Exact Phase-Space Analytical Solution for the Power-Law Damped Contact Oscillator

TL;DR

This paper derives an exact analytical solution for the phase-space behavior of power-law damped contact oscillators, generalizing previous results and providing explicit formulas for key dynamic quantities.

Contribution

It introduces a transformation that maps the nonlinear oscillator to a linear system, enabling closed-form solutions for phase portrait, restitution coefficient, and maximum penetration.

Findings

Phase portrait $v( ext{delta})$ obtained in closed form.

Coefficient of restitution $e$ is independent of initial velocity.

Universal calibration formula for damping coefficient $eta$ derived.

Abstract

We present an exact phase-space analytical treatment of the power-law damped contact oscillator governed by $m\ddot{\delta} + \alpha\sqrt{mk_H}\,\delta^{(p-1)/2}\dot{\delta} + k_H\delta^p = 0$, valid for all force-law exponents $p \geq 1$ and all initial impact velocities $v_0$. The central result is the transformation $\delta = Ax^{2/(p+1)}$, where $A = [(p+1)/2]^{1/(p+1)}$, which maps the nonlinear phase-space equation $v\,dv/d\delta + \dots = 0$ exactly onto a linear spring-dashpot (LSD) system with effective damping ratio $\alpha_\text{eff} = \frac{\alpha}{\sqrt{2(p+1)}}$. The phase portrait $v(\delta)$, coefficient of restitution $e$, and maximum penetration $\delta_\text{max}$ follow in closed form. The physical time-domain solution $(\delta(t), v(t))$ is obtained parametrically via a single quadrature, which evaluates analytically for $p=1$ and at negligible numerical cost for all other $p$. We prove that $e$ is exactly independent of $v_0$ for all $p \geq 1$ and derive the universal calibration formula: $\alpha = \sqrt{2(p+1)}\cdot\frac{-\ln e}{\sqrt{\pi^2 + \ln^2 e}}$. This generalises the known results for $p=1$ (linear spring-dashpot) and $p=3/2$ (Hertz contact, Antypov and Elliott, 2011) to the entire power-law family. A closed-form estimate for the critical timestep of explicit time integration is also derived, exhibiting universal scaling with impact velocity and force-law exponent.