throw-simulator

Perform simulation of a ball throwing

Disclaimer

All the formulas used in this project are obtained from AI (ChatGPT and Claude) and have not been verified with appropriate sources. In theory, they all might be completely wrong; however, visually the results look correct. Nevertheless, I do not recommend blindly trusting all the calculations from this project.

This project assumes that simulations are happening in a vacuum, or air resistance is negligible, as there is no account for air or any environmental resistance during calculations.

During the simulation, possible bending of the cylinder where the ball is attached is calculated; however, any vibrations or other effects on the part of the arm positioned under the motor attachment point are not calculated and are ignored.

The end result of the simulation is provided in the form of a trajectory graph for easy comparison between different input parameters.

Solution Explanation

In order to calculate the trajectory and distance traveled of the ball attached to the rod and thrown via an engine, I divided the calculations into three parts.

1 - Calculating Actual Angular Speed at the Moment of Launch

Because the motor is moving from the idle position to a set angle that is smaller than a single revolution, there are no guarantees that the speed of the motor will be its maximum possible speed at the moment of ball separation. To calculate the actual speed of the motor at the angle of separation, $\omega$ was calculated using formulas from 5 to 9. It is important to note that $\omega$ is capped at the engine's maximum speed, as it is possible to have parameters that will exceed it.

2 - Calculating Actual Ball Launch Angle

While the motor will stop and signal for the ball separation at the required angle, in reality, it will not be exactly that angle. There are two factors that need to be taken into account:

a) During the arm's movement, some centrifugal forces will affect the rod holding the ball, making it bend backward slightly. The amount of deformation is determined by the rod material and ball weight.

b) While the rod will be stopped at the angle of the engine, the actual ball is attached to the rod in the direction of the movement. As a result, the distance to the ball from the engine is slightly different and is calculated using formula 4.

In the end, the actual ball launch angle is calculated using formulas from 10 to 12.

3 - Calculation of the Ball Trajectory

After the ball launch angle is known and its position in the world is calculated for the ball, the trajectory is calculated until the ball reaches the ground, assumed at y=0. That is calculated using formulas from 13 to 17.

Lastly, the maximum travel distance is calculated for the ball using formula 18 as a simple distance check along the x-axis between the starting position where the ball was separated from the rod and when it hit the ground (y=0).

Detailed formulas for each step were acquired in comprehensive chats with the aforementioned AI, including follow-up conversations, as the AI was unable to include all details mentioned here and it was prompted to check if some specific cases could be a factor or not

Used Formulas

1. Ball radius: $$r_{ball} = \frac{d_{ball}}{2}$$
2. Mass of the rod: $$m_{rod} = \rho_{rod} \pi (\frac{d_{cylinder}}{2})^2 l_{cylinder}$$
3. Mass of the ball: $$m_{ball} = \rho_{ball} \pi r_{ball}^3 (\frac{4}{3})$$
4. Distance from the motor to the center of the ball in normal position: $$d_{ball} = \sqrt{(l_{arm} - r_{ball})^2 + (\frac{d_{cylinder}}{2} + r_{ball})^2}$$
5. Moment of inertia of the rod and the ball: $$I_{total} = \frac{1}{3} m_{rod} l_{cylinder}^2 + (\frac{2}{5} m_{ball} r_{ball}^2 + m_{ball} d_{ball}^2)$$
6. Total gravitational torque at the end angle: $$T_{gravity} = m_{rod} g (\frac{l_{arm}}{2} + (l_{cylinder} - l_{arm})) \sin(\theta_{launch}) + m_{ball} g d_{ball} \sin(\theta_{launch})$$
7. Total torque at the end angle: $$T_{total} = T_{max} - T_{gravity}$$
8. Angular acceleration: $$\alpha = \frac{T_{total}}{I_{total}}$$
9. Angular speed: $$\omega = \min(\sqrt{2 \alpha (\theta_{launch} - \theta_{start})}, \omega_{max})$$
10. Centrifugal force: $$F_{centrifugal} = m_{ball} d_{ball} \omega^2$$
11. Rod deformation: $$\delta_{rod} = \frac{F_{centrifugal} l_{cylinder}^3}{3 E_{rod} I_{total}}$$
12. Actual launch angle: $$\theta_{actual} = \theta_{launch} - \arctan(\frac{\delta_{rod}}{d_{ball}})$$
13. Ball position in world space: $$x_{ball} = x_{engine} + d_{ball} \cos(\theta_{actual})$$ $$y_{ball} = y_{engine} + d_{ball} \sin(\theta_{actual})$$
14. Initial velocity of the ball: $$v_0 = d_{ball} \omega$$
15. Components of initial velocity: $$v_{0x} = v_0 \cos(\theta_{actual} + \frac{\pi}{2})$$ $$v_{0y} = v_0 \sin(\theta_{actual} + \frac{\pi}{2})$$
16. Time of flight: $$t_{flight} = \frac{v_{0y} + \sqrt{v_{0y}^2 + 2 g y_{ball}}}{g}$$
17. Trajectory equations: $$x(t) = x_{ball} + v_{0x} t$$ $$y(t) = y_{ball} + v_{0y} t - \frac{1}{2} g t^2$$
18. Travel distance: $$d_{travel} = |x(t_{flight}) - x(0)|$$

Project details

Project is implemented using python and django framework.

Project comes with Docker file and docker-compose.yml file. In order to run the project use docker-compose.yml file.

docker-compose build

docker-compose up

You can then access project at localhost:8010