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。     

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。     

带硬涂层圆盘构件的固有振动特性的精确解法

研究了在广义弹性简支边界条件下的具有硬涂层的圆盘构件的自由振动的量纲一固有频率的精确解.首先利用多铁性多层圆盘的解析分析的多层板弹性理论,导出带硬涂层的圆盘结构的状态方程,其中以位移、电势、磁势、应力、电位移和磁感应强度为状态变量.利用有限Hankel变换和传播矩阵法,得到考虑压电和压磁效应的带硬涂层的圆盘的量纲一固有频率的精确解.根据算例结果,比较了压电、压磁两类硬涂层材料在单面涂层、双面涂层和不同涂层厚度的结构配置下的固有频率变化规律.     

水轮发电机定子拉紧螺杆动态特性分析

采用求解弹性梁横向振动微分方程的方法计算拉紧螺杆固有频率，考虑了把合轴向力对固有频率的影响。     

考虑齿轮腹板柔性的齿轮-转子横向振动特征研究

航空用齿轮具有结构轻量化和低刚性的特点,在齿轮副内部激励作用下容易发生横向振动,研究航空齿轮横向振动固有特性对航空齿轮设计十分必要。考虑齿轮腹板柔性、轴承刚度、齿轮轮齿啮合柔性,利用转子动力学有限元软件Samcef建立齿轮-转子有限元模型,计算出高速齿轮转子系统的固有频率、振型以及临界转速。基于Timoshenko节点动力学模型,计算出高速齿轮转子系统的固有频率、振型以及临界转速。两种模型计算出的齿轮-转子系统固有特性进行对比表明:考虑齿轮腹板柔性的Samcef有限元方法,可以得到齿轮-转子系统齿轮横向振动的频率值及振型,没有考虑齿轮柔性的Timoshenko梁单元方法,不能得到齿轮横向振动的频率值及振型。     

Timoshenko模型轴向运动梁的横向振动特性分析

通过对梁微单元体的受力分析,导出Timoshenko模型的轴向运动梁横向振动的运动方程, 并利用复模态分析方法及半解析半数值方法, 研究两端铰支条件下轴向运动梁横向振动的振动模态及固有频率.文中还讨论运动梁前两阶固有频率随轴向运动速度变化的情况.最后利用数值算例对Timoshenko梁、Euler梁、Rayleigh梁及剪切梁的固有频率进行比较, 分析转动惯量及剪切变形的影响.     

细长杆车削系统的动力学建模

根据等截面梁的横向振动理论，建立细长杆系统横向自由振动的偏微分方程，求出细长杆的振型函数。当设定装夹情况为卡盘—顶尖装夹时，求得细长杆的一阶固有频率。从激励及切削力的分析知，切削力对细长杆的激励可简化为两个相互垂直的简谐力激励，此简谐力的频率就是细长杆的转速。所以，要减小细长杆车削中的振动，转速必须远离细长杆系统的一阶固有频率，即避免产生共振。     

轴向运动Timoshenko梁固有频率的求解方法研究

研究两端铰支边界条件下Timoshenko模型轴向运动梁的横向振动问题,分别利用复模态分析方法和Galerkin方法求解系统的固有频率;讨论轴向运动梁前两阶固有频率随轴向运动速度的变化情况;最后给出数值算例,分析复模态分析方法、二阶Galerkin截断和四阶Galerkin截断方法对固有频率结果精确度的影响.     

摩擦离合器摩擦片的热弹耦合振动分析

基于弹性薄板小挠度理论和考虑变形影响的热传导方程,建立了摩擦离合器摩擦片的热弹耦合圆环板模型和相应的运动微分方程,采用微分求积法离散运动微分方程和边界条件,得到了离合器摩擦片在横向温度变化影响下前3阶无量纲固有频率与无量纲角速度和热弹耦合系数之间的关系曲线。研究(计算)结果表明,摩擦片的前3阶无量纲固有频率随着无量纲角速度和无量纲热弹耦合系数的增大而增大,不同的边界条件对摩擦片横向振动固有频率的增大幅度有一定的影响。该结论为摩擦离合器的设计与性能分析提供了一定的理论基础。     

基于Cosserat理论的微梁振动特性的尺度效应

不少微观实验已经证实,微尺度领域材料的力学性能存在尺度效应.采用偶应力理论(又称Cosserat理论)研究微梁振动特性(主要是固有频率)的尺度效应.文中首先对偶应力理论进行简介,然后采用Hamilton变分原理推导基于Cosserat理论的微梁无阻尼自由振动的微分方程,分析微梁固有频率对微尺度的依赖性.结果表明,当微梁的厚度减小到可以和材料的本征长度相比时,微梁的固有频率将显著增大.     

Natural frequencies,modes and critical speeds of axially moving Timoshenko beams with different boundary conditions

In this paper, natural frequencies, modes and critical speeds of axially moving beams on different supports are analyzed based on Timoshenko model. The governing differential equation of motion is derived from Newton's second law. The expressions for various boundary conditions are established based on the balance of forces. The complex mode approach is performed. The transverse vibration modes and the natural frequencies are investigated for the beams on different supports. The effects of some parameters, such as axially moving speed, the moment of inertia, and the shear deformation, are examined, respectively, as other parameters are fixed. Some numerical examples are presented to demonstrate the comparisons of natural frequencies for four beam models, namely, Timoshenko model, Rayleigh model, Shear model and Euler–Bernoulli model. Finally, the critical speeds for different boundary conditions are determined and numerically investigated.     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.     

Instability and vibration of a rotating Timoshenko beam with precone

A rotating blade with a precone angle is usually designed, but little literature has investigated the effect of the precone angle on vibration. This paper investigates divergence instability and vibration of a rotating Timoshenko beam with precone and pitch angles. It uses Hamilton's principle to derive the coupled governing differential equations and boundary conditions for a rotating Timoshenko beam. Analytical solution of an inextensional Timoshenko beam without taking into account the Coriolis force effect can be derived. Some simple relations among the parameters of rotating Timoshenko beams are revealed. Based on these relations, one can predict the natural frequencies and parameters of other systems from those of known systems. Moreover, the mechanism of divergence instability (tension buckling) is investigated. Finally, the effects of the parameters on natural frequencies, and the phenomenon of divergence instability are investigated.     

Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform

This study investigates the vibration problems of an axially loaded non-uniform spinning twisted Timoshenko beam. First, using the Timoshenko beam theory and Hamilton's principle, we derive the governing equations and boundary conditions of the beam. Secondly, the differential transform method is used to solve these equations with appropriate boundary conditions. Finally, the effects of the twist angle, spinning speed, and axial force on the natural frequencies of a non-uniform Timoshenko beam are investigated and discussed.     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

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.     

Natural frequencies of Euler-Bernoulli beam with open cracks on elastic foundations

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Winkler foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.     

The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.     

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.