Mechanical networks comprising one damper and one inerter

This paper solves the following problem: what is the most general class of mechanical admittances (or impedances) which can be realised using one damper, one inerter and 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) realisation 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 parametrised in terms of five circuit arrangements each containing four springs.