28 Applying Uncertainty Quantification to Structural Systems: Parameter Reduction for Evaluating Model Complexity 251 Fig. 28.9 Two damping models, F (j) bs,d and F (j) bs,e , were fit toFbs, and both models were calculated using the (a) piecewise linear polynomials Fks,a when j =a, (b) cubic polynomial Fks,b when j =b, and (c) piecewise power functions Fks,c when j =c where [M], [B], [K], and{F} are the mass matrix, damping matrix, stiffness matrix, and force vector, respectively. The mass, damping, and stiffness matrices and force vector were composed, such that M=⎡ ⎣ mu 0 0 ml ⎤ ⎦ B=⎡ ⎣ bs( ˙zr) − bs( ˙zr) − bs( ˙zr) bs( ˙zr) + bef ⎤ ⎦ K=⎡ ⎣ ks(zr) − ks(zr) − ks(zr) ks(zr) +kef(zl) ⎤ ⎦{ F}= ⎧ ⎨ ⎩ g(mu +mn) gml ⎫ ⎬ ⎭ , (28.12) where g represents the gravitational acceleration constant (i.e. 9.81 ms2 ), and mn represents the additional payload ranging between 0 and 100 kg added to the upper mass of the 2DOF system. The index (t + t) in Eq. (28.11) indicates the future time step values for the acceleration { ¨z}, velocity { ˙z}, and displacement {z} vectors, which are unknown and must be solved using numerical integration. In this study, Newmark-β numerical integration was leveraged to solve the 2DOF equation of motion in Eq. (28.11) and, therefore, the dynamic outputs of interest in Eq. (28.2). Because the initial conditions of the system are known (see Eq. (28.1)), the future time step value (t + t) for the acceleration, velocity, and displacement vectors must be calculated. Equations (28.13)–(28.15) indicate how the future acceleration, velocity, and displacement vector values are calculated as a function of the present vector values. { ¨z}t+ t =e0{ z}−e1{ ˙z}t −e2{ ¨z}t (28.13) { ˙z}t+ t ={ ˙z}t +e3{ ¨z}t +e4{ ¨z}t+ t (28.14) {z}t+ t ={z}t +{ z} (28.15)
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