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Performance Benefits in Passive Vehicle Suspensions
Employing Inerters1
Malcolm C. Smith2 and Fu-Cheng Wang3
2 Department

of Engineering, University of Cambridge, Cambridge CB2 1PZ, U.K.

Tel: +44 (0)1223 332745, Fax: +44 (0)1223 332662, e-mail: mcs@eng.cam.ac.uk
3

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan.
Abstract
A new ideal mechanicalone-port network element named the inerter was recently
introduced, and shown to be realisable, with the property that the applied force is proportional to the relative acceleration across the element. This paper makes a comparative
study of several simple passive suspension struts, each containing at most one damper
and inerter as a preliminary investigation into the potential performance advantagesof the
element. Improved performance for several different measures in a quarter-car model is
demonstrated here in comparison with a conventional passive suspension strut. A study
of a full-car model is also undertaken where performance improvements are also shown
in comparison to conventional passive suspension struts. A prototype inerter has been
built and tested. Experimental results arepresented which demonstrate a characteristic phase advance property which cannot be achieved with conventional passive struts
consisting of springs and dampers only.

Running head: Benefits in Suspensions using Inerters

1 Introduction
In [1] an alternative to the traditional electrical-mechanical analogies was proposed in the
context of synthesis of passive mechanical networks. Specifically,a new two-terminal element
called the inerter was introduced, as a substitute for the mass element, with the property
that the force across the element is proportional to the relative acceleration between the
terminals. It was argued in [1] that such an element is necessary for the synthesis of the
full class of physically realisable passive mechanical impedances. Indeed, the traditionalsuspension strut employing springs and dampers only and avoiding the mass element has
dynamic characteristics which are greatly limited in comparison. The consequence is that,
1 This

works was published in: Vehicle System Dynamics, 2004, vol. 42, no. 4, pp. 235–257. The published

version contains several small errors (especially in the labelling of Figure 7) which are corrected here. Thiswork was supported in part by the EPSRC.

1

potentially, there is scope to improve the vehicle dynamics of a passively suspended vehicle by
using suspension struts employing inerters as well as springs and dampers. It is the purpose of
the present paper to give more detailed consideration to these possible performance benefits
using some standard performance measures for quarter-car andfull-car vehicle models. In
addition, some experimental test results on a prototype inerter will be reported.

2 Background on the inerter
The force-current analogy between mechanical and electrical networks has the following
correspondences:
force

↔

velocity ↔

mechanical ground ↔

spring ↔

damper ↔

current
voltage
electrical ground
inductor
resistor.

Additionally themass element has always been taken as the analogue of the capacitor, even
though it has been appreciated [2, p. 111], [3, p. 10-5] that the mass is strictly analogous
only to a capacitor with one terminal connected to ground. This is due to the fact that
Newton’s Second Law refers the acceleration of the mass to a fixed point in an inertial
frame, i.e. mechanical ground. The restrictive nature ofthe mass element in networks has
the disadvantage that electrical circuits with ungrounded capacitors do not have a direct
spring-mass-damper analogue. This imposes a restriction on the class of passive mechanical
impedances which can be physically realised. A further problem is that the suspension strut
needs to have small mass compared to that of the vehicle body and wheel hub, which...