44 Developments in the Prediction of Full Field Dynamics in the Nonlinear Forced Response of Reduced Order System Models 483 Beam A Beam B [Mn A] , [C n A] , [Kn A] , [Un A] [Mn B] , [C n B] , [Kn B] , [Un B] [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = AB nM [Mn A] [Mn B] [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = AB nC [Cn A] [Cn B] [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = AB nK [Kn A] [Kn B] [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = AB nU [Un A] [Un B] Fig. 44.1 Sample components arranged into common matrix space. The mass, stiffness, and damping matrices of each component are assembled for the combined system as well as the resulting mode shapes (Un) of the two beam structure The two beams shown in Fig. 44.1 are completely uncoupled and will respond independent of one another when excited. A system model of the uncoupled components is generated by simply writing the variables in common matrix space, as shown in the diagram. To generate a coupled system model, specific coupling terms must be introduced at the desired locations. To include the spring(s) in the system modeling, either a modal or physical approach can be employed. The modal approach involves using Structural Dynamic Modification (SDM) and Component Mode Synthesis (CMS). The physical approach involves using a physical tie matrix to couple the beams. Both approaches involve the use of a mode contribution matrix to determine the appropriate number of component modes that contribute to the system modes. For the results presented here, physical system modeling techniques were used to generate databases for the various configurations. 44.2.2.1 Physical Space System Modeling To form a physical system model, the mass and stiffness matrices of each component (A and B) are assembled in stacked form into the system mass and stiffness matrices. In physical space, these are coupled with a stiffness tie matrix; a mass tie can also be included if desired but not included in this work. MA MB fR xgC KA KB C KTIE fxgDfFg (44.5) This can be cast in a modal space representation as 2 4 2 4 h M Ai hM Bi 3 5 3 5 ˚ R pA ˚ RpB C 2 4 hK Ai hK Bi 3 5 ˚ pA ˚ pB C ŒU T Œ K ŒU ˚ pA ˚ pB Df 0g (44.6) where M and K are diagonal matrices and with the mode shapes of each component stacked as ŒU D UA UB (44.7) Equation (44.5) is a general equation of motion in physical space; Eq. (44.6) is the modal space equation used for the eigensolution.
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