弹性地基上转动功能梯度材料Timoshenko梁自由振动的微分变换法求解

基于Timoshenko梁理论研究弹性地基上转动功能梯度材料（FGM）梁的自由振动。首先确定功能梯度材料Timoshenko梁的物理中面,利用广义Hamilton原理推导出该梁在弹性地基上转动时横向自由振动的两个控制微分方程。其次采用微分变换法（DTM）对控制微分方程及其边界条件进行变换,计算了弹性地基上转动功能梯度材料Timoshenko梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种不同边界条件下横向自由振动的量纲一固有频率,与已有文献的计算结果进行比较,退化后结果一致。最后讨论了不同边界条件、转速、弹性地基模量和梯度指数对功能梯度材料Timoshenko梁自振频率的影响。结果表明：功能梯度材料Timoshenko梁的量纲一固有频率随量纲一转速和量纲一弹性地基模量的增大而增大;在量纲一转速和量纲一弹性地基模量一定的情况下,梁的量纲一固有频率随着功能梯度材料梯度指数的增大而减小。     

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.     

轴向运动简支-固支梁的横向振动和稳定性

研究一端简支一端固支轴向运动梁的横向振动和稳定性。提出在给定边界条件下确定一匀速运动梁固有频率和模态函数的方法。当轴向运动速度在其常平均值附近作简谐波动时,应用多尺度法给出轴向变速运动梁参数共振时的不稳定条件。用数值仿真说明相关参数对固有频率和不稳定边界的影响。     

Dynamic Behaviors of Axially Moving Viscoelastic Plate with Varying Thicknessn

Structural components of varying thickness draw increasing attention these days due to economy and light-weight considerations. In view of the absence of research in vibration analysis of viscoelastic plate with varying thickness, this study devotes to investigate the dynamic behaviors of axially moving viscoelastic plate with varying thickness. Based on the thin plate theory and the two-dimensional viscoelastic differential constitutive relation, the differential equation of motion of the axially moving viscoelastic rectangular plate is derived, the plate constituted by Kelvin-Voigt model has linearly varying thickness in the y-direction. The dimensionless complex frequencies of axially moving viscoelastic plate with four edges simply supported are calculated by the differential quadrature method, curves of real parts and imaginary parts of the first three-order dimensionless complex frequencies versus dimensionless moving speed are obtained, the effects of the aspect ratio, thickness ratio, the dimensionless moving speed and delay time on the dynamic behaviors of the axially moving viscoelastic rectangular plate with varying thickness are analyzed. When other parameters keep constant, with the decrease of thickness ratio, the real parts of the first three-order natural frequencies decrease, and the critical divergence speeds of various modes decrease too, moreover, whether the delay time is large or small, the frequencies are all complex numbers.     

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.     

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.     

A further study about the behaviour of foundation piles and beams in a Winkler–Pasternak soil

The natural vibrations and critical loads of foundation beams embedded in a soil simulated with two elastic parameters through the Winkler–Pasternak (WP) model are analysed. General end supports of the beam are considered by introducing elastic constraints to transversal displacements and rotations. The solution is tackled by means of a direct variational methodology previously developed by the authors who named it as whole element method. The solution is stated by means of extended trigonometric series. This method gives rise to theoretically exact natural frequencies and critical loads. A particular behaviour arises from the analysis of the lateral soil influence. It is found that the boundary conditions of the beam are influenced by the soil at the left and right sides of the beam. The possible alternatives are that the soil be cut or dragged by the non-fixed ends of the beam. In the standard WP model, the lateral soil influence is not considered. Natural frequencies and critical load numerical values are reported for beams and piles elastically supported and for various soil parameters. The results are found with arbitrary precision depending on the number of terms taken in the series. Some unexpected modes and eigenvalues are found when the different alternatives are studied. It should be noted that this special behaviour is present only when the Pasternak contribution is taken into account.     

Free vibration analysis of circularly curved multi-span Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the free vibration analysis of circularly curved multi-span Timoshenko beams. Each section of the beam has its own basis functions, and the continuity conditions at the intermediate supports as well as the boundary condition are treated as the constraints of the basis functions so that the number of degrees of freedom matches the number of the pseudospectral expansion coefficients. The computed natural frequencies are compared with those of existing literature, where it is shown that they are in good agreement. Numerical examples are provided for pinned-pinned, clamped-clamped and free-pinned boundary conditions for different numbers of sections and for different thickness-to-length ratios.     

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.     

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.     

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。     

Flow-induced vibration and stability analysis of multi-wall carbon nanotubes

The free vibration and flow-induced flutter instability of cantilever multi-wall carbon nanotubes conveying fluid are investigated and the nanotubes are modeled as thin-walled beams. The non-classical effects of the transverse shear, rotary inertia, warping inhibition, and van der Waals forces between two walls are incorporated into the structural model. The governing equations and associated boundary conditions are derived using Hamilton’s principle. A numerical analysis is carried out by using the extended Galerkin method, which enables us to obtain more accurate solutions compared to the conventional Galerkin method. Cantilevered carbon nanotubes are damped with decaying amplitude for a flow velocity below a certain critical value. However, beyond this critical flow velocity, flutter instability may occur. The variations in the critical flow velocity with respect to both the radius ratio and length of the carbon nanotubes are investigated and pertinent conclusions are outlined. The differences in the vibration and instability characteristics between the Timoshenko beam theory and Euler beam theory are revealed. A comparative analysis of the natural frequencies and flutter characteristics of MWCNTs and SWCNTs is also performed.     

Effect of axial force on free vibration of Timoshenko multi-span beam carrying multiple spring-mass systems

The situation of structural elements supporting motors or engines attached to them is usual in technological applications. The operation of machine may introduce severe dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli–Euler single-span beams carrying a number of spring-mass system and Bernoulli–Euler multi-span beams carrying multiple spring-mass systems are plenty, but that of Timoshenko multi-span beams carrying multiple spring-mass systems with axial force effect is fewer. This paper aims at determining the exact solutions for the first five natural frequencies and mode shapes of a Timoshenko multi-span beam subjected to the axial force. The model allows analyzing the influence of the shear and axial force effects and spring-mass systems on the dynamic behavior of the beams by using Timoshenko Beam Theory (TBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The calculated natural frequencies of Timoshenko multi-span beam by using secant method for non-trivial solution for the different values of axial force are given in tables. The mode shapes are presented in graphs.     

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.     

Eigenvalue analysis of double-span timoshenko beams by pseudospectral method

The pseudospectral method is applied to the free vibration analysis of double-span Timoshenko beams. The analysis is based on the Chebyshev polynomials. Each section of the double-span beam has its own basis functions, and the continuity conditions at the intermediate support as well as the boundary conditions are treated separately as the constraints of the basis functions. Natural frequencies are provided for different thickness-to-length ratios and for different span ratios, which agree with those of Euler-Bernoulli beams when the thickness-to-length ratio is small but deviate considerably as the thickness-to-length ratio grows larger.     

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.     

Rotary inertia and temperature effects on non-linear vibration, steady-state response and stability of an axially moving beam with time-dependent velocity

Free non-linear transverse vibration of an axially moving beam in which rotary inertia and temperature variation effects have been considered, is investigated. The beam is moving with a harmonic velocity about a constant mean velocity. The governing partial-differential equations are derived from the Hamilton's principle and geometrical relations. Under special assumptions, the two partial-differential equations can be mixed to form one integro-partial-differential equation. The multiple scales method is applied to obtain steady-state response. Elimination of secular terms will give us the amplitude of vibration. Additionally, the stability and bifurcation of trivial and non-trivial steady-state responses are analyzed using Routh-Hurwitz criterion. Eventually, numerical examples are presented to show rotary inertia, non-linear term, temperature gradient and mean velocity variation effects on natural frequencies, critical speeds, bifurcation points and stability of trivial and non-trivial solutions.     

横向磁场中轴向运动条形导电板的组合共振

给出了磁场中轴向运动条形导电薄板的动能、应变能以及电磁力表达形式,应用哈密顿变分原理,推导出了轴向运动导电板的非线性磁弹性振动微分方程。通过位移函数的设定并应用伽辽金积分法,得到横向磁场中对边简支边界约束轴向运动条形板的达芬型磁弹性振动方程。利用多尺度法进行求解,得到组合共振发生时确定共振幅值的幅频响应方程,并给出定常稳定解的判定条件。通过数值算例,得到轴向运动速度、磁感应强度、激励力和轴向拉力等参量不同时的振幅变化规律曲线图以及系统振动的时程响应图和相图,分析了不同参量对共振幅值和非线性特征的影响,并对系统呈现的概周期和混沌运动行为变化规律进行了分析。     

In-plane free vibration analysis of curved Timoshenko beams by the pseudospectral method

The pseudospectral method is applied to the analysis of in-plane tree vibration of circularly curved Timoshenko beams. The analysis is based on the Chebyshev polynomials and the basis functions are chosen to satisfy the boundary conditions. Natural frequencies are calculated for curved beams of rectangular and circular cross sections under hinged-hinged, clampedclamped and hinged-clamped end conditions and the results are compared with those by transfer matrix method. The present method gives good accuracy with only a limited number of collocation points.     

Explicit formulation for natural frequencies of double-beam system with arbitrary boundary conditions

In this paper, free transverse vibration of two parallel beams connected through Winkler type elastic layer is investigated. Euler-Bernoulli beam hypothesis has been applied and it is assumed that boundary conditions of upper and lower beams are similar while arbitrary without any limitation even for non-ideal boundary conditions. Material properties and cross-section geometry of beams could be different from each other. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli-Fourier method. Explicit expressions are derived for the natural frequencies. In order to verify accuracy of results, the problem once again solved using modified Adomian decomposition method. Comparison between results indicates excellent accuracy of proposed formulation for any arbitrary boundary conditions. Derived explicit formulation is simplest method to determine natural frequencies of double-beam systems with high level of accuracy in comparison with other methods in literature.