Technical Briefs

Nonsmooth Dynamics by Path Integration: An Example of Stochastic and Chaotic Response of a Meshing Gear Pair

[+] Author and Article Information
E. Mo

Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim NO-7491, Norway

A. Naess

Department of Mathematical Sciences, Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, Trondheim NO-7491, Norway

J. Comput. Nonlinear Dynam 4(3), 034501 (May 20, 2009) (4 pages) doi:10.1115/1.3124780 History: Received August 23, 2007; Revised August 28, 2008; Published May 20, 2009

The probability density function (PDF) of the solution process of a nonlinear stochastic differential equation (SDE) is found in this paper using the path integration technique. The SDE is a piecewise linear system representing a model of an imperfectly mounted spur gear pair with a small stochastic noise added to the driving force. It is known that the system model for a particular choice of parameters shows chaotic behavior (Kahraman and Singh, 1990, “Non-Linear Dynamics of a Spur Gear Pair,” J. Sound Vibrat., 142(1), pp. 49–75). The PDF is compared with the Poincaré map of the deterministic system and it is shown that the stochastic and deterministic attractors are very similar. Then it is shown that although the stochastic attractor appears clearly after just a few iterations, the probability density over the attractor depends on the initial condition. The system does converge to one unique periodic PDF eventually but the convergence is fairly slow. However, the transient is almost periodic with a period that is twice that of the forcing, which can be utilized to obtain a much higher convergence rate. The advantage of using a SDE to study this rattling problem is that it can provide a very detailed picture of the dynamics and the most likely states of the system can immediately be identified.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

Schematic illustration of the meshing gears

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Figure 2

The backlash function B for β=6×10−4

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Figure 3

Snapshot of the calculated PDF after ten whole periods from a small initial distribution

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Figure 4

Snapshot of the calculated PDF after 11 periods

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Figure 5

A Poincaré image of the deterministic system sampled at every full period

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Figure 6

The constructed “final” PDF; a snapshot of the system that just repeats itself after each full period

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Figure 7

The same PDF as in Fig. 6, but scaled and plotted in 3D to give a visual impression of the full structure

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Figure 8

Four different measures of convergence for the PDF. The successive L2 difference is measured by sampling at each whole period (dT=1) and every second period (dT=2). The same measure is found from the time averages over the periods and the distance to an estimated “final” PDF.




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