Twinkling Phenomena in Snap-Through Oscillators

Oscillatory behavior in a chain of masses connected by springs with continuous but nonmonotonic spring forces can be induced under quasistatic loading. An insight into the birth of this behavior is obtained from a single mass system. A bifurcation study shows the potential for equilibrium jumps between multiple equilibria. As such, the transients occurring under quasistatic loading do not converge to the static loading case. Transients during dynamic loading show sensitivity to the loading parameters.

[1]  N. Lebovitz,et al.  Exchange of Stabilities in Autonomous Systems , 1975 .

[2]  Richard Haberman,et al.  Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems , 1979 .

[3]  Anthony N. Kounadis,et al.  Dynamic stability criteria of nonlinear elastic damped/undamped systems under step loading , 1990 .

[4]  Li,et al.  Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential. , 1985, Physical review letters.

[5]  Huw G. Davies,et al.  Nonstationary response near generic bifurcations , 1996 .

[6]  Andrej Cherkaev,et al.  Transition waves in bistable structures. II. Analytical solution: wave speed and energy dissipation , 2005 .

[7]  Lev Truskinovsky,et al.  Rate independent hysteresis in a bi-stable chain , 2002 .

[8]  Lev Truskinovsky,et al.  Mechanics of a discrete chain with bi-stable elements , 2000 .

[9]  Andrej Cherkaev,et al.  Transition waves in bistable structures , 2004 .

[10]  A. Cherkaev,et al.  The waves of damage in elastic-plastic lattices with waiting links: design and simulation , 2006 .

[11]  L. Truskinovsky,et al.  A mechanism of transformational plasticity , 2002 .

[12]  Andrej Cherkaev,et al.  Dynamics of chains with non-monotone stress–strain relations. II. Nonlinear waves and waves of phase transition , 2001 .

[13]  Philip Holmes,et al.  Strange Attractors and Chaos in Nonlinear Mechanics , 1983 .

[14]  Philip Holmes,et al.  A magnetoelastic strange attractor , 1979 .

[15]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[16]  G. J. M. Marée,et al.  Slow Passage Through a Pitchfork Bifurcation , 1996, SIAM J. Appl. Math..

[17]  A. Cherkaev,et al.  Transition waves in bistable structures. I. Delocalization of damage , 2005 .

[18]  Earl H. Dowell,et al.  Routes to escape from an energy well , 1995 .

[19]  Anil K. Bajaj,et al.  ON THE SLOW TRANSITION ACROSS INSTABILITIES IN NON-LINEAR DISSIPATIVE SYSTEMS , 1996 .

[20]  A. Bajaj,et al.  On the non-stationary passage through bifurcations in resonantly forced Hamiltonian oscillators , 1998 .

[21]  Andrej Cherkaev,et al.  Protective Structures with Waiting Links and Their Damage Evolution , 2005 .

[22]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[23]  Andrej Cherkaev,et al.  Dynamics of chains with non-monotone stress-strain relations , 2001 .

[24]  N. Lebovitz,et al.  Exchange of Stabilities in Autonomous Systems—II. Vertical Bifurcation , 1977 .

[25]  M. H. Lock,et al.  Snapping of a shallow sinusoidal arch under a step pressure load. , 1966 .