PSI - Issue 44

Alberto Castellani et al. / Procedia Structural Integrity 44 (2023) 19–26 A.Castellani / Structural Integrity Procedia 00 (2022) 000–000

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A is a matrix 8  29, which does not change with time. Vector u z (t)

� � � � � � � � � 00, � � � � � � � � 12, � � � � � � � � 06, � � � � � � � � 01, � � � � � � � � 07, � … � �� � � � � � 012, � � �

(7)

For each instant of time, the matrix arrangement is: A common time scale has been considered for all stations. 5. Results

� , , � � � � � �

(8)

Equation (8) has been solved 1500 times, for t from zero to 15 sec, step of 0.01 seconds, unknown d 1 d 2 .. d 8 , to create u z (t) equation (1). Such function is only necessary to check the validity of the procedure, as it is shown for instances in figure 4. The required two rotation functions are computed as: ψ � � �  � � � 2 � � � � 2 � � � � (9) ψ � � �  � � � 2 � � � � � � � 2 � (10) The coefficients of "best fit", d 1 d 2 , ...d 8 ...are function of the instant of time t. For each instant of time there are more equations, 29, than unknowns, 8. The best fit is sought, the one which comes closest to satisfying all equations simultaneously. Closeness is defined in the least square sense, i.e., that to sum of the squares of the differences between the left and the right-hand side of equation 1 be minimized. Then the over-determined linear problem reduces to a usually solvable linear problem, called the linear least-square problem. See, for more detail, Castellani (2017). In fig. 4 the vertical acceleration at a station of the first circle, I06, is shown. A short time window is reproduced, of 2 seconds duration. However, the quality of fitting is similar for the entire time duration, and for the other stations of the inner circle of 200 m radius. The fitting requirement is less respected for the distant stations, at the circle of radius 2000 m. The procedure implied by Eq. 8 ignores the time correlation because the equations are solved independently instant by instant. However, as it will be seen, displacements and rotations appear to be continuous functions of time, and the time correlation present in the original data is kept in the rotations, fig 5. The procedure to obtain d 1 , d 2 ,.. is repeated instant by instant, independently from one instant to the other. An important worth of the procedure is to find out rotations as continuous functions of time.