482 S.E. Obando and P. Avitabile The novel part of the expansion and prediction at all DOF is that the response is essentially being extrapolated from the embedded information in the reduction process by using a linear combination of the mode shapes of the structure as building blocks to all possible configurations in the system. A summary of the salient results of these efficient reduced order modeling techniques is presented here. The methodology is applied to cantilever and multiple beam assemblies. 44.2 Theoretical Background The techniques summarized here rely on the use of the mode shapes (eigenvectors) of the system as building blocks of all possible dynamic behavior of the structure and utilize well-established modal modeling (SDM [25]) and reduction techniques (Guyan Condensation [26] and SEREP [27]). An overview of the relevant theory of reduced order modeling and forced response is presented with references for more in depth treatment of each topic. 44.2.1 Equations of Motion for Multiple DOF System The general equation of motion for a multiple degree of freedom system written in matrix form is ŒM1 fRxgCŒC1 ˚ : x CŒK1 fxgDfF.t/g (44.1) Assuming proportional damping, the eigensolution is obtained from ŒŒK1 ŒM1 fxgDf0g (44.2) The eigensolution yields the eigenvalues (natural frequencies) and eigenvectors (mode shapes) of the system. The eigenvectors are arranged in column fashion to form the modal matrix [U1]. Often times, only a subset of modes is included in the modal matrix to save on computation time and due to the fact that only certain modes actually contribute to the response. Exclusion of modes results in truncation error which can be serious if key modes are excluded. Truncation error will be discussed in further detail in the structural dynamic modification section. The physical system can be transformed to modal space using the modal matrix as ŒU1 T ŒM1 ŒU1 fRp1gCŒU1 T ŒC1 ŒU1 ˚ : p1 CŒU1 T ŒK1 ŒU1 fp1gDŒU1 T fF.t/g (44.3) Scaling to unit modal mass yields 2 664 : : : I1 : : : 3 775fRp1gC 2 664 : : : 2 !n : : : 3 775 ˚: p1 C 2 664 : : : 2 1 : : : 3 775fp1gD Un 1 T fF.t/g (44.4) where [I1] is the diagonal identity matrix, [ 1 2] is the diagonal natural frequency matrix and [2 !n] is the diagonal damping matrix (assuming proportional damping). More detailed information on the equation development is contained in [25]. 44.2.2 System Modeling and Mode Contribution The approach used for the prediction of the nonlinear system response uses concepts from system modeling. In order to create multi-component models, various techniques are available for the coupling of several component models into a single system model. These system modeling techniques are used to define the various states that the system will undergo when the different nonlinear contact connections occur. The system modeling can be performed in physical space, modal space, or a combination of both physical and modal space. Consider two beams that are completely independent of one another, as illustrated in Fig. 44.1.
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