Extending the Method of Multiple Scales to Strongly Nonlinear Vibration Problems

摘要

The paper considers an extension of the Multiple Scales method to strongly non-linear vibration systems. It is well known that similarly to other perturbation methods, the method of Multiple Scales produces good accuracy results when the perturbation parameter is small which is not the case in many practical applications. A two-step hybrid technique aimed at dealing with strong nonlinearities has originally been developed by Noor. It is based on combining a perturbation method with a direct variational procedure. The full potential of such a hybrid technique has not yet been fully realized in solving non-linear vibrational problems. This paper examines the hybrid technique of combining the Multiple Scale method with the Galerkin procedure. Firstly, a perturbation solution is generated assuming a perturbation parameter is small. The next step involves using the computed perturbation functions as the coordinate (or approximation) modes whose amplitudes are computed by applying Bubnov-Galerkin conditions. The effectiveness of the hybrid Multiple Scales-Galerkin procedure is demonstrated on a 2-degree-of-freedom autoparametric vibration absorber by comparing the solutions computed by the method of Multiple Scales and by the hybrid method with the numerical results.