Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts

Bifurcation Analysis of a Rigid Impact Oscillator with Bilinear Damping

“…Compared to the research on the rigid impact oscillators with linear damping in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], the bilinear damping was considered and the mapping method was adopted to investigate the dynamics of the rigid impact oscillator with bilinear damping focusing on grazing bifurcations, which can be utilized for analyzing this type of nonsmooth dynamical systems. Compared to the research on piecewise linear dampers in [19][20][21][22][23][24][25][26][27][28][29], numerical methods including bifurcation diagrams, phase-portraits, and basin of attraction were carried out to show the influence of the forcing frequency and amplitude on the dynamics of the system.…”

“…In the following, the mapping methods proposed in [8,17,27,[39][40][41] will be used to study the stability and the bifurcation of the impact oscillator (3). In order to describe the segments of the impact oscillator, we divide the phase space into four subspaces Π 1 , Π 2 , Π 3 , and Π 4 , where…”

“…In general, such impact can be studied using the rigid impact oscillator, which is modelled using the coefficient of restitution rule assuming the instantaneous reversal of velocity for the collision body. The rigid impact oscillator is a nonsmooth dynamical system which can exhibit complex dynamical behavior, so the stability and the bifurcation of the rigid impact oscillator have received great attention; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In [1], Shaw and Holmes observed the chaotic and long period motions in a class of periodically forced linear oscillators with impacts.…”

“…In [7], Luo and Xie observed Hopf bifurcations in a vibroimpact system at two strong resonance cases. Luo and Chen [8] investigated the mapping dynamics of periodic motions for a class of rigid impact oscillators. Dankowicz and Zhao [9,10] studied different bifurcation scenarios associated with the switching between impacting and nonimpacting motions of impact actuators near grazing.…”

The Nonsmooth Vibration of a Relative Rotation System with Backlash and Dry Friction

“…In [16], the idea of mapping structure was used to investigate a periodic forced piecewise linear system. In addition, the investigations by Nordmark [17], Błazejczyk et al [18], Blazejczyk-Okolewska et al [19], Luo et al [20], di Bernardo et al [21,22], and Luo et al [23][24][25][26] can also provide us with a plenty of meaningful conclusions.…”