LAUSR

Abstract This paper presents a derivation of a hinged–hinged Euler–Bernoulli beam including cubic and quintic nonlinearities. Then, a three-mode Galerkin discretization technique has been utilized to generate a system of ordinary differential equations governing the temporal deflections of the first three modes of the studied beam. The pioneering work of Nayfeh and Mook (1995) has shown the absence of internal resonances among the modes of a hinged–hinged beam with cubic nonlinearities despite there are commensurable linear natural frequencies. In this work, extra quintic nonlinearities are involved to show the presence of internal resonances among such modes. Approximate solutions of the resulted system have been approached by the method multiple scales to get the modulations governing the amplitudes and phases of the first three modal temporal deflections. Their stability is investigated via Routh–Hurwitz criterion and a shaded region of unstable solutions has been plotted to conclude picture where the stable solutions are. Different response curves are plotted to explore the effects of beam parameters on the nonlinear dynamical behavior of such beam. Finally, a simulated response of the modal temporal deflections and the overall spatial–temporal deflection are portrayed to show the accurate behavior of the beam at different initial conditions.