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An impedance head combines a force transducer and an accelerometer in one housing for a driving-point vibration test. It measures applied force and acceleration at nearly the same location and along the intended excitation direction. The raw channels yield mechanical accelerance or its reciprocal mechanical dynamic mass; deriving mechanical driving-point mobility or mechanical driving-point impedance additionally requires the acceleration-to-velocity conversion.^[ref]

In the manufacturer's Types8000/8001 example, a piezoelectric acceleration sensor sits above a force gauge in a stiff housing, with separate electrical outputs. The arrangement aims to preserve the phase relation needed for a complex response ratio. This solves a practical apparatus problem: a separate exciter and response sensor can otherwise occupy different points, particularly on a small structure.

An impedance-head mass correction accounts for moving material between the force-sensing plane and the specimen. In a rigid lumped approximation, choose a common positive axis and let that material have effective mass $m_h$ and acceleration complex amplitude $A$. Then Newton's second law givesFor an original in-phase example, let $m_h=0.020\mathrm{kg}$, $A=10m/{\mathrm{s}}^2$ and gauge force $1\mathrm{N}$. Specimen force is $0.80\mathrm{N}$, so specimen accelerance is $12.5\mathrm{k}{\mathrm{g}}^{-1}$ rather than the uncorrected value10. With general signals the correction is a complex subtraction, not subtraction of unsigned amplitudes.

This force-plane correction does not undo every effect of attaching the instrument: added mass and mounting stiffness can change the specimen's actual vibration. The test must also state mounting, calibration, usable frequency range and the motion of any second terminal. Manufacturer guidance distinguishes measurements requiring mass compensation and cases where separate force/acceleration sensors are preferable.^[ref]

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.

Mechanical driving-point mobility is the frequency response relating motion to excitation at the same mechanical port. Using relative velocity $v$ and signed entering force $F$, its zero-initial-state Laplace transform convention iswhere $Z$ is the force/velocity mechanical driving-point impedance. Mobility has units $m/(\mathrm{N}\mathrm{s})$. The measured direction, port and boundary conditions are part of its specification.^[ref]

For an inertial element $b$, dashpot coefficient $c$ and spring stiffness $k$ sharing a port,For $e^{i\omega t}$ complex amplitudes, mobility tends to $i\omega /k$ at low frequency when $k>0$ and to $1/(i\omega b)$ at high frequency when $b>0$. A spring therefore prevents a nonzero steady drift response to constant force. This differs from the steady mechanical mobility of a body under pure linear drag, which has no elastic restoring term.

Take $b=2\mathrm{kg}$, $c=4\mathrm{N}s/m$ and $k=200N/m$. At $\omega =10rad/s$ the reactive terms cancel and $Y=0.25m/(\mathrm{N}\mathrm{s})$. A peak force amplitude $0.8\mathrm{N}$ gives peak velocity $0.20m/s$ and mean entering power $0.08\mathrm{W}$. These statements concern the same port; a response measured elsewhere is a transfer mobility.

A response/force ratio using acceleration instead is mechanical accelerance, $A/F=sY$; using displacement gives mechanical receptance, $X/F=Y/s$. Confusing these ratios introduces a frequency-dependent magnitude and phase error. With several coupled ports, matrix inversion rather than elementwise reciprocation is needed to exchange complete impedance and mobility descriptions.

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.