Abstract

Play is one of the most common nonlinearities in mechanical networks and it is often described in input-output notation. It is the purpose of this seminar to show that the treatment of play as an operator in mechanical networks leads to solutions which are unsatisfactory from a physical point of view. It will be explained that the common operator definition is not invariant to changes in the inertial frame or to terminal reversal, and that it shows inconsistencies with respect to the applied force. Therefore, the question of modelling, well-posedness and design of mechanical networks with the nonlinearity play is addressed.

A behavioural definition of play will be given which contrasts with the conventional play operator. With this definition, play cannot be treated in isolation as an input-output relation. Consequently, methods of nonlinear feedback systems are not readily applicable for establishing well-posedness of solutions of mechanical networks incorporating this ideal play element.

Mechanical network examples including springs, dampers, masses and inerters will be discussed in conjunction with the behavioural definition of play. Some networks require approximating solutions to recover well-posedness leading to a semi-ideal definition of play, which can be shown to be well-posed for piecewise monotone continuous inputs.

Finally, we apply the play element arrived at to optimal vehicle suspension networks and illustrate its effect on ride comfort and tyre grip.