= Mechanical accelerance
{title2=$A(s)/F(s)$}
<Mechanical accelerance> is the acceleration-to-force <frequency response>, $G(s)=A(s)/F(s)$ under zero <initial conditions>. It has units $\mathrm{kg^{-1}}$. It is related to <mechanical receptance> and <mechanical driving-point mobility> by
$$
G(s)=s^2H(s)=sY(s).
$$
Thus an acceleration response cannot be labeled mobility without a frequency-dependent conversion. Its inverse is <mechanical dynamic mass>, which is generally complex and frequency dependent.<Impedance Heads: Types 8000 and 8001 by Brüel and Kjær>[Printed p.2, footnote: direct force/acceleration outputs give dynamic mass and accelerance, not mobility or impedance]{ref}
For $m\ddot x+c\dot x+kx=F$,
$$
G(s)=\frac{s^2}{ms^2+cs+k},\qquad
G(i\omega)=\frac{-\omega^2}{k-m\omega^2+i c\omega}.
$$
For a pure mass, $G=1/m$, independent of frequency in the ideal rigid model. A supported mass is different: spring and damping forces also enter its response. If $k>0$, accelerance tends to zero as $\omega$ tends to zero, while its high-frequency limit is $1/m$ within the model's useful range.
With $m=2\,\mathrm{kg}$, $c=4\,\mathrm{N\,s/m}$, $k=200\,\mathrm{N/m}$ and $\omega=10\,\mathrm{rad/s}$, the result is $G=2.5i\,\mathrm{kg^{-1}}$. A peak <force> $0.8\,\mathrm N$ gives acceleration amplitude $2\,\mathrm{m/s^2}$, leading force by a quarter cycle. The reciprocal response is $-0.4i\,\mathrm{kg}$: it describes a dynamic ratio, rather than a negative physical amount of inertial material.
An <impedance head> measures force and acceleration together, making this ratio directly accessible after <measurement calibration> and signal processing. Driving-point accelerance uses coincident force and response directions. A response at another point is a transfer measurement; a small or vanishing transfer response can arise from the mode shapes and does not establish that the driven structure has infinite physical mass.
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