Investigating the internal resonance and energy exchange between the vibration modes of a cracked beam

The equations of nonlinear motion of clamped-hinged beam with an open crack were extracted and through solving them, the internal resonance in the cracked beam was studied. To this end, the crack was modeled as a torsional spring and the cracked beam was considered as two beam segments connected by a torsional spring. The equations of motion of the cracked beam were extracted considering the geometrical nonlinearity. Then, using the Galerkin’s method, these equations were changed to a set of nonlinear differential equations for vibration modes which were solved by the perturbation method. Since the mechanical energy of the beam in each mode depends on the instantaneous amplitude of vibration of the beam at the corresponding mode, so to analyze the influence of the crack on the energy exchange between the modes, the instantaneous amplitudes of the vibration modes were obtained. The results show that in the cracked beam the magnitude of the energy exchanged between the modes is less and the frequency is more than that in the intact beam. Also, by increasing the crack depth the frequency of energy exchange between the modes increases. The Vibration response obtained for the cracked beam with various amounts of the damping ratios shows that the frequency and the amplitude of energy exchange between the modes are independent of the system damping. To validate the results by the perturbation method, the equations of motions are also solved by a numerical method and the obtained results are in agreement with the results of the analytical method.