A new analytical formulation for investigating in modern engineering for the harmonic distortion occurring at large vibration amplitudes of clamped-clamped beams: Explicit Solutions

This work is a contribution to the numerical modeling and computer implementation of geometrically non-linear vibrations of the thin beams. The spatial distribution over the beam span of the harmonic distortion induced by large vibration amplitudes has been examined, and an analytical investigation has been elaborated to describe this aspect of non-linear vibration. This model allowed us to obtain the explicit analytical expressions for the non-linear response, including the contributions of various spatial functions, associated to the first and higher time harmonics. In the present work, devoted to this particular but practically very important aspect of non-linear vibration, a review is made of some important experimental and theoretical works on the subject. The model is based on an expansion of the transverse displacement function as a sum of series, each series being the product of a given harmonic time function by a series of chosen basic spatial functions, multiplied by the unknown contribution coefficients, to be determined. The explicit analytical expressions obtained for the function contributions corresponding to the first time harmonic are identical to those obtained in the previous works above assuming harmonic motion, which allows one to consider that the present model is a generalization of the previous ones. Also, the results of the model presented here, corresponding to the higher harmonics, are in a very close agreement with each other. They are also in a qualitative agreement with previously published numerical results, based on the hierarchical finite element method.