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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.

Mechanical network synthesis constructs an arrangement of mechanical elements to realize a specified relation between applied force and motion. In the linear two-terminal setting, a natural specification is the mechanical impedancewhere $F(s)$ and $V(s)$ are Laplace transforms of terminal force and relative velocity, with zero initial conditions. Choosing elements and their connections to obtain a desired $Z(s)$ is the synthesis problem; calculating $Z(s)$ for an already chosen arrangement is the corresponding analysis problem. Malcolm C. Smith's 2002 work introduced the inerter as a genuinely two-terminal inertial element, allowing a broader class of passive mechanical networks than a construction restricted to grounded masses.^[ref]

A linear spring with stiffness $k$, a dashpot with coefficient $c$, and an inerter with mechanical inertance $b$ have respective mechanical impedancesFor elements attached between the same two moving terminals, the relative velocity is common and their forces add, so their mechanical impedances add. Consequently the targetcan be realized by a $20\mathrm{kg}$ inerter, a $100\mathrm{N}s/m$ dashpot and a $2000N/m$ spring sharing those terminals. The units of the resulting mechanical impedance are $\mathrm{N}s/m$. At angular frequency $\omega =10rad/s$, the sinusoidal mechanical impedance is $Z(i\omega )=100\mathrm{N}s/m$: the spring and inerter contributions are equal and opposite imaginary terms. A sinusoidal relative velocity of amplitude $1m/s$ therefore needs a terminal force amplitude of $100\mathrm{N}$ in this ideal example.

The apparent cancellation does not remove stored energy. With relative displacement $x$ and $v=\dot{x}$, the terminal power satisfiesThe inerter and spring exchange stored energy, while the dashpot dissipates it. This balance verifies that the example is a passive mechanical network for nonnegative $b,c,k$. A practical inerter can use a rack and pinion to drive a flywheel; friction, backlash and finite travel limit the frequency and displacement range over which the synthesized response matches the ideal mechanical impedance. Thus synthesis gives a testable target for an apparatus, rather than a guarantee that a real mechanism has no parasitic effects.^[ref]

A mechanical supply rate specifies the rate at which energy enters a mechanical system through its stated ports. For a translational mechanical power port, entering force $F$ and its conjugate velocity $v$ give $w=Fv$; for rotation, torque $\tau$ and angular velocity $\omega$ give $w=\tau \omega$. The supply rate has units of power, while its time integral is supplied work.^[ref]

A two-terminal element makes the sign convention explicit. If forces on the element are $F$ and $-F$ at terminal velocities $v_1,v_2$, total entering power isOnly when terminal2 is stationary may one replace relative velocity by $v_1$. For an original example, $F=100\mathrm{N}$, $v_1=0.030m/s$ and $v_2=0.010m/s$ give $w=2\mathrm{W}$. Over two seconds the first terminal supplies $6\mathrm{J}$ and the second returns $2\mathrm{J}$, leaving net input $4\mathrm{J}$. Ignoring terminal2 would overstate the net supply.

For several translational and rotational ports, sum all entering power terms:The mechanical dissipation inequality then requires $\dot{E}\le w$ for an appropriate nonnegative mechanical energy storage function. Negative instantaneous $w$ is permitted when stored energy returns through the ports; it does not automatically contradict a passive mechanical network.

The chosen supply rate also defines the question being certified. A mathematical quadratic supply used in dissipativity can encode another input/output property, but it represents physical mechanical power only when its variables, units and sign convention match the actual energy exchange. This is why port bookkeeping must precede interpreting a fitted response model as passive.

Modal damping coupling is the interaction between undamped modal coordinates produced by off-diagonal entries of the modal damping matrix. With mass-normalized modes and $D={\Phi }^{T}C\Phi$, the equations areThus $D_{ij}{\dot{q}}_j$ contributes to mode $i$'s force when $j\ne i$. Distinct undamped natural frequencies fix the modal directions up to scaling and ordering; repeated-frequency subspaces admit additional basis rotations and must be treated before interpreting an off-diagonal entry as unavoidable nonclassical damping.^[ref]

For an original example, takeIts eigenvalues are $0.5$ and $0.1$, so it is positive definite. At modal velocities $\dot{q}=(1,-2{)}^T$, total dissipated power isA diagonal-only approximation would give1.5 instead, despite leaving both diagonal coefficients unchanged.

The individual modal energies $E_i=\frac{1}{2}({\dot{q}}_i^2+{\omega }_i^2{q}_i^2)$ obey, with no external forcing,For these velocities, ${\dot{E}}_1=+0.1$ while ${\dot{E}}_2=-0.8$: one modal energy grows temporarily, even though the total falls at rate0.7. This modal energy transfer is compatible with a passive mechanical network; it is not evidence that an external energy source drives the growing coordinate.

For harmonic forcing, the complete response solvesOff-diagonal effects therefore depend on forcing frequency and the other dynamic terms, not solely on the size of $D_{ij}$ relative to $D_{ii}$. Neglecting them is an approximation whose error can be enhanced near nearby resonances. Solving the coupled equations or the damped quadratic eigenvalue problem retains those interactions; assigning independent scalar damping ratios does not, by itself, justify dropping them.

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.