Seismic Response Analysis of a Semi-active-controlled Base ...

the total deformation x is divided between both elements leading to the following equations:kMaxwellF = Fdamper = Fspringand x = xdamper+ xspring⇒ F& = kMaxwellx&− F (1)cMaxwellConsidering the particular case when Maxwell dampers are only installed at the base isolation layer,the state equation of the system can be restated as:with⎡MO 0⎤⎡1⎤A X&+ BX =⎢ ⎥⎢⎥&⎢O O 0⎥⎢0⎥x& g(2)⎢⎣0 0 0⎥⎦⎢⎣0⎥⎦⎡OA =⎢⎢M⎢⎣0− kMCmaxwell× δT10⎤0⎥⎥,1⎥⎦⎡⎢−M⎢B = ⎢ O⎢⎢0⎣OK004δkc1maxwellmaxwell⎤⎥⎥⎥⎥⎥⎦and⎡ x&⎤X =⎢⎢x ,⎢ ⎥ ⎥⎥ ⎣F⎦⎡1⎤⎢ ⎥⎢0δ ⎥1= (3)⎢M⎥⎢ ⎥⎣0⎦where M, C and K are the mass, damping, and stiffness matrices, respectively; x and & x& grelative displacement vector and the ground acceleration.Direct Identification Methodare theRather than identifying the natural periods and damping ratios of the system, it is possible to directlyestimate the structural parameters by defining a measure of the error e between outputs time historiesof the original system y and the estimated system y m . As story masses are supposed to be correctlyestimated at the design stage, and only inter story stiffness k and damping coefficients c are updated.During strong ground motions, primary concern is given to the maximum value of drift andacceleration; thus the error e is defined to give much importance to the approximation of peak values.outputtime11e = ∑ ∑ ( yij− ym,ij ) yij(4)n n mean( y )outputni= 1 time ij j=1jConsequently the values of k and c are obtained by minimizing e. This measure can also be used tocompare this identification method with an ARX model and the N4SID method. The natural periodsand damping ratios corresponding to the direct identification are given in appendix.Updated ParametersThe updated structural parameters are also obtained by using the ARX and N4SID results. The updateis realized by optimizing K and C matrices, so that the natural periods and damping ratios calculatedfrom the complex eigenvalues analysis (Eq. 5) of the A-B system of Eq. (3) are similar to the results ofthe ARX and the N4SID identifications.ΩA + B = 0 , n = Ωnω , n = − Re( Ω n ) Ω nUpdated stiffness and damping coefficients are depicted on Figs. 8 and 9.nξ (5)1031