{title2=$\dot E\leq Fv$}

The <mechanical dissipation inequality> states that a <mechanical energy storage function> can increase no faster than energy supplied through the mechanical terminals. With one port and entering <power> $w=Fv$, its differentiable form and interval form are
$$
\dot E\le Fv,\qquad
E(t_1)-E(t_0)\le\int_{t_0}^{t_1}Fv\,dt.
$$
The second form remains meaningful when a pointwise time derivative is unavailable. The <mechanical supply rate> $w$ records the external input; the difference $Fv-\dot E\ge0$ records irreversible loss in the model. This is the mechanical specialization of <dissipativity (control theory)>, not a demand that entering power be positive at every instant.<Dissipative Dynamical Systems, Part I: General Theory by Jan C. Willems>[Time-varying extension following Definition1′, storage inequality in paragraph2; stationary mechanical specialization]{ref}

For a <spring>, <inerter> and <dashpot> sharing a port,
$$
F=kx+b\dot v+cv,\qquad v=\dot x,\qquad
E=\frac12kx^2+\frac12bv^2,
$$
and multiplication by $v$ gives the exact balance
$$
Fv-\dot E=cv^2\ge0\quad(c\ge0).
$$
A <spring> may return energy with $Fv<0$ while $E$ decreases; that does not violate the inequality. Conversely, a negative viscous coefficient fails this storage balance by supplying energy to motion.

For an original free-decay example, omit the spring and take grounded inertial coefficient $b=2\,\mathrm{kg}$, $c=1\,\mathrm{N\,s/m}$ and initial velocity $v_0=1\,\mathrm{m/s}$. With zero applied port force, $b\dot v+cv=0$, so
$$
v(t)=v_0e^{-ct/b},\qquad E(t)=\frac12bv_0^2e^{-2ct/b}.
$$
At $t=(b/(2c))\ln2\approx0.6931\,\mathrm s$, stored energy has fallen from $1\,\mathrm J$ to $0.5\,\mathrm J$; the missing $0.5\,\mathrm J$ equals $\int_0^t cv^2dt$. No external work entered during that interval.

For several ports, use $\sum_jF_jv_j$ as the total supply. When passive elements are connected with power-conserving internal constraints, their internal supply terms cancel, so the sum of their storage functions satisfies the same external inequality. This explains why passive interconnections can be checked element by element, provided every power-carrying terminal is accounted for.


 Mechanical dissipation inequality (source code)