For the first motion stage (free fall), the following equations govern the motion of the particle:

(2–6)

(2–7)

(2–8)

where y is the center of the particle with respect to the wall and g is the gravitational acceleration. The particle velocity v and position y are given by

(2–9)

(2–10)

When the sphere's center position is equal to its radius, the free fall stage ends and the second motion stage begins. The particle-wall collision is treated using the linear spring-dashpot model such that force balance on the particle during contact is given by

(2–11)

(2–12)

(2–13)

where and are the system natural frequency and the damping ratio, respectively. The damping ratio is determined directly by the relation shown in the figure below. Refer to Rocky DEM Technical Manual to see this relation formulation.

Figure 2.16: Relation between the damping ratio and the restitution coefficient e.

The normal spring coefficient is and is the particle mass. The initial particle velocity is obtained when the particle center position is equal to its radius. The velocity and position of the particle center during contact, regarding an underdamped system with , are given by

(2–14)

(2–15)

where is the damped natural frequency.

The third and last motion stage correspond to the ball rebound, which begins when the particle center position is equal to its radius. The force balance on the particle results in the following

(2–16)

(2–17)

(2–18)

Therefore, the sphere's velocity v and center position y are given by

(2–19)

(2–20)