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Mechanical available storage is the supremum of net work that can be extracted from a system's current state through its allowed mechanical terminals. If $Fv$ is entering power and the initial state is $x_0$, defineThe zero-duration choice gives $S_a\ge 0$. Any nonnegative mechanical energy storage function satisfying the mechanical dissipation inequality bounds the extracted work, so $S_a(x_0)\le E(x_0)$. Available storage is thus an operational recoverability quantity; it need not equal every form of internal stored energy. The general supply/storage construction was developed by Jan C. Willems.^[ref]

Consider a spring and dashpot in parallel, with $F=kx+c\dot{x}$ and initially $x=x_0$. Unload to $x=0$ at constant speed over duration $T$, so $\dot{x}=-x_0/T$. The outgoing work isThe first term is released elastic potential energy; the second is lost in the dashpot. For $k=1000N/m$, $c=10\mathrm{N}s/m$ and $x_0=0.10\mathrm{m}$, the initial spring energy is $5\mathrm{J}$. A one-second unloading recovers $4.90\mathrm{J}$ and a ten-second unloading recovers $4.99\mathrm{J}$. With unrestricted duration and suitable terminal control, these paths approach the energy upper bound, proving $S_a=5\mathrm{J}$ for this ideal example. A supremum need not be attained in finite time.

The allowed operations are essential. If the same example must finish at zero extension within one second, the Cauchy-Schwarz inequality gives ${\int }_0^T{\dot{x}}^{2}dt\ge x_0^2/T$, so the best recoverable work under that extra requirement is $4.90\mathrm{J}$. A physically inaccessible internal motion may store energy without allowing its recovery at the chosen terminals. Thus stating the initial state, usable ports, time and actuator limits is part of an available-storage claim, not an optional qualification.

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 isThis state-based accounting includes energy present before the measurement begins. A temporary outward power is compatible with release of stored energy.^[ref]

For an ideal linear spring, $E_k=\frac{1}{2}k{x}^2$ is elastic potential energy. For an ideal inerter, $E_b=\frac{1}{2}b{v}^2$ is kinetic energy associated with terminal relative motion. An ideal dashpot need not store mechanical energy: its entering power $c{v}^2$ is dissipated. If these elements share terminals, their state variables obey $v=\dot{x}$ andThis equality proves the inequality for $c\ge 0$. 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 $k=800N/m$, $b=5\mathrm{kg}$ and $c=10\mathrm{N}s/m$. Start at $x=0.050\mathrm{m}$ and $v=0.20m/s$, 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 $c{v}^2\Delta t=0.040\mathrm{J}$, 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.