18 Modal Parameter Uncertainty Quantification Using PCR Implementation with SMIT 187 H0 DR†S T D R1 R2 †1 0 0 0 S1 S2 DPpQq (18.3) In Eq. 18.3, the non-zero singular values are used to construct the possible set of block matrices, Pp DR1˙1 1/2 and Qq D˙1 1/2S1 T. Accordingly, the transition state matrix is determined as: ADPp 1H1Qq 1 (18.4) The eigenvalue decomposition of A is used to obtain the modal parameters of the system assuming that the physical system is equivalent to the numerical model. The essence of ERA based SID method is to create the impulse response from the measured vibrations, to assemble unique Hankel matrices depending on the model order specification. 18.2.1 ERA-NExT Natural Excitation Technique (NExT) algorithm uses correlation functions to identify modal parameter for output-only systems [4]. The numerical expression for the auto and cross correlation function infers the combination of impulse responses. The correlation functionRij,k between response at nodes i andj in terms of time lag k is expressed as: Rij;k D N X rD1 r i Qr j mr!d r exp. r!n rk/sin.!d rk C r/ (18.5) In Eq. 18.5, the superscript r denotes a particular mode from a total of N modes; r i is the ith ordinate of the rthmode shape; mr is the rth modal mass, Qr j is a constant associated with response at node j; r and !n r are the rth mode damping ratio and natural frequency, respectively; !d r D!n rq1 . r/2 is the rth mode damped natural frequency; and r is the phase angle associated with the rth modal response. The sequentially arranged Markov parameters Rk 2< m m substitute the impulse responses in Eq. 18.2. 18.2.2 ERA-NExT-AVG A modified NExT algorithm, known as ERA-NExT-AVG, is introduced in [13], which uses a coded average of row vectors in each Markov parameter for original ERA-NExT. The coding coefficient vector is designed to avoid eliminating the special cases for which the simple average results in zero. The element of coded correlation functions b Rk 2< m 1 is determined as: b Ri;k D 1 m m X jD1 ˛jRij;k D N X rD1 2 4 0 @ 1 m m X jD1 ˛jQr j 1 A r i mr!d r exp. r!n rk/sin.!d rk C r/ 3 5 (18.6) In Eq. 18.6, the subscript i is varied from 1 to m, and ˛j is a coding coefficient associated with the jth column of the correlation functionRk. The coded correlation functions b Rk substitute the impulse response in ERA and the rest procedures follow ERA method. The main advantage of ERA-NExT-AVG is to reduce the computational cost by reducing the size of Markov parameter while maintaining an accuracy equivalent with ERA-NExT. 18.2.3 ERA-OKID-OO The observer Kalman filter is used to remedy the lack of initial conditions in ERA method and extends its applicability to the general input/output systems [3]. The method is further extended to the identification of output only systems by separating the response to the sum of deterministic and stochastic contributions. For the structural systems experiencing non-measurable
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