202 C. Zang et al. the measured data in the frequency range of interest. The model updating problem is essentially an optimization, where the prediction error is given as g=min x WiR(x) 2 2, where R(x)={fe}−{fa(x)}, xl ≤x ≤xu, (20.4) g is the objective function, Wi is the weighting matrix, and R is the residual vector. The vector x represents the design parameters that have specified upper and lower bounds. fe and fa denote vectors of the test and predicted dynamic properties, respectively. Using the first-order method [18], an unconstrained objection function outlined can be formulated from Eq. (20.4) as Q(x,q)=g/g0+ n ∑i=1 Px(xi)+q m1 ∑i=1 Pf (fi)+ m2 ∑i=1 Ph(hi)+ m3 ∑i=1 Pw(wi) (20.5) Where Qdenotes a dimensionless and unconstrained objective function, g0 represents the reference prediction error selected from the current group of design parameter sets. Px are penalties applied to the design variables, i.e. the Young’s modulus and density of the structure and Pf , Ph, Pw are penalties based on the state variables, for instance, natural frequencies and mode shapes. The constraint satisfaction is controlled by a response surface parameter, q. The penalty functions are defined as Pg(gi)= gi gi +αi 2λ (20.6) Where λis an integer and αi denotes a tolerance number. To solve Eq. (20.5), an iterative method is used. At each optimization iteration, j, the design parameters are given by the vector x(j). A search direction vector, d(j), is calculated, and the estimates of the design parameters are obtained as x(j+1) =x(j) +s jd (j) (20.7) where the line search parameter, sj, corresponds to the minimum value of Qin the direction of d(j). This line search can be solved using a combination of a golden-section algorithm and a local quadratic fitting technique. The search parameter sj is usually limited to the range given by 0≤sj ≤ smax 100 s∗j (20.8) where s∗j is the largest possible step size for the line search at the current iteration, j, and smax is the maximum line search step size. 20.2.3 Joint Modeling The components are updated and validated with the test data in the first step. The second step is to model the joints and update the joints parameters for the assembly of the valid components using the measurement data tested from the assembled structure. In this paper, thin layer elements are employed to model the joints. For simplicity, the thin layer elements are often formulated using a linear elastic constitutive relation, which generally has the form ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ σxx σyy σzz σxy σyz σxz ⎫ ⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎭ = ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ c11 c12 c13 c14 c15 c16 c22 c23 c24 c25 c26 c33 c34 c35 c36 c44 c45 c46 c55 c56 sym c66 ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ εxx εyy εzz εxy εyz εxz ⎫ ⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎭ (20.9) Where σand εrepresent the stress and strain respectively, and ci j, i, j =1,...,6, represents the stiffness at each degree of freedom. The behavior of the joint is governed mainly by the normal and shear stiffness, whilst the coupling terms have a secondary effect [16]. Hence the coupling between the normal and shear stiffness will be neglected. Thus, the constitutive relation further simplifies to
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