Brief Overview

This project simulates a planer multi-body dynamics of a jack in the box. In the simulation, external forces are applied to the box. While the jack is falling freely, it will then experience some collisions with the box’s walls.

Video demo

System Schematic

In the figure shown above, the system is a planar model, which means the two objects can only move in x and y directions. There are 9 frames in total:

• w: the world frame (reference frame)
• b: the box frame at the center of the box
• j: the jack frame at the center of the box
• bs1,bs2,bs3,bs4: the frames of the four corners of the box
• j1,j2,j3,j4: the frames of the 4 coners of the jack

System Parameters

• Mass of box: 20
• Mass of jack: 1
• Length of box: 10
• Length of jack: 1

Euler-Lagrange Equations

1. Calculate the inertia tensor I for both the box and dice.
2. Compute body angular velocities (Vb) for each object using respective world to box center {gw_b} and world to jack center {gw_j} frames.
3. Calculate rotational kinetic energy (KE) of the system.

Lagrangian and Euler-Lagrange Equations

1. Compute potential energy (V) using y-values of {gw_bc} and {gw_dc}.
2. Generate the Lagrangian L = KE - V.
3. Use the Lagrangian to calculate Euler-Lagrange (EL) equations.

System Configuration and Impacts

• The system configuration q includes x-coordinate, y-coordinate, and theta rotation for each object.
• External forces: Vertical force (Fyb) and horizontal force (Fthetab) on the box.

Impact Calculation

• Define impacts as corners of the die coming into contact with a side of the box.
• Calculate impacts using transforms.
• Check for impacts along the y-axis of the box corner frame.
• 16 total impacts to check (4 sides of the box for each of the 4 die corners).

Simulation Results

• Trajectory generation with impacts.
• Plot and animate the simulation.
• The die bounces repeatedly up and down, gradually decreasing in overall y-value.
• Proper impacts on all sides of the box.
• Applied forces on the box (Fyb and Fthetab) may slightly alter box dynamics.

Conclusion

The simulation successfully captures the dynamics of the box and dice system, considering impacts and external forces. Further analysis may be needed to fine-tune the applied forces on the box for more accurate simulation results.