Hexagon-Simulation

Overview

In many applications, studying the rotational dynamics of irregularly shaped objects can be approximated using more familiar and symmetric shapes. However, in systems exhibiting the tennis racket effect [link to tennis racket effect], the accuracy of moment of inertia tensor calculations plays a crucial role. This dependency becomes even more pronounced when an object is highly symmetric about two or three axes.

Sensitivity of the Moment of Inertia Tensor in the Tennis Racket Effect

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Consider an object with moment of inertia tensor $I$ that has eigenvalues $\lambda_1 < \lambda_2 < \lambda_3$ corresponding to the principal axes. For a small perturbation $\delta I$, first-order perturbation theory gives the change in the eigenvalues as:

where $\psi_i$ is the eigenvector associated with $\lambda_i$.

Now, consider the relative error for the intermediate moment $\lambda_2$. Its sensitivity is characterized by the ratio:

assuming that $\lambda_3 - \lambda_2$ is small.

In the context of the tennis racket effect, Euler’s equations for rotation are:

A small miscalculation in $\lambda_2$ can therefore lead to a large error in predicting the stability of rotation about the intermediate axis. This demonstrates the high sensitivity of rotationally asymmetric objects, where the eigenvalues do not exhibit simple ratios.

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We investigate the rotational dynamics of hexagonal prisms and configurations of multiple hexagonal prisms to understand the evolution of a system with these inertial properties exhibiting the tennis racket effect.

Features

• Simulation of hexagonal prism rotations

• Visualization of the tennis racket effect in 3D

• Customizable parameters for moment of inertia

• Analysis of stability and eigenvalue sensitivity

Usage

[Add details on how to use here]

Results

The project generates visual outputs showing the rotation dynamics and key numerical results related to the moment of inertia sensitivity.

License

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