A computational algebraic geometry technique for determining nonlinear normal modes of structural systems

Abstract

A computational algebraic geometry technique is developed for determining nonlinear normal modes (NNMs) of multi-degree-of-freedom (MDOF) nonlinear dynamic systems. Specifically, resorting to computational algebraic geometry concepts and tools, such as Gröbner basis, it is shown that the complete solution set of a nonlinear algebraic system of coupled multivariate polynomial equations is determined exactly, even for a relatively large number of unknowns. Herein, these unknowns represent expansion coefficients for approximating NNMs of nonlinear dynamic systems. In this regard, the exact nature of the determined coefficient vector, which is one the significant advantages of the technique, implies increased confidence in approximating NNMs more accurately by utilizing higher-order expansions with additional terms. Several numerical examples referring to various MDOF nonlinear dynamic systems are considered for demonstrating the efficacy and potential of the solution technique. Also, note that although the technique is developed herein by relying on the classical Shaw–Pierre definition of NNMs, this is not necessarily a restriction of the methodology. In fact, the proposed computational algebraic geometry solution framework can be potentially used, in a rather straightforward manner, in conjunction with alternative NNM definitions and problem formulations requiring the solution of nonlinear algebraic equations of polynomial form.