9 On Dynamic Substructuring of Systems with Localised Nonlinearities 109 The link solutions for each of the substructures are then found, these represent the effect on the state of the substructure of the interface forces due to coupling. These link solutions are found by first finding the state rate due to the interface forces as in equation 9.14 and then the link state using this state rate as in equation 9.15. ˙Y P,L n+1 =D−1 P LP n+1 ˙Y N,L n+1 =D−1 N LN n+1 (9.14) Y P,L n+1 =γ T ˙Y P,L n+1 Y N,L n+1 =γ T ˙Y N,L n+1 (9.15) Finally the global solution for each of the substructures taking into account the coupling between them is found as in equation 9.16 by summing the free and link solutions for each of the substructures. YP n+1 =Y P,F n+1 +Y P,L n+1 YN n+1 =Y N,F n+1 +Y N,L n+1 (9.16) 9.2.3 With Sub-cycling In order to ameliorate the problem of actuator smoothness in hybrid simulations, wherein the numerical integration time step is necessarily larger than the control frequency of the actuator, a set of methods exist in which the physical system is controlled at a smaller time step than the numerical system, with coupling enforced at the larger numerical time step [16]. In connection with the above described algorithm, the sub-cycling method can be implemented in the calculation of the free solution of the physical substructure. Firstly, a number of sub-cycles is decided upon, defined as the ratio of the numerical time step to the physical time step. The integration scheme with sub-cycling operates similarly as described previously except that the free solution for the physical substructure, as described by equations 9.8, 9.9, 9.10, and 9.11 above, is now replaced by the following procedure. The sub-cycling loop runs over the number of sub-cycles ss. It is initiated by the coupled solution at the previous numerical system time step. The loop ends by finding the free solution at the next numerical time step. Equation 9.17 shows the prediction step of the trapezium rule in which TP is the time step of the physical system. ˜Y P,F n+ j ss =Y P,F n+ j−1 ss +(1−γ) TP ˙Y P,F n+ j−1 ss (9.17) Equation 9.18 calculates the state rate prediction in which it is worth noting that the Lagrange multipliers from the last coupling time step are now included in the solution and that the D matrix is formed as in equation 9.10 albeit with the finer time step of the physical system used in the assembly. ˙Y P,F n+ j ss =D−1 P (FP n+ j ss −R( ˜Y P,F n+ j ss ) +LPλ(1− j ss )) (9.18) Finally, the corrected free solution is calculated as in equation 9.19. Y P,F n+ j ss = ˜Y P,F n+ j ss +γ TP ˙Y P,F n+ j ss (9.19) The sub-cycling loop then ends after ss sub-cycles whereby the n+1 prediction of the free state of the physical system is found and the coupling procedure can then be followed as described in the previous section. 9.3 Case Study The case study considered herein is a steel vehicle-like frame considered to be linear elastic, which is mounted on four nonlinear suspension structures. Computer models of the frame and suspension structures were created based on physical specimens. The frame structure model, along with the mounting points are presented in Fig. 9.1. The frame structure is considered to be linear elastic and is meshed using a combination of quadrilateral shell elements and hexahedral solid elements with a total of 45,564 DOFs. The material properties of the frame are a Youngs modulus of
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