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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 impedance measurement determines the amplitude and phase of driving force relative to motion at a specified port. With the convention $Z=F/V$, a driving-point test measures the force and velocity at the same location and in the same direction. Both terminal motions must be included if the second terminal moves. Ole Døssing's manufacturer guide describes this need for coincident excitation/response measurements and the use of an impedance head combining force and acceleration sensing.^[ref]

A typical apparatus uses a vibration exciter or instrumented impact to drive the structure, a force transducer in the excitation path, an accelerometer near the driving point, and synchronized acquisition of both signals. The measurement calibration, mounting, excitation range and fixture motion belong in the test description. A transfer response at a remote point is useful but answers a different question from driving-point mechanical impedance.

For a steady sinusoid with convention $e^{i\omega t}$, acceleration complex amplitude $A$ gives $V=A/(i\omega )$, henceFor example, at $\omega =100rad/s$ let the applied force amplitude be $1\mathrm{N}$ and acceleration amplitude be $10m/{\mathrm{s}}^2$ in phase with the force. Then $|V|=0.10m/s$ and $Z=i10\mathrm{N}s/m$. This is the response of a pure $0.10\mathrm{kg}$ mass: force leads velocity by a quarter cycle. Reporting only the two amplitudes would lose that distinction from a dissipative response.

For broadband excitation, estimate the relevant frequency response from the synchronized force and response records, with suitable error analysis. A response peak alone can come from a peak in the applied force, so the response spectrum by itself does not establish structural resonance. Transducer mass can modify a small structure, and fixture inertia can contribute measured force; both require an explicit measurement model. Repeating at another excitation amplitude tests the assumed linear regime rather than treating a changed response ratio as the same invariant system property.^[ref]

Mechanical receptance is a displacement-to-force frequency response. With zero initial conditions, the Laplace transform convention is $H(s)=X(s)/F(s)$, with units $m/N$. Mechanical driving-point mobility is $Y=sH$ and mechanical accelerance is $s^{2}H$. The port, direction and boundary conditions must accompany the ratio.^[ref]

For a single-coordinate model $m\ddot{x}+c\dot{x}+kx=F$,Its inverse is mechanical dynamic stiffness. If $k>0$, the zero-frequency limit $H(0)=1/k$ agrees with static mechanical compliance. At finite frequency, inertia and damping change both magnitude and phase; replacing the denominator by $k$ would discard them.

For an original example, use $m=2\mathrm{kg}$, $c=4\mathrm{N}s/m$ and $k=200N/m$. At $\omega =10rad/s$, stiffness and inertia cancel, giving $H=-0.025im/N$. A peak force amplitude $0.8\mathrm{N}$ therefore gives displacement amplitude $0.020\mathrm{m}$, lagging force by a quarter cycle. Static displacement under the same force magnitude would be only $0.004\mathrm{m}$.

For sufficiently high frequency, this model has $H(i\omega )\sim -1/(m{\omega }^2)$, revealing an inertia-dominated response with a phase reversal. A mechanical impedance measurement can infer receptance from acceleration by dividing the acceleration/force response by $-{\omega }^2$ for nonzero $\omega$. This conversion becomes sensitive to low-frequency measurement errors and is not a substitute for specifying the actual displacement measurement or integration model. With multiple coupled coordinates, the receptance matrix is the inverse of the complete dynamic-stiffness matrix, rather than a collection of elementwise reciprocal entries.