模态局部化及频率曲线转向现象研究的几何理论

根据复平面映射理论，详细讨论了结构振动系统的固有频率曲线在复平面的几何特点．研究表明，频率曲线重合和转向现象实际上可进行统一的几何描述．复参数平面中支点的形成是解释局部化现象和曲线转向现象的基础．利用近频摄动理论中的缩聚方法将系统降维，大大拓宽了这一几何方法的使用范围．计算实例验证了理论分析所得的结果．该文是对Ｍ．Ｓ，Ｔｒｉ－ａｎｔａｆｙｌｌｏｕ等近期工作的改进与推广．     

关于含参变量特征值轨迹偏转现象的探讨

含参变量的特征值轨迹的偏转现象出现在许多力学和理论物理问题中。本文通过两个实例,对特征值轨迹偏转现象进行了深入的探讨。先讨论了一个两维平板振动问题,通过求特征值的上、下界的方法,确定了该问题的精确特征值呈现了曲线(含一个参变量)偏转和曲面(含两个参变量)偏转现象;然后讨论了一个双震荡器的振动问题,发现当某个参数发生变化时的特征值曲线是由相交变为偏转的。且当一对特征值曲线偏转前后,两个特征向量几乎互相改变了方向;而当一对特征值曲面偏转前后,对应的特征向量的分量也发生偏转。     

Static and dynamic response of elastic suspended cables with thermal effects

Cable structures are often subjected to severe and variable environmental conditions, and their mechanical behavior is known to be particularly sensitive to different ambient factors. The paper analyzes temperature effects on the static and dynamic response of suspended inclined cables through a continuous monodimensional model including geometric nonlinearities. Uniform temperature changes are introduced through a non-homogeneous constitutive law for the material linear elasticity. Exact and approximate solutions of the equations governing the cable static equilibrium under self-weight are achieved, and the significance of the temperature-dependent variation of tension and sag are parametrically investigated. The spectral properties characterizing the free dynamics are obtained in a closed-form fashion for shallow parabolic cables within the low frequency vibration range. The sensitivity of the linear frequencies to temperature changes is discussed, outlining two thermal effects, which are distinguished by their different origins, geometric or static. For a generic temperature change, the geometric effect produces a systematic increment or reduction of all the frequencies, for both symmetric and anti-symmetric modes. The static effect stiffens or softens only the symmetric modes, and may prevail over the competing geometric effect, depending on the cable Irvine parameter. Finally, the thermal effects on the frequency veering and modal hybridization phenomena, which characterize quasi-resonant shallow cubic cables, are analyzed.     

航空发动机涡轮盘裂纹萌生寿命的估算方法

针对航空发动机涡轮盘裂纹萌生寿命的估算问题，给出一个全新的损伤参量。该损伤参量结合了分析多轴疲劳的临界面方法，“应力功”的概念和疲劳破坏的微观机制。依据该损伤参量，利用有限元程序和自编程序计算得出损伤参量和疲劳循环次数的关系，建立以这一损伤参量为基础的疲劳损伤累积模型，进而估算涡轮盘裂纹萌生寿命。     

Problem of interaction of thin rigid needle-shaped inclusions located between different elastic materials

We study a piecewise-homogeneous elastic plane composed of two half-planes with different elastic parameters and two thin rigid needle-shaped inclusions located between them. One inclusion is rigidly connected with the environment, and the other inclusion is not, while contacting with it like a smooth rigid punch. We consider the plane deformed state generated by stresses given at infinity. The problem is reduced to a combination of a matrix Riemann boundary-value problem from the theory of analytic functions and a matrix Hilbert problem, which can be solved in terms of integrals through the reduction to two separate scalar Riemann boundary-value problems on a twosheeted Riemann surface.We explicitly obtain the complex potentials of the composite elastic plane, the stress intensity factors near the tips of the inclusion, and the rotation angles of the inclusions. We also present numerical examples illustrating how the stresses near the inclusions depend on the elastic and geometric parameters of the problem.     

考虑尺度效应的夹层楔形多孔梁动力学分析

基于修正偶应力理论, 研究了具有大范围旋转中心刚体-功能梯度夹层Euler-Bernoulli楔形多孔柔性微梁系统的动力学特性.楔形梁是中间层为不完全功能梯度层, 两表层为均质材料的功能梯度夹层结构, 它可以减小传统夹层结构由于层与层之间材料属性的不同导致脱粘类型损伤的影响.采用假设模态法描述变形, 考虑具有捕捉动力刚化效应的非线性耦合项, 计及von Kármán几何非线性应变, 运用第二类Lagrange方程, 导出了适用于较大变形的高次刚柔耦合动力学方程.对在平面内做大范围运动的中心刚体-功能梯度夹层Euler-Bernoulli楔形多孔微梁的动力学特性进行了详细研究.研究表明: 功能梯度夹层楔形梁表层结构高度、旋转角速度、功能梯度幂指数、尺度参数、孔隙度以及各层结构的体积分数对系统的动力学特性都有很大的影响; 功能梯度夹层楔形梁综合了功能梯度直梁和楔形梁的特性, 其相对于功能梯度直梁的固有频率增大, 同时使得孔隙度对结构固有频率变化趋势的影响不再与功能梯度直梁相同; 由于柔性梁变形能中具有横向与轴向的耦合势能, 系统在稳态下的平衡位置发生了迁移现象; 系统随着尺度参数的变化发生了频率转向与振型转换.      

Solving the problem of linear viscoelasticity for multiply connected anisotropic plates

This paper describes the method for solving the problems of linear viscoelasticity for thin plates under the influence of bending moments and transverse forces. The small parameter method was used to reduce the original problem to a sequence of boundary-value problems solved via complex potentials of the bending theory of multiply connected anisotropic plates. The general representations of complex potentials and boundary conditions for their determination are obtained. The method for determining the stress state of the plate at any time with respect to complex approximation potentials is developed by replacing the powers of the small parameter by the Rabotnov operators. The problem of a plate with elliptical holes is solved. The numerical calculation results in the case of a plate with one or two holes are given. The variation of bending moments in time until stationary condition is reached is studied, and the influence of geometric characteristics of the plate on these variable is described.     

挠性联结双体航天器的稳定性与分岔

研究圆轨道内受万有引力矩作用的挠性联结双体航天器在轨道平面内的姿态运动,讨论其相对轨道坐标系统平衡状态的稳定性与分岔。提出判平衡方程非平凡解存在性的几何方法,并应用Ｌｉａｐｕｎｏｖ直接法、Ｌｉａｐｕｎｏｖ－Ｓｃｈｍｉｄｔ约化方法和奇异性理论导出解析形式的稳定性与分岔的充要条件,从而对系统的全局运动性态作出定性的描述。     

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

Static and dynamic response of elastic suspended cables with damage

In cable-stayed structures cables are subjected to potential damage, mainly due to fatigue and galvanic corrosion. The paper presents an analysis of damage effects on the statics and dynamics of suspended cables. An elastic continuous monodimensional model for damaged cables, including geometric nonlinearities, is formulated for the purpose. The damage is described as a diffused reduction of the cable axial stiffness, and defined through its intensity, extent and position. Exact solutions of the equations governing the cable static equilibrium under self-weight are achieved, and the significance of the tension loss and sag augmentation resulting from damage are investigated under variation of practically significant parameters. The system spectral properties characterizing the free undamped dynamics are obtained in a closed-form fashion for shallow cables within the low frequency range. The sensitivity of the frequencies to the intensity and extent of damage is discussed, outlining two damage effects, which alternatively stiffen or soften the cable modes, whose respective static and geometric origin is recognized. Finally, the symmetry-breaking induced by damage on the static profile is verified to destroy the crossing phenomenon (crossover) characterizing the frequency loci of undamaged cables, which degenerates into a narrow frequency veering phenomenon.     

Formability of Linearly Elastic Structures with Volume‐Type Actuation

Formation theory concerns the modification of the geometric configuration of an elastic structure by means of attached and/or embedded actuators. In this paper we consider "volume" type actuation, which involves application of an isotropic expansive/contractive stress to the elastic medium. The question of "formability", i.e., whether or not a given modified geometric configuration for the elastic body can be achieved with actuation of this type, is considered at length, in both the two- and three-dimensional contexts, along with related questions of optimal formability, expressed in terms of the L² norm of the volume controller employed. In two dimensions, with the aid of the Airy "stress" function, we establish connections between optimal formation, in the "L"² norm sense, and the standard theory of conformal mapping of simply-connected regions in the complex plane. Further results are presented for multiply-connected domains, including a complete discussion of the case of an annulus.     

Influence of couple-stresses on stress concentrations around the cavity

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2010城市地质环境与可持续发展论坛（二号通知）

用峰值振幅比定义局部化度,用平尾刚度与垂尾刚度的比值定义耦合度. 基于T 尾结构的质量失调模型,从模态峰值振幅比、失调耦合比、常规摄动和近频摄动4个角度, 提出4个不同的局部化判据来预测T尾结构模态局部化的发生. 对一个T尾结构模型局部化 振动的数值分析结果表明: (1) T尾结构系统一般具有弱耦合性,小量的失调就可以使T尾 结构发生模态局部化; (2) T尾结构一旦发生模态局部化,不但使对称一弯模态和反对称一 弯模态的振型发生较大改变,而且其模态频率也将改变,模态频率的改变在失调量的正负区 间内具有唯一性; (3) 算例验证了4个模态局部化判据的可行性和有效性,为T尾结构的模 态局部化分析和设计提供了依据.     

两空间耦合下齿轮传动系统多稳态特性研究

通过将系统参数定义为参数变量, 构成参数空间,研究齿轮传动系统在参数空间和状态空间耦合下的非线性全局动力学特性,以及多参数、多初值和多稳态行为之间的关联特性.首先设计了一个两空间耦合下非线性系统多稳态行为的计算和辨识方法.其次,基于该方法并结合相图、Poincaré映射图、分岔图、最大Lyapunov指数、吸引域等,研究齿轮传动系统在不同参数平面上多稳态行为的存在区域和分布特性,以及多稳态行为在状态平面上的分布特性,揭示了参数平面和状态平面上系统可能隐藏的多稳态行为和分岔,并分析了多稳态行为的形成机理. 结果发现,两空间耦合下系统在参数平面上存在大量多稳态行为并呈"带状"分布, 状态平面上多稳态行为出现两种不同的侵蚀现象, 即内部侵蚀和边界侵蚀.分岔点或分岔曲线对初值的敏感性导致多稳态行为的出现.当齿侧间隙和误差波动在较小的范围内变化时,系统全局动力学特性受间隙和误差扰动的影响较小,受啮合频率的影响较大.两空间耦合下系统全局动力学特性变得丰富和复杂.      

Embedding formulae for Laplace-Beltrami problems on the sphere with a cut

We consider the problem of diffraction of a plane wave by a quarter-plane (a flat cone) with Dirichlet boundary conditions. The most efficient approach to this problem is the technique of Smyshlyaev’s formulae or a modified Smyshlyaev’s formula, both of which are representations of the diffraction coefficient as contour integrals over a complex parameter. These representations have been proven independently. Here we are demonstrating a link between these classes of formulae. The link is established by developing an embedding procedure (in the very special sense of diffraction theory) on the unit sphere.     

Direct and inverse problems of oscillations of an elastoliquid layered medium

A plane problem of forced oscillations of an ideal compressible liquid bounded from above by an elastic layer with a rough lower surface and an inverse geometric problem of determining the shape of the rough lower surface of an elastic layer from the wave characteristics on the upper surface are considered. Three methods are used to solve the direct problem: the small parameter method, the boundary element method, and the Born approximation. Solving the inverse problem is reduced to solving the integral Fredholm equation of the first kind. Results of a numerical experiment are presented.     

Rays and modes for plane wave coupling into a large open-ended circular waveguide

Generalizing an earlier study of high-frequency plane wave coupling into a wide semi-infinite parallel-plane waveguide, this investigation is concerned with coupling into a large semi-infinite circular waveguide. Emphasis is placed on the interplay between ray fields and mode fields, on the excitation mechanism of each, and on their properties when the field incidence angle is strongly oblique, thereby introducing marked azimuthal asymmetries. The tools of analysis are the geometrical theory of diffraction (GTD), physical optics (PO) and hybrid ray-mode theory. The results reveal that coupling from edge-diffracted rays into a waveguide mode occurs when such rays emanating from the flash points on the edge line up with rays in the modal congruence. However, the diffracted ray caustics between initial reflections do not coincide with the cylindrical caustic of the mode, and only after many reflections is there synchronism between the edge-diffracted and modal rays. Complete synchronism throughout obtains for each angular harmonic constituent of the field, for which the incident wave is conical. Under these conditions, one may also distinguish a multiple reflected truncated conical geometric optical beam, the analog of the geometric optical sheet beam in plane parallel geometry. A variety of other ray and modal characteristics are elucidated as well from different perspectives, thereby granting insight into the ray-mode phenomenology in the circular wave-guide environment. Such insight is expected to be useful when one attempts to deal with nonideal but strongly guiding configurations.     

旋转薄板的一种高次动力学模型与频率转向

对固结于转动刚体上柔性薄板的刚柔耦合动力学和频率转向特性进行了深入研究,建立了系统的高次刚柔耦合动力学模型,该动力学模型计入了由于横向变形而引起的面内纵向缩短项,即非线性耦合变形量,并且完整保留了与非线性耦合变形量相关的所有项. 研究表明,高次耦合模型不仅适用于小变形问题,而且还适用于大变形问题,弥补了一次近似耦合模型在处理大变形问题上的不足. 旋转悬臂薄板相邻两阶模态间既有柔和的频率转向现象也有剧烈的频率转向现象. 柔和的频率转向伴随着的振型转换的过程是连续的,而剧烈的频率转向伴随着的振型转换的过程则是不连续的. 相隔多阶模态间存在传递性频率转向,并伴随着振型转移.      

Surface effect on the biaxial buckling and free vibration of FGM nanoplate embedded in visco-Pasternak standard linear solid-type of foundation

In this study, nonlocal elasticity theory in conjunction with Gurtin–Murdoch elasticity theory is employed to investigate biaxial buckling and free vibration behavior of nanoplate made of functionally graded material (FGM) and resting on a visco-Pasternak standard linear solid-type of the foundation. The material characteristics of simply supported FGM nanoplates are assumed to be varied continuously as a power law function of the plate thickness. Hamilton’s principle is implemented to derive the non-classical governing equations of motion and related boundary conditions, which analytically solved to obtain the explicit closed-form expression for complex natural frequencies and buckling loads. Finally, attention is focused on considering the influences of various parameters on variation of damped natural frequency and buckling load ratio such as nonlocal parameter, surface effects, geometric parameters, power law index and properties of visco-Pasternak foundation and it is clearly demonstrated that these factors highly affect on vibration and buckling behavior.     

Dynamics of mechanical systems involving impact and friction using an efficient contact detection algorithm

In this work, some new results are presented on the dynamics of a class of multibody mechanical systems, involving contact and friction. The main contribution refers to the development of a systematic, accurate and efficient method for detecting contact among the components of a system of solid bodies. For some simple geometries, this task is achieved by employing analytical means. For systems possessing components with complex geometric shapes a more involved numerical methodology is developed. In both cases, once a potential contact point is detected, the common tangent plane and normal vector are located and the penetration depth is calculated, leading to determination of the force arising between the contacting bodies. This information is then passed to a solver, providing the full dynamic response of the system. The validity and numerical efficiency of the methodology developed is first demonstrated by considering a number of examples with relatively small geometric complexity but large traditional value and interesting dynamic response. Some new results are obtained and presented on the dynamics of these systems. Finally, the same methodology is also tested in a more complicated and demanding mechanical application.