In between the experimental and numerical damping

In this paper, following a two-stage methodology, the differential quadrature element model of frame structures is updated using the particle swarm inspired multi-elitist artificial bee colony algorithm. In the first stage, the mass and stiffness matrices are updated using the experimental natural frequencies. Then, having the updated mass and stiffness matrices, the structural damping matrix is updated to minimize the error between the experimental and numerical damping ratios. To model the welded joints of the frame, some geometrical and elemental parameters are utilized to model the flexibility of these joints. To investigate whether the selected design parameters are proper ones, the global and local sensitivities of the natural frequencies to the design parameters are evaluated. The optimum values of the selected design parameters are obtained by updating the DQE model using the experimental modal parameters obtained through modal testing. Considering the robustness of the evolutionary algorithms in the model updating practice, the particle swarm inspired multi-elitist artificial bee colony (PS-MEABC) algorithm is used to solve the problem. This approach benefits from the advantages of both the particle swarm optimization and the artificial bee colony algorithms. The accuracy of the results confirms the effectiveness of the presented approach. Keywords: Particle Swarm Inspired Multi-elitist Artificial Bee Colony Algorithm; Model Updating; Differential Quadrature Element Method; Vibrations; Frame; Structural Damping.1.    IntroductionThe differential quadrature element method (DQEM) introduced by Chen 1, is a robust and computationally efficient numerical method that utilizes the differential quadrature approximation 2 to discretize the differential equations, continuity and boundary conditions governing each element in the numerical model of structures. In the last two decades, the DQEM was successfully applied to solve a variety of engineering problems including vibration modeling of different structures 3-9. Since there are always some uncertainties in physical and elemental parameters of the numerical models of structures, different model updating schemes are utilized to estimate the uncertain parameters based on experimentally obtained data. These methods can be classified as direct and iterative ones or in another classification as gradient-based methods and non-gradient ones with random computations. A good review of different updating procedures can be found in ref. 10. All the updating methods are aimed to reduce the difference between the outputs of the numerical model and those of the real structure, but each method has its own pros and cons. The direct methods are computationally efficient and therefore are proper for large complicated models; however, the updated matrices may not be physically correct. Among the iterative methods, selection of the design parameters can help to estimate uncertainties in the model but the convergence of the gradient-based approaches highly depends on the initial guess for the design parameters. On the contrary, the success of the random iterative algorithms does not depend on the initial model which in fact is selected randomly from the search space, but the drawback of these methods is their computational costs where the best result is usually obtained after lots of iterations. Proper selection of the design parameters plays an important role in the success of all the updating procedures. A sensitivity analysis can be performed and the design parameters which the output of the model is insensitive to them should be avoided. Different techniques for sensitivity analysis can be utilized to estimate local and global sensitivities of the model output to the design parameters. A review of the available techniques for parameter sensitivity analysis was presented by Hamby 11. When a global representation of structures, e.g. a frame structure, is of interest, it is unnecessary, and computationally expensive, to have a model with many details. In such cases, one-dimensional elements are commonly used; however, one of the facing challenges is how to model the joints efficiently. Since the welded joints are not completely rigid in practice, one way to deal with flexibility of the joint is using rotational and/or translational springs, but it is not an efficient model mainly because the modal parameters of the structure are almost insensitive to the associated stiffness parameters of the springs 12. In the following work, to model the flexibility of the welded joints in a frame structure, the effective lengths of the members connected by the joints are changed and the effect of using these geometrical design parameters on the accuracy of the updated model is also studied. The optimum values of these parameters in a specific range are obtained by solving aiscussed the need to develop the identification methodologies of general damping models and addressparameters of a three-story frame and utilizing an iterative random evolutionary algorithm e.g.  the PS-MEABC algorithm, the DQE model of the frame is updated. In the first stage of the updating, to update the mass and stiffness matrices, Young’s modulus, density, and some geometrical and elemental design parameters to recover the flexibility of the joints, are considered as design parameters. To verify the suggested design parameters, local and global sensitivities of the natural frequencies to the design parameters are evaluated. By choosing different sets of design parameters, it is shown that how selection of the design parameters will affect the accuracy of the results. Moreover, for damping identification, two different structural damping models are used.  In the first one, the damping matrix is assumed to be diagonal while in the second model, a general damping model is used and all the elements of the damping matrix are identified. The robustness and simplicity of the evolutionary algorithms especially in the case of a problem with high dimensions, e.g. identifying all the elements of the damping matrix, makes them a good candidate for engineering optimization problems. Their drawback of being computationally expensive can also be tackled by high-performance computers. 2.    The differential quadrature element model of the frameIn the case