14 Evaluation of On-Line Algebraic Modal Parameter Identification Methods 149 a11 D. 1/ 2nZ .2n/ t0 . t/2nxi .t/ a12 D 1 X kD0 . 1/2n k .2n/Š.1/Š kŠ.2n k/Š.1 k/Š Z .2n 1Ck/ t0 . t/2n kxi .t/ : : : a1n 1 D 2n 2 X kD0 . 1/2n k .2n/Š.2n 2/Š kŠ.2n k/Š.2n 2 k/Š Z .2Ck/ t0 . t/2n kxi .t/ a1n D 2n 1 X kD0 .2n/Š.2n 1/Š kŠ.2n k/Š.2n 1 k/Š s 1 k d2n k ds2n k xi .s/ b1 D 2n X kD0 . 1/2n k .2n/Š .2n/Š kŠ.2n k/Š.2n k/Š Z .k/ t0 . t/2n kxi .t/ By solving Eq. 14.11 one obtains the parameter vector as ™ DA 1BD 1 2 6 6 6 6 6 4 1 2 : : : n 1 n 3 7 7 7 7 7 5 (14.12) Then, we propose the following algebraic identifiers to estimate the coefficientsak of the characteristic polynomial without problems of singularities when the determinant Ddet(A(t)) crosses by zero: b ak D Z j k 1j Z j j ; k D1; 2;:::;2n 1 (14.13) In the above expression we can apply more integrations on the numerator and denominator to get smoother estimations but introducing slower responses. Thus, one could implement the algebraic identifiers (14.13) using only those available position measurements xi of any specific mass carriage. From the estimated coefficients b ak, one can obtain the roots of the characteristic polynomial as follows b i D b i Cj b !di ; b i Db i j b !di ; i D1;2; : : : ;n (14.14) where b i and b !di are estimates of the damping factors and damped natural frequencies of the mechanical system, respectively. Hence, the estimates of the natural frequencies and damping ratios are given by b !ni D q b 2 i Cb ! 2 di ; b i D b i q b 2 i Cb ! 2 di (14.15) Note that, this identification approach can also be extended to the case of acceleration sensors, instead of displacement measurements. Specifically, one could simply multiply Eq. 14.6 bys2 and describe an acceleration output as yi .s/ D n X jD1 ij p0;js 2 Cp1;js 3 s2 C2 j!js C!2 j (14.16) where yi(s)Ds 2x i(s) is some acceleration output variable described in complex s domain, and then follow the proposed algebraic identification methodology.
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