Nonlinear free vibration of an elastically-restrained beam with a point mass via the newton-harmonic balancing approach

In this paper, a new harmonic balance approach for solving the large amplitude nonlinear vibration of an elastically-restrained beam with a point mass is introduced. The coupling of Newton's method with harmonic balancing takes the advantage of reducing the deficiency and complexity of the classical harmonic balance method in dealing with the nonlinear systems. Using the approach, accurate higher-order approximate analytical expressions for period and periodic solution are established. Results in non-dimensional forms are presented graphically to illustrate the effects of the base stiffness, position and magnitude of lumped mass on the variation of oscillation period and amplitude. Comparison with published results and numerical integration solutions shows that the accuracy of this new Newton-harmonic balance (NHB) approach is excellent while some published numerical solutions for specific examples are rather unsatisfactory.