# Inerter (mechanical networks)

Inerter device being used to mechanically isolate vibration at a given frequency.

In the study of mechanical networks in control theory, an inerter is a two-terminal device in which the forces applied at the terminals are equal, opposite, and proportional to relative acceleration between the nodes. Under the name of J-damper the concept has been used in Formula 1 racing car suspension systems.

It can be constructed with a flywheel mounted on a rack and pinion. It has a similar effect to increasing the inertia of the sprung object.

## Discovery

Malcolm C. Smith, a control engineering professor at the University of Cambridge, first introduced inerters in a 2002 paper.[1] Smith extended the analogy between electrical and mechanical networks (the mobility analogy). He observed that the analogy was incomplete, since it was missing a mechanical device playing the same role as an electrical capacitor. The analogy makes mass the analogy of capacitance, but the capacitor representing a mass always has one terminal connected to ground potential. In a real electrical network, capacitors can be connected between any two arbitrary potentials, they are not limited to ground. Noticing this, Smith set about finding a mechanical device that was a true analog of a capacitor. He found that he could construct such a device using gears and flywheels, one of several possible methods.

The constitutive equation is,

${\displaystyle F=b({\dot {v}}_{2}-{\dot {v}}_{1})}$,

where the constant b is the inertance and has units of mass.

## Construction

A linear inerter can be constructed by meshing a flywheel with a rack gear. The pivot of the flywheel forms one terminal of the device, and the rack gear forms the other.

A rotational inerter can be constructed by meshing a flywheel with the ring gear of a differential. The side gears of the differential form the two terminals.

## Applications

Shortly after its discovery, the inerter principle was used under the name of J-damper in the suspension systems of Formula 1 racing cars. When tuned to the natural oscillation frequencies of the tires, the inerter reduced the mechanical load on the suspension. McLaren Mercedes began using a J-damper in early 2005, and Renault shortly thereafter.[2] J-dampers were at the center of the 2007 Formula One espionage controversy which arose when Phil Mackereth left McLaren for Renault.

Researchers are developing new vibration-control devices based on inerters to build high-rise skyscrapers which can withstand high winds.[3][4]

## References

1. ^ Smith, M. C. (2002). "Synthesis of mechanical networks: The inerter". IEEE Transactions on Automatic Control. 47 (10): 1648–1662. doi:10.1109/TAC.2002.803532.
2. ^ Chen, M.; Papageorgiou, C.; Scheibe, F.; Wang, F. C.; Smith, M. (2009). "The missing mechanical circuit element". IEEE Circuits and Systems Magazine. 9: 10–26. doi:10.1109/MCAS.2008.931738.
3. ^ Petrini, Francesco; Giaralis, Agathoklis; Wang, Zixiao (2020). "Optimal tuned mass-damper-inerter (TMDI) design in wind-excited tall buildings for occupants' comfort serviceability performance and energy harvesting" (PDF). Engineering Structures. 204: 109904. doi:10.1016/j.engstruct.2019.109904.
4. ^