Non-stationary response of a beam to a moving random force

The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force.     

Bending and springback theory of metal-polymer sandwich laminates

In this report an analysis is made of the behavior of sandwich beams in which the core polymer is laminated on both sides with surface metal sheets, each of which has a different thickness and mechanical properties when they are loaded with a uniform bending moment which is then released resulting in springback of the bent sandwich beam. It is assumed that the polymer behaves elastically because the bending strain in the core is small and its elastic limit is much larger than that of metals. Sandwich beams have various elastoplstic stess distributions when bent depending on the mutual relationships between their dimensions, the mechanical moduli, and the applied bending moment. Further, residual curvatures, shifting position of neutral axis, and residual stress distributions in sandwich beams variously elastoplastically stressed initially on the decrease of applied bending moment are analyzed.     

Transient response in flexure to general uni-directional loads of variable cross-section beam with concentrated tip inertias immersed in a fluid

This paper presents a method for solving problems of transient response in flexure due to general unidirectional dynamic loads of beams of variable cross section with tip inertias. An elastodynamic theory which includes effects of continuous mass and rigidity of the beam has been applied. In the analysis the general dynamic load is expanded into a Fourier series and the beam is divided into many small uniform thickness segments. The equation of motion of each segment is mapped onto the complex domain by use of the Laplace transform method. The solutions of each set of adjoining segments are related to each other at the boundaries by the use of the transfer matrix method. The displacement, the bending slope, the bending moment and the shearing force at each boundary and at arbitrary time are obtained from the Laplace transform inversion integral by using the residue theorem. The theoretical results given in this paper are applicable to problems of dynamic response due to arbitrary loads varying with time of beams of arbitrary shape with concentrated tip inertias. As applications of the present theoretical results, numerical calculations have been carried out for two cases: a uniform beam with a tip inertia and a non-uniform beam (a truncated cone) with a tip inertia. Both are immersed in a fluid and subjected to large waves such as cnoidal waves.     

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.     

Prediction capabilities of classical and shear deformable beam models excited by a moving mass

In this paper, a comprehensive assessment of design parameters for various beam theories subjected to a moving mass is investigated under different boundary conditions. The design parameters are adopted as the maximum dynamic deflection and bending moment of the beam. To this end, discrete equations of motion for classical Euler-Bernoulli, Timoshenko and higher-order beams under a moving mass are derived based on Hamilton's principle. The reproducing kernel particle method (RKPM) and extended Newmark-β method are utilized for spatial and time discretization of the problem, correspondingly. The design parameter spectra in terms of the beam slenderness, mass weight and velocity of the moving mass are introduced for the mentioned beam theories as well as various boundary conditions. The results indicate the existence of a critical beam slenderness mostly as a function of beam boundary condition, in which, for slenderness lower than this so-called critical one, the application of Euler-Bernoulli or even Timoshenko beam theories would underestimate the real dynamic response of the system. Moreover, there would be a roughly linear relation between the weight of the moving mass and the design parameters for a certain value of the moving mass velocity in most cases of boundary conditions.     

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.     

Vehicle-passenger-structure interaction of uniform bridges traversed by moving vehicles

An investigation into the dynamics of vehicle-occupant-structure-induced vibration of bridges traversed by moving vehicles is presented. The vehicle including the driver and passengers is modelled as a half-car planar model with six degrees-of-freedom, and the bridge is assumed to obey the Euler-Bernoulli beam theory with arbitrary conventional boundary conditions. Due to the continuously moving location of the variable loads on the bridge, the governing differential equations become rather complicated. The numerical simulations presented here are for the case of vehicle travelling at a constant speed on a uniform bridge with simply supported end conditions. The relationship between the bridge vibration characteristics and the vehicle speed is rendered, which yields into a search for a particular speed that determines the maximum values of the dynamic deflection and the bending moment of the bridge. Results at different vehicle speeds demonstrate that the maximum dynamic deflection occurs at the vicinity of the bridge mid-span, while the maximum bending moment occurs at ±20% of the mid-span point. It is shown that one can find a critical speed at which the maximum values of the bridge dynamic deflection and the bending moment attain their global maxima.     

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.     

Improved beam theory for multilayer graphene nanoribbons with interlayer shear effect

The bending of multilayer graphene nanoribbons incorporating the effect of interlayer shear is analyzed in this Letter. An improved beam theory is adopted and extended in which the in-plane extension of each layer is also taken into account. The governing equations for bilayer and trilayer graphene nanoribbons subjected to bending are presented as illustrative examples. Exact solutions for cantilever multilayer graphene nanoribbons are derived. Compared with the molecular dynamics (MD) simulations, the present beam model predicts much better results than the previous beam model in which the in-plane extension is ignored. The current study provides a strong evidence to include the in-plane extension effect in the continuum modeling of multilayer graphene structures.     

On the correct representation of bending and axial deformation in the absolute nodal coordinate formulation with an elastic line approach

In finite element methods that are based on position and slope coordinates, a representation of axial and bending deformation by means of an elastic line approach has become popular. Such beam and plate formulations based on the so-called absolute nodal coordinate formulation have not yet been verified sufficiently enough with respect to analytical results or classical nonlinear rod theories. Examining the existing planar absolute nodal coordinate element, which uses a curvature proportional bending strain expression, it turns out that the deformation does not fully agree with the solution of the geometrically exact theory and, even more serious, the normal force is incorrect. A correction based on the classical ideas of the extensible elastica and geometrically exact theories is applied and a consistent strain energy and bending moment relations are derived. The strain energy of the solid finite element formulation of the absolute nodal coordinate beam is based on the St. Venant-Kirchhoff material: therefore, the strain energy is derived for the latter case and compared to classical nonlinear rod theories. The error in the original absolute nodal coordinate formulation is documented by numerical examples. The numerical example of a large deformation cantilever beam shows that the normal force is incorrect when using the previous approach, while a perfect agreement between the absolute nodal coordinate formulation and the extensible elastica can be gained when applying the proposed modifications. The numerical examples show a very good agreement of reference analytical and numerical solutions with the solutions of the proposed beam formulation for the case of large deformation pre-curved static and dynamic problems, including buckling and eigenvalue analysis. The resulting beam formulation does not employ rotational degrees of freedom and therefore has advantages compared to classical beam elements regarding energy-momentum conservation.     

Performance of a cantilever piezoelectric energy harvester impacting a bump stop

Piezoelectric cantilever beam energy harvesters are commonly used to convert ambient vibration into electrical energy. In practical applications, energy harvesters are subjected to large shocks which can shorten the service life by causing mechanical failure. In this work, a bump stop is introduced into the design of a piezoelectric cantilever beam energy harvester to limit the maximum displacement of the cantilever and prevent excessively high bending stresses developing as a result of shocks. In addition to limiting the maximum displacement of the beam, it is inevitable that the deflected shape of the beam and the electrical output are modified. A theoretical model for a piezoelectric cantilever beam harvester impacting against a stop is derived, which aims to develop an understanding of the vibration characteristics of the cantilever and quantify how the electrical output of the harvester is affected by the stop. An experiment is set up to measure the dynamics and the electrical output of a bimorph energy harvester and to validate the theoretical model. Numerical simulation results are presented for energy harvesters with different initial gaps and different stop locations, and it is found that the reduction in maximum bending stress is at the expense of the electrical power of the harvester.     

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.     

An iterative approach to curved ray reconstruction of birefringent fields

The transmission of polarized light through non-homogeneous birefringent fields is examined, and an iterative approach for incorporating ray curvature into the analysis of tomographic data is proposed. The method extends strain-gradient theory and applies iterative algorithms which have been used by other researchers to reconstruct fields which are non-homogeneous but optically Isotropic. Optical anisotropy is incorporated by using a light propagation model which includes both bending and separation of initially coincident light ray pairs. To illustrate the method, data are generated computationally and are analysed to determine the bending moment in a prismatic beam and the forces acting on a diametrically loaded disc.     

A modified transfer matrix method for the study of the bending vibration band structure in phononic crystal Euler beams

We introduce a modified transfer matrix (MTM) method for the calculation of the bending vibration band structure of one-dimensional phononic crystal (PC) Euler beams. A particular combination of hyperbolic functions and triangular functions is introduced to transform the state parameters of the transfer matrix (TM) method into four initial parameters, which have the explicit meanings of the displacement, rotation angle, bending moment and shear force at one beam end. The method is used to calculate the band structures of two PC Euler beams constructed from aluminum–Lucite and 100 kinds of materials. The effectiveness and high efficiency of the MTM method are demonstrated by the results. Several advantages make it a proper choice for the calculation of the bending vibration band structure of PC Euler beams.     

托卡马克聚变堆中环向磁场超导线圈的磁弹性弯曲与稳定性

本文针对托卡马克装置中超导载流磁体的磁弹性弯曲与稳定性问题，运用Ｂｉｏｔ－Ｓａｖａｒｔ定律和曲梁弯曲理论，给出了其环向磁场的超导载流线圈结构在自身电流产生的磁力作用下的磁弹性力学模型。所得到的控制方程反映了磁场与线圈变形之间的非线性耦合作用，全面描述了结构的轴向拉伸、绕轴扭转、面内弯曲和面外弯曲等各种变形模式。本文采用半解析半数值方法对控制方程进行了定量求解，获得了有关线圈形变和内力分布的定量结果。通过其面外弯曲变形与外加电流的非线性关系，应用Ｓｏｕｔｈｗｅｌ图，给出了线圈在磁弹性相互作用下发生磁弹性失稳的临界电流值，并讨论了临界电流随环向磁场线圈个数变化的规律     

光弹材料载荷过程中红外辐射特征机理的计算分析

基于材料的热力学关系和能量守恒定律,给出弹性体形变过程中红外辐射温度改变量和辐出度变化的物理计算方程.并结合有限元方法进行物理建模,对光弹材料三点弯载荷实验过程中的红外辐射出射度的分布进行数值计算和模拟.结果与实验结果吻合较好,表明模型及其相关理论是合理的,可以用来量化揭示实验中的红外辐射特征.从而得到一种能够利用计算机量化分析固体材料载荷过程中红外辐射特征机理的计算方法.     

The effect of bending vibrations on the dipole moment of a linear polyatomic molecule: A theoretical treatment of the vibrational changes and transition moments

The vibrational effect on the dipole moment of a linear molecule is theoretically considered from the aspects of the dipole moment changes with the excitation of bending vibrations and the transition moments for the overtone, combination, and difference bands associated with bending modes. Such dipole moment changes and transition moments consist of two components, one depending on the first dipole moment derivatives with respect to bond lengths and the other depending on the second dipole moment derivatives with respect to bond angles. We show that the first component normally contributes little, and propose an approximation in which only the second component is retained. This approximation is practically important because the second component can be calculated without the anharmonic force constants. We derive formulas for the dipole moment changes and transition moments to facilitate a simultaneous analysis of different isotopic species. We introduce the concept of the equivalent mode, by which we may readily understand the correlation between the dipole moment change for a bending mode and the transition moment for a vibrational band.     

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.     

Local and global vibration of thin-walled members subjected to compression and non-uniform bending

The local-plate, distortional and global vibration behaviour of thin-walled steel channel members subjected to compression and/or non-uniform bending is studied. This investigation is carried out by means of a very recently developed Generalised Beam Theory (GBT) formulation, which takes into account the geometrically nonlinear stiffness reduction caused by the presence of (i) longitudinal stress gradients and (ii) the ensuing shear stresses. Taking advantage of the GBT modal features, one analyses the effect of the applied load and bending moment gradient on the small amplitude vibration behaviour of the loaded members (beam-columns). For validation purposes, some GBT-based results are compared with values yielded by either shell finite element analyses, performed in commercial codes, or experimental results available in the literature.