Abstract
A computational method is presented for determining backbone curves of arbitrary geometry. As an example, beam vibration is considered. The beam response is expanded into a truncated Fourier series with respect to time. After starting from the integral of action, the variational approach and the finite element method are used to formulate the non-linear eigenvalue problem. The continuation method is adopted to solve the resulting non-linear eigenvalue problem and to obtain the non-linear frequencies and modes of vibration. Numerical results for various beams are presented and compared with available results to demonstrate the accuracy and applicability of the method. Moreover, the bifurcation points on some beams' backbone curves are found and reported for the first time.
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