= Mechanical inertance
{title2=$b$}
\b[Mechanical inertance] $b$ is the coefficient relating the terminal <force> of an ideal <inerter> to relative terminal <acceleration>:
$$
F_1=b(\ddot x_1-\ddot x_2),\qquad F_2=-F_1.
$$
The forces here are applied to the device. Its mechanical inertance has <SI> units $\mathrm{N\,s^2\,m^{-1}}=\mathrm{kg}$. The constitutive law was introduced by <Malcolm C. Smith> as a two-terminal mechanical-network element.<Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith>[Abstract and pp1648–1651, ideal element and realizations]{ref}
Mechanical inertance is a response coefficient and need not equal the assembly's translating <mass>. For an ideal <rack and pinion> driving a <flywheel>, let relative terminal speed $v$ produce <angular velocity> $\omega=v/r$. Comparing stored energies gives
$$
\frac12bv^2=\frac12J\omega^2,\qquad b=\frac{J}{r^2},
$$
where $J$ is the flywheel's <moment of inertia> and $r$ is the pinion's pitch radius. In an original example, $J=0.0125\,\mathrm{kg\,m^2}$ and $r=0.005\,\mathrm m$ give $b=500\,\mathrm{kg}$. A relative acceleration of $0.30\,\mathrm{m\,s^{-2}}$ then requires150newtons. Halving $r$ with $J$ fixed multiplies $b$ by four; it does not multiply the physical mass by four.
At relative speed $0.20\,\mathrm{m\,s^{-1}}$, the same ideal element stores10joules. If both terminals instead have identical acceleration, the ideal inertial force is zero regardless of their common acceleration. This two-terminal behavior distinguishes the coefficient from simply attaching a mass to a fixed reference.
A measurement can estimate mechanical inertance from the slope of force against relative acceleration over a range where the ideal model is adequate. <Friction>, <backlash> and <mechanical compliance> can produce additional force components, so one force-to-acceleration ratio at an arbitrary operating point need not identify a constant inertance.
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