Mathematical Sciences: Modeling, Analysis and Control Of Multilayer Plates

这项研究涉及多层板的建模和分析，其中层内剪切变形的影响可能是重要的。具体的例子包括层合板、约束层板和表面安装驱动器的板。该项目将(1)开发新的动态多层板模型并完善现有模型；（ii）确定各模型相对于模型中的设计参数的有效性范围；（iii）确定如何设计控制和稳定反馈，使其相对于模型中的参数不确定性具有鲁棒性。利用剪切变形板的Reissner-Mindlin理论，得到了允许每一层独立剪切变形的初始模型。然后通过摄动技术获得更适合计算和模拟的低阶近似模型。各种模型的有效性范围将通过分析初始模型的解相对于适当的扰动参数的连续性来研究。一旦开发了各种系统解的连续性，就可以分析关于参数和不确定性的鲁棒性和灵敏度问题。本研究涉及多层板的建模和分析，其中层内剪切变形的影响可能很重要。具体的例子包括层合板、约束层板和表面安装驱动器的板。该研究适用于多层复合材料设计和控制中出现的大量建模问题。这些材料的应用范围很广，从体育用品到船体，通常需要建造以满足各种设计标准，例如最佳强度质量比，或具有某些最佳阻尼特性。在许多应用中，主动反馈控制将被用来提高材料在其预期应用中的性能。本研究将为此类材料及其动力学特性的建模提供统一、严谨的数学基础，并将发展模型的各种控制理论特性。为了跟上传感器和执行器快速增长的速度并充分利用现代制造能力，需要这样的理论基础。＊＊＊

9623144 Hansen This research involves the modeling and analysis of multilayer plates in which the effects of shear deformation within layers may be important. Specific examples include laminated plates, constrained layer plates and plates with surface-mounted actuators. The project will (i) develop new dynamic multilayer plate models and refine existing ones; (ii) determine the range of validity of the various models with respect to design parameters in the models; and (iii) determine how to design controls and stabilizing feedbacks that are robust with respect to parameter uncertainties in the models. The Reissner-Mindlin theory of shear-deformable plates is used to obtain an initial model that allows independent shear deformations in each layer. Lower order approximate models which are more amenable to computations and simulations will then be obtained by perturbation techniques. The range of validity of the various models will be investigated by analyzing the continuity properties of the solutions of the initial model with respect to the appropriate perturbation parameters. Once the continuity properties of the solutions of the various systems are developed, issues of robustness and sensitivity with regard to parameters and uncertainties will be analyzed. %%% This research involves the modeling and analysis of multilayer plates in which the effects of shear deformation within layers may be important. Specific examples include laminated plates, constrained layer plates and plates with surface-mounted actuators. The research is applicable to a large class of modeling problems that arise in the design and control of multilayer composite materials. Such materials are used in a wide range of applications, from sporting goods to ship hulls, and often need to be constructed to meet various design criteria, such as optimal strength-to-mass ratio, or to have certain optimal damping properties. In many applications, active feedback control will be utilized to enhance the performance of the material in its intended application. This research will provide a unified, rigorous, mathematical foundation for the modeling of such materials and their dynamic properties, and will develop various control-theoretic properties of the models. Such a theoretical foundation is needed in order to keep pace with the rapidly increasing speed of sensors and actuators and to take full advantage of modern manufacturing capabilities. ***