彈性支撐複合材料板結構系統參數之識別

Abstract

由於複合材料在相同重量條件下，具有很高之強度和剛性。已廣泛使用於航空和太空工業與其他尖端科技產業上，製造出各種高性能之結構物。對於不同的製造或成型方式都會對其構件之材料性質造成影響；而且結構物通常是由許多部分所組成，複合材料平板可能只是其中一個構件，通常可藉由不同型式的接點和其他結構件結合在一起，而在力學上可以用等效之彈性支撐方式來模擬其接點處之性質。且若要精確預知此結構物之力學行為，則先要有真實的材料常數與邊界支撐之性質。因此，如何有效且正確的反算出其結構系統參數是非常重要的。 本文首先建立彈性支撐複合材料平板的振動分析，利用里茲法將板之變形以級數方式表示，而變形特徵函數則採用雷建德正交多項式函數組成。建立彈性支撐平板之總勢位能，並對其變形函數未定係數變分，可以建立振動特徵方程式，再求解此方程式，即可得到板之自然頻率、模態。文中主要探討兩大類複材板：一是複合材料積層板；另一則是複合材料三明治板。在複合材料積層板部分，考慮積層薄板和積層厚板兩種，而其所使用之理論分別為古典積層板理論與一階剪變形理論；而複合材料三明治板部分則是使用分層理論，亦即多層一階剪變形理論。複材板的邊界是以長條狀墊片彈性支撐，積層薄板部分是考慮四個邊界全部彈性支撐，或同時板中央還有彈簧支撐，還是部分彈性支撐等情形；積層厚板時，則考慮單邊彈性支撐；而複合材料三明治板，則考慮四邊全部彈性支撐。 接著利用限制性總域極小化程序，以廣義拉格蘭吉乘子方法，將原先有限制條件之最小化問題，轉變成無限制條件之最小化問題。配合振動實驗測量與里茲方法理論分析，建立實驗與理論之自然頻率差值最小平方為目標函數，並採隨機多起始點搜尋、設計變數單位化及貝氏分析法，以非破壞性方式識別彈性支撐複合材料板結構之系統參數。並藉由識別各種彈性支撐複合材料積層板與三明治板試片為範例，說明本方法之可行性與精確性。本文之研究方法，將可輕易應用於其他不同型式彈性支撐結構之系統參數識別。

Due to their high strength/stiffness to weight ratios, the fiber reinforced composite materials have become important in weight-sensitive applications like aeronautical and aerospace industry as well as many other fields of modern technology to fabricate high performance structures. As well known, there are many methods for manufacturing laminated composite components and different manufacturing or curing processes may produce different mechanical properties for the components. And the composite plates in these structures are connected to other members using different joining methods. One popular way to analyze the mechanical behaviors of the composite plates is to consider the plates being supported at the boundary by equivalent elastic restraints. The attainment of the actual behavioral predictions of the flexibly supported composite plates, however, depends on the correctness of the system parameters such as the elastic constants of the plates and the spring constants of the elastic restraints at the plate boundaries. Therefore, the determination of realistic material and spring constants of laminated composite components has become an important topic of research. In this paper several methods are proposed for vibration analysis of elastically restrained rectangular symmetrically laminated composite thin or thick plates and the laminated composite sandwich plates. The methods are constructed based on the Rayleigh-Ritz method in which the deformation characteristic functions are expressed as the Legendre’s orthogonal polynomials. The displacement models of the thin or thick laminated composite plates and laminated sandwich plate are constructed on the basis of the classical laminate plate theory (CLPT) or the first-order shear deformation theory (FSDT) and the layer-wise linear displacement theory, respectively. Extremization of the functional, the total potential energy, with respect to the displacement coefficients leads to the eigenvalue problem. The solution of above equation gives the theoretical natural frequencies of flexibly supported composite plates. The Rayleigh-Ritz method is then used to study the free vibration of different kinds of plates with various supporting conditions such as thin laminated composite plates supported by strip-type elastic pads around the peripheries of the plates with or without center elastic supports, thin plates partially supported by edge elastic restraints, thick laminated composite plates partially supported by edge elastic restraints, and laminated sandwich composite plates supported by strip-type elastic pads around the peripheries of the plates. In this paper, a constrained minimization method is presented for the nondestructive parameters identification of flexibly supported composite plate structures. A frequency discrepancy function is established to measure the sum of the differences between the experimental and theoretical predictions of natural frequencies of the elastically restrained laminated composite plates. The use of a multi-start global minimization method to identify the elastic constants by making the frequency discrepancy function a global minimum, and a design variables normalization technique for expediting the convergence of the search of the global minimum. The accuracy and applications of the proposed method are demonstrated by means of several examples. The present method can be extended without difficultly to the material and spring constants identification of other types of structures.