sensitivity of the parameters to update the model. The error between analytical and experimental results is set as an objective function to be minimized by changing a set of physical parameters in the numerical model. Usually, structural model parameter estimation problems based on measured data are often formulated as weighted least-squares problems. To enhance the convergence rate and simplify the optimization, modal metrics, measuring the residuals between measured and model predicted modal data, are built up into a single metric formed as a weighted average of the multiple modal metrics. However, the values assigned to the weights cannot be known a priori and strongly influence the results of the optimization [4]. For these reasons, parameter identification has been also formulated as a multi-objective problem, allowing the minimization of the multiple modal metrics but eliminating the need for using arbitrary weighting factors. The set of multiple optimal solutions will results in a Pareto front from which the user will select the more adequate. Developing models using the Multibody Simulation (MBS) approach allows the designer to analyze the behavior of complex mechanisms working in a multiphysic environment. Besides analyzing the kinematic and dynamic behavior of the system, they can be integrated with other simulation tools to study, e.g., the durability, aerodynamic, or acoustic performances. They are also widely used to test and design control strategies and more in general to develop mechatronic systems. To take into account for flexibility in a multibody model, three approaches are available. The first one consists in modeling the flexible bodies using the FE method, reduce them using Component Mode Synthesis and linking them to the structure using joints and constraints [6]. For such models, the model updating techniques described above can be directly applied. Another approach is to write the equations of motion of the bodies using a finite element coordinate system approach, in which flexibility is directly taken into account [7]. The last approach deals with purely rigid bodies connected using massless elastic force elements. Using this method has clearly some limitations, since it can be applied only to relatively simple geometries and the accuracy depends on the number of elements used to discretize the component, but on the other hand it allows reducing the complexity of the structure and making it computationally lighter. If model updating is to be applied to this class of models, no dedicated methods were developed. The aim of this study is the development of a methodology to update multibody model with experimental data, using as automation tool a commercial optimization environment. The method is developed to be as general as possible, and will be here applied to a full scale wind turbine model with unknown blade properties and for which experimental modal analysis results are available. Modeling wind turbine with multibody simulation tools is very important, being possible to analyze the response of the structure to dynamic loads, such as aerodynamic and gravitational forces, varying with time as the blades rotate [8]. An accurate determination of the modes and natural frequencies in different operating conditions is mandatory to obtain an adequate insight of the system dynamics and optimize the performances by separating the turbine natural frequencies from the rotor speed’s harmonics. Another common solution for reducing dynamic loads is the definition of control laws to actively damp component vibrations. In order to design efficient controls, the model must accurately predict the dynamic behavior for a wide range of operating conditions. The paper is organized as follows: in section 2, the proposed model updating methodology based on modal data and the correlation indices selection are presented. Section 3 will describe the CART3 wind turbine, the experimental campaign and the multibody modeling. Section 4 and 5 will then present the obtained results for the single and multi-objective optimization problems. Finally some conclusion will be drawn. 2. MODEL UPDATING BASED ON MODAL DATA In this paper, a methodology to update multibody simulation models based on modal data is presented. The results of an Experimental Modal Analysis campaign on a structure consist of the FRFs between the input and the output points, which can be processed to obtain the natural frequencies and mode shapes. Let define { } { } { } N M 1 2 N N 1 2 N C ψ ,..., ψ, ψ Ψ C , ω ,..., ω, ω ω D × ∈ = ∈ = = (1) the set of measured modal data from a structure, consisting of natural frequencies ω and modal matrix Ψ, with N the number of identified modes observed at M degrees of freedom. Usually modal parameters are considered to be real, but for generality they will be considered here as complex. Obtaining the modal parameters for a multibody model is not as straightforward as for finite elements models: the model needs to be linearized at a given instant in time and the state space formulation derived, where the state variables are the degrees of freedom (DOF) of each body, considered in its center of gravity. Being q the vector of the generalized coordinates and M the generalized mass matrix, the equation of motion can be written as: ( ) Mq f q,q,t & &&= (2) At the selected instants in time, the model is frozen and perturbed around its configuration to obtain the state space formulation: 350
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