= Mechanical receptance
{title2=$X(s)/F(s)$}
<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 $\mathrm{m/N}$. <Mechanical driving-point mobility> is $Y=sH$ and <mechanical accelerance> is $s^2H$. The port, direction and boundary conditions must accompany the ratio.<Structural Testing Part I: Mechanical Mobility Measurements by Ole Døssing>[“Mobility Measurements — Definitions”, printed p.16, displacement-, velocity- and acceleration-response functions]{ref}
For a single-coordinate model $m\ddot x+c\dot x+kx=F$,
$$
H(s)=\frac1{ms^2+cs+k},\qquad
H(i\omega)=\frac1{k-m\omega^2+i c\omega}.
$$
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 $k=200\,\mathrm{N/m}$. At $\omega=10\,\mathrm{rad/s}$, stiffness and inertia cancel, giving $H=-0.025i\,\mathrm{m/N}$. 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.
Back to article page
