局部损伤梁动力问题的近似计算方法

应用模态摄动法求解局部损伤梁动力特性。这一方法是在无损伤梁的模态子空间内实施,将含局部损伤梁的微分方程的求解转化为代数方程求解。通过算例,表明这一方法可有效的简化计算,同时计算结果具有较高的精度。     

Subharmonic and ultra-subharmonic response of nonlinear elastic beams subjected to harmonic excitation

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Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments

Modeling and nonlinear vibration analysis of graphene-reinforced composite (GRC) laminated beams resting on elastic foundations in thermal environments are presented. The graphene reinforcements are assumed to be aligned and are distributed either uniformly or functionally graded of piece-wise type along the thickness of the beam. The motion equations of the beams are based on a higher-order shear deformation beam theory and von Kármán strain displacement relationships. The beam–foundation interaction and thermal effects are also included. The temperature-dependent material properties of GRCs are estimated through a micromechanical model. A two-step perturbation approach is employed to determine the nonlinear-to-linear frequency ratios of GRC laminated beams. Detailed parametric studies are carried out to investigate the effects of material property gradient, temperature variation, stacking sequence as well as the foundation stiffness on the linear and nonlinear vibration characteristics of the GRC laminated beams.     

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.     

挠性根部梁的动力学建模

挠性根部梁具有整体平动和转动自由度，其传统模型只适宜根部挠性很小的梁．采用柔性多体系统的建模方法建立了挠性根部Euler—Bernoulli梁的非线性动力学模型及线性耦合模型，所建模型不受根部挠性大小的限制；既可描述挠性根部梁的耦合振动，也可分别退化为固支梁或刚性梁的动力学模型；且线性耦合模型可线性变换为挠性根部梁传统模型．作为算例，采用假设模态法分析了两类线性模型的振动特性，表明线性耦合模型优于挠性根部梁传统模型．     

Simply supported elastic beams under parametric excitation

In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.     

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.     

腹筋对连续深梁剪切破坏影响的有限元分析

我国现行混凝土结构设计规范关于腹筋对深梁抗剪承载力影响的考虑,主要基于简支深梁的试验数据和计算模型,相关规定对于连续深梁的适用性有待验证。腹筋对连续深梁抗剪性能的影响机理十分复杂,系统的试验研究较少,试验结果比较分散。为了弥补试验研究的不足,采用非线性有限元方法模拟钢筋混凝土连续深梁的受力性能。利用已有的试验数据,验证了基于ATENA软件建立的有限元模型,并以此为基础,对以腹筋配筋率和剪跨比为变量的64根连续深梁进行分析,研究腹筋对连续深梁受剪承载力的影响。有限元分析结果与规范公式和压杆-拉杆模型(STM)及最新提出的三参数机动法(3PKT)的比较表明,中国规范高估了水平腹筋对构件受剪承载力的贡献,STM和3PKT方法也存在各自的不足。     

On the Problem of Torsion in Finite Elasticity:Part I,The General Formulation in the Case of Torsion with Variable Twist

ABSTRACT

The problem of nonuniform torsion of beams with solid cross-sections is examined in the finite displacement theory of elasticity. Two approximate theories are formulated by applying Reissner's theorem for finite elastic deformations. A perturbation solution scheme is presented, to study results implied by the two approximate formulations.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性,同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应,利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析,在不同的条件下,该方程在相平面上存在同宿轨道或异宿轨道, 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法,对该非线性方程进行 求解,得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解,讨论了这些解存 在的必要条件,这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程,从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

Unified proof to oscillation property of discrete beam

The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.     

Stability of axially accelerating viscoelastic beams: asymptotic perturbation analysis and differential quadrature validation

An asymptotic perturbation method is proposed to investigate stability of an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The stability condition can be determined via the asymptotic perturbation method. The differential quadrature scheme is developed to solve numerically the equation of axially accelerating viscoelastic beams with simple supports. The stability boundaries are numerically located in the summation parametric resonance and the principal parametric resonance. Numerical examples show the effects of the beam viscoelasticity and the mean axial speed. The numerical calculations validate the analytical results in the principal parametric resonance.     

轴向变速运动黏弹性梁稳定性渐近分析和数值验证

研究了轴向变速运动黏弹性梁参数振动的稳定性.对黏弹性本构关系采用物质时间导数,轴向速度用关于恒定平均速度的简单谐波变化来描述.发展浙近摄动法确定稳定性条件.应用微分求积法数值求解简支边界条件下的轴向变速运动黏弹性梁方程,并进而确定次谐波参数共振的稳定性边界.数值结果显示了梁的黏性阻尼和轴向平均速度的影响并验证了次谐波共振的解析结果.     

On the first order change of bifurcation buckling loads due to structural geometry perturbations

It is shown that a simple method may be applied in order to determine the first order change of eigenmodes and critical loads at bifurcation buckling of slender structures when cavities or reinforcements are introduced. The perturbation procedure developed is applicable whenever a characteristic diameter of the structural geometry perturbation is one order less than the wave length of the unperturbed eigenmode and necessary computations involve at most the solution of two linear boundary value problems. Situations when the procedure may be further simplified are discussed and statements are made regarding requisite conditions for the buckling load definitely to increase or decrease due to geometry perturbations. Application of the method is illustrated by means of four cases of engineering interest, which involve perforated and reinforced beams and plates with holes and cracks.     

大型空间结构的热-动力学耦合问题及其有限元分析

论文对辐射换热条件下闭口薄壁杆件与单枝开口薄壁杆件的瞬态温度场问题,提出了一种一维傅立叶温度有限元,克服了传统一维温度单元只能计算薄壁杆截面平均温度的缺点,通过增加结点摄动温度自由度的方法,该一维单元能计算杆截面的温度分布.在此一维温度单元与梁位移单元相协调的基础上,进一步发展了大型空间结构热诱发振动稳定性判据与热颤振响应有限元计算方法.对于柔性空间结构发展了考虑几何非线性的热-结构动力学耦合有限元计算方法,成功地对这类结构的热动力屈曲问题进行了数值模拟.     

Dynamic response of an infinite Timoshenko beam on a nonlinear viscoelastic foundation to a moving load

The present paper investigates the dynamic response of infinite Timoshenko beams supported by nonlinear viscoelastic foundations subjected to a moving concentrated force. Nonlinear foundation is assumed to be cubic. The nonlinear governing equations of motion are developed by considering the effects of the shear deformable beams and the shear modulus of foundations at the same time. The differential equations are, respectively, solved using the Adomian decomposition method and a perturbation method in conjunction with complex Fourier transformation. An approximate closed form solution is derived in an integral form based on the presented Green function and the theorem of residues, which is used for the calculation of the integral. The dynamic response distribution along the length of the beam is obtained from the closed form solution. The derivation process demonstrates that two methods for the dynamic response of infinite beams on nonlinear foundations with a moving force give the consistent result. The numerical results investigate the influences of the shear deformable beam and the shear modulus of foundations on dynamic responses. Moreover, the influences on the dynamic response are numerically studied for nonlinearity, viscoelasticity and other system parameters.     

Exact deflection solutions of beams with shear piezoelectric actuators

Exact deflection models of beams with n actuators of shear piezoelectric are developed analytically. To formulate the models, the first-order and higher-order beam theories are used. The exact solutions are obtained with the aid of the state-space approach and Jordan canonical form. A case study is presented to evaluate the performance of the authors’ previously reported models. Through a demonstrative example, a comparative study of the first-order and higher-order beams with two shear piezoelectric actuators is attained. It is shown that the first-order beam cannot predict the beam behavior when compared with the results of the higher-order beam. Further applications of the solutions are presented by investigating the effects of actuators lengths and locations on the deflected shapes of beams with two piezoelectric actuators. Some interesting deflection curves are presented. For example, the deflection curve of a H–H beam resembles saw teeth that rotate clockwise about the central location with the increase of actuators lengths. The presented exact solutions can be used in the design process to obtain detailed deformation information of beams with various boundary conditions. Moreover, the presented analysis can be readily used to perform precise shape control of beams with n actuators of shear piezoelectric.     

The asymptotic properties of high frequencies for bars,beams and membranes

In this paper, a uniform analysis of the asymptotic properties of high frequencies of non-uniform bars, beams and circular membranes is given by using perturbation method, and the case of discontinuous physical parameters is also discussed.     

Dispersion and localisation in structured Rayleigh beams

This paper brings a comparative analysis between dynamic models of couple-stress elastic materials and structured Rayleigh beams on a Winkler foundation. Although physical phenomena have different physical origins, the underlying equations appear to be similar, and hence mathematical models have a lot in common. In the present work, our main focus is on the analysis of dispersive waves, band-gaps and localised waveforms in structured Rayleigh beams. The Rayleigh beam theory includes the effects of rotational inertia which are neglected in the Euler–Bernoulli beam theory. This makes the approach applicable to higher frequency regimes. Special attention is given to waves in pre-stressed Rayleigh beams on elastic foundations.