G Bashir Asdaque

R K Behera

S Bala Murugan

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73 
 
Research Article 
International Journal of Current Engineering and Technology  
ISSN 2277 - 4106 
© 2014 INPRESSCO. All Rights Reserved. 
Available at http://inpressco.com/category/ijcet  
Vibration Analysis of Convergent-Divergent Shaft-Rotor System 
 
P M G Bashir Asdaque
Ȧ*
, R K Behera
Ȧ
 and S. Bala Murugan
Ȧ 
 
ȦDepartment of Mechanical Engineering, N.I.T. Rourkela, Rourkela, India 
 
Accepted 18 March 2014, Available online 01 April 2014, Special Issue-3, (April 2014) 
 
 
Abstract 
  
Cantilevered shaft-rotor systems consisting of disk and profiled shafts are considered. The procedures for the 
determination of the deflection, slope, shear force and bending moment at the extremities of the shaft are used. Transfer 
Matrix Method is used for the computation of the resonance, critical speed or whirling frequency conditions. Here shaft-
rotor has three portions namely convergent, neck and divergent portion.  For particular profiles, shaft length and rotor 
speeds, the response of the system are determined for the establishment of the dynamic characteristics. A built-in shaft-
rotor system with one disk and 1N force on the disk is investigated for illustration purposes. Hence the analysis of the 
shaft whose radius is the function of length has been done. 
 
Keywords:Convergent, Shaft, Rotor, Frequency, Whirling, Response. 
 
 
1. Introduction 
 
1
 Rotating Shafts are employed in industrial machines like 
steam and gas turbines, internal combustion engines, 
compressors and pumps for power transmission. With 
increasing demand for power and high speed 
transportation, study of vibratory motion becomes 
essential. Rotor Dynamics deal with dynamics of rotating 
machinery. It is different from structural vibrations 
analysis because of the gyroscopic moments. Basic idea of 
Transfer Matrix Method was first put forth by Holzer for 
finding natural frequencies of torsional systems. Later 
adapted by Myklestad for computing natural frequencies 
of airplane wing. Prohl applied it to rotor-bearing systems 
and included gyroscopic moments in his computations. 
Lund [4] showed how system damping could be accounted 
for including self-exciting influences, such as oil whip and 
internal frictions. The above developments leads to the 
method came to be known as “The Transfer Matrix 
Method” (TMM).  The design of shaft-rotor systems 
includes computation of the critical speed. An improved 
method for calculating critical speeds and rotor stability is 
done by Murphy and Vance. Whalley and Abdul Ameer 
used frequency response analysis for profiled shafts to 
study dynamic response of distributed-lumped shaft rotor 
system. They studied the system behaviour for the shafts 
with diameters which are functions of their lengths. They 
derived an analytical method which uses Euler-Bernoulli 
beam theory in combination with transfer matrix method 
(TMM). Here, profiled shaft with two disks have been 
considered with the method TMM for Frequency 
                                                          
*Corresponding author: P M G Bashir Asdaque 
Response Calculation and then is validated with Whalley 
and Ameer for a single disk. 
 
2. Mathematical modeling 
 
2.1 Shaft Model 
 
Input and output relationship for the distributed parameter 
shaft model (Whalley and Ameer, 2009) is given by, 
 
(             ) ᵀ = F(s) (             ) ᵀ 
 
2 3
32 4
2 3
2
34 2
2
2
3 4 2
2
23
32 4
3 2
(0) (0)
1 2 (0) 2 (0) 2 (0)
(0)(0) (0)
12 2 (0) 2 (0)
(0) (0)
12 (0) 2 (0)2 (0)
(0)(0) (0)
122 (0) 2 (0)
( )
c l ff c l f
c l ff c lf
l
f f f
c ll c
ff f
lc l l c
f
f
F s
f
f
  

 
 

 
 

 

 
 
 
 
 
 
 
  
 
 
The elements of F(s) are- 
 
   = 
(cosh ( ) cos ( )) / 2l l 
   
   = 
(sinh ( ) sin ( )) / 2l l 
   
   =
(cosh ( ) cos ( ))l l 
&   
   = 
(sinh ( ) sin ( ))l l 
   
 
Where ( ) ( )x l x   , l = length of the distributed 
parameter shaft, 
1 1 1
2 2 4( ) ( ( ) ( ))x s L x c x  and 
P M G Bashir Asdaque et al                                                  International Journal of Current Engineering and Technology, Special Issue-3, (April 2014) 
 
74 
 
1
( )
( )
c x
EI x

 
 
2.2 Rigid disk model 
 
The output vector from the shaft will become the input for 
the rigid rotor model. i.e., for single disk and shaft model, 
we have 
 
  ( )    ( )     
  ( )    ( )     
   ( )        ( )     ( )  
   ( )    
   ( )     ( )   
 
Hence 
 
(Y3(s), θ3(s), My3(s), Qy3(s))
T 
= 
      R(s) (Y2(s), θ2(s), My2(s), Qy2(s)) 
 
Where, 
 
R(s)= 
2
1 0 0 0
0 1 0 0
0 1 0
0 0 1
J s
ms
 
 
 
  
 
 
 
 
3. Numerical Results 
 
3.1 Cantilever Shaft-rotor system with a disk 
 
A cantilever rotor shaft system with one disc at free end is 
shown for illustration purposes in Fig. 1. Effects of 
bearings are neglected. As we proceeded in above 
sections, in the same way for the system illustrated in 
Fig.3 can be formulated as- 
 
H(s) =. ( )  ( ) 
 
(  ( )   ( )) (  ( )   ( )) (   ( )    ( ))  and 
(   ( )    ( )) are the deflections, slopes, bending 
moments and shear forces at the fixed and free end 
respectively. 
 
11 12
21 22
( )
H H
H s
H H
 
  
 
 
 
Input-output vectors relationship is given by: 
 
3 1
3 1
3 1
3 1
( )
y y
y y
y y
H s
M M
Q Q
 
   
   
   
   
   
   
 
 
After applying the boundary conditions, we getdeflection 
3y  at the free end of the system, so we will ultimately get 
the Transfer Function. For example, TF will be obtained 
for NN=20, 40 and 60 respectively.  
 
 
Fig. 1 Cantilever rotor with a disc 
 
Validation: The profile equation for the shaft-rotor 
(Whalley and Ameer, 2009) is given by- 
 
2( ) 0(1 ( ))r x r NN x   
 
For NN=25, r0=0.005 and 10,000 rpm, for single disk we 
get approximately the same results as in (Whalley and 
Ameer, 2009).  
 
Fig. 2 The Bode plot for profiled shaft with single disk 
 
3.2 Convergent-divergent shaft rotor system 
 
 
Fig. 3 Convergent divergent shaft rotor system with 1 N 
force at the tip 
 
Fig. 3 shows the convergent-divergent shaft-rotor system 
for the illustration purpose. The profile equation for the 
shaft-rotor is given by- 
2( ) 0(1 ( ))r x r NN x  
The „ ‟ sign indicates the decrement and then increment 
in the cross-sectional area of the shafts.  A unit impulse is 
P M G Bashir Asdaque et al                                                  International Journal of Current Engineering and Technology, Special Issue-3, (April 2014) 
 
75 
 
applied on the disk at the free end to produce excitations in 
the shaft-rotor system. 
 For convergent part the diameter of the shaft is 
continuously decreasing while after the neck portion, i.e., 
for the divergent part the diameter is continuously 
increasing till the complete length of the shaft-
rotor.Values of different parameters of the convergent-
divergent shaft-rotor system are given in Table 1. 
 
Table 1 Various parameters of convergent-divergent 
shaft-rotor system as shown in Fig. 3. 
 
Parameters Values 
Length of the shaft-rotor, 1 2,l l  (m) 
0.1, 0.1 
Mass of the disk, m (Kg) 0.75 
Diameter of the disk, D (m) 0.09 
Young‟s Modulus of Elasticity, E (GPa) 209 
Density of the material,  ( 3/Kg m ) 7800 
Rotational Speed, N (rpm) 10000 
Profile Value, NN (Constant) 15 
 
The values for the „neck-radius‟ for different values of NN 
is given in Table 2. 
 
Table 2 Neck radius for various profile values (NN) for 
convergent-shaft length of 0.1 m 
 
Profile Value NN Neck radius (m) of the shaft-rotor 
shown in Fig 4 
15 0.0043 
25 0.0037 
40 0.0030 
 
The transfer function obtained for the system shown in 
Fig. 3, for the values given in Table 1, is given as, 
 
3 2 5 9
1.333 s + 7090
s  + 5317 s  + 7.673 10  s + 1.044 10 
 
 
3.2.1 Varying Profiles 
 
Fig. 4 Bode plot for varying profiles 
3.2.2 Different lengths 
 
Fig. 5 Bode plot for varying length with profile value 
N=15 
 
3.2.3 Changing rotor speed 
 
Fig. 6 Bode plot for varying rotor speeds 
 
3.2.4 Summary 
 
Bode plots for various profile values, shaft lengths and 
rotor speed has been plotted in Figs. 4, 5 and 6 
respectively. The results obtained from the Bode plots are 
tabulated in Table 3. 
 
Table 3 Results obtained from Bode plots for convergent-
divergent shaft-rotor system 
 
1l
 (m) 
2l  
(m) 
Speed 
(rpm) 
Value 
of NN 
Critical 
Frequency 
(rad/sec) 
Amplitude 
(dB) 
0.10 0.10 10000 15 447 -91.3 
0.10 0.10 10000 25 370 -91.0 
0.10 0.10 10000 40 273 -89.9 
0.15 0.15 10000 15 161 -76.4 
0.20 0.20 10000 15 47 -62.4 
0.10 0.10 1000 15 449 -71.3 
0.10 0.10 2500 15 449 -79.3 
0.10 0.10 4000 15 448 -83.4 
P M G Bashir Asdaque et al                                                  International Journal of Current Engineering and Technology, Special Issue-3, (April 2014) 
 
76 
 
Conclusion 
 
The establishment of the vibrational characteristics of 
profiled shaft elements present challenging problems. The 
vibration analysis via bode plot has been done for the 
convergent-divergent rotor system. Here, disk-rotor 
system is shown for the illustration purposes. For a 
particular speed but value of NN is varied, as shown in 
Fig. 4,  the resonant frequency decreases with the increase 
in the value of NN while amplitude increases for values of 
NN from 15 to 40.  For different lengths, we obtain a bode 
plot as shown in Fig. 5, and it depicts that with slight 
increase in length of convergent and divergent part, the 
resonance frequency decreases drastically while the 
amplitude increases. For changing rotor speeds, we notice 
that the critical frequency remains the same while the 
amplitude of vibration decreases due to increasing effect 
of gyroscopic couple with increasing speed. Hence, such 
type of complicated systems can be analysed with 
simplicity using the approach discussed here in. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
References 
 
N. O. Myklestad, (1944), A new method for calculating natural 
modes of uncoupled bending vibration of airplane wings, 
Journal of Aeronautical Science, 153–162. 
N. O. Myklestad, (1945), New Method of Calculating Natural 
Modes of Coupled Bending-Torsion Vibration of Beams, 
Trans. ASME, 61-67. 
M. A. Prohl, (1945), A General Method for Calculating Critical 
Speeds of Flexible Rotors, Trans. ASME, 66, A-142-A-148. 
J. W. Lund, (1974) , Stability and damped critical speeds of a 
flexible rotor in fluid-film bearings, Journal of Engineering 
for Industry, Trans. ASME, SeriesB, 96(2), 509-517. 
B. T. Murphy, J. M. Vance, (1983), An improved method for 
calculating critical speeds and rotor dynamic stability of turbo 
machinery, ASME Journal of Engineering for Power, 105,  
591-595. 
R. Whalley, A. Abdul Ameer, (2009), Whirling prediction with 
geometrical shaft profiling, Applied Mathematical Modeling, 
33, 3166-3177. 
 
 
 
 


Vibration Analysis of Convergent-Divergent Shaft-Rotor System

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Vibration Analysis of Convergent-Divergent Shaft-Rotor System

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