part shows
vibration after finishing the torque, and its altitude is
related to the situation that torque comes to the end.
1500
2
x 10
-4
1.5
1
theta1(rad)
Torque (N.m)
1000
500
0
0.5
0
-0.5
-1
-1.5
-500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-2
0.8
0
0.1
0.2
0.3
t(second)
Figure 10- Delta force function
0.4
t(second)
0.5
0.6
0.7
0.8
Figure 13- Time response of disk vibration under delta
force function
1500
1.5
x 10
-3
1
0.5
theta3(rad)
Torque (N.m)
1000
500
0
-0.5
0
-1
-500
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-1.5
0
0.4
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
t(second)
t(second)
Figure 11- Step force function
Figure 14– Time response of free point vibration of
shaft under step force function
1500
2000
1500
1000
500
T0(N)
Torque (N.m)
1000
500
0
-500
0
-1000
-1500
-500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t(second)
-2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t(second)
Figure 12– Ramp-step force function
Figure 15–Time response of torque at the clamped point
of shaft under ramp-step force function
Conclusions
This paper shows how the DLMT can be used to
analyze torsional vibration of a complex rotating system
for investigating natural frequencies, time, and
frequency responses.
The frequencies computed by DLMT are compared with
those obtained by finite element and Holzer's methods,
in the case of clamped-clamped and clamped-free
boundary conditions. As it is shown in Tables 2 and 3,
the three methods are differed less than 9 percent which
confirms the DLMT results. Since the analytical
equations of motion are solved exactly by the
distributed-lumped technique, the achieved results are
of high accuracy.
It is also shown that DLMT can also be used to compute
the frequency and time responses to different forces and
torques, which can be used for condition monitoring of
vibrating systems. The DLMT can be used for other
kinds of vibration, for instance, longitudinal and
transverse vibrations. It is shown that while the new
method brings highly accurate results, its simplicity and
accuracy brings proper application on industrial
systems. The present method is straightforward and
general, and can readily be used in developing a more
advanced theory such as transverse vibration of
Timoshenko beams.
List of Symbols
E
k
Modulus of elasticity
Stiffness
References
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