Bouncing dynamics of a spring

• We propose a model for the dynamics of a deformable bouncing object.
• We obtain the equation of motion and simulate numerically the bouncing dynamics.
• Bifurcation diagrams illustrated the influence of the deformations in the dynamics.
• The bouncing threshold curve is analyzed theoretically and numerically.

We consider the dynamics of a deformable object bouncing on an oscillating plate and we propose to model its deformations. For this purpose, we use a spring linked to a damper. Elastic properties and viscous effects are taken into account. From the bouncing spring equations of motion, we emphasize the relevant parameters of the dynamics. We discuss the range of parameters in which elastic deformations do not influence the bouncing dynamics of this object and compare this behavior with the bouncing ball dynamics. By calculating the spring bouncing threshold, we evidence the effect of resonance and prove that elastic properties can make the bounce easier. This effect is for example encountered in the case of bouncing droplets. We also consider bifurcation diagrams in order to describe the consequences of a dependence on the frequency. Finally, hysteresis in the dynamics is presented.