Stability for Leipholz's Type of Laminated Box Columns with Nonsymmetric Lay-Ups on Elastic Foundation

The stability behavior of the Leipholz's type of laminated box columns with nonsymmetric lay-ups resting on elastic foundation is investigated using the finite element method. Based on the kinematic assumptions consistent with the Vlasov beam theory, a formal engineering approach of the mechanics of the laminated box columns with symmetric and nonsymmetric lay-ups is presented. The extended Hamilton's principle is employed to obtain the elastic stiffness and mass matrices, the Rayleigh damping and elastic foundation matrices, the geometric stiffness matrix due to distributed axial force, and the load correction stiffness matrix accounting for the uniformly distributed nonconservative forces. The evaluation procedures for the critical values of divergence and flutter loads with/without internal and external damping effects are briefly presented. Numerical examples are carried out to validate the present theory with respect to the previously published results. Especially, the influences of the fiber angle change and damping on the divergence and flutter loads of the laminated box columns are parametrically investigated.     

On the shear coefficient in Timoshenko's beam theory

Some existing formulations for the shear coefficient in Timoshenko's beam theory are discussed, especially through evaluation of the accuracy to which natural frequencies of simply supported, prismatic, thin walled beams can be obtained. The main conclusion drawn is that if a consistent expression for the shear coefficient, such as those given by Cowper [1] or Stephen [2], is used in Timoshenko's beam theory, then very high accuracies can be expected for the natural frequencies, even for wavelengths of the same magnitude as the transverse dimension of the beam. It is noted that no reduction of the moment of inertia due to shear lag effects should be made as these effects are included in the consistent formulas for the shear coefficient. Finally, some apparently paradoxical results indicating that a reduction in shear stiffness occurs in rare cases if more material is added to a section are discussed and explained as resulting from the use of integrated rather than pointwise deflection measures in the derivation of consistent shear coefficient expressions. The results are discussed in the light of the importance of the shear stiffness of the hull girder in ship hull vibration analysis.     

Vibrations and buckling of a beam on a variable winkler elastic foundation

Two methods for solving the eigenvalue problems of vibrations and stability of a beam on a variable Winkler elastic foundation are presented and compared. The first is based on using the exact stiffness, consistent mass, and geometric stiffness matrices for a beam on a variable Winkler elastic foundation. The second method is based on adding an element foundation stiffness matrix to the regular beam stiffness matrix, for vibrations and stability analysis. With these matrices, it is possible to find the natural frequencies and mode shapes of vibrations, and buckling load and mode shape, by using a small number of segments. It is concluded that the use of the element foundation stiffness approach yields better convergence at lower computation costs.     

Vibration analysis of horizontal self-weighted beams and cables with bending stiffness subjected to thermal loads

This paper aims at proposing an analytical model for the vibration analysis of horizontal beams that are self-weighted and thermally stressed. Geometrical nonlinearities are taken into account on the basis of large displacement and small rotation. Natural frequencies are obtained from a linearization of equilibrium equations. Thermal force and thermal bending moment are both included in the analysis. Torsional and axial springs are considered at beam ends, allowing various boundary conditions. A dimensionless analysis is performed leading to only four parameters, respectively, related to the self-weight, thermal force, thermal bending moment and torsional spring stiffness. It is shown that the proposed model can be efficiently used for cable problems with small sag-to-span ratios (typically , as in Irvine's theory). For beam problems, the model is validated thanks to finite element solutions and a parametric study is conducted in order to highlight the combined effects of thermal loads and self-weight on natural frequencies. For cable problems, solutions are first compared with existing results in the literature obtained without thermal effects or bending stiffness. Good agreement is found. A parametric study combining the effects of sag-extensibility, thermal change and bending stiffness is finally given.     

Instability of vibrations of a mass that moves uniformly along a beam on a periodically inhomogeneous foundation

The stability of vibrations of a mass that moves uniformly along an Euler-Bernoulli beam on a periodically inhomogeneous continuous foundation is studied. The inhomogeneity of the foundation is caused by a slight periodical variation of the foundation stiffness. The moving mass and the beam are assumed to be always in contact. With the help of a perturbation analysis it is shown analytically that vibrations of the system may become unstable. The physical phenomenon that lies behind this instability is parametric resonance that occurs because of the periodic (in time) variation of the foundation stiffness under the moving mass. The first instability zone is found in the system parameters within the first approximation of the perturbation theory. The location of the zone is strongly dependent on the spatial period of the inhomogeneity and on the weight of the moving mass. The larger this period is and/or the smaller the mass, the higher the velocity is at which the instability occurs.     

Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass

In this paper, a method of analysis of a beam that is continuously supported on a linear nonhomogeneous elastic foundation and subjected to a harmonically excited mass is presented. The solution is obtained by decomposing the nonhomogeneous foundation properties and the beam displacement response into double Fourier summations which are solved in the frequency–wavenumber domain, from which the space–time domain response can be obtained. The method is applied to railway tracks with step variation in foundation properties. The validity of this method is checked, through examples, against existing methods for both homogeneous and nonhomogeneous foundation parameters. The effect of inhomogeneity and the magnitude of the mass are also investigated. It is found that a step variation in foundation properties leads to a reduction in the beam displacement and an increase in the resonance frequency for increasing step change, with the reverse occurring for decreasing step change. Furthermore, a beam on nonhomogeneous foundation may exhibit multiple resonances corresponding to the foundation stiffness of individual sections, as the mass moves through the respective sections along the beam.     

Mono-coupled periodic systems with a single disorder: Response to convected loadings

This paper presents a general theory of the forced response under convected loading of mono-coupled periodic systems with a single disorder. The general expressions derived have been used to study the response of an infinite periodic beam on simple supports with one of the support spacings different from all the others. Convected harmonic pressure fields and frozen random pressure fields have been considered. Computer studies are presented showing the moment response at supports and the space-time-averaged responses in the disorder and in the nearby periodic beam elements. High response levels can occur due to (i) resonances of the beam length disorder against the stiffness of the attached periodic systems and (ii) hydrodynamic coincidence vibration occurring in the periodic beam. The frequency zones in which these high responses may occur are identified. The high response due to the resonance (ii) is restricted to the vicinity of the disorder, whereas that due to coincidence occurs throughout the system. Computed results show that the highest response levels do not necessarily occur in the beam length disorder, but may occur in one of the nearby periodic beam elements. The dependence of the maximum response levels on the magnitude of the disorder has been investigated. The conditions under which small disorders may be neglected have been pointed out.     

New Insights on the Deflection and Internal Forces of a Bending Nanobeam

Nanowires, nanofibers and nanotubes have been widely used as the building blocks in micro/nano-electromechanical systems,energy harvesting or storage devices,and small-scaled measurement equipment. We report that the surface effects of these nanobeams have a great impact on their deflection and internal forces. A simply supported nanobeam is taken as an example. For the displacement and shear force of the nanobeam, its dangerous sections are different from those predicted by the conventional beam theory, but for the bending moment, the dangerous section is the same. Moreover, the values of these three quantities for the nanobeam are all distinct from those calculated from the conventional beam model. These analyses shed new light on the stiffness and strength check of nanobeams, which are beneficial to engineer new-types of nano-materials and nano-devices.     

Random vibration of an initially stressed thick plate on an elastic foundation

The mean-square bending moment of a thick rectangular plate excited by a uniform distribution of stationary random forces that are uncorrelated in space is calculated. The plate has in-plane compressive or tensile stresses. In addition, the plate is mounted on an elastic foundation. Numerical results are given for plates with uniform initial stress when the temporal correlation function of the excitation possesses an exponential decay. In general it can be said that the position on the plate where the mean-square moment takes on a maximum value depends upon the relative values of the initial stress, the stiffness of the foundation and the aspect ratio of the plate. The mean-square response amplitude of the plate on a foundation never exceeds that of the plate without a foundation, regardless of the intensity of the initial stress or the geometrical configuration of the plate.     

Modal coupling effects in the free vibration of elastically interconnected beams

The problem of free vibration of a uniform beam elastically interconnected to a cantilevered beam, representing an idealized launch vehicle aeroelastic model in a wind tunnel, is studied. With elementary beam theory modelling, numerical results are obtained for the frequencies, mode shapes and the generalized modal mass of this elastically coupled system, for a range of values of the spring constants and cantilevered beam stiffness and inertia values. The study shows that when the linear springs are supported at the nodal points corresponding to the first free-free beam mode, the modal interaction comes primarily from the rotational spring stiffness. The effect of the linear spring stiffness on the higher model modes is also found to be marginal. However, the rotational stiffness has a significant effect on all the predominantly model modes as it couples the model deformations and the support rod deformations. The study also shows that through the variations in the stiffness or the inertia values of the cantilever beam affect only the predominantly cantilever modes, these variations become important because of the fact that the cantilevered support rod frequencies may come close to, or even cross over, the predominantly model mode frequencies. The results also bring out the fact that shifting of the support points away from the first mode nodal points has a maximum effect only on the first model mode.     

The role of non-linear theories in transient dynamic analysis of flexible structures

Transient dynamic analysis of flexible structures undergoing large motions is considered. For rotating structures, it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non-linear (at least second-order) beam theory. Use of a first-order (linearized) linear beam theory results in a spurious loss of bending stiffness. For a rotating plane beam, a set of linear partial differential equations of motion—that includes all inertia effects (Coriolis, centrifugal, acceleration of revolution) and coupling between extensional and flexural deformations—is derived from the fully non-linear beam theory by consistent linearization. The analysis is subsequently extended to the more general case of a plate, accomodating shear deformation, and undergoing a general three-dimensional rotating motion. The discretization process of the resulting linear equations of motion for the beam and the plate is also discussed.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.     

Semi-Inverse Solution of a Pure Beam Bending Problem in Gradient Elasticity Theory: The Absence of Scale Effects

The semi-inverse solutions of pure beam bending problems within the three-dimensional formulation of gradient elasticity theory as exact tests for the problem of estimating the efficient bending stiffness of so-called scale-dependent thin beams and plates due to the necessity of modeling sensing devices are presented. It is shown that the solutions within the gradient elasticity theory give classic beam bending stiffnesses and demonstrate the invalidity of the widespread results and estimates obtained in the past 15 years during study of scale effects within the gradient beam theories, according to which the relative bending stiffness grows by a hyperbolic law with decreasing thickness.     

Comparison of Two Theories for the Relativistic Self Focusing of Laser Beams in Plasma

In the present paper, self‐focusing of laser beams in relativistic plasmas is studied by the moment theory approach. The equilibrium beam radius of the self‐trapped laser beams is also derived. Results are compared with the paraxial ray theory. It is observed from the analysis that at higher intensities, the equilibrium beam radius increases in case of the paraxial ray theory, whereas it becomes independent of the beam intensity in case of the moment theory. Analysis also confirms the role of relativistic electrons travelling with the light pulse in (3D PIC) simulation studies (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of two theories during self-focusing of Gaussian laser beam in thermal conduction-loss predominant plasmas

In the present paper, self-focusing phenomenon occurring as a result of non-linear interaction of intense laser beam with thermal conduction-loss predominant plasmas is studied by following both approaches viz. paraxial theory approach and moment theory approach. Non-linear differential equations for the beam width parameters of laser beam have been set up and solved numerically in both cases to study the variation of beam width parameters with normalized distance of propagation. Effects of laser intensity as well as plasma density on the beam width parameters have also been analyzed. It is observed from the analysis that in case of moment theory approach, strong self-focusing of laser beam is observed as compared to paraxial theory approach.     

A general procedure for the dynamic analysis of finite and infinite beams on piece-wise homogeneous foundation under moving loads

Transversal vibrations induced by a load moving at a constant speed along a finite or an infinite beam resting on a piece-wise homogeneous visco-elastic foundation are studied. Special attention is paid to the amplification of the vibrations which arise as the point load traverses a foundation discontinuity. The governing equations of the problem are solved by the normal-mode analysis. The solution is expressed in the form of an infinite sum of orthogonal natural modes multiplied by the generalized displacements. The natural frequencies are obtained numerically exploiting the concept of the global dynamic stiffness matrix. This ensures that the frequencies obtained are accurate. The methodology is neither restricted by load velocity nor damping and is simple to use, though obtaining the numerical expression of the results is not straightforward. A general procedure for numerical implementation is presented and verified. There is no restriction for finite structures, however, for infinite structures, validity of the results is restricted to a “region of interest” of finite length. To illustrate the methodology, the probability of exceeding an admissible upward displacement is determined when the load travels at a certain velocity according to the normal distribution. In this problem, the given structure has an intermediate part of adaptable foundation stiffness, which is optimized in a parametric way, enabling to draw important conclusions about the optimum intermediate stiffness. The results obtained have direct application on the analysis of railway track vibrations induced by high-speed trains crossing regions with significantly different foundation stiffness.     

A scheme for the interpretation of data from instrumented offshore platforms

A procedure is presented for estimating the key parameters associated with the dynamic behaviour of deepwater gravity platforms. Efficient modelling of the coupled soil/structure/fluid system is achieved by the method of component modes. This permits accurate analysis of the dynamic behaviour of the platform with an idealization having only a few co-ordinates. Full-scale measurements (in the form of direct and cross spectral densities of water surface elevation, overturning moment, deck displacement, etc.) may then be used to obtain “best fit” estimates of the unknown stiffness and damping parameters. Computed results demonstrating the technique are presented, in which simulated data from a platform are used to back-figure the stiffness and damping associated with structure and foundation.     

半导体激光器的M2因子可以小于1

根据 Porras的非傍轴矢量矩理论 ,对双异质结半导体激光器的光束质量进行了研究 .结果表明 ,在有源层厚度 da远小于波长 λ的条件下 ,半导体激光器的 M2 因子可以小于 1 ,并且没有下限 .     

非傍轴矢量拉盖尔-高斯光束的二阶矩表示

基于Porras提出的光传输的非傍轴矢量矩理论，推导出初始圆偏振的非傍轴矢量拉盖尔-高斯(LG)光束的特征参数，包括束宽、远场发散角和M²因子等的公式，并表示为级数求和形式.非傍轴矢量高斯光束公式作为特例给出.研究表明，基于二阶矩定义的束宽按双曲线规律传输，当w₀/λ→0(w₀为束宽，λ为波长)时，远场发散角θ趋于90°，大于非傍轴标量理论预示的值63.435°.非傍轴矢量LG光束的M²因子不仅与模指数p有关，而且还与w₀/λ有关.最后，对非傍轴矢量LG光束和非傍轴标量LG光束的传输作了比较，结果表明在w₀/λ较小时，矢量效应对远场发散角的影响十分显著.对θ→90°引起的问题和非傍轴矢量矩理论的适用范围，以及解决问题的可能途径作了分析和讨论. 关键词： 非傍轴矢量拉盖尔-高斯光束 圆偏振 非傍轴矢量矩理论 光束参数     

Higher order tapered beam finite elements for vibration analysis

In this paper, explicit for mass and stiffness matrices of two higher order tapered beam elements for vibration analysis are presented. One possesses three degrees of freedom per node and the other four degrees of freedom per node. The four degrees of freedom of the latter element are the displacement, slope, curvature and gradient of curvature. Thus, this element adequately represents all the physical situations involved in any combination of displacement, rotation, bending moment and shearing force. The explicit element mass and stiffness matrices eliminate the loss of computer time and round-off-errors associated with extensive matrix operations which are necessary in the numerical evaluation of these expressions. Comparisons with existing results in the literature concerning tapered cantilever beam structures with or without an end mass and its rotary inertia are made. The higher order tapered beam elements presented here are superior to the lower order one in that they offer more realistic representations of the curvature and loading history of the beam element. Furthermore, in general the eigenvalues obtained by employing the higher order elements converge more rapidly to the exact solution than those obtained by using lower order one.