228 S. Vettori et al. amplitude of the virtual force by the vector norm of the actual excitation force vector, as shown in Eq. (23.6). |Fv (ω)| =.|F1 (ω)| 2 +|F2 (ω)| 2 (23.6) The virtual acceleration is then computed as in Eq. (23.8) taking into account that Eq. (23.7), which is obtained stating the equivalence between the complex power of the virtual input Fv and the one of the actual inputs F1 andF2, can be also written substituting acceleration to velocity in case of harmonic vibrations. |Fv (ω)|Vv (ω) = |F1 (ω)|V1 (ω) +|F2 (ω)|V2 (ω) (23.7) Yv (ω) = | F1 (ω)|Y1 (ω) +|F2 (ω)|Y2 (ω) |Fv (ω)| (23.8) For what concerns the virtual force computation, in terms of both amplitude and phase, it is possible to state that, in case of symmetric excitation, the virtual force weighted by its own amplitude is equal to the sum of the two applied forces, each of them weighted by their respective amplitudes. The resulting formula for the virtual force in case of symmetric excitation is described by Eq. (23.9). Fv (ω) = | F1 (ω)|F1 (ω) +|F2 (ω)|F2 (ω) |Fv (ω)| (23.9) When the excitation is not symmetric and a generic phase lag ϕ between the inputs is used, a phase correction, whose mathematical proof is not reported here, must be introduced to bring the delayed input F2 to have the same phase of the reference one in order to be able to sum them up in the Fv computation. Therefore, Eqs. (23.10) and (23.11) are the generic expressions for the virtual force and acceleration that can be applied when anyϕ between the inputs is used. The introduced phase correction turns into a summation when the excitation is symmetric and a subtraction when it is antisymmetric. Fv (ω) = | F1 (ω)|F1 (ω) +e−jϕ|F2 (ω)|F2 (ω) |Fv (ω)| (23.10) Yv (ω) = | F1 (ω)|Y1 (ω) +e−jϕ|F2 (ω)|Y2 (ω) |Fv (ω)| (23.11) After its computation, the virtual accelerationYv (ω) is appended to the vector of the measured responses {Y (ω)}, and a new response vector of No+1 elements, expressed in Eq. (23.12), is obtained. {Y (ω)}new = ⎧⎪⎪⎨ ⎪⎪⎩ Y1 (ω) . . . YNo (ω) Yv (ω) ⎫⎪⎪⎬ ⎪⎪⎭ (23.12) For the single sweep a new data set can be considered: the applied force is only one and it is represented by Fv (ω), while the response vector is {Y (ω)}new. From this data set, a column matrix of virtual FRFs can be computed simply using Eq. (23.13). {H (ω)}v ={Y (ω)}new 1 Fv (ω) (23.13) The virtual driving point FRF is the last element of the column matrix{H (ω)}v. Once the virtual FRFs matrices are computed for each single sweep, the modal analysis process can start. From the two FRFs sets, {H(ω)}vsym from the symmetric sweep and{H (ω)}vant from the antisymmetric sweep, two Polymax stabilization diagrams [7] are defined and two groups of system poles, and therefore two different mode sets, are extracted. The system poles, and so the mode shapes, are identified correctly according to their symmetric or antisymmetric nature: symmetric
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