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6.3.5 Generalized- Method for Rigid Body Dynamics
In the generalized- method, we evaluate forces and moments at time and the time rate of change of linear and angular momenta at time , where and may be evaluated from the spectral radius for an infinite time step, (Section 3.9↑). For second-order systems these parameters may be evaluated from  Then, the Newmark parameters are given by Accordingly, to solve numerically for over the time domain, we express (6.3.3-4) in the discretized time domain as or equivalently, Thus, the residual vector is given by where According to the Newmark integration scheme, where and .
The nonlinear system is solved using a Newton scheme that requires linearizing along increments and . Thus, The increments and are evaluated at and the iterative Newton scheme requires updates of the form where represents the Newton iteration. At each Newton iteration, the current value of is used to perform the update until convergence is achieved.
In practice, it is convenient to store and in quaternions, recognizing that where is the unit vector along and . Thus, it is which is stored in the quaternion, instead of .
In the linearization of , the contributions from the rate of change of linear momentum reduce to To evaluate the contributions from the rate of change of angular momentum , we start from . Then, it becomes necessary to transform the variables to the material frame, It follows that Then, using the relations in Section 22.214.171.124↑, it can be shown that Alternatively, we may use the discretization in (6.3.5-7) to produce where so that Therefore, the contribution to from the linear and angular momenta produces a stiffness matrix called the mass matrix,
Consider that the moments about the rigid body center of mass are produced by the forces according to where is the moment arm at , where is the moment arm in the reference configuration. Note that Thus, and