Impact Oscillations: Linear Theory Of Stability And Bifurcations

Corner bifurcations in non-smoothly forced impact oscillators

“…Now, using (22) and (23) we can expand (25) and (26) about ε = 0, and look at terms in successive powers of ε.…”

“…The dynamics of impacting mechanical systems has been the subject of much recent investigation, as it is known that even very simple systems can have very rich dynamics [1,2,3,4,5,11,15,16,18,22,25,30,31,32,35,36,37,39,38]. An example of such a simple system is the (so-called) single degree of freedom oscillator.…”

Dynamics of an impact oscillator near a degenerate graze

“…A simple sketch of the graph of the cubic function on the right hand side of (29) shows that this equation has three solutions m 1 , m 2 , m 3 with…”

“…The resulting bifurcation phenomena have been studied by many authors including Budd et al [14], Chin et al [17], Dankowitz et al [18], Foale et al [21], Ivanov [29], Nordmark [33], [34], Szalai et al [42], Molenaar et al [32], Zhao et al [53].…”

Bifurcation Analysis of a Rigid Impact Oscillator with Bilinear Damping

“…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.…”