The Eigensensitivity Iterative Method on the Model Updating of Structural Model

結構設計和分析過程，需要合理的有限元素分析模型來掌握所模擬結構物的動態特性，例如:模態參數(自然頻率及模態振型)。但由於結構的複雜性，常會先對幾何條件或材料特性的模型變數（例如：二次慣性矩及楊氏係數)作合理的假設而予以簡化。這些未確定的變數值常需有試驗來驗證，分析模型如有誤差則需要經過有效的修正，使模型分析與真實結構的動態特性之差異最小。 本文將「特徵靈敏度矩陣」迭代法與逐次二次規劃法應用在結構分析模型的更新。特徵靈敏度矩陣的組成為各模態之特徵值與模態振型對待修正模型變數的一階導函數，用來控制變數修正的迭代方向。透過模型變數的調整來更新結構的質量矩陣與勁度矩陣，使分析模型之「自然頻率」及相對「模態振型」逼近模態試驗結果。 本文模型更新方法之重點為:(一)本文採用Frobenius模數與相關係數(Correlation Coefficients)為模型更新的收斂評估標準。(二)本文採用有限元素分析模型之靈敏度矩陣評估試驗模型中各自由度位置的重要性，依自由度的重要性由高至低之次序列入模態能量(Modal Kinetic Energy)的累計總和值作計算，當值達0.9時的自由度數目採為「模態試驗」的量測點數目。(三)本文先由「殘留力向量」的分量為較大者，其相對應的自由度所連結的元素認為是誤差較大的位置，第一階段列為可能的「待修正元素」。第一階段篩選可能的待修正元素當中，當元素的兩個「模型變數」之第一個模態自然頻率的靈敏度均相較其他元素為大者，則該等元素作為第二階段選定的「待修正元素」，為兩階段方法。配合選定的「待修正之元素」，本文結合逐次二次規劃法，可調整各待修正元素的模型變數使其模態參數能逼近模態試驗結果。 本文以「門型梁結構」來說明有限元素模型更新的程序，其結果顯示：有依靈敏度矩陣與模態能量(MKE)的累計總合值作量測佈設之量測資料，本文的兩階段方法能排除確認待修正之元素。當待修正元素的模型變數之間的值存有數量級差異，兩階段方法可偵得待修正元素。又如果有相鄰元素為待修正元素，兩階段方法仍可偵得該等元素。 另本文以國家地震工程研究中心「五層樓縮尺鋼結構」的試驗結果為比較範例。針對一般結構初步設計過程採用等效「剪力梁」模型估算等效勁度可能存在的模型未確定性來進行結構模型變數的更新。本文比較「自然頻率」與有無「模態振型」之靈敏度進行迭代計算。結果顯示考慮模態振型之更新效果較佳。另外以各層樓的等效勁度作為模型變數，分析得到前五個模態的特徵值對各模型變數之靈敏度，當前五個模態各挑選對其特徵值的靈敏度最高的前三個模型變數進行更新，可有良好的更新效果。 「三維有限元素」結構模型的分析結果與模態試驗結果作比較仍有誤差，預期剪力梁結構模型的誤差更大。但是等效「剪力梁」模型更新後的分析結果可逼近模態試驗結果，仍適合用來作進一步的結構動態反應分析。由等效「剪力梁」模型之「模型變數」的更新結果，可得剪力梁模型如何作合適的等效勁度分佈之調整，如五層樓結構的各樓層應如何調整等效勁度大小，可得正確的結構系統動態特性。此種模型更新方式可應用於常用的簡化結構分析模型，藉由調整勁度以提高模型的動態特性準確性。During structural design and analysis, an appropriate finite element analysis model is required to capture the dynamic characteristics of structures, such as modal parameters (natural frequencies and mode shapes). However, owing to the structural complexity, model variables related to geometry and material properties (for example, moment of inertia and Young’s modulus) should be reasonably simplified. In order to minimize the difference in dynamic characteristics between simulation and actual phenomena, these variables need to be verified through tests or modified if errors exist. In this study, an iterative method of the eigensensitivity matrix, which is an iterative method, is used for updating the structural analysis model. The eigensensitivity matrix is composed of first-order partial differentials of the model variables for natural frequency and mode shape of each modes, which were used to control the regulating of variables in iteration and convergence. The mass matrix and stiffness matrix of the structure were corrected by updating model variables of the analysis model through the iterative procedure, making the analysis results close to the results in the modal test. First, in the present study, a portal beam structure was used to illustrate the updating procedure for the finite element model. The results: (I)Regardless of the noise contamination in the test results (natural frequencies and mode shapes), the model updating effects considering the two convergence indexes (Frobenius norm and Correlation Coefficients) were good. Moreover, the influence of the contaminated test results on the updating effects can be evaluated using model variables as examples. (II)The modal kinetic energy (MKE) in the finite element analysis model was applied to evaluate the appropriate number of measuring points of the modal test; eigensensitivity matrix was used to analyze and obtain the importance order of degrees of freedom in test model; and measured data was set as updating targets, a better updating effect was obtained. (III) We have combined the “residual force vector method” and the “eigensensitivity method” and proposed a two-phase evaluation method to determine the location of elements to be modified in the finite element model. In this study, the test results of a “five-story scaled steel structure” from the National Center for Research on Earthquake Engineering are regarded as an equivalent “shear beam” analysis model to be updated. Based on the sensitivities of characteristic values of the five modes to “model variables,” choosing model variables to which the characteristic value of a mode are more sensitive has better updating effects than using the“same” model variables for each mode as the updating target. Compared with the analysis results of “3D finite element” structure model, the results of the equivalent shear beam structure model after the update show better agreement with the test results. Furthermore, the updated results of model variables meet the engineering requirements. Therefore, the equivalent “shear beam” structure model and its updated results of “model variables” can be used for structural design and analysis