Dynamic Stiffness Formulation for Plates with Arbitray Boundary Conditions through the Solution of the Biharmonic Equation

飞机结构通常被建模为薄壁结构单元的装配。特别是，机翼的顶部和底部蒙皮、扭力箱、肋骨和腹板被理想化为板或板组件。这类结构的模态分析在航空设计中占有重要地位。该分析还方便了气动弹性和响应计算。通常采用的有限元方法是进行此类分析的常用方法。然而，有限元是一种数值密集的近似方法，需要大量的计算资源和建模工作。随着单元数目的增加，有限元计算的结果通常收敛到精确的结果，但结果的准确性并不总是得到保证。这在高频下的模态分析中尤其如此，当有限元分析变得不可靠时。在有限元分析中，有一种强有力的替代方法，那就是动力刚度法。DSM的建模能力远远优于有限元方法，所需的计算资源也少得多，但重要的是，DSM的结果精度始终得到保证。这种方法提供了精确的结果，因为单元的属性是从单元运动控制方程的精确解推导出来的。这与有限元中用于推导单元属性的假定形函数形成鲜明对比。因此，有限元中的离散误差在DSM中是不存在的。对于梁单元，DSM已经得到了很好的发展，但对于板单元，该方法目前还存在一定的不足，因为到目前为止，只研究了板简支的约束情况。这一限制阻碍了需求侧管理在更广泛的背景下的一般用途。本项目的目的是消除这一限制，发展任意边界条件的板的动力分析方法，使复杂飞机结构的精确模态分析成为可能。这项研究将使DSM成为一个多功能的工具。这将是结构力学的一项重大突破。在一般情况下推导板单元的动力刚度(DS)矩阵是困难的，因为控制板的动力特性的双调和方程不易得到闭合形式的解析解。因此，具有一般边界条件的板的DS展开依赖于双调和方程的成功解，而双调和方程是一个高度复杂的数学问题。特别是两位著名数学家(他们将参与该项目)最近取得的进展为拟议的需求侧管理的发展提供了巨大的前景。由于DS矩阵将通过求解双调和方程来建立，PI及其研究团队与上述两位数学家的互动将在该项目中发挥重要作用。最初的注意力将集中在各向同性的板材料上，但稍后也将考虑各向异性的板材料。一旦建立了DS矩阵，就可以使用Wittrick-Williams算法作为求解技术来计算复杂航空结构的固有频率和振型。结果将通过大量的案例研究得到广泛的验证，包括带纵梁的翼箱，以及使用有限元和文献中的其他结果。将开发使用Fortran和MatLab的计算机程序并将其记录在案，同时提供用户手册。该项目将积累的新知识将对国民经济产生相当大的影响，它将通过应用需求侧管理对计算机辅助结构分析和设计的新设计方法产生未来的投资。从长远来看，对社会的影响将体现在航空和其他行业的更低燃料消耗以及由于更高效的轻质结构设计而减少的碳足迹方面。

Aircraft structures are generally modelled as an assembly of thin-walled structural elements. In particular, the top and bottom skins, torsion box, ribs and webs of the wing are idealised as plates or plate assemblies. The modal analysis of such structures plays an important role in aeronautical design. The analysis also facilitates aeroelastic and response calculations. The usually adopted finite element method (FEM) is generally used to carry out such analysis. However, the FEM is an approximate method which is numerically intensive, requiring considerable computational resources and modelling efforts. The results from the FEM generally converge to exact results with increasing number of elements, but the accuracy of results cannot be always guaranteed. This is particularly true in modal analysis at high frequencies when the FEM can become unreliable. There is a powerful alternative to the FEM in modal analysis which is that of the dynamic stiffness method (DSM). The DSM has far superior modelling capability than the FEM and it requires much less computational resources, but importantly, the accuracy of results in the DSM is always guaranteed. The method provides exact results because the element properties are derived from the exact solution of the governing differential equation of motion of the element. This is in sharp contrast with assumed shape functions used in the FEM to derive the element properties. Thus, the discretisation error in the FEM is non-existent in the DSM. The DSM is well developed for beam elements, but for plate elements, the method is still somehow deficient at present because only the restricted case when the plate is simply supported has been investigated to date. This restriction has prevented a general purpose use of the DSM in a wider context. The purpose of this project is to remove this restriction and develop the DSM for plates with arbitrary boundary conditions so that modal analysis of complex aircraft structures in an exact sense becomes possible. The research will make the DSM a versatile tool. It will be a major break-through in structural mechanics.The difficulty to derive the dynamic stiffness (DS) matrix of a plate element for the general case arises from the fact that the bi-harmonic equation which governs the dynamic behaviour of plates is not easily amenable to closed form analytical solution. The DS development of a plate with general boundary conditions thus relies on the successful solution of the biharmonic equation which is a highly complex mathematical problem. Recent progresses made by two eminent mathematicians in particular (who will take part in the project) offer great prospects for the proposed DSM development. As the DS matrix will be developed through the solution of the biharmonic equation, the interaction of the PI and his research team with the above two mathematicians will play an important role in this project. Initially, attention will be focused on isotropic plate materials, but later, anisotropic plate materials will also be considered. Once the DS matrix is developed, the Wittrick-Williams algorithm will be used as solution technique to compute the natural frequencies and mode shapes of complex aeronautical structures. The results will be extensively validated by a number of case studies including a wing-box with stringers and by using the FEM and other results in the literature. Computer programs using Fortran and Matlab will be developed and documented with the provision of a user manual. The new knowledge that will accrue from the project will have considerable impact upon the national economy by creating future investments in new design methodologies in computer aided structural analysis and design through the application of the DSM. In the long run, the impact on the society will be felt in terms of lower fuel consumption in aviation and other industry and reduced carbon footprint as a result of more efficient design of light weight structures.