Energy-conserving intermittent-contact motion in complex models

Some mechanical systems, that are modeled to have inelastic collisions, nonetheless possess energy-conserving intermittent-contact solutions, known as collisionless solutions. Such a solution, representing a persistent hopping or walking across a level ground, may be important for understanding animal locomotion or for designing efficient walking machines. So far, collisionless motion has been analytically studied in simple two degrees of freedom (DOF) systems, or in a system that decouples into 2-DOF subsystems in the harmonic approximation. In this paper we extend the consideration to a N-DOF system, recovering the known solutions as a special N=2 case of the general formulation. We show that in the harmonic approximation the collisionless solution is determined by the spectrum of the system. We formulate a solution existence condition, which requires the presence of at least one oscillating normal mode in the most constrained phase of the motion. The developed general framework is validated by finding a collisionless solution for a rocking motion of a biped with an armed standing torso. •Persistent passive level-ground walking is possible, despite inelastic collisions.•An oscillating mode in the constrained phase is required for collisionless motion.•The impact time equations can be expressed through the energy spectra alone.•The impact equations can always be expressed in terms of the impact phases.