44 B. Moldenhauer et al. substructuring within Dassault Systèmes®SIMULIA™ Abaqus finite element analysis software suite. The following sections provide a brief overview of experimental substructuring and the mathematical basis for the TS method, an overview of the process required to implement this in Abaqus, and numerical and experimental test cases that evaluate the accuracy and practicality of this method relative to what is typically done in MATLAB. 4.2 Background and Theory As the scale and complexity of a design increases, its structural dynamics become progressively more challenging to simulate. An unreasonable amount of computational power or time may be necessary to directly solve complicated FEMs, and intricate physical systems can require an unrealistic number of sensors to experimentally model. These difficulties may be avoided by dividing the structure into subcomponents that can be analyzed individually and combined to reproduce the total system dynamics. The techniques that are employed to reassemble these subcomponent models are referred to as dynamic substructuring, and the term experimental substructuring is used when the subcomponents are a combination of analytical and experimental models [3]. The work detailed herein is an implementation of the TS method, which is a procedure for efficiently and accurately performing experimental substructuring. In more traditional substructuring approaches, two components are assembled by defining constraints between the physical DOF at their interface. This is extremely difficult to accomplish with experimental subcomponents as it requires rotational DOF to be measured at the interface; a tedious and error prone task. The TS method circumvents this by introducing a third component, referred to as the TS, which exists in the experimental subsystem at its interface. Instead of directly assembling the experimental and analytical subsystems, a FEM of the TS is constrained such that it is decoupled from the experimental subsystem and coupled to the analytical subsystem. The TS then acts as a distributed interface that satisfies the system constraints in terms of its own dynamics. This softens the constraints, making them less sensitive to experimental noise and less likely to induce locking, which is an unrealistic increase in stiffness at the interface [2]. This process is demonstrated in Fig. 4.1, in which the experimental subsystem is a cantilever beam with a perpendicular beam segment attached to the tip, the TS is a FEM of that attached segment, the analytical subsystem is a FEM of a free-free beam, and the resulting model is simply a longer cantilever beam. In this scenario the TS is decoupled from the experimental subsystem, removing its physical presence from the beam tip while preserving the effects of a mass-loaded interface in the remaining cantilever. This provides an improved basis for coupling to the analytical subsystem, yielding a more accurate final assembled system. A more comprehensive discussion of the TS method and its strengths and weaknesses can be found in [2, 4, 5]. Mathematically, this process is based on enforcing compatibility of displacements and force equilibrium at the interface between the subsystems. This is typically done in either a primal or dual formulation if the displacements or forces, respectively, are used as unknowns. In this work, constraints are defined in a primal formulation and used to assemble Fig. 4.1 Visual demonstration of the TS method. The tip section of the cantilever beam is effectively removed and replaced by a second beam, yielding a longer cantilever
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