Jack in Box Dynamics Simulation

Objective: Simulated the constrained dynamics of a multi-body planar system involving impacts, external forcing, and rigid-body motion using symbolic and numerical methods.

Requirements: Model a 2D system with at least two rigid bodies, rotational inertia, collisions, and external inputs. Derive equations using Euler-Lagrange mechanics, enforce impact constraints, and visualize the simulation with an animated trajectory.

Top Skills: dynamics modeling, Python (SymPy, NumPy), symbolic computation, Lagrangian mechanics, constrained systems

Overview

This ME314 final project involved simulating a jack bouncing inside a box (a planar system with six degrees of freedom, impacts, and external forcing). I modeled the system using rigid-body transformations and Euler-Lagrange equations, enforced impact constraints via energy and momentum conservation and visualized the simulation in Colab.

System Setup

• Bodies:

  • A square box with four walls (modeled as a rigid body)

  • A jack with four protruding arms ("knobs")

• Generalized Coordinates (6):

  • [x_box, y_box, θ_box, x_jack, y_jack, θ_jack]

• Modeling Approach:

  • Rigid body transformations defined jack and wall points

  • Homogeneous coordinates used for all position mappings

  • Signed distances tracked to detect contact events

Dynamics & Forces

• Lagrangian:

  • Built from kinetic and potential energy of both the box and jack

• External Forcing Applied:

  • Horizontal force: Fₓ = 300·sin(2πt/3)

  • Torque: τ = 45000·sin(2πt/3)

  • Constant lift to counteract gravity

• Impact Update Logic:

  • Contact detection when distance φ < 1e−1

  • Velocity jump handled via Lagrange multiplier

• Energy conservation enforced using the Hamiltonian

Simulation Highlights

• Jack begins at rest towards the center of the box

• External shaking causes the jack to bounce and rotate upon impact

• Impacts correctly handled through constrained momentum updates

• Accurate physics with visible bouncing, sliding, and torque-induced rotation

• Configured symbolic expressions in SymPy and used lambdify  function to evaluate dynamics over time

Key Takeaways

This project gave me experience with constrained multibody dynamics and impact modeling using conservation laws. I built confidence working with symbolic mechanics, rigid-body transforms, and dynamic visualization in Python. If I had more time, I would have explored a more complex aerospace-related system, such as simulating aircraft landing gear dynamics with a spring-mass-damper model.