Study On The Bifurcation And Control Of Vibro-Impact Between Two Elastic Cantilever Beams

Non-smooth dynamics are widely found in our practical production and always play a vital role, therefore, more attention has been draw to the non-smooth system and lots of related research works have been done. This paper mainly focuses on one of vibro-impact models, that is derived from the collisions between two elastic cantilever beams.Firstly, based on the cantilever beam model obtained by using Extended Hamilton Principle, the vibro-impact model between two elastic cantilever beams has been established in collaboration with Newton's Law of Restitution. And with the help of numerical and experimental methods, the responses of the vibro-impact system with same cantilever beam parameters have been carefully studied, and k = p /l type periodic motions, bifurcations and chaos have been observed. Meanwhile, using the global Jacobian matrix based on the discontinuous mapping, the stability and Lyapunov exponents spectrums of the vibro-impact system can be easily obtained. Next, further analysis on the influence of system parameters (clearance, damping and non-linear item) to the responses has been done. Clearance-induced Hopf bifurcation and non-linear-item-induced jump phenomenon have been found through the analysis.Secondly, based on the simplified linear dimensionless vibro-impact system, the analytical solutions of k = 1/1 type have been thoroughly done. The existence of k = 1/1 type analytical solutions and its stability under different ratioes of two cantilever beam natural frequencies have been carefully studied. The instability of triangle area where exists co-existence of k = 1/1 type solutions and period-double bifurcation boundary have been found through the analysis. Finally, based on the energy-based control method and improved OGY method, the impact-induced chaos control has been carefully analyzed. Through the numerical simulations, the energy-based chaos control under un-changed control parameter shows its control effect in frequency band and has good robustness. For the improved OGY, it also shows its flexibility to the impact-induced chaos control, and makes the chaos orbit converge quickly to the targeted unstable periodic orbit.