= Normal contact penalty time scale
The <normal contact penalty time scale> describes the fast dynamics introduced by a stiff normal-contact approximation. For a compressed linear penalty spring with <stiffness> $k$ acting on a point <mass> $m$, small oscillations about equilibrium satisfy $m\ddot x+kx=0$. Their <angular frequency>, reference time and period are
$$
\omega_c=\sqrt{k/m},\qquad t_c=\sqrt{m/k},\qquad T_c=2\pi t_c.
$$
The reference time $t_c$ is not the full oscillation period. Rescaling time by $s=t/t_c$ gives $d^2x/ds^2+x=0$, so a numerical step is controlled by its dimensionless size $h/t_c$, not just its duration in seconds. The primary compliant-contact treatment highlights the numerical <stiffness> caused by such springs.<Underactuated Robotics by Russ Tedrake>[Multi-Body Dynamics, Compliant Contact Models]{ref}
A modeling and numerical budget interact. Limiting equilibrium overlap under a specified maximum supported load $F_{\max}$ to $d_{\max}$ requires $k\ge F_{\max}/d_{\max}$ in this linear model. For an original choice $F_{\max}=10\,\mathrm N$ and $d_{\max}=10\,\mathrm{\mu m}$, choose $k=10^6\,\mathrm{N\,m^{-1}}$. At $m=0.5\,\mathrm{kg}$,
$$
t_c\approx0.7071\,\mathrm{ms},\qquad T_c\approx4.4429\,\mathrm{ms}.
$$
A chosen resolution target of twenty steps per contact oscillation would set $h\le0.2221\,\mathrm{ms}$. This is a resolution target, not a universal accuracy or stability theorem for every integrator.
Increasing the <stiffness> by a factor of 100 reduces the same-load overlap by 100 but reduces these times by only ten. Keeping the old step then gives roughly two samples per new oscillation instead of twenty. The parent <penalty approximation of unilateral contact> also shows why the <Forward Euler method>'s undamped-oscillator instability cannot be cured merely by calling the step small.
For a multibody contact, the <effective mass in a normal-contact penalty mode> replaces an arbitrary total body <mass>. Additional damping and nonlinear <force> laws can introduce other time scales. A convergence study should therefore check <force> histories and contact events as well as maximum overlap; smaller geometric penetration alone does not establish a more accurate simulation.
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