 Contact-mode numerical phase error

Contact-mode numerical phase error is the difference between the oscillation phase produced by a time integrator and the phase of the modeled contact dynamics. It can grow while the numerical solution remains bounded. Contact-mode timestep stability controls a different property; stable integration alone does not ensure accurate force timing or contact-event timing. The physical motivation is the fast oscillation introduced by a stiff compliant contact model.^[ref]

For a frozen undamped contact mode $\ddot{x}+{\omega }^{2}x=0$, central differences givePut $x_n=e^{in\theta }$ and use $e^{i\theta }+e^{-i\theta }=2\cos \theta$. On the stable branch $0<h\omega <2$,The numerical frequency exceeds the modeled frequency. A Taylor series givesFor the trapezoidal rule, its amplification-factor phase instead givesThat scheme preserves modal amplitude in this undamped linear setting while lagging in phase. These formulas isolate a single mode with fixed frequency and consistent initialization; general nonlinear contacts require additional analysis.

Accumulated contact-mode phase drift after a duration $T$ is $\Delta \varphi =T(\tilde{\omega }-\omega )$ at time-grid points. For an original example, let $\omega =1000rad/s$ and $T=0.10\mathrm{s}$. With $h=1\mathrm{ms}$, the central-difference frequency is about $1047.20rad/s$, producing $4.720\mathrm{rad}$ of phase lead. The step satisfies the strict stability inequality, yet the phase lead exceeds half a cycle.

Reducing $h$ to $50\mu \mathrm{s}$ gives about $0.01042\mathrm{rad}$ of lead over the same duration. At $25\mu \mathrm{s}$ it is about $0.002604\mathrm{rad}$, approximately one quarter as large, matching the leading second-order error. For a small desired phase budget ${\varphi }_{\max }$, the central-difference expansion suggestsWith ${\varphi }_{\max }=0.01\mathrm{rad}$, this estimate is about $49.0\mu \mathrm{s}$, much smaller than the $2\mathrm{ms}$ stability boundary. Check the exact frequency formula when the small-step expansion is not sufficiently accurate.

In a contact experiment or simulation, a shifted oscillation phase changes when force maxima and gap crossings are predicted. Damping, changing tangent stiffness and a contact opening can invalidate the fixed-frequency model, so the phase estimate is not by itself an event-time error certificate. A proposed convergence check compares force histories, dominant frequency and event times at successively halved steps, while keeping contact geometry and constitutive parameters fixed. Agreement in maximum penetration alone can miss a phase error that accumulates over many cycles.