46 B. Moldenhauer et al. ˆK−λCON ˆM CON =0 ; TSM= ⎡ ⎣ EX,F 0 0 0 TS,F 0 0 0 FE,F ⎤ ⎦ L CON ; ωn,TSM= √λCON (4.7) Equations (4.1)–(4.7) define the standard procedure for implementing the TS method. This is usually done in MATLAB, as the calculations are then trivial to complete. However, this process cannot be directly implemented in Abaqus as its eigensolvers, Lanczos, subspace iteration and automatic multi-level substructuring, are designed to accept FEM matrices that are always symmetric and positive definite or positive semidefinite, for mass and stiffness respectively [6]. To decouple the TS from the experimental subsystem, it is effectively given negative mass and stiffness, making both negative definite and incompatible with the Abaqus eigensolvers. To bypass this, this work divides the simultaneous decoupling and coupling operations into two separate computations. The TS is first decoupled from the experimental subsystem in MATLAB, and the resulting modal model is imported into Abaqus and coupled to the analytical subsystem. To decouple the TS from the experimental subsystem in a standalone operation, the constraint equation matrix in Eq. (4.5) and the subsystem matrices in Eq. (4.6) are simply truncated to the terms enforcing compatibility between the TS and the experimental subsystem, as given in Eqs. (4.8) and (4.9). BD = † TS EX −I ; LD =null (BD) (4.8) ˆMD =LT D IEX 0 0 −ITS LD ; ˆKD =LT D ⎡ ⎣ \ω 2 n,EX\ 0 0 −\ω 2 n,TS\ ⎤ ⎦ LD (4.9) Typically, removing the TS from the experimental subsystem yields areas of residual negative mass and/or stiffness in the resultant decoupled system matrices, ˆMD and ˆKD, making them not positive definite or positive semidefinite, respectively, and thus incompatible with Abaqus. A method for correcting this is to essentially add a set of point masses to the system to make the mass matrix positive definite [7, 8]. While not done in the authors’ prior works, one can similarly add a set of grounded springs to the stiffness matrix to make it positive semidefinite. For the mass matrix, and similarly for the stiffness matrix, this is done by first computing its eigensolution and creating a vector corresponding to the amount of mass to add to each eigenvector according to Eq. (4.10), where λk are the eigenvalues and is a small, nonzero value. λk = 0 λk >0 −λk + λk ≤0 (4.10) This is then put into diagonal matrix, as in Eq. (4.11), where the only nonzero entries will be negative eigenvalues with a flipped sign and a numerically small amount added. ˆΛ= \ λ\ (4.11) The modeshapes corresponding to the negative eigenvalues, φk, are then collected into a matrix φMD , as in Eq. (4.12). φMD = φk · · · φn (4.12) The mass matrix is then made positive definite in Eq. (4.13), which essentially adds to the negative eigenvalues to make them equal to . ˆMD = ˆMD+φ T MD ˆ ΛφMD (4.13) Ideally, very little mass would need to be added to make the matrix positive definite. This amount can be quantified by computing the ratio of the matrix norms as given in Eq. (4.14). nratio = !! ! φ T MD ˆΛφMD!! ! !! ! ˆMD!! ! (4.14)
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