Dynamics of one-dimensional granular arrays with pre-compression

Bifurcations of periodic orbits and band zones of a one-dimensional granular array are numerically investigated in this study. A conservative two-bead system is first considered without any dissipation or external forcing. By using the Hertzian contact model, and confining the system's total energy to a certain level, changes in the in-phase periodic orbit are studied for various pre-compression levels. At a certain pre-compression level, symmetry breaking and period doubling simultaneously occur, and an asymmetric period-two orbit emerges from the in-phase periodic orbit. Floquet analysis is conducted to study the stability of the in-phase periodic solution and to detect the bifurcation location. Although the trajectory of the period-two orbit is close to the in-phase orbit at the bifurcation point, the asymmetry of the period-two orbit becomes more pronounced away from the bifurcation point. Pre-compression is found to affect the periodic orbit frequencies, which in turn result in changes in the wave propagation band zones. These changes are illustrated by studying vibration transmission through a granular chain at different frequencies to ascertain the band zone limits.