= Mechanical driving-point mobility
{title2=$Y(s)=V(s)/F(s)$}
<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 is
$$
Y(s)=\frac{V(s)}{F(s)}=\frac1{Z(s)},
$$
where $Z$ is the force/velocity <mechanical driving-point impedance>. Mobility has units $\mathrm{m/(N\,s)}$. The measured direction, port and boundary conditions are part of its specification.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[“Mobility Measurements — Definitions”, printed p.16]{ref}
For an inertial element $b$, <dashpot> coefficient $c$ and <spring> stiffness $k$ sharing a port,
$$
Z(s)=bs+c+\frac{k}{s},\qquad
Y(s)=\frac{s}{bs^2+cs+k}.
$$
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 $k=200\,\mathrm{N/m}$. At $\omega=10\,\mathrm{rad/s}$ the reactive terms cancel and $Y=0.25\,\mathrm{m/(N\,s)}$. A peak <force> amplitude $0.8\,\mathrm N$ gives peak <velocity> $0.20\,\mathrm{m/s}$ 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.
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