Inverse Eigenvalue Problem For Mass-Spring-Inerter Systems

TL;DR

This paper addresses the inverse eigenvalue problem for mass-spring-inerter systems, providing conditions for assigning natural frequencies with specified multiplicities, extending classical results to systems involving inerters.

Contribution

It introduces a necessary and sufficient condition for natural frequency assignment in mass-spring-inerter systems with arbitrary eigenvalue multiplicities.

Findings

Derived a condition on eigenvalue multiplicities for system design.

Extended classical eigenvalue assignment results to inerter systems.

Provided a theoretical framework for inverse eigenvalue problems with inerters.

Abstract

This paper has solved the inverse eigenvalue problem for "fixed-free" mass-chain systems with inerters. It is well known that for a spring-mass system wherein the adjacent masses are linked through a spring, the natural frequency assignment can be achieved by choosing appropriate masses and spring stiffnesses if and only if the given positive eigenvalues are distinct. However, when we involve inerters, multiple eigenvalues in the assignment are allowed. In fact, arbitrarily given a set of positive real numbers, we derive a necessary and sufficient condition on the multiplicities of these numbers, which are assigned as the natural frequencies of the concerned mass-spring-inerter system.