58 W. Witteveen qn Dqn-1 ChPqn-1 C h2 2 Œ.1-2“/ Rqn-1 C2“Rqn Pqn D Pqn-1 ChŒ.1-”/ Rqn-1 C”Rqn (5.7) ’2 1 3 ;0 I ” D 1 2’ 2 I “ D .1 ’/2 4 (5.8) The derivative of the state and the velocity with respect to the acceleration are needed later and can be given as @qn @Rqn D“h2 DkD @Pqn @Rqn D h” DkV (5.9) One of the key ideas of the HHT method is to evaluate a “damped” equation of motion in the form of M N .qi / Rqi n C.1C’/ i Pi i 1;i Ci PiC1 i;iC1 Qi n ’ i Pi i 1;i Ci PiC1 i;iC1 Qi n-1 D 0 DW ei (5.10) Note that the equation looks bit different in case of i D1 and i DN. Due to numeric the final solution will not lead to a zero sum but to a so called residuum which should be close to zero. The symbol of the residuum for the i-th body is ei. Under the general assumption of the presence of state depended inertia forces, a Newton iteration can be performed. The relevant equations are: @.ei/ @Rqi Rqi D ei Rqi;n D Rqi;n C Rqi;n (5.11) The Jacobian can be given as @.ei/ @Rqi D 1 1C’ M N .qi / C“h 2 0 BB @ 1 1C’ @ M N .qi/ Rqi @qi @ŒQ.qi; Pqi / @qi 1 CC A ”h2 @ŒQ.qi; Pqi / @Pqi D J Ni (5.12) At the end of the “body iteration” the residuum of all N equations of motion (5.10) are below a certain, user defined limit. Note that all N bodies can be executed by its own. Therefore a massively parallel implementation is thinkable. 5.2.2 Constraint Update After the “body iteration” (5.1) and (5.4) ((5.5)) are, in general, not fulfilled. The corresponding errorscj andpj are defined as Cj Dcj (5.13) and Pj C N j qj T œj Dpj (5.14) In a next step, the Lagrange multipliers and the constraint forces have to be updated, so that the error is decreasing. This leads to a update equation in the form of: 2 66 64 @pj @Pj @pj @œj @cj @Pj @cj @œj 3 77 75 Pj œj D pj cj (5.15)
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