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
Your Session has timed out. Please sign back in to continue.



Grahic Jump Location
Figure 1

Schematic illustration of the meshing gears

Grahic Jump Location
Figure 2

The backlash function B for β=6×10−4

Grahic Jump Location
Figure 3

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

Grahic Jump Location
Figure 4

Snapshot of the calculated PDF after 11 periods

Grahic Jump Location
Figure 5

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

Grahic Jump Location
Figure 6

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

Grahic Jump Location
Figure 7

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

Grahic Jump Location
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.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In