Nonlinear effect of forced harmonic oscillator subject to sliding friction and simulation by a simple nonlinear circuit

TL;DR

This paper investigates the nonlinear dynamics of a mass-spring system with dry friction using a simplified electrical circuit model, deriving analytical solutions and demonstrating hysteresis and stick-slip phenomena.

Contribution

It introduces a simple nonlinear circuit model that accurately simulates the nonlinear behaviors of a dry friction damped oscillator, including analytical solutions and experimental validation.

Findings

Series solutions describe resonant and superharmonic responses.

Experimental results match theoretical predictions.

Hysteresis and stick-slip phenomena are observed and analyzed.

Abstract

We study the nonlinear behaviors of mass-spring systems damped by dry friction using simulation by a nonlinear LC circuit damped by anti-parallel diodes. We show that the differential equation for the electric oscillator is equivalent to the mechanical one's when a piecewise linear model is used to simplify the diodes I-V curve. We derive series solutions to the differential equation under weak nonlinear approximation which can describe the resonant response as well as amplitudes of superharmonic components. Experimental results are consistent with series solutions. We also present the phenomenon of hysteresis. Theoretical analysis along with numerical simulations are conducted to explore the stick-slip boundary. The correspondence between the mechanical and electric oscillators makes it easy to demonstrate the behaviors of this nonlinear oscillator on a digital oscilloscope. It can be used to extend the linear RLC experiment at the undergraduate level.