= Mechanical driving-point impedance
{title2=$Z(s)=F(s)/V(s)$}
<Mechanical driving-point impedance> describes the relation between a driving <force> and motion at the same mechanical port. Here the convention is
$$
Z(s)=\frac{F(s)}{V(s)},
$$
where $F,V$ 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.<The Inerter: A Retrospective by Malcolm C. Smith>[Accepted manuscript §17, p.25, mechanical impedance defined as transformed velocity divided by force]{ref}
For a <spring>, <dashpot> and <inerter> sharing the same terminals, relative <velocities> agree and <forces> add. Hence
$$
Z(s)=\frac{k}{s}+c+bs.
$$
At a sinusoidal <angular frequency> $\omega$, using <complex amplitudes> proportional to $e^{i\omega t}$,
$$
Z(i\omega)=c+i\left(b\omega-\frac{k}{\omega}\right).
$$
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 $\tfrac12\operatorname{Re}(Z)|V|^2$.
For an original example, choose $b=10\,\mathrm{kg}$, $c=30\,\mathrm{N\,s/m}$ and $k=1000\,\mathrm{N/m}$. At $\omega=10\,\mathrm{rad/s}$, the inertial and elastic terms cancel, giving $Z=30\,\mathrm{N\,s/m}$. A relative <velocity> amplitude $0.020\,\mathrm{m/s}$ 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>.
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