Non-harmonic resonance of viscoelastic beam under time decreasing dynamic transverse load, with compressive axial load

In this article, the analysis of a simply supported viscoelastic Bernoulli beam under a dynamic transverse loading combined with a compressive axial load was discussed. The applied dynamic transverse load was considered a non-harmonic load with an exponential decreasing function over time, and the applied axial load was considered constant compressive. This study investigated the effect of a specific decreasing transverse load and introduced, formulated, and proved the phenomenon of non-harmonic resonance of the viscoelastic beam, for the first time. The stress-strain relationship of the linear viscoelastic material was expressed according to the Boltzmann integral law. The relaxation function of the viscoelastic material was defined based on two terms of the Prony series. The displacement field was approximated by the product of two functions, a known spatial function and a time function with unknown coefficients. Using the principle of virtual work, the formulation of the viscoelastic beam was extracted. Then, utilizing the Laplace transform, the unknown coefficients of the time function were calculated. Finally, the displacement field of the viscoelastic beam was expressed in the time and space domains. To validate the results, Abaqus software was used for beam modeling. In the numerical results section, the effect of different viscoelastic material parameters and different loads on the displacement of the viscoelastic beam was investigated. Also, the occurrence time of the non-harmonic resonance, and the ratio of the maximum displacement at the resonance time to the displacement at time zero were calculated