Modeling, Modal Properties, and Mesh Stiffness Variation Instabilities of Planetary Gears

Planetary gear noise and vibration are primary concerns in their applications in helicopters, automobiles, aircraft engines, heavy machinery, and marine vehicles. Dynamic analysis is essential to the noise and vibration reduction. This work analytically investigates some critical issues and advances the understanding of planetary gear dynamics. A lumped-parameter model is built for the dynamic analysis of general planetary gears. The unique properties of the natural frequency spectra and vibration modes are rigorously characterized. These special structures apply for general planetary gears with cyclic symmetry and, in practically important case, systems with diametrically opposed planets. The special vibration properties are useful for subsequent research. Taking advantage of the derived modal properties, the natural frequency and vibration mode sensitivities to design parameters are investigated. The key parameters include mesh stiffnesses, support/bearing stiffnesses, component masses, moments of inertia, and operating speed. The eigensensitivities are expressed in simple, closed-form formulae associated with modal strain and kinetic energies. As disorders (e.g., mesh stiffness variation, manufacturing and assembling errors) disturb the cyclic symmetry of planetary gears. their effects on the free vibration properties are quantitatively examined. Well-defined veering rules are derived to identify dramatic changes of natural frequencies and vibration modes under parameter variations. The knowledge of free vibration properties, eigensensitivities and veering rules provide important information to effectively tune the natural frequencies and optimize structural design to minimize noise and vibration. Parametric instabilities excited by mesh stiffness variations are analytically studied for multi-mesh gear systems. The discrepancies of previous studies on parametric instability of two-stage gear chains are clarified using perturbation and numerical methods.

[1]  Teruaki Hidaka,et al.  Analysis of Dynamic Tooth Load on Planetary Gear , 1979 .

[2]  Subhash C. Sinha,et al.  A HYBRID FORMULATION FOR THE ANALYSIS OF TIME-PERIODIC LINEAR SYSTEMS VIA CHEBYSHEV POLYNOMIALS , 1996 .

[3]  Takeshi Ishida,et al.  Effects of errors of elements on load distribution in planetary gears with various load equalizing mechanisms. , 1986 .

[4]  Peter L. Balise,et al.  Nonlinear Differential Equations , 1962 .

[5]  Ahmet Kahraman,et al.  Experiments on Nonlinear Dynamic Behavior of an Oscillator With Clearance and Periodically Time-Varying Parameters , 1997 .

[6]  Teruaki Hidaka,et al.  Dynamic Behavior of Planetary Gear : 2nd Report, Displacement of Sun Gear and Ring Gear , 1976 .

[7]  Rajendra Singh,et al.  Analytical solution for modulation sidebands associated with a class of mechanical oscillators , 1995 .

[8]  R T.E.,et al.  DYNAMIC ANALYSIS OF A REVERSE-IDLER GEAR PAIR WITH CONCURRENT CLEARANCES , 2022 .

[9]  Teruaki Hidaka,et al.  Dynamic Behavior of Planetary Gear : 1st Report Load Distribution in Planetary Gear , 1975 .

[10]  M. Botman.  Vibration Measurements on Planetary Gears of Aircraft Turbine Engines , 1980 .

[11]  Robert G. Parker,et al.  NATURAL FREQUENCY VEERING IN PLANETARY GEARS* , 2001 .

[12]  A. Seireg,et al.  Factors Influencing Instability and Resonances in Geared Systems , 1981 .

[13]  J. D. Smith,et al.  Dynamic Tooth Loads in Epicyclic Gears , 1974 .

[14]  J. Ginsberg,et al.  On the Relationship Between Veering of Eigenvalue Loci and Parameter Sensitivity of Eigenfunctions , 1992 .

[15]  Kazuteru Nagamura,et al.  Dynamic Behavior of Planetary Gear : 6th Report, Influence of Meshing-Phase , 1978 .

[16]  S. Sinha,et al.  Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems , 1996 .

[17]  Raouf A. Ibrahim,et al.  Parametric Vibration: Part I: Mechanics of Linear Problems , 1978 .

[18]  V. V. Bolotin,et al.  The Dynamic Stability of Elastic Systems , 1966 .

[19]  R. Fox,et al.  Rates of change of eigenvalues and eigenvectors. , 1968 .

[20]  Teruaki Hidaka,et al.  On the Relation between the Run-Out Errors and the Motion of the Center of Sun Gear In a Stoeckicht Planetary Gear , 1978 .

[21]  P. Seshu,et al.  On the principle of cyclic symmetry in machine dynamics , 1990 .

[22]  S. Theodossiades,et al.  NON-LINEAR DYNAMICS OF GEAR-PAIR SYSTEMS WITH PERIODIC STIFFNESS AND BACKLASH , 2000 .

[23]  G. V. Tordion,et al.  Dynamic Stability of a Two-Stage Gear Train Under the Influence of Variable Meshing Stiffnesses , 1977 .

[24]  D. L. Seager.  Conditions for the Neutralization of Excitation by the Teeth in Epicyclic Gearing , 1975 .

[25]  Robert G. Parker,et al.  Structured vibration characteristics of planetary gears with unequally spaced planets , 2000 .

[26]  Ahmet Kahraman,et al.  Effect of Involute Contact Ratio on Spur Gear Dynamics , 1999 .

[27]  Robert G. Parker,et al.  Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems , 2002 .

[28]  A. Bajaj,et al.  FREE AND FORCED RESPONSE OF MISTUNED LINEAR CYCLIC SYSTEMS: A SINGULAR PERTURBATION APPROACH , 1998 .

[29]  Timothy L. Krantz.  Mechanical Systems Technology Branch research summary, 1985 - 1992 , 1993 .

[30]  A. Kahraman,et al.  Steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non-linearity , 1995 .

[31]  N. N. Bogoli︠u︡bov,et al.  Introduction to non-linear mechanics , 1943 .

[32]  Donald R. Houser,et al.  Mathematical models used in gear dynamics—A review , 1988 .

[33]  R. H. Badgley,et al.  Reduction of Vibration and Noise Generated by Planetary Ring Gears in Helicopter Aircraft Transmissions , 1973 .

[34]  P. Velex,et al.  Dynamic Response of Planetary Trains to Mesh Parametric Excitations , 1996 .

[35]  Herbert W. Muller,et al.  Epicyclic Gear Trains: Analysis, Synthesis and Applications , 1983 .

[36]  S. Vijayakar.  A combined surface integral and finite element solution for a three‐dimensional contact problem , 1991 .

[37]  A. Pintz,et al.  Dynamic Load Factors in Internal Spur Gear Drives , 1985 .

[38]  Teruaki Hidaka,et al.  Dynamic Behavior of Planetary Gear : 4th Report, Influence of the Transmitted Tooth Load on the Dynamic Increment Load , 1978 .

[39]  Ahmet Kahraman,et al.  Effect of Involute Tip Relief on Dynamic Response of Spur Gear Pairs , 1999 .

[40]  Vimal Singh,et al.  Perturbation methods , 1991 .

[41]  Rajendra Singh,et al.  Non-linear dynamics of a geared rotor-bearing system with multiple clearances , 1991 .

[42]  K. G. Lindh,et al.  Infinite determinant methods for stability analysis of periodic-coefficient differential equations , 1970 .

[43]  Durbha V. Murthy,et al.  Aeroelastic modal characteristics of mistuned blade assemblies - Mode localization and loss of eigenstructure , 1991 .

[44]  Robert G. Parker,et al.  SENSITIVITY OF PLANETARY GEAR NATURAL FREQUENCIES AND VIBRATION MODES TO MODEL PARAMETERS , 1999 .

[45]  Ahmet Kahraman,et al.  Natural Modes of Planetary Gear Trains , 1994 .

[46]  W. C. Mills-Curran,et al.  CALCULATION OF EIGENVECTOR DERIVATIVES FOR STRUCTURES WITH REPEATED EIGENVALUES , 1988 .

[47]  Christophe Pierre,et al.  Localization Phenomena in Mistuned Assemblies with Cyclic Symmetry Part I: Free Vibrations , 1988 .

[48]  Ling-Cheng Zhao,et al.  MODE LOCALIZATION DUE TO ASYMMETRIC STRUCTURAL CHANGES , 1997 .

[49]  R. Parker,et al.  Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model , 2000 .

[50]  Ahmet Kahraman,et al.  Load sharing characteristics of planetary transmissions , 1994 .

[51]  C. D. Mote,et al.  Comments on curve veering in eigenvalue problems , 1986 .

[52]  C. Q. Liu,et al.  Sensitivity analysis of discrete structural systems , 1995 .

[53]  Oddvar O. Bendiksen,et al.  Numerical study of vibration localization in disordered cyclic structures , 1992 .

[54]  Peter Lynwander,et al.  Gear Drive Systems , 1983 .

[55]  C. Hsu,et al.  Further Results on Parametric Excitation of a Dynamic System , 1965 .

[56]  F. B. Oswald,et al.  Vibration in Planetary Gear Systems with Unequal Planet Stiffnesses , 1982 .

[57]  Robert G. Parker,et al.  A PHYSICAL EXPLANATION FOR THE EFFECTIVENESS OF PLANET PHASING TO SUPPRESS PLANETARY GEAR VIBRATION , 2000 .

[58]  Takeshi Ishida,et al.  Vibration of Solar-Type Planetary Gears. , 1993 .

[59]  Ahmet Kahraman,et al.  Planetary gear train dynamics , 1994 .

[60]  J. W. Evans,et al.  An Extended Model for Determining Dynamic Loads in Spur Gearing , 1981 .

[61]  C. Hsu.  On the Parametric Excitation of a Dynamic System Having Multiple Degrees of Freedom , 1963 .

[62]  A. Seireg,et al.  The Application of the Ritz Averaging Method to Determining the Response of Systems with Time Varying Stiffness to Harmonic Excitation , 1980 .

[63]  S. T. Noah,et al.  A Generalized Hill’s Method for the Stability Analysis of Parametrically Excited Dynamic Systems , 1982 .

[64]  Anil K. Bajaj,et al.  A Singular Perturbation Analysis Of Eigenvalue Veering And Modal Sensitivity In Perturbed Linear Periodic Systems , 1993 .

[65]  Teruaki Hidaka,et al.  Dynamic Behavior of Planetary Gear : 3rd Report, Displacement of Ring Gear in Direction of Line of Action , 1977 .

[66]  A. Seireg,et al.  Normal Mode Uncoupling of Systems with Time Varying Stiffness , 1980 .

[67]  Ahmet Kahraman,et al.  Static Load Sharing Characteristics of Transmission Planetary Gear Sets: Model and Experiment , 1999 .

[68]  A. K. Rakhit.  An Improved Epicyclic Gearbox for Reduced and Controllable Subsynchronous Vibrations in Gas Turbogenerator Applications , 1997 .

[69]  Michael I. Friswell,et al.  The Derivatives of Repeated Eigenvalues and Their Associated Eigenvectors , 1996 .

[70]  Timothy L. Krantz.  Gear Tooth Stress Measurements of Two Helicopter Planetary Stages , 1992 .

[71]  Ahmet Kahraman,et al.  Dynamic Analysis of a Multi-Mesh Helical Gear Train , 1994 .

[72]  H. N. Özgüven,et al.  Dynamic analysis of high speed gears by using loaded static transmission error , 1988 .

[73]  Kazuteru Nagamura,et al.  Dynamic Behavior of Planetary Gear : 7th Report, Influence of the Thickness of the Ring Gear , 1978 .

[74]  L. Meirovitch.  Analytical Methods in Vibrations , 1967 .

[75]  Robert G. Parker,et al.  Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration , 1999 .

[76]  Marco Amabili,et al.  DYNAMIC ANALYSIS OF SPUR GEAR PAIRS: STEADY-STATE RESPONSE AND STABILITY OF THE SDOF MODEL WITH TIME-VARYING MESHING DAMPING , 1997 .

[77]  Ahmet Kahraman,et al.  INTERACTIONS BETWEEN COMMENSURATE PARAMETRIC AND FORCING EXCITATIONS IN A SYSTEM WITH CLEARANCE , 1996 .

[78]  Teik C. Lim,et al.  Dynamic Analysis of Multi-Mesh Counter-Shaft Transmission , 1999 .

[79]  Kazuteru Nagamura,et al.  Dynamic Behavior of Planetary Gear : 5th report, Dynamic Increment of Torque , 1978 .

[80]  Anil K. Bajaj,et al.  A singular perturbation perspective on mode localization , 1991 .

[81]  R. Lane Dailey,et al.  Eigenvector derivatives with repeated eigenvalues , 1989 .

[82]  M. Botman,et al.  Epicyclic Gear Vibrations , 1975 .

[83]  A. Seireg,et al.  Simulation of Resonances and Instability Conditions in Pinion-Gear Systems , 1978 .

[84]  Hsiang Hsi Lin,et al.  Prediction of Gear Dynamics Using Fast Fourier Transform of Static Transmission Error , 1993 .

[85]  Michael L. Bernard,et al.  Modal expansion method for eigensensitivity with repeated roots , 1994 .

[86]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[87]  S. Sinha,et al.  An efficient computational scheme for the analysis of periodic systems , 1991 .

[88]  N. Mclachlan.  Theory and Application of Mathieu Functions , 1965 .

[89]  P. Velex,et al.  SIMULATION OF THE DYNAMIC BEHAVIOUR OF SINGLE AND MULTI-STAGE GEARED SYSTEMS WITH SHAPE DEVIATIONS AND MOUNTING ERRORS BY USING A SPECTRAL METHOD , 1999 .

[90]  C. Hsu,et al.  Steady-State Response of a Dynamical System Under Combined Parametric and Forcing Excitations , 1974 .

[91]  R. August,et al.  Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System , 1986 .

[92]  Robert G. Parker,et al.  NON-LINEAR DYNAMIC RESPONSE OF A SPUR GEAR PAIR: MODELLING AND EXPERIMENTAL COMPARISONS , 2000 .

[93]  Ahmet Kahraman,et al.  On the response of a preloaded mechanical oscillator with a clearance: Period-doubling and chaos , 1992 .

[94]  R. August,et al.  Gear mesh stiffness and load sharing in planetary gearing , 1984 .

[95]  James D. Smith,et al.  Gears and Their Vibration, A Basic Approach to Understanding Gear Noise , 1983 .

[96]  C. Pierre.  Mode localization and eigenvalue loci veering phenomena in disordered structures , 1988 .

[97]  A. Leissa.  On a curve veering aberration , 1974 .

[98]  M. Botman,et al.  Load Sharing in a Planetary Gear Stage in the Presence of Gear Errors and Misalignment , 1985 .

[99]  C. Padmanabhan,et al.  Linear dynamic analysis of multi-mesh transmissions containing external, rigid gears , 1995 .