双层纳米板结构的非线性动力学特性分析

发布时间：2019-02-11 19:21

【摘要】：自从人类发现和成功制备碳纳米管以来,纳米科学和技术迅速发展成为当今科学研究的一个重要领域,大批专家和学者开展了广泛而深入的研究。由于纳米材料和结构具有出色的化学、电学以及力学等特性,被广泛应用于微纳电机系统(MEMS/NEMS)、生物传感器、原子力显微镜以及场滤波器中。这些元器件的设计和制造依赖于纳米结构力学性质的研究,因此对纳米结构力学特性的研究很有意义。本文基于非局部板理论,研究了双层纳米板结构在不同边界条件下的非线性自由振动和主共振性质,得到了结构非线性振动频率以及幅频响应关系的表达式,并且采用全局摄动方法研究了结构的同宿和混沌动力学特性。主要研究工作如下:研究了双层纳米板结构的非线性弯曲振动特性。考虑结构在振动中的几何大变形,采用非线性应变位移关系建立结构的非线性动力学方程,在简支和固支可动边界条件下,利用多尺度法解析地分析求得结构的主共振幅频响应关系式,以及非线性自由振动的解析表达式,分析讨论了非局部效应及结构参数对幅频响应曲线和非线性振动频率的影响,并发现双层纳米板结构不存在内共振的振动形式。研究了嵌入弹性基础中双层纳米板结构的非线性振动特性,同时比较了四种边界条件下结构非线性自由振动的频率特性。考虑结构所处弹性介质环境,讨论了弹性基础的刚度系数对结构非线性主共振幅频响应曲线和自由振动频率的影响。并且注意到结构非线性振动特性与边界是否可动有关,因此在简支可动、简支不可动、固支可动、固支不可动这四种边界条件下,讨论了纳米板的长宽比与结构一阶和二阶非线性振动频率比的关系。研究了在面内载荷作用下,屈曲双层纳米板结构的同宿和混沌动力学行为。双层纳米板结构在面内载荷作用下经历同步屈曲和异步屈曲状态,在这两种不同屈曲状态下,发展并利用改进的高维Melnikov方法研究了结构的同宿和混沌运动,建立横截同宿轨道是否存在的判别条件,在异步屈曲条件下,划分了结构发生不同混沌运动和超混沌运动的参数区域,计算了这些参数区域中结构振动的Lyapunov指数和Lyapunov维数。并且对比了经典连续介质理论和非局部模型的计算结果,在两种屈曲条件下,讨论了非局部参数即小尺度效应对结构发生同宿和混沌运动的影响规律。研究受面内静态载荷以及横向简谐激励的双层纳米板结构的同宿运动。利用双模态Galerkin截断方法建立结构的非线性动力学方程,建模过程中发现结构的转动惯量可以破坏未扰振动系统的Hamilton辛对称结构,而根据Reddy和Amabili的文献,在之后的非线性动力学分析中可以忽略转动惯量的影响。利用改进的高维同宿Melnikov方法讨论了结构在四种屈曲状态下的同宿现象,给出结构在不同相平面上发生同宿运动的解析判定条件,并分析了小尺度效应以及边界条件对结构同宿运动的影响。研究参数激励下双模态屈曲双层纳米板结构的同宿现象和混沌运动。在结构一阶和二阶模态分别经历同步和异步屈曲的条件下,利用推广的Melnikov方法在八维相空间中分析了结构的同宿和混沌运动,将分析所得结果与Lyapunov指数谱以及公开发表的文献中采用分子动力学方法的计算结果加以对比,并分析了小尺寸效应等因素对结构同宿运动等非线性动力学行为的影响规律。
[Abstract]:Since the discovery and successful preparation of carbon nanotubes, the rapid development of nano-science and technology has become an important field of scientific research in the present day, and a large number of experts and scholars have conducted extensive and in-depth research. Due to the excellent chemical, electrical and mechanical properties of the nano-materials and the structure, it is widely used in the micro-nano-motor system (MEMS/ NEMS), the biosensor, the atomic force microscope and the field filter. The design and manufacture of these components depend on the study of the mechanical properties of the nano-structure, so the research on the mechanical properties of the nano-structure is of great significance. Based on the non-local plate theory, the nonlinear free vibration and the main resonance properties of the two-layer nano-plate structure under different boundary conditions are studied, and the expressions of the nonlinear vibration frequency and the amplitude-frequency response relation of the structure are obtained. The homoclinic and hybrid dynamics of the structure are studied by the global perturbation method. The main research work is as follows: the nonlinear bending vibration characteristics of the double-layer nano-plate structure are studied. considering the geometric large deformation of the structure in the vibration, the nonlinear dynamic equation of the structure is established by using the nonlinear strain displacement relation, and the main common amplitude frequency response relation of the structure is analyzed by using the multi-scale method under the dynamic boundary conditions of the simple branch and the solid support, The influence of the non-local effect and the structure parameters on the amplitude-frequency response curve and the nonlinear vibration frequency is discussed, and the vibration form of the inner resonance is not found in the double-layer nano-plate structure. The non-linear vibration characteristics of the two-layer nano-plate structure embedded in the elastic foundation are studied, and the frequency characteristics of the nonlinear free vibration of the structure under the four boundary conditions are compared. Considering the elastic medium environment at the structure, the influence of the stiffness coefficient of the elastic foundation on the nonlinear main-amplitude-frequency response curve and the free-vibration frequency of the structure is discussed. The relationship between the aspect ratio of the nano-plate and the structure-order and the second-order non-linear vibration frequency ratio is discussed. In this paper, the behavior of the co-sink and mixed kinetics of the two-layer plate structure with buckling under the effect of internal load is studied. The double-layer nano-structure experiences synchronous buckling and asynchronous buckling under the action of the internal load, and under the two different buckling conditions, the improved high-dimensional Melnikov method is used to study the homoclinic and hybrid motion of the structure, and the judging conditions of the existence of the cross-section and the sink track are established. Under the condition of asynchronous buckling, the parameters of the structure's different mixed motion and the super-mixed motion are divided, and the Lyapunov exponent and the Lyapunov dimension of the structural vibration in these parameters are calculated. The influence of the non-local parameters, i.e. the small-scale effect on the structure and the sink and the hybrid motion, is discussed in the two buckling conditions. In this paper, the homoclinic motion of the two-layer nano-plate structure excited by the static load in the plane and the transverse simple harmonic excitation is studied. The nonlinear dynamic equation of the structure is established by using the double-mode Galerkin truncation method, and the rotational inertia of the structure is found in the modeling process, so that the Hamiltonian symmetries of the unperturbed vibration system can be destroyed, and according to the literature of Reddy and Amabili, The effect of the moment of inertia can be neglected in the subsequent nonlinear dynamics analysis. In this paper, the homoclinic phenomenon of the structure in four kinds of buckling is discussed with the improved high-dimensional and sink Melnikov method, and the analytical judgment condition of the structure on the plane of different phases is given, and the effect of the small-scale effect and the boundary condition on the structure and the sink movement is also analyzed. In this paper, the homoclinic phenomenon and the mixed motion of the double-mode buckling double-layer nano-plate structure under the excitation of the parameters are studied. under the condition of synchronous and asynchronous buckling of the first and second order modes of the structure, the homoclinic and the mixed motion of the structure are analyzed in the eight-dimensional phase space by using the generalized melnikov method, The results of the analysis and the Lyapunov exponent and the published literature are compared with the results of the molecular dynamics method, and the influence of the factors such as the small-size effect on the nonlinear dynamic behavior of the structure and the sink is analyzed.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O327

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