横向载荷作用下轴向运动梁的屈曲失稳以及非线性振动特性

梁的轴向运动会诱发其产生横向振动并可能导致屈曲失稳,对结构的安全性和可靠性产生重大的影响。本文重点研究了横向载荷作用下轴向运动梁的屈曲失稳及横向非线性振动特性。基于Hamilton变分原理,建立了横向载荷作用下轴向运动梁的动力学方程,获得了梁的后屈曲构型。使用截断Galerkin法,将控制方程改写成Duffing方程的形式。用同伦分析方法确定载荷作用下轴向运动梁的非线性受迫振动的封闭形式的表达式。结果表明,后屈曲构型对轴向速度和初始轴向应力有明显的依赖性。通过同伦分析法得出非线性基频的显式表达式,获得了初始轴向力会影响非线性频率随初始振幅和轴向速度的线性关系。另外,轴向外激励的方向也会改变系统固有频率。     

Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading

Free transverse vibration and buckling of a double-beam continuously joined by a Winkler elastic layer under compressive axial loading with the influence of rotary inertia and shear are considered in this paper. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli?CFourier method. The boundary value and initial value problems are solved. The natural frequencies and associated amplitude ratios of an elastically connected double-beam complex system and the analytical solution of the critical buckling load are determined. The presented theoretical analysis is illustrated by a numerical example, in which the effect of physical parameters characterizing the vibrating system on the natural frequency, the associated amplitude ratios and the critical buckling load are discussed.     

Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium

A mathematical model for nonlocal vibration and buckling of embedded two-dimensional(2 D) decagonal quasicrystal(QC) layered nanoplates is proposed. The Pasternak-type foundation is used to simulate the interaction between the nanoplates and the elastic medium. The exact solutions of the nonlocal vibration frequency and buckling critical load of the 2 D decagonal QC layered nanoplates are obtained by solving the eigensystem and using the propagator matrix method. The present three-dimensional(3 D) exact solution can predict correctly the nature frequencies and critical loads of the nanoplates as compared with previous thin-plate and medium-thick-plate theories.Numerical examples are provided to display the effects of the quasiperiodic direction,length-to-width ratio, thickness of the nanoplates, nonlocal parameter, stacking sequence,and medium elasticity on the vibration frequency and critical buckling load of the 2 D decagonal QC nanoplates. The results show that the effects of the quasiperiodic direction on the vibration frequency and critical buckling load depend on the length-to-width ratio of the nanoplates. The thickness of the nanoplate and the elasticity of the surrounding medium can be adjusted for optimal frequency and critical buckling load of the nanoplate.This feature is useful since the frequency and critical buckling load of the 2 D decagonal QCs as coating materials of plate structures can now be tuned as one desire.     

DYNAMIC RESPONSES OF VISCOELASTIC AXIALLY MOVING BELT

Based on the Kelvin viscoelastic differential constitutive law and the motion equation of the axially moving belt, the nonlinear dynamic model of the viscoelastic axial moving belt was established. And then it was reduced to be a linear differential system which the analytical solutions with a constant transport velocity and with a harmonically varying transport velocity were obtained by applying Lie group transformations. According to the nonlinear dynamic model, the effects of material parameters and the steady-state velocity and the perturbed axial velocity of the belt on the dynamic responses of the belts were investigated by the research of digital simulation . The result shows:1) The nonlinear vibration frequency of the belt will become small when the relocity of the belt increases . 2) Increasing the value of viscosity or decreasing the value of elasticity leads to a deceasing in vibration frequencies. 3) The most effects of the transverse amplitudes come from the frequency of the perturbed veloc     

Effects of shear stresses and rotary inertia on the stability and vibration of sandwich cylindrical shells with FGM core surrounded by elastic medium

The vibration and stability of axially loaded sandwich cylindrical shells with the functionally graded (FG) core with and without shear stresses and rotary inertia resting Pasternak foundation are investigated. The dynamic stability is derived based on the first order shear deformation theory (FSDT) including shear stresses. The axial load and dimensionless fundamental frequency for FG sandwich shell with shear stresses and rotary inertia and resting on the Pasternak foundation. Finally, the influences of variations of FG core, elastic foundations, shear stresses and rotary inertia on the fundamental frequencies and critical axial loads are investigated.     

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

THE TRANSVERSE NONLINEAR FREE VIBRATION AND THE BUCKLING OF A COMPRESSIVE BAR ON AN ELASTIC FOUNDATION

基于Hamilton 原理,运用假设时间模态法,得到了弹性基础上压杆的横向非线性自由振动与屈曲的位移型常微分控制方程. 考虑一端固定另一端可移简支边界条件,采用打靶法得到了结构第一至第三阶结构频率与一阶屈曲载荷的数值结果. 结果表明：随轴心压力增加,结构频率减小;随弹性基础刚度增加,结构频率与屈曲载荷均增加;弹性基础刚度对结构频率的影响随振型阶数增加在减小;在小振幅的情形下,不同振型对一阶屈曲载荷的影响很小.     

Finite amplitude,free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler–Bernoulli assumptions together with Von Karman’s strain–displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He’s variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the effect of vibration amplitude, elastic coefficients of foundation, axial force, and material inhomogenity are presented for future references.     

Coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments

Delaminations in structures may significantly reduce the stiffness and strength of the structures and may affect their vibration characteristics. As structural components, beams have been used for various purposes, in many of which beams are often subjected to axial loads and static end moments. In the present study, an analytical solution is developed to study the coupled bending-torsion vibration of a homogeneous beam with a single delamination subjected to axial loads and static end moments. Euler–Bernoulli beam theory and the "free mode" assumption in delamination vibration are adopted. This is the first study of the influences of static end moments upon the effects of delaminations on natural frequencies, critical buckling loads and critical moments for lateral instability. The results show that the effects of delamination on reducing natural frequencies, critical buckling load and critical moment for lateral instability are aggravated by the presence of static end moment. In turn, the effects of static end moments on vibration and instability characteristics are affected by the presence of delamination. The analytical results of this study can serve as a benchmark for finite element method and other numerical solutions.     

Analytical study of the dynamic response of an embedded railway track to a moving load

Summary In this paper, the steady-state dynamic response of an embedded railway track to a moving train is investigated theoretically. The model for the track consists of a flexible plate performing vertical vibrations, two beams that are connected to the plate by continuous visco-elastic elements and an elastic foundation that supports the plate. Two harmonic loads that move uniformly along the beams describe the train load. The plate, the beams and the elastic foundation are employed to model a concrete slab of an embedded track, the rails and the ground reaction, respectively. The problem is studied by employing the Fourier integral transforms in the following way. Firstly, the dispersion analysis of waves that may propagate along the system is accomplished in the frequency-wavenumber domain. On the basis of this analysis, critical velocities of the loads are found both for the in-phase and anti-phase vibrations of the loads. Secondly, the vertical displacement of the rails and the slab, along with the stresses in the slab, are investigated as functions of the velocity and frequency of the loads. Finally, the response of the two-dimensional model is compared to that of a simplified one-dimensional model.     

Free vibration analyses of axially loaded laminated composite beams based on higher-order shear deformation theory

The dynamic stiffness matrix method is introduced to solve exactly the free vibration and buckling problems of axially loaded laminated composite beams with arbitrary lay-ups. The Poisson effect, axial force, extensional deformation, shear deformation and rotary inertia are included in the mathematical formulation. The exact dynamic stiffness matrix is derived from the analytical solutions of the governing differential equations of the composite beams based on third-order shear deformation beam theory. The application of the present method is illustrated by two numerical examples, in which the effects of axial force and boundary condition on the natural frequencies, mode shapes and buckling loads are examined. Comparison of the current results to the existing solutions in the literature demonstrates the accuracy and effectiveness of the present method.     

轴向受载的Bernoulli—Euler薄壁梁固有振动和稳定性的理论研究

通过直接求解轴向受载的单对称均匀Bernoulli-Euler薄壁梁单元弯扭耦合振动的运动微分方程,推导了其动态传递矩阵,讨论了轴向载荷的变化对薄壁梁弯扭耦合振动固有频率的影响,并由此得到零频率振动（弹性屈出）发生时相应的轴向载荷,数值结果表明本文方法在其适应范围内是精确有效的。     

端部切向载荷作用下的压杆失稳分析及其应用

细直杆件在压应力作用下会产生横向屈曲即失稳．直杆撞击刚性平面或拉断卸载后将形成压缩波，因承载压缩载荷的长度增加可以引起失稳．冲击速度转换的压应力沿着杆件切线方向，该处弯矩和剪力为零；而众多文献设定的失稳段固支边界条件并不准确．基于精确的杆件变形曲率方程得到端部载荷指向杆件中固定点时的受压失稳条件，得到其极限状态即载荷沿杆端切向作用时失稳长度相当于两端简支的1.5 倍．对于钢丝绳拉断形成的冲击失稳，载荷恒定而长度增加，可以产生高阶屈曲即在侧向出现多次曲折，并基于尼龙-橡胶带的模拟试验给出了定性说明．     

考虑分布轴向力的细长杆横向振动与失稳分析

为量化梁、杆、柱的自重(下称分布轴向力)对静力失稳和动力横向振动的影响,在《材料力学》和《机械系统动力学》教材的基础上,建立了分布轴向力下的杆柱失稳和横向振动的力学、数学模型.采用有限差分法、伽辽金法和数值积分法获取计算结果.结果表明:考虑分布轴向力的杆柱横向振动固有频率随杆长增加而减小,杆柱失稳时一阶固有频率为0;分...     

高速移动荷载下黏弹性半空间体的动力响应

分别以移动荷载和黏弹性半空间体模拟运动列车荷载和地基,分析了地基在运动列车作用下的动力响应.首先采用Green函数法求解黏弹性半空间体在各种移动荷载模式作用下的动力响应的解析解,包括恒常和简谐移动点源、线源和面源荷载.然后采用IFFT算法和自适应数值积分算法计算解析解中的二维积分,得到了包括低音速、跨音速和超音速移动荷载作用下位移的数值结果.最后分析了速度对位移的分布和最大值的影响,发现当速度大于Rayleigh波速时,位移发生显著变化.     

STUDY ON AMPLIFICATION FACTOR OF FLEXURAL BUCKLING CRITICAL LOAD OF AXIAL COMPRESSION BARS1)

为了建立一般条件下轴压构件屈曲临界载荷的计算理论,首先对轴心受压构件发生屈曲时的总势能方程进行了推导,然后采用Rayleigh-Ritz法并基于势能驻值原理得到了4种不同端部约束条件下轴压构件的屈曲临界载荷,对比欧拉临界载荷,给出了临界载荷放大系数 的计算式,全面考虑了构件长细比、压缩变形、剪切变形以及截面形状系数对临界载荷的影响,推导的计算式可用于较小长细比轴压构件发生屈曲时临界载荷的计算.圆截面和双轴对称工字形截面轴压构件屈曲临界载荷的分析表明构件长细比是影响放大系数的主导因素。     

Nonlinear dynamic responses of high-speed railway vehicles under combined self-excitation and forced excitation considering the influence of unsteady aerodynamic loads

The dynamic characteristics of a railway vehicle system under unsteady aerodynamic loads are examined in this study. A dynamic analysis model of the railway vehicle considering the influences of aerodynamic loads was established. The model not only considers the forced excitation effect of unsteady aerodynamic loads but also accounts for the effect of unsteady aerodynamic loads on the change of the wheel–rail contact normal forces as well as changes of the wheelset creep coefficients and creep forces/moments. Therefore, this model also considers the influences of unsteady aerodynamic loads on the self-excited vibration characteristics of the vehicle system. The time-history curves, phase trajectory diagrams, Poincaré sections, and Lyapunov exponents of the vehicle system running on a smooth straight track under unsteady aerodynamic loads were determined. The results show that when the critical speed is exceeded, the vehicle system usually performs quasi-periodic motion under unsteady aerodynamic loads, which is significantly different from the periodic motion under steady aerodynamic loads. In different cases, the amplitude and phase of motion are significantly different. The amplitude of the motions can be increased by more than 159%, and the difference of phase can be up to 173°. (The phase is almost reversed.) The dynamic responses of the vehicle system under unsteady aerodynamic loads contain abundant frequency components, including the frequency of the self-excited vibration, the frequency of the forced excitation, and combinations of their integer multiples. The vibration forms corresponding to the main harmonic components under unsteady and steady aerodynamic loads were compared, and the self-excited vibration component of the vehicle system under unsteady aerodynamic loads was identified. The variations in the critical speed with various parameter combinations were computed. The variation range of the critical velocity can reach 73%.

     

Exploiting the subharmonic parametric resonances of a buckled beam for vibratory energy harvesting

The potential of harvesting vibratory energy via a bistable beam subjected to subharmonic parametric excitations is investigated. The vibrating structure is a buckled beam with two stable equilibria separated by a potential barrier. The beam is subjected to a superposition of a static axial load beyond its buckling load and a harmonic axial excitation whose frequency is around twice the frequency of the buckled beam’s first vibration mode. A macro-fiber composite patch is attached to one side of the beam to convert the strain energy resulting from the beam’s oscillation into electricity. The study considers two regimes of excitations: an amplitude sweep and a frequency sweep. In the first regime, the amplitude of excitation is quasi-statically varied while the excitation frequency is tuned at twice the natural frequency of the first vibration mode. In the second regime, the excitation frequency is swept forward and backward around the subharmonic resonant frequency while the amplitude of excitation is kept constant. A theoretical model which governs the electromechanical coupling of the transverse vibrations of the beam and the output voltage is used to monitor the response as the excitation parameters are changed. An experimental setup is also built and a series of tests is performed to validate the theoretical findings. It is shown that, depending on the amplitude and frequency of excitation, the harvester can perform small-amplitude periodic intra-well motion, intra- and inter-well chaotic motions, as well as periodic inter-well motions. Experimental results also show that, as compared to the classical linear resonance, utilizing the sub-harmonic resonance of a bistable energy harvesters can result in a broadband frequency response.     

移动随机荷载作用下梁桥非平稳随机振动的PEM-FFT方法

针对移动随机载荷作用下桥梁结构非平稳随机振动问题,建立了一种基于频域分析的虚拟激励-傅里叶变换方法(PEM-FFT)。与通常的非平稳随机振动的时频分析方法不同,提出的方法完全在频域上执行。其主要特色是能够给出随机输入与随机输出的频域关系,表明移动载荷作用下结构非平稳随机振动分析仅需对载荷的确定性移动函数项进行计算。数值算例考虑了等截面/变截面简支梁受移动随机载荷作用的问题,首先与通常的时频分析方法进行对比,验证了PEM-FFT方法的正确性和有效性,进一步讨论了不同结构形式以及不同载荷移动速度对响应演变统计量的影响。