Realization of Three-port Spring Networks with Inerter for Effective Mechanical Control

This note is concerned with the passive network synthesis problem of one-port networks consisting of one inerter, one damper, and at most three springs. To solve the problem, a necessary and sufficient condition is first derived for the realization of a three-port resistive network containing at most three elements, utilizing graph theory and several existing results of n-port resistive networks. By extracting the damper and the inerter, a necessary and sufficient condition is obtained for the realization of one-port networks containing one damper, one inerter, and at most three springs under an assumption that the admittance of three-port networks containing only springs is well-defined. The covering networks are also presented. Based on properties of circuit topology, a realizability condition is derived for the special case when the earlier assumption does not hold. Combining the two conditions when the assumption holds or not, the final realizability condition is obtained.