Dynamics of a harmonically excited impact damper: Bifurcations and chaotic motion

On the approximate solution of a piecewise nonlinear oscillator under super-harmonic resonance

“…The four constants 1 A , 1 B , 2 A , 2 B , and two crossing times 0 t and 1 t in equations (4) and (6) can be numerically determined by implementing the following set of matching conditions:…”

“…Inserting equation (9) into equation (2) and then solving the resultant equations leads to the following standard form of the equations governing ) (t a and ) (t θ :…”

“…The dynamics of piecewise linear systems has been the subject of many studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. However, piecewise linear models may not always be well representative of some physical systems.…”

Periodic Solutions, Bifurcations, Chaos and Vibration Quenching in Impact Damper

“…Bapat and Sankar (8) , Popplewell and Liao (9) showed the optimum vibration quenching condition, assuming that the system had only this type of solution. On the other hand, Masri (10) , Popplewell et al (11) , Sung and Yu (12) showed the method to obtain a solution which had any amount of collisions per period and the stability analysis of such a solutions. However, their methods are approximate and complicated, and the solution obtained had at most five collisions per period (11) .…”

“…The main reason is the methods used to obtain solutions. Moreover, in the recent research on impact system, period doubling bifurcation (12) , Hopf bifurcation (13) and chaos generation by grazing bifurcation (14) were reported from the viewpoint of the bifurcation theory and chaos. However, as the Lyapunov exponents of the impact damper system have not been calculated, the research on chaos and grazing bifurcation that generate in the impact system is not enough.…”