= Mechanical power port
A <mechanical power port> specifies the variables and sign convention through which a mechanical system exchanges <energy>. For translational motion these variables are an entering <force> $F$ and the corresponding signed <velocity> $v$; entering <power> is $p=Fv$. A rotational port uses entering <torque> $\tau$ and <angular velocity> $\omega$, giving $p=\tau\omega$. The pairing matters: multiplying a <force> by a displacement gives <work>, not its rate. A <mechanical supply rate> sums the powers of all specified ports.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[Storage/supply framework and discussion of power-neutral interconnections]{ref}
A two-terminal element makes <mechanical power-port orientation> explicit. Equal and opposite terminal <forces> $F,-F$ acting at velocities $v_1,v_2$ give
$$
p=Fv_1-Fv_2=F(v_1-v_2).
$$
Thus the port's flow variable is relative <velocity>, unless the second terminal is fixed. Reversing both chosen directions leaves the physical <power> unchanged. For $F=100\,\mathrm N$, $v_1=0.030\,\mathrm{m/s}$ and $v_2=0.010\,\mathrm{m/s}$, net entering <power> is $2\,\mathrm W$; the separate terminal contributions are $3\,\mathrm W$ and $-1\,\mathrm W$.
At an ideal rigid interconnection, connected terminals have the same <velocity>, while their entering <forces> sum to zero by <Newton's third law>. Their port powers therefore cancel in the combined system. This cancellation explains why internal connections disappear from an overall <mechanical energy> balance. A <spring> at the remaining external port obeys $p=kx\dot x=d(kx^2/2)/dt$; a <dashpot> obeys $p=cv^2$. One stores <energy>, while the other dissipates it.
For sinusoidal peak <complex amplitudes>, mean entering <power> is $\overline p=\tfrac12\operatorname{Re}(FV^*)$. A pure <mass> can have nonzero force and velocity amplitudes yet zero mean <power>, because their phases differ by a quarter cycle. Measuring magnitudes alone therefore does not determine energy transmission or establish the passivity of a <mechanical network synthesis> realization.
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