Theory Manual Version 3.4
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Subsubsection 6.3.1.3: Linearization Along Rotational Increment Up Section 6.3: Rigid Body Dynamics Subsection 6.3.3: Rigid Body Momentum Balance

### 6.3.2 General Rigid Body Motion

If the point is connected to a rigid body, its motion is given by where is the position of the rigid body center of mass and is the body's rotation tensor, which satisfies at the initial time ; here, is the distance of the point from the body's center of mass, and is the initial position. The velocity of that point is where Here, is an antisymmetric tensor with axial vector which represents the spatial angular velocity vector; similarly, is an antisymmetric tensor with axial vector (the material angular velocity), such that and We may now rewrite so that the acceleration of the point is where is the spatial angular acceleration vector, is the material angular acceleration vector. As shown below, the time discretization is performed in the material frame.
Subsubsection 6.3.1.3: Linearization Along Rotational Increment Up Section 6.3: Rigid Body Dynamics Subsection 6.3.3: Rigid Body Momentum Balance