Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

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118 C. Nataraj and K. Kappaganthu

−1.5 −1 −0.5 0 0.5 1 1.5

x 10

−7

x 10

−7

−1.5

−1

−0.5

0.5

1.5

Fig. 10.18 Zoomed view of the chaotic orbit

x 10

−7

−1

−0.5

0.5

270 271 272 273 274 275

Fig. 10.19 Zoomed view of the variation of v

D

.w

 ˝



 ˝

C .v

C ˝



3=2

; (10.22)

.w

 ˝

v/.v

C ˝



 ˝

C .v

C ˝



3=2

; (10.23)

D

˝

 ˝



 ˝

C .v

C ˝



3=2

; (10.24)

10 Nonlinear Response in a Rotor System With a Coulomb Spline 119

˝

 ˝

v/.v

C ˝



 ˝

C .v

C ˝



3=2

; (10.25)

.w

 ˝

v/.v

C ˝



 ˝

C .v

C ˝



3=2

; (10.26)

D

.v

C ˝



 ˝

C .v

C ˝



3=2

: (10.27)

The trace of the A matrix is given by (10.28). Since the second part of the

equation is less than zero, according to the comparison lemma [10], the volume re-

lation is given by (10.29). Hence, the volume implodes with time and the response

is chaotic.

tr.A/ D4 





 ˝

C .v

C ˝



1=2

(10.28)

V./ < V.0/e

4

(10.29)

Finally, Fig. 10.20 shows the convergence of the largest Lyapunov exponent for

the parameter values listed in Table 10.1; the converged value is 0.1483.

0 200 400 600 800 1000 1200 1400 1600

Iteration Number

Largest Lyapunov Exponent

−12

−10

−8

−6

−4

−2

Fig. 10.20 Largest Lyapunov exponent

120 C. Nataraj and K. Kappaganthu

10.4 Conclusion

This chapter analyzed the nonlinear behavior of a system with two rigid rotors

connected by a Coulomb spline and mounted on isotropic bearings. The ﬁxed points

and limit cycles were identiﬁed analytically. The transient chaos exhibited near the

limit cycle indicated crisis and chaos. Owing to the higher dimensionality of the

system numerical analysis alone was performed. The strange attractor at the origin

was analyzed and chaos has been established satisfactorily within the limitations of

numerical simulations.

References

1. Nataraj C, Nelson HD, Arakere N (1985) Effect of coulomb spline on rotor dynamic response.

Washington, DC, pp 225–233

2. Grebogi C, Ott E, Yorke JA (1983) Crisis, sudden changes in chaotic attractors and transient

chaos. Physica D 7:181–200

3. Grebogi C, Ott E, Yotke JA (1986) Critical exponent of chaotic transients in nonlinear dynam-

ical systems. Phys Rev Lett 57(11):1284–1287

4. Bartuccelli M, Christiansen PL (1986) Horseshoe chaos in the space-independent double sine-

gordon system. Wave Motion 8:581–594

5. de Souza SLT, Caldas IL (2001) Basins of attraction and transient chaos in a gear-rattling

model. J Vib Control 7(6):849–860

6. He K (2003) Critical phenomenon, crisis and transition to spatiotemporal chaos in plasmas.

Space Sci Rev 107:475–494

7. Stouboulos IN, Miliou, AN, Valaristos AP, Kypriandis IM, Anagnostopoulos AN (2007) Crisis

induced intermittency in a fourth-order autonomous electric circuit. Chaos Solitons Fractals

33(4):1256–1262

8. Nataraj C (1987) Periodic oscillations in mechanical systems with nonlinear components. PhD

thesis, Arizona State University

9. Strogatz SH (2000) Nonlinear Dynamics and Chaos. Westview Press, Boulder County, CO

10. Khalil HK (2002) Nonlinear systems. Prentice Hall, Upper Saddle River, NJ

Chapter 11

The Inﬂuence of the Cross-Coupling Effects

on the Dynamics of Rotor/Stator Rubbing

Zhiyong Shang, Jun Jiang, and Ling Hong

Abstract In this chapter, the inﬂuence of cross-coupling effects on the rubbing-

related dynamics of rotor/stator systems is investigated. The model considered in

this chapter is a four-dof rotor/stator system, which takes into account the dynamics

of the stator and the deformation on the contact surface as well as the cross-coupling

effects. The stability of the synchronous full annular rub solution of the model is ﬁrst

analyzed. Then, the cross-coupling effects on the stability of the system at different

system parameter planes are studied. It is found that the cross-coupling damping of

the stator beneﬁts the synchronous full annular rubs and that of the rotor has a little

inﬂuence on the response. While the cross-coupling stiffness of the stator always

reduces the stability domain of the response, the cross-coupling stiffness of the rotor

may either increase or decrease the stability domain depending upon its value.

11.1 Introduction

In order to improve the efﬁciency, the gap between the rotor and the stator of a

rotating machine is setting smaller and smaller, while the rotor/stator rubbing is be-

coming more and more common. Rotor/stator rub is a serious malfunction in the

operation of a rotating machine, which can seriously degrade the machine perfor-

mance and can even lead to disastrous consequences of the machine. There are a

large amount of works on the rub-related phenomena in the rotor/stator systems in

order to get deep insights into the dynamical behaviors and their relationship with

system parameters. It has now become well known that rotor-to-stator rubbing can

induce periodic synchronous full annular rubs [1–4], sub and superharmonic mo-

tions [5,6], quasiperiodic partial rubs [7,8], chaotic responses [9,10]aswellasthe

destructive self-excited dry friction backward whirl [2,11,12]. Since rotor-to-stator

J. Jiang (



)

MOE Key Laboratory for Strength and Vibration, Xi’an Jiaotong University,

Xi’an 710049, People’s Republic of China

e-mail: jun.jiang@mail.xjtu.edu.cn

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

11,

 Springer Science+Business Media, LLC 2010

121

122 Z. Shang et al.

rubbing is usually caused by other malfunctions that induce large rotor deﬂections

covering the clearance between the rotor and the stator, an effective way to prevent

from the rotor-to-stator rubbing is to suppress the vibration amplitude of the rotor,

i.e., through active magnetic bearing or active controlled journal bearings [13–15].

When the rotor/stator rubbing is unavoidable, i.e., without the vibration suppres-

sion measures or due to the failure of the suppression measures in the extreme

running condition, it is still highly expected that the rotating machine can adapt

to the changing environments and optimize its performance to reduce the rub-

bing induced degradation and, especially, to avoid the occurrence of the destructive

instability. In this chapter, the inﬂuences of the cross-coupling effects on the rotor-

to-stator rubbing will be investigated. The aims are twofolds: at one hand, to take

into account more realistic effects in the rotor/stator model and study their inﬂuence

on the rub-related behaviors of the system; on the other hand, to examine the possi-

bility to use these cross-coupling effects to develop new control approaches in order

to reduce the rubbing severity. The chapter is arranged as follows. In Sect. 11.2,the

model of the rotor/stator system is introduced. The stability of the synchronous full

annular rub solution is carried out in Sect. 11.3. The inﬂuence of cross-coupling

effects on the stability of the synchronous full annular rub solution is studied in

Sect. 11.4. Finally in Sect. 11.5, the conclusions of this work are drawn.

11.2 The Rotor/Stator Model with Cross-Coupling Effects

The schematic of the rotor/stator model studied in this chapter is shown in Fig. 11.1.

A weightless shaft supported by two ideal bearings has effective transverse stiffness

and rotates at an angular speed !.Arigiddiskofmassm

is mounted at the

midpoint of the shaft. Concentric with the disk is an annular stator (an auxiliary

bearing) of mass m

. The stator is elastically supported by a symmetrical set of

springs with isotropic radial stiffness k

. A clearance ı exists between the rotor

and the stator. To consider the deformation at the contact surface between the rotor

and stator, a symmetrical set of ﬁctive springs with isotropic radial stiffness k

Fig. 11.1 Left: the schematic plot of the rotor-to-stator system; Right: the section view on the

plane of the rotor and the stator ring

11 Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing 123

assumed being laid in the inner ring of the stator to model the contact stiffness. The

rotor also possesses a mass eccentricity of e.

The equations that govern the motion the rotor/stator system in the complex form

can be written as:

C .c

 j

/Pr

C .k

 jQ

C F D m

j!t

C .c

 j

/Pr

C .k

 jQ

 F D 0

F D k

.1 C j/



 r

 ı

r



; (11.1)

where r

D y

C jz

and r

D y

C jz

are the complex deﬂections and c

and c

the

damping of the rotor and the stator, respectively. F represents the resultant contact

force on the contact surface.  is the friction coefﬁcient. Besides, the cross-coupling

effects represented by the cross-coupling damping terms, with 

and 

for the rotor

and the stator, respectively, and the cross-coupling stiffness terms, with Q

and Q

for the rotor and the stator, respectively, are also included in the model. Additionally,

we deﬁne  D 1,if

 r

 ı and  D 0,if

 r

<ı.

Equation (11.1) can be formulated into the nondimensional form as

C .2

 j

/Or

C .1  j

/Or C 

F D ˝

j

;

C .2

 j

/Or

C .ˇ

 j

/Or

 

F D 0;

F D ˇ

.1 C j/



Or

 

Or



; (11.2)

where

represents the differentiation with respect to the nondimensional time

 D !

t with !

being the natural frequency of the rotor. The other

nondimensional variables are deﬁned as following:

; Or

;

F D

;ˇ

;D

;˝ D

;



;

;



;



; 

11.3 The Solution and the Stability Analysis

It is well known that for a linear rotor system without rubbing, the cross-coupling

stiffness may induce the instability of the rotor when the cross-coupling stiffness

goes above a critical value. It has been shown that for a rotor system with rubbing,

the dry friction at the contact surface may induce the instability of the rotor/stator

system when the dry friction exceeds a critical value [12]. It is, therefore, of great

interest to investigate the interaction effect between the cross-coupling effects and

the dry friction on the rubbing behaviors of the rotor/stator system.

124 Z. Shang et al.

When  D 1 in (11.1)or(11.2), the governing equation has a steady-state

periodic solution with constant amplitude and frequency that equals to the rotating

speed of the rotor. This solution is called a synchronous full annular rub solution.

As known that only a stable solution corresponds to a physical observable response,

so the stability of the solution is of great interest. It is known that the destructive dry

friction backward whirl of a rotor/stator system always occurs after the synchronous

full annular rub loses stability. The stability analysis will provide useful information

for the design of the rotor/stator system.

11.3.1 Synchronous Full Annular Rub Solution

To carry out the stability analysis, the explicit form of the solution must be ﬁrst avail-

able. By doing so, an equivalent form of the resultant contact force is adopted [16]

F D ˇ

.1 C j/

Or

 

1  j

1 C 

: (11.3)

Since the synchronous full annular rub solution has a frequency equal to the rotating

speed, the stability analysis of the solution will become easier when (11.2)istrans-

formed into a rotating coordinate system with the frequency equal to the rotating

speed of the rotor. To do so, we introduce the transformations

D 

jt

; Or

D 

jt

;

F D ˚e

jt

: (11.4)

After substituting (11.4)into(11.2), we get the equations in the rotating coordi-

nates as



CŒ2

C j.2˝  

/

CŒ1  ˝

C˝

C j.2˝

 

/

D ˝

 ˚;



CŒ2

Cj.2˝M

 

/

CŒˇ

 ˝

C˝

Cj.2˝

 k

/

D ˚;

˚ Dˇ

.1 C j/





 

 

1j

1C

j˚j



; (11.5)

where 

D 

.UsingC

und C

to denote the dynamical stiffness of the

rotor and the stator, C

D 1  ˝

C ˝

C j.2˝

 

/; C

D ˇ

 ˝

˝

C j.2˝

 

Since the amplitude of the solution we seek is constant, the terms containing

differentiation with respect to  will be cancelled. In this way the algebraic equations

on the complex amplitudes of the rotor and the stator as well as the corresponding

resultant contact force are obtained as

11 Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing 125



D ˝

 ˚



D ˚

.1 C j/ˇ





 

 

1j

1C

j˚j



D ˚: (11.6)

It is seen that the equations governing the motion of the rotor and the stator are

linear. However, the system with rubbing becomes nonlinear due to the presence of

the resultant contact force. In this case, the complex amplitudes of the rotor and the

stator are the functions of the resultant contact force. From the ﬁrst two equations

of (11.6), we derive



D .˝

 ˚/=C



D ˚=C

: (11.7)

After substituting (11.7) back into the third equation of (11.6) and doing some

manipulation, the magnitude of the resultant contact force can be written as

.1 C R

/K ˙

Œ.1 C R

C I

.R

C I

/  K

.1 C R

C I

; (11.8)

where K D ˇ



1 C 

is a real number and



.1 C j/ˇ

C C





.1 C j/ˇ



;



.1 C j/ˇ

C C





.1 C j/ˇ



with <./ and =./ standing for the real and the imaginary part of a complex number.

The resultant contact force is then given by

˚ D

.1 C j/ˇ

1 C

C .1 C j/ˇ

C C

: (11.9)

Substituting (11.9)into(11.7), the complex amplitudes of the rotor, 

,andthe

stator, 

, can be determined. The synchronous full annular rub solution is then

obtained by substituting 

and 

into (11.4), which should also meet the condition



 

 .

11.3.2 Stability Analysis

To carry out the stability analysis on the synchronous full annular rub solution, the

complex amplitude of the rotor and the stator will be ﬁrst written in their component

form as:



D Y

C jZ

D B

exp.j˛/; 

D Y

C jZ

D B

exp.jˇ/:

126 Z. Shang et al.

The synchronous full annular rub solutions are in the form

D 

j

D B

j.C˛/

D y

./ C jz

./; Or

D 

j

D B

j.Cˇ/

D y

./ C jz

./

Deﬁning the state vector in the inertial coordinate system

./ D

Equation (11.2) can be written as ﬁrst-order differential equations in the state space

D Ax C g.x;/ D G.x;/; (11.10)

where A and g(x

; ) are the coefﬁcient matrix and the nonlinear vector as

A D

00 0 01000

00 0 00100

00 0 00010

00 0 00001

.1 C ˇ

/ .

 ˇ

/ˇ

ˇ

2





 ˇ

.1 C ˇ

/ˇ



2

B

.B

C B

/ .K

 B

/0 02„

G

B

 B

.B

C B

/0 0 G

2„

and g.x;/D

0000ˇ

C˝

cos ˝ ˇ

C˝

sin ˝ B

B



The variables above are deﬁned as followings

;„



;„









Œ.y

 y

/  .z

 z

/; H



Œ.y

 y

/ C .z

 z

/

with R D

 y

C .z

 z

After linearizing (11.10) about the synchronous full annular rub solution, which is

denoted by

cos '

sin '

cos '

sin '

 ˝B

sin '

˝B

cos '

 ˝B

sin '

˝B

cos '



;

with '

D ˝C˛; '

D ˝Cˇ, a set of time-variant linear differential equations

is obtained

D ŒJ.x

;/ıx; (11.11)

11 Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing 127

where



J.x

;/



88

xDx

D A C

xDx

is the so-called Jacobian matrix.

ıx

D x  x

is the perturbation to the synchronous full annular rub solution.

Since the Jacobian matrix is periodic time-dependent, it cannot be directly used

to derive the information of stability of the analyzed solution. We thus make the

following transformation:

ıx

D ŒT ıu; (11.12)

where

ŒT D

Œt

 000

0 Œt

 00

00Œt

 0

000Œt



;Œt

 D



cos '

sin '

sin '

cos '



;Œt

 D



sin '

cos '

cos '

sin '



Substituting (11.12)into(11.11) and after some simple manipulation, we get

ıu

D ŒJ

ıu; (11.13)

where [J

] D ŒT

1

.ŒJ.x

;£/ŒT  ŒT

The stability of the synchronous full annular rub solution can now be determined

through the examination of the sign of the real parts of the characteristic roots of the

Jacobian matrix, , which is solved from

ŒŒJ

  I

88

 D 0: (11.14)

The solution is stable if all the real parts of its characteristic roots are less than zero.

Otherwise it is unstable.

11.4 The Cross-Coupling Effects on the Stability

To demonstrate the inﬂuence of the cross-coupling effects on the stability of the

synchronous full annular rub solution, the stability domains of the solution are

drawn on different parameter planes at different cross-coupling coefﬁcients. In the

following analysis, some system parameters are ﬁxed: 

D 0:05; 

D 0:05;

D 200:0;  D 2:0. In the following ﬁgures, two solid curves SN

and SN

deﬁne the lower and the upper existence boundaries of the synchronous full annular

rub solution. Outside the region enclosed by SN

and SN

, the synchronous no-rub

response always exists. Inside the region enclosed by SN

and SN

, the dotted curves

with different symbols show the stability boundaries (Hopf bifurcation) of the syn-

chronous full annular rub solution at different cross-coupling coefﬁcients. Outside

each stability domain of the synchronous full annular rub solution, the quasiperi-

odic partial rub response or the dry friction backward whirl response with heavy or

destructive rubbing severity may appear.