= Mechanical supply rate
{title2=$w=Fv$}
A <mechanical supply rate> specifies the rate at which energy enters a mechanical system through its stated ports. For a translational <mechanical power port>, entering <force> $F$ and its conjugate <velocity> $v$ give $w=Fv$; for rotation, <torque> $\tau$ and <angular velocity> $\omega$ give $w=\tau\omega$. The supply rate has units of <power>, while its time integral is supplied <work>.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[Storage/supply formulation of dissipativity and power-neutral interconnections]{ref}
A two-terminal element makes the sign convention explicit. If forces on the element are $F$ and $-F$ at terminal velocities $v_1,v_2$, total entering power is
$$
w=Fv_1-Fv_2=F(v_1-v_2).
$$
Only when terminal2 is stationary may one replace relative velocity by $v_1$. For an original example, $F=100\,\mathrm N$, $v_1=0.030\,\mathrm{m/s}$ and $v_2=0.010\,\mathrm{m/s}$ give $w=2\,\mathrm W$. Over two seconds the first terminal supplies $6\,\mathrm J$ and the second returns $2\,\mathrm J$, leaving net input $4\,\mathrm J$. Ignoring terminal2 would overstate the net supply.
For several translational and rotational ports, sum all entering power terms:
$$
w=\sum_jF_jv_j+\sum_k\tau_k\omega_k.
$$
The <mechanical dissipation inequality> then requires $\dot E\le w$ for an appropriate nonnegative <mechanical energy storage function>. Negative instantaneous $w$ is permitted when stored <energy> returns through the ports; it does not automatically contradict a <passive mechanical network>.
The chosen supply rate also defines the question being certified. A mathematical quadratic supply used in <dissipativity (control theory)> can encode another input/output property, but it represents physical mechanical power only when its variables, units and sign convention match the actual energy exchange. This is why port bookkeeping must precede interpreting a fitted response model as passive.
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