As we watch the baseball game or table tennis game, we can see the spinning ball moving with a tricky trajectory. That is because the Magnus effect gives an additional force to the ball. As a result, the ball drops rapidly.

After evaluating the free body diagram of a spinning ball, there are three external force exerting on the ball, which are gravity, drag and force from the Magnus effect. And we can get the following equations

$$\ddot{x}=\dfrac{F_{D_x}}{m}+\dfrac{F_{L_x}}{m}$$

$$\ddot{y}=\dfrac{F_{D_y}}{m}+\dfrac{F_{L_y}}{m}$$

$$\ddot{z}=\dfrac{F_{D_z}}{m}+\dfrac{F_{L_z}}{m}-g$$

$F_{D}$ is drag force and $F_{L}$ is Magnus effect force. Their relation with some fluid attributes are shown below: $$F_{D}=\dfrac{1}{2} \rho U^{2}AC_{D}$$ $$F_{L}=\dfrac{1}{2}\rho U^2AC_{L}$$ where

• $A$ is the cross-area of the ball,
• $\rho$ is the fluid density,
• $U$ is the relative velocity between the ball and the fluid,
• $C_{D}$ is the drag coefficient which is related to the reynolds number,
• $C_{L}$ is the lift coefficient which is related to the reynolds number as well.

To solve this problem, Runge–Kutta method is used to calculate the state of variables.

1. Download the Spinball project through git.

   git clone https://github.com/zurzeit/SpinBall.git

2. Create conda environment with name "spinball" and activate it.

   conda create -n spinball python=3.9
   conda activate spinball
   

3. Install requirements with the following command:

   make setup

Use makefile to build the spinball package.

make all

Now, we are ready to use the Spinball functions!

1. import _spinball

   • Import the spinball package to solve the following problem.

2. spinball_obj = _spinball.Spinball(ball_mass, ball_radius, density_of_air, max_x_distance=inf)

   • This function will create the Spinball instance, with the input arguments of ball_mass (kg), ball_radius (m), density_of_air (rad), max_x_distance (m). The max_x_distance means that the following result will only show the ball states whose x positions are still smaller than the max_x_distance.

3. spinball_obj.RK4_main(total_time, time_inc, init_state, omega)

   • Return the states using the Runge–Kutta method to calculate the positions and the velocity in a given time interval.
     • total_time (sec): means the time length we want the ball states to be calculated.
     • time_inc (sec): means the time increment when using the Runge–Kutta method. The smaller time_inc is expected to produce a more precise result.

4. spinball.print_states()

   • This function will print the states info with the last calculated states.

5. spinball_obj.states

   • Return the list with the length of timestamps. Each list element contains a 6-element length list, which indicates the states of the moment. (e.g. [x,vx,y,vy,z,vz])

6. spinball_obj.M

   • Return the mass of the ball.

7. spinball_obj.R

   • Return the radius of the ball.

8. spinball_obj.Max_x_distance

   • Return the limit of Max_x_distance of this spinball instance.

9. spinball_obj.Omega

   • Return a list, each element is the angular velocity of each axis. (i.e. [wx, wy, wz])

10. spinball_obj.Num_state

    • Return a int, which means the number of the state variable we calculate in this class. In this case, it is 6. (state variables: [x, vx, y, vy, z, vz])

11. spinball_obj.Total_timestamp

    • Return a integer, it means the total frames of the states including the initial states(i.e. t = 0).

12. spinball_obj.Done_timestamp

    • Return a integer, it means the index of frame which the states finish the calculation. The initial states is defined with index 0.

13. spinball_obj.Final_timestamp

    • Return a integer, it means the index of the end frame. The initial states is defined with index 0.

This section is about the visualization part. In the file "viz_example.py" showcases some of the examples.

Let's try to run a simulation!

make viz

1. viz = viz_tool.viz_tool_baseball(spinball_obj,list_of_states_dict): Create the instance to prepare for the simulation.
2. viz.viz_video(): Play the simulation video.
3. viz.viz_static(): Show the final state of the simulation.

1. Magnus effect Wiki: https://en.wikipedia.org/wiki/Magnus_effect
2. Table Tennis and Physics: https://www.intechopen.com/online-first/83844
3. Chapter 6 - Rotating Cylinders, Annuli, and Spheres https://www.sciencedirect.com/science/article/pii/B9780123820983000068