“Introduction:

To fix ideas, we will consider an automobile with independent, four-wheel suspension driving over a sinusoidal road.

Assuming the car body to be rigid, this is a problem in receiver isolation of a six degree-of-freedom system. Since forward and transverse displacement and rotation about the vertical axis are controlled by the driver, it is usual to consider only the three remaining degrees-of-freedom: bounce (vertical displacement); pitch (rotation about the transverse axis; and roll (rotation about the forward axis).

For a long time, passenger comfort was based on a quarter car model, i.e. a single degree-of-freedom (SDOF) system consisting of a quarter of the car body mass riding on a spring and damper representing the suspension of a single wheel, see Fig. 1. This is where we will start our quantitative discussion. Later we will consider a half car model, which will allow us to investigate the interaction of bounce and pitch. We shall close by making some qualitative remarks about the full car model. It will be left to the reader to make the necessary connections between these car models and his/her resiliently supported structure or piece of equipment.

SDOF VIBRATION ISOLATION

This is the elementary isolation problem: a single, rigid mass mounted through a massless linear spring-damper combination to an inelastic base, see Fig 2.

If, as in Fig 2(a), the excitation is due to a force acting on the mass, it is the source isolation problem; if, as in Fig. 2 (b), the excitation is due to motion of the base, it is the receiver isolation problem. In both cases, the excitation will be assumed to harmonic with a frequency of Ω.

This elementary problem is treated in every vibration textbook, from the early days of the 1934 edition of Den Hartog, ref [1], to the 1998 edition of Thomson, ref [2]. The metric of interest is the transmissibility; force transmissibility in the case of Fig 2(a),

TF = T̂ F / T̂

And motion transmissibility in the case of Fig 2(b),

TM = x̂ / ŷ

The notation (^) denotes the amplitude of the appropriate sinusoidal quantity used and, as such, TF and TM will always be positive. All textbooks point out that TF= TM but only a few show that this equality is not fortuitous but is a consequence of the dynamic reciprocity inherent in all linear systems. From a discussion of this important concept see Ref. [3].”