= Passive mechanical network
{title2=$\Delta E\leq\int Fv\,dt$}
A <passive mechanical network> can store and dissipate <energy>, but cannot supply more net <work> than was previously stored or delivered through its terminals. For a two-terminal mechanical element, choose signed <force> $F$ and relative <velocity> $v$ so that $p=Fv$ is <power> entering the network. With a nonnegative <mechanical energy storage function> $E$, passivity requires
$$
E(t_1)-E(t_0)\le\int_{t_0}^{t_1}F(t)v(t)\,dt.
$$
In particular, the net extracted <energy> over that interval cannot exceed $E(t_0)$. This permits temporary outward <power>: a stretched <spring> can release stored <elastic potential energy>. A requirement that $Fv$ be nonnegative at every instant would incorrectly exclude such a <spring>.<The Inerter: A Retrospective by Malcolm C. Smith>[Accepted manuscript, §17, p. 24, equation (30): integrated terminal-power definition of passivity; the displayed storage inequality makes the initial stored energy explicit]{ref}
For a <spring>, <dashpot> and <inerter> connected between the same two terminals, let $x$ be relative displacement and $v=\dot x$. Their constitutive equations give
$$
F=kx+cv+b\dot v,\qquad
E=\frac12kx^2+\frac12bv^2,
\qquad Fv=\dot E+cv^2.
$$
For $k,b,c\ge0$, integration proves the passivity inequality: the extra term $\int cv^2dt$ is dissipated <energy>. The <inerter> stores <kinetic energy> associated with relative motion, so a <passive mechanical network> need not be a device that merely resists displacement or converts all input <work> into heat. This is central to <mechanical network synthesis> using <springs>, <dashpots> and <inerters>.<Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith>[Abstract and pp. 1648–1651: passive networks and the two-terminal inerter]{ref}
For example, a <spring> with $k=1000\,\mathrm{N/m}$ initially extended by $0.10\,\mathrm m$ stores $5\,\mathrm J$. It can deliver up to $5\,\mathrm J$ while relaxing, despite being passive. A purely viscous element with $c=-10\,\mathrm{N\,s/m}$ would instead have input <power> $-10v^2$: maintaining $v=1\,\mathrm{m/s}$ for one second would extract $10\,\mathrm J$ even from an initially unenergized model. An apparatus producing that sustained behavior needs another energy source.
An experimental check records terminal <force> and relative <velocity> simultaneously and integrates their product, accounting for initial <energy> and every power-carrying connection. A brief interval of negative measured <power> is compatible with stored-energy release. For a linear zero-initial-state network, the <mechanical driving-point impedance> provides a complementary frequency description; a nonnegative real part at measured sinusoidal frequencies checks mean absorbed <power> there, but a finite frequency sweep alone does not establish the full time-domain passivity inequality.
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