Nonlinear Dynamics Behaviors of a Rotor Roller Bearing System with Radial Clearances and Waviness Considered

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Chinese Journal of Aeronautics 21(2008) 86-96

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Nonlinear Dynamics Behaviors of a Rotor Roller Bearing System with Radial Clearances and Waviness Considered

Wang Liqin*, Cui Li, Zheng Dezhi , Gu Le

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China Received 30 June 2007; accepted 3 December 2007

Abstract

A rotor system supported by roller bearings displays very complicated nonlinear behaviors due to nonlinear Hertzian contact forces, radial clearances and bearing waviness. This paper presents nonlinear bearing forces of a roller bearing under four-dimensional loads and establishes 4-DOF dynamics equations of a rotor roller bearing system. The methods of Newmark-P and of Newton-Laphson are used to solve the nonlinear equations. The dynamics behaviors of a rigid rotor system are studied through the bifurcation, the Poincar e maps, the spectrum diagrams and the axis orbit of responses of the system. The results show that the system is liable to undergo instability caused by the quasi-periodic bifurcation, the periodic-doubling bifurcation and chaos routes as the rotational speed increases. Clearances, outer race waviness, inner race waviness, roller waviness, damping, radial forces and unbalanced forces—all these bring a significant influence to bear on the system stability. As the clearance increases, the dynamics behaviors become complicated with the number and the scale of instable regions becoming larger. The vibration frequencies induced by the roller bearing waviness and the orders of the waviness might cause severe vibrations. The system is able to eliminate non-periodic vibration by reasonable choice and optimization of the parameters.

Keywords: roller bearing; rotor system; nonlinear bearing force; dynamics behaviors; bifurcation; chaos

ELSEVIER

1 Introduction

In recent years, as high-speed rotary machines with roller bearings found wide applications, their dynamics properties were extensively studied. As an ever-increasing demand is posed to their running accuracy and speed, much more attention is paid to the vibration analysis of the rotors supported by roller bearings. Moreover, as a source of vibration, the roller bearings attract strong academic concerns because of their nonlinearity due to the Hertzian force deformation relationship, the radial clearances and the bearing waviness.

Corresponding author. Tel.: +86-451-86402012. E-mail address: lqwang@hit.edu.cn

Foundation items: National Natural Science Foundation of China (50575054); 973 Program (2007CB607602)

Yamamoto[1] introduced non-linearity into the Jeffcott equation to consider the effects of the bearing clearances. The conclusion drawn by Yamamoto is that the maximum amplitude at the critical speed decreases as the radial clearance increases. Fukata and Kim[2-3] analyzed nonlinear dynamics characteristics of bearings using a numerical method, in which the single disk-like rigid rotor supported by symmetrical bearings was simplified into a single bearing with rotational loads. Sankaravelu[4] determined the dynamics parameters of a rotor system with bifurcation through an experiment. The operational and the structural parameters of bearings are essential to the dynamics properties of a rotor system. Tiwari[5-6] studied the nonlinear behaviors of a balanced rotor by taking into account the effects of

internal clearances of a roller bearing, and analyzed the dynamics characteristics of an unbalance rotor. Harsha[7] simulated some dynamics responses of a rotor supported by ball bearings using a 2-DOF model with clearances and waviness, and found that nonlinear dynamic responses were associated with the ball passage frequency and the severe vibration occurred when the number of the balls equaled that of waves of an outer race. In China, the dynamic behaviors of the rolling element bearing and the rotor system have been studied for years[8-11]. However, all the studies mentioned above were only applicable to ball bearings. Recently, Harsha[12-13] investigated the dynamics behaviors of a rotor system supported by roller bearings and constructed a rotor dynamics model using the Lagrange's equation. Nevertheless, in order to simplify the model, he adopted some hypotheses such as neglect of skewing and bending in the rollers and the waviness of races, which made the model inapplicable to the roller bearings that work under complex loads.

Focusing on the dynamics behaviors of a rigid rotor system supported by roller bearings, this paper introduces a 4-DOF dynamics model to study its dynamics performances in consideration of skewing, declining, corners of the rollers and waviness of the bearings. Since roller bearings work under 4-dimen-sional loads, the dynamics behaviors of a rotor system supported by roller bearings should be investigated considering the effects of different parameters. As a result, the paper provides bases for optimization and reasonable choice of bearing's structural and operational parameters.

2 Dynamics Model

2.1 Contact forces of roller bearings

In order to investigate the dynamics characteristics of a rotor bearing system, the nonlinear contact forces should be determined. Fig.1 shows the half cross section of a roller bearing with a roller, an inner race and an outer race. In Fig.1, Dm is a pitch diameter of the bearing, Dr1 and Dr2 are diameters of the outer race and inner race respectively, S is a

radial clearance of the bearing.

Fig.1 Half cross section of a roller bearing model.

Assuming that the bearing displacement is {Y2, Z2, 6y, 6z} under a 4-dimensional load {Fy, Fz, My, Mz}, rj a radial displacement of the jth roller, a and Pj are a skewing angle and a declining angle of the roller, respectively. The inner ring and rollers decline under the bending moment My as shown in Fig.2. Dw is a diameter of the roller. For accurate

Fig.2 Geometry model of roller bearing.

computation, the roller with round corners is divided into n round pieces. Let le be the length of the roller, ls the length of the linear part of the roller, and w the thickness of a round piece of the roller. For the kth round piece of the roller, radius modification is expressed as[14]

l -1 l + l Ck — 0 -S—*- < (k - 0.5)w < -s—*-

Ck — (R2 - ^)05 - (R2 - xk2 )0 5 Other

xk — -^ + ( k - 0.5) w

The additional deformation caused by My in the region where the kth round piece stays in contact with outer ring and inner ring can be obtained from

jkß -

D2 Oy cos Ф

2 jkß

xk + —tan(-

D2 Oy cos (¡)j

tan(±ßj ) (3)

xk +—tan(

tan(Oy cos фф + ßj )

2n , .

фф =— ( j - 1) + (ct

where N is the number of rollers. The rotational speed of cage is expressed as

(c —-

(ф1 + Dw ) + (2(1 - Dw ) 1 Dm 2 Dm

where co1 is rotational speed of the outer ring, co2 rotational speed of the inner ring.

If cos fy is positive, the upper sign in the formula should be used; otherwise, neither sign should be done.

The inner ring and rollers will skew under the bending moment Mz. The additional deformation caused by Mz in the region where the kth round piece stays in contact with the outer ring and inner ring can be represented by

Л jka

J2 jka

D j -(xk sin(Oz sin ф j - aj ))2

D] - (xk sin(Oz sin ф] - CCj ))2

- f (7)

- f (8)

If the waviness of periodic lobes, an important source of vibration in roller bearings, is described by a sinusoidal function, the local defects could be expressed by appropriate superposition of several sinusoidal functions. The radial waviness of the

outer race, inner race and rollers, as shown in Fig.3, can be expressed as[15]

P1j = Z A1n c0s n ((c -(1 )t + 'N + %1n

n—1 L ^

P2j =ZA2n c0s n((c-(2)t +N

n—1 L N

where A is the amplitude of race waviness in radial direction, n the waviness order, % the initial phase angle of waviness.

Fig.3 Waviness model of roller bearing.

The roller waviness in contact with the outer and inner races can be expressed as

W1 j = Z Cnj c0s (bt + %bj )

W2 j = Z Cnj c0s L n(( bt + n) + %bj ]

where C is the amplitude of race waviness in radial direction. Taking the effects of internal radial clearances, waviness, additional deformation and oil film into account, the deformation caused by the contact of the jth roller with the outer ring and inner ring

can be expressed as $

$1 jk = rj - P1 j - W1 j - Ck - h1 Jk + ¿1 Jka + $1 jkp (I3)

$2 jk = Y2 sin fy + Z2 cos fy - Tj - P2 j -W2j -

Ck -h2jk +^2jka +^2jkß

where h1jk and h2jk are the thickness of oil films with elasto-hydrodynamic lubrication, which can be obtained by using the Dowson-Higginson's formula.

A positive contact deformation means that the contact force could be calculated using the Hertzian contact theory, while a negative one implies that no load is transmitted. The contact force can be expressed as

nE/e0.9

MX + CX + KX - f ( X, t)

Qi jk -

Q2 jk -

OV )++1

(82 ]k )+

where "+" represents a positive contact force if Sjk is positive, Ei and E2 are modulus of elasticity of outer ring and inner ring respectively.

While considering the centrifugal force of the roller, the equilibrium of the jth roller in the z direction is represented by

1J (Qjk - 61 jk) +1 m®c2Dm = 0 (17) nk=1 2

where m is the mass of the roller. Given the bearing displacements, the contact forces could be computed from Eqs.(1)-(17) with the Newton-Laphson method. Then the nonlinear bearing forces can be expressed as

?2 jk sin <

Mry Mrz

1Z Zö

n j-1 k-1

1Z Z q

n j-1 k-1

1 N n Z Q2 jkxk

- Z (Z Q2jk • ^-)cos Ij

j-1 k-1

, N n Z Q2 jkxk

- Z (Z Q2 k • -)sin Ij

j-1 k-1

Z Q2 jk k-1

where Qry, Qrz are the bearing forces in y direction and z direction, Mry, Mrz are the bearing moments in y direction and z direction.

2.2 Dynamics model of a rigid rotor system

The equations of the motion of a rigid rotor roller bearing system may be written as

where M is mass vector of the rotor and inner ring of bearing, C is damping vector of the system, K is stiffness vector of the system, X is displacement vector of the rotor, the force vector f could include the nonlinear bearing forces, external load, gravity load and unbalance load.

For a 4-DOF system, the equations of motion can be represented by

mY2 + cY2 + Q = Fy + mea2 cos at

mZ2 + cZ2 + Qrz - FZ - mg + mem sin at Iydy -Iza0z + My - MY

+ Iyaûy + Mrz - Mz

Where {FY, FZ, MY, MZ} are loads acting on the rotor, Cy and Cz are damping of the system in y direction and z direction, Iy and Iz are inertia moment of the system in y direction and z direction, me a2 indicates the unbalance load due to the rotor mass eccentricity e. Using the Newmark-P method, the differential equations of motion can be solved and the transient responses at every time increment are obtained.

3 Calculation and Discussion

Taking the rigid rotor supported by roller bearing as an example, the mass of the rotor bearing system is 10 kg; damping factor of the bearing is 250 N-s/m; radial load of z direction is 5 000 N and the rotor mass eccentricity e is 0. In order to investigate the dynamic behaviors, the geometric parameters of the bearing are given in Table 1.

Table 1 Geometric parameters of the bearing

Parameters Value

Pitch/mm 183

Diameter of a roller/mm 14

Length of roller/mm 20

Number of rollers 36

Radial clearance/mm 0.063 5

Next, the responses of the system with different operational and structural parameters will be studied.

3.1 Effects of rotational speed

The displacement bifurcation in z direction as a function of rotational speed is given in Fig.4, which shows some unstable regions including non-periodic and periodic-doubling bifurcation regions.

Speed/(10 r-min

Fig.4 Displacement bifurcation as a function of speed.

Fig.5 shows the dynamics behaviors at 1 700 r/min, 4 300 r/min and 9 300 r/min respectively. The time-dependent displacement responses are characterized by a stochastical nature; the Poincarè map with fractal structure and the noise in the frequency spectrum is at 1 700 r/min, which is a strong indication of chaotic nature of the system. In Fig.5, the displacement responses have a periodic-doubling bifurcation nature at 4 300 r/min and the system has a quasi-periodic nature at 9 300 r/min.

From the responses, it is seen that the system vibration behavior tends to lose its stability while the rotational speed increases to 2 000 r/min. The system becomes unstable from 4 000 to 4 500 r/min

because of period doubling bifurcation. It shows quasi-dynamic behavior from 9 000 to 9 500 r/min and period doubling bifurcation from 13 100 to 13 600 r/min.

From the above analysis, it is clear that chaos, period doubling bifurcation and quasi-dynamic nature might arise in the rotor roller bearing system as the rotational speed increases.

3.2 Effects of radial clearance

Radial clearances between rollers and bearing races are essential to roller bearings. In order to investigate the vibration characteristics under the influence of clearances, the speed bifurcation at 9 300 r/min is shown in Fig.6, which indicates the appearance of some unstable regions including non-periodic and periodic-doubling bifurcation regions as the clearance increases. The speed responses have a nature of periodic-doubling bifurcation when the radial clearance varies from 26 to 30 jam, and have non-periodic nature in the range of 58-67 jm.

Fig.6 Speed bifurcation versus speed.

Fig.7 shows the displacement bifurcation in z direction as a function of rotational speed at 45 jm and 75 jm. From it, only a small number of a small non-periodic regions and one periodic-doubling bifurcation region exist when the clearance is equal to 45 jm. However, the responses at 75 jm becomes complicated due to the nonlinear nature. Fig.7 also demonstrates that as the clearance increases, more and wider non-periodic and periodic-doubling bifurcation regions appear.

Fig.5 Dynamic behavior at different speeds.

1.5(30ac - ac). In the case of 33 waves, an occurrence of noise is evidenced by the spectrum of a

Fig.7 Displacement bifurcation as a function of speed.

3.3 Effects of waviness

The inner race, outer race and rollers of a roller bearing are supposed to have the waviness amplitude of 10-6 m. The displacement responses with different waviness orders are analyzed at 6 000 r/min. At this speed, the rotational frequency of inner race (2 and of cage (c are 100 Hz and 46.2 Hz respectively; the variable compliance frequency is (vc = (N = 1 662 Hz.

Fig.8 shows the spectrum and axis orbit without waviness. The peak vibration amplitude appears at (VC = 1 662 Hz.

The spectrums due to the outer race waviness of different orders are shown in Fig.9. When the number of waves amounts to 27, the peak vibration amplitudes appear at 600, 1 200 and 1 800 Hz corresponding to 0.5(27(c - (c), (27 (c- (c) and 1.5(27(c - (c). In the case of 30 waves, the peak vibration amplitudes at 670, 1 340 and 2 001 Hz, corresponding to 0.5(30(c - (c), (30(c - (c) and

Frequency/Hz

Fig.8 Spectrum of roller bearing without waviness.

Fig.9 Spectrums due to outer race waviness of different orders.

broadband shape, and when the number increases to 37, severe vibration occurs together with the maximum noise.

Based on the above analysis, it is concluded that the waviness order Np1 and the vibration frequency cop1 due to outer race waviness abide by

p>1 = (pNpi -p) (21)

The waviness order for the severe vibration is at or near N + 1.

The spectrums due to inner race waviness of different orders are shown in Fig.10. With the wave number of 20, the peak vibration amplitudes appear at 1 123 Hz and 2 246 Hz corresponding to [20(p -p) + p] and 2[20(p - p) + p] respectively. With the wave number of 25, the peak vibration amplitude appears at 696, 1 392, 2 088 and 2 784 Hz, corresponding to 0.5[20(p - p) + p], [20(p -p) + p], 1.5[20(p - p) + p] and 2[20(p - p) + p] respectively. In the case of 30 waves, the severe vibration occurs with the maximum noise, and when the number is 36, the peak vibration amplitude appears at 496, 992, 1 488, 1 984 and 2 480 Hz, corresponding to 0.25[36(p - p) + p], 0.50[36(p -p) + p], 0.75[36(p - p) + p], [36(p - p) + p] and 1.25[36(p - p) + p] respectively.

The above analysis shows that the waviness order Np2 and the vibration frequency p2 due to inner race waviness abide by

p>2 = (p - p)Np2 + p (22)

The waviness order for severe vibration is at or (Dm - Dw)(N-1)

(Dm + D w)

Fig.10 Spectrums due to inner race waviness of different orders.

The spectrums due to roller waviness of different orders are presented in Fig.11, which shows when the wave number amounts to 3, the peak vibration amplitude appears at 138.5 Hz and 1 662.3 Hz corresponding to 3ac and 36 ac respectively. The vibrations are caused by roller waviness and passage vibration.

Fig.11 Spectrums due to roller waviness of different orders.

Based on the above analysis, the waviness order Nw and the vibration frequency cw due to roller waviness obey

Cw = Cc Nw (23)

The waviness order for severe vibration is at or near N.

3.4 Effects of damping coefficient

Fig.12 shows the displacement bifurcation in z direction as a function of the damping coefficient at 4 300 r/min. The system has non-periodic nature before the damping coefficient reaching 100 N-s/m. Then it has two-periodic nature in the range of 100350 N-s/m, and later periodic nature as the damping coefficient increases.

Fig.12 Displacement bifurcation as a function of damping coefficient.

3.5 Effects of radial force

The speed bifurcation in z direction as a function of radial forces at 4 300 r/min is given in Fig.13, which shows the system has non-periodic nature when the radial force increases up to 300 N, periodic-doubling bifurcation nature when it ranges from 3 200 to 3 600 N, two-periodic nature from 4 700 to 5 600 N, and afterwards periodic nature as the damping increases.

Fig.13 Speed bifurcation as a function of radial forces.

3.6 Effects of unbalanced forces

Let the rotor mass eccentricity e be 20 jm, the displacement bifurcation as a function of rotational speed is shown in Fig. 14. It clearly demonstrates that the non-periodic regions of the system increase when the unbalanced forces are taken into account, which indicates the importance of the unbalanced forces in a rotor system.

Fig. 15 shows the spectrum at 6 000 r/min, in which the peak amplitudes appear at (2 = 100 Hz,

CyC = 1 662.3 Hz) and OyC ± c2. The vibration behavior with unbalanced forces considered appears very complicated, and the role the unbalanced forces play increases as rotational speed increases.

| Mill

; ¡1 iM* 11

JSfttl*

'lj .....

\ H 11

1 I§1M if f I ! 1 fp ■ :

0 2 4 6 8 10 Speed/(103r-min_l)

Fig. 14 Displacement bifurcation.

Frequency/Hz Fig.15 Spectrum at 6 000 r/min.

4 Conclusions

An expression of nonlinear contact forces of a roller bearing under 4-dimensional loads is deduced and a 4-DOF transient dynamics model of roller bearings is presented to investigate the vibration behaviors of the rotor roller bearing system, which takes radial clearances and waviness of the bearing into account. The following conclusions could be drawn:

(1) A rotor roller bearing system may have chaos, period doubling bifurcation, and quasi-periodic nature as rotational speed increases.

(2) As the clearance increases, the nonlinear nature possessed by the system becomes complicated, and both the non-periodic and the periodic-doubling bifurcation regions increase and

widen.

(3) The vibration frequency due to outer race waviness can be expressed by cp1 = o)cNp1 - cc. The waviness order for severe vibration is at or near N + 1. The vibration frequency due to inner race waviness can be expressed by cp2 = (c2 - cc)Np2 + cc. The waviness order for severe vibration is at or near

(Dm Dw)(N 1) . The vibration frequency due to

(Dm + D w)

roller waviness can be expressed by cow = cocNw. The waviness order for severe vibration is at or near N.

(4) Damping coefficient, radial forces and unbalanced forces exert significant influences on the stability and vibration behaviors of a rotor roller bearing system.

(5) Close attention should be paid to the effective choice of structural and operational parameters of the bearing in designing a rotor bearing system.

References

[1] Yamamoto T. On the vibration of a shaft supported by bearing having radial clearance. Transactions of the Japanese Society of Mechanical Engineering 1955; 21(103): 182-192. [in Japanese]

[2] Fukata S, Gad E H, Tamura H. On the radial vibration of ball bearings (computer simulation). Bulletin of the JSME 1985; 28(239): 899-904.

[3] Kim Y B, Noah S T. Bifurcation analysis for a modified Jeffcott rotor with bearing clearances. Nonlinear Dynamics 1990; 13: 221241.

[4] Sankaravelu A, Noah S T, Burger C P. Bifurcation and chaos in ball bearings. Nonlinear and Stochastic Dynamics 1994; 192(78): 313-325.

[5] Tiwari M, Gupta K. Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. Journal of Sound and Vibration 2000; 238(5): 723-756.

[6] Tiwari M, Gupta K. Dynamic response of an unbalanced rotor supported on ball beaings. Journal of Sound and Vibration 2000; 238(5): 757-779.

[7] Harsha S P, Sandeep K, Prakash R. The effect of speed of balanced rotor on nonlinear vibrations associated with ball bearings. Mechanical Sciences 2003; 45(4): 725-740.

[8] Tang Y B, Gao D P, Luo G H. Non-linear bearing force of the rolling ball bearing and its influence on vibration of bearing system. Journal of Aerospace Power 2006; 21(2): 366-373. [in Chi-

[9] Yuan R, Zhao L Y, Wang S M. Analysis of the nonlinear dynamic behaviors of a rolling bearing-rotor system. Mechanical Science and Technology 2004; 24(10): 1175-1177. [in Chinese]

[10] Bai C Q, Xu Q Y, Zhang X L. Nonlinear stability of balanced rotor due to the effect of ball bearing internal clearance. Applied Mathematics and Mechanics 2006; 27(2): 159-169. [in Chinese]

[11] Bai C Q, Xu Q Y. Dynamic model of ball bearings with internal clearance and waviness. Journal of Sound and Vibration 2006; 294: 23-48.

[12] Harsha S P. Nonlinear dynamic analysis of an unbalanced rotor supported by roller bearing. Chaos, Solitons and Fractals 2005; 26(1-2): 47-66.

[13] Harsha S P. Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect. Mechanism and Machine Theory 2006; 41(6): 688-706.

[14] Harris T A. Rolling bearing analysis. 4th ed. New York: John Wiley & Sons Inc, 2001.

[15] Jang G H, Jeong S W. Nonlinear excitation model of ball bearing waviness in a rigid rotor supported by two or more ball bearings considering five degrees of freedom. ASME Journal of Tribology 2002; 124(1): 82-90.

Biographies:

Wang Liqin Born in 1964, he received Ph.D. degree from Harbin Institute of Technology in 1994, and then became a teacher there. His main research interests include non-conventional tribology under extreme working conditions, ceramic bearing, and so on. E-mail: lqwang@hit.edu.cn

Cui Li Born in 1981, he received M.S. from Harbin Institute of Technology in 2005. Currently he is a Ph.D. candidate in the same university. His main research interests include rotor dynamics and bearing dynamics. E-mail: mechcui@163.com