23 Development and Validation of Data Processing Techniques for Aircraft Ground Vibration Testing 229 mode shapes of the system are well detected only by the symmetric sweep, while the antisymmetric mode shapes are well identified only by the antisymmetric sweep. In order to reconstruct a single and complete mode set for the system, symmetric and antisymmetric modes, respectively extracted from{H (ω)}vsym and{H (ω)}vant , must be grouped together. Mode shapes that are part of the final mode set are vectors, as the one expressed by Eq. (23.14), of No+1 components, since in the response vector that has been used to compute the FRFs, the additional component Yv (ω) related to the virtual response is present. {Ψ}r = ⎧⎪⎪⎨ ⎪⎪⎩ Ψ1r . . . ΨNor Ψvr ⎫⎪⎪⎬ ⎪⎪⎭ (23.14) The Ψvr component of the mode shapes, i.e. their last element, must be deleted because it represents a non physical DOF which, depending on the scaling scheme that is used, can also have influence on the values of the modal displacements related to the other DOFs. After the last element of the mode shapes has been deleted, the resulting mode shapes are scaled again and all the modal parameters are correctly defined. 23.4 Validation Cases The VDP method has been at first applied to a three DOFs discrete system in order to investigate the possible results it could lead to and to compare them to the theoretical ones computable through a simple eigenvalue problem. Afterwards, several GVT data sets have been post-processed with the purpose of validating the proposed approach. Each application allowed to identify different specific advantages of the method. The main achieved results are described in the following sections. 23.4.1 Three DOFs Discrete System The three DOFs discrete system depicted in Fig. 23.2a, symmetric in terms of masses, stiffness and damping (m1 =m2 = m3 =0.05kg, k1 =k2 =k3 =k4 =10,000N/m, c1 =c2 =c3 =c4 =0.2 kg/s), has been numerically simulated applying two forces F1 and F3 to masses 1 and 3, with the intention of simulating the symmetric position that the shakers usually have during GVT. The two forces have been simulated as two linear sine sweeps at 1 Hz/s, symmetric during a first run and antisymmetric during the second. a b Fig. 23.2 Three DOFs system. (a) Three DOFs system scheme. (b) Actual and virtual driving points FRFs
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