 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 impedancewhere $F(s)$ 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.^[ref]

A linear spring with stiffness $k$, a dashpot with coefficient $c$, and an inerter with mechanical inertance $b$ have respective mechanical impedancesFor elements attached between the same two moving terminals, the relative velocity is common and their forces add, so their mechanical impedances add. Consequently the targetcan be realized by a $20\mathrm{kg}$ inerter, a $100\mathrm{N}s/m$ dashpot and a $2000N/m$ spring sharing those terminals. The units of the resulting mechanical impedance are $\mathrm{N}s/m$. At angular frequency $\omega =10rad/s$, 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 $1m/s$ 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 satisfiesThe 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.^[ref]