An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems

Abstract

The nonlinear normal modes of a class of one-dimensional, conservative, continuous systems are examined. These are free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. During a nonlinear normal mode, the motion of an arbitrary particle of the system is expressed in terms of the motion of a certain reference point by means of a modal function. Conservation of energy is imposed to construct a partial differential equation satisfied by the modal function, which is asymptotically solved using a perturbation methodology. The stability of the detected nonlinear modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and analyzing the resulting set of linear differential equations with periodic coefficients by Floquet analysis. Applications of the general theory are given by computing the nonlinear normal modes of a simply-supported beam lying on a nonlinear elastic foundation, and of a cantilever beam possessing geometric nonlinearities.