附有滑动质量的旋转柔性梁动力学特性研究

提出用弹簧-质量系统抑制旋转柔性梁的振动,建立Euler-Bernoulli梁的动力学模型.对方程进行无量纲化,并在质量为慢时变运动、旋转角速度和角加速度为梁变形同阶小量时,对非线性方程进行近似简化,分析弹簧-质量系统对梁振动特性的影响机理.利用多尺度方法对非线性方程近似求解,在主共振、内共振条件下,研究梁和质量运动幅值随质量位置和调协参数的变化趋势.     

A cross-sectional analysis of composite beams based on asymptotic framework

This paper presents the accurate prediction of static behavior of composite beams with arbitrary cross-sections. The asymptotic recursive formulation is reviewed first, where the initial three-dimensional problems are split into the macroscopic 1D problems and the microscopic 2D problems. The finite element formulation for the microscopic 2D problems is then presented in order to find the crosssectional warping solutions. The warping solutions obtained contribute the cross-sectional properties to the macroscopic 1D problems. The end effect of the 1D beam problem is also considered via the kinematic correction for a displacement prescribed boundary. The approach presented is applied to the beams with relatively complicated material distributions and cross-sectional geometry. As numerical test-beds, a three-layered sandwich beam and a composite beam with the multi-cell cross-section are taken to analyze the local deformation. A parametric study is also carried out to investigate the significance of shear deformation due to the cross-sectional orthotropic characteristics. The cross-sectional deformation is predicted based on the asymptotic framework. The accuracy of the present approach is assessed by comparing the results obtained with the 3D FEM solutions obtained by ANSYS.     

Vibration analysis of non-uniform beams having multiple edge cracks along the beam's height

Bending vibration of non-uniform rectangular beams with multiple edge cracks along the beam's height is investigated. These cracks are called height-edge cracks in this paper. The energy based method is used for defining the vibration of height-edge cracked beams. The opening form of the height-edge crack is determined when the external moment is assumed to be applied for stretching the beam's width. Strain energy increase is obtained by calculating the strain change at the stretched surface by taking into account the effect of angular displacement of the beam due to the bending. The Rayleigh-Ritz approximation method is used in the analysis. The cases of multiple cracks are analysed in the method by using the approach based on the definition of strain disturbance variation along the beam. Examples are presented on a fixed-fixed beam and several cantilever beams having different taper factors. When the results are compared with the results of a commercial finite element program, good agreement is obtained. The effects of taper factors, boundaries and positions of cracks on the natural frequency ratios are presented in graphics.     

Dynamic responses and fuzzy control of a simply-supported beam subjected to a moving mass

This paper deals with the active vibration control of a simply-supported beam traversed by a moving mass using fuzzy control. Governing equations for dynamic responses of a beam under a moving mass are derived by Galerkin’s mode summation method, and the effect of forces (gravity force, Coliolis force, inertia force caused by the slope of the beam, transverse inertia force of the beam) due to the moving mass on the dynamic response of a beam is discussed. For the active control of dynamic deflection and vibration of a beam under the moving mass, the controller based on fuzzy logic is used and the experiments are conducted by VCM (voice coil motor) actuator to suppress the vibration of a beam. Through the numerical and experimental studies, the following conclusions were obtained. With increasing mass ratioy at a fixed velocity of the moving mass under the critical velocity, the position of moving mass at the maximum dynamic deflection moves to the right end of the beam. With increasing velocity of the moving mass at a fixed mass ratioy, the position of moving mass at the maximum dynamic deflection moves to the right end of the beam too. The numerical predictions of dynamic deflection of the beam have a good agreement with the experimental results. With the fuzzy control, more than 50% reductions of dynamic deflection and residual vibration of the tested beam under the moving mass are obtained.     

Dynamic behaviors of an elastically restrained beam carrying a moving mass

Dynamic responses of a simply supported beam with a translational spring carrying a moving mass are studied. Governing equations of motion including all the inertia effects of a moving mass are derived by employing the Galerkin’s mode summation method, and solved by using the Runge-Kutta integral method. Numerical solutions for dynamic responses of a beam are obtained for various cases by changing parameters of the spring stiffness, the spring position, the mass ratio and the velocity ratio of a moving mass. Some experiments are conducted to verify the numerical results obtained. Experimental results for the dynamic responses of the test beam have a good agreement with numerical ones.     

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.     

A reduced time-varying model for a long beam on elastic foundation under moving loads

Dynamics of a long beam on the elastic foundation subjected to moving loads is studied in the present paper. The sliding window technique is used to dynamically truncate the long beam and a reduced time-varying beam system is obtained. The Hamilton’s principle is employed to establish the equations of motion of the reduced system. The variable separation method is adopted to solve dynamical responses of the reduced system. Examples of a long simply supported Timoshenko beam on the nonlinear foundation subjected to a single moving load and multiple loads are included. Numerical results of the reduced model compared with the ones obtained from the moving element model adapted in literature are carried out to show the validity and the good efficiency of the method proposed in the present paper.     

An approximate analysis of the behaviour of laterally loaded, fully fixed beams and plates

An approximate method is developed for the elasto-plastic analysis of rotationally fixed and axially restrained rectangular beams and long plates subjected to lateral loading. The boundary conditions of the problem are satisfied exactly, but equilibrium is only satisfied at one point along the beam length. The load deflection behaviour can be determined by hand calculation. The behaviour of beams tested by others is analysed and the method is found to closely predict the behaviour observed experimentally. The effect of initial imperfection and axial movement at the beam ends is investigated. The possibility of beam instability is pointed out, and it is found that small movements at the beam ends can significantly affect its behaviour.     

Forced responses of cracked cantilever beams subjected to a concentrated moving load

An analytical method is developed to present the dynamic response of a cracked cantilever beam subject to a concentrated moving load. The cracked beam system is modeled as a two-span beam and each span of the continuous beam is assumed to obey Euler–Bernoulli beam theory. The crack is modeled as a rotational spring with sectional flexibility. Considering the compatibility requirements on the crack, the relationships between these two spans can be obtained. By using the analytical transfer matrix method, eigensolutions of this cracked system are obtained explicitly. The forced responses can be obtained by the modal expansion theory using the determined eigenfunctions. Some numerical results are shown to present the crack effects (crack extent, location of the crack) and are studied for different speeds of the moving load.     

受多点横向非定常约束梁的稳态与瞬态响应

研究受有多点横向非定常约束梁的动态响应问题。利用横向非定常约束为周期函数梁的微分代数方程的线性特点,确定出多基频周期性非定常约束激励下梁稳态响应的解析解;利用截断模态的微分代数方程确定出受有横向非定常约束梁的瞬态响应。为此,首先利用无阻尼截断模态微分代数方程的齐次形式,将欧拉-伯努利梁的等效多跨梁模态表示为简单边界条件下模态函数的线性组合,再利用得到的模态函数与位移影响函数重新将微分代数方程表示为常微分方程的形式,进而得到横向非定常约束作用下梁的瞬态响应解的积分形式。在此过程中,研究了微分代数方程齐次式的特征值及其数量、求解方法等相关问题。通过单点多频率的横向非定常约束作用的梁及多点单频横向非定常约束作用梁的算例,重点分析了非定常约束位置与基频对梁稳态响应的影响。结果表明：在欠阻尼下,两种梁模态响应的极大值均在等效多跨梁的各个主频附近;单点横向非定常约束作用响应极小值在简单边界条件梁的各个主频附近,而多点横向非定常约束作用梁响应极小值的分布比较复杂。算例也说明了所提出的方法是正确、有效的。     

FRP 层压复合梁结构屈曲变形在线监测方法

为了实现梁结构屈曲变形在线监控,提出了一种FRP层合梁屈曲变形重构方法。首先,依据高阶剪切变形理论,提出了一种复合梁结构变形场描述方法,并基于冯卡门应变梯度理论,推导出了中性轴应变表述方式。然后,利用最小二乘变分法建立了位移重构模型。其中,利用四次B-样条基函数构造了屈曲变形位移插值函数,推导了理论中性轴应变计算公式。并基于少量应变测量,提出了非线性项应变解耦方法,建立了测量应变与实测中性轴应变转换关系。最后,为了验证所提方法的精确性,以25层碳纤维复合梁为样件,搭建固定-简支梁试验平台,对其进行数值计算和试验论证。结果表明,建立的屈曲变形重构模型在不同轴向载荷作用时,位移场重构误差均小于8%。     

移动质量激励下梁动态响应分析与试验研究

以移动质量激励下的梁为研究对象,建立了移动质量-简支梁和移动质量-悬臂梁耦合振动模型.考虑了移动质量的重力、Corilous加速度和惯性加速度对梁横向振动的影响,利用得到的级数解编制程序分析了移动质量参数变化对梁动态响应的影响.建立了移动质量激励下的梁动态响应试验平台,通过试验得到了各种规格质量块以不同运动速度激励作用下的梁动态响应曲线.试验结果表明了移动质量激励下梁动态响应模型的正确性.     

Free and forced vibration analysis of FG beam considering temperature dependency of material properties

This paper presents a finite element method (FEM) free and forced lateral vibration analysis of beams made of functionally graded materials (FGMs). The temperature dependency of material properties along with damping had not previously been taken into account in vibration analysis. In the present study, the material properties were assumed to be temperature-dependent, and were graded in the thickness direction according to a simple power law distribution of the volume fractions of the constituents. The natural frequencies were obtained for functionally graded (FG) beams with various boundary conditions. First, an FG beam was assumed to be isotropic (metal rich) and the results were compared with the analytical solution and the results for ANSYS and NASTRAN software. Finally, dynamic responses were obtained for damped and un-damped systems. Numerical results were obtained to show the influences of the temperature dependency of the materials properties, the boundary conditions, the volume fraction distribution (the index of power law, N) and the geometrical parameters.     

Stick-slip vibration of a moving oscillator on an axially flexible beam

This study investigates the stick-slip vibration between an axially flexible beam fixed at both ends and an oscillator moving on the beam. After deriving the equations of motion for the stick and slip states, the stick-slip vibrations between the oscillator and the beam are analyzed. In addition, to obtain the irregularly changed contact position due to the axial deformation of the beam and oscillator movement, a mathematical expression for the contact position is derived. It is found that the long-period stick-slip vibration is influenced mainly by the oscillator and the short-period vibration is influenced mainly by axial deformation of the beam. Furthermore, the dynamic responses show that even if a high damping ratio is applied to the oscillator, stick-slip vibration due to axial deformation of the beam can occur. Finally, the analysis shows that a kind of the internal resonance occurs between the oscillator and the beam when the harmonics of the natural frequency of the oscillator match the natural frequencies of the beam.

     

Moving-load dynamic problems: A tutorial (with a brief overview)

This tutorial is dedicated to the study of structural dynamics problems caused by moving loads. Through a simple example of a simply supported beam traversed by a moving mass, several fundamental concepts peculiar to moving-load problems are introduced. The necessary mathematics involved is presented. The analytical procedure is also presented for a circular plate excited by a rotating oscillator. Then numerical results of a circular beam spinning about its longitudinal axis excited by an axially moving surface load are provided. A variety of moving-load problems are briefly reviewed with some published papers and books to help readers quickly get into problems of their interests. Readers are expected to get a flavour of what moving-load problems are about, what general methods are available and what research has been done from studying this tutorial. Knowledge of partial differential equations and vibration theory of beams and plates is required in order to understand this tutorial.     

移动质量-梁耦合系统的动态响应分析

在建立了移动质量-梁耦合系统振动方程的基础上,针对时变力学系统采用Newmarkβ逐步积分法进行求解.对不同运动状态移动质量作用下的简支梁动态响应进行研究,得到在移动质量的初速度和加速度两个运动参数变化的情况下梁的挠度变化规律.数值模拟结果表明:移动质量作用下,梁的挠度曲线是以一定频率围绕静挠度线的一种类正弦波,移动质量的惯性作用对梁动态响应的影响不能忽略.     

Isogeometric bending analysis of composite plates based on a higher-order shear deformation theory

This research paper presents an isogeometric plate finite element formulation for analysis of thick composite plates. Isogeometric finite element method which is based on non-uniform rational B-splines (NURBS) basis functions, is a novel numerical procedure developed to bridge the gap between CAD and FEM modeling of structures. In order to investigate the behavior of isogeometric plate elements under static loading, plate kinematics is based on third order shear deformation theory (TSDT) of Reddy, which is free from transverse shear locking. This paper discusses accurate transverse stress recovery procedures for TSDT isogeometric finite elements. Numerical experiments with quadratic, cubic and quartic elements are presented and obtained results are compared to other available ones.     

In-plane and out-of-plane buckling of arches made of FGM

The mechanical buckling of curved beams made of functionally graded materials is studies in this paper. The equilibrium and stability equations of curved beams under mechanical loads are derived. Using proper approximate functions for the displacement components, the stability equations are employed to obtain the related eigenvalues associated with the buckling load of the curved beam. Closed-form solutions are obtained for mechanical buckling of curved beams with doubly symmetric cross section subjected to uniform distributed radial load and pure bending moment. The results are validated with the known data in the literature for beams with isotropic materials.     

Vibration of initially stressed beam with discretely spaced multiple elastic supports

Vibration behavior of an initially stressed beam on discretely spaced multiple elastic supports has been studied and a theoretical formulation of the system is derived using the variational principle. Unlike beams on an elastic foundation, discretely spaced supports can distort the beam mode shapes when the supports have rather large stiffness, i.e. usually expected beam modes cannot be obtained, but rather irregular mode shapes are observed. Conversely, irregular modes can be recovered by changing initial stress. Since support location is closely associated with the dynamic characteristics, this work also discusses eigenvalue sensitivity with respect to the support position and some numerical examples are investigated to illustrate the above findings.     

Safety and collapse of elastic–plastic beams against dynamic loads

The dynamic load-bearing capacity of elastic–plastic beam structures is analysed by the apparatus of shakedown theory. The reduced kinematic formulation for bending beams, which is equivalently deduced from Koiter’s kinematic theorem, combined with the plastic collapse’s method of hinge mechanisms appears effective in solving practical problems. The safety limits on the quasiperiodic dynamic loads as well as respective collapse mechanisms for a number of practical beams are determined.