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A flywheel stores kinetic energy in a rotating body. For fixed-axis rotation with moment of inertia $J$ and angular velocity $\omega$, its energy and axial angular momentum areAn applied torque changes the angular momentum; a drive can put energy into the flywheel, and a load can receive energy as it slows. Energy varies quadratically with speed, while angular momentum varies linearly.

For an original idealized calculation, take $J=0.20\mathrm{kg}{\mathrm{m}}^2$. Slowing from $300$ to $200\mathrm{rad}{\mathrm{s}}^{-1}$ releasesThe wheel still retains4000joules at the lower speed. Delivering the5000joule difference uniformly over10seconds would supply an average power of500watts before losses. This example distinguishes usable energy over a speed interval from the energy at the initial speed.

A NASA laboratory demonstration combined two flywheels to regulate electrical power while supplying a commanded torque. Its G2 and D1 units used magnetically levitated rotating wheels, were mounted on an air-bearing table, and operated between20000and60000revolutions per minute. A load cell measured table force so that output torque could be inferred. The apparatus connects energy exchange with the reaction torque exerted on the surrounding structure.^[ref]

A flywheel can also produce mechanical inertance through a rack and pinion. If relative terminal speed $v$ gives $\omega =v/r$, the ideal stored energy is $J{v}^2/(2r^2)$ and the equivalent inertance is $b=J/r^2$. Bearing losses, friction and material stresses constrain real devices; the ideal energy equation alone is not an operating-speed specification.

A gear is a toothed mechanical element that transmits motion and torque through contact with another gear or a rack and pinion. The teeth maintain a prescribed relation between the motions: unlike a smooth friction drive, the ideal gear ratio is set by geometry and tooth counts. The gear pitch circle supplies an imaginary rolling geometry for describing that relation; it is not the circle through the tooth tips.^[ref]

For two external gears on fixed parallel axes, let $N_1,N_2$ be their tooth counts and let signed angular velocities ${\omega }_1,{\omega }_2$ use the same orientation. Each tooth on one gear must pass a tooth space on the other, so counting tooth passages givesThus a 20-tooth input driving a 40-tooth output makes the output turn half as fast in the opposite direction. An input of $1200$ revolutions per minute gives an output of $600$ revolutions per minute in magnitude. These signs describe an external pair; a rack and pinion instead converts rotation into translation.

The useful trade is between angular speed and torque. In an ideal steady transmission, conservation of energy equates input and output power, $|{\tau }_1{\omega }_1|=|{\tau }_2{\omega }_2|$. The same 20-tooth/40-tooth pair therefore turns an input torque of $2\mathrm{N}\mathrm{m}$ into an output torque of $4\mathrm{N}\mathrm{m}$ while halving angular speed. This does not create energy: more torque accompanies less angular displacement per input revolution. Real gears dissipate energy through friction, and backlash permits a small lost motion on reversal; tooth strength and bearings, rather than the gear ratio alone, determine allowable load.

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.

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.

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]