非对称截面Kirchhoff弹性细杆模型简化方法研究

研究弹性细杆Kirchhoff模型及其相关演化系统, 是深入考察宏观、微观柔性体拓扑结构与稳定性问题的重要依据. 以DNA弹性细杆数学模型为背景, 考虑截面非对称性特征的影响, 构造新的复数形式Kirchhoff系统. 在此基础上, 结合复变量扭矩设解形式, 获得了非对称截面系统的有效抗弯刚度; 并通过相关理论在高维系统简化过程中的应用, 得到了对应于原有系统的单变量二阶常微分方程. 此外, 将DNA分子具备的抗弯刚度周期变化特征转化为针对有效抗弯刚度的周期摄动形式, 以期从总体上减少理论分析对于数值积分的依赖, 为后续定量分析工作提供新的思路.     

受曲面约束弹性细杆的平衡问题

作为DNA等一类生物大分子的力学模型，弹性细杆的非线性力学再次受到关注，形成一个力学与分子生物学的交叉学科.除了不受外界约束的自由弹性细杆外，受曲面约束的弹性细杆静力学具有重要的应用背景.在分析约束、约束方程和约束力的基础上建立了受曲面约束的圆截面弹性细杆的平衡微分方程，即曲面上的Kirchhoff方程，它是以截面主矢和截面姿态坐标以及中心线的Descartes坐标为变量的微分/代数方程.作为应用，讨论了约束是圆柱面的情形.此时平衡的无量纲方程仅含的物理参数是截面对形心的抗扭刚度和对主轴的抗弯刚度的比值，与几何参数无关.由此导出方程的螺旋杆特解.数值计算表明，对弹性细杆中心线的几何形状有显著影响的是截面主矢和姿态坐标及其导数的起始值，而不是物理参数. 关键词： 弹性细杆 DNA超螺旋 曲面约束 螺旋杆     

受拉扭弹性细杆超螺旋形态的定性分析

基于弹性杆的Kirchhoff模型讨论受拉扭弹性细杆的超螺旋形态.导出细长螺旋杆的等效抗弯和抗扭刚度.分析受拉扭弹性细杆的稳定性和分岔,且利用等效刚度概念将弹性杆的稳定性条件应用于对细长螺旋杆稳定性的判断.在扭矩不变条件下增加拉力至极限值时,直杆平衡状态失稳转为螺旋杆状态.继续增加拉力,直螺旋杆平衡状态失稳卷绕为超螺旋杆.从而对Thompson/Champney实验中受拉扭弹性细杆形成超螺旋形态的多次卷绕现象作出定性的理论解释. 关键词： 弹性细杆 Kirchhoff动力学比拟 等效刚度 超螺旋形态     

弹性细杆弯曲的Kirchhoff方程的违约校正求解

采用Euler四元数表示的Kirchhoff方程来研究受力挤压作用下的弹性细杆的拓扑构形，进一 步研究弹性细杆的力学性质；将得到的微分方程与约束条件组成微分代数方程后再转化为微 分方程规范形式以便求解；为满足边界条件，应用数值打靶法求解边值条件，并将弹性细杆 在力作用下的拉压过程用Matlab仿真出来.同时对由于误差导致的违约现象进行处理，并针 对欧拉参数的特征，选取合适的修正系数以保持方程的稳定性. 关键词： DNA Euler四元数 Kirchhoff方程 弹性细杆 违约修正     

基于高斯原理的Cosserat弹性杆动力学模型

在动力学普遍原理中, 高斯最小拘束原理的特点是可通过寻求函数极值的变分方法直接得出运动规律, 而无须建立动力学微分方程. Kirchhoff动力学比拟方法以刚性截面的姿态表述弹性细杆的几何形态, 并发展为以弧坐标s和时间t为自变量的弹性杆分析力学. 由于截面姿态的局部微小改变沿弧坐标的积累不受限制, Kirchhoff模型适合描述弹性杆的超大变形. Cosserat弹性杆模型考虑了Kirchhoff模型忽略的截面剪切变形、中心线伸缩变形和分布力等因素, 是更符合实际弹性杆的动力学模型. 建立了基于高斯原理的Cosserat弹性杆的分析力学模型, 导出拘束函数的普遍形式, 以平面运动为例进行讨论. 关于弹性杆空间不可自相侵占的特殊问题, 给出相应的约束条件对可能运动施加限制, 以避免自相侵占情况发生.     

弹性压扭直杆的Greenhill公式对精确模型的推广

将圆截面Kirchhoff弹性压扭直杆的Greenhill公式推广到精确模型.基于平面截面假定,在弯扭的基础上增加了拉压和剪切变形,将弹性杆的位形表达为截面的弧坐标历程.由弹性杆精确模型的平衡微分方程,得到了两端受力螺旋作用时对应于直线平衡状态的特解,导出了线性化扰动方程及其通解,再根据两端为铰支时的边界条件以及积分常数存在非零解的条件导出弹性直杆精确模型的Greenhill公式.结果表明,由力螺旋表示的稳定域为一对称的封闭区域,拉压和剪切对稳定性的影响取决于拉压柔度与剪切柔度之差、抗弯刚度和杆长这三个因素.     

Kirchhoff弹性直杆在力螺旋作用下的稳定性

研究受力螺旋作用的圆截面Kirchhoff弹性直杆在各种边界条件下的稳定性问题. 用直角坐标和Cardano角表示截面的形心位置和姿态. 由Kirchhoff方程得到弹性细杆的直线平衡特解,导出线性化扰动方程及其通解. 根据边界条件确定积分常数的非零解存在条件,讨论了各种边界条件,如两端铰支、两端固定、一端铰支一端固定以及一端固定一端自由的弹性细杆直线平衡状态的稳定性,导出了临界载荷的表达式,绘制了稳定域,将Greenhill公式推广到其他边界条件,并且使压杆的Euler 公式成为其特例. 关键词： Kirchhoff弹性杆 稳定性 力螺旋 Greenhill公式     

弹性细杆平衡的动态稳定性

从动力学观点讨论弹性细杆的平衡稳定性问题.建立了弹性杆的动力学方程，导出了杆的弯扭度与截面角速度之间的运动学关系式.对具有弧坐标s和时间t双重自变量的离散动力系统扩充了Lyapunov 稳定性定义.以具有初扭率的非圆截面直杆的平衡稳定性为例，应用一次近似方法证明了当静力学意义下的稳定性条件得到满足时，动力学意义下的稳定性条件必同时满足. 关键词： 弹性杆动力学方程 Kirchhoff理论 Lyapunov稳定性     

圆截面弹性螺旋杆的稳定性与振动

基于Kirchhoff理论讨论圆截面弹性螺旋杆的动力学问题.以杆中心线的Frenet坐标系为参考系，建立用欧拉角描述的弹性杆动力学方程.讨论其在端部轴向力和扭矩作用下保持的无扭转螺旋线平衡状态.在静力学和动力学领域内讨论其平衡稳定性问题.还讨论了弹性杆平衡的Lyapunov稳定性和欧拉稳定性两种不同稳定性概念之间的区别和联系.在一次近似意义下证明了螺旋杆在空间域内的欧拉稳定性条件是时域内Lyapunov稳定性的必要条件.导出了解析形式螺旋杆三维弯曲振动的固有频率，为螺旋线倾角和受扰挠性线波数的函数. 关键词： 弹性螺旋杆 Kirchhoff动力学比拟 Lyapunov稳定性 欧拉稳定性     

Kirchhoff弹性杆动力学建模的分析力学方法

以杆的横截面为研究对象，讨论了其自由度，给出了截面虚位移定义，并定义变分和偏微分运算对独立坐标服从交换关系. 给出了曲面约束的基本假设，讨论了约束对截面自由度的影响以及加在虚位移上的限制方程. 从D'Alembert原理出发结合虚功原理，建立了弹性杆动力学的D'Alembert-Lagrange原理，当杆的材料服从线性本构关系时，化作Euler-Lagrange形式、Nielsen形式和Appell形式. 由此导出了Kirchhoff方程以及Lagrange方程、Nielsen方程和Appell方程，得到 关键词： 超细长弹性杆 分析力学方法 Kirchhoff动力学比拟 变分原理     

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract

We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.     

Vibrations of oblique shear-deformable plates

The proposed numerical analysis of moderately thick plates subject to rather general boundary conditions is based on the direct boundary element method (BEM) in the frequency domain. First order shear-deformation theory of the Reissner-Mindlin-type is considered. A step forward in efficiency is obtained when the force and double force with moment Green's functions of the rectangular simply supported base plate of the same stiffness are applied. The time-reduced equations of hard-hinged polygonal plates correspond to those of a background Kirchhoff plate having frequency-dependent effective parameters like mass, lateral and in-plane load, and is further forced by imposed fictitious curvatures. This analogy holds even for the quasi-static shear forces and bending moments, i.e., when inertia effects become negligible. Furthermore, it can be shown that, in the static case, these stress resultants for certain groups of Reissner-type shear-deformable plates are identical with those resulting from the Kirchhoff theory in the background. Since this analogy is restricted to hard-hinged supports of straight edges, it is necessary to apply, e.g., the direct BEM of analysis to the plate of general planform and boundary conditions. The main effort is thus to study the properties and effective representations of the Green's dyadics and their singularities, in view of their proper integration. Similarly as for Kirchhoff plates, the strong singularity of the infinite domain is identified for the rectangular plate and subject to indirect integration. The resulting direct BEM proves to be efficient, robust and, in connection with proper pre- and post-processors, becomes an effective tool of engineering analyses just within the frequency limits given by the first two of the three spectral branches.     

变厚度弹性圆薄板问题

一.引言 变厚度的圆薄板常常在机器零件的设计中遇到,例如蒸汽涡输机和活塞的膜片便是。 用平板小挠度理论求解变厚度平板问题并不是容易的事,因此到目前为止,这类的实际问题已解决的为数并不多。     

带强迫项变系数组合KdV方程的无穷序列复合型类孤子新解

为了获得变系数非线性发展方程的无穷序列复合型新解,研究了G′(ξ)G(ξ)展开法.通过引入一种函数变换,把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题.在此基础上,利用Riccati方程解的非线性叠加公式,获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解.借助这些复合型新解与符号计算系统Mathematica,构造了带强迫项变系数组合KdV方程的无穷序列复合型类孤子新精确解.     

一个基于Chen系统的新混沌系统的分析与同步

通过在Chen系统的第一个方程中加入一个可变系数的乘积项, 构造了一个新的三维自治混沌系统.新系统可通过调节其可变系数实现不同系数组合下系统的混沌产生或混沌抑制, 即调节该乘积项的可变系数, 可使不出现混沌的Chen系统产生混沌现象, 同时也可使产生混沌运动的Chen系统不再产生混沌现象.详细分析了新系统的特性, 研究了新系统的混沌同步问题, 并给出了相应的仿真结果.     

Wavelength tuning characteristics of a fabry-perot semiconductor laser with an external short cavity

Wavelength tuning characteristics of a Fabry-Perot semiconductor laser with an external short cavity are analyzed, in which the oscillation wavelength can be changed by slightly altering the external cavity length. Analysis is based on rate equations for an optical power and carrier density. It is shown that the wavelength tuning range is dominated by a change of carrier density through the effect of carrier-induced refractive index change in the active layer of a laser diode. This depends on effective coupling coefficients of the optical field iteratively reflected back to the laser diode by the external mirror, and on reflection coefficients of an anti-reflection coated laser facet and the external mirror which compose the external cavity. The effective coupling coefficient is also derived using the waveguide theory and Kirchhoff ’s theory. Finally, an unstable condition which may limit a stable wavelength tuning range is shown by results of a linear stability analysis of rate equations.     

A basic hybrid finite element formulation for mid-frequency analysis of beams connected at an arbitrary angle

When beams are connected at an arbitrary angle and subjected to an external excitation, both longitudinal and bending waves are generated in the system. Since longitudinal wavelengths are considerably longer than bending wavelengths in the mid-frequency region, the number of bending wavelengths in the beams is considerably larger than the number of longitudinal wavelengths. In this paper, plannar beams connected at arbitrary angles are considered. The energy finite element analysis (EFEA) is employed for modelling the bending behavior of the beams and the conventional finite element analysis (FEA) is utilized for modelling the longitudinal vibration in the beams. Thus, a basic hybrid FEA formulation is presented for mid-frequency analysis of systems that contain two types of energy. The bending vibration is associated with the long members in the system and the longitudinal vibration is associated with the short members. The long members are considered to have high modal overlap and to contain several wavelengths within their dimension, and uncertainty effects are present. The short members contain a small number of wavelengths, and exhibit a low modal overlap. Due to the low modal overlap the resonant frequencies are spaced far apart in the frequency domain, therefore the short members exhibit resonant or non-resonant behavior depending on the frequency of the excitation.In this work, the bending and the longitudinal vibration within the same beam member are treated as a long and as a short member, respectively. A hybrid joint formulation is developed between long and short members. Power reflection and transmission coefficients are derived for each joint. The distribution of the energy throughout the system demonstrates a strong dependency on the power transfer coefficients. Several systems are analyzed by the hybrid FEA and by analytical solutions, and good correlation between them is observed.     

Analysis of multiple scattering from two-dimensional dielectric sea surface with iterative Kirchhoff approximation

<正>An iterative method in the Kirchhoff approximation is proposed for high frequency multiple electromagnetic scattering from two-dimensional dielectric sea surface.The multiple interaction of the scattering field is characterized with the corrected electromagnetic currents of the wind-driven sea surface.The actual surface currents are approximated with the iterative solution of the corrected currents.A newly developed sea spectrum,Elfouhaily spectrum,is utilized to build the sea surface model.The shadowing correction is improved by the Depth-Buffer algorithm.The validity of the iterative Kirchhoff approximation is verified by the agreement of backscattering coefficients with the measured data.     

Impulse response of an infinitely long thick strip plate

This paper is a study of the dynamic response of an infinitely long thick strip plate subjected to an impulsive load. The plate is simply supported along the edges and resting on an elastic foundation. The problem is studied on the basis of a plate theory in which the effects of rotatory inertias and shear deformations are retained. Governing equations are solved by applying the methods of the Laplace transform with respect to time and the Fourier transform with respect to a longitudinal space variable. Dynamic coefficients (maximum dynamic displacement/static displacement, maximum dynamic bending moment/static bending moment) are calculated numerically for plates subjected to a step line load and shown graphically for various values of the parameters included.