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2 H K Yadav et al, D H Pandya received his PhD in Vibration and Noise Control from IIT. Roorkee His research areas are machine design vibration acoustics NDT. controls and nonlinear dynamics He has published more than 20 papers in. various referred journals, S P Harsha received his PhD in Nonlinear Dynamics and Chaos from BITs. Pilani and Post Doctoral from Villanova University Philadelphia USA. His research areas are vibrations and control nonlinear dynamics and chaos. rolling element bearings design of experiments fault diagnosis and prognosis. unmanned surface air vehicles and CNT based mass sensor He has published. more than 96 papers in various referred journals and 81 in international. conference He has contributed to explain term waviness for rolling element. bearing which is also cited in Wikipedia the free Encyclopedia. 1 Introduction, Rolling element bearings are one of the key components in rotating machinery and their. good condition is vital for the machine performance Any bearing in operation will. invariably fail at some point with risk of machine breakdown as a result Allowing a. machine to break down before repairing is expensive as production time is lost and the. bearing defect may propagate to other machine components which will also need to be. replaced The stiffness rotational accuracy and vibration characteristics of a high speed. shaft are partly controlled by the ball bearings that support it In the rotor bearing. assembly supported by perfect ball bearings the vibration spectrum is dominated by the. vibrations at the natural frequency and the varying compliance frequency The vibrations. at this later frequency are called parametric vibrations. For the particular bearing the internal radial clearance IRC cannot be changed after. manufacturing Clearance in mechanical components introduces very strong nonlinearity. Clearance which provided in the design of bearing to compensate the thermal expansion. is also a source of vibration and introduces the nonlinearity in the dynamic behaviour. Sunnersjo 1978 studied the varying compliance vibrations theoretically and. experimentally taking inertia and damping forces into account Fukata et al 1985 first. took up the study of varying compliance 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 compared with Sunnersjo s 1978 works as regimes of. super harmonic sub harmonic and chaotic behaviour are discovered The studies. undertaken by Day 1987 and Kim and Noah 1996 considered the effect of unbalanced. force only but not a varying compliance effect In the present analysis all the three. effects i e the unbalanced rotor the varying compliance and the radial internal. clearance are studied in addition to nonlinearity due to Hertzian contact Harris 2001. considered two factors for structural vibrations of ball bearings one of these is contact. load from the balls which deform the races into a polyhedral shape and the other is the. motion of balls relative to the line of action of the radial load which fluctuates the rigidity. of the bearing, The varying compliance effect was studied theoretically by Perret 1950 considering. a deep groove ball bearing with the elastic deformation between race and balls modelled. by the Hertzian theory and no bending of races Perret studied the bearing at the instant. Nonlinear vibration signature analysis of a rotor supported ball bearings 3. when the balls are arranged symmetrically around the load line i e with either a ball or a. ball gap directly under the load In the intermediate cage position however the balls are. non symmetrically arranged which means that when loaded vertically the centre of the. inner ring will undergo a horizontal as well as vertical displacement Meldau 1951. studied theoretically the two dimensional motion of shaft centre Both Perret and Meldau. performed a quasi static analysis since inertia and damping force were not taken into. Mevel and Guyader 1993 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 1985 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 the sub harmonic route and the quasi periodic. route Datta and Farhang 1997 developed a nonlinear model for structural vibration in. the rolling bearings by considering the stiffness of the individual region where the. elements contact each other but in this model distributed defects are not considered. Tiwari et al 2000 has studied the effect of radial internal clearance the appearance of. sub harmonics and Hopf bifurcation is seen theoretically where as the shift in the peak. response is also observed experimentally, Harsha et al 2003 analysed the nonlinear behaviour of a high speed horizontal.

balanced rotor supported by a ball bearing The conclusion of this work shows that the. most severe vibrations occur when the varying compliance frequency VC and its. harmonics coincide with natural frequency Harsha 2005a has studied the effects of. radial internal clearance and rotor speed The appearance of periodic sub harmonic. chaotic and Hopf bifurcation is seen theoretically But Harsha considered only nonlinear. stiffness He studied the effects of radial internal clearance for both balanced and. unbalanced rotor speed Harsha 2005b 2006a The appearance of periodic. sub harmonic chaotic and Hopf bifurcation is seen theoretically But he has considered. only nonlinear stiffness Harsha 2006b has studied the effects of rotor speed with. geometrical imperfections The results are from a large number of numerical integrations. and are mainly presented in the form of Poincar maps and frequency spectra. The effect of fluctuation of the speed of the rotor has been studied and from the. analysis performed it was concluded that even a minimum fluctuation of the rotor speed. may result in major changes of the system dynamics indicating that speed fluctuations of. the rotor are a governing parameter for the dynamic behaviour of the system Cao and. Xiao 2008 have developed the comprehensive mathematical model for the spherical. roller bearing This paper represents that the larger the radial clearance the higher the. modal density and the higher the response at the roller passing frequency and its super. harmonics But overall the benefit of smaller radial clearance is limited in reducing the. displacements of inner race, Upadhyay et al 2010 developed the mathematical model for the bearing by. considering the nonlinear spring along with contact damping at the ball race contact. Effect of IRC along with unbalanced rigid rotor has been explained Period doubling and. mechanism of intermittency have been observed that lead to chaos The outcomes. illustrate the appearance of instability and chaos in the dynamic response as the speed of. the rotor bearing system is changed Also it has been shown by Ghafari et al 2010 that. the bearing having the clearance more than 4 5 m has more than one equilibrium point. noted as strange attractor The system vibrates around these strange attractors randomly. 4 H K Yadav et al, It has been reported that bearing having clearance more than 12 m has chaotic nature at. 1 000 RPM onwards, The practical occurrences of the nonlinear phenomena such as periodic subharmonic. chaotic and Hopf bifurcation are explained in the different books of authors Nayfeh and. Balachandran 1995 Moon 1987 Steven 1994 The authors want to give more. importance to all these books because a different tool to identify the nonlinear behaviour. of the system has been explained very nicely with plots of experiments. 2 Problem formulation, A schematic diagram of rolling element bearing is shown in Figure 1 For investigating. the structural vibration characteristics of rolling element bearing a model of bearing. assembly can be considered as a spring mass damper system Elastic deformation. between races and balls gives a nonlinear force deformation relation which is obtained. by Hertzian theory In the mathematical modelling the rolling element bearing is. considered as spring mass damper system and rolling elements act as nonlinear contact. spring as shown in Figure 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 into play when the instantaneous spring length is shorter. than its unstressed length otherwise the separation between balls and the races takes. place and the resultant force is set to zero The excitation is because of the varying. compliance vibrations of the bearing which arise because of the geometric and elastic. characteristics of the bearing assembly varying according to the cage position. Figure 1 The flexibility of the rolling contacts in a rolling element bearing is represented by. nonlinear springs and nonlinear damping, Nonlinear vibration signature analysis of a rotor supported ball bearings 5.

Figure 2 Mass spring damper of rolling element bearing see online version for colours. 2 1 Equation of motion, The controlling equations of motion describing the dynamic behaviour of the complete. model can be derived from a variational principle as Euler Lagrange equations. The equations of motion that describe the dynamic behaviour of the complete model can. be derived by using Lagrange s equation for a set of independent generalised coordinates. dt p p p p, where T V p and f are kinetic energy potential energy vector with generalised. degree of freedom DOF coordinate and vector with generalised contact forces. respectively and Pd represent the dissipation energy due to damping 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. kinetic energy and potential energy contributed by the inner race outer race balls. 6 H K Yadav et al, rotor and springs can be differentiated with respect to the generalised coordinates. j j 1 2 Nb xin and yin to obtain the equations of motion For the generalised. coordinates j where j 1 2 Nb the equations are,kir contact ir i kor contact or. j g sin j j 2,1 kir contact ir,1 kor contact or,2m j j 2m j j.

2m j Cin Kir contact ir j,or K or contact or3 2 j q 0 j 1 2 N b. For the generalised coordinate xin the equation is. xir ir contact,mrotation j 1,q j Fu sin s t,2mrotation Cir K ir contact ir3 2 j. xir mrotation, For the generalised coordinate yin the equation is. mrotation k,ir contact ir 3 2,q j W Fu cos s t,2mrotation Cin K ir contact ir j. yir mrotation,j xor xir j cos j,j yor yir j sin j,1 j j j xor xir cos j yor yir sin j.

j j j yor yir cos j xor xir sin j,j xir xor j cos j. j yir yor j sin j,where mrotation minner mrotor, Nonlinear vibration signature analysis of a rotor supported ball bearings 7. 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 j 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 unbalanced rotor force Fu is set to be zero. In the present paper the authors have written the equation of motion directly The. derivation of the equation of motion has been explained in detail in Upadhyay et al. 3 Methods of solution, The coupled nonlinear second order differential equations 3 to 5 are solved by. numerical integration technique which is a time domain approach The non analytic. nature of the stiffness term renders the system equations difficult for analytical solution. 3 1 Numerical integration, The equations of motion 3 to 5 are solved by using the explicit type numerical. integration technique Runge Kutta fourth order method to obtain radial displacement. velocity and acceleration of the rolling elements For performing numerical integration. the system equations are transformed into first order form by state variable method. z1 xir z2 xir z3 yir z4 yir,z1 xir z2 z2 xir z3 yir z4 z4 yir.

z1 xir z1 z2 xir,2 ir 2 2 ir,z and z 12,z3 yir z3 z4 yir. z4 yir z4 z4 yir, So finally following four first order differential equation cab be obtained in the form of. state variable which can be solved by first order RK method. z2 ir contact,mrotation j 1,q j Fu sin s t,2mrotation Cin K ir contact ir3 2 j. xir mrotation,8 H K Yadav et al,mrotation k,ir contact ir 3 2. q j W Fu cos s t,2mrotation Cin K ir contact ir j,yir mrotation.

Also replacing xin and yin by z1 and z3 respectively we can rewrite the equations 6 to. 10 as following 17 to 21 Here it is important to note that xin and yin are variable. because they are the generalised coordinates of inner race mass centre which is moved. with shaft While xout and yout are the constant because mass centre of outer race is not. Vibration Laboratory Mechanical and Industrial Engineering Department Indian Institute of Technology Nonlinear vibration signature analysis of a rotor supported ball bearings 3 when the balls are arranged symmetrically around the load line i e with either a ball or a ball gap directly under the load In the intermediate cage position however the balls are non symmetrically arranged

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