Restricted Complexity Network Realizations for Passive Mechanical Control

This paper considers a realization problem of restricted complexity arising in an approach to passive control of mechanical systems. This approach is based on synthesizing a positive-real admittance or impedance function using springs, dampers and inerters. This paper solves the following problem: what is the most general class of mechanical admittances which can be realized if the number of dampers and inerters is restricted to one in each case, while allowing an arbitrary number of springs and no transformers (levers)? The solution uses element extraction of the damper and inerter followed by the derivation of a necessary and sufficient condition for the one-element-kind (transformerless) realization of an associated three-port network. This involves the derivation of a necessary and sufficient condition for a third-order non-negative definite matrix to be reducible to a paramount matrix using a diagonal transformation. It is shown that the relevant class of mechanical admittances can be parametrized in terms of five circuit arrangements each containing four springs.