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

纳米管和纳米管被认为在生物传感器、药物载体、谐振器、人造微血管、纳米开关和纳米功率发生器的建模中有着广泛的应用。几乎所有纳米结构的力学和电学性质都受到其非线性振动特征和共振频率的很大影响。本研究的目的是建立一个可靠的模型来一致地模拟弹性包围的纳米管和纳米棒的大振幅非线性振动。 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的类型和强度是根本不同的，因此寻求一种全面的研究来研究纳米结构的非线性振动的未知方面。在这个研究计划中，我将在一个统一的框架下研究四个主要问题：a)为纳米管和纳米棒的振动分析引入精确的模型和新的解析解，b)考虑不同类型的非线性对纳米结构参数的灵敏度分析，c)纳米结构振动的稳定性分析和性能研究，d)纳米结构中潜在的混沌行为的混沌预测和控制。 这一建议的基本基础是基于我最近在非线性振动系统的建模和分析方面的贡献，以及与该领域其他研究人员最近报道的实验结果的性能验证。这项研究的目的之一是研究不同类型的非线性在纳米尺度上的相互作用，并确定共振放大或抵消的潜力。对于直的、弯曲的和起皱的纳米棒，将开发一种基于变分的通用方法。利用非均匀有理基样条(NURBS)理论对起皱的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诸如边缘效应和卡西米尔效应等其他非线性现象相结合时，纳米结构的未被发现的动力学行为被识别出来。我们将模拟这种纳米结构的自由振动以及简谐和随机振动。考虑了不同类型的可能的共振条件，包括一次共振和二次共振，以及内部共振情况。得到了系统的主次谐波、亚谐和超谐响应。进行了分叉分析，研究了纳米棒和纳米管的周期解和准周期解的稳定性。将深入研究相互作用的非线性力、尺寸、表面和几何条件对这些结构振动行为的影响。 这项研究的成果不仅将提高更现实的模型和解决方案的知识前沿，而且将极大地惠及加拿大和国际行业，特别是在电路、组织工程和再生医学这三个类别的行业。

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.