A ball drops to the floor, squashes for an instant, and flies back up. How much energy it returns depends on the material: rubber bounces back to nearly the same height, while a lump of clay just splats and stays put.
Engineers used to calculate such bounces accurately only for two idealized cases: when the ball behaves like a spring (compression force grows proportionally to squish) or like a Hertzian steel ball (force grows faster). For the full variety of real materials (foam, plastic, living tissue) they had to rely on approximations.
The new work provides a general recipe. The authors noticed that if you replot the graph of ball velocity versus compression in special coordinates, any material turns into an ordinary spring with friction. You just need to choose the right friction coefficient in this imaginary system. Then the bounce is computed with one short formula that involves only two quantities: material stiffness and its "stickiness" (energy lost during compression).
The practical benefit is direct: computer games and engineering simulations can compute collisions faster and more accurately. Measure the two material constants once, and the machine will honestly predict the bounce at any speed.
as written by the authors arXiv:2603.27764
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.
If you compress a rubber ball to half its diameter, the compression force for many materials grows not linearly but as the square of the compression. That's why the general formula uses an unusual change of variable, as if we're looking at the world through a magnifying glass that straightens curved graphs.