2 1 (2 ) 0 c OI I I IO I IO T T T M C M (9) Taking the eigenvector to be the one for which 1 U and, using the fact for 1 U one has [6] 2 1 O \ \ \ \ T T j j j j j M C (10) Therefore, substituting Eq.10 into eq.9 one gets 2 1 0 Oc \ \O TM (11) from where it follows, adding the subscript j for specificity, that 2 1 Oc U I I O j j T j j j M (12) On the premise that only two tests are available the derivative of the eigenvalue has to be estimated with the forward difference and one has 2 O O U # I ' I O j j j T j j j M (13) Numerical Examinations This section presents numerical results illustrating the performance of RBN and contrasts the performance with the sensitivity based solution. The structure selected is a 16-DOF chain system with a mass distribution [1 2 1 2 .. etc] and a uniform stiffness equal to 5000, in some consistent set of units. The mass change distribution consists of three equal masses located at coordinates {4 8 and 13}, with the magnitude varied to achieve different eigenvalue shifts. Sensors are placed at coordinates {4,8,13 and 16}. The damping matrix is defined by prescribing 2% modal damping and subsequently adding external dashpots at DOF #1 and #2, with constants such that the diagonal entries at these two DOF are increased by a factor of 10. Table#1 lists the weighted modal collinearity index (mpcw) for the first 8 modal pairs [7], as well as the poles, the magnitude of the poles, and the damping ratios, defined in standard fashion as the real component of the pole divided by its magnitude. For interest the table also shows the exact undamped natural frequencies, which are not equal to the magnitude of the poles because the system is not classically damped. Table 1. Some parameters of the 16-DOF structure used in the numerical evaluations Mode # (pair) Pole Absolute Value of Pole (rad/sec) Exact undamped frequency (rad/sec) % Damping Ratio mpcw 1 -0.184+ 5.443i 5.446 5.440 3.38 99.94 2 -0.932+16.374i 16.401 16.247 5.68 97.16 3 -1.997+27.227i 27.300 26.847 7.32 84.30 4 -3.236+37.482i 37.622 37.082 8.60 60.53 5 -3.995+46.342i 46.514 46.761 8.59 42.44 6 -3.289+54.558i 54.657 55.613 6.02 48.27 7 -2.346+62.442i 62.486 63.186 3.75 72.44 8 -1.686+68.397i 68.418 68.641 2.46 92.00 395
RkJQdWJsaXNoZXIy MTMzNzEzMQ==