1. Introduction

Ball bearings are very important in numerous rotating machinery systems. The prediction and analysis of the dynamic behavior of rotor systems are important because their rotating components possess amounts of energy that can be transformed into vibrations. These vibrations not only affect the performance of rotor system, but may also cause serious damage to the rotating machinery systems. Hence this field has become more challenging since nonlinear analysis is far more difficult compared to the analysis of the linear phenomena. Nonlinearities in rotor systems can be caused by many reasons [1] such as Hertzian contact force, internal radial clearance, surface waviness [2], and stiffness coefficient. Waviness is defined as the geometric imperfection of inner or outer race in ball bearing, and it is also considered one of the sources of vibration [3].

The aim of this study is to develop a numerical model for investigating the dynamic properties of rotor system supported by ball bearings under the effect of bearing running waviness. In the analytical formulation, the waviness of rolling elements is modeled by sinusoidal function. The Hertzian contact theory is applied to calculate the elastic deflection and non-linear contact force. A two-degree of freedom system is considered with the assumption that there is no friction between the rolling elements and raceways of ball bearings. The rolling elements are positioned symmetrically such that their motion is synchronized. The 4th order Runge-Kutta numerical integration technique has been applied to solve the nonlinear differential equations. The results are presented in form of time displacement response, Poincarè map, and frequency spectra. The analysis demonstrates that the surface waviness is an important parameter affecting on the dynamic characteristics of rotor bearing system [4, 5]. The model can also be used as a tool for predicting nonlinear dynamic behavior of rotor system under different operating conditions.

2. The Problem Formulation

Since a real rotor bearing system is very complicated and difficult to model [6-10], the analysis of the dynamic behavior of the system is done based on the following assumptions while developing the mathematical model as follows:

1. Torsional vibration of rotor and gyroscopic effects may be neglected and only transverse vibration of rotor should be considered. 2. The outer race of the ball bearing is fixed to a rigid support. The inner race is fixed rigidly to the shaft. There is no slipping of balls. 3. Constant vertical radial force acts on the bearing. 4. Elastic deformation between race and ball gives a nonlinear force deformation relation, which is obtained by using Hertzian theory.

Zeillinger and K Köttritsch [11] noted that additional damping usually occurs between the bearing outer race and its housing. It is known that ball bearing have very low inherent damping. This damping would be effective only for small vibration amplitude [12]. Krämer [13] has provided as estimation of the bearing damping and [14] also the bearing coefficient of ball bearing is well within the range of 33.75 Ns/m to 337.5 Ns/m. A value of c = 200 Ns/m was chosen.

2-1. Waviness model

Waviness is defined as the geometric imperfection of inner or outer race in a ball bearing, and it is considered one of the sources of vibration. These are global sinusoidal shaped imperfections on the outer race surface of the bearing components as in Fig. 1. Waves can be described in terms of two parameters: wavelength (λ), and its amplitude (A). The amplitude of wave at the contact angle can be expressed as follows: where Ai is initial wave amplitude and A0 is maximum amplitude of wave in radial direction of outer race; L is the arc length of wave of inner race,

Fig. 1.Race waviness model.

where Nw is number of wave lobes and θj is the position of jth ball; r is inner race radius.

Because the inner race is moving at the speed of the shaft and the ball center at the speed of the cage so the contact angle (θj) is given by

2-2. Nonlinear ball bearing force

As shown in Fig. 2, the tangent velocity of the contact point between the ball and the inner race VA, outer race VB, respectively, can be given by

Fig. 2.Schematic diagram of ball bearing.

Since the outer race is assumed to be stationary, VA = 0. Therefore, the tangent velocity of the cage is

The inner race is fixed to the shaft, ωB = ωrotor. Then, the angular velocity of the cage is written by

The varying compliance frequency or the ball passage frequency can be expressed in terms of the cage speed times the number of balls, N.

According to the Hertzian contact theory [Hertz, 1896], the local Hertzian contact force Fj between the jth ball and the race is given as follows [15]:

The contact stiffness coefficient K, can be given by the stiffness coefficient between the ball and each race [16], ki and k0 in series as follows:

where ki and k0 can be determined by elastic modulus and Poisson’s ratio and curvature sum of the contact points as from Harris [15]. The geometrical properties of the system is in Table 1, K = 7.055 × 109 (N/mm1.5).

Table 1.Geometrical properties of the system

Based on the Hertzian contact force between inner and outer race and the ball, the total restoring force is the sum of restoring force from each of the rolling elements. The “+” sign subscript in the equation (13, 14) signify this step change in the restoring force expression and models the clearance non-linearity. If the expression inside the bracket is greater than zero, then the ball at the angular location θj is within the angular contact zone and it is loaded giving rise to a restoring force. If the expression within the brackets is negative or zero, then the ball is not in the load zone, and restoring force is set to zero. So the contact force between the jth ball and inner race can be expressed as,

The mathematical model takes into account the sources of nonlinearities in rotor bearing system. After assembling the inertia, ball bearing force, damping force, race waviness and constant vertical force acting on the inner race, the dynamic equations of the system are established as follows:

3. Results and Discussion

The nonlinear governing equations of motion (15) are solved by the 4th order Runge-Kutta numerical integration technique to obtain and radial displacement of the rolling element and the shaft. The initial conditions are set to x0 = 10−6 (m), y0 = 10−6 (m), x’0 = 0 and y’0 = 0; K = 7.05 × 109 (N/m1.5). In order to observe the effect of race waviness, time displacement response, Poincarè map, and frequency domain vibrations of shaft rotor bearing system are obtained to determine the dynamic behavior of system [4].

The amplitude of race waviness is one of the key factors affecting on the dynamic characteristics of a rotor system. Fig. 3(a, b, c) show the response at a speed of 6,000 rpm without waviness in inner and outer race. For horizontal displacement response, the natural frequency coincides with the varying compliance frequency (ωvc = 354 Hz). The peak amplitude is 0.0846 μm on the horizontal direction. The major peaks amplitude of vibration appear at 3ωvc/2 = 531 Hz, 2ωvc = 708 Hz. The amplitude of peaks are 0.003 μm and 0.0058 μm at 3ωvc/2 = 531 Hz and 2ωvc = 708 Hz, respectively. For horizontal displacement response without outer race waviness and with amplitude of inner race waviness, Ai = 0.1 μm as in Fig. 3(d, e, f). The peak amplitude of vibration is 0.0867 μm at ωvc = 354 Hz. The major peaks are at 3ωvc/2 = 531 Hz, and 2ωvc = 708 Hz. The Poincarè maps showed in Fig. 3(b) and 3(e), give an indication of a quasi-periodic response because of “net” structure.

Fig. 3.a), b), c) Displacement response with time, Poincarè map for displacement response, and FFT for horizontal displacement response at 6,000 rpm, respectively; when Ai = 0, A0 = 0. sd), e), f) Displacement response with time, Poincarè map for displacement response, and FFT for horizontal displacement response at 6,000 rpm, respectively; when Ai = 0.1, A0 = 0.

Fig. 4(a, b, c) show the response with amplitude of inner race waviness, Ai = 0.1 μm and amplitude of race waviness, A0 = 0.3 at 6,000 rpm. The natural frequency coincides with the varying compliance frequency (ωvc = 354 Hz). The peak amplitude is 0.1018 μm on horizontal direction. The major peaks amplitude of vibration appear at ωvc/4 = 90 Hz, 5ωvc/2 = 890 Hz. Fig. 4(d, e, f) show the response with amplitude of inner race waviness, Ai = 0.1 μm and amplitude of race waviness, A0 = 0.5 at 6,000 rpm. The natural frequency coincides with the varying compliance frequency (ωvc = 354 Hz). The peak amplitude is 0.111 μm on horizontal direction. The major peaks amplitude of vibration appear at 3ωvc/2 = 531 Hz, 2ωvc = 708 Hz, and 5ωvc/2 = 890 Hz. With the orbits of a dense structure in the Poincarè maps, which give indication of chaotic response, as shown in Fig. 4(b) and 4(e).

Fig. 4.a), b), c) Displacement response with time, Poincarè map for displacement response, and FFT for horizontal displacement response at 6,000 rpm, respectively; when Ai = 0.1, A0 = 0.3. d), e), f) Displacement response with time, Poincarè map for displacement response, and FFT for horizontal displacement response at 6,000 rpm, respectively; when Ai = 0.1, A0 = 0.5.

It is seen from the solutions of the effect of race waviness by certain conditions, the amplitude response are modulation. Without waviness, the peak amplitudes of vibrations at the varying compliance frequency are insignificant. When increasing amplitude of outer race waviness, the peak amplitudes of vibrations at the varying compliance frequency are more significant. This is also demonstrated by Poincarè maps.

4. Conclusions

In this paper, a two-degree of freedom nonlinear model of a rotor-ball bearing system has been developed to obtain the nonlinear vibration response by varying amplitude of race waviness. Nonlinear analysis of this model was performed numerically using 4th order Runge-Kutta integration method.

The results show that the dynamic characteristics behavior of the system is sensitive to small variation of the system parameters. The race waviness of ball bearing is an important parameter for vibration analysis of rotor bearing system and should be considered at the design stages of rotating machinery systems. The model can be used as a tool for predicting nonlinear dynamic behavior of rotor ball bearing system under different operating conditions. Moreover, the study may contribute to a further understanding of the nonlinear dynamics of rotor bearing system.

Nomenclature

m : Mass of rotor, kg c : Equivalent viscous damping factor, Ns/m Fx/Fy : Hertzian contact force on horizontal/vertical direction, N L : Arc length of wave of surface waviness, μm Fu : Force due to unbalance rotor, N K : Hertzian contact stiffness, N/m3/2 N/Nw : Number of balls / wave lobes RB : Inner race radius, mm RA : Outer race radius, mm r : Inner race radius, mm t : Time, s Ai : Initial wave amplitude, μm A0 : Maximum wave amplitude, μm Vcage : Translational velocity of cage center, mm/s Va : Translational velocity of the inner race, mm/s Vb : Translational velocity of the outer race, mm/s W : Radial load, N γ : Internal radial clearance, μm λ : Wavelength of surface waviness, μm ωrotor : Angular speed of rotor, rad/s ωcage : Angular speed of the cage, rad/s ωA : Angular speed of the inner race, rad/s ωB/ωb : Angular speed of the outer race / ball, rad/s θj : Angular location of jth rolling element, rad/s