4 Implementing Experimental Substructuring in Abaqus 47 The modal properties of the decoupled system, D and ωn, D, are then found by transforming the eigensolution of the corrected mass and stiffness matrices as in Eq. (4.15), where EX, F and TS, F are matrices of the subsystem basis modeshapes at all DOF. If this continues to yield negative eigenvalues, they must simply be removed from the model. ˆKD−λCON,D ˆMD CON,D =[0] ; D = EX,F 0 0 TS,F LD CON,D ; ωn,D ="λCON,D (4.15) The decoupled model may then be imported into Abaqus as a set of modal DOF with unit mass and a stiffness equal to the square of the natural frequency. To constrain these to the analytical subcomponent in Abaqus, the compatibility condition is formed as shown in Eq. (4.16), where xFE corresponds to a set of physical DOF in the Abaqus FEM to constrain to. † TS D I qD xFE =[0] (4.16) The constraint equations are then as given in Eq. (4.17), which can be implemented in Abaqus as a set of linear equation multipoint constraints. BABQ = † TS D † TS (4.17) 4.3 Implementation The general process for implementing the TS method in Abaqus as presented in the previous section is detailed in the following steps. This procedure will require the use of MATLAB and Abaqus and assumes that work has already been completed in designing the subcomponents, measuring experimental data, and creating the necessary FEMs. This data is then imported into MATLAB where constraint DOF are identified and the TS is decoupled from the experimental subsystem. The resultant decoupled model is then imported into Abaqus and constrained to the analytical subsystem. 4.3.1 Gather Subsystem Data and Import into MATLAB The substructuring process begins with designing the experimental subsystem, the TS, and the analytical subsystem; several guidelines for creating a suitable TS are given in [5]. From a modal test of the experimental component and a FEM of the TS, the linear natural frequencies and modeshapes of both may be determined; damping is optional. This data is then imported into MATLAB, along with node locations and DOF directions for every subsystem, including the analytical component. 4.3.2 Identify Constraint DOF In order to define the substructuring constraint equations, the DOF to be constrained between each subsystem must be determined. After ensuring that all components are defined within the same global coordinate system, or performing any transformations required to make them so, this can be done by first locating node pairs between the components and then matching any common DOF directions at each pairing. Typically, the constraints are defined in terms of the experimental subsystem DOF, as in pairings are only found between it and the other subsystems. However, since the TS and analytical component are FEMs, it is possible to define the constraints at every viable node pair between them. While this may yield improved results, it can also drastically increase the number of terms in each constraint equation. The Abaqus documentation advises against this, as implementing long equations can severely degrade solver performance [6].
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