Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect

Mechanism and Machine Theory 41 (2006) 688–706

Mechanism and Machine Theory www.elsevier.com/locate/mechmt

Nonlinear dynamic response of a balanced rotor supported by rolling element bearings due to radial internal clearance effect S.P. Harsha

*

Mechanical Engineering Group, Birla Institute of Technology and Science, Pilani 333031, India Received 23 July 2005; received in revised form 3 August 2005; accepted 7 September 2005 Available online 2 November 2005

Abstract This paper presents a model for investigating structural vibrations in rolling element bearings due to radial internal clearance. The mathematical formulation accounted for tangential motions of rolling elements as well as inner and outer races with sources of nonlinearity such as Hertzian contact force and internal radial clearance resulting transition from no contact to contact state between rolling elements and races. The contacts between rollers and races are treated as nonlinear springs and the springs act only in compression to simulate contact deformation and resulting force. The nonlinear stiffness is obtained by application of equations for the Hertzian elastic contact deformation theory. The effect of radial internal clearance for rotor bearing system in which rolling element bearings show periodic, quasi-periodic and chaotic behavior are analyzed. Time response, rotor trajectories, Poincare` maps and power spectra are used to elucidate and to illustrate the diversity of system behavior.  2005 Elsevier Ltd. All rights reserved. Keywords: Nonlinear dynamic response; Internal radial clearance; Chaotic vibration; Poincare` maps; Ball passage frequency

1. Introduction The stiffness, rotational accuracy and vibration characteristics of a high-speed shaft are partly controlled by the ball bearings that support it. An analysis of ball bearing dynamic behavior is important to predict the system vibration responses. The behavior of nonlinear systems often demonstrates unexpected behavior patterns that are extremely sensitive to initial conditions. This is well illustrated by the complex behavior of Lorentz equation (for example), which can be shown to posses a large range of phenomena including strange attractors. When rolling element bearings are operated at high speed, they generate vibrations and noise. The principle forces, which drive these vibrations, are time varying nonlinear contact forces, which exist between the various components of the bearings: rolling elements, races and shafts. In the rotor bearing assembly *

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0094-114X/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2005.09.003

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Nomenclature bp ball passage frequency, Hz I moment of inertia of each rolling element Icage moment of inertia of the cage Iin moment of inertia of the inner race Iout moment of inertia of the outer race L arc length, mm min mass of the inner race, kg mj mass of the rolling elements, kg mout mass of the outer race, kg mrotor mass of the rotor, kg N number of wave lobes Nb number of balls R radius of outer race r radius of inner race rin position of mass center of inner race rout position of mass center of outer race T kinetic energy of the bearing system Tcage kinetic energy of the cage Ti_race kinetic energy of the inner race To_race kinetic energy of the outer race Troll_e kinetic energy of the rolling elements V potential energy of the bearing system Vcage potential energy of the cage Vi_race potential energy of the inner race Vo_race potential energy of the outer race Vroll_e potential energy of the rolling elements Vsprings potential energy of the springs xin, yin center of inner race xout, yout center of outer race d deformation at the point of contact at inner and outer race _ ð/Þ angular velocity of inner race in _ ð/Þ angular velocity of outer race out c internal radial clearance k wave length, mm xcage angular velocity of cage qj radial position of the rolling element qr radius of each rolling element hj angular position of rolling element vj position of jth rolling element from the center of inner race BPF ball passage frequency

supported by perfect ball bearings, the vibration spectrum is dominated by the vibrations at the natural frequency and the ball passage frequency (BPF). The vibrations at this later frequency are called ball passage vibrations. The first studies on the ball passage vibrations were done by Perret [1] and Meldau [2] as a static running accuracy problem. They suggested that an increase in number of balls in a bearing reduces its untoward effects. Gustafson et al. [3] studied the effects of waviness and pointed out that lower order ring waviness affects the amplitude of the vibrations at the ball passage frequency. They observed that vibrations at higher harmonics

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of the ball passage frequency are also present in the vibration spectrum and their amplitudes depend on the radial load, radial clearance, rotational speed and the order of harmonics. The same conclusion is proved theoretically by Meyer et al. [4] for perfect radial ball bearings with linear modeling of the spring characteristics of balls. Gad et al. [5] showed that resonance occurs when BPF coincides with frequency of the system and they also pointed out that for certain speeds, BPF can exhibit its sub and super-harmonic vibrations for rotor bearing system. Aktu¨rk and Gohar [6] performed a theoretical investigation of effect of varying the preload on the vibration characteristics of a rotor bearing system and also suggested that by taking correct number of balls and amount of preload in a bearing untoward effect of the BPV can be reduced. They also showed that ball passage vibrations are very sensitive to the number of rolling elements and the amount of axial preload. Harsha et al. [7] analyzed the nonlinear behaviors of high-speed horizontal balanced rotor supported by ball bearing. The conclusion of this work shows that most sever vibrations occur when the ball passage frequency (BPF) and its harmonics coincide with the natural frequency. Clearance, which is provided in the design of bearing to compensate the thermal expansion, is also a source of vibration and introduces the nonlinearity in the dynamic behavior. The study of the effect of clearance nonlinearity on the response of rotors has gained a lot of attention lately because of the development of high-speed rotors such as space shuttle main engine turbo-pump rotor. Clearance nonlinearity is different from most of other nonlinearities because it cannot be approximated by a mathematical series. The early work in rotor dynamics by Yamamoto [8] introduces nonlinearity to the Jeffcott equation by including the effect of bearing clearances (or dead bands). The conclusion of this work shows that the maximum amplitude at critical speed decreases with increasing radial clearance and critical speed disappears under the condition beyond a marginal clearance, which depends on the amount of unbalance. Sunnersjo [9] studied the ball passage vibrations theoretically and experimentally, taking inertia and damping forces into account. Fukata et al. [10] first took up the study of ball passage vibrations and the nonlinear dynamic response for the ball bearing supporting a balanced horizontal rotor with a constant vertical force. It is a more detailed analysis as compared to SunnersjoÕs [9] work as regimes of super-harmonic, sub-harmonic and chaotic behavior are found out. Mevel and Guyader [11] have developed a theoretical model of a ball bearing supporting a balanced horizontal rigid rotor, with a constant vertical radial force. This is similar to the work done by Fukata et al. [10] but more results have been reported for parametric studies undertaken and routes to chaos traced out. Chaos in this model of bearing has been reported to come out of sub-harmonic route and quasi-periodic route. Sankaravelu et al. [12] have reported that the arc length continuation method takes less computation time as compared to direct integration, and the method obtains steady state response and stability analysis simultaneously. The eigenvalues of the Floquet matrix are obtained with the shooting technique, which gives the bifurcation points. Once the stability sets in numerical integration, it is used to obtain the response. Tamura and Tsuda [13] have theoretically estimated the stiffness of the ball bearing subjected to a constant radial load. Garguilo [14] has developed a new set of equations for providing initial estimates of stiffness of rolling element bearings. Yamauchi [15] developed a numerical harmonic balance method using the FFT algorithm for multiple degree of freedom rotor systems, including nonlinear bearings and couplings. Saito [16] calculated the nonlinear unbalance response of horizontal Jeffcott rotors with radial clearance. Both studies were concerned only with the harmonic response. Childs and Moyer [17] presented an explanation for the sub-harmonic response of rotors in presence of bearing clearance and side load. Choi and Noah [18] analyzed the coherence of superand sub-harmonic in a rotor bearing model-using harmonic balancing method along with discrete Fourier transform procedure. For multi disk rotor systems, Nataraj and Nelson [19] developed a periodic solution method based on a collocation approach for the response of rotor. They utilized a sub-system approach to reduce the size of the resulting system of algebraic equations. The dynamic responses of rotors in high-speed rotors with bearing clearance have been studied by Ehrich [20–22]. These studies by Ehrich had shown the appearance of high sub-harmonic and chaotic response in the rotor. Apart from super- and sub-harmonic responses, aperiodic whirling motions in a high-pressure oxygen turbo pump of the space shuttle main were also reported by Kim and Noah [23]. Wensing [24] developed a mathematical model using dynamics of ball bearing. The dynamic analysis includes the effects of surface waviness and cage run-out in his study.

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In this work, the effect of radial internal clearance has been studied. The appearance of periodic, sub-harmonic, chaotic and Hopf bifurcation is seen theoretically. The results presented here have been obtained from a large number of numerical integrations; mainly presented in form of time response, phase trajectories, Poincare` maps and frequency spectra. 2. The problem formulation A schematic diagram of rolling element bearing is shown in Fig. 1. For investigating the structural vibration characteristics of ball bearing, a model of bearing assembly can be considered as a spring mass system, in which the outer race of the bearing is fixed in a rigid support and the inner race is fixed rigidly with the shaft. Elastic deformation between races and rollers give a nonlinear force deformation relation, which is obtained by Hertzian theory. In the mathematical modeling, the ball bearing is considered as spring mass system and balls act as nonlinear contact spring as shown in Fig. 2. Since, the Hertzian forces arise only when there is contact deformation, the springs are required to act only in compression. In other words, the respective spring force comes in to play when the instantaneous spring length is shorter then its unstressed length, otherwise the separation between balls and the races takes place and the resultant force is set to zero. 2.1. Contact stiffness Hertz considered the stress and deformation in the perfectly smooth, ellipsoidal, contacting elastic solids. The application of the classical theory of elasticity to the problem forms the basis of stress calculation for machine elements as ball and roller bearings. Therefore the point contact between the race and ball develop into an area contact which has the shape of an ellipse with a and b as the semi major and semi minor axes, respectively. The curvature sum and difference are needed in order to obtain the contact force of the ball. P The curvature sum q is obtained as from Harris [25] is expressed as:

Fig. 1. A schematic diagram of a rolling element bearing.

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Fig. 2. Mass–spring model of the rolling element bearing.

X

q ¼ qI1 þ qI2 þ qII1 þ qII2 ¼

1 1 1 1 þ þ þ rI1 rI2 rII1 rII2

ð1Þ

The curvature difference F(q) is expressed as: F ðqÞ ¼

ðqI1  qI2 Þ þ ðqII1  qII2 Þ P q

ð2Þ

The parameters rI1, rI2, rII1, rII2, qI1, qI2, qII1, qII2 are given dependent upon calculations referring to the inner and outer races as shown in Fig. 3. If the inner race is considered, rI1 ¼ D=2; qI1 ¼ 2=D;

rI2 ¼ D=2; rII1 ¼ d=2; rII2 ¼ r and qI2 ¼ 2=D; qII1 ¼ 2=d; qII2 ¼ 1=r

ð3Þ

If outer race is considered, they are given as: rI1 ¼ D=2; qI1 ¼ 2=D;

rI2 ¼ D=2; rII1 ¼ D=2; rII2 ¼ R and qI2 ¼ 2=D; qII1 ¼ 2=D; qII2 ¼ 1=R

ð4Þ

As per the sign convention followed, negative radius denotes a concave surface. Using Table 2 one can calculate all the parameters including curvature difference at inner and outer race. For the contacting bodies being made of steel, the relative approach between two contacting and deforming surface is given by [26] X 1=3 d ¼ 2:787  108 Q2=3 q d ðmÞ ð5Þ where d* is a function of F(q). Hence, the contact force (Q) is:   X 1=2  3=2 7 Q ¼ 3:587  10 q ðd Þ d3=2 ðNÞ The elastic modulus for the contact of a ball with the inner race is:   X 1=2 N qi K i ¼ 3:587  107 ðdi Þ3=2 mm3=2

ð6Þ

ð7Þ

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Plane 1 Plane 2 Body I r I2 r I1 r II1 r II2

Plane 1 0

90

Plane 2

Fig. 3. Geometry of contacting bodies.

And for the contact of a ball with the outer race is:   X 1=2 N 3=2 qo K o ¼ 3:587  107 ðdo Þ mm3=2

ð8Þ

Then the effective elastic modulus K for the bearing system is written as: K¼

1 1=n ð1=K i

n

þ 1=K 1=n o Þ

;

where n ¼ 3=2

ð9Þ

In Eqs. (7) and (8), the parameters di and do can be attained from Table 1, if the values of F(q)i and F(q)o are available with using of Table 2. The effective elastic modulus (K) for bearing system by using geometrical and physical parameters is written as:   pffiffiffi N 5 K ¼ 7:055  10 d ð10Þ mm3=2 2.2. Derivation of governing equations of motion A real rotor bearing system is generally very complicated and difficult to model, so for effective and simplified mathematical model following assumptions are made: 1. Deformations occur according to the Hertzian theory of elasticity. Small elastic deformations of the rolling elements and the races are considered but plastic deformations are neglected. 2. The rolling elements, the inner and outer races and the rotor have motions in the plane of bearing only. 3. The rollers in a rolling element bearing are assumed to have no angular rotation about their axes, i.e. no skewing. Hence there is no interaction of the corners of the rollers with the cage and the flanges of the races. 4. There is no slipping of balls as they roll on the surface of races. Since there is perfect rolling of the balls on the surface of races and the two points of ball touching the races have different linear velocities, the center of the ball has a resultant translational velocity.

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Table 1 Dimensional contact parameters from [25] F (q)

d*

0 0.1075 0.3204 0.4795 0.5916 0.6716 0.7332 0.7948 0.83595 0.87366 0.90999 0.93657 0.95738 0.97290 0.983797 0.990902 0.995112 0.997300 0.9981847 0.9989156 0.9994785 0.9998527 1

1 0.997 0.9761 0.9429 0.9077 0.8733 0.8394 0.7961 0.7602 0.7169 0.6636 0.6112 0.5551 0.4960 0.4352 0.3745 3176 0.2705 0.2427 0.2106 0.17167 0.11995 0

Table 2 Geometric and physical properties used for the rolling element bearings Mass of rolling element (mj) Mass of the inner race (min) Mass of the outer race (mout) Mass of rotor (mrotor) Radius of inner race with point of contact with the rolling element (r = d/2) Radius of outer race with point of contact with the rolling element (R = D/2) Radius of each rolling element (qr) Internal radial clearance (c) No. of rolling elements (Nb) Initial radial position of jth rolling element (qj)

0.06 kg 0.09 kg 0.09 kg 6 kg 23 mm 31 mm 4 mm 1, 6, 12, 20 lm 8 27 mm

5. The damping of a ball bearing is very small. This damping is present because of friction and small amount of lubrication. The estimation of damping of ball bearing is very difficult because of the dominant extraneous damping which swamps the damping of the bearing. Kramer [26] has provided an estimation of the bearing damping. The element race vibratory system is reduced to a linear spring–mass damper system for which xeq, the frequency at which the amplitude of damping force equals the spring force when a harmonic motion is given to the system, is evaluated. Hence, xeq ¼ ð0:4–4Þ105 s1 feq ¼ 2p c ¼ ð0:25–2:5Þ105 k ðNs=lmÞ; ð11aÞ where k is stiffness of the ball baring. 6. The cage ensures the constant angular separation (b) between rolling elements, hence there is no interaction between rolling elements. In addition, at any given instant, some of the rolling elements will contact both races. Hence,

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b¼

2p Nb

695

ð11bÞ

The equations of motion that describe the dynamic behavior of the complete model can be derived by using LagrangeÕs equation for a set of independent generalized coordinates, as: d oT oT oV  þ ¼ ff g dt ofp_ g ofpg ofpg

ð12Þ

where the T, V, p and f are kinetic energy, potential energy, vector with generalized degree-of-freedom (DOF) coordinate and vector with generalized contact forces respectively. The kinetic and potential energies can be subdivided into the contributions from the various components i.e. from the rolling elements, the inner race, the outer race and the rotor. The total kinetic energy (T) of the rotor bearing system is the sum of the rolling elements, inner and outer races and the rotor, as T ¼ T r.e. þ T i

þ To

race

race

þ T rotor

ð13Þ

The subscripts i_race, o_race, rotor and r.e. refers to, respectively, the inner race, the outer race, the rotor and the rolling elements. The potential energy is provided by deformations of the balls with the races and deformations occur according to Hertzian contact theory of elasticity. Potential energy formulation is performed taking datum as the horizontal plane through the global origin. The total potential energy (V) of the bearing system is the sum of the balls, inner and outer races, springs and the rotor, as: V ¼ V r. e . þ V i

race

þV0

race

þ V springs þ V rotor

ð14Þ

where Vr.e., Vi_race, Vo_race and Vrotor are the potential energies due to elevation of the rolling elements, inner and outer races and the rotor, respectively. Vsprings is potential energy due to nonlinear spring contacts between rollers and the races. 2.2.1. Contribution of the inner race Apart from local deformations in the contacts, the inner race is considered as a rigid body. The kinetic energy of the inner race about its center of mass is evaluated in x- and y-frames. The position of the origin of the moving frame relative to the reference frame is described by transitional DOF x_ in and y_ in . The kinetic energy expression for the inner race is:  1 2 1 !_ ! _ T i race ¼ min r in  r in þ I in /_ in ð15Þ 2 2 The displacement vector showing the location of inner race center with respect to that of outer race center is then given by ! r in

¼ r out þ r in

! r in

!  !  ! ! ¼ x in þ x out ^i þ y in þ y out ^j

or,

!

!

ð16Þ

out

ð17Þ

Differentiating rin with respect to time (t) and put that value in Eq. (15) gives:

1 2 1 T i race ¼ min x_ 2in þ y_ 2in þ I in /_ in ð18Þ 2 2 Since the position of the inner race is defined from the outer race center, hence the potential energy for the inner race is: Vi

race

¼ min gðy in

out

þ y out Þ

ð19Þ

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2.2.2. Contribution of the outer race The outer race is also considered as a rigid body and it is assumed that the outer race is stationary. Hence, r_ out ¼ 0 and /_ out ¼ 0. The kinetic energy expression for the outer race is zero. The potential energy of the outer race is: Vo

race

¼ mout gy out

ð20Þ

2.2.3. Contribution of the rolling elements The rolling elements are also considered as rigid bodies. For the determination of their contribution to the kinetic energy, the position of the jth-rolling element is describe by two transitional DOF,

q_ j þ r_ out and /_ j The kinetic energy due to the rolling elements to be obtained as a summation of those from each element as: T r.e. ¼

Nb X

ð21Þ

Tj

j¼1

The position of the center of roller is defined with respect to the outer race center. Hence, the kinetic energy of the rolling elements may be written as:  !  1 2 1 !_ ! ! _ _ _ T j ¼ mj q j þ r out  q j þ r out þ I j /_ j ð22Þ 2 2 The displacement vector showing the location of jth rolling elements is: !  !  ! q j ¼ q j cos hj ^i þ q j sin hj ^j

ð23Þ

And for the outer race center is: !

!

^

!

^

r out ¼ x out i þy out j

ð24Þ

The summation of Eqs. (23) and (24) after differentiation with respect to time (t) leads to the following expression: !  !  2 ! !_ _ _ _ q j þ r out  q j þ r out ¼ q_ 2j cos2 hj þ q2j sin2 hj  h_ j  2q_ j  qj  h_ j cos hj sin hj   2 þ x_ 2out þ 2_xout q_ j cos hj  qj sin hj  h_ j þ q_ 2j sin2 hj þ q2j cos2 hj  h_ j   þ 2q_ j  qj  h_ j cos hj sin hj þ y_ 2out þ 2y_ out q_ j cos hj  qj sin hj  h_ j ð25Þ The outer race is assumed to be stationary, hence x_ out ¼ 0 and y_ out ¼ 0. Therefore Eq. (25) becomes, !  !  !_ ! _ _ _ q j þ r out  q j þ r out ¼ q_ 2j cos2 hj þ q2j sin2 hj  h_ 2j þ q_ 2j sin2 hj þ q2j cos2 hj  h_ 2j or;

!  !    ! ! _ _ _ _ q j þ r out  q j þ r out ¼ q_ 2j þ q2j  h_ 2j

From the Eq. (22), we get,  1 2 2 1  T j ¼ mj q_ 2j þ q2j  h_ j þ I j /_ j 2 2

ð26Þ

ð27Þ

ð28Þ

It is assumed that there is no slip, hence the relative transitional velocity of outer race and rolling element must be same and in opposite direction. Therefore, the contact equation for jth-rolling element and the outer race can be written as     R /_ out  h_ j ¼ qr /_ j  h_ j ð29Þ

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Since the outer race is stationary, hence /_ out ¼ 0

ð30Þ

The rotation of jth rolling element about its center of mass is,   _/ ¼ h_ j 1 þ R j qr Now the kinetic energy of the rolling elements can be written as, 2 Nb  1 2 X 2 1  2 R mj q_ j þ q2j  h_ j þ I j h_ j 1 þ T r. e . ¼ 2 2 qr j¼1

ð31Þ

ð32Þ

The position of the center of roller is defined with respect to the outer race center. Hence, the potential energy of the rolling elements may be written as: V r. e . ¼

Nb X

mj gðqj sin hj þ y out Þ

ð33Þ

j¼1

or; V r.e. ¼ mgN b y out þ

Nb X

ðmj gqj sin hj Þ

ð34Þ

j¼1

2.2.4. Contribution of the rotor The kinetic energy of the rotor is calculated by assuming that its center remains coincident with the inner race. Hence, the kinetic energy of the rotor is:

1 2 1 T rotor ¼ mrotor x_ 2in þ y_ 2in þ I rotor h_ rotor 2 2

ð35Þ

The rotor center is coinciding with inner race center and position of the inner race center is defined with respect to outer race center. Hence, the potential energy of the rotor is expressed as: V rotor ¼ mrotor gðy in

out

þ y out Þ

ð36Þ

2.2.5. Contribution of the contact deformation The contacts between rolling elements and races are treated as nonlinear springs, whose stiffnesses are obtained by Hertzian theory of elasticity. The expression of potential energy due to the contact deformation of the springs is: V springs ¼

Nb Nb X X 1 1 k in d2in þ k out d2out 2 2 j¼1 j¼1

ð37Þ

where kin and kout are the nonlinear stiffness due to Hertzian contact effects and evaluated by Eqs. (51) and (52). The deformation at contact points between the jth rolling element and inner race is: din ¼ ½fr þ qr g  vj 

ð38Þ

In this expression, if {r + qr}ivj, compression takes place and restoring force acts. If {r + qr}hvj, no compression and restoring force is set equal to zero. Similarly, at the outer race the deformation at the contact points is: dout ¼ ½R  fqj þ qr g In this expression, if R < {qj + qr}, compression takes place and restoring force act.

ð39Þ

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If R > {qj + qr}, no compression and restoring force is set equal to zero. The radial internal clearance (c0) is the clearance between the imaginary circles, which circumscribe the rolling elements and the outer race. Hence with the consideration of radial internal clearance (c0), the contact deformations at the inner and outer races are: din ¼ ½fr þ qr þ c0 g  vj 

ð40Þ

dout ¼ ½R  fqj þ qr þ c0 g

ð41Þ

2.3. Equations of motion The kinetic energy and potential energy contributed by the inner race, outer race, balls, rotor and springs, can be differentiated with respect to the generalized coordinates qj (j = 1, 2, . . ., Nb), xin, and yin to obtain the equations of motion. For the generalized coordinates qj, where j = 1, 2, . . ., Nb, the equations are: ovj 2 1 ok in 2 €j þ mj g sin hj þ mj qj h_  ðk in Þ½din þ mj q þ ðk out Þ½dout þ þ ½din þ 2 oqj oqj þ

1 ok out 2 ½dout þ ¼ 0; 2 oqj

j ¼ 1; 2; . . . ; N b

ð42Þ

For the generalized coordinate xin the equation is: ðmin þ mrotor Þ€xin 

Nb X

k in ½din þ

j¼1

ovj ¼ F u sinðxtÞ oxin

ð43Þ

For the generalized coordinate yin the equation is: ðmin þ mrotor Þ€y in þ ðmin þ mrotor Þg 

Nb X

k in ½dout þ

j¼1

ovj ¼ W þ F u cosðxtÞ oy in

ð44Þ

This is a system of (Nb + 2) second order, nonlinear differential equations. There is no external radial force is allowed to act on the bearing system and no external mass is attached to the outer race. The ‘‘+’’ sign as subscript in these equations signifies that if the expression inside the bracket is greater than zero, then the rolling element at angular location hj is loaded giving rise to restoring force and if the expression inside bracket is negative or zero, then the rolling element is not in the load zone, and restoring force is set to zero. For the balanced rotor condition, the unbalance rotor force (Fu) is set to be zero. The deformation of spring at inner race vj (Fig. 2) can be obtained as: xin þ vj cos hx ¼ xout þ qj cos hj

ð45Þ

y in þ vj sin hx ¼ y out þ qj sin hj

ð46Þ

From these two equations, the expression for vj is obtained as: vj ¼ ½ðxout  xin Þ2 þ q2j þ 2qj ðxout  xin Þ cos hj þ 2qj ðy out  y in Þ sin hj þ ðy out  y in Þ2 1=2

ð47Þ

Now the partial derivatives of vj with respect to qj, xin and yin are: ovj qj þ ðxout  xin Þ cos hj þ ðy out  y in Þ sin hj ¼ oqj vj

ð48Þ

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699

ovj ðxout  xin Þ  qj cos hj ¼ oxin vj

ð49Þ

ovj ðy out  y in Þ  qj sin hj ¼ oy in vj

ð50Þ

The nonlinear stiffness of spring for the inner and outer race is from Eq. (20): 1=2

ð51Þ

½d1=2 out

ð52Þ

ðk in Þ ¼ 7:055  105 ½din ðk out Þ ¼ 7:055  10

5

ðok in Þ 1=2 ovj ¼ 3:5725  105 ½din oqj oqj

ð53Þ

ðok out Þ 1=2 ¼ 3:5725  105 ½dout oqj

ð54Þ

2.4. Ball passage frequency When the rotor is rotating, applied loads are supported by a few balls restricted to a narrow load region and the radial position of the inner race with respect to outer race depends on the elastic deflections at the ball to raceways contacts. Balls are deformed as they enter the loaded zone where the mutual convergence of the bearing races takes place and the balls rebound as they move to unloaded region. Time taken by rotor to regain its initial position is: t¼

time for a complete rotation of cage Nb

ð55Þ

As the time needed for a complete rotation of the cage is 2p/xcage, the rotor will be excited at the frequency of (Nb · xcage) known as ball passage frequency. Since outer is assumed to be stationary, the ball passage frequency (xbp) is:  1 db xbp ¼ N b xinner 1  ð56Þ 2 dm Vibrations associated with the ball passage frequency are known as ball passage vibration (BPV) or the elastic compliance vibrations. The effect of ball passage frequency can be worst when it coincides with a natural frequency of the rotor bearing system. 3. Computational solution of the equations The equations of motion (42)–(44) are solved by modified Newmark-b method to obtain the radial displacement and velocity of the rolling elements. The longer the time to reach steady state vibrations, the longer CPU time needed and hence the more expensive the computation. To study the stability of the system, parameters of rolling bearing are selected as shown in Table 2. 3.1. Initial conditions For numerical solution, the initial conditions and step size are very important for successive and economic computational solution. Particularly for nonlinear systems, different initial conditions mean a totally different system and hence different solutions. The larger the time step, Dt, faster the computation. On the other hand the time step should be small enough to achieve an adequate accuracy. Also, very small time steps can increase

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the truncation errors. Therefore an optimization should be made between them. The time step for the investigation is assumed as Dt = 105 s. At time t = 0 the following assumptions are made: (i) The shaft is held at the center of the bearing and all balls are assumed to have equal axial preload. (ii) The shaft is then given initial displacements and velocities. For fast convergence the initial displacements are set to the following values: x0 = 106 m and y0 = 106 m. The initial velocities are assumed to be zero: x_ 0 ¼ 0 and y_ 0 ¼ 0. (iii) When t > Dt the initial conditions have already passed and the normal procedure commences. Rolling elements are radially preloaded in order to ensure the continuous contact of all rolling elements and the raceways, otherwise a chaotic behavior might be observed. 3.2. Time responses, phase trajectories, power spectra and poincare` maps The time history of the unbalance response has been examined for periodic behavior. This is done by examining the time series out put, once per cycle, for sufficiently long segments so that multiple periodic and aperiodic behaviors could be discerned from the post transient solutions. When motion is periodic, the phase plane orbits traces out a closed curve. Chaotic motions have orbits that never close or repeat. Thus trajectories of the orbits in the plane will tend to fill up the section of the phase space. Aperiodic behavior in a deterministic dynamical system is characterized by broadband frequency spectra. In sub-synchronous frequencies the significant energy shows the aperiodic nature of the response. Poincare` maps are produced by plotting one of the variables of the system, e.g. the vertical or horizontal displacement against its derivative, once per rotational periodic of the system. For synchronous limit cycle a single point in the plane is repeated every cycle, while nth sub-harmonic is revealed by n and only n repeated points. However, Poincare` maps of chaotic system have a fractal structure, which can be used to identify chaotic states. The stiffness of the rolling element bearing has been estimated for point contact with the vertical force of 6 N by using Hertzian contact theory. From Fig. 4, it can be seen that the stiffness remains practically constant till a radial clearance of 20 lm for ball bearing. In the paper the time responses, power spectra, rotor trajectories and Poincare` maps are analyzed for different internal radial clearances as at 1 lm, 6 lm, 12 lm and at 20 lm at the rotor speed 1500 rpm. Therefore, it is observed that at a particular speed with increase in radial clearance, the regions of unstable and chaotic response are wider. The time response, power spectra, rotor trajectories and Poincare` maps at the rotor speed 1500 rpm with internal radial clearance 1 lm are shown in Fig. 5, respectively. Fig. 5(a) shows time response with horizontal and vertical displacement. Fig. 5(b) shows power spectral density and the peak amplitude of vibration appears in the spectrum at the ball passage frequency xbp = 90 Hz. Others peaks of lower amplitude appear at 2xbp = 180 Hz and at 3xbp = 270 Hz. Fig. 5(c) and (d) show the phase plot and Poincare` map. The system losses its periodicity and shows the quasi-periodic nature.

900 Natural Frequency (Hz)

800 Vertical, W = 6N

700

m= 0.6 kg

600 500 400 300 200 Horizontal, W = 6N

100 0 0

5

10 γ (μm)

15

20

Fig. 4. Vertical and horizontal critical frequency.

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Fig. 5. Responses at 1500 rpm with radial clearance = 1 lm. (a) Time displacement response, (b) FFT, (c) phase trajectories and (d) Poincare` maps.

The time response, power spectra, rotor trajectories and Poincare` maps at the rotor speed 1500 rpm with internal radial clearance 6 lm are shown in Fig. 6, respectively. Fig. 6(a) shows time response with horizontal and vertical displacement. The time series out put, once per cycle is sub-harmonic of second order. Fig. 6(b) shows power spectral density and the peak amplitudes of vibration appear in the spectrum at the ball passage frequency xbp = 90 Hz, at 2xbp = 180 Hz and at 3xbp = 180 Hz. Other peaks of small amplitude appear in the vibration spectrum at 4xbp = 360 Hz and at 5xbp = 450 Hz. The phenomenon of sub-harmonic motions, in

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Fig. 6. Responses at 1500 rpm with radial clearance = 6 lm. (a) Time displacement response, (b) FFT, (c) phase trajectories and (d) Poincare` maps.

which the ration of the forced frequency and the response frequency becomes rational, is called phase locking or mode locking. Fig. 6(c) and (d) show the phase plot and Poincare` map. Two and only two repeated points reveal second order sub-harmonic response. The time response, power spectra, rotor trajectories and Poincare` maps at the rotor speed 1500 rpm with internal radial clearance 12 lm are shown in Fig. 6, respectively. Fig. 7(a) shows time response with horizontal and vertical displacement. The time series out put, once per cycle is onset of chaos. Fig. 7(b) also shows the dense spectrum between ball passage frequency and its multiples, which is an indication of onset of chaos. The peak amplitudes of vibration appear in the spectrum at the ball passage frequency xbp = 90 Hz, at

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Fig. 7. Responses at 1500 rpm with radial clearance = 12 lm. (a) Time displacement response, (b) FFT, (c) phase trajectories and (d) Poincare` maps.

2xbp = 180 Hz and at 3xbp = 270 Hz. Other peaks with lower amplitude appear in the spectrum at 4xbp = 360 Hz and at 7xbp = 630 Hz Fig. 7(c) and (d) shows the phase plot and Poincare` map. The time response, power spectra, rotor trajectories and Poincare` maps at the rotor speed 1500 rpm with internal radial clearance 20 lm are shown in Fig. 8, respectively. The system shows chaotic nature. Fig. 8(a) shows time response with horizontal and vertical displacement. The time series out put, once per cycle is chaotic. Fig. 7(b) shows power spectral density and the peak amplitudes of vibration appear in the spectrum at the ball passage frequency xbp = 90 Hz and at 1/2xbp = 45 Hz. Fig. 8(c) and (d) shows the phase plot and Poincare` map also show the chaotic nature of the system. Poincare` map is shown its fractal structure, which is

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Fig. 8. Responses at 1500 rpm with radial clearance = 20 lm. (a) Time displacement response, (b) FFT, (c) phase trajectories and (d) Poincare` maps.

a strong indication of chaotic nature of the system and frequency spectra at this speed also show broadband spectrum i.e. significant energy at this speed exist. 4. Conclusion The nonlinear response of a balanced rotor has been demonstrated to be chaotic for some specific combination rotational speed combined with varying internal radial clearance to provide sufficient nonlinearity. For

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cases, which are stable to free motion and not close to the neutral stability line, a limited range of chaos can be detected. Varying the internal radial clearance bound the region of quasi-periodic, sub-harmonic and chaotic for ball bearing excited by various defects. Decrease in radial clearance increase the linear characteristics of the system. There is no chaos appears as clearance decreases. It is observed from the obtained power spectra that peak amplitude of vibration always appear in the spectrum only at the ball passage frequency (BPF). Based on the characteristics of the dynamic behaviors of the system, the nonlinear dynamic responses are categorized as: First the system responses are periodic and are not sensitive to initial conditions or small variations of system parameters. This is well behaved region (for the range of internal radial clearance 0.2–1 lm) permits the designer to predict the trends accurately and without ambiguity. Second the system responses are having weak attractors of chaotic, which are quite near to quasi-periodic or sub-harmonic in nature and are extremely sensitive to small variations of system parameters and operating conditions i.e. either rotor speed or radial internal clearance. The contact force significantly fluctuates as the internal radial clearance changes. For example, as the clearance is changes from 6 lm to 12 lm, the corresponding response changes as sub-harmonic to onset of chaos. For these responses, the hidden danger is periodicity. The periodic response may lead designers to overlook its large sensitivity to small variations of system parameters or operating conditions. Third the responses are unpredictable, either periodic or chaotic and is extremely sensitive to both initial conditions and small variations system parameters. When chaotic, the predicted contact force is sensitive to initial conditions. 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