Stochastic Evaluation of the Run-Time of a Stick-Slip Oscillator Problem

Abstract

In this article, the run-times associated to a Monte Carlo simulation are studied from a stochastic perspective for a problem consisting of an oscillator with uncertain dry friction that can exhibit stick-slip vibrations. Some of the novelties of the paper are that the run-times are modeled as random variables, which is not a common approach; and that a comparison of the run-times obtained with two different simulation strategies is included to assess their performance. In one of the simulations, the Monte Carlo (MC) is used in combination with a Runge-Kutta numerical integration scheme. In another, the MC method combined with an analytical approximation obtained with the Multiple Scales method is used. The hypotheses that some characteristics of the friction force and the stick-phase duration may have a direct influence on the run-time are investigated. The dependency of the run-times on the magnitude of the dynamic friction and the first stick-duration is explored, and the histograms for the run-times are constructed. The results, for 5 e5 realizations, show that the run-times where an analytical approximation was used are, in mean, 3.8 times faster than those of the numerical integration, and they are also less sensitive to variations considering that the standard deviation is 2.7 times smaller. The improvement observed in the run-time obtained with the analytical approximation can be a determining factor for the feasibility of a stochastic study, given that the Monte Carlo approach requires a large number of realizations to calculate a statistical model with accuracy.