31 Online Systems Parameters Identification for Structural Monitoring Using Algebraic Techniques 253 with x.t/ D‰q.t/ (31.4) where !i and i denote the natural frequencies and damping ratios associated to the i-th vibration mode, respectively, and‰ is the so-called 6 6 modal matrix given by ‰D 2 66 66 64 11 12 : : : 15 16 21 22 : : : 25 26 : : : : : : : : : : : : : : : 51 52 : : : 55 56 61 62 : : : 56 66 3 77 77 75 (31.5) In notation of Mikusin´ski operational calculus [5, 6], this modal model is then described as s2 C2 i!is C! 2 i qi.s/ Dp0;i Cp1;is C 1if1 C 2if2 C C 6if6 (31.6) where po,i are constants depending on the system initial conditions at the time t0 0. From (31.3) and (31.5), one then obtains that xi.s/ D n XjD1 ij p0;j Cp1;js s2 C2 j!js C!2 j (31.7) Therefore, the physical displacements xi are given by pc.s/ xi.s/ Dr0;i Cr1;is C Cr2n 2;is 2n 2 Cr 2n 1;is 2n 1 (31.8) with pc.s/ Ds 2n Ca 2n 1s 2n 1 C Ca 1s Ca0 (31.9) where pc(s) is the characteristic polynomial of the mechanical system and ri,j are constants which can be easily calculated by using the values of the system initial conditions as well as the modal matrix components ij. Naturally, the roots of the characteristic polynomial (31.9) provide the damping factors and damped natural frequencies, and hence the most descriptive information about the structure and its status. 31.3 Online Structural Monitoring The proposed online algebraic monitoring scheme shown in Fig. 31.2 works in conjunction with the building like structure shown in Fig. 31.1 for the illustrative cases to be considered in this work. The ARM (Advanced Risc Machine) takes samples of the acceleration of the sixth floor (in the horizontal axis direction) of the structure at a precisely fixed sample rate of 1Khz, and then, those samples are sent to a standard PC running under Windows 7® and Matlab®/Simulink to finally perform the online identification scheme. We perform an on-line algebraic identification approach to estimate the modal parameters of the mechanical system through the real-time estimation of the coefficients ak of the system’s characteristic polynomial as reported in [5, 6] using only acceleration measurements of any floor of the structure. The application of the online algebraic identification scheme is performed using cumulative trapezoidal numerical integration with fixed sampling time of 1 ms. The algebraic identification scheme applied here is described on detail in [4] where is shown that by solving the algebraic equation (31.9) also detailed in [4–7] one obtains the parameter vector as: ™ DA 1BD 1 2 66 66 64 1 2 : : : n 1 n 3 77 77 75 (31.10)
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