= + = + y C x D u x A x B u & (3) where ( ) q,q x & = is the vector of the states. The modal data are now the eigenvalues and eigenvectors of the A matrix in equation 3. Usually, some outputs points need to be defined on the multibody model on the locations where the sensors on the actual structure were positioned. The relation between the states and the outputs is expressed by the C matrix in equation 3 hence, by multiplying the modal matrix for the C matrix, the mode shapes can be projected from the generalized coordinates to the output points, allowing a direct comparison with the experimental results. Being Ψ ~ the modal matrix obtained from the multibody model and referred to the generalized coordinates, the modal vectors referred to the output points are: Ψ = ×Ψ ~ ~ C m (4) Given a set of uncertain parameters δof the multibody model that need to be updated, the numerical modal set can be expressed as: ( ) ( ) ( ) { } N M N m m C Ψ ~ , C ω ~ D ~ × ∈ = ∈ δ δ δ m (5) By properly comparing equation 1 with equation 5, the correlation between the experimental and numerical models can be evaluated. To compare the mode shapes, the well-known Modal Assurance Criterion is applied: [ ] [ ]k 1,N i 1,N ψ ψ ψ ~ ψ ~ ψ ψ ~ MAC m k H k i H i 2 k H i i,k = = = (6) Where N is the number of modes identified from the experimental results and Nm the number of calculated mode shapes. By analyzing the MAC matrix, the pairs of modes with the higher correlation are identified and the errors between the corresponding natural frequencies are evaluated as follows: , k 1,N ω ω ~ ω k = − = k k k ε (7) In this study, the problem of identifying the model parameter values δ that give the best fit with the experimental results set D is formulated as an optimization problem. Due to the non-linear relation between the parameters and the modal response of the model, many local extrema could exist for the chosen objective function. To prevent the convergence toward one of those local points, gradient-based algorithm should be avoided; by using evolutionary-based algorithms, on the contrary, the robustness of the solution can be guaranteed. Depending on the combination of the correlation indices in equations (6) and (7), different optimization problems can be defined. Initially, a single-objective optimization problem is defined, by using the objective function: ( ) ( ) ( ) ∑ = + ⋅ − = N i i i w w MAC δ J 1 2 1 1 1 ε (8) In equation (8), i identifies the pairs of more correlated modes in term of MAC values and w1 and w2 are the weights for the two indices. The solution of a single-objective optimization problem is relatively fast, but the main disadvantage is that it strongly depends on the values assigned to the weights. The weights should be selected with respect to the adequacy of the numerical model and the accuracy of the experimental data obtained. Hence, their selection cannot be made a priori since modeling and measurement errors cannot be easily estimated and their selection must rely on the analyst’s experience [4]. The definition of the parameter identification as a multi-objective problem eliminates the need of arbitrary weighting factors. Another advantage is that all admissible solutions in the parameter space are obtained which constitute model trade-off in fitting the different modal properties. The solution of the optimization problem results in a Pareto front, which is a collection of points in the design space that represent a limit beyond which the design cannot be improved without causing degradation in one of the other objectives [9]. Points belonging to the Pareto frontier are usually called dominating solutions; the concept of Pareto dominance of point A over point B can be defined as follow: ( ) ( ) ( ) ( ) ( ) ( ) F A F B F A F B A j j j i i i P < ∩ ∃ ≤ ⇔ ∀ : : Bf (9) 351
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