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Euler's Equations for Rigid Body Rotations

Key Takeaways

  • A rigid body is one in which the distance between two internal points remains unchanged when force is applied to it.

  • Rigid body motion can be expressed in two coordinate frames - the space-fixed inertial frame and the body-fixed non-inertial frame.

  • Except for a few exceptional cases, such as torque-free rigid body and integrable cases solved by Euler, Lagrange, and Kovalevskaya, solutions are yet to be evaluated for Euler’s equations.  

 Satellite

A satellite is an example of a rigid body

You may not realize it, but rigid body motion is something you probably encounter on a daily basis. For example, the dumbbell that you use for workouts is a rigid body. Pyramids, planets, and satellites are also rigid bodies. Rigid bodies have six degrees of freedom for movement and their motion can be translational, rotational, or both.

In mechanics, rigid body motion is mathematically described using Euler’s equations. Euler’s equations for rigid body motion include three non-linear equations coupled with differential equations. Solving Euler’s equations is a challenging task and engineers have yet to find a complete general analytical solution.

In this article, we will explore rigid body motion and look at Euler’s equations as a way to express this motion.

Rigid Body Motion

A rigid body is one in which the distance between two internal points remains unchanged when force is applied to it. In terms of the deformation aspect, a body that does not change its shape under the influence of force is a rigid body. When considering rigid bodies as the collection of different particles of mass held in places by massless bonds of length, the particles are unaffected by vibrations, stress, and strain. Particles in a rigid body do not have any internal degrees of freedom, and the distance between any two particles remains fixed all the time. This approach to describing a rigid body helps in expressing the dynamics as the cumulative sum over the particles. In ordinary solids, dynamics can be expressed by integrals over the continuous mass distribution. 

Two Reference Frames for Expressing Rigid Body Motion

Rigid body motion can be expressed in two coordinate frames. The frame of reference of rigid body motion can be in:

  1. Space-fixed inertial frame - The reference frame in which Newton’s second law holds. Unless the rigid body is spherical, it is difficult to get far with the inertial frame, as the inertia seen in this coordinate system varies with time. It is hard to define rigid body motion in the inertial frame, especially when it is rotating. 
  2. Body fixed non-inertial frame - The inertia tensor is a known value and is a constant. It is simpler to calculate the equations of motion of the rigid body in a body-fixed principal frame.

Euler’s Equations for Rigid Body Motion

In this section, we will express the rotational motion of a rigid body in a body-fixed frame, ignoring the translational motion for simplicity.

According to Newton’s second law, the external torque N can be expressed as the following, where L is the angular momentum:

 Space-fixed frame equation

The above equation is in the space-fixed inertial frame and needs to be transformed into the body-fixed frame. Transformed, the equation can be written as:

Space-fixed frame to body-fixed frame transformation

The body axis is chosen to be the principal axis such that: 

Angular momentum equation

Ii is the principal moments of inertia about principal axis ‘ωi’ and i is the angular velocity about the principal axis  ‘i’.

In the body-fixed coordinate frame, the equations of motion of a rigid body can be expressed as:

Euler’s equation matrix form and expansion

The components in the body-fixed axes can be given by: 

Euler’s equation for rigid body motion

The above equations represent Euler's equations for the rigid body under applied external torque N in the body-fixed reference frame.

Euler's equations of rigid body rotation are hard to solve, and engineers are still trying to find solutions. Except for a few exceptional cases, such as torque-free rigid body and integrable cases solved by Euler, Lagrange, and Kovalevskaya, solutions are yet to be evaluated. 

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