94 M. El-Kafafy et al. Fig. 9.2 The different steps of the automatic tracking procedure of the drivetrain modes [14] with b SYoYo .!k/ the auto power spectra, b SYoYref .!k/ the cross-power spectra calculated between the different channels and the channels taken as references, !k the circular frequency inr/s at frequency linek, Nb the number of the overlapped blocks, Yw ob .!k/ the DFT of thebth windowed time-domain data block of theoth channel, andYw Refb .!k/ the DFT of thebth windowed time-domain data block of reference channel. In the second step, the well-known operational pLSCF (Polymax) modal parameter estimator [17, 18] is applied to the calculated power spectra matrix to estimate the resonance frequencies, the damping ratios, and the mode shapes of the different physical modes in the selected frequency band. In the operational pLSCF modal parameter estimator, the following so-called right matrix-fraction model is assumed to represent the measured power spectra matrix: SYY.¨k/ D n X rD0 r k Œ“r n X rD0 r k Œ’r ! 1 2CNo Nref (9.3) where Œˇr 2 RNo Nref are the numerator matrix polynomial coefficients, Œ˛r 2 RNref Nref are the denominator matrix polynomial coefficients, n is the polynomial order, No is the number of measured outputs, and Nref is the number of the channels taken as reference signals. The operational pLSCF (Polymax) estimator uses a discrete time frequency domain model (i.e. z-domain model) with k De j!kTs (!k is the circular frequency in r/s and Ts is the sampling time). Equation (9.3) can be written for all the values of the frequency axis of the power spectra data. The unknown model coefficients [ˇr] and [˛r] are then found as the least-squares solution of these equations. Once the denominator coefficients [˛r] are determined, the poles i and the right eigenvectors Vi 2CNref 1 are retrieved from these coefficients as the eigenvalues and the eigenvectors of their companion matrix. An nth order right matrix-fraction model yields nNref poles. For a displacement output quantity, the full power spectrum can be written as a function of the modal parameters as follows [8, 18]: b SYY .!k/ D Nm X iD1 ig Ti j!k i C i g Hi j!k i C gi T i j!k i C g i Hi j!k i (9.4) with gi 2 C1 No are the so-called operational reference factors, which replace the modal participation factors in case of output-only data, i 2 CNo 1 the mode shapes, and Nm the number of the estimated modes. The operational reference factors gi is a complex function of the spectral density matrix b SYY .!k/of the random input force and the modal parameters of the structure under test, and they are simply the right eigenvectors Vi that are obtained from the companion matrix of the denominator coefficients [˛r]. After getting the poles and the right eigenvectors from the denominator coefficients, the mode shapes can be calculated from Eq. (9.4) by solving a linear least-squares problem. When the number of the reference channels Nref is not equal to the number of the outputs No a special implementation of Eq. (9.4) is needed to obtain the mode shapes.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==