29 Static Torsional Stiffness from Dynamic Measurements Using Impedance Modeling Technique 311 where ŒHn is the Compliance and/or the FRF matrix for component n, it includes all the points of interest on any given component n. The system matrixŒHs is generated simply by constructing a matrix of all the components. The component matrices are simply put into a system matrix whose size is equal to the summation of the sizes of the component matrices. The components are arranged along the diagonal of the system matrix similar to what was done in the lumped mass modeling, ŒHs D 2 66 64 ŒH1 0 0 : : : 0 0 ŒH2 0 : : : 0 0 : : : : : : : : : 0 0 0 0 : : : ŒHn 3 77 75 (29.13) The Dynamic Stiffness Matrix is the inverse of the Frequency Response Function matrix as mentioned in lumped mass modeling or, ŒKs mxm DŒHs 1 mxm ŒHs C1 mxm (29.14) The components FRF matrix ŒHn may be partially measured and as a result the inverse of the system matrix ŒHs may not exist. In fact, ŒHs is normally ill conditioned and, in general, a pseudo inverse solution can be used. The pseudo inverse will generate a system dynamic stiffness matrix that can be used predict the response of the modified system only at the measured points of interest. The measured FRFs require driving-point measurement at all connection points and at points where external forces might act on the system model. Cross measurement are required between the connection/external force points and response points of interest. The primary reason for choosing this matrix formulation is that it is very simple to develop the modification matrix where the measurement DOF corresponds to the row or column space of the Dynamic Stiffness matrix. This modification matrix is also the same as the matrix used in the Lumped Modeling and Modal Modeling processes. The modification dynamic stiffness matrixŒ Kdyn in terms of ŒM , ŒC andŒK matrices is, Œ Kdyn D ! 2Œ M Cj!Œ C CŒ K (29.15) where !is the frequency measured in rad=s. The modified dynamic stiffness matrixŒKsmod is equal to system dynamic stiffness matrix plus the modification dynamic stiffness matrix or: ŒKsmod DŒ Ks CŒ Kdyn (29.16) and the modified Compliance/Frequency Response Function matrix ŒHmod is the inverse or the pseudo inverse of the Modified Dynamic Stiffness Matrix. ŒHmod DŒKsmod 1 ŒKsmod C1 (29.17) This is important, because the modification matrix will be common between the Impedance and Modal Modeling cases, which means that it is simple to program systems with both analytical and experimentally based components. The system matrices ŒH will always be square and the system matrix will be generated by placing the component matrices along the diagonal. As a result, the matrix size of the system matrix will be equal to the sum of all the component matrices sizes. This tremendously reduces the book-keeping for the keeping track of the modifications. Hence it is relatively easy using indexing to keep track of information. The same indexing can be used in the Impedance, Model Modeling and analytical modeling scripts. This method is simple to program and on analytical datasets will estimate the modifications of the system to the numerical accuracy of the computer. For example, in Matlab simulations, the accuracy of solution is to 15 decimal places. 29.3 Experimental Validation Impedance modeling is a valuable technique for predicting the result of a structural modification to the system. The proposed method to evaluate the static torsional stiffness involves the use of the dynamic stiffness method. The technique is demonstrated to estimate the static torsional stiffness of a rectangular steel plate structure. The addition of stiffness constraints to the measured FRF data for three perturbed mass configurations will be utilized to numerically model the fixed-free configurations utilized in the static testing test-rigs.
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