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.     

A study of the effect of laser beam geometries on laser bending of sheet metal by buckling mechanism

Laser beam forming has emerged as a new and very promising technique to form sheet metal by thermal residual stresses. The objective of this work is to investigate numerically the effect of rectangular beam geometries, with different transverse width to length aspect ratio, on laser bending process of thin metal sheets, which is dominated by buckling mechanism. In this paper, a comprehensive thermal and structural finite element (FE) analysis is conducted to investigate the effect that these laser beam geometries have on the process and on the final product characteristics. To achieve this, temperature distributions, deformations, plastic strains, stresses, and residual stresses produced by different beam geometries are compared. The results suggest that beam geometries play an important role in the resulting temperature distributions on the workpiece. Longer beam dimensions in the scanning direction (in relation to its lateral dimension) produce higher temperatures due to longer beam–material interaction time. This affects the bending direction and the magnitude of the bending angles. Higher temperatures produce more plastic strains and hence higher deformation. This shows that the temperature-dependent yield stress plays a more dominant role in the deformation of the plate than the spread of the beam in the transverse direction. Also, longer beams have a tendency for the scanning line to curve away from its original position to form a concave shape. This is caused by buckling which develops tensile plastic strains along both ends of the scanning path. The buckling effect produces the opposite curve profile; convex along the tranverse direction and concave along the scanning path.     

Second Harmonic Generation of Self Focused Cosh‐Gaussian Laser Beam in Collisionless Plasma

This paper presents an investigation of self‐focusing of a Cosh‐Gaussian (ChG) laser beam and its effect on second harmonic generation in collisionless plasma. In the presence of ChG laser beam the carriers get redistributed from high field region to low field region on account of ponderomotive force as a result of which a transverse density gradient is produced in the plasma which in turn generates an electron‐plasma wave at pump frequency. Generated plasma wave interacts with the incident laser beam and hence generates its second harmonics. Moment theory has been used to derive differential equation governing the evolution of spot size of ChG laser beam propagating through collisionless plasma. The differential equation so obtained has been solved numerically. The effect of decentered parameter, intensity of ChG laser beam and density of plasma on self‐focusing of the laser beam and second harmonic yield has been investigated. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Effect of High Spatial Harmonics in RF Linac

The effect of high spatial harmonics on beam dynamics in a axisymmetric radio frequency linear accelerator (RF linac) is examined. The high spatial harmonics forms the second order ponderomotive focusing for the relativistical particle beam when the transverse motion is averaged over the period of RF structure, the beam transverse envelop equation is obtained including the ponderomotive focusing force which is useful for the design of RF gun and linac. Under a certain of high spatial harmonics coefficients, the high spatial harmonics increase the energy spread due to the stochasitical beam energy diffusion.     

Sound beams with shockwave pulses

The beam equation for a sound beam in a diffusive medium, called the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, has a class of solutions, which are power series in the transverse variable with the terms given by a solution of a generalized Burgers’ equation. A free parameter in this generalized Burgers’ equation can be chosen so that the equation describes an N-wave which does not decay. If the beam source has the form of a spherical cap, then a beam with a preserved shock can be prepared. This is done by satisfying an inequality containing the spherical radius, the N-wave pulse duration, the N-wave pulse amplitude, and the sound velocity in the fluid.     

Nonlinear free transverse vibration of an axially moving beam: Comparison of two models

Nonlinear free transverse vibration of an axially moving beam is investigated. A partial-differential equation governing the transverse vibration is derived from the Newton's second law. Under the assumption that the tension of beam can be replaced by the averaged tension over the beam, the partial-differential reduces to a widely used integro-partial-differential equation for nonlinear free transverse vibration. The method of multiple scales is applied directly to two equations to evaluate nonlinear natural frequencies. Numerical examples are presented to demonstrate the analytical results and to highlight the difference between two models. Two models yield the essentially same results for the weak nonlinearity, the small axial speed and the low mode, while the difference between two models increases with the nonlinear term, the axial speed, and the order of mode.     

Forced response of neutrally buoyant inflated viscoelastic cantilevers to ocean waves

The dynamics of the neutrally buoyant inflated viscoelastic cantilevers constituting a submarine detection system is investigated. Thin shell theory is used to account for the stresses arising due to the internal pressure. A significant feature of the analysis is the use of the reduced shell equation which is similar in form to that for a vibrating beam with rotary effects. The forcing function in the form of surface wave excitation consists of a fundamental frequency and its second harmonic. Both the effects of apparent inertia and viscous drag are accounted for. The highly complicated non-linear, coupled equations are analyzed numerically. Use of the reduced form of the shell equations appears to avoid the problems of numerical instability and convergence reported by several investigators. The amount of information generated is rather enormous; however, for conciseness, only a few of the typical data, sufficient to establish trends, are presented. The results suggest that for the case of simple harmonicexcitation, the non-linear hydrodynamic drag introduces no superharmonic components into the response. The analysis provides valuable information concerning the system parameters leading to critical response and hence should prove useful in the design of inflatable structural members.     

Acceleration and focusing of intense ion beams in rf undulator structures

Bunching, acceleration, and transverse focusing of intense ion beams in an undulator linac are considered. Such an accelerator features the absence of an rf field harmonic synchronous with the beam. A 3D equation of motion in the Hamiltonian form is derived in the smooth approximation, and the general conditions for ion beam acceleration and transverse focusing in the undulator linac are formulated. Basic analytical results are compared with the results of numerical simulation of the beam dynamics in the polyharmonic field of an accelerating cavity.     

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.     

高增益自由电子激光饱和态分析的三维统计理论

考虑电子束横向发射度和电子β振荡,将2005年国际上提出的单通过高增益自由电子激光饱和状态分析的统计物理方法发展到三维情形。首先建立一种描述电子三维运动的归一化简化模型,推导了一维光场下包含电子横向运动的Vlasov方程。在螺旋型波荡器情形下通过引入横向运动守恒量发展了三维统计物理分析方法,并编写了相应计算程序,计算自由电子激光达到饱和时系统的光强增益、聚束因子。作为对比验证,编写包含N个电子自由电子激光系统的三维直接数值模拟程序,结果表明数值模拟和统计计算结果相一致。对比文献中一维模拟和一维统计理论计算结果,所得结果反映了电子束横向发射度以及电子在波荡器中的横向β振荡对饱和点参数的影响。     

Second Harmonic Generation of Self‐Focused Cosh‐Gaussian Laser Beam in Thermal Quantum Plasma by Excitation of an Electron Plasma Wave

This paper presents a scheme for second harmonic generation (SHG) of an intense Cosh‐Gaussian (ChG) laser beam in thermal quantum plasmas. Moment theory approach in W.K.B approximation has been adopted in deriving the differential equation governing the propagation characteristics of the laser beam with distance of propagation. The effect of relativistic increase in electron mass on propagation dynamics of laser beam has been incorporated. Due to relativistic nonlinearity in the dielectric properties of the plasma, the laser beam gets self‐focused and produces density gradients in the transverse direction. The generated density gradients excite electron plasma wave (EPW) at pump frequency that interacts with the incident laser beam to produce its second harmonics. Numerical simulations have been carried out to investigate the effects of laser parameters on selffocusing of the laser beam and hence on the conversion efficiency of its second harmonics. Simulation results predict that within a specific range of decentered parameter the ChG laser beams show smaller divergence as they propagate and, thus, lead to enhanced conversion efficiency of second harmonics. (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam

An axially moving nested cantilever beam is a type of time-varying nonlinear system that can be regarded as a cantilever stepped beam. The transverse vibration equation for the axially moving nested cantilever beam with a tip mass is derived by D’Alembert?s principle, and the modified Galerkin?s method is used to solve the partial differential equation. The theoretical model is modified by adjusting the theoretical beam length with the measured results of its first-order vibration frequencies under various beam lengths. It is determined that the length correction value of the second segment of the nested beam increases as the structural length increases, but the corresponding increase in the amplitude becomes smaller. The first-order decay coefficients are identified by the logarithmic decrement method, and the decay coefficient of the beam decreases with an increase in the cantilever length. The calculated responses of the modified model agree well with the experimental results, which verifies the correctness of the proposed calculation model and indicates the effectiveness of the methods of length correction and damping determination. Further studies on non-damping free vibration properties of the axially moving nested cantilever beam during extension and retraction are investigated in the present paper. Furthermore, the extension movement of the beam leads the vibration displacement to increase gradually, and the instantaneous vibration frequency and the vibration speed decrease constantly. Moreover, as the total mechanical energy becomes smaller, the extension movement of the nested beam remains stable. The characteristics for the retraction movement of the beam are the reverse.     

A conserved quantity and the stability of axially moving nonlinear beams

Free nonlinear transverse vibration is investigated for an axially moving beam modeled by an integro-partial-differential equation. Based on the equation, a conserved quantity is defined and confirmed for axially moving beams with pinned or clamped ends. The conserved quantity is applied to demonstrate the Lyapunov stability of the straight equilibrium configuration in transverse nonlinear of beam with a low axial speed.     

旋转柔性悬臂梁动力学的Bezier插值离散方法研究

在旋转柔性梁变形场描述中,引入Bezier插值离散方法.首先构建旋转运动悬臂梁物理模型,接着采用第二类Lagrange动力学方程和Bezier插值离散方法,在计入柔性梁横向弯曲变形引起的纵向缩短的情况下,推导了旋转柔性梁的刚柔耦合动力学方程,并编制旋转柔性梁的动力学仿真软件,然后通过仿真算例对系统的动力学问题进行研究.最后将仿真结果与有限元法、假设模态法进行分析比较,验证了提出的Bezier插值离散方法的正确性,并得出Bezier插值离散法的计算效率较高;计算精度符合工程实际需要,高速时计算精度大于假设模态法;Bezier插值离散方法在处理大柔性问题时比假设模态法合理.因此在多体系统动力学领域具有优良性能和应用价值的Bezier插值离散方法将具有推广价值.     

Solitons in PT-symmetric potential with competing nonlinearity

We investigate the effect of competing nonlinearities on beam dynamics in PT-symmetric potentials. In particular, we consider the stationary nonlinear Schrödinger equation (NLSE) in one dimension with competing cubic and generalized nonlinearity in the presence of a PT-symmetric potential. Closed form solutions for localized states are obtained. These solitons are shown to be stable over a wide range of potential parameters. The transverse power flow associated with these complex solitons is also examined.     

The refined theory of deep rectangular beams for symmetrical deformation

Based on elasticity theory, various one-dimensional equations for symmetrical deformation have been deduced systematically and directly from the two-dimensional theory of deep rectangular beams by using the Papkovich-Neuber solution and the Lur’e method without ad hoc assumptions, and they construct the refined theory of beams for symmetrical deformation. It is shown that the displacements and stresses of the beam can be represented by the transverse normal strain and displacement of the mid-plane. In the case of homogeneous boundary conditions, the exact solutions for the beam are derived, and the exact equations consist of two governing differential equations: the second-order equation and the transcendental equation. In the case of non-homogeneous boundary conditions, the approximate governing differential equations and solutions for the beam under normal loadings only and shear loadings only are derived directly from the refined beam theory, respectively, and the correctness of the stress assumptions in classic extension or compression problems is revised. Meanwhile, as an example, explicit expressions of analytical solutions are obtained for beams subjected to an exponentially distributed load along the length of beams. Supported by the National Natural Science Foundation of China (Grant Nos. 10702077, 10672001, and 10602001), the Beijing Natural Science Foundation (Grant No. 1083012), and the Alexander von Humboldt Foundation in Germany     

Transverse spectra of waves in random media

Abstract

We consider the statistics of the transverse spectra of forward-propagating waves in a stationary random medium. A short-range perturbation solution is used to derive the difference equations that govern the long-range evolution of the ensemble-averaged transverse wave spectrum and coherence. The conditions under which these equations may be approximated by differential and integro-differential equations are given, and it is shown that the approximation is valid for the treatment of beam propagation provided that the transverse dimension of the beam is sufficiently large, and at ranges where the transverse coherence length of the beam remains larger than a wavelength. The equations that are derived are not limited by the parabolic approximation, and are amenable to numerical solution by marching techniques. We use the equation that governs the spectral density of the total energy flux, and also the propagation of waves which are statistically homogeneous in transverse planes, to show the conditions under which previously studied approximations derive from the present formulation, and to illustrate the numerical solution of the problem.     

Reynolds stress models applied to canonical shock-turbulence interaction

Shock waves in high-speed flows can drastically alter the nature of Reynolds stresses in a turbulent flow. We study the canonical interaction of homogeneous isotropic turbulence passing through a normal shock, where the shock wave generates significant anisotropy of Reynolds stresses. Existing Reynolds stress models are applied to this canonical problem to predict the amplification of the stream-wise and transverse normal Reynolds stresses across the shock wave. In particular, the efficacy of the different models for the rapid pressure–strain correlation is evaluated by comparing the results with available direct numerical simulation (DNS) data. The model predictions are found to be grossly inaccurate, especially at high-Mach numbers. We propose physics-based improvement to the Reynolds stress-transport equation in the form of shock-unsteadiness effect and enstrophy amplification for turbulent dissipation rate . The resulting model is found to capture the essential physics of Reynolds stress amplification, and match DNS data for a range of Mach numbers. Numerical error encountered at shock waves are also analysed and the model equations are cast in conservative form to obtain physically consistent results with successive grid refinement. Finally, the proposed model for canonical shock-turbulence interaction is generalised to multi-dimensional flows with shock of arbitrary orientation.     

光束在尾流气泡中传输的复散射效应

为了了解光束在尾流气泡中的传输特性,为前向光尾流的探测提供理论依据,研究了光束在尾流中传输时传播方向上和横截面方向上的辐射强度分布特性.基于辐射传输方程的小角度近似解,得到了探测截面上的约化强度和漫射强度的表达式,其中漫射强度表征了复散射的强弱|针对典型的尾流气泡分布,通过数值计算分析了光束传输方向上的约化强度和漫射强度与接收视场角、光学厚度和光束大小的关系,也计算分析了光束横截面方向上的辐射强度随光束大小和横向距离的变化关系.结果表明,光束在尾流气泡中传输时复散射效应明显,且复散射的强弱与接收视场角、光束直径、光学厚度和横向距离密切相关.