Vibrational control and resonance of a nonlinear tilted cantilever beam under multi-harmonic low and high-frequency excitations

Abstract

This paper investigates the vibrational resonance and control of the response of a nonlinear tilted cantilever beam under combined multi-harmonic low and non-resonant, high-frequency excitations. The basic system considers the first mode of vibration of a nonlinear tilted beam under parametric and direct base excitation with the frequency components locked in 2:1 ratio. Such a system is susceptible to parametric amplification of resonance, which may be detrimental for structural health in many mechanical systems. The primary objective of the research is to investigate the non-trivial effects of non-resonant high-frequency excitation on parametric amplification, vibrational resonance and chaos in the system, which, to the best of the authors’ knowledge, is not reported previously in the literature. The slow dynamic equations of the system are obtained using the method of direct partition of motion (MDPM), which is then analysed using the harmonic balance method (HBM). It has been shown that the high-frequency base excitation considerably influences the effective damping and stiffness of the system under consideration. The results of the research demonstrate that high-frequency excitation can effectively control the parametric amplification by shifting the resonance curve in the frequency axis and eliminating undesirable bifurcations by increasing the effective damping. The phenomenon of vibrational resonance has been found for specific parameter values in the presence of high-frequency base excitation. Detail parametric studies are performed to reveal the system’s rich nonlinear dynamics and bifurcations, like multiple jumps in response, dual peaks, isolated resonance response (isola), loop formation in frequency response plot, etc. The numerical results reveal that high-frequency excitation transforms chaotic responses into regular periodic responses. Direct numerical simulations are performed in MATLAB SIMULINK to validate the analytical results.