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Mechanical driving-point impedance describes the relation between a driving force and motion at the same mechanical port. Here the convention iswhere $F,V$ are Laplace transforms with zero initial conditions, and $v$ is the signed relative velocity of the two terminals. Choose signs so $Fv$ is entering power. The units of this mechanical impedance are $\mathrm{N}s/m$. Its inverse is the mechanical driving-point mobility. Conventions must be checked explicitly: in The Inerter: A Retrospective by Malcolm C. Smith, the force–current analogy calls $V/F$ the mechanical impedance and $F/V$ the admittance.^[ref]

For a spring, dashpot and inerter sharing the same terminals, relative velocities agree and forces add. HenceAt a sinusoidal angular frequency $\omega$, using complex amplitudes proportional to $e^{i\omega t}$,The real part determines mean absorbed power, while the imaginary part records the out-of-phase reactive response. With peak velocity amplitude $|V|$, the mean entering power is $\frac{1}{2}\mathrm{Re}(Z)|V{|}^2$.

For an original example, choose $b=10\mathrm{kg}$, $c=30\mathrm{N}s/m$ and $k=1000N/m$. At $\omega =10rad/s$, the inertial and elastic terms cancel, giving $Z=30\mathrm{N}s/m$. A relative velocity amplitude $0.020m/s$ requires force amplitude $0.60\mathrm{N}$, and mean absorbed power is $0.006\mathrm{W}$. The individual spring and inerter still store and exchange energy.

A mechanical impedance measurement drives the chosen port and records synchronized force and motion there. A load cell measures force; displacement or an accelerometer can supply motion. For an acceleration complex amplitude $A$, $V=A/(i\omega )$, so the phase as well as magnitude matters. Fixture inertia, sensor measurement calibration and the second terminal's motion belong in the measurement model. Using a response at a different point instead defines a transfer relation, not this driving-point relation. Initial stored motion also prevents interpreting an arbitrary transient quotient as a zero-initial-state transfer function.

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.

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 requiresIn 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.^[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 giveFor $k,b,c\ge 0$, integration proves the passivity inequality: the extra term $\int c{v}^{2}dt$ 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.^[ref]

For example, a spring with $k=1000N/m$ 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 $v=1m/s$ 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.