Abstract
In this paper, a symbolic/numeric method is developed to compute nonlinear normal modes (NNMs) in conservative, two-degree-of-freedom (2-DoF) systems. Based upon the notion of NNMs, periodic motions are sought during which the two coordinates ‘vibrate-in-unison’. By parameterizing the response of one coordinate with respect to the response of the other (reference) coordinate and by imposing conservation of energy, we obtain a nonlinear, singular ordinary differential equation. Approximate solutions for these modal functions are obtained, for a given energy level, via truncated power-series expansions. The coefficients of the expansion, along with the maximum and minimum reference displacements, are then computed by (i) symbolically evaluating the singular differential equation at various (distinct) reference displacements, and then (ii) numerically solving the resulting set of nonlinear algebraic equations. Since the approximate solution inherently depends upon the order of the expansion, convergence studies must be performed in order to ensure sufficient accuracy. Note that even though the formulation presented herein is based on 2-DoF systems, the methodology is quite general and can readily be extended to higher-order discrete systems. Moreover, since it does not rely upon any ‘small-quantity’ assumptions, it can be used to investigate the dynamics of coupled, strongly nonlinear systems.
