纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

轴向运动功能梯度梁的横向振动

新型非均匀复合材料,功能梯度材料具有防止脱层和减缓热应力等优良性能,将其应用于功能梯度梁的结构有着非常重要的工程应用价值。基于Euler-Bernoulli梁理论和Hamilton原理,建立轴向运动功能梯度梁横向自由振动的运动微分方程,其中假设功能梯度梁的材料特性沿梁厚度方向按各组分材料体积分数的幂函数连续变化;再对运动微分方程和边界条件进行量纲一处理,采用微分求积法对其进行离散化,导出系统的广义复特征方程,然后计算分析轴向运动功能梯度简支梁横向振动复频率的实部和虚部随量纲一轴向运动速度、梯度指标等参数的变化情况,并讨论量纲一轴向运动速度和梯度指标对功能梯度梁的横向振动特性以及失稳形式的影响。     

A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect

A single-walled nanotube structure embedded in an elastic matrix is simulated by the nonlocal Euler-Bernoulli, Timoshenko, and higher order beams. The beams are assumed to be elastically supported and attached to continuous lateral and rotational springs to take into account the effects of the surrounding matrix. The discrete equations of motion associated with free transverse vibration of each model are established in the context of the nonlocal continuum mechanics of Eringen using Hamilton's principle and an efficient meshless method. The effects of slenderness ratio of the nanotube, small scale effect parameter, initial axial force and the stiffness of the surrounding matrix on the natural frequencies of various beam models are investigated for different boundary conditions. The capabilities of the proposed nonlocal beam models in capturing the natural frequencies of the nanotube are also addressed.     

Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity

This paper presents a new approach for analyzing transverse bending and vibration of circular cylindrical beams with radial nonhomogeneity. The radial nonhomogeneity may be continuous or piecewise-constant, corresponding a functionally graded circular cylinder or a multi-layered circular cylinder, respectively. Different from the Euler-Bernoulli and Timoshenko theories of beams, our analysis considers shear deformation, but does not need to introduce a shear correction factor. Using the shear-stress-free condition at the surface of the cylinder, coupled governing equations for deflection and rotation angle are derived, and then converted to a single governing equation. The influences of gradient index on the deflection and stress distribution for cantilever and simply-supported beams are studied. Natural frequencies of free vibration of a cylindrical beam with circular cross-section are calculated for different power-law gradients. In particular, a circular cylindrical shell may be taken as a special case of a bi-layered cylinder where the material properties of the inmost cylinder vanish. For this case, the natural frequencies for simply-supported and clamped-clamped cylindrical shells are evaluated and compared with those using three-dimensional theory. Our results coincide well with the previous.     

The mechanical analysis of a mass-spring load supported on a beam system

The static deflection and dynamic characteristics of a mass-spring system supported on beam systems are investigated in this paper. In statics, it shows that the maximum deflection is reduced considerably when a clamped-free beam is replaced by a beam system which consists of a primary beam one end of which is clamped and the other end is supported by a subsidiary beam. The addition of a subsidiary beam leads to axial forces in both beams, the primary one in tension and the subsidiary in compression. The dynamic characteristic shows that the natural frequency of the mass-spring system decreases. In some cases it becomes imaginary, because buckling occurs in the subsidiary beam. This means that the effects of the addition of a subsidiary beam are not always of a positive nature, with respect to the stiffness of the whole system. At low frequencies, the response of the mass is larger than that of the mass supported on a motionless foundation. At high frequencies the dynamic characteristics of the foundation influence the vibration of the mass only a little; i.e. it moves as if the foundation were motionless.     

Use of a generalized beam/spring element to analyze natural vibration of rail track and its application

This paper describes the formulation of a generalized beam/spring track element to obtain the natural vibration characteristics of a railway track modeled as a periodic elastically coupled beam system on a Winkler foundation. The rail/tie beams are described by either the Timoshenko beam theory or the Bernoulli-Euler beam theory. The rail beam is assumed to be discretely coupled to the cross-track ties through the coupling spring elements at the periodic rail/tie intersections. The generalized beam/spring element consists of a rail span beam segment, two adjacent tie beams, the coupling spring elements and the ultimate foundation stiffness. The entire track/beam system is then discretized into an assembly of periodic structural units. An equivalent frequency-dependent spring coefficient representing the resilient, flexural and inertial characteristics of the track substructure unit is formulated to establish the dynamic stiffness matrix of the generalized element. The eigenvalue problem of the track/beam system is solved by employing a comprehensive and efficient numerical routine. Solutions are provided for the natural frequencies of the track and the mode shapes of the rail/tie beams under transversely (cross-track) symmetric vibration. The natural vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

大挠度后屈曲倾斜梁结构的非线性力学特性

基于弹性梁的几何非线性大挠度屈曲理论,建立两端固定对称倾斜支撑梁结构的大挠度后屈曲控制微分方程,采用几何非线性隐式变形协调关系来表达强非线性超静定边值问题,得到描述倾斜梁大挠度后屈曲行为的精确解析解.采用数值方法求解含有第一、二类椭圆积分的强非线性微分方程,给出不同倾角梁结构从初始屈曲到后屈曲并发生两态跳转过程中的位形曲线及非线性刚度.根据最小能量原理和挠曲线拐点个数,分析对称屈曲模态与非对称屈曲模态之间相互跳转的内在联系及其对结构非线性刚度突变的影响,得到了屈曲模态之间的转换条件.跳转过程的数值仿真表明,倾斜支撑梁结构发生大挠度后屈曲时具有明显的双稳态特性且只出现低阶(1、2阶)屈曲模态,仿真计算结果与试验结果相一致.     

Vibration of two cantilever beams clamped at one end and connected by a rigid body at the other

This study presents the pure bending and coupled bending-torsional vibration characteristics of a beam structure which consists of two cantilever beams and a rigid body at their free ends. This structure is available in many mechanical structures such as robots, space constructions, and optical pickup actuators in optical disc drives (ODDs). In order to depict the vibration of the beam structure originating from the deflection and torsion of two beams, the motion equations and continuity conditions are analytically induced by using energy conservation. In the process that the free vibration problem is solved, two independent characteristic equations are obtained. The former is an equation for the pure bending vibration of two beams, and the latter is for their coupled bending-torsional vibration. It is proved that these characteristic equations are exact by comparing natural frequencies obtained from FEM. As natural frequencies are described in many dimensional variations, the relation between vibration characteristics and the dimensions of the given system is also investigated. Finally, resonant frequencies from test results are presented to confirm the validation of this study for a new type optical pickup actuator. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Kyung Taek Lee received a Ph.D. degree in Mechanical Engineering from Yonsei University, Seoul, Korea in 2003. He joined LG Electronics, Seoul, Korea, in 1989, where he has worked on precise mechanical structures and microactuating systems for optical information storage devices, as a Research Engineer. His current interests include microactuators for position control, haptic elements for mobile devices, and etc.     

基于固有频率的呼吸式裂纹梁损伤识别方法

将呼吸裂纹梁简化为由扭转弹簧连接的两段弹性梁,在假定振动响应随振幅变化的基础上推导出呼吸裂纹梁的固有频率方程;考虑振动过程中呼吸裂纹的开合情况,假定裂纹梁的刚度是振幅的非线性函数,建立了呼吸裂纹梁的多项式刚度模型;结合等高线裂纹识别理论和方法,提出了一种基于固有频率的呼吸裂纹梁损伤识别方法,算例验证了方法的可行性与有效性。研究表明,该方法的识别精度取决于实验固有频率的精度。     

基于欧拉梁的准零刚度隔振系统动力特性分析

将两端受轴向压力的欧拉梁和线性弹簧并联,组成一个具备高静刚度和低动刚度的准零刚度隔振器。通过对隔振系统进行静力分析,给出系统具备准零刚度特性所需的条件。利用谐波平衡法求解系统的振动微分方程,分析系统的幅频特性,给出了系统的力传递率,讨论了阻尼、激励力等参数对系统传递率的影响。最后分析了该隔振系统的跳跃频率。研究结果表明：激励力以及初始偏移量的增大会使系统的隔振效果变差,因此要控制激励力的大小并尽量避免超载;阻尼比的选取需要综合考虑高频的隔振效果和有效隔振频率范围。     

Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads

A finite-length tensioned beam on a damped elastic foundation is acted upon by an infinite series of equally spaced and steadily moving concentrated transverse loads. The deflection response of the beam is obtained by an expansion in terms of the normal modes of vibration. Numerical results are determined for various values of the load-spacing, beam tension, foundation stiffness and damping, and for a range of load-speeds. It is found that the critical velocities for repetitive loading exist at significantly lower speeds than would be expected based upon the well-known critical speed for a single moving load. An interpretation in terms of forced vibration response is given.     

Dynamic responses of composite H-beams with different elastic couplings

Governing equations are derived via Hamilton’s principle for composite thin-walled H-type open cross-section beams that show a number of non-classical effects such as transverse shear, primary and secondary warping, and anisotropy of constituent materials. The vibration characteristics of composite thin-walled beams with different elastic couplings, such as circumferentially asymmetric stiffness (CAS) and circumferentially uniform stiffness (CUS) configurations are investigated with respect to the bending-transverse shear coupling and the bending-twist coupling resulting from the directional properties of fiber reinforced composite materials. The dynamic responses of anisotropic thin-walled beams to harmonic and exponentially time-dependent loads are also investigated. It was revealed that transverse shear, elastic couplings, and warping effects greatly influence the free vibration and dynamic response characteristics of composite H-type open cross-section beams.     

轴压下弹性支承夹芯连续梁固有横振

研究了轴压下弹性支承夹芯连续梁的固有横振,利用Ｌａｐｌａｃｅ变换导出了其振型函数及频率方程。     

Natural frequencies of stepped thickness rectangular plates

This paper presents the natural frequencies of stepped thickness square and rectangular plates together with the mode shapes of vibration. The transverse deflection of a stepped thickness plate is written in a series of the products of the deflection functions of beams parallel to the edges satisfying the boundary conditions, and the frequency equation of the plate is derived by the energy method. By use of the frequency equation, the natural frequencies (the eigenvalues of vibration) and the mode shapes are calculated numerically in good accuracy for square and rectangular plates with edges simply supported or elastically restrained against rotation, having square, circular or elliptical stepped thickness, from which the effects of the stepped thickness on the vibration are studied.     

Zigzag theory for buckling of hybrid piezoelectric beams under electromechanical loads

A new efficient coupled one-dimensional (1D) geometrically nonlinear zigzag theory is developed for buckling analysis of hybrid piezoelectric beams, under electromechanical loads. The potential field is approximated layerwise as piecewise linear. The deflection is approximated to account for the normal strain due to electric field. The axial displacement is approximated as a combination of a global third-order variation and layerwise linear variation. It is expressed in terms of three primary displacement variables and a set of electric potential variables by enforcing exactly the conditions of zero transverse shear stress at the top and bottom and the conditions of its continuity at the layer interfaces. The governing coupled nonlinear field equations and boundary conditions are derived using a variational principle. Analytical solutions for buckling of simply supported beams under electromechanical loads are presented. Comparisons with the exact 2D piezoelasticity solution establish that the present zigzag theory is very accurate for buckling analysis.     

利用局部振动频率识别框架结构构件刚度参数

鉴于依据结构整体动力反应很难准确识别结构局部构件的物理参数,提出了结构局部激振检测方法,既对结构局部进行激振,依据结构局部激振反应提取其固有频率,再利用此固有频率运用方法识别结构局部构件的物理参数。本文将框架结构中的梁柱简化为两端有弹性约束、具有等刚度和等分布质量的杆件模型,推导了这一模型的频率特征方程,并进一步分析了两端弹性约束对其振动固有频率的影响。仿真算例表明：在固有频率误差较小的情况下,本文方法可准确地给出构件抗弯刚度和边界条件。     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

An exact solution for the vibration of helical springs using a Bernoulli-Euler model

A solution is obtained for the steady-state vibration behaviour of uniform helical springs in which the following effects are ignored; deformation due to shear and axial loads, the rotational inertias of the cross-section, and static loads applied to the spring. A theory is developed for the evaluation of the dynamic stiffness matrix. Natural frequencies are found using an adaption of the Wittrick-Williams algorithm. Details of the method of calculation are discussed. Results are presented for natural frequencies, and are compared with values obtained using other methods and assumptions. The method of this paper is shown to be particularly efficient.     

考虑翘曲影响的Bernoulli-Euler薄壁梁的弯扭耦合振动

通过直接求解均匀薄壁梁单元弯扭耦合振动的运动偏微分方程，推导其自由振动时的精确动态传递矩阵。采用考虑翘曲影响的Bernoulli-Euler梁理论，且假定薄壁梁单元的横截面是单对称的。动态传递矩阵可以用于计算薄壁梁集合体的精确固有频率和模态形状。针对两个薄壁梁算例，采用自动Muller法和结合频率扫描法的二分法求解频率特征方程，并讨论翘曲刚度对弯扭耦合：Bernoulli-Euler薄壁梁固有频率的影响。数值结果验证了本文方法的精确性和有效性，并指出翘曲刚度可以显著改变薄壁开口截面梁的固有频率。