Assignment of natural frequencies by an added mass and one or more springs

Abstract

The problem of assigning natural frequencies to a multi-degree-of-freedom undamped system by an added mass connected by one or more springs is addressed. The added mass and stiffnesses are determined using receptances from the original system. The modifications required to assign a single natural frequency may be obtained by the non-unique solution of a polynomial equation. If more than one frequency is to be assigned, then a system of non-linear multivariate polynomial equations must be solved. Such a modification involves not only an added mass and one or more stiffness terms, but also an added coordinate. The paper presents a methodology, using Groebner bases for the solution of the multivariate polynomials, together with examples of natural frequency assignment. Realistic modifications are found to be bounded within certain frequency ranges. The effect of the modification on the natural frequencies not assigned and the antiresonances is explained.