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

Abstract

We present an exact phase-space analytical treatment of the power-law damped contact oscillator governed by $m\ddot{\hat{I}}´+\hat{I}\pm \sqrt{m{k}_H}\hat{I}{´}^{(p-1)/2}\dot{\hat{I}}´+k_H\hat{I}{´}^p=0$, valid for all force-law exponents $p\ge 1$ and all initial impact velocities $v_0$. The central result is the transformation $\hat{I}´=A{x}^{2/(p+1)}$, where $A=[(p+1)/2{]}^{1/(p+1)}$, which maps the nonlinear phase-space equation $vdv/d\hat{I}´+\cdots =0$ exactly onto a linear spring-dashpot (LSD) system with effective damping ratio $\hat{I}{\pm }_{\text{eff}}=\frac{\hat{I}}{\pm }\sqrt{2(p+1)}$. The phase portrait $v(\hat{I}´)$, coefficient of restitution $e$, and maximum penetration $\hat{I}{´}_{\text{max}}$ follow in closed form. The physical time-domain solution $(\hat{I}´(t),v(t))$ 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 $p\ge 1$ and derive the universal calibration formula: $\hat{I}\pm =\sqrt{2(p+1)}\cdot \frac{-\ln e}{\sqrt{\ddot{I}{€}^2+{\ln }^{2}e}}$. 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.