168 B. Yin and H. Gavin For ith(i D1;2; : : : n) component, the constrained optimization problem is expressed as min Qa.i/ J.i/ D 1 2Q a.i/TKQa.i/ (17.19) such that Qa.i/ 1 D Ny .i/ 1 Qa.i/ m D Ny.i/ m where Qa.i/ D ŒQa .i/ 1 I Qa .i/ 2 I I Qa .i/ m ; a.i/ D ŒNy .i/ 1 I Ny .i/ m . Lagrange multipliers can be applied to modify the objective function through the addition of terms describing the constraints. The augmented objective function is then expressed as, J.i/ A D 1 2Q a.i/TKQa.i/ C .i/T .AQa.i/ a.i/ / (17.20) The optimization problem can be written in terms of the augmented objective function as, max min Qa.i/ J.i/ A (17.21) such that j 08j. The following conditions define the optimal design variables such that the constraints are satisfied. @J .i/ A @Qa.i/ D0 @J .i/ A @ .i/ D0 (17.22) The optimal principal components at every floor Qa.i/ D ŒQa .i/ 1 I Qa .i/ 2 I I Qa .i/ m ; i D .1;2; ; n/ obtained from this interpolation are then used to recover the base shear force, which is expressed as follows: fB D m X jD1 M.jj/ n X iD1 Qa.i/ j (17.23) The method presented above is based on one dimensional motion. Several minor changes are necessary if we consider the two orthogonal translational motion and torsional motion. First, output measurements from available sensors are decoupled into x; ytranslation and rotation; then these are decomposed into principal components respectively using SSA method. Second, principal components for different directions are assembled together as coupled. Third, the coupled principal components are directly used for the acceleration interpolation of each floor without other manipulations. In order to numerically verify the proposed algorithm, a series of simple numerical models were constructed in which the nonlinearities resulting from impact or friction were incorporated in the isolation-level. Figure 17.5 shows one of the numerical examples conducted to numerically verify the proposed methodology. The SSA technique plays the role of an effective filter. In the proposed approach, there is no presumed parametric model to fit measured responses. The prior information of excitation is not necessary. Moreover, the SSA based method accompanied with principal components interpolation is not only able to handle linear system with full sensors and limited sensors, but also nonlinear system with full sensors and limited sensors. 17.3 Numerical Simulations In this section, a number of numerical simulations are conducted to verify the proposed SSA technique and principal components interpolation on several MDOF mass-spring-damper models with base isolation. A series of earthquake ground accelerations are used as nonstationary random excitation, which are modeled as independent, enveloped, filtered white-noise process. A fixed-step fourth-order Runge-Kutta solver is used to obtain the time series of transient system responses with a sampling frequency of 200 Hz.
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