The dynamics of the rubbing rotor system is analyzed by applying harmonic balance method. The relationship between harmonic components in the response of the rubbing rotor system and the dynamic stiffness matrix of the fault free rotor system is revealed, based on which a new model based method for rubbing identification is presented. By applying this method, the fault location and rubbing forces of the single rubbing rotor system can be identified by using vibration data of only two nodes, the rubbing locations and rubbing forces of the double rubbing rotor system can be identified by using vibration data of three nodes. The numerical simulations and experiments on the rotor test-rig are carried out to verify the efficiency of the present method

Bangchun  Wen

Hongliang  Yao

Qi Xu

Zhaohui  Ren

English

Vibroengineering PROCEDIA

  ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533 1 
The Quantitative Diagnosis Method of Rubbing Rotor System 
Hongliang Yao, Qi Xu, Zhaohui Ren and Bangchun Wen 
THE QUANTITATIVE DIAGNOSIS METHOD OF RUBBING ROTOR SYSTEM.  
HONGLIANG YAO, QI XU, ZHAOHUI REN AND BANGCHUN WEN 
Hongliang Yao, Qi Xu1, Zhaohui Ren and Bangchun Wen 
School of Mechanical Engineering and Automation, Northeastern University 
No. 11, Lane 3, WenHua Road, HePing District, Shenyang, Liaoning, China 110819 
 
E-mail: xuqi_010904@126.com 
Abstract. The dynamics of the rubbing rotor system is analyzed by applying harmonic balance method. The 
relationship between harmonic components in the response of the rubbing rotor system and the dynamic stiffness 
matrix of the fault free rotor system is revealed, based on which a new model based method for rubbing 
identification is presented. By applying this method, the fault location and rubbing forces of the single rubbing 
rotor system can be identified by using vibration data of only two nodes, the rubbing locations and rubbing 
forces of the double rubbing rotor system can be identified by using vibration data of three nodes. The numerical 
simulations and experiments on the rotor test-rig are carried out to verify the efficiency of the present method. 
1. Introduction 
Rotor-to-stator rub is a common fault in rotor system, which can cause catastrophic failure if 
undetected. The early detection of the existing of rubbing, and then obtaining the exact information 
about the fault location and the fault extent can significantly improve the safety issue in the operation 
of the rotating machinery.   
Extensive researches have been done on the forecast and detection of the rotor-to-stator rub in the 
past few decades. In general, there are mainly two types of diagnosis methods for rotor-to-stator rub: 
one is the signal-based method, and the other is the model-based method. The signal-based method 
uses vibration response signals and signal processing techniques such as FFT, STFT, CWT for the 
detection of rubs. A lot of researches have been conducted in this area, such as references [1-3]. 
Enough signal-based diagnosis tools have been provided to judge the existence of the rubbing fault, 
but the existing signal-based identification methods are difficult to detect the rubbing location and the 
fault extent. 
Model-based identification methods, on the contrary, can provide quantitative information about 
the fault, which includes the rubbing location and the fault extent. Excellent examples can be seen in 
references [4-7]. In this literatures, only the fundamental harmonic components are used in the model 
based diagnosis process. The sub and super harmonic components commonly exist in the response of 
the rubbing rotor systems, but are rarely utilized.  
So in this paper the relationship between the sub or super harmonic components in the response of 
the rubbing rotor system and the dynamic stiffness matrix of the fault free rotor system is revealed, 
based on which a new model based method for rubbing identification is presented. By applying this 
method, the location and rubbing forces of the single rubbing rotor system can be identified by using 
sub or super harmonic components of only two node’s responses, the rubbing locations and the 
rubbing force of the double rubbing rotor system can be identified by using sub or super harmonic 
responses of three nodes. To validate the efficiency of the present method, numerical studies and 
experiments on rotor test-rig are carried out. 
                                                     
1 Corresponding author 
THE QUANTITATIVE DIAGNOSIS METHOD OF RUBBING ROTOR SYSTEM.  
HONGLIANG YAO, QI XU, ZHAOHUI REN AND BANGCHUN WEN 
2 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533  
2. The mathematical model of the locally rubbing rotor system 
 
Figure 1. The finite element 
model of the rotor segment. 
 
The rotor is made up of elastic segments and each segment has two translational and two rotational 
degrees of freedom for bending mode at each node. The generalized displacement of each element is:  
  
T
r A A yA xA B B yB xB
x jy j x jy jθ θ θ θ = − − − − y   (1) 
The translating mass matrix, rotating mass matrix, stiffness matrix and gyroscopic matrix are 
shown in reference [8]. The finite element equation is 
  
r r r
+ + =MR CR KR Uɺɺ ɺ   (2) 
where, M  is the mass matrix, ω= +C D G  is the damping and gyroscopic matrix, ω  is the rotating 
speed, K  is the stiffness matrix, 
r
R  is the response vector, U  is the exciting force vector. The 
dimensions of the matrixes are 2J×2J and the dimensions of the vectors are 2J×1. 
Considering the stimulation caused by unbalance, the expression of U  is: 
  2 11
T
1 2 1
0 ... 0J
j tj t
J
Ae A e
ω ϕω ϕ
−
++
−
 =  U  (3) 
where, ( )1, ...,iA i J=  is the eccentricity concentrated on node i, and ( )1, ...,i i Jϕ =  is the phase of the 
unbalance force. 
Assuming the rub happens on node 
1
L  of the rotor system, the dynamic model of the fault rotor 
system is [9]: 
  ( ) ( )1 1
j t
r r r x y
j e
ω ξ−
+ + = − +MR CR KR U T P Pɺɺ ɺ  (4) 
where, 

1
T
2 2
0... 0 1 0... 0
L − 
 =
  
T  and 
1x
P , 
1y
P  are force components in x and y direction of the rubbing 
force at node
1
L . 
(4) - (2) obtains:  
  ( ) ( )1 1
j t
r r r x y
j e
ω ξ−
∆ + ∆ + ∆ = − +M R C R K R T P Pɺɺ ɺ  (5) 
3. The characteristics of the harmonic components of the equivalent system 
Consider the steady state situation of the rubbing rotor system, the responses of the equivalent system 
are in steady state, and the responses can be decomposed in to harmonic components 
  2 3
0 1 2 3
...
j t j t j t
e e e
ω ω ω
∆ = ∆ + ∆ + ∆ + ∆ +R R R R R  (6) 
The equivalent forces, which are all in the right side of equation (5), can be expanded to 
  ( ) ( ) ( )2 31 1 0 1 2 3 ...
j t j t j t j t
x y
j e A Ae A e A e
ω ξ ω ω ω−
− + = + + + +T P P T  (7) 
By applying the harmonic balance criteria from equation (5), for each harmonic component, one 
THE QUANTITATIVE DIAGNOSIS METHOD OF RUBBING ROTOR SYSTEM.  
HONGLIANG YAO, QI XU, ZHAOHUI REN AND BANGCHUN WEN 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533 3 
gets the following equation 
  ( ) ( )
1
1
0 0 2 1 0
0 0
L
A A A
−
−
∆ = = =R K T E T E  
  ( ) ( )
1
1
2
1 0 0 2 1 0L
j A j A j Aω ω ω ω
−
−
 ∆ = − + + = = R M C K T E T E  (8) 
  ( ) ( )
1
1
2 2
0 2 1 0l l L
l j l A jl A jl Aω ω ω ω
−
−
 ∆ = − + + = = R M C K T E T E  
where, ( )
1
2 2jl j l lω ω ω
−
 = + − E K C M  and 12 1( )LE jlω−  is the 12 1L −
th column of ( )jlωE . 
Assuming the lth harmonic component response of node i is jl t
i
e
ω
∆r , and that of node k is jl t
k
e
ω
∆r . 
It can be deduced from equation (8) that: 
  
( )
( )
1
1
,2 1
,2 1
k Lk
i i L
E jl
E jl
ω
ω
−
−
∆
=
∆
r
r
 (9) 
This means that the ratio of the harmonic components responses of any two nodes equals to the 
ratio of the corresponding elements in the 
1
2 1L −
th column of the dynamics stiffness matrix of the 
pre-fault rotor system.  
Equation (10) can be obtained from equation (9): 
  
( )
( )
2 1,
2 1,k
, 1, 2, ...,
n ik
n
i n
E jl
n J
E jl
ω
δ
ω
−
−
∆
= = =
∆
r
r
 (10) 
( )abs Lδ  will be the minimum of all the ( )abs nδ . So the rubbing location can be found when the 
minimum of ( )abs
n
δ  be found. 
After the location 
1
L  has been identified, the lth harmonic component response of every node of the 
system can be obtained as:  
  ( ) ( )
( )
( )1
1
2 1
,2 1
i
L
i L
jl
jl jl
E jl
ω
ω ω
ω
−
−
∆
∆ =
r
R E  (11) 
Assuming there are Nth order harmonic components that are evident in the response, then the 
rubbing force can be deduced as:  
  ( ) ( )
1
2 1
0
N
L
l
jl jlω ω
−
=
∆ = ∆∑F H R  (12) 
where ( )
1
2 1L
jlω
−
H  is the 
1
2 1L −
th row of ( ) 2 2jl l j lω ω ω = − + + H M C K . 
Sometimes there are two rubbing locations exist in the system, which may happen on node 
1
L  and 
2
L . In this case, the following equation can be obtained from equation (8): 
  
( ) ( )
( ) ( )
1 2
1 2
,2 1 ,2 1
,2 1 ,2 1
k L k Lk
i i L i L
E jl E jl
E jl E jl
α ω β ω
α ω β ω
− −
− −
+∆
=
∆ +
r
r
 (13) 
α  and β  can be obtained by:  
  
( ) ( )
( ) ( )
1 1
1 1
,2 1 ,2 1
,2 1 ,2 1
k L k L k
ii L i L
E jl E jl
E jl E jl
ω ωα
β ω ω
− −
− −
  ∆  
=     
∆      
r
r
 (14) 
THE QUANTITATIVE DIAGNOSIS METHOD OF RUBBING ROTOR SYSTEM.  
HONGLIANG YAO, QI XU, ZHAOHUI REN AND BANGCHUN WEN 
4 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533  
So in this case, harmonic component 
h
∆r  of node h is needed for the rubbing location 
identification:  
 
( ) ( )
( ) ( )
1 2
1 2
1 2
,2 1 ,2 1
1 2
,2 1 ,2 1
, 1, 2, ..., , 1, 2, ...,
h L h Lh
n n
i k L k L
E jl E jl
n J n J
E jl E jl
α ω β ω
δ
α ω β ω
− −
− −
+∆
= − = =
∆ +
r
r
 (15) 
4. The identification method based on super-harmonic components 
A simple rubbing position and forces identification method is presented based on the property of the 
rubbing rotor system. The procedures are as follow: 
(1) Obtain the vibration response at any two nodes of the rubbing rotor system, for example, the 
nodes at the bearings position. Obtain one sub or super harmonic components of the two signals, for 
example lth harmonic components, mark it as ( ) ( )i ix iyy l jy lω ω∆ = −r  and ( ) ( )k kx kyy l jy lω ω∆ = −r . 
(2) Obtain the dynamic stiffness matrix ( )jlωE  of the former linear system. It can be calculated 
from structure parameters or measure from model testing. 
(3) Build the following equation: 
  
( )
( )
( )2 1,
2 1,k
, 1, 2, ...,
n ik
n
i n
E jl
n J
E jl
ω
δ
ω
−
−
∆
= − =
∆
r
r
 (16) 
and calculate 
n
δ  when 1,  2,  ,n J= …  . 
(4) Obtain the minimum 
n
δ , then n is the position where rotor-to-stator rub occurs. 
(5) Obtain the rubbing forces using equation (12). 
The procedures for the identification of two rubbing are almost the same, except that vibration 
responses of three nodes are needed, and equation (15) should be used instead of equation (16). 
5. The numerical example of the identification method 
The rotor system used to test the identification method is shown as follows: 
 
 
Figure 2. Sketch and finite element 
model of the rotor system. 
 
The diameter of the shaft is 10mm and the length is 420mm. The diameters of the discs are 80mm 
with the thickness of 20mm respectively. The rotor system is divided into 23 segments with 24 nodes. 
Assuming the elastic modulus of the steel material is 2.1×1011Pa, the stiffness of each supporting is 
7×106N/m, the eccentricity of the rotor system is at node 9, which is 
r
m =5×10-6N/m in magnitude.  
Assuming the rubbing occurs at node 12. The contact stiffness is 2×105N/m and the clearance is 
e=2×104N/m. Using the present identification method, when the rotating speed is ω =300rad/s, the 
change of ( )abs
n
δ  is shown in Fig. 3(a). It can be seen from Fig. 3(a) that the minimum ( )abs
n
δ  is at 
node 12, which means that rub occurs at node 12. The identified rubbing forces are compared with the 
actual rubbing forces in Fig. 3(b) and (c), in which the actual forces are drawn in real lines and the 
identified results are drawn in dashed lines. It can be seen that the identified results confirm the actual 
forces very well. This proved the validity of the present method. 
The method is still effective when two rubs happen at the same time. Assuming the rubs occur at 
node 12 and node 18, the rubbing type is partial rub and the rubbing angle are 
1
0 45α° ≤ ≤ ° , 
2
0 45α° ≤ ≤ °  respectively. Fig. 4(a) is the contour plot of the ( )
1 2
abs
n n
δ  when ω =600rad/s, and 
THE QUANTITATIVE DIAGNOSIS METHOD OF RUBBING ROTOR SYSTEM.  
HONGLIANG YAO, QI XU, ZHAOHUI REN AND BANGCHUN WEN 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533 5 
Fig. 4(b) and (c) are the identified rubbing forces at node 12. It can be seen that the identified results 
confirm the actual results very well, which proved the validity of the present method again. 
 
 
(a) 
 
(b) 
 
(c) 
Figure 3. The identification results when ω =300rad/s. 
 
 
(a) 
 
(b) 
 
(c) 
Figure 4. The identification results when ω =600rad/s. 
6. Experiment results 
The experimental work are carried out on the Bently rotor test-rig. The general view of the rotor rig is 
shown in Fig. 5. The displacement sensors are installed on nodes 4 and 18 and are used to detect 
displacements, and the collector is B&K3560B. Rotor-to-stator rub will happen when the rubbing rod 
and the shaft contacts. So the rubbing is of partial rub type. 
 
 
Figure 5. General view of 
the experimental rotor rig. 
 
 
(a) 
 
(b) 
 
(c) 
Figure 6. The identification results when ω =20Hz. 
 
Firstly the vibration signals of nodes 4 and 18 are measured when the rotating speed is 
1
ω =20Hz. 
By adopting the identification method presented in this paper and the 3X super-harmonic component 
as a reference, the change of ( )abs
n
δ  is shown in Fig. 6(a). It can be seen from Fig. 6(a) that the 
( )12abs δ  is the minimum, which means that the rubbing occurs at node 12. Fig. 6(b) and (c) are the 
identified results of the rubbing forces. It’s difficult to measure the actual rubbing forces, so the 
THE QUANTITATIVE DIAGNOSIS METHOD OF RUBBING ROTOR SYSTEM.  
HONGLIANG YAO, QI XU, ZHAOHUI REN AND BANGCHUN WEN 
6 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. NOVEMBER 2013. VOLUME 2. ISSN 2345-0533  
identified results cannot be compared with the actual forces. But the identified results have the 
characteristic of rubbing forces such as periodicity, and the dwell period can be seen clearly on the 
figures. 
7. Discussion 
The efficiency of the present method has been verified by numerical studies and experiments. But the 
present method is only fit for rotor systems with steady state responses. This is due to that the sub or 
super harmonic components can only be obtained from steady state responses. For the various motion 
types such as quasi-periodic and chaotic motion, the method is invalid.  
The advantage of the method is that only two sensors are enough to detect the location of rubbing 
fault. For the large-scale rotating machinery, the sensors can only be installed near bearings or on very 
few other places. The present method is very convenience in these cases. 
8. Conclusions 
In this paper, a novel approach has been proposed for rubbing location detection in rotor systems. This 
new approach was developed based on the properties of the high-order harmonic components of the 
system response. By applying this method, the single rubbing location and rubbing force can be 
identified by using vibration data of only two nodes, the double rubbing locations and the rubbing 
forces on each location can be identified by vibration data of three nodes. So the proposed method is 
much simpler than the existing model-based identification method. Both simulation studies and 
experimental work based on a rubbing rotor system have verified the effectiveness of the new 
approach. 
The present approach could also be applied to detect the rub or impact locations of other practical 
MDOF structures such as beams, truss and so on. Only one time sinusoidal excitation and two places 
of vibration measurements are required. 
Acknowledgements 
The authors would like to gratefully acknowledge the Natural Science Foundation of China (Grant 
No. 51005042, 50775028) and 973 Program (2011CB706504) for the financial support for this study. 
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The Quantitative Diagnosis Method of Rubbing Rotor System

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The Quantitative Diagnosis Method of Rubbing Rotor System

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