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TECHNICAL PAPERS

Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems

[+] Author and Article Information
Jian Lin

John Deere Product Engineering Center, Waterloo, IA 50704-8000

Robert G. Parker

Department of Mechanical Engineering, The Ohio State University, 206 W. 18th Ave, Columbus, OH 43210e-mail: parker.242@osu.edu

J. Vib. Acoust 124(1), 68-76 (Sep 01, 2001) (9 pages) doi:10.1115/1.1424889 History: Received August 01, 2000; Revised September 01, 2001
Copyright © 2002 by ASME
Topics: Gears , Stiffness
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References

Benton,  M., and Seireg,  A., 1978, “Simulation of Resonances and Instability Conditions in Pinion-Gear Systems,” ASME J. Mech. Des., 100, pp. 26–30.
Kahraman,  A., and Blankenship,  G. W., 1997, “Experiments on Nonlinear Dynamic Behavior of an Oscillator with Clearance and Periodically Time-Varying Parameters,” ASME J. Appl. Mech., 64, pp. 217–226.
Kahraman,  A., and Singh,  R., 1991, “Interactions Between Time-varying Mesh Stiffness and Clearance Non-linearities in a Geared System,” J. Sound Vib., 146, pp. 135–156.
Blankenship,  G. W., and Kahraman,  A., 1995, “Steady State Forced Response of a Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Non-linearity,” J. Sound Vib., 185, pp. 743–765.
Kahraman,  A., and Blankenship,  G. W., 1996, “Interactions Between Commensurate Parametric and Forcing Excitations in a System with Clearance,” J. Sound Vib., 194, pp. 317–336.
Parker,  R. G., Vijayakar,  S. M., and Imajo,  T., 2000, “Nonlinear Dynamic Response of a Spur Gear Pair: Modeling and Experimental Comparisons,” J. Sound Vib., 236, pp. 561–573.
Ibrahim,  R. A., and Barr,  A. D. S., 1978, “Parametric Vibration Part-I: Mechanics of Linear Problems,” Shock Vib. Dig., 10, pp. 15–29.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley, New York.
Bollinger,  J. G., and Harker,  R. J., 1967, “Instability Potential of High Speed Gearing,” J. of Industrial Mathematics, 17, pp. 39–55.
Benton,  M., and Seireg,  A., 1981, “Factors Influencing Instability and Resonances in Geared Systems,” ASME J. Mech. Des., 103, pp. 372–378.
Nataraj, C., and Whitman, A. M., 1997, “Parameter Excitation Effects in Gear Dynamics,” ASME Design Engineering Technical Conferences, Paper No. DETC97/VIB-4018, Sacramento, CA.
Nataraj, C., and Arakere, N. K., 1999, “Dynamic Response and Stability of a Spur Gear Pair,” ASME Design Engineering Technical Conferences, Paper No. DETC99/VIB-8110, Las Vegas, NV.
Amabili,  M., and Rivola,  A., 1997, “Dynamic Analysis of Spur Gear Pairs: Steady-State Response and Stability of the SDOF Model With Time-Varying Meshing Damping,” Mech. Syst. Signal Process., 11, pp. 375–390.
Tordion,  G. V., and Gauvin,  R., 1977, “Dynamic Stability of a Two-Stage Gear Train Under the Influence of Variable Meshing Stiffnesses,” ASME J. Eng. Ind., 99, pp. 785–791.
Benton,  M., and Seireg,  A., 1980, “Normal Mode Uncoupling of Systems with Time Varying Stiffness,” ASME J. Mech. Des., 102, pp. 379–383.
Kahraman,  A., and Blankenship,  G. W., 1999, “Effect of Involute Contact Ratio on Spur Gear Dynamics,” ASME J. Mech. Des., 121, pp. 112–118.
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Lin,  J., and Parker,  R. G., 2002, “Planetary Gear Parametric Instability Caused by Mesh Stiffness Variation,” J. Sound Vib., 249, pp. 129–145.

Figures

Grahic Jump Location
Two-stage gear system with (a) four gears and (b) three gears. kL1,kL2 denote mesh stiffnesses and Z2,Z4 denote number of gear teeth. kL0 is the torsional stiffness of the anchored shaft.
Grahic Jump Location
Modeling of mesh stiffnesses ki(t)=kgi+kvi(t).ci are contact ratios, kgi are average mesh stiffnesses, and piTi are phasing angles.
Grahic Jump Location
Instabilities regions when Ω12=Ω,ε12=ε; — analytical solution;* * * numerical solution. The parameters are from Table 1 and c1=c2=1.5,h=0.
Grahic Jump Location
Vibration modes for the time-invariant system with parameters in Table 1
Grahic Jump Location
Comparison of instability regions for various contact ratios and mesh phasing. The parameters are in Table 1. ---c1=c2=1.5,h=0.5; —c1=1.1,c2=1.9,h=0.9; -⋅-⋅-⋅- c1=c2=1.5,h=0.
Grahic Jump Location
Instabilities regions when Ω1=RΩ212=ε. (a) R=3/5, (b) R=1/2. The parameters are in Table 1 and c1=c2=1.5,h=0.* * * denotes numerical solutions.
Grahic Jump Location
Instabilities regions when Ω12. (a) Ω1 versus ε1 and ε2=C=0.3. (b) Ω1 versus ε12 and the solid line indicates vanishing of the combination instability. The parameters are in Table 1 and c1=c2=1.5,h=0.
Grahic Jump Location
Comparison of instability regions from different analyses. The parameters are from Table 1, c1=1.47,c2=1.57, and phasing (a) h=0, (b) h=0.4. —Perturbation method;* * * Numerical method; ---Tordion and Gauvin (1977); ⋅⋅⋅ Benton and Seireg (1980).
Grahic Jump Location
Free responses for Ω=4.2,ka=ε=0.3 (point A in Fig. 8) and the parameters of (a) Fig. 8(a) and (b) Fig. 8(b). The initial conditions are x1=x2=x3=0.1,ẋ1=ẋ2=ẋ3=0.

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