Dynamics and Impacts of a 2-D Jack and Box system

Jyothi Swaroop | Jan 3, 2026

Code

Overview

This project models the dynamics and impacts of a Jack in a hollow box system floating without gravity. The jack(blue square) is given an initial velocity in the x-direction and the box(red square) is given a rotational velocity along its z-axis. All the impacts are assumed to be elastic impacts(i.e. energy is not lost as a result of the collision.)

The dynamics and impacts are modelled using the Lagrangian Dynamics specifically the Euler-Lagrange equations.

Setup

The state of the system has 6 variables they are :

$$q = (x_{jack}, y_{jack}, \theta_{jack},x_{box}, y_{box}, \theta_{box})$$

The steps to taken to model the system :

  • Setup Euler-Lagrange Equations.
  • Impact equations setup.
  • Impact updates.
  • Plotly Animation.

All of the programming was done using the python library sympy

Equations

Lagrangian Formulation

For any dynamics system , the Lagrangian of the system is defined as :

$$ L = T - V $$

where

  • T : Total Kinetic Energy of the system.
  • V : Total Potential Energy of the system.

Euler-Lagrange Equation

The Euler-Lagrange equations are used to solve for the dynamics of the system(i.e. $q , \dot{q}$)

$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = F $

The resulting expressions for $\ddot{q}$ are solved symbolically and lambdified into fast numerical functions for time integration.

Constraint Equations

The constraints for the jack are the equations which should represent that the box cannot move out of the box.

For the constraints:

  • Four corners of the jack
  • Four edges of the box

It is considered that only one collision can occur at a time and it will be point collision and not a surface to surface collision.

Simulation Loop

The simulator uses fixed-step RK4 for continuous dynamics. At each step:

  1. Evaluate constraint values $\phi_k $
  2. Detect an impending impact if $\phi_k $is near contact and decreasing
  3. If impact occurs, apply an impulsive update $\dot{q}^- \to \dot{q}^+ $
  4. Continue integration with updated velocities

This produces piecewise-smooth trajectories with discrete jump events.

Acknowledgements

This project was developed as a final project for the ME-314 Theory of Machines : Dynamics taught by Professor Todd Murphey.