= Mechanical power-port orientation
<Mechanical power-port orientation> specifies the signs of the conjugate variables used to describe entering <power> at a <mechanical power port>. For a translational port with signed <force> $F$ and <velocity> $v$, use $p=Fv>0$ for net supply into the chosen system. Reversing the coordinate axis reverses both variables, so $(-F)(-v)=Fv$. Reversing just one variable instead changes the entering-versus-leaving convention and must be stated explicitly. This bookkeeping makes a <mechanical supply rate> meaningful in a <mechanical dissipation inequality>.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[Storage/supply formulation and power-neutral interconnections]{ref}
For an element acted on by terminal <forces> $F,-F$ and terminal <velocities> $v_1,v_2$,
$$
p=F(v_1-v_2).
$$
Exchanging terminal labels gives $F'=-F$ and $v'_1-v'_2=-(v_1-v_2)$, preserving <power>. For $F=100\,\mathrm N$, $v_1=0.030\,\mathrm{m/s}$ and $v_2=0.010\,\mathrm{m/s}$, both descriptions give $2\,\mathrm W$. Taking force from the exchanged convention while retaining the original relative <velocity> would incorrectly report $-2\,\mathrm W$.
More general coordinate changes require the corresponding conjugate-force change. If physical displacement coordinates satisfy $\mathbf x=\mathbf T\mathbf q$, then $\dot{\mathbf x}=\mathbf T\dot{\mathbf q}$. Equality of <virtual work>, or instantaneous <power>, requires
$$
\mathbf f_q=\mathbf T^{\mathsf T}\mathbf f_x,\qquad
\mathbf f_x^{\mathsf T}\dot{\mathbf x}=\mathbf f_q^{\mathsf T}\dot{\mathbf q}.
$$
For the scalar coordinate $q=2x$, the generalized <force> is $F_q=F_x/2$. Thus $F_x=10\,\mathrm N$ and $\dot x=0.10\,\mathrm{m/s}$ correspond to $F_q=5\,\mathrm N$ and $\dot q=0.20\,\mathrm{m/s}$, both giving1W. Merely rescaling the motion channel without its conjugate <force> changes the apparent <power>. At an ideal connection, equal terminal <velocities> and opposite entering <forces> cancel internal port powers, enabling the combined <mechanical energy> balance.
Back to article page
