Mechanical inertance ($b$)

Mechanical inertance $b$ is the coefficient relating the terminal force of an ideal inerter to relative terminal acceleration:The forces here are applied to the device. Its mechanical inertance has SI units $\mathrm{N}{\mathrm{s}}^2{\mathrm{m}}^{-1}=\mathrm{kg}$. The constitutive law was introduced by Malcolm C. Smith as a two-terminal mechanical-network element.^[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 giveswhere $J$ is the flywheel's moment of inertia and $r$ is the pinion's pitch radius. In an original example, $J=0.0125\mathrm{kg}{\mathrm{m}}^2$ and $r=0.005\mathrm{m}$ give $b=500\mathrm{kg}$. A relative acceleration of $0.30\mathrm{m}{\mathrm{s}}^{-2}$ then requires150newtons. Halving $r$ with $J$ fixed multiplies $b$ by four; it does not multiply the physical mass by four.

At relative speed $0.20\mathrm{m}{\mathrm{s}}^{-1}$, 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.