 Inerter

An inerter is an ideal two-terminal mechanical element whose equal and opposite terminal forces are proportional to the relative acceleration of its terminals. With displacements $x_1,x_2$, define the forces applied to the device asThe parameter $b$ is its mechanical inertance and has units of mass. Both terminals can move; if their accelerations are identical, this ideal constitutive law gives zero terminal force. Malcolm C. Smith introduced the element in a 2002 mechanical-network synthesis paper.^[ref]

The power and stored-energy relation explains its role. Let $v={\dot{x}}_1-{\dot{x}}_2$. ThenAn ideal inerter stores and returns energy; this relation is not viscous dissipation. For an original force calculation, $b=100\mathrm{kg}$ and relative acceleration $0.2\mathrm{m}{\mathrm{s}}^{-2}$ give a 20-newton terminal-force magnitude.

A rotational realization also shows why inertance need not equal the assembly's translational mass. In an idealized rack and pinion driving a flywheel, suppose relative terminal speed $v$ produces angular speed $\omega =v/r$. A flywheel with moment of inertia $J$ has energyso its ideal equivalent inertance is $b=J/r^2$. Choosing $J=0.02\mathrm{kg}{\mathrm{m}}^2$ and $r=0.01\mathrm{m}$ gives $b=200\mathrm{kg}$. This is a transmission-based example, not an assertion that the physical flywheel has a 200-kilogram mass.^[ref]

Friction, backlash, compliance and finite travel alter real-device behavior. An inerter can be combined with a spring or viscous damper, but their constitutive roles remain different: relative acceleration, displacement and velocity enter different element laws.