Tennis Racket Effect

Ruiqi Feng¹, Cuiqin Bai

Introduction

TRE is the phenomenon where a rigid body's unstable rotation around its second inertial axis. The dynamics equation of a free rigid body is

When takes an intermediate value and , the rigid body can be observed to flip while to rotate about the z-axis. A typical image of the angular velocity in the z axis when the TRE does or does not happen is shown as below:

Numerical Simulation using COMSOL

Essentially the free rotation of a rigid body can be described using a system of 3 ODEs as is shown above. There are mature numerical methods which are not hard to implement. But we are using COMSOL anyways. The version of COMSOL used in this tutorial is 5.6.

Physics Field Setting

• Open COMSOL to create a new project. Select Model Wizard and then 3D since we are interested in the object's 3-dimensional motion.
• Select Multibody Dynamics as the physics field. Choose Time Dependent for Study.

Rigid Body Setting

• Include the geometric bodies you want in Component 1/Geometry. We will take the T-shaped wrench as an example.
  • Add to cylinders and set their sizes.
• Choose Form Union in Component 1/Geometry and press Build All to form multiple geometric bodies into one.
  • Check how they look in the window on the right.
  • 
• Select the material. Since it is a rigid body, it is only the density that matters. Moreover, if the density distribution is homogeneous, even the density will not have any effect on the motion.

Simulation Settings

• Under Component 1/Multibody Dynamics, add Rigid Domain and add the geometric body we just created into this domain.

  • Click on Multibody Dynamics
  
  • Add Rigid Domain in the right-click menu
  

• Choose Study 1/Step 1: Time Dependent and set the simulation step size and simulation duration. We are trivially solving an ODE so no mesh grid is required.

  • Select in the menu on the left
  
  • Adjust in the setting
  

• Set initial conditions in Multibody Dynamics. We need to ensure the the angular momentum along the second inertia axis is far greater then the other axes.

  • Set Initial Values to Locally Defined in Rigid Domain
  
  • Set the initial conditions in the submenu
  

• Start simulation and examine the results in Results. You can export the data for further analysis.

Experiment using Phyphox

• Download Phyphox App on your smartphone
• Select to record the angular velocity
• Throw your smartphone in the air, trying your best to make it rotate around its second inertia axis.
• Catch your phone to avoid extra reparation expenses.
• Export the data and analyze it.

Additional Materials

• My data analysis data can be found on github
• The COMSOL project I created can be also found on github
• Another nice demo of TRE simulation can be found in 示例, to which our project refers.

Footnotes

1. Please contact me if you would like to leave comments or spotted mistakes here. Email: rqfeng20@fudan.edu.cn ↩