Realization of a Special Class of Admittances with One Damper and One Inerter for Mechanical Control

In this note, we investigate the realization problem of a special class of positive-real admittances, which is common in vehicle suspension designs. The number of inerters and dampers is restricted to one in each case and the number of the springs is arbitrary. To solve the problem, we first convert a previous result by

to a more direct form. A necessary and sufficient condition for realizability is then derived and explicit circuit arrangements are provided by assuming that the three-port network consisting of only springs after extracting the damper and the inerter has a well-defined impedance. To remove the assumption on the existence of a well-defined impedance, a condition is established on the topological property of the
$n$
-port network without a well-defined impedance to obtain an equivalent class of such networks so that the realizability condition is derived with realization. By combining the conditions with and without a well-defined impedance, the final realization result is obtained.