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Research Papers

Periodic Motions With Impacting Chatter and Stick in a Gear Transmission System

[+] Author and Article Information
Albert C. J. Luo1

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026aluo@siue.edufff

Dennis O’Connor

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL 62026

1

Corresponding author.

J. Vib. Acoust 131(4), 041013 (Jul 14, 2009) (11 pages) doi:10.1115/1.3142868 History: Received March 28, 2007; Revised February 15, 2009; Published July 14, 2009

In this paper, an impact model with possible stick between the two gears is proposed for gear transmission systems, which includes the piecewise backlash model and the traditional impact model for the first time. The new model presented in this paper possesses a time-varying boundary for two dynamical systems either to switch or to impact. Such a model can catch impacting chatter and stick phenomena in gear transmission systems. Based on the new model, periodic impacting chatter and stick in a gear transmission system can be investigated. For doing so, switching sets on the time-varying boundaries are introduced to define basic mappings. Mapping structures based on basic mappings are developed for characterizing motions in gear transmission systems, and from such mapping structures, periodic motions with impacting chatter and stick in such a gear transmission system are predicted analytically. Numerical simulations are performed for illustration of periodic motions with impacting chatter and stick phenomena.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

One-stage gear chattering: (a) physical model and (b) possible sequence of impacts

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Figure 2

A mechanical model for a gear transmission

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Figure 3

Three regions for two independent systems in a gear transmission system

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Figure 4

Basic mappings: (a) impacting chatter only and (b) with stick switching. The straight lines with arrow represent impacts on the boundaries.

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Figure 5

Mappings between switching planes for stick and impacting

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Figure 6

Mapping structures for (a) impacting chatter and (b) impacting chatter with stick motion of two gear systems

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Figure 7

The analytical prediction of a periodic motion of impacting chatter P652322: (a) switching displacement and (b) switching velocity of the first gear, and (c) switching velocity of the second gear, and (d) switching phase (m1=2, m2=1, r1=r2=0.6, k1=30, k2=20, Q0=35, d=1.0, and e=0.7)

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Figure 8

The analytical prediction of a periodic motion of impacting chatter with stick P645m312m(m=24,25,…,28): (a) switching displacement and (b) switching velocity of the first gear, and (c) switching velocity of the second gear and (d) switching phase (m1=2, m2=1, r1=r2=0.6, k1=30, k2=20, Q0=35, d=1.0, and e=0.7).

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Figure 9

The periodic motion of impacting chatter P652322: (a) displacement time history, (b) velocity time history, (c) trajectories of the first gear, and (d) trajectories of the second gear in phase plane (m1=2, m2=1, r1=r2=0.6, k1=30, k2=20, Q0=35, d=1.0, e=0.7, and Ω=2.72). The initial conditions are t0≈2.0559169, x0(1)≈1.0505790, x0(0)≈0.5505790, y0(1)≈1.68706528, and y(2)≈4.5219068.

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Figure 10

The periodic motion of impacting chatter with stick P5243122464: (a) displacement time history, (b) velocity time history, (c) trajectories of the first gear, and (d) trajectories of the second gear in phase plane. (m1=2, m2=1, r1=r2=0.6, k1=30, k2=20, Q0=35, d=1.0, e=0.7, Ω=0.5). The initial condition is t0≈4.8888960, x0(1)≈−0.7387947, x0(2)≈−0.2387947, y(1)=y0(2)≈−0.2754161.

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