Numerical and experimental study of a rotor–bearing–seal system

Mechanism and Machine Theory

Mechanism and Machine Theory 42 (2007) 1043–1057

www.elsevier.com/locate/mechmt

Numerical and experimental study of a rotor–bearing–seal system Mei Cheng *, Guang Meng, Jianping Jing State Key Laboratory of Vibration, Shock & Noise, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240, People’s Republic of China Received 22 September 2005; received in revised form 20 March 2006; accepted 11 April 2006

Abstract The non-linear dynamic behaviors of a rotor–bearing–seal coupled system are investigated. The influence of parameters, such as the rotation speed, seal clearance and eccentricity of rotor are analyzed by state trajectory, Poincare´ maps, frequency spectra and bifurcation diagrams. Various non-linear phenomena compressing periodic and quasi-periodic motion in the rotor–bearing–seal system are investigated. The comparison between the analytical results of the numerical and the experimental findings shows that the predictive results agree with the experimental results. It is indicated that this study may contribute to a further understanding of the non-linear dynamics of such a rotor–bearing–seal coupled system.  2006 Elsevier Ltd. All rights reserved. Keywords: Non-linear vibration; Rotor; Bearing; Seal

1. Introduction Recently the demand for more powerful and efficient turbomachinery has led to higher operating speed and working pressure, result in the need of reliable components such as bearings and seals. Not only in bearing, but also in seals, the shaft motion can induce rotordynamic forces that may lead to self-excited vibration of the shaft. Therefore, more detailed rotordynamics analysis is required. Literature related to rotor–bearing and rotor–seal phenomena is abundant. Many theoretical studies, numerical calculations and measurements have been carried out to determine the effect of the self-excited vibration due to seals or bearings, for example, in papers by Qin et al. [1], Jing et al. [2], Jiao et al. [3], Xia et al. [4], Akhmetkhanov et al. [5], Hua et al. [6], Ding et al. [7] and Li et al. [8]. However, in the published papers, only a single factor, bearing or seal fluid force, is taken into account in studying the dynamics of a rotor system. The method will cause the results to deviate from the fact. Therefore, it is necessary to consider the rotor, bearings and seals as a system and model them in one numerical model. The dynamic behavior of this model can then reflect the rotor–bearing–seal coupled system features. In addition, linearized stiffness and *

Corresponding author. Tel.: +086 21 54747990. E-mail address: [email protected] (M. Cheng).

0094-114X/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2006.04.010

1044

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

Nomenclature c1 c2 cm cz D1 D e fX1 fY1 fx1 fy1 FX2 FY2 h1 K K L1 m1 m2 mf O1 O2 Om p1 r R1 t x1 y1 x2 y2 X1 Y1 X2 Y2 z1 l1 s s x

damping coefficient of the bearing damping coefficient of the disk radial clearance of the seal radial clearance of the bearing diameter of bearing coefficient of the fluid film damping qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi relative eccentricity of rotor and e ¼ X 22 þ Y 22 =cm ¼ x22 þ y 22 oil-film force in X1-direction oil-film force in Y1-direction non-dimensional oil-film force in x1-direction non-dimensional oil-film force in y1-direction sealing force in X2-direction sealing force in Y2-direction non-dimensional oil-film thickness rotor stiffness coefficient of the fluid film radial stiffness length of bearing mass of bearing mass of disk coefficient of the fluid film inertia center of bearing center of disk center of disk mass non-dimensional oil-film pressure eccentricity of rotor radius of bearing time non-dimensional displacement of bearing in x1-direction non-dimensional displacement of bearing in y1-direction non-dimensional displacement of disk in x2-direction non-dimensional displacement of disk in y2-direction displacement of bearing in X1-direction displacement of bearing in Y1-direction displacement of disk in X2-direction displacement of disk in Y2-direction non-dimensional axial coordinate viscosity of the lubricant oil fluid average circumferential velocity ratio non-dimensional time and s = xt rotating speed of rotor

damping coefficients are widely used to approach the dynamic characteristics of bearings and seals. However, the observed phenomena indicate that bearing and seal fluid force have strong non-linearity. Therefore, considering the non-linear oil-film force under short bearing assumption, a dynamic model of the non-linear oil-film force of the bearing [9] with better accuracy and convergence is adopted in this study. As for the seal model, the Muszynska model [10,11] not only reflects the non-linear factors in seals but also has a clear physical meaning. And it is employed in this study.

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1045

Therefore, the rotor system is modeled by considering the non-linear oil film force and the non-linear Muszynska seal force simultaneously to analyze the coupling complicated non-linear vibrations. The paper examines the results of the simulation and the experiment, including dynamic trajectories, frequency spectra, Poincare´ maps and bifurcation diagrams which are applied to validate the model and analyze the features of the rotor–bearing–seal system in various parameters. Finally, some conclusions are drawn to summarize the effects of the structure parameters of the rotor–bearing–seal system on the dynamic behavior. 2. Test facility and apparatus The tests were performed in the steam excitation test rig, as illustrated in Fig. 1. The test rotor is supported by two oil lubricating bearings. The seal rig is designed in the middle of the rotor. High-pressure fluid enters the center of the stator and discharges axially across the test seals. There are three types of seals that can be chosen for test use. However, only the plain annular seals (see Fig. 2) are studied in this paper. The experimental test facility is designed to measure the rotor dynamic characteristic under a series of conditions, such as a comprehensive set of various rotation speeds. These experimental results can be used to validate the result from numerical simulation. Data acquisition equipment consists of Bently 208P, a key phase electronic sensor and four current displacement sensors. The key phase electronic sensor is fixed on a machine foundation to measure the rotating speed. The shaft flexural vibrations in two lateral direction is measured by two current displacement sensors, e.g., one located vertically and the other horizontally, as shown in Fig. 1.

Fig. 1. Test facility: (a) test map and (b) test rig.

Stator

Fluid Rotor Fig. 2. Plain annular seals.

1046

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

The experimental rotor is made of steel, elasticity modulus E is 2 · 1011 N/m2, density q is 7800 kg/m3, rotor length is 1800 mm, bearing length-to-diameter ratio is 1, lubricating oil is 32 turbine oil and seal radius is 53 mm. 3. Mathematical model The key point is to analyze the influence of the non-linear oil-film forces and non-linear seal forces in the paper. As a result, it makes an assumption that the torsional vibration of rotor and the gyroscopic effects may be neglected and only the transverse vibration of rotor should be considered. Thus the rotor–bearing–seal system can be modeled as a Jeffcot rotor system, in which the rotor is simplified to one disk with two transverse stiffness and the masses of the shaft being equivalent to the disk and two bearings. The model of rotor–bearing–seal system is shown in Fig. 3 where O1 is the bearing center, O2 is the disk center, Om is the disk mass center, fX1 and fY1 are oil forces, FX2 and FY2 are sealing forces, m1 is the bearing mass, m2 is the disk mass, K is the stiffness of rotor. 3.1. Governing equations The mathematical model of the rotor–bearing–seal system takes into account four degrees of freedom— horizontal and vertical displacements of the rotor at the disk location (X2, Y2) and at the journal (X1, Y1), correspondingly. Then the dynamic equations of system are established as follows: 8 m1 X€ 1 þ c1 X_ 1 þ K2 ðX 1  X 2 Þ ¼ fX 1 > > > > < m Y€ þ c Y_ þ K ðY  Y Þ ¼ f  m g 1 1 1 1 1 2 Y1 1 2 ð1Þ 2 > € _ > X X þ c þ KðX  X Þ ¼ F þ m rx cos xt m 2 2 2 2 2 1 X 2 2 > > : € m2 Y 2 þ c2 Y_ 2 þ KðY 2  Y 1 Þ ¼ F Y þ m2 rx2 sin xt  m2 g 2

where c1 is damping coefficient of the bearing, c2 is damping coefficient of the disk, X1 and Y1 are displacement components of the bearing, X2 and Y2 are displacement components of the disk, r is eccentricity of the rotor, x is rotation speed and t is time. 3.2. Non-linear oil-film force Considering the non-linear oil-film force model under short bearing theory [9], a dynamic model of the nonlinear oil-film force is established for better accuracy and convergence. The model non-dimensional Reynolds equation is listed as follows:  2   R1 o 3 op 1 h ð2Þ ¼ x1 sin h  y 1 cos h  2ðx_ 1 cos h þ y_ 1 sin hÞ L1 oz1 1 oz1

Y2

Y2

Seal

FY2 Y1

Y1

K/2 f X1

O1 X1 f Y1

01

K/2 X2

m2 g m1g

0m

O2

f X1

O1

m1g FX2

X1 Seal

f Y1

Fig. 3. A rotor–bearing–seal system.

X2

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1047

where R1 is the radius of bearing, L1 is the length of bearing, z1 is non-dimensional axial coordinate and z1 ¼ z1 =L1 ; z1 is displacement of bearing in z1 direction, h1 is non-dimensional oil-film thickness and h1 ¼ h1 =cz ;  h1 is the oil-film thickness, cz is the radial clearance of the bearing, p1 is non-dimensional oil film pressure, x1 and y1 are non-dimensional displacement components of the bearing, x01 and y 01 are non-dimensional velocity components of the bearing. The non-dimensional oil-film pressure equation is given by Eq. (2) and is written as follows: p1 ¼

 2 1 L1 ðx1  2y_ 1 Þ sin h  ðy 1 þ 2_x1 Þ cos h 2 ð4z1  1Þ 3 2 D1 ð1  x1 cos h  y 1 sin hÞ

ð3Þ

The non-dimension oil-film force equation of f x1 , f y 1 can be written as "

fx1

# ¼

fy1

  ½ðx1  2_y 1 Þ2 þ ðy 1 þ 2_x1 Þ2 1=2 3x1 V ðx1 ; y 1 ; aÞ  sin aGðx1 ; y 1 ; aÞ  2 cos aSðx1 ; y 1 ; aÞ 1  x21  y 21 3y 1 V ðx1 ; y 1 ; aÞ þ cos aGðx1 ; y 1 ; aÞ  2 sin aSðx1 ; y 1 ; aÞ

ð4Þ

where V ðx1 ; y 1 ; aÞ ¼ Sðx1 ; y 1 ; aÞ ¼

2 þ ðy 1 cos a  x1 sin aÞGðx1 ; y 1 ; aÞ 1  x21  y 21 x1 cos a þ y 1 sin a 2

1  ðx1 cos a þ y 1 sin aÞ " # 2 p y 1 cos a  x1 sin a þ arctg Gðx1 ; y 1 ; aÞ ¼ 1=2 ð1  x21  y 21 Þ1=2 2 ð1  x21  y 21 Þ   y þ 2_x1 p y þ 2_x1 p a ¼ arctan 1  sgn 1  sgnðy 1 þ 2_x1 Þ 2 x1  2y_ 1 2 x1  2_y 1 3.3. Non-linear seal force For small motion about a centered position, the rotordynamical equation for seals is usually defined by the following linearized force–displacement model:  

F x2 F y2



 ¼

K k

k K



  x2 C þ c y2

c C



x_ 2 y_ 2

 ð5Þ

However, the shaft motion can induce rotordynamic forces in seals that may lead to the self-excited vibration of the shaft. Therefore, there are non-linear factors in seal force. However, non-linear analysis of the seal fluid forces is still rare due to the difficulties in obtaining the analytical non-linear model of seal fluid force from the complicated fluid dynamics. To overcome this difficulty, a simple model of non-linear fluid dynamic forces generated in the seal based on the results of a series of experiments was proposed by Muszynska [11–13]. It is assumed that when the shaft is rotating, centered fully developed fluid flow is established in the circumferential direction: that is to say, on average, the fluid is rotating at the rate sx, where x is the shaft rotation speed and s is the fluid average circumferential velocity ratio whose value is close to a half [11]. The fluid average velocity varies for different seals. The seal force will have the form: 

F X2 F Y2

"

 ¼

K  mf s2 x2

sxD

sxD

K  mf s2 x2

#

X2 Y2

"

 

D

2sxmf

2sxmf

D

#"

X_ 2 Y_ 2

#

 

mf

0

0

mf

"

X€ 2 Y€ 2

# ð6Þ

1048

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

In Eq. (6), K, D and s are non-linear functions of the radial displacement and were introduced in Ref. [11,14]: K ¼ K 0 ð1  e2 Þn ;

D ¼ D0 ð1  e2 Þn ;

s ¼ s0 ð1  eÞb

where n, b, s0 vary for various types of seals. In Ref. [15], the characteristics of K0, D0 and mf are described in detail. For the convenience of calculating composition, the dimensionless transformations are given as follows: xt ¼ s;

X1 Y1 ; y1 ¼ ; cz cz 2 d2 2 d ¼ x ; dt2 ds2

x1 ¼

d d ¼x ; dt ds

y2 ¼

Y2 ; cm

X2 ; cm

x2 ¼

and define x0 ¼

dx ; ds

y0 ¼

dy ; ds

x00 ¼

d2 x ; ds2

y 00 ¼

d2 y ds2

ð7Þ

Eqs. (4), (6) and (7) are substituted into Eq. (1), Eq. (1) become: x001 þ

c1 0 K Kcm r x1 þ x1  x2 ¼ fx 2 2 2m1 x m1 cz x2 1 m1 x 2m1 cz x

y 001 þ

c1 0 K Kcm r 1 y1 þ y1  y2 ¼ fy 1  g 2 2 2 2m1 x m1 cz x c1 x2 m1 x 2m1 cz x

x002 þ

c2 K Kcz x0 þ x2  x1 ðm2 þ mf Þx 2 ðm2 þ mf Þx2 ðm2 þ mf Þcm x2

¼ y 002 þ ¼

ð8Þ

1 m2 r F x2 þ cos s 2 ðm2 þ mf Þx ðm2 þ mf Þcm c2 K Kcz y0 þ y  y ðm2 þ mf Þx 2 ðm2 þ mf Þx2 2 ðm2 þ mf Þcm x2 1 1 m2 r m2 g F y2 þ sin s  2 ðm2 þ mf Þx ðm2 þ mf Þcm ðm2 þ mf Þcm x2

where "

F x2 F y2

#

" ¼

K  m f s2 x 2

sxD

sxD

K  mf s2 x2

#

x2 y2

"

 x

D

2sxmf

2sxmf

D

#

  2  2 x02 R1 L1 ; r ¼ l1 xR1 L1 0 cz 2R1 y2

4. Experimental results The main test parameters are written as follows (see Table 1). Fig. 4 is rotor center trajectory and the frequency spectrum by experiment where seals have no fluid, with different rotation speeds. It can be found that the frequency response of the rotor center has two peaks at Table 1 Test parameters m1 (kg)

m2 (kg)

K (N m1)

l1

r (mm)

cz (mm)

cm (mm)

R1 (mm)

28

70

8.6 · 106

18 · 103

0.06

0.2

1

50

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1049

Fig. 4. Experiment results without fluid: (a) orbit of rotor center at x = 5250 rpm, (b) spectrum of frequency at x = 5250 rpm, (c) orbit of rotor center at x = 5600 rpm, (d) spectrum of frequency at x = 5600 rpm.

frequency 45 Hz and 97.5 Hz. 97.5 Hz dominates in the system frequency spectrum (Fig. 4(b)). When the rotation speed is increased to 5600 rpm, the frequency spectrum (Fig. 4(d)) shows that the amplitude at 48 Hz is much higher than that at 92 Hz. That is to say, the system demonstrates the oil whip phenomenon at rotation speed 5600 rpm. Fig. 5 is the rotor center trajectory and the frequency spectrums by experiment where seals have fluid, with different rotation speeds. It is evident notice that the oil whip phenomenon occurs at rotation speed 6325 rpm. Comparison between Figs. 4 and 5 reveals that the rotation speed of the oil whip phenomenon of the latter is higher than that of the former. This indicates that the seal fluid dynamic may delay the oil whip phenomenon and improve the stability of the rotor–bearing–seal system. 5. Numerical analysis The responses of the rotor–bearing–seal system become more complicated when combining the non-linear oil-film force with the non-linear seal force. Due to the strong non-linearity of the system, numerical analysis is

1050

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

Fig. 5. Experiment results with fluid: (a) orbit of rotor center at x = 5606 rpm, (b) spectrum of frequency at x = 5606 rpm, (c) orbit of rotor center at x = 6325 rpm, (d) spectrum of frequency at x = 6325 rpm.

carried out by fourth-order Runge–Kutta method. In order to validate the model in this study, the condition of numerical simulation is chosen to be the same as those of the experiment. 5.1. Analysis method To illustrate the numerical results, the following plots are used: Bifurcation diagram. The bifurcation diagram exhibits continuous changes of motion states for the dynamic system. It provides a summary of the essential dynamics and is a useful way to observe the non-linear dynamic behavior. The control parameter varies according to a constant step. The variation of return points in the Poincare´ map of dimensionless time is then plotted against the control parameter, which generates the bifurcation diagram. Poincare´ map. A Poincare´ section is a stroboscopic picture of motion in a phase plane and it consists of the time series at a constant interval of T (T = 2p/x). The point on the Poincare´ section is referred to as a return point. The projection of a Poincare´ section on the X(nT)  Y(nT) plane is referred to as the Poincare´ map of

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1051

Table 2 Motion characteristic in Poincare´ map Motion state

Poincare´ map

Synchronous KT-periodic Quasi-periodic Chaotic

Single point K discrete points Closured curve formed by infinite points Dense points with the geometrically fractal structure

the motion, which indicates the nature of the motion. Characteristics of different typical motion state in the Poincare´ map is shown in Table 2. Frequency spectrum. The frequency spectrum of the Y-coordinate is computed. Trajectory map. By the numerical integration of the equations of rotor motion, the Y-coordinate is plotted versus the X-coordinate as time T increase. 5.2. Numerical results and analysis Rotational speed is one of the key parameters affecting the dynamic characteristics of a rotor system. The bifurcation diagram of the rotor center without considering seal force is established and the results are shown in Fig. 6. It can be observed that the response of the system varies with rotation speed. The system possesses the periodic motion at x < 5350 rpm, only one point is correspondingly shown in the bifurcation map for every rotation speed. Fig. 7 plots the bifurcation of the rotor system considering seal force, in the rotation 2

y

1

0

-1

2000

4000 w rpm

6000

Fig. 6. Bifurcation diagram without considering seal force.

0.2

y

0.1

0

-0.1

-0.2

2000

4000 w rpm

6000

Fig. 7. Bifurcation diagram considering seal force.

1052

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

speed range 1146–6688 rpm. It can be found that the dynamic motion of the coupled system is synchronous with period one at x < 6115 rpm. The period-one motion loses its stability at x = 6115 rpm. Among the above-described dynamic phenomena, there is a clear distinction related to whether the seal force is taken into account or not. Fig. 7 shows that the seal force will extend stability region. The conclusion is in agreement with the experiment results in Section 4. Fig. 8 depicts the trajectory of rotor center, the projection of Poincare´ section and the frequency spectrum at x = 5599 rpm, considering seal force. There is an isolated point in the Poincare´ section, a limited circle in the trajectory map and one discrete frequency component in the frequency spectrum. As the rotor rotational speed continues to increase, the motion appears quasi-periodic. As the rotation speed is increased to x = 6325 rpm, there is a appearance of the quasi-periodic motion in Fig. 9. And there exist two discrete frequency components in the frequency spectrum and a closed circle in the projection of Poincare´ section. The rotor center trajectory is irregular. All of these prove that the motion is quasi-periodic. Via the comparison of the value from experiments (see Section 4) with the numerical computation of the rotor–bearing–seal model obtained in this study, it is shown that the predictive result agrees with the experimental result. Evidently, this model is effective for further study of non-linear rotordynamics of the rotor–bearing–seal system. Seal clearance is also one of the key factors affecting dynamic characteristics of the system. So it is very essential to study its influences on dynamic characteristics of the system. Fig. 10 shows the bifurcation diagram of the system with increasing in the seal clearance. The bifurcation characteristics of x = 5733 rpm were shown in Fig. 10(a). It can be found from Fig. 10(a) that there is only period-one movement among the whole range of seal clearance. When rotation speed x is 6114 rpm(Fig. 10(b)), the period-one motion is limited. Fig. 10(c) shows the bifurcation characteristics of x = 6306 rpm. It is evident the one-periodic motion is limited gradually. And the system presents obviously complex dynamic characters.

0

1 3

-0.05 2

y

y

0

-0.1

1

-0.15

-1

-0.2 -0.05

0

0.05

(a)

0.1

0.15

x

-0.04

(b)

-0.03

x

-0.02

-0.01

0 0

1

2

(c)

3

4

5

4

5

f/w

Fig. 8. Periodic motion at x = 5599 rpm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

0.2

10

0.1

0.1

8 0

6

y

y

0 -0.1

-0.1

4

-0.2

2 -0.2

-0.3 -0.2

(a)

-0.1

0

0.1

x

0.2

0.3

-0.2

(b)

-0.1

0

x

0.1

0.2

0 0

(c)

1

2

3

f/w

Fig. 9. Quasi-periodic motion at x = 6325 rpm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1053

Fig. 10. Bifurcation diagrams with increasing seal clearance: (a) rotation speed x = 5733 rpm, (b) rotation speed x = 6114 rpm, (c) rotation speed x = 6306 rpm.

1

0.2

12 0

4

-0.4

-1

-0.6 -0.2

(a)

8

y

y

0 -0.2

0

0.2

x

0.4

-1

0.6

(b)

0

x

1

0 0

(c)

1

2

3

4

5

f/w

Fig. 11. Periodic motion at cm = 0.22 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

Fig. 12. Quasi-periodic motion at cm = 0.8 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

And the responses (see Figs. 11–14) are further analyzed by the projections of Poincare´ section, the rotor trajectory and the frequency spectrum. When the seal clearance varies from 0.1 mm to 6.5 mm (see Figs. 11–14), one point in the Poincare´ map is separated gradually into a closed curve. Fig. 11 illustrates the rotor center trajectory, the projection of Poincare´ section and the frequency spectrum for seal clearance cm = 0.22 mm. There exists an isolated point in the Poincare section, one discrete frequency component in the frequency spectrum and a limited circle in the trajectory map, which represents period-one motion. As

1054

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057 0.1

6

0

0

4

-0.1

2

y

y

0.1

-0.1

-0.2 -0.2

(a)

-0.1

0

x

0.1

0.2

(b)

-0.2 -0.1

0

x

0.1

(c)

0 0

1

2

f/w

3

4

5

Fig. 13. 6 T periodic motion at cm = 1.6 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

Fig. 14. Quasi-periodic motion at cm = 3 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

the seal clearance continues to increase, the motion shows different phenomena. Fig. 12 displays the rotor center trajectory, the projection of Poincare´ section and the frequency spectrum for seal clearance cm = 0.8 mm. Two discrete frequency components in the frequency spectrum appear and a closed circle is observed on the Poincare´ map. The rotor center trajectory is irregular. All of these prove that the motion is quasi-periodic. Fig. 13 depicts the rotor center trajectory, the projection of Poincare´ section and the frequency spectrum for cm = 1.6 mm. There are sixteen isolated points in the Poincare´ map and two discrete frequency components in frequency spectrum. The rotor trajectory shows 16 T periodic motion. Fig. 14 is the rotor center trajectory, the projection of Poincare´ section and the frequency spectrum for cm = 3 mm. Its characteristics are similar to those in Fig. 12. Namely, quasi-periodic motion appears in Fig. 14. So it can be concluded from the above-described dynamic phenomena, when x < 5733 rpm, in that range of rotating speed, the system has only period-one movement with different seal clearance. However, as the rotating speed increases further, such as x = 6306 rpm, more complex motions are expected. Eccentricity of rotor is also one of the main factors affecting dynamic characteristics of the coupled system. Fig. 15 provides an illustration as to how the eccentricity of rotor r influences the coupled system dynamics when the rotating speed x = 6114 rpm. The bifurcation diagram in the figure indicates that initially quasiperiodic motion transits to a period-one motion then into a range r 2 [0.59 2.2] mm where the solution shows quasi-periodic. Finally the motion goes back to period one. This implies that proper eccentricity of rotor values may improve the stability of the coupled system. These bifurcations can be illustrated by the rotor trajectory, the projections of Poincare´ section and the frequency spectrum, as depicted in Figs. 16–19. Fig. 16(a) demonstrates the orbit of quasi-periodic motion while a closed curve on Poincare´ map is depicted in Fig. 16(b) and two discrete frequency components appear in Fig. 16(c). At r = 0.3 mm, a period-one motion is shown by the regular orbit in Fig. 17(a), one isolated point

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1055

Fig. 15. Bifurcation diagrams with increasing eccentricity of disk r.

0 -0.08

2

-0.1

1

y

y

-0.04

-0.08

-0.12

-0.16 -0.04

(a)

0

0.04

0.08

0.12

-0.12 -0.03

(b)

x

-0.015

0

0.015

0.03

0 0

1

2

3

4

5

f/w

(c)

x

Fig. 16. Quasi-periodic motion at r = 0.04 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

0.4

15

1

10

y

y

0 0

5 -0.4

-0.4

(a)

0

x

0.4

(b)

-1 -1

0

x

1

0 0

(c)

1

2

f/ w

3

4

5

Fig. 17. Periodic motion at r = 0.3 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

in Fig. 17(b) and one discrete frequency component in Fig. 17(c). Then there is a reappearance of quasi-periodic motion in Fig. 18. Finally, it is evident from Fig. 19 that the orbit of the rotor center is regular and one isolated point in the Poincare´ map. It indicates that the system represents a period-one motion.

1056

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057 30

1

0

15

y

y

-0.4

-0.6

(a)

-1 -1

0

1

x

(b)

-0.8 -0.6

-0.4

-0.2

x

0 0

(c)

1

2

f/w

3

4

5

4

5

Fig. 18. Quasi-periodic motion at r = 0.7 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

1

40

0

20

y

y

0

-1

-1 -1

0

(a)

x

1

-2

(b)

-1

0

x

0 0

1

(c)

1

2

3

f/ w

Fig. 19. Periodic motion at r = 2.3 mm: (a) orbit of rotor center, (b) Poincare´ map, (c) spectrum of frequency.

6. Conclusions In this paper, a model considering non-linear oil-film forces and non-linear seal forces is put forward to analyze the complicated non-linear vibrations of a rotor–bearing–seal system. The model is validated by experiments and it can be employed for further studies of the rotor–bearing–seal system. The conclusions drawn from the study can be summarized as follows: (1) The dynamic phenomena of the rotor–bearing–seal system are clearly distinct depending on whether the seal force is taken into account or not. Computational and experimental results show that the seal force will increase the stability region of the system. (2) Some non-linear analysis methods are employed to explore dynamic behaviors of the discussed system, including bifurcation diagram, dynamic orbit, Poincare´ map and frequency spectrum. The results show that the system exhibits rich forms of periodic and quasi-periodic motions. And the quasi-periodic motion is the route to the chaotic motion. It indicates that the system has the potential for chaotic motion. (3) Seal clearance and eccentricity of rotor are the key factors affecting dynamic characteristics of rotor– bearing–seal system. Numerical analysis show the coupled system will exhibit more complex motion with these parameters changing.

Acknowledgements The National Natural Science Foundation of China (No. 10572087) and the project was supported by China ‘‘863’’ Project (No. 2002AA52613-8).

M. Cheng et al. / Mechanism and Machine Theory 42 (2007) 1043–1057

1057

References [1] P. Qin, Y. Shen, J. Zhu, H. Xu, Dynamic analysis of hydrodynamic bearing–rotor system based on neural network, International Journal of Engineering Science 43 (2005) 520–531. [2] J.P. Jing, G. Meng, Y. Sun, S.B. Xia, On the nonlinear dynamic behavior of a rotor–bearing system, Journal of Sound and Vibration 274 (2004) 1031–1044. [3] Y.H. Jiao, Z.B. Chen, X.Q. Qu, Study of nonlinear dynamic characteristics of rotor–bearing systems, Journal of Harbin Institute of Technology (New Series) 11 (5) (2004) 493–497. [4] S.B. Xia, X.J. Zhang, X.H. Wu, G.F. Xu, Study on nonlinear dynamic characters of rotor–bearing system, Proceedings of the ASME Design Engineering Technical Conference 6 (2001) 2843–2848. [5] R. Akhmetkhanov, L. Banakh, A. Nikiforov, Flow-coupled vibrations of rotor and seal, Journal of Vibration and Control 11 (7) (2005) 887–901. [6] J. Hua, S. Swaddiwudhipong, Z.S. Liu, Q.Y. Xu, Numerical analysis of nonlinear rotor–seal system, Journal of Sound and Vibration 283 (2005) 525–542. [7] Q. Ding, J.E. Cooper, A.Y.T. Leung, Hopf bifurcation analysis of a rotor/seal system, Journal of Sound and Vibration 252 (5) (2002) 817–833. [8] S.T. Li, Q.Y. Xu, F.Y. Wan, X.L. Zhang, Stability and bifurcation of unbalance rotor/labyrinth seal system, Applied Mathematics and Mechanics (English Edition) 24 (11) (2003) 1290–1301. [9] G. Adiletta, A.R. Guido, C. Rossi, Chaotic motions of a rigid rotor in short, Journal Bearings Nonlinear Dynamics 10 (6) (1996) 251– 269. [10] A. Muszynska, Model testing of rotor/bearing system, The International Journal of Analytical and Experimental Model Analysis 1 (3) (1986) 15–34. [11] A. Muszynska, D.E. Bently, Frequency-swept rotating input perturbation techniques and identification of the fluid force models in rotor/bearing/seal systems and fluid handling machines, Journal of Sound and Vibration 143 (1) (1990) 103–124. [12] A. Muszynska, Whirl and whip-rotor/bearing stability problems, Journal of Sound and Vibration 110 (1986) 443–462. [13] A. Muszynska, Improvements in lightly loaded rotor/bearing and rotor/seal models, Journal of Vibration, Acoustics, Stress and Reliability in Design 110 (1988) 129–136. [14] L.T. Tam, Numerical and analytical study of fluid dynamical forces in seals and bearings, Journal of Vibration, Acoustics, Stress and Reliability in Design 110 (1988) 315–325. [15] W. Zhang, The Theoretical Base of Rotordynamic, Science Press, Beijing, 1990.