Exploring oscillations with a nonlinear restoring force

Abstract

Abstract When exploring the oscillation of an object for small displacements from the equilibrium position, the magnitude of the applied force is approximately considered to be proportional to the object’s displacement. However, for bigger displacements, studied in this paper, the magnitude of the applied force is best approximated to an equation of the form F = S|x| n , n > 0. In this paper accurate and approximate equations regarding the period of the oscillation and also approximate functions regarding the object’s position with respect to time are being derived. The interesting result is the simplicity of the equation that relates the period to the exponent n and the oscillation’s amplitude at the domain 0 < n ⩽ 2. It is also shown that the force data in many well-known cases can be fitted to the aforementioned equation; typical examples are the polynomial force law, the impact of a sphere on a surface, the free oscillations of an atomic force microscopy tip at the end of a cantilever, the simple pendulum’s free oscillation, etc. Thus, the methodology and the findings presented in this paper can be applied to a variety of different situations under the restriction that the force data can be approximately described by an equation of the form F = S|x| n , n > 0.