For the sake of illustration, let us now simplify the above assembled equations for the case where the boundary DoF of both substructures are ordered equally. In this case L(1) b , L (2) b and B (2) b are identity matrices, while B (1) b is minus identity. If we reorder the DoF sets such that the Lagrange multipliers are associated to component 1 and the interface displacement field is associated to component 2, we can write the assembled equations of motion according to: ⎡ ⎢⎣ Z (1) ii Z (1) ib 0 Z (1) bi Z (1) bb −I 0 −I −Y (2) bb ⎤ ⎥⎦ ⎡ ⎣ u (1) i uγ λ ⎤ ⎦ =⎡ ⎢⎣ f (1) i f (1) b −p (2) b ⎤ ⎥⎦ (16) Fromthe above equation one can clearly see the way the two systems interact. In addition to the external excitations, component 2 is excited by interface displacements fromcomponent 1 through its boundary DoF, while component 1 feels and additional forces fromcomponent 2 through its interface. The mixedassembly expression in (15) will formthe basis for the alternative “inverse free” FBSmethod, although we could also take eq. (12) as a starting point. The mixed assembly expression is somewhat more general in the sense that it can also handle frequency dependent properties in the dynamic stiffness matrix. For cases in which we deal with measured receptance FRFmatrices only, or the FEmodeled substructures can be converted to receptance FRFs (e.g. through modal synthesis), one could however also start from(12). The method for solving the assembled equations outlined in the next section is equally valid for both representations. 3 Solving the Assembled Equations As can be seen fromsection 2, the systemthat has to be solved (i.e. eq. 15) contains a dynamic stiffness part, a receptance part and a number of off-diagonal coupling terms. Fromthis systemof coupled subsystemequations we, in general, want to find a set of assembled eigenfrequencies and modes or a set of coupled FRFs. One approach would be to either transformthe mixed equations into a stiffness or flexibility form, but in order to do so we eventually need to invert the flexibility matrix of the interface of the measured system. This will then result in exactly the same sensitivity to measurement errors as found with the classical experimental substructuring techniques. Another approach would be to try to obtain the eigenfrequencies andmodes directly fromthe mixedassembled of equations. One such method was first presented Wittrick and Williams in the 1970’s [29]. With this method, which will be explained in this section, one is able to determine the natural undamped frequencies of any linearly elastic structure if either its dynamic stiffness or receptance FRFmatrix is known. 3.1 Identification of Eigenfrequencies For the readers’ general understanding, the concept and basic algorithmof the method will be briefly described in this section. The algorithmis based on a theorempostulated by Rayleigh his classic work on the Theory of Sound [25]: “If one constraint is imposed upon a linearly elastic structure, whose natural frequencies of vibration, arranged in ascending order of magnitude, are ωr, the natural frequencies ˜ωr of the constrained structure are such that: ωr < ˜ωr < ωr+1 r =1,2,3,..”. This can be reformulated as: “If one constraint is removed froma structure, the number of natural frequencies which lie below some fixed chosen frequency either remains unchanged ore increases by one”. Ed En E1 >1 >d >n ?n ?1 ?d Fig. 7: mass & springmodel S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 338
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