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

飞机结构通常建模为薄壁结构元件的组装。特别是，机翼的顶部和底部蒙皮、扭转箱、肋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具有重要的作用。该分析还有助于气动弹性和响应计算。通常采用有限元法（FEM）进行此类分析。然而，有限元法是一种近似方法，需要大量的计算资源和建模工作。随着单元数的增加，有限元计算结果普遍收敛于精确结果，但计算结果的准确性并不能得到保证。这在高频模态分析中尤其如此，此时有限元法可能变得不可靠。在模态分析中有一种强有力的替代方法，即动刚度法（DSM）。DSM的建模能力远远优于FEM，所需的计算资源也少得多，但重要的是，DSM的计算结果的准确性始终得到保证。该方法提供了精确的结果，因为单元的性质是由单元的控制运动微分方程的精确解导出的。这与有限元法中用于导出元件特性的假设形状函数形成鲜明对比。因此，有限元中的离散化误差在DSM中不存在。对于梁单元，DSM方法已经得到了很好的发展，但对于板单元，由于迄今为止只研究了板简支时的限制情况，因此目前该方法还存在一定的不足。这一限制阻碍了DSM在更广泛背景下的通用使用。该项目的目的是消除这一限制，并开发具有任意边界条件的板的DSM，以便在精确意义上对复杂飞机结构进行模态分析成为可能。这项研究将使DSM成为一个通用的工具。这将是结构力学的一个重大突破。一般情况下，板单元的动刚度矩阵难以求出，这是因为控制板动力特性的双谐波方程不容易得到封闭形式的解析解。因此，具有一般边界条件的平板的DS发展依赖于双调和方程的成功解，这是一个高度复杂的数学问题。特别是两位杰出的数学家（他们将参与该项目）最近取得的进展为拟议的DSM发展提供了巨大的前景。由于DS矩阵将通过解双调和方程来发展，因此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。首先，我们将关注各向同性的板材材料，但之后，各向异性的板材材料也将被考虑。一旦建立了DS矩阵，Wittrick-Williams算法将成为计算复杂航空结构固有频率和模态振型的求解技术。这些结果将通过一些案例研究得到广泛的验证，包括一个带弦的翼箱，并通过使用FEM和文献中的其他结果。将使用Fortran和Matlab开发计算机程序，并提供用户手册。从该项目中积累的新知识将对国民经济产生相当大的影响，因为它将通过应用DSM在计算机辅助结构分析和设计的新设计方法方面创造未来的投资。从长远来看，对社会的影响将体现在航空和其他行业的燃料消耗降低以及由于更有效的轻量化结构设计而减少的碳足迹。

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.