Hidden Harmonic Structure, Universal Damping, and Stability Bounds in Nonlinear Contact Dynamics

Abstract

Nonlinear contact dynamics are widely regarded as intrinsically nonlinear systems whose behaviour depends strongly on geometry and impact conditions. Here we show that any one-dimensional conservative contact system satisfying monotone energy-consistent conditions admits two complementary structures: (i) a canonical action-angle representation in physical time, and (ii) an exact harmonic oscillator representation under an energy-based coordinate transformation combined with time reparametrisation. This reveals a hidden linear structure underlying nonlinear contact interactions. Building on this result, we derive a unique universal damping law that preserves linear dissipative dynamics in the transformed harmonic space, and establish a rigorous, closed-form lower bound for the critical timestep in numerical simulations. The framework generalises classical power-law contact models and provides a unified basis for restitution control across arbitrary geometries, recovering known exact solutions as explicit monomial special cases.

Figures (1)

• Figure 1: (Color online) Exact harmonic regularisation of a highly nonlinear contact oscillator (an ellipsoid impacting a flat wall with $\alpha = 0.5$ volumetric regularisation and stiffness $K_n = 10^8$ Pa). Dashed lines denote the strictly conservative interaction ($C_0 = 0$), while solid lines denote the dissipative extension governed by the universal damping law ($C_0 = 0.5$). The colours correspond to the initial impact velocities: $v_0 = 0.50$ m/s (blue), $0.99$ m/s (red), and $1.50$ m/s (yellow). (a) In the physical phase space, the conservative trajectories are symmetric but non-elliptical, whereas the dissipative trajectories are heavily skewed and asymmetric. (b) Under the exact energy-based coordinate transformation $x(\delta)$ and time reparametrisation $\tau$ (using virtual scaling constants $K=1$ and $M=0.75$), the identical physical trajectories map perfectly onto the symmetric semi-ellipses (conservative) and exact logarithmic spirals (dissipative) of a linear harmonic oscillator. (c) The physical time-domain waveforms illustrate the corresponding nonlinear penetration pulses and energy loss.