Inerter (source code)
= Inerter
{title2=$F=b(\ddot x_1-\ddot x_2)$}
An \b[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 as
$$
F_1=b(\ddot x_1-\ddot x_2),\qquad F_2=-F_1,\qquad b>0.
$$
The 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.<Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith>[Abstract and pp1648–1651, two-terminal element and realizations]{ref}
The power and stored-energy relation explains its role. Let $v=\dot x_1-\dot x_2$. Then
$$
F_1\dot x_1+F_2\dot x_2=b\dot v\,v=\frac{d}{dt}\left(\frac12bv^2\right).
$$
An ideal inerter stores and returns energy; this relation is not viscous dissipation. For an original force calculation, $b=100\,\mathrm{kg}$ and relative acceleration $0.2\,\mathrm{m\,s^{-2}}$ 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 energy
$$
E=\frac12J\omega^2=\frac12\frac{J}{r^2}v^2,
$$
so its ideal equivalent inertance is $b=J/r^2$. Choosing $J=0.02\,\mathrm{kg\,m^2}$ 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.<Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith>[Mechanical realizations and flywheel-based element]{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.
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