On the discretization of spatially continuous systems with quadratic and cubic nonlinearities

Methods for the study of weakly nonlinear continuous (distributed-parameter) systems are discussed. Approximate solution procedures treating reduced-order models of systems with quadratic and cubic nonlinearities obtained with the Galerkin procedure are contrasted with direct application of the method of multiple scales to the governing partial-differential equations and boundary conditions. By means of several examples and an experiment, it is shown that low-order reduced models of nonlinear continuous systems can lead to erroneous results. A method for producing reduced-order models that overcome the shortcomings of the Galerkin procedure is discussed. Treatment of these models yields results in agreement with those obtained experimentally and those obtained by directly attacking the continuous system. Convergence of the reduced order models as the order increases is also discussed.