Theory Manual Version 3.4
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Subsection 6.3.2: General Rigid Body Motion Up Section 6.3: Rigid Body Dynamics Subsection 6.3.4: Time Discretization

### 6.3.3 Rigid Body Momentum Balance

For a rigid body, the conservation of linear momentum is given by where is the mass of the rigid body, is the velocity of the center of mass, is the linear momentum, and represents the sum of external forces acting on the body. Here, is constant for a rigid body. There are typically four contributions to : Body forces (where represents the body force per mass, such as gravitational acceleration), other user-prescribed forces (which act at the center of mass), forces produced by rigid body connectors (such as revolute and prismatic joints, or contact forces), and forces produced by rigid-flexible connections (where deformable materials interface with the rigid body), in which case is evaluated from the traction over that interface, with representing the stress in the deformable material.
The conservation of angular momentum is similarly given by where is the rigid body mass moment of inertia about its center of mass, is its angular velocity, is its angular momentum, is the rigid body angular acceleration, and is the sum of moments acting on the rigid body. External moments include contributions from user-prescribed moments/torques , from rigid body connectors, where is the connector insertion relative to the rigid body center of mass, and rigid-flexible interfaces, where is the position of the interface point relative to the rigid body center of mass. Since body forces and user-prescribed forces act at the center of mass, they do not contribute to . Note that where is the mass moment of inertia about the center of mass in the reference configuration and is the rotation tensor representing the orientation of the rigid body at time , with in the reference configuration.
The virtual work statement is given by where is the virtual velocity of the center of mass and is the virtual angular velocity of the rigid body.
Subsection 6.3.2: General Rigid Body Motion Up Section 6.3: Rigid Body Dynamics Subsection 6.3.4: Time Discretization