COMSOL 6.4 - Dynamics of a Hopping Hoop

In this conceptual example, a hopping hoop is modeled using the Multibody Dynamics interface. The model setup is based on a problem described in Ref. 1 with slight modifications. The original problem discusses the dynamics of an ideal, massless, rigid ring with a point mass attached to its perimeter. In this example, the different types of motion are investigated, which a thin rolling ring-point mass system can exhibit. It is shown, that under certain conditions, the rolling ring can disconnect from the ground and lift into the air.

Model Definition

As shown in Figure 1, the 2D model geometry consists of a thin ring placed at one end of a fixed, flat surface.

Figure 1: Geometry of the ring with a point mass at the top, placed on a rectangular block.

Both the ring and the plane are modeled as rigid bodies. A point mass is attached to the top point of the ring. Unlike the original problem, where the ring is considered as massless, a small fraction of the ring’s total mass is uniformly distributed along its perimeter. The mass ratio of the system is γ. Thus, if m is the total mass of the system, a mass γm is assigned to the point mass and the remaining mass, (1 − γ)m, is assigned to the ring. Using the special case γ = 1, the configuration of the original problem can be recovered. Gravity is acting in the vertical direction.

The physical parameters used in the model are listed in Table 1. The fundamental model parameters are set to unity, as the exact numerical values are irrelevant for the general dynamics of the system.

Table 1: Physical properties.
Property	Symbol	Unit	Value
Radius of ring	Rc	m	1
Total mass	m	kg	1
Mass ratio	γ	1	0.75

The contact between the rolling ring and the plane is modeled using the node, and frictional forces are included with the subnode. The coefficient of friction between the ring and the plane is set to μ = 1.

The motion starts with the ring being in an unstable equilibrium, where the point mass is at the top of the ring. To nudge the ring into a rolling motion, initial linear and angular velocities are applied. A time-dependent study is run for different initial velocities.

Results and Discussion

When the ring is pushed forward with an initial velocity, it begins to roll. The mass at the top of the ring moves toward the ground plane accelerated by gravity. The translational velocity of the system along the plane increases. With the continued rotation of the ring, the point mass reaches the bottom most position and then moves up. While the point mass moves upward, the ring’s translational and rotational velocity decreases. When the point mass reaches top again, the ring velocity is at a minimum. Thus, during the revolution of the ring, the rotational speed of the system varies substantially.

Figure 2 shows the velocity of the ring for four different initial velocities. The cycloid path of the point mass and the center of gravity of the system are also plotted. For different initial velocities, the system exhibits different types of motions. For a small initial velocity (v0 = 0.1 m/s), the ring exhibits a pure rolling motion without any slipping or hopping. For a slightly increased initial velocity (v0 = 2.8 m/s), the ring rolls on the plane with some slipping. As the initial velocity is increased further (v0 = 3.1 m/s and v0 = 3.5 m/s), the ring initially rolls and then starts to hop. This is illustrated in Figure 2, where the trace of the center of gravity is colored red when the hoop is airborne. This part of the curve forms a projectile motion parabola.

Figure 2: Velocity of the ring at t = 5.2 s for four different initial velocities. The path of the point mass and center of gravity are also shown. The arrows proportional to contact forces.

Below, we look at some key features of the system for the different types of motions.

Pure Rolling

For a small initial velocity, the ring-mass system undergoes pure rolling without slipping or hopping. Figure 3 plots the variation of translational velocity of the ring center and the normalized angular velocity for two different initial velocities. As evident from the plot, the translational velocity of the center is equal to the angular velocity of the system for pure rolling. As the initial velocity increases, the velocity of the system increases.

Figure 3: Velocity as a function of rotation angle for v0 = 0.1 m/s and 2 m/s.

Figure 4: Energy as a function of rotation angle for v0 = 0.1 m/s.

In the case of pure rolling, the potential and kinetic energies of the system vary harmonically as shown in Figure 4. As the ring does not hop for low of initial velocities, the height of the center of mass of the ring does not change. Hence, the vertical position of the point mass is the only contributing factor for the harmonic variation of the energies.

The variations of normal and frictional contact forces are plotted in Figure 5. The motion of the system is driven by the horizontal force from friction, which causes the system to accelerate or decelerate. When the ring is in contact with the surface, the vertical contact force would be positive. When contact is lost, the normal contact force becomes zero. You can also see that as the initial velocity increases, the minimum contact force decreases.

Another condition for pure rolling is that the friction force cannot exceed the normal force. The ratio of friction force to normal force, also called friction force utilization factor, is plotted in Figure 6. The figure shows that for v0 = 0.1 m/s, only 38% of the available friction is utilized in the pure rolling case. This can also be interpreted as the minimum friction coefficient needed to maintain pure rolling. As the initial velocity increases, the friction force utilization factor also increases reaching a maximum value of 1, thus utilizing a larger fraction of the available friction force.

Figure 5: Contact and friction forces as a function of rotation angle for v0 = 0.1 m/s and 2 m/s.

Figure 6: Friction utilization factor as a function of rotation angle for v0 = 0.1 m/s and 2 m/s.

Rolling with slipping

As the initial velocity is increased further, the ring starts to slip. The translational and angular velocities for v0 = 2.8 m/s are plotted in Figure 7. As shown in figure, the translational and angular velocities are no longer identical for this case. This is also evident from the friction utilization factor shown in Figure 10, which levels out at 1. As shown in Figure 7, the peak velocity is decreasing with each cycle. This system could reach a pure rolling state, when enough energy has dissipated. The system energy as a function of rotation angle is plotted in Figure 8 (note that the total energy in this plot is defined as the sum of potential and kinetic energies).

Figure 7: Velocity as a function of rotation angle for v0 = 2.8 m/s.

Figure 8: Energy as a function of rotation angle for v0 = 2.8 m/s.

Figure 9: Contact and friction forces as a function of rotation angle for v0 = 2.8 m/s.

Figure 10: Friction utilization factor as a function of rotation angle for v0 = 2.8 m/s.

Rolling with Hopping

For the initial velocity of v0 = 3.1 m/s, the ring hops just before completing one revolution. The plot of translational and angular velocity (Figure 11) shows that the angular velocity is constant when the hoop is airborne. The system energy as a function of rotation angle is plotted in Figure 12. Between 350° and 390°, the potential energy of the system is larger than zero, causing the kinetic energy to be larger than the total energy. This is one consequence of the upward motion of the ring’s center.