Flexible Beam - Column System Repeat The Collision Response Of Nonlinear Analysis

The problem of structure vibro-impact is of wide significance in the field ofengineering, especially the repeated impacts will cause the high amplitude of vibration,fatigue, noise and other negative influences. The common characters of vibro-impactproblem is that the multi-freedom system should be introduced to reduce thevibro-impact system due to that the most of the vibro-impact systems are composed offully continuous bodies. However, the multi-freedom results in high dimensions and theshort collision time of the system will cause complicate dynamical responses. Meanwhile,the change of the system parameters will bring the jump of the dynamical responses anda series of non-linear phenomenon. The dynamical behaviors of the vibro-impact systemcomposed of continuous bodies need to be more comprehensive understanding.A full continuous model of the Bernoulli-Euler beam impact with elastic rod-like hasbeen represented to simulate the impacts between the valve reed and the valve seat inengineering in the present paper, the formulas of the propagation of the transient wavealong beam-rod structures are derived by the use of the eigenfunctions. The exactsolution theory analysis and numerical simulation are given for the description of thephenomenon of the repeated impacts between the beam-rod.The focus of the present paper is to analyze the effects of the system parameters, suchas the frequency of the period excitation and the mode number of beam-rod, for thelong-time dynamics behavior of the repeated impacts vibrating responses. The phasetrajectory of the beam end responses, the corresponding frequency spectrum have beendescribed and compared in detail and the phenomenon of phase locking has been foundfor different degrees of freedom. The more the degrees of freedom in system are, themore complicate the dynamics responses of the system will be. The displacementresponses on the end of the beam are sensitive to the frequency of the period excitationstrongly. Some tiny change will cause the rich non-linear dynamic behaviors, such asjump phenomenon, dynamic bifurcation, periodic window, period doubling bifurcation,similar chaotic character et. al. The frequency regions of corresponding non-linearphenomenon mentioned in above have been obtained, by studying the non-lineardynamical behavior during the repeated impacts between the beam and elastic rod-like.The conclusion obtained in the present paper can offer certain value for reference inengineering application.