Nonlinear forced response of non-uniform beams carrying point masses using the harmonic balance method with an arc-length continuation scheme

This work examines the forced vibrations of a non-uniform Euler-Bernoulli beam carrying point masses and subjected to a concentrated transverse harmonic force. By the means of Newton’s second law of motion, the integro-partial differential equation of motion is derived. The Galerkin approach is applied, and the normalized mode shapes of the non-uniform base beam are used to derive the nonlinear governing equation of motion that contains cubic and quintic nonlinearities. The method of harmonic balance method (HB) in conjunction with the pseudo arc-length continuation scheme are employed to obtain the amplitude-frequency curves. Non-uniform beams with two different boundary conditions are considered in this study; cantilever and simply supported beams. To examine the accuracy and validity of the proposed method and solution, some results obtained by the suggested approach have been compared with those found through numerical integration, where a good agreement was achieved. The influences of several factors such as the non-uniformity parameter, the magnitude of the point masses, and the boundary conditions on the steady state amplitude have been examined. It is observed that these parameters have significant effects on the dynamic behavior of the non-uniform beams. For generality and convenience, the results are displayed in dimensionless forms.