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

• Published in: American journal of physics Vol. 87; no. 2; pp. 116 - 124
• Main Authors: Xu, Qian, Fan, Wenkai, Luo, Yao, Wang, Sihui, Jiang, Hongjian
• Format: Journal Article
• Language: English
• Published: Woodbury American Institute of Physics 01.02.2019
• Subjects: Differential equations; Diodes; Friction; Linear equations; Mechanical properties; Nonlinear systems; Oscillators; Simulation; Theoretical analysis
• ISSN: 0002-9505; 1943-2909
• DOI: 10.1119/1.5082534

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 that of the mechanical system 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. The experimental results are consistent with the series solutions. We also present the phenomenon of hysteresis. A theoretical analysis along with numerical simulations is 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.