PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Rotor Dynamics Analysis under Uncertainty in Lubricant Film
Xiaodong SUN1; Kheirollah SEPAHVAND1; Zhe LIU1; Steffen MARBURG1Chair of Vibro-Acoustics of Vehicles and Machines, Technical University of Munich, Germany

ABSTRACT
Predicting vibrations in rotor dynamics under the impact of uncertainty is still a challenging issue. Research
work published in this area is usually focused on the stochastic dynamics of rotor- bearing systems byconsidering the stiffness and damping coefficients of simplified bearing models as random parameters rather
than the original characteristics of the oil film. In this paper, the characterizing parameters of the lubricant
film such as the dynamic viscosity are considered as uncertain parameters to investigate their influences on
the rotor dynamics response. To this end, the modified Jeffcott rotor model with offset disc is establishedconsidering the gyroscopic effect. At the same time, the parameters of the bearing are obtained by solving
the dynamic characteristics of oil film. Then, the dynamic response of the rotor is numerically calculated. Inorder to investigate the influence of the parameters of the lubricant film, the response is modeled as astochastic process with unknown coefficients by using the generalized polynomial chaos expansion. The
coefficients are then determined from some realizations of the response on a set of collocation points. Finally,
the accuracy of the method is compared with Monte Carlo Simulation.
Keywords: Uncertainty quantification, rotor dynamics, lubricant film

1. INTRODUCTION
In recent years, rotor dynamics problem has drawn a wide attention due to the constant increasing
application of rotating machine in industry. As a consequence, numerous investigations about rotor dynamics
have appeared, and most of them showed a great significance on improving the reliability of rotating machine.
Although many research work have been done for rotor dynamics, for example, the critical speed,
unbalance response and even some nonlinear behavior of rotor have been investigated (1, 2), most of theseresearches are under deterministic analysis. However. there are some uncertainties in rotor system, such as
the geometry parameters, material properties, and even some parameters in the related parts like bearing.
These uncertain parameters can directly affect dynamics behavior, such as changing the critical speed which
can affect the efficiency of rotating machine. Furthermore, some uncertain parameters can affect the stability
problem, for instance, the oil whirl and whip phenomenon which can affect the stability of rotor may occur
when considering the influence of oil-film bearing. Since the great influence on rotor dynamics caused byuncertainty has been commonly recognized, some research related to this problem has been done. For
example, Sepahvand investigated the influence of random geometrical parameters of bladed disc on natural
frequency of rotor system (3); Sinou chose the young modulus and stiffness of shaft to study their effect on
dynamic response of rotor (4); Ma viewed the stiffness, damping coefficients of support as randomparameters and studied their influence on dynamic response of rotor (5); Fu observed the influence of
uncertainties from unbalance magnitude, crack depth, Young’s modulus and bearing stiffness on dynamic
response of a crack rotor (6). Although many work about rotor dynamics with uncertainty have been done,
most of them only considered the random parameters of the rotor itself and ignored the influence of theconnecting components. Even if someone considered the random parameters of connected components, such
as Ma and Fu considered the parameters of support, but it should be noted that these parameters like stiffness
in each direction cannot be changed independently in reality. Therefore, stochastic rotor dynamics
considering the original properties of connecting parts, like viscosity of oil-film bearing, is still challenging.
As for the methods used for uncertainty problems, Monte Carlo (MC) method, a kind of sampling method,is widely used in many engineering problems. MC method can reach a better accuracy especially with more1xiaodong.sun@tum.de3193

samples, but it usually causes a high computational cost. In order to deal with this problem, a non-sampling
method called the generalized Polynomial Chaos (gPC) expansion has been introduced to address theuncertain problem (7, 8). Lots of research show that, compared with the traditional sampling-based method
such as MC method, gPC expansion is more efficient.
In this paper, the random parameters of short oil-film bearing are considered to investigate their influenceon rotor dynamics based on gPC expansion. Rotor-bearing model is explained in section 2, and the gPC
expansion method is introduced in section 3, lastly, the numerical analysis is presented in section 4.

2. ROTOR-BEARING MODEL
2.1 Rotor Model
Jeffcott rotor with offset disc is usually described as a model including a massless shaft and a rigid
disc, and its motion can be written as follows:
MU
U + (C + ZG )U + KU = Ff
(1)
Where M, C, G, K are global mass, damping, gyroscopic and stiffness matrices; U, U, U are
acceleration, velocity and displacement vector, respectively; and Ff is external force, which include
the unbalance force vector.
The natural frequencies and the response of the system can be calculated by assuming the solution
Otof U as the form of U U 0 e . In addition, since the rotor is supported by bearings, the dynamics
characteristics of the rotor system will be affected by the properties of bearing. Therefore, the bearing
is usually taken into consideration of rotor dynamics.
2.2

Bearing Model
y
ξ

hmax

φ
η

e
Oj
hmin

Ob

x

Ωθ

Figure 1 –Scheme of a hydrodynamic bearing
The scheme of hydrodynamic bearing can be seen in Figure 1. It can be seen that the thickness ofoil-film h is a function of eccentricity e:
h c  e cos(M )(2)Where c is the clearance between journal and bearing shell, and M represent the coordinate incylindrical direction.
Reynolds’ Equation of cylindrical oil-film bearing can be written as:
1 w § 3 wp · w § 3 wp ·¨h¸  ¨h¸
R 2 wM © wM ¹ wz © wz ¹

ª wh
º
6P «:
i M »
 2 e cos M  eT sin
w
M
¬
¼

(3)

Where R is radius, p is pressure, P is viscosity, : is rotating speed. For a relative short bearing,
the first term in the left side of Eq. (3) can be ignored (9). Thus, the analytical solution of oil-filmpressure can be described by integration considering the boundary condition:
3 l
H sin Mp  ( )2z2  1(4)
2 D (1  H cos M )33194

In which H is the ratio of eccentricity to clearance. By integrating Eq. (4) on the length direction,
the oil-film force can be obtained. And then, this force can be expressed as the form of vertical andhorizontal force applied to shaft:Fx FH cosT  FT sin T(5)Fy

FH sin T  FT cosT

(6)

ª§
º
2T · 2H 2
H 1  2H 2 »
FH =FK «¨1  ¸S«©
: ¹ 1 H 2 2: 1 H 2 5 2 »
¬
¼

(7)

ª §
º
S
2T ·
H
H
4H »
FT =FK «- ¨1  ¸(8)« 2©
: ¹ 1 H 2 3 2 : 1 H 2 2 »¬¼
The journal bearing connects the support structure and rotating shaft, and it is usually simplified
as a model with stiffness and damping coefficients, as shown in Figure 2. The stiffness and damping
coefficients of bearing in this model can be obtained by partially differentiating the vertical andhorizontal oil-film forces by displacements x, y and velocities x, y :kijwFi wFi wH wFi wT=+
wj wH wj wT wj(i, j = x, y )

(9)

dij

wFi wFi wH wFi wT=+
wj wH wj wT wj(i, j = x, y)(10)Bearing shell(kxx,cxx)

Journal(kyx,cyx)(kxy,cxy)(kyy,cyy)

Figure 2 – Simplified model of journal bearing
For the global stiffness matrix K in Eq. (1), it only includes the stiffness of rotating shaft whenusing rigid supports, but when considering the support as flexible one, this stiffness matrix will bereconstructed by adding the parameters of supports (10).

3. UNCERTAINTY ANALYSIS
3.1 Polynomial Chaos Expansion Based on Collocation Method
The basic idea of gPC expansion is to project the variables of system onto a stochastic spaceyspanned by a set of complete orthogonal polynomials
which are the function of random variable, forexample, an uncertain parameter F˖: o can be represented as:
i1

f

f

i1 1

i1 1 i2 1

f

i1

i2

F =x0 < 0  ¦ xi1 <1 ([i1 )  ¦¦ xi1i2 < 2 ([i1 , [i2 )  ¦¦¦ xi1i2i3 < 3 ([i1 , [i2 , [i3 ) (11)

i1 1 i2 1 i3 1

Usually, it can be written as:f

F =¦ xi <1 ([i )

(12)

i 1

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In which xi are unknown coefficients of polynomial <i , while <i are a set of multidimensional
polynomials with respect to the multidimensional random variables [ with the orthogonality relationship,
and in practical simulation, the series in Eq. (6) is commonly truncated to a finite number of terms anddenoted by N. In addition, the selection of polynomial <i is depend on the distribution type of randomvariable [ , for example, the Legendre orthogonal polynomial are the optimal basis for the random variablewith uniform distribution.
In some cases, the stochastic process of system response can be represented by directly substituting therandom parameters into the specific expression between input and output. However, the expressions of somesystems are very complex, and even some systems can not be described by a series of analytical equations.
In such situation, the stochastic process of system can be constructed by collocation method, which directly
represent the output parameters as the gPC expansion.
3.2 Coefficients of Polynomials
In gPC expansion based on collocation method, the polynomials of output are determined by the
distribution type of random variables [ . For example, if [ ={[1 [2 } , and [1 N (0, 1) which best
polynomial is Hermite polynomial + j [1 ) while [2 U (1, 1) which best polynomial is Legendre
polynomial L j [ 2 ) , then polynomial of output can be assumed as:< j [ )=+ j [1 )L j [2 )(13)
After that, the gPC expansion of output can be viewed as the combination of polynomials in Eq. (13) withseveral unknown coefficients. These coefficients can be solved by the least square method, which need anumber of sample points, and the number of points should be more than the number of unknown coefficients.
Meanwhile, the process of solving these points can be viewed as a black-box.
Details of gPC expansion based on collocation method can be seen from Figure 3:
Output y under uncertainty

N order gPC expansion withunknown coefficients
Collocation points{ξ}1, {ξ}2,Ă,{ξ}n)Optimalpolynomial for ξ1
Optimalpolynomial for ξ2

Ă
Optimalpolynomial for ξn

Polynomial
for output
Ψi(ξ)

Black-box
Value ofΨi(ξ)

Output{y1, y2, Ă,yn}

The least square method
CoefficientsgPC expansion
Compare with othermethod, eg. MC

Add the order ofgPC expansion N

Satisfied theaccuracy
Output by gPC expansion

Figure 3 – Flowchart of gPC expansion based on collocation method

4. NUMERICAL ANALYSISThe analyzed model is a kind of modified Jeffcott rotor model with offset disc, and the parametersof this model are provided in Table 1.

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Table 1 Parameters for analyzed rotor-bearing modelParameter

Value

Length of shaft, m

0.88

Distance from end of shaft to disc, m

0.3

Diameter of shaft, m

0.03

Diameter of disc, m

0.24

Thickness of disc, m

0.04

Viscosity of oil, Pa·s

0.074

Length of bearing, m

0.06

Damping coefficients (N/m·s)

Stiffness coefficients (N/m)

4.1 Deterministic Analysis
It can be concluded from Eq. (9) – (10) that the stiffness and damping coefficients of bearing are
the function of rotating speed, which can be illustrated in Figure 4. Then, the unbalance response in
x-direction is solved for the rotor model with rigid support. The result is shown in Figure 5. It can benoted that the critical speed is 36.17 Hz. When considering the oil-film bearing, the response in xdirection is different compared with the rigid support situation, which is shown in Figure 6. Comparing
these two types of supports, the critical speed is different, and the amplitude of response under flexible
support is significantly lower than that under rigid support.
6
6 10
kxx
kxy
4
kyx
kyy
2
0
-2

0

10

20

30
40
50
Rotating speed (Hz)

60

6
15 10

70

dxx
dxy
dyx
dyy

10
5
0
0

10

20

30
40
50
Rotating speed (Hz)60

70

Amplitude (m)Figure 4 – Stiffness and damping coefficients of oil-film bearing
0.02
0.015
0.01
0.005
0

0

10

20

30
40
50
60
70
Rotating speed (Hz)Figure 5 – Unbalance response in x direction for rotor with rigid support

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10-3

Amplitude (m)

1
0.75
0.5
0.25
0

0

10

20

30
40
50
Rotating speed (Hz)60

70

Figure 6 – Unbalance response for rotor with short bearing model
4.2 Random Parameters and Uncertainty Analysis
In this work, the parameters of oil-film are viewed as the random parameters. The length of bearing
and clearance are assumed as uniform distribution, and the viscosity is assumed as normal distribution.
The detailed information can be seen in Table 2.
Table 2 Random parameters and distribution
Random Parameter

Distribution

Viscosity of oil, Pa·sP

N (0.074(0.074, 0.003)

Length of bearing, m

Lb

U (0.05, 0.07)

c U (0.000(0.0008, 0.0012)
Clearance, m
The choice of collocation points is significant for gPC expansion. It can affect the convergenceand convergent rate of the construction of gPC expansion. Here, the combination of specific roots ofhigher order polynomial are used to build the sample space, and the corresponding response underthese sample points can be obtained by deterministic analysis. Three order gPC expansion is built inthis work. The response under these points are shown in Fig ure 7. It can be seen that the responses
are different due to the different input parameters, which means the input parameters like viscosity
can affect the characteristic of oil-film so that can affect the response of rotor.

Amplitude (m)

1.2

10-3

0.8

0.4

0
26

28

30
32
Rotating speed (Hz)

34

36

cp1
cp2
cp3
cp4
cp5
cp6
cp7
cp8
cp9
cp10
cp11
cp12
cp13
cp14
cp15
cp16

Figure 7 – Unbalance response of sample points
After these responses obtained, the gPC expansion can be built by choosing the optimal
polynomials and calculating the related coefficients. In order to give a comparison, the maximum
amplitude and corresponding rotating speed are extracted to construct their gPC expa nsion. To
validate the precise of constructed gPC expansion, a comparison with the MC simulation is carriedout. The PDF of maximum amplitude and corresponding rotating speed under gPC expansion and MC
simulation are obtained and shown in Figure 8 and Figure 9. The PDF obtained by gPC expansion are
well fitted with that by MC simulation.

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4000
2000

0

6

1.5

gPC
MC
PDF

PDF

6000

gPC
MC

1
0.5

8
10
12
-4
Maximun amplitude (m) 10

Figure 8 – PDF of maximum amplitude

0
30

31
32
Critical speed (Hz)33

Figure 9 – PDF of critical speed

5. CONCLUSIONS
Rotor dynamics analysis under the impact of oil-film bearing is conducted considering theuncertainties in bearing. In this work, a short bearing model is employed to study its impact to rotor
dynamics. The pressure of oil-film is calculated by solving Reynolds’ Equation, and the oil-film force
is obtained by integrating the pressure on length direction. Then, the dynamics coefficients including
stiffness and damping are calculated by differencing the oil-film force. After that, the rotor-bearing
model is established by taking the parameters of bearing into consideration. It can be concluded thatthe flexible bearing has a significant influence on dynamics response of rotor.
Parameters of oil-film such as viscosity are viewed as random parameter to study the influence on
rotor dynamics caused by their uncertainties. The gPC expansion based on collocation method isdeveloped to study the uncertain problem in this rotor-bearing model. Results show that the stochastic
process of response can be well described by the constructed gPC expansion in a higher accuracy
compared with MC simulation. It was also shown that the selected random parameters can affect the
response of rotor system.

ACKNOWLEDGEMENTS
The authors acknowledge the support from China Scholarship Council (File No. 20180 6450023).

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