Forced response of double beams with initial geometric imperfections and connected by a nonlinear elastic layer

This study investigates the forced vibrations of double simply supported Euler-Bernoulli beams with initial deflections connected through a distributed linear and nonlinear elastic layer. It is assumed that the upper beam is subjected to a concentrated transverse harmonic force located at its midspan. The integro-partial differential equations of motion are introduced. The Galerkin approach is utilized, and the mode shapes of uniform simply supported beams are used to establish the nonlinear governing equations of motion that incorporate quadratic and cubic nonlinearities. The harmonic balance method (HB) in conjunction with the pseudo arc-length continuation scheme are applied to obtain the amplitude-frequency curves. To assess the accuracy and validity of the proposed method and solution, some findings obtained by the suggested approach have been compared with those found through numerical integration, where a strong concordance was demonstrated. The influences of several factors such as the linear and nonlinear stiffness parameters, and the initial deflections of the beams on the steady state amplitudes have been examined. It is observed that these factors have considerable influences on the dynamic responses of the double beams. For the sake of generality and convenience, the results are displayed in dimensionless forms.