Impulse Based Substructuring Unravelled; Simulation and Coupling of Structural Dynamics in the Time Domain

Abstract

One way of deriving the dynamics of a structure, is by combining the dynamics of its substructures. This concept is named ’Dynamic Substructuring’ and it allows us to cope with the increasing complexity of models by dividing them into substructures and deriving their structural dynamics independently. This allows an improvement in computational efficiency. Substructuring in the frequency domain is well established nowadays, but it is not per definition best suited for simulations containing impact-like load cases. Impulse Based Substructuring (IBS) has recently been proposed allowing analysis of the high-frequency dynamics induced by these load cases more efficiently than the so-called Frequency Based Substructuring (FBS). Unfortunately IBS is not yet as mature as its frequency based counterpart. Performing a stable substructuring operation without non-physical side-effects using experimentally obtained models is at least as extensive as when done in the frequency domain. This research is performed in order to make IBS a worthy alternative to FBS. The focus of this research is twofold. First, methods on how to obtain a structure’s dynamics using its Impulse Response Functions (IRFs) are discussed. It is derived how structural dynamics can be obtained by the convolution product between the IRF and force loading history. It is discussed how this convolution product can be discretised and it is shown that an algorithm exists which assumes piecewise linear behaviour for both the IRF and the force loading history. This results in only a third order error in the obtained response compared to the response obtained by the original convolution. Obtaining these IRFs is a challenge on its own. IRFs can be obtained either numerically, analytically or experimentally. It is shown how the IRFs of a multiple Degree of Freedom system is derived using Modal Superposition and how this relates to obtaining the system of IRFs numerically using two Newmark time integration methods. The errors made when obtaining IRFs experimentally are discussed and it is shown what effect they have on the simulated dynamics of the structure for varying load cases. Since solving the convolution product for lengthy load cases becomes computational extensive, techniques to enhance computational performance are discussed. Among those a matrix recurrence procedure for modal contributions is proposed. Secondly, the research focusses on the coupling procedure itself. It is explained how the convolution product is expanded to satisfy equilibrium and compatibility between the coupled substructures. It is shown that the main challenge is to accurately determine the forces acting on the interfaces between the substructures such that compatibility is maintained. Three coupling methods are discussed. The first method amounts an analytical procedure using the Laplace domain to obtain the interface forces. Secondly, the classical discrete coupling method is discussed, which satisfies compatibility explicitly to obtain the interface force every time step. Finally an inverse IRF filter approach is proposed. This approach uses the predicted uncoupled behaviour of the system to obtain the required interface forces. Next, it is discussed how the contribution of the interface forces in the convolution product relates to the contribution of the excitation on the coupled structural dynamics. It is shown how the contributions of the interface forces are constantly compensating the uncoupled structural dynamics induced by solely the excitation, such that their combined contributions show the coupled structural dynamics. This process is very prone to errors in the IRFs. Different effects as a result of these errors cause unstable and incomplete coupling behaviour. These effects are discussed and eventually a summary is given on which criteria an IRF should satisfy in order to guarantee stable and clean substructuring. Finally, the classical discrete approach and the inverse IRF filter approach have been tested on a case consisting of the coupling of two numerical models of a linear bar. It was seen that the used IRFs are required to be causal in order to result in a stable substructuring procedure. Furthermore it is seen that the inverse IRF filter method seems to underestimate the interface forces resulting in incomplete coupling.