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7.9.1 Virtual Work
The virtual work of a connector represents the work of external forces on the rigid body; it is given by where is the virtual velocity of the joint origin and is the virtual angular velocity of rigid body . Using the above relations for and , it reduces to The analysis thus returns the values of the reaction force and moment acting on rigid body . The virtual velocities at the joint may be evaluated from (7.9-1) as where is the skew-symmetric tensor whose dual vector is , such that for any vector . Substituting this expression into (7.9.1-2) allows us to express the virtual work in terms of the virtual velocities of the centers of mass and the virtual angular velocities of the rigid bodies,
When using time discretization in the interval , the external forces and moments may be evaluated at the intermediate time point using We solve for using Newton's method in the usual manner, by evaluating the linearization of along increments in the rigid body degrees of freedom at , Assuming that and it follows that It becomes immediately apparent that the stiffness matrix for a rigid connector is not symmetric. Therefore, rigid body dynamics should be analyzed using non-symmetric solvers.