Free vibrations of skirt supported pressure vessels — an engineering application of timoshenko beam theory

Timoshenko beam theory is applied to the study of the free vibrations of skirt supported pressure vessels in this paper; such systems are used in the process and power generation industries as well as aboard nuclear powered vessels. It is shown that the analysis is not significantly more complicated than the analysis of skirt-vessel combinations by Euler-Bernoulli beam theory. This latter analysis is provided in an appendix. Two sets of boundary conditions are considered: namely, the cases of (a) a cantilevered system and (b) a fixed-pinned system. The first two natural frequencies of nine typical cases are calculated and compared with the corresponding results obtained from Euler-Bernoulli beam theory. The numerical differences are significant so that if a beam theory is adequate to model the system, it is clear that Timoshenko beam theory is the appropriate one to use. In addition, the first two mode shapes for a particular case are presented for comparison with the corresponding mode shapes predicted by Euler-Bernoulli beam theory. Finally, some comments on the modeling and analysis of specific, real systems are made. It is emphasized that the purpose of the paper is to demonstrate that Timoshenko beam theory is not unduly difficult to apply to problems of engineering interest when a beam theory model is suitable.     

The refined theory of deep rectangular beams based on general solutions of elasticity

The problem of deducing one-dimensional theory from two-dimensional theory for a homogeneous isotropic beam is investigated. Based on elasticity theory, the refined theory of rectangular beams is derived by using Papkovich-Neuber solution and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beam under transverse loadings are derived directly from the refined beam theory and are almost the same as the governing equations of Timoshenko beam theory. In two examples, it is shown that the new theory provides better results than Levinson’s beam theory when compared with those obtained from the linear theory of elasticity.     

衍射理论对局域空心光束及无衍射光束重建的描述

利用衍射理论导出了局域空心光束的传输表达式及光强分布,给出了局域空心光束的精细结构,详细分析了其演变过程。讨论了聚焦透镜的焦距f对径向暗域最大尺寸及轴上暗域长度的影响。结果表明,径向暗域最大尺寸及轴上暗域长度都随着f的增大而增大。通过轴棱锥-透镜系统获得局域空心光束,用体视显微镜和CCD照相机组成的系统拍摄光束强度分布。结果表明应用衍射理论可以较精确地描述局域空心光束的演变过程。找出了其应用中的不利因素,更清晰地展现无衍射光束的重建现象。这种描述方法弥补了几何理论和干涉理论的不足。     

漂移对聚焦高斯光束闪烁影响的数值模拟

 采用非自适应坐标变换对聚焦高斯光束在湍流大气中的传输进行了数值模拟,结果显示轴闪烁指数并没有出现如Rytov理论所预言的随初始光束半径的增大而明显减小的现象,其原因在于Rytov近似理论未考虑大尺度湍涡产生的漂移效应对闪烁的贡献。对比数值模拟结果与漂移理论结果以及相关实验结果,三者相吻合,表明未考虑漂移效应的Rytov近似理论不能完全准确地描述聚焦光束的闪烁特征,在研究聚焦光束的闪烁时,应当考虑漂移的影响。     

A new rectangular beam theory

A new theory for beams of rectangular cross-section which includes warping of the cross-sections is presented in the present work. By satisfying the shear-free conditions on the lateral surfaces of the beam a pair of coupled equations of motion are obtained such that no arbitrary shear coefficient is required. It is shown that the uncoupled equation for the transverse displacement is the same as the corresponding equation in Timoshenko beam theory provided that for the Timoshenko equation the shear coefficient is taken to be $\text{5}\text{6}$; this value lies within the range of values, 0·822–0·870, appearing in the literature for the beam of rectangular cross-section. Results for two typical static examples are given for both the new theory and Timoshenko beam theory. These results are compared with the solutions of the comparable problems in the linear theory of elasticity. For the end loaded cantilever beam the new theory predicts the same result for the neutral surface deflection as does the linear theory of elasticity while Timoshenko beam theory underestimates the shear correction term by 20%. For the uniformly loaded and simply supported case both beam theories provide the same overestimate of the central deflection when compared with the theory of elasticity solution.     

Effect of turbulent atmosphere on the on-axis average intensity of Pearcey–Gaussian beam

The propagation characteristics of the Pearcey–Gaussian(PG) beam in turbulent atmosphere are investigated in this paper.The Pearcey beam is a new kind of paraxial beam,based on the Pearcey function of catastrophe theory,which describes diffraction about a cusp caustic.By using the extended Huygens–Fresnel integral formula in the paraxial approximation and the Rytov theory,an analytical expression of axial intensity for the considered beam family is derived.Some numerical results for PG beam propagating in atmospheric turbulence are given by studying the influences of some factors,including incident beam parameters and turbulence strengths.     

The refined theory of transversely isotropic piezoelectric rectangular beams

The problem of deducing one-dimensional theory from two-dimensional theory for a transversely isotropic piezoelectric rectangular beam is investigated. Based on the piezoelasticity theory, the refined theory of piezoelectric beams is derived by using the general solution of transversely isotropic piezoelasticity and Lur’e method without ad hoc assumptions. Based on the refined theory of piezoelectric beams, the exact equations for the beams without transverse surface loadings are derived, which consist of two governing differential equations: the fourth-order equation and the transcendental equation. The approximate equations for the beams under transverse loadings are derived directly from the refined beam theory. As a special case, the governing differential equations for transversely isotropic elastic beams are obtained from the corresponding equations of piezoelectric beams. To illustrate the application of the beam theory developed, a uniformly loaded and simply supported piezoelectric beam is examined.     

一种用于光束整形的衍射光学元件设计算法

在光束整形衍射光学元件的设计中,为同时减小输出光束的均方根误差和顶部不均匀度值,提出了模糊控制迭代算法(IAFC)。在盖师贝格-撒克斯通(Gerchberg-Saxton,G-S)算法的基础上,提出了平滑修正法,可有效改善输出光束的顶部均匀度,但却增大了均方根误差值。模糊控制迭代算法依据模糊控制理论,通过有效结合盖师贝格-撒克斯通算法和平滑修正法来同时降低均方根误差和顶部不均匀度值。计算机设计的结果表明,利用模糊控制迭代算法可以得到非常理想的输出光束,其均方根误差和顶部不均匀度值分别为0.75%和0.46%,能量转换效率可达94.91%。为光束整形衍射光学元件的设计提供了一种有效的新算法。     

无衍射 Mathieu光束自重建特性的理论和实验研究

首次对无衍射Mathieu光束的自重建特性进行理论和实验研究, 利用Mathieu-Hankel波理论分析了Mathieu光束的自重建机理. 基于菲涅尔衍射积分理论, 推导出了高斯吸收型圆形障碍物部分遮挡后的Mathieu光场重建后的解析表达式, 并数值模拟了无衍射Mathieu光束的经圆形障碍物部分遮挡后光场的自重建过程. 采用柱透镜-轴棱锥组合光学系统产生近似零阶无衍射Mathieu光束, 实验分别验证了轴上和轴外障碍物遮挡时近似零阶无衍射Mathieu光束的自重建特性. 理论模拟与实验结果相符.     

泵浦光束质量对受激拉曼散射的影响

主要讨论泵浦光束质量对低气压拉曼种子源特性的影响。文中给出了一阶斯托克斯光泵浦阈值、输出能量和光束质量的实验数据。用Ｍ＾２因子分析了泵浦光和一阶斯托克斯光的光束质量。根据考虑泵浦聚焦和泵浦光束质量影响的近似拉曼散射理论计算了泵浦阈值的理论曲线，并与实验数据和有关文献进行了比较。     

Dynamics of finite-length tubular beams conveying fluid

The dynamics and stability of short tubes conveying fluid is re-examined by means of Timoshenko beam theory for the tube and a three-dimensional fluid-mechanical model for the fluid flow, rather than the plug-flow model utilized heretofore. The tubes considered are either clamped at both ends or cantilevered; in the latter case, special “outflow models” were introduced to describe the boundary conditions on the fluid exiting from the free end. By comparison with experiments, it is shown that this refined theory is necessary for describing adequately the dynamical behaviour of extremely short tubes, although Timoshenko beam theory together with a plug-flow model are quite satisfactory for relatively longer short tubes; for long tubes, Euler-Bernoulli beam theory and a plug-flow model are perfectly adequate.     

Measurements of Beam Shape Coefficients in Generalized Lorenz-Mie Theory and the Density-Matrix Approach. Part 2: The Density-Matrix Approach

In Part 1, it was demonstrated that intensity measurements on an actual beam in the laboratory allow one to determine so-called density matrices related to the beam shape coefficients encoding the beam and used in many expressions of the generalized Lorenz-Mie theory. In this paper, expressions in the generalized Lorenz-Mie theory are rewritten and it is shown that they can be expressed in terms of the density matrices rather than in terms of beam shape coefficients, leading to what is called the density matrix approach to the generalized Lorenz-Mie theory. The possibility of rewriting the generalized Lorenz-Mie theory in terms of quantities describing the illuminating beam and experimentally measurable in the laboratory offers new opportunities in optical characterization.     

径向渡越时间振荡器互作用特性的流体理论分析

 用线性流体理论分析了径向渡越时间振荡器谐振腔中电子束与高频场的相互作用，通过数值求解得到了其中束 场能量交换等随直流渡越角的变化规律。与单电子模型的分析结果相比较，流体模型所得结果更具有普适性。     

Dynamic coefficient of a two-layered thick beam with imperfect bonding

This paper is concerned with the impulse response of a two-layered prestressed beam with flexible bonding resting on an elastic foundation. Two dissimilar layers of the beam are assumed to bend according to the Timoshenko beam theory. Numerical results are given for simply supported beams subjected to a uniformly distributed step load. The dynamic coefficient (= dynamical maximum moment/static moment in the layer) is calculated as a function of the shear bond stiffness (ranging from zero to infinity) for several values of the axial prestress and the foundation stiffness parameters. The results are also compared with those from the Euler beam theory.     

外开槽同轴波导大轨道回旋自谐振脉塞放大器

 运用大轨道电子束,对外开槽同轴波导回旋自谐振放大器进行了线性动力学理论分析和非线性自洽模型模拟。由线性动力学理论分析和非线性模拟得到的结果在线性增长部分吻合得很好。为了有效抑制绝对不稳定性,对电子束流的选择进行了讨论。计算结果表明：对于电压为700 kV, 电流为100 A, 电子束纵横向速度比为0.8的大轨道电子束, TE₅₁外开槽同轴波导回旋自谐振脉塞放大器的峰值功率和峰值效率分别可以达到6.27 MW和8.96%。     

Free vibration analysis of a cracked shear deformable beam on a two-parameter elastic foundation using a lattice spring model

The free vibration of a shear deformable beam with multiple open edge cracks is studied using a lattice spring model (LSM). The beam is supported by a so-called two-parameter elastic foundation, where normal and shear foundation stiffnesses are considered. Through application of Timoshenko beam theory, the effects of transverse shear deformation and rotary inertia are taken into account. In the LSM, the beam is discretised into a one-dimensional assembly of segments interacting via rotational and shear springs. These springs represent the flexural and shear stiffnesses of the beam. The supporting action of the elastic foundation is described also by means of normal and shear springs acting on the centres of the segments. The relationship between stiffnesses of the springs and the elastic properties of the one-dimensional structure are identified by comparing the homogenised equations of motion of the discrete system and Timoshenko beam theory.     

氢原子束在大气长程传输中自剥离效应研究

氢原子束在大气传输时,束流粒子与大气粒子碰撞电离形成的大气剥离效应,以及和大气剥离产生次级粒子碰撞电离形成的自剥离效应,是造成氢原子束能量损失的重要机制。考虑到自剥离效应成因复杂,虽然目前已有一些理论方面的研究结果,但对其发生机理和对束流损失效果尚未有实验或数值模拟方面的工作,因此,通过对自剥离效应的发生机理和对束流损失的影响进行分析,进一步完善了自剥离效应理论,在通过束流传输方程验证了粒子云网格-蒙特卡罗法对氢原子束大气传输仿真模拟适用的基础上,将仿真结果与自剥离理论进行了对比,验证了自剥离效应理论的适用性。模拟结果表明,自剥离效应是由束流被大气电离产生的带电次级粒子团在地磁场的影响不停地穿越束流导致的,且自剥离效应的强弱与原子束的密度有关,束流密度越大,自剥离效应越强,对束流的影响越大。     

Investigation of intense sheet electron beam transport using the macroscopic cold-fluid model and the single-particle orbit theory

The focusing and the stable transport of an intense elliptic sheet electron beam in a uniform magnetic field are investigated thoroughly by using the macroscopic cold-fluid model and the single-particle orbit theory.The results indicate that the envelopes and the tilted angles of the sheet electron beam obtained by the two theories are consistent.The single-particle orbit theory is more accurate due to its treatment of the space-charge fields in a rectangular drift tube.The macroscopic cold-fluid model describes the collective transport process in order to provide detailed information about the beam dynamics,such as beam shape,density,and velocity profile.The tilt of the elliptic sheet beam in a uniform magnetic field is carefully studied and demonstrated.The results presented in this paper provide two complete theories for systemically discussing the transport of the sheet beam and are useful for understanding and guiding the practical engineering design of electron optics systems in high power vacuum electronic devices.     

Self-consistent field theory of a helix traveling wave tubeamplifier

A self-consistent relativistic field theory of a helix traveling wave tube (TWT) is presented for a configuration in which a thin annular beam propagates through a sheath helix enclosed within a loss-free wall. A linear analysis of the interaction is carried out, subject to the boundary conditions imposed by the beam, helix, and wall. A detrimental dispersion equation is obtained which implicitly includes beam space-charge effects without recourse to a heuristic model of the space-charge field. The equation is valid for arbitrary azimuthal mode number and is solved numerically for the azimuthally symmetric case. The coupled-wave Pierce theory is recovered in the near-resonant limit. Numerical comparisons between the complete dispersion equation and the Pierce model are described. A discrepancy is found between the Pierce and the field theory even for low currents in the nominally ballistic regime, owing to the dielectric effect of the beam on the helix modes     

双轴棱锥产生长距离近似无衍射光的新技术

提出了一种利用会聚透镜、轴棱锥等简单光学元件产生长距离近似无衍射光的新技术. 分别利用几何光学和衍射理论分析了该方法产生长距离近似无衍射光束的原理, 通过软件模拟了长距离近似无衍射光束的形成过程,得出了该光束在不同距离处的横向光强分布. 模拟结果显示该光束在较长距离处的横向光强分布满足Bessel分布. 从实验上获得了传播距离长达80 m、中心光斑发散角约为0.12 mrad的近似无衍射光束, 相比于国外学者最近的研究成果(Belyi et al. 2010 Opt. Exp. 18 1966)将传播距离延长了50多米,而光束发散角压缩了22倍.实验中, 对光束沿光轴传播时在不同距离处的光斑进行了拍摄,所得实验结果与理论分析基本符合.