= Mechanical energy storage function
{title2=$E$}
A <mechanical energy storage function> assigns a nonnegative stored-energy quantity $E$ to the state of a mechanical model so that its increase never exceeds entering <work>. For a one-port <passive mechanical network> with entering <power> $Fv$, the <mechanical dissipation inequality> is
$$
E(t_1)-E(t_0)\le\int_{t_0}^{t_1}Fv\,dt.
$$
This state-based accounting includes energy present before the measurement begins. A temporary outward <power> is compatible with release of stored <energy>.<The Inerter: A Retrospective by Malcolm C. Smith>[Accepted manuscript §17, p.24, equation(30), integrated terminal power and passivity]{ref}
For an ideal linear <spring>, $E_k=\tfrac12kx^2$ is <elastic potential energy>. For an ideal <inerter>, $E_b=\tfrac12bv^2$ is <kinetic energy> associated with terminal relative motion. An ideal <dashpot> need not store mechanical <energy>: its entering <power> $cv^2$ is dissipated. If these elements share terminals, their state variables obey $v=\dot x$ and
$$
E=\frac12kx^2+\frac12bv^2,\qquad
Fv=\dot E+cv^2.
$$
This equality proves the inequality for $c\ge0$. In a larger network, summing element balances cancels internal transfers, leaving power through the external terminals and total dissipation.
For an original numerical check, let $k=800\,\mathrm{N/m}$, $b=5\,\mathrm{kg}$ and $c=10\,\mathrm{N\,s/m}$. Start at $x=0.050\,\mathrm m$ and $v=0.20\,\mathrm{m/s}$, so $E=1.10\,\mathrm J$. Maintain that relative <velocity> for $0.10\,\mathrm s$: displacement reaches $0.070\,\mathrm m$ and stored <energy> becomes $2.06\,\mathrm J$. Dissipation is $cv^2\Delta t=0.040\,\mathrm J$, so input <work> is $0.96+0.04=1.00\,\mathrm J$. Counting all entering work as newly stored energy would overestimate storage.
The <mechanical available storage> is the maximum net work that could be recovered under allowed future terminal motions; it is a related operational quantity, rather than automatically every form of internal physical energy. A model's storage function is a certificate that must respect its accessible states and port variables. With zero external <power>, a positive-definite storage function can supply a <Lyapunov function>, but nonincreasing <energy> alone does not establish that every motion decays to rest: a lossless <spring>–<inerter> oscillation keeps a constant nonzero stored <energy>.
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