# RD-E: 0902 Collision between Two Balls

Two balls are now considered in order to study the behavior of impacting spherical balls.

## Input Files

The input files used in this example include:
Collision study

## Model Description

Study on Trajectories

The balls' behavior is described using the parameters (angles and velocities) shown in Figure 1. The numerical results are compared with the analytical solution, assuming a perfect elastic rebound (coefficient of restitution is equal to 1).
Initial Values
V1
0.7m.s-1
V2
1m.s-1
$\theta$1
40°
$\theta$2
30
massball
44.514g

### Model Method

The balls and the table have the same properties, previously defined for a pool game. The dimensions of the table are 900 mm x 450 mm x 25 mm and the balls' diameter is 50.8 mm. The balls and the table are meshed with 16-node thick shell elements for using the TYPE16 Lagrangian interface.
The initial translational velocities are applied to the balls in the /INIV Engine option. Velocities are projected on the X and Y axes.

Gravity is considered for the balls (0.00981 mm.ms-2 ).

The ball/ball and balls/table contact is modeled using the TYPE16 interface (secondary nodes/main 16-node thick shells contact). The interface defining the ball/ball contact is shown in Figure 4.

### Analytical Solution

Take two balls, 1 and 2 from masses ${m}_{1}$ and ${m}_{2}$, moving in the same plane and approaching each other on a collision course using velocities ${V}_{1}$ and ${V}_{2}$, as shown in Figure 5.
Velocities are projected onto the local axes $n$ and $t$. To obtain the velocities and their direction after impact, the momentum conservation law is recorded for the two balls:(1) $-{m}_{1}{V}_{1n}+{m}_{2}{V}_{2n}={m}_{1}{V}_{1n}^{\text{'}}-{m}_{2}{V}_{2n}^{\text{'}}$
Or(2) $-{m}_{1}{V}_{1}\mathrm{sin}{\theta }_{1}+{m}_{2}{V}_{2}\mathrm{sin}{\theta }_{2}={m}_{1}{V}_{1}^{\text{'}}\mathrm{sin}{\theta }_{1}^{\text{'}}-{m}_{2}{V}_{2}^{\text{'}}\mathrm{sin}{\theta }_{2}^{\text{'}}$
The shock is presumed elastic and without friction. Maintaining the translational kinetic energy is respected as there is no rotational energy:(3) $\frac{1}{2}{m}_{1}\left({V}_{1n}^{\text{'}2}+{V}_{1t}^{\text{'}2}\right)+\frac{1}{2}{m}_{2}\left({V}_{2n}^{\text{'}2}+{V}_{2t}^{\text{'}2}\right)=\frac{1}{2}{m}_{1}\left({V}_{1n}^{2}+{V}_{1t}^{2}\right)+\frac{1}{2}{m}_{2}\left({V}_{2n}^{2}+{V}_{2t}^{2}\right)$

Such equality implies that the recovering capacity of the two balls corresponds to their tendency to deform.

This condition equals one of the elastic impacts, with no energy loss. Maintaining the system's energy gives:(4) $\left({V}_{2n}^{\text{'}}-{V}_{1n}^{\text{'}}\right)=-\left({V}_{2n}-{V}_{1n}\right)$

This relation means that the normal component of the relative velocity changes into its opposite during the elastic shock (coefficient of restitution value e is equal to the unit).

The following equations must be checked for normal components:(5) $\begin{array}{l}{V}_{2n}^{\text{'}}={V}_{1n}^{\text{'}}=-\left({V}_{2n}-{V}_{1n}\right)\\ {m}_{2}{V}_{2n}^{\text{'}}+{m}_{1}{V}_{1n}^{\text{'}}={m}_{2}{V}_{2n}+{m}_{1}{V}_{1n}\end{array}$
The equations system using V'1 and V'2 as unknowns is easily solved:(6) $\begin{array}{l}{V}_{2n}^{\text{'}}=\left(\frac{{m}_{2}-{m}_{1}}{{m}_{2}+{m}_{1}}\right){V}_{2n}+\left(\frac{2{m}_{1}}{{m}_{1}+{m}_{2}}\right){V}_{1n}\\ {V}_{1n}^{\text{'}}=\left(\frac{{m}_{1}-{m}_{2}}{{m}_{1}+{m}_{2}}\right){V}_{1n}+\left(\frac{2{m}_{2}}{{m}_{1}+{m}_{2}}\right){V}_{2n}\end{array}$

It should be noted that these relations depend upon the masses ratio.

As the balls do not suffer from velocity change in the t-direction, maintaining the tangential component of each sphere's velocity provides:(7) $\begin{array}{l}{V}_{1t}^{\text{'}}={V}_{1t}={V}_{1}\mathrm{cos}{\theta }_{1}\\ {V}_{2t}^{\text{'}}={V}_{2t}={V}_{2}\mathrm{cos}{\theta }_{2}\end{array}$
The norms of velocities after shock result from the following relations.(8) ${V}_{1}^{\text{'}}={\left({\left({V}_{1n}^{\text{'}}\right)}^{2}+{\left({V}_{1t}^{\text{'}}\right)}^{2}\right)}^{1/2}$ (9) ${V}_{2}^{\text{'}}={\left({\left({V}_{2n}^{\text{'}}\right)}^{2}+{\left({V}_{2t}^{\text{'}}\right)}^{2}\right)}^{1/2}$

In this example, balls have the same mass: m1 = m2.

Therefore, ${V}_{2}^{\text{'}}={V}_{1n}$ and ${V}_{1n}^{\text{'}}={V}_{2n}$.

The norms of the velocities are given using the following relations, depending on the initial velocities and angles. Used to determine the analytical solutions (angles and velocities after collision):(10) ${V}_{1}^{\text{'}}={\left({\left({V}_{2}\right)}^{2}{\mathrm{sin}}^{2}{\theta }_{2}+\left({V}_{1}\right){\mathrm{cos}}^{2}{\theta }_{1}\right)}^{1/2}$ (11) ${V}_{2}^{\text{'}}={\left({\left({V}_{1}\right)}^{2}{\mathrm{sin}}^{2}{\theta }_{1}+\left({V}_{2}\right){\mathrm{cos}}^{2}{\theta }_{2}\right)}^{1/2}$
By recording the projection of the velocities, directions after shock can be evaluated using relation. Used to determine the analytical solutions (angles and velocities after collision):(12) ${\theta }_{1}^{\text{'}}=\mathrm{arcsin}\left(\frac{{V}_{2}}{{V}_{1}}\mathrm{sin}{\theta }_{2}\right)$ (13) ${\theta }_{2}^{\text{'}}=\mathrm{arcsin}\left(\frac{{V}_{1}}{{V}_{2}}\mathrm{sin}{\theta }_{1}\right)$

## Results

### Numerical Results Comparison with the Analytical Solution

Figure 6 shows the trajectories of the balls' center point obtained using numerical simulation before and after collision.
For given initial values of ${V}_{1}$, ${V}_{2}$, $\theta$1 and $\theta$2, simulation results are reported in Table 1.
Table 1. Comparison of Results for After Collision
Numerical Results Analytical Solution
$\theta$ 1' 42.27° 44.72°
$\theta$ 2' 26.75° 26.48°
V 1' 0.731 m/s .731 m/s
V 2' 0.969 m/s 0.977 m/s

### Conclusion

The simulation corroborates with the analytical solution. The 16-node thick shells are fully-integrated elements without hourglass energy. This modeling provides a good transmission of momentum. However, the TYPE16 interface does not take into account the quadratic surface on the secondary side (ball 2), due to the node to thick shell contact. Accurate results are obtained for a collision without penetrating the quadratic surface of the secondary side in order to confirm impact between the spherical bodies.

A fine mesh could improve the results.