3 Using Hybrid Modal Substructuring with a Complex Transmission Simulator to Model an Electrodynamic Shaker 25 Fig. 3.1 Transmission simulator method schematic and subsystems completed. Essentially, from an initial experimental modal model of the shaker and half cube, the half cube will be removed and replaced by a FEM of the half cube with the cantilever beam, yielding a hybrid model predicting the response of the complete assembly. While the response of a cantilever beam is not necessarily complex, the dynamics of the half cube are. This work seeks to assess if the half cube response can be preserved through the decoupling and coupling process, and if that will affect how accurately the beam is added to the system. This will be done by comparing the substructuring model to experimental truth data of the half cube and the beam on the shaker. 3.2.3 Mathematical Basis To combine the subsystems, the equations of motion for each are first formulated in terms of modal coordinates, as given in Eq. (3.1) [7], where the damping is left out for brevity. In this equation, ωis frequency, I is an appropriately dimensioned identity matrix, η are generalized modal coordinates vectors, [ωn] are diagonal matrices of subsystem natural frequencies, φ T are transposed subsystem mode shape matrices, and Fare vectors of applied forces. The terms representing the TS are negative to signify that this subsystem is being removed, while the experimental and analytical FEM subsystems are being coupled. −ω 2 ⎡ ⎣ IEX 0 0 0 −ITS 0 0 0 IAN ⎤ ⎦ ⎡ ⎣ ηEX ηTS ηAN ⎤ ⎦+ ⎡ ⎢⎢ ⎢⎣ ω 2 n,EX 0 0 0 −ω 2 n,TS 0 0 0 ω 2 n,AN ⎤ ⎥⎥ ⎥⎦ ⎡ ⎣ ηEX ηTS ηAN ⎤ ⎦= ⎡ ⎣ T EX 0 0 0 − T TS 0 0 0 T AN ⎤ ⎦ ⎡ ⎣ FEX FTS FAN ⎤ ⎦ (3.1)
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