Summary
This research extends the Enriched Multiple Scales method to nonlinear partial differential equations (PDEs), illustrating the procedure using a beam with mid-plane stretching. Enriched multiple scales was originally formulated for predicting the periodic response and stability of weakly and, in some cases, strongly nonlinear non-autonomous ordinary differential equations (ODEs). It combines multiple time dilatation and asymptotic expansions found in the conventional Multiple Scales (MS) method with a continuation, or homotopy, of the original equation’s governing operator using an embedding parameter, commonly denoted by p. The central principle of the method, as applied to ODEs, requires that the undamped linear problem homotopic to the nonlinear problem (recovered by setting p = 0) be chosen to respond at the forcing frequency (or a sub- or super-harmonic thereof), leading to secular term cancellation using the excitation frequency. For ODEs, establishing the homotopy is straight-forward since the natural frequency derives from mass-like and stiffness-like coefficients. In contrast, such a simple procedure cannot be employed for PDEs where the natural frequency is dependent not only on the coefficients found in the field equation, but also the explicit form of the imposed boundary conditions. To surmount this difficulty, we chose herein to omit the embedding parameter from the boundary conditions, and instead introduce an additional stiffness parameter in the linear problem, which is ultimately determined as part of the solution procedure. It is this stiffness parameter which guarantees the linear problem responds at the forcing frequency or a sub- or super-harmonic thereof. We demonstrate the procedure for a nonlinear beam vibration problem with mid-plane stretching and hinged-hinged boundary conditions. Additionally, we develop a local stability analysis and apply it to the periodic solutions predicted for the nonlinear beam. Comparisons to finite element simulations document strong agreement with results obtained using the enriched multiple scales procedure, even for forcing resulting in strongly nonlinear response.
