Constrained geometric mechanics: theory and applications

约束几何力学：理论与应用

• 批准号: 435827-2013
• 负责人: Putkaradze, Vakhtang
• 金额: $ 1.68万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=533335
• 关键词: Constrained; geometric; mechanics; theory; applications

This proposal focuses on the applications of geometric methods to constrained dynamics of elastic structures. In the first part, we shall apply the methods of geometric mechanics, and corresponding constraints, to systems involving point wise, non-holonomic constraints, with a particular example of elastic rods in contact. This problem will be tackled using the recently developed methods of geometric Lagrange-d'Alambert's methods. The proposal will explore interesting and new directions provided by this theory, such as highly complex dynamics of rods caused by the contact, generalizations to discrete rods, symmetry reductions, steady and unsteady exact solutions, and others. In the second part, we will explore tree-like structures with linked branches, or, alternatively, an array of linked tree-like branches, evaluated from the point of view of geometry. These problems will be treated with methods geometric mechanics of iterative semidirect product groups with holonomic constraints. We shall study the effect of constraints on the geometric structures recently derived by the PI for such objects, such as conservation laws and Poisson brackets. The problem of tree-like structure dynamics will also be used in the development of a nanomechanical sensor and a macroscale tree-like energy harvesting device. Mathematical questions outlined above play a crucial role in the design of both devices. The sensor will be developed together with the PI's collaborators at CSU and will utilize a direct visualization of nanoscale motion using EUV pulsed laser. The tree-like energy harvester is being developed with the PI's collaborators in Japan (U of Kyoto and Shiga Prefecture). The PI's expertise in the dynamics and modeling, nano-fabrication facilities at NINT on campus, and collaboration with the EUV laser group at CSU and Japan makes this project likely to succeed. The project will focus heavily on the training of graduate students and postdocs in the development of tools of modern mathematics and subsequent application of these tools to modern, practical problems. While the project is closely involved with experiments, support is sought only for the theory part.

该提案重点是几何方法在约束弹性结构动力学上的应用。在第一部分中，我们将应用几何力学的方法和相应的约束方法，涉及涉及点明智的非全面约束的系统，并提供了弹性杆的特定示例。该问题将使用最近开发的几何拉格朗日 - d'alambert方法解决。 该提案将探索该理论提供的有趣和新的方向，例如由接触引起的棒的高度复杂动力学，离散杆的概括，对称性减少，稳定且不稳定的精确解决方案等。 在第二部分中，我们将探索具有链接的分支的类似树状结构，或者从几何学的角度来评估，或者是链接的树状分支的数组。这些问题将通过具有自动限制的迭代半领产品组的几何方法来处理。我们将研究约束对PI最近针对此类物体（例如保护法和泊松支架）的几何结构的影响。 树状结构动力学的问题也将用于开发纳米​​力学传感器和宏观树状能量收集设备。 上面概述的数学问题在两个设备的设计中都起着至关重要的作用。传感器将与PI在CSU的合作者一起开发，并将使用EUV脉冲激光直接可视化纳米级运动。 PI在日本的合作者（京都和志加县）开发了类似树状的能量收割机。 PI在校园NINT的动态和建模，纳米制作设施方面的专业知识，以及与CSU和日本的EUV激光集团合作，使该项目可能成功。 该项目将重点关注研究生和博士后的培训，以开发现代数学工具，并随后将这些工具应用于现代实用问题。 虽然该项目与实验密切相关，但仅针对理论部分寻求支持。