Effect of nonlinear inertia and Duffing-type nonlinearity on the dynamic behaviour of nonlinear parametrically excited cantilever beams with tip mass

The dynamic response of parametrically excited cantilever beams with a tip mass (PECBM) is investigated in this paper. Employing the Method of Multiple Scales (MMS) and the Method of Varying Amplitudes (MVA), the frequency-response equations are obtained and the stability of the response is analysed. MVA results show that, for frequency ranges around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing-type nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations. These results are supported by numerical simulations obtained by direct integration (DI) of the equation of motion, while the MMS fails to predict the critical frequency at which this phenomenon occurs. Furthermore, based on the MVA, a criterion for determining the hardening and softening characteristics of PECBM is developed, showing that both Duffing nonlinearity and nonlinear inertia terms contribute to determining this characteristic. To verify the results, experimental measurements of the response of a PECBM are presented, showing good agreement with analytical and numerical results. It is shown that the mass added at the cantilever tip can change the characteristics of the system's dynamics, enhancing the softening behaviour of the PECBM.