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220.127.116.11 Newmark Integration for Rigid Body Dynamics
Let and represent consecutive time points. According to the Newmark integration scheme, the rigid body center of mass velocity and acceleration at may be expressed in terms of their values at as where and are Newmark parameters that satisfy and .
Let the rigid body rotation tensor be expressed as , and is the material rotation of the rigid body from its reference configuration. Thus, and respectively represent the rigid body rotation tensors at and . (In practice, is stored as a quaternion to facilitate the multiplication of rotation tensors.) These tensors are related by the incremental spatial rotation or material rotation from to according to Here, it should be understood that the material frame for this incremental rotation is the configuration at time , while the spatial frame is the configuration at . For rotational motion, the Newmark scheme is applied in the material frame as Then, using the relations and at and , along with (18.104.22.168-2), we may express these relations in the spatial frame as
In a nonlinear solution scheme we solve for incrementally. According to (22.214.171.124-3), the linearization of along an increment is given by so that The linearizations of and , as given in (126.96.36.199-3), along an increment requires us to first evaluate . According to Puso , where Thus,