Dynamic Analysis of Planetary Gears With Bearing Clearance

Yi Guo1, Robert G. Parker2
1Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210
2Ohio State University, Distinguished Professor Chair and Executive Dean, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, China

Tóm tắt

This study investigates the dynamics of planetary gears where nonlinearity is induced by bearing clearance. Lumped-parameter and finite element models with bearing clearance, tooth separation, and gear mesh stiffness variation are developed. The harmonic balance method with arc length continuation is applied to the lumped-parameter model to obtain the dynamic response. Solution stability is analyzed using Floquet theory. Rich nonlinear behavior is exhibited, consisting of nonlinear jumps, a hardening effect induced by the transition from no bearing contact to contact, and softening induced by tooth separation. Bearings of the central members (sun, carrier, and ring) impact against the bearing races near resonances, which leads to coexisting solutions in wide speed ranges, grazing bifurcation, and chaos. Secondary Hopf and period-doubling bifurcations are the routes to chaos. Input torque can suppress some of the nonlinear effects caused by bearing clearance.

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Tài liệu tham khảo

Oswald, F., Zaretsky, E., and Poplawski, J., 2012, “Effects of Internal Clearance on Load Distribution and Life of Radially-Loaded Ball and Roller Bearings,” STLE Tribol. Trans., Preprint.

Kahraman, Nonlinear Dynamics of a Geared Rotor-Bearing System With Multiple Clearances, J. Sound Vib., 144, 469, 10.1016/0022-460X(91)90564-Z

Gurkan, Interactions Between Backlash and Bearing Clearance Nonlinearity in Geared Flexible Rotors, ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference

Tiwari, Effect of Radial Internal Clearance of a Ball Bearing on the Dynamics of a Balanced Horizontal Rotro, J. Sound Vib., 238, 723, 10.1006/jsvi.1999.3109

Bai, Nonlinear Stability of Balanced Rotor Due to Effect of Ball Bearing Internal Clearance, Appl. Math. Mech., 27, 175, 10.1007/s10483-006-0205-1

Kim, Bifurcation Analysis for a Modified Jeffcott Rotor With Bearing Clearances, Nonlinear Dyn., 1, 221, 10.1007/BF01858295

Foale, Bifurcations in Impact Oscillators, Nonlinear Dyn., 6, 285, 10.1007/BF00053387

Hinrichs, Dynamics of Oscillators With Impact and Friction, Chaos, Solitons Fractals, 8, 535, 10.1016/S0960-0779(96)00121-X

Nordmark, Non-Periodic Motion Caused by Grazing Incidence in an Impact Oscillator, J. Sound Vib., 145, 279, 10.1016/0022-460X(91)90592-8

Peterka, Bifurcations and Transition Phenomena in an Impact Oscillator, Chaos, Solitons Fractals, 7, 1635, 10.1016/S0960-0779(96)00028-8

Nordmark, Existence of Periodic Orbits in Grazing Bifurcations of Impacting Mechanical Oscillators, Nonlinearity, 14, 15171542, 10.1088/0951-7715/14/6/306

Luo, Grazing and Chaos in a Periodically Forced, Piecewise Linear System, ASME J. Vibr. Acoust., 128, 28, 10.1115/1.2149390

Chin, Grazing Bifurcation in Impact Oscillators, Phys. Rev. E, 50, 4427, 10.1103/PhysRevE.50.4427

Halse, Coexisting Solutions and Bifurcations in Mechanical Oscillators With Backlash, J. Sound Vib., 305, 854, 10.1016/j.jsv.2007.05.010

Ellermann, The Motion of Floating Systems: Nonlinear Dynamic in Periodic and Random Waves, ASME J. Offshore Mech. Arct. Eng., 131, 041104, 10.1115/1.3160649

Blazejczyk-Okolewska, Co-Existing Attractors of Impact Oscillator, Chaos, Solitons Fractals, 9, 328, 10.1016/S0960-0779(98)00164-7

Kahraman, Experiments on Nonlinear Dynamic Behavior of an Oscillator With Clearance and Periodically Time-Varying Parameters, ASME J. Appl. Mech., 64, 217, 10.1115/1.2787276

Parker, Non-Linear Dynamic Response of a Spur Gear Pair: Modelling and Experimental Comparisons, J. Sound Vib., 237, 435, 10.1006/jsvi.2000.3067

Seaman, Component Inertial Effects on Transmission Design, SAE Tech. Pap. Ser., 841686, 6.990, 10.4271/841686

Botman, Epicyclic Gear Vibrations, ASME J. Eng. Ind., 97, 811, 10.1115/1.3439034

Lin, Planetary Gear Parametric Instability Caused by Mesh Stiffness Variation, J. Sound Vib., 249, 129, 10.1006/jsvi.2001.3848

Parker, Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model, ASME J. Mech. Des., 122, 304, 10.1115/1.1286189

Sun, Nonlinear Dynamics of a Planetary Gear System With Multiple Clearances, Mech. Mach. Theory, 38, 1371, 10.1016/S0094-114X(03)00093-4

Al-shyyab, A Non-Linear Dynamic Model for Planetary Gear Sets, Proc. Inst. Mech. Eng. Part K: J. Multi-Body Dyn., 221, 567, 10.1243/14644193JMBD92

Ambarisha, Nonlinear Dynamics of Planetary Gears Using Analytical and Finite Element Models, J. Sound Vib., 302, 577, 10.1016/j.jsv.2006.11.028

Bahk, Analytical Solution for the Nonlinear Dynamics of Planetary Gears, ASME J. Comput. Nonlinear Dyn., 6, 021007, 10.1115/1.4002392

Zhu, Non-Linear Dynamics of a One-Way Clutch in Belt-Pulley Systems, J. Sound Vib., 279, 285, 10.1016/j.jsv.2003.11.031

Blankenship, Steady State Forced Response of a Mechanical Oscillator With Combined Parametric Excitation and Clearance Type Non-Linearity, J. Sound Vib., 185, 743, 10.1006/jsvi.1995.0416

Al-shyyab, Non-Linear Dynamic Analysis of a Multi-Mesh Gear Train Using Multi-Term Harmonic Balance Method: Subharmonic Motions, J. Sound Vib., 279, 417, 10.1016/j.jsv.2003.11.029

Nayfeh, Applied Nonlinear Dynamics, 10.1002/9783527617548

Raghothama, Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method, J. Sound Vib., 226, 469, 10.1006/jsvi.1999.2264

Blair, Harmonic Balance and Continuation Techniques in the Dynamic Analysis of Duffing’s Equation, J. Sound Vib., 202, 717731, 10.1006/jsvi.1996.0863

Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, 10.1007/978-1-4419-1740-9

Lin, Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration, ASME J. Vibr. Acoust., 121, 319, 10.1115/1.2893982

Wu, X., and Parker, R. G., 2012, “Parametric Instability of Planetary Gears with Elastic Continuum Ring Gears,” ASME J. Vibr. Acoust.10.1115/1.4005836

Guo, Dynamic Modeling and Analysis of a Spur Planetary Gear Involving Tooth Wedging and Bearing Clearance Nonlinearity, Eur. J. Mech. A/Solids, 29, 1022, 10.1016/j.euromechsol.2010.05.001

Vijayakar, S. M. , 2005, Calyx User’s Manual, http://ansol.com.

Vijayakar, A Combined Surface Integral and Finite-Element Solution for a 3-Dimensional Contact Problem, Int. J. Numer. Methods Eng., 31, 525, 10.1002/(ISSN)1097-0207

Vijayakar, A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem, Int. J. Numer. Methods Eng., 31, 524, 10.1002/(ISSN)1097-0207

Kahraman, Effect of Internal Gear Flexibility on the Quasi-Static Behavior of a Planetary Gear Set, J. Mech. Des., 123, 408, 10.1115/1.1371477

Thomsen, Vibration and Stability: Advanced Theory, Analysis, and Tools, 2nd ed.

Grippo, A Nonmonotone Line Search Technique for Newton’s Method, SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 23, 707, 10.1137/0723046

Seydel, Practical Bifurcation and Stability Analysis, 10.1007/978-1-4419-1740-9

Friedmann, Numerical Methods for the Treatment of Periodic Systems With Applications to Structural Dynamics and Helicopter Rotor Dynamics, Comput. Struct., 35, 329, 10.1016/0045-7949(90)90059-B

Lin, Structured Vibration Characteristics of Planetary Gears With Unequally Spaced Planets, J. Sound Vib., 233, 921, 10.1006/jsvi.1999.2581

Meirovitch, Principles and Techniques of Vibrations

Blankenship, Gear Dynamics Experiments, Part I: Characterization of Forced Response, ASME Power Transmission and Gearing Conference

Blankenship, Steady State Force Response of a Mechanical Oscillator With Combined Parametric Excitation and Clearance Type Non-Linearity, J. Sound Vib., 185, 743, 10.1006/jsvi.1995.0416

Swift, Suppress of Period Doubling in Symmetric Systems, Phys. Rev. Lett., 52, 705, 10.1103/PhysRevLett.52.705