= Mechanical network synthesis
<Mechanical network synthesis> constructs an arrangement of mechanical elements to realize a specified relation between applied <force> and motion. In the linear two-terminal setting, a natural specification is the <mechanical impedance>
$$
Z(s)=\frac{F(s)}{V(s)},
$$
where $F(s)$ and $V(s)$ are <Laplace transforms> of terminal <force> and relative <velocity>, with zero initial conditions. Choosing elements and their connections to obtain a desired $Z(s)$ is the synthesis problem; calculating $Z(s)$ for an already chosen arrangement is the corresponding analysis problem. <Malcolm C. Smith>'s 2002 work introduced the <inerter> as a genuinely two-terminal inertial element, allowing a broader class of <passive mechanical networks> than a construction restricted to grounded masses.<Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith>[Abstract and pp. 1648–1651: passive one-port synthesis, two-terminal inerter and mechanical realizations]{ref}
A linear <spring> with stiffness $k$, a <dashpot> with coefficient $c$, and an <inerter> with <mechanical inertance> $b$ have respective <mechanical impedances>
$$
Z_k(s)=\frac{k}{s},\qquad Z_c(s)=c,\qquad Z_b(s)=bs.
$$
For elements attached between the same two moving terminals, the relative <velocity> is common and their <forces> add, so their <mechanical impedances> add. Consequently the target
$$
Z(s)=20s+100+\frac{2000}{s}
$$
can be realized by a $20\,\mathrm{kg}$ <inerter>, a $100\,\mathrm{N\,s/m}$ <dashpot> and a $2000\,\mathrm{N/m}$ <spring> sharing those terminals. The units of the resulting <mechanical impedance> are $\mathrm{N\,s/m}$. At <angular frequency> $\omega=10\,\mathrm{rad/s}$, the sinusoidal <mechanical impedance> is $Z(i\omega)=100\,\mathrm{N\,s/m}$: the <spring> and <inerter> contributions are equal and opposite imaginary terms. A sinusoidal relative <velocity> of amplitude $1\,\mathrm{m/s}$ therefore needs a terminal <force> amplitude of $100\,\mathrm N$ in this ideal example.
The apparent cancellation does not remove stored <energy>. With relative displacement $x$ and $v=\dot x$, the terminal <power> satisfies
$$
Fv=\frac{d}{dt}\left(\frac12bv^2+\frac12kx^2\right)+cv^2.
$$
The <inerter> and <spring> exchange stored <energy>, while the <dashpot> dissipates it. This balance verifies that the example is a <passive mechanical network> for nonnegative $b,c,k$. A practical <inerter> can use a <rack and pinion> to drive a <flywheel>; <friction>, <backlash> and finite travel limit the frequency and displacement range over which the synthesized response matches the ideal <mechanical impedance>. Thus synthesis gives a testable target for an apparatus, rather than a guarantee that a real mechanism has no parasitic effects.<Synthesis of Mechanical Networks: The Inerter by Malcolm C. Smith>[pp. 1648–1651: ideal-element assumptions, inerter and mechanical implementations]{ref}
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