Asymptotic trends in time-varying oscillatory period for a dual-staged torsional system

Abstract

The primary goal of this article is to propose a new analysis tool that estimates the asymptotic trends in the time-varying oscillatory period of a non-linear mechanical system. The scope is limited to the step-response of a torsional oscillator containing a dry friction element and dual-staged spring. Prior work on the stochastic linearization techniques is extended and modified for application in time domain. Subsequently, an instantaneous expected value operator and the concept of instantaneous effective stiffness are proposed. The non-linear system is approximated at some instant during the step-response by a linear time-invariant mechanical system that utilizes the instantaneous effective stiffness concept. The oscillatory period of the non-linear step-response at that instant is then approximated by the natural period of the corresponding linear system. The proposed method is rigorously illustrated via two computational example cases (a near backlash and near pre-load non-linearities), and the necessary digital signal processing parameters for time domain analysis are investigated. Finally, the feasibility and applicability of the proposed method is demonstrated by estimating the softening and hardening trends in the time-varying oscillatory period of the measured response for two laboratory experiments that contain clearance elements and multi-staged torsional springs.