Nonlinear Vibration and Stability Analysis of Nano-scale Elastic Beam-type Structures

纳米束和纳米管被认为在生物传感器、药物载体、谐振器、人造微小血管、纳米开关和纳米发电机的建模中有着广泛的应用。几乎所有纳米结构的力学和电子性能都受到其非线性振动特征和共振频率的高度影响。本研究旨在建立一个可靠的模型，以一致地模拟弹性包围纳米管和纳米梁的大振幅非线性振动。由于纳米结构的非线性振动的类型和强度与宏观结构的非线性振动有着根本的不同，因此需要对纳米结构的非线性振动进行全面的研究。在本研究计划中，我将在统一的框架下研究四个主要问题：a)引入精确的建模和新的分析方法来分析纳米管和纳米梁的振动；b)考虑不同类型非线性的纳米结构参数的敏感性分析；c)振动纳米结构的稳定性分析和性能研究；d)纳米结构中潜在混沌行为的混沌预测和控制。*本建议的基本基础是基于我最近在非线性振动系统建模和分析方面的贡献，以及该领域其他研究人员最近报告的实验结果的性能验证。本研究的目标之一是研究不同类型的非线性在纳米尺度上的相互作用，并确定共振放大或消除的潜力。一种通用的基于变分的方法将被开发用于直的、弯曲的和褶皱的纳米梁。利用非均匀有理基样条（NURBS）理论对皱褶纳米结构进行建模是本研究的另一个目标和贡献。当几何非线性倾向于与其他非线性现象（如条纹效应和卡西米尔效应）相结合时，纳米结构的未被发现的动力学行为被识别出来。将模拟这种纳米结构的自由振动、谐波振动和随机振动。考虑了不同类型的可能谐振条件，包括主谐振和次谐振，以及内部谐振情况。得到了主谐波、次谐波和超谐波响应。进行了分岔分析，并研究了纳米梁和纳米管的周期解和准周期解的稳定性。相互作用的非线性力，尺寸，表面和几何条件对这些结构的振动行为的影响将被彻底检查。*这项研究的结果不仅将增强知识的前沿，以获得更现实的模型和解决方案，而且将极大地有利于加拿大和国际工业，特别是在电路，组织工程和再生医学三个类别。

Nanobeams and Nanotubes are considered nowadays to have wide range of applications in modeling of biosensors, medicine carriers, resonators, artificial tiny blood vessels, Nano-switches and Nano-power generators. Mechanical and electronic properties of almost all the nanostructures are highly influenced by their nonlinear vibration signatures and resonant frequencies. This research is aimed to develop a reliable model to consistently simulate the large-amplitude nonlinear vibration of elastically surrounded nanotubes and nanobeams. *A comprehensive study is sought to investigate the unexplored aspects of nonlinear vibration of nanostructures, since the type and strength of nonlinearities are fundamentally different from those of the macro-scale structures. In this research program I will investigate four main issues under a unified framework of: a) Introducing accurate modeling and new analytical solutions for vibration analysis of nanotubes and nanobeams, b) Sensitivity analysis of nanostructure parameters considering different types of nonlinearities, c) Stability analysis and performance study of vibrating nanostructures, and d) Chaos prediction and control of potential chaotic behavior in nanostructures. *The essential foundation of this proposal is based on my recent contributions in the modeling and analysis of nonlinear vibrating systems and also performance verification with recent reported experimental results by other researchers in this field. One of the goals of this research is to investigate the interactions of different types of nonlinearities in Nano-scale and also determine the potential of either resonance magnification or cancelation. A general variational-based approach will be developed for the straight, curved and wrinkled nanobeams. Modeling of a wrinkled nanostructures employing the theory of non-uniform rational basis spline (NURBS) is another objective and contributions of this research. The undiscovered dynamic behavior of nanostructures is identified when the geometrical nonlinearities tend to combine with other nonlinear phenomena such as the fringing and Casimir effects. Free vibration, as well as, the harmonic and stochastic vibrations of such nanostructures will be simulated. Different types of possible resonant conditions, including the primary and secondary resonances, as well as the internal resonant cases are considered. The main harmonics together with the sub-harmonic and super-harmonic responses are obtained. A bifurcation analysis is carried out and the stability of periodic and quasi-periodic solutions for nanobeams and nanotubes will be investigated. The effects of interactive nonlinear forces, size, surface and the geometrical conditions on the vibration behaviors of these structures will thoroughly be examined. *The outcome of this research will not only enhance the frontier of knowledge for more realistic models and solutions, but also greatly benefit the Canadian and international industries especially in three categories of electric circuits, tissue engineering and regenerative medicine.