Natural frequency assignment for mass-chain systems with inerters

This paper studies the problem of natural frequency assignment for mass-chain systems with inerters. This is the problem to determine whether an arbitrary set of positive numbers may be assigned as the natural frequencies of a chain of n masses in which each element has fixed mass and is connected to its neighbour by a parallel combination of a spring and inerter. It is proved that mass-chain systems with inerters may have multiple natural frequencies, which is different from conventional mass-chain systems (without inerters) whose natural frequencies are always simple. It is shown that arbitrary assignment of natural frequencies including multiplicities is not possible with the choice of n inerters and n springs. In particular, it is shown that an eigenvalue of multiplicity m may occur only if n⩾2m-1. However, it is proved that n-1 inerters and n springs are necessary and sufficient to freely assign an arbitrary set of distinct positive numbers as the natural frequencies of an n-degree-of-freedom mass-chain system.