The spring-mass model is often used to describe bouncing gaits. Although at first inspection the mechanical system appears simple, the solution to the motion cannot be derived easily. An analytical solution would provide a fast and intuitive method to determine the kinetic and kinematics of the centre of mass of terrestrial animals during over-ground steady state locomotion. Here, an analytical...

Abstract. We investigate the dynamics of a simple pendulum coupled to a horizontal mass-spring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass-spring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system...

Here, • m > 0 is the mass. • k > 0 is the spring constant and the spring force −ku(t) is modeled to be proportional to u(t) (i.e. Hooke’s law). The minus sign shows that it is always a restoring force which pulls the mass back towards the equilibrium. • γ ≥ 0 is the damping constant. The damping force may come from the medium (e.g. air, water, or honey), the spring or other sources. It has a mi...