# Model Updating Using Frequency Response Functions Based on Sherman-Morrison Formula
## 1 Introduction
The above scholars have done a lot of research on the model updating based on FRFs. However, the selection of parameters plays a significant role in the progress of model updating. The sensitivity of the FRFs has great potential for a vibration analysis, which is not involved in the above methods. Reasonable selection of parameters can effectively improve the quality and efficiency of model updating, which is inseparable from the sensitivity analysis of the parameters, so model updating method based on the sensitivity of FRFs has more important significance. The model updating based on the sensitivity of FRFs attracted wide attention in the early 1990's [23]. [PERSON] et al. [24] proposed a new FE model updating method that uses numerical sensitivity instead of analytical sensitivity to solve inverse problems, which does not need model reduction and data extension, and has good robustness, stability and fault tolerance; [PERSON] et al. [25] proposed a new iterative substructuring method to accurately obtain the characteristic solution and characteristic sensitivity of the structure, which is suitable for model updating of large and complex structures; [PERSON] et al. [26] expressed the thermal dependent elastic constants and thermal expansion coefficients as intermediate function by temperature independent variables, and used the perturbation method to calculate the sensitivity for parameter identification; On the basis of the basic concept of unconstrained optimization problem and three regularization solutions, [PERSON] et al. [27] proposed a novel sensitivity based FE model updating method, and simultaneously updated the element mass matrix and stiffness matrix of the FE model; [PERSON] et al. [28] proposed a FE model updating method using incomplete strain data in frequency domain to detect variations in stiffness and mass parameters. The common sensitivity analysis methods using FRFs are the finite difference method [29] or the combined approximations (CAs) method [30]. These methods are based on obtaining the stiffness matrix, damping matrix, and mass matrix of the structures. The disadvantage of these methods is that repeated analysis is required in the process of structural parameter identification, and they are not applicable under the premise of unknown dynamic stiffness matrix. For example, the modal parameters of the structure can be obtained through experiments, but the complete information of the dynamic stiffness matrix is difficult to obtain. The Sherman-Morrison formula was first proposed by [PERSON] and [PERSON] and it can be used to represent the varying of matrix; [PERSON] et al. [31] used the Sherman-Morrison formula of singular value decomposition to calculate the optimal perturbation of the sensitivity value of the extreme value and the design variable, thus achieving the response change that is difficult to obtain with the response derivatives; [PERSON] et al. [32; 33; 34; 35] used the Sherman-Morrison formula for mass elimination of multi-transducers, and the revised FRFs results were in good agreement with the target values. In this study, a model updating method using FRFs based on Sherman-Morrison formula is proposed, only the initial FRFs and parameter perturbations are employed to calculate the sensitivity avoiding repeated FE analyses and calculations, which improves the computational efficiency of model updating. The detail formulation of sensitivity analysis and model updating based on Sherman-Morrison formula is introduced in Section 2, the implementation procedure of the model updating is described in Section 3, and the numerical examples of a planar truss and a solar wing and the experiment of aluminum solar panel frame are used to verify the proposed method in Section 4. The conclusions are summarized in Section 5.