时变张力作用下轴向运动梁的分岔与混沌

轴向运动结构的横向参激振动一直是非线性动力学领域的研究热点之一. 目前研究较多的是轴向速度摄动的动力学模型,参数激励由速度的简谐波动产生. 但在工程应用中,存在轴向张力波动的运动结构较为广泛,而针对轴向张力摄动的模型研究较少. 本文研究了时变张力作用下轴向变速运动黏弹性梁的分岔与混沌. 考虑随着时间周期性变化的轴向张力,计入线性黏性阻尼,采用Kelvin模型的黏弹性本构关系,给出了梁横向非线性 振动的积分--偏微分控制方程. 首先应用四阶Galerkin截断方法将控制方程离散化,然后采用四阶Runge-Kutta方法计算系统的数值解,进而确定其动力学行为. 基于梁中点的横向位移和速度的数值结果,仿真了梁沿平均轴速、张力摄动幅值、张力摄动频率以及黏弹性系数变化的倍周期分岔与混 沌运动,并且通过计算系统的最大李雅普诺夫指数来识别其混沌行为. 结果表明:较小的平均轴速有助于梁的周期运动,梁在临界速度附近容易发生倍周期分岔与混沌行为. 随着张力摄动幅值的增大,梁的振动幅值的混沌区间不断增大. 较小的黏弹性系数和张力摄动频率更容易使梁发生混沌运动. 最后,给出时程图、频谱图、相图以及Poincaré 映射图来确定梁的混沌运动.      

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

NONLINEAR VIBRATIONS OF AXIALLY ACCELERATING VISCOELASTIC TIMOSHENKO BEAMS

研究了轴向加速黏弹性Timoshenko梁的非线性参数振动。参数激励是由径向变化张力和轴向速度波动引起的。引入了取决于轴向加速度的径向变化张力,同时还考虑了有限支撑刚度对张力的影响。应用广义哈密尔顿原理建立了Timoshenko梁耦合平面运动的控制方程和相关的边界条件。黏弹性本构关系采用Kelvin模型并引入物质时间导数。耦合方程简化为具有随时间和空间变化系数的积分-偏微分型非线性方程。采用直接多尺度法分析了Timoshenko梁的组合参数共振。根据可解性条件得到了Timoshenko梁的稳态响应,并应用Routh-Hurvitz判据确定了稳态响应的稳定性。最后通过一系列数值例子描述了黏弹性系数、平均轴向速度、剪切变形系数、转动惯量系数、速度脉动幅值、有限支撑刚度参数以及非线性系数对稳态响应的影响。     

横向磁场中轴向运动铁磁梁的磁弹性主共振

研究在外激励力与磁场作用下轴向运动铁磁梁的磁弹性非线性主共振问题．基于弹性理论和电磁理论,给出梁的动能和弹性势能表达式,根据哈密顿原理,推导出磁场中轴向运动铁磁梁的磁弹性双向耦合非线性振动方程．通过伽辽金积分法进行离散,得出两端简支边界条件下铁磁梁磁弹性非线性强迫振动方程．应用多尺度法对方程进行求解,得出幅频响应方程．最后通过算例,给出铁磁梁的幅频特性曲线、振幅-磁感应强度和振幅-外激励力曲线并进行分析．结果显示,在幅频响应曲线中铁磁梁的轴向运动速度、外激励力、轴向拉力越大,共振振幅越大;而磁感应强度越大,振幅越小．     

Investigation of two oscillation types in a beam generated plasma

High frequency oscillations are observed in a beam generated plasma subjected to an axial uniform magnetic field. The oscillation frequency is measured as a function of the cyclotron frequency. The system is in a condition just below total glow.The oscillations are classified by comparing the experimental results with theoretical analyses of beam plasma systems in literature. Two types of oscillation are found. They result from (A) the interaction between the forward plasma wave and the slow space charge wave of the beam and (B) the interaction between the backward plasma wave and the slow space charge wave of the beam. The oscillation frequency of type A is practically independent of the cyclotron frequency, whereas the frequency of oscillation B increases linearly with it. For type A oscillations to occur a minimum value of the cyclotron frequency is required.Nomenclature phase constant - B ₀ static axial magnetic induction - e electron charge - f _c cyclotron frequency - f _cc critical cyclotron frequency - f _h hybrid frequency - f _osc oscillation frequency - f _p plasma frequency - I _b beam current - I _c collector current - m electron mass - N _e electron density per unit beam length - _b beam charge density - _p plasma charge density - r _p plasma radius - S beam pervenance - V _ak anode-cathode potential - V _b beam voltage - Units SI-units are employed     

横向载荷作用下轴向运动梁的屈曲失稳以及非线性振动特性

梁的轴向运动会诱发其产生横向振动并可能导致屈曲失稳,对结构的安全性和可靠性产生重大的影响。本文重点研究了横向载荷作用下轴向运动梁的屈曲失稳及横向非线性振动特性。基于Hamilton变分原理,建立了横向载荷作用下轴向运动梁的动力学方程,获得了梁的后屈曲构型。使用截断Galerkin法,将控制方程改写成Duffing方程的形式。用同伦分析方法确定载荷作用下轴向运动梁的非线性受迫振动的封闭形式的表达式。结果表明,后屈曲构型对轴向速度和初始轴向应力有明显的依赖性。通过同伦分析法得出非线性基频的显式表达式,获得了初始轴向力会影响非线性频率随初始振幅和轴向速度的线性关系。另外,轴向外激励的方向也会改变系统固有频率。     

THE STABILITY OF AN AXIALLY ACCELERATING BEAM ON SIMPLE SUPPORTS WITH TORSION SPRINGS

The axially moving beams on simple supports with torsion springs are studied. The general modal functions of the axially moving beam with constant speed have been obtained from the supporting conditions. The contribution of the spring stiffness to the natural frequencies has been numerically investigated. Transverse stability is also studied for axially moving beams on simple supports with torsion springs. The method of multiple scales is applied to the partialdifferential equation governing the transverse parametric vibration. The stability boundary is derived from the solvability condition. Instability occurs if the axial speed fluctuation frequency is close to the sum of any two natural frequencies or is two fold natural frequency of the unperturbed system. It can be concluded that the spring stiffness makes both the natural frequencies and the instability regions smaller in the axial speed fluctuation frequency-amplitude plane for given mean axial speed and bending stiffness of the beam.     

Nonlinear Vibration Analysis of Timoshenko Beams Using the Differential Quadrature Method

This paper addresses the large-amplitude free vibration of simplysupported Timoshenko beams with immovable ends. Various nonlineareffects are taken into account in the present formulation and thegoverning differential equations are established based on theHamilton Principle. The differential quadrature method (DQM) isemployed to solve the nonlinear differential equations. Theeffects of nonlinear terms on the frequency of the Timoshenkobeams are discussed in detail. Comparison is made with otheravailable results of the Bernoulli–Euler beams and Timoshenkobeams. It is concluded that the nonlinear term of the axial forceis the dominant factor in the nonlinear vibration of Timoshenkobeams and the nonlinear shear deformation term cannot be neglectedfor short beams, especially for large-amplitude vibrations.     

An analytical study on the nonlinear vibration of functionally graded beams

Nonlinear vibration of beams made of functionally graded materials (FGMs) is studied in this paper based on Euler-Bernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through thickness direction. Galerkin procedure is used to obtain a second order nonlinear ordinary equation with quadratic and cubic nonlinear terms. The direct numerical integration method and Runge-Kutta method are employed to find the nonlinear vibration response of FGM beams with different end supports. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FGM beams are discussed. It is found that unlike homogeneous beams, FGM beams show different vibration behavior at positive and negative amplitudes due to the presence of quadratic nonlinear term arising from bending-stretching coupling effect.     

经典梁热过屈曲问题的解析解

基于非线性经典梁理论,建立了控制轴向和横向变形的基本方程,将两个非线性方程化简为一个关于横向挠度的四阶非线性积分-微分方程。对于本文所考虑的三类边界条件,该方程与相应的边界条件构成了微分特征值问题;直接求解该问题,得到热过屈曲构形的解析解,该解是外加热载荷的函数。为考察热载荷以及边界条件的影响,根据得到的解析解给出了一些数值算例,讨论了梁过屈曲行为的性质。本文得到的解析解可用于验证或改进各类近似理论和数值方法。     

Errors in particle sizing by LDA due to turbidity in the incident laser beams

Measurements have been made of the effect of flow turbidity on the visibility and pedestal amplitude of an anemometer signal when incident laser beams are interrupted by particulate flow. The purpose is to assess the likely accuracy of particle sizing and the reliability of discrimination between continuous and particulate phase velocities. Optical depths of field were varied between 2.5 × 10^–2 and 14 × 10^–2 mm the diameter of the interrupting particles ranged between 14 and 800 m in six discrete ranges and the corresponding void fractions lay between 0.003% and 0.378%. The incident beam diameter was approximately 400 m.The measured size is subject to both systematic and random errors when inferred from measurements of pedestal amplitude: the random error increases as the ratio of the incident beam diameter to that of the particulate phase decreases. Systematic errors corresponding to a 10% underestimation of diameter occur at void fractions of 0.003%, 0.01% and 0.018% for particles below 40 m 75 m and 105 m respectively over a 5 cm depth of field. The r.m.s. error is smaller than 7% for particles below 40 m for all conditions studied but increases with increasing diameter and exceeds 10% at void fractions greater than 0.1% for particles above about 100 m. The random error in measured diameter derived from measurements of visibility is influenced mostly by the flow turbidity over the 5 cm of the incident beams closest to the measuring volume. For interrupting particles smaller than about 100 m the r.m.s. error is similar to that for measurements based on the pedestal amplitude.Discrimination of the velocity signal between the particulate and dispersed phase, based on the separation of pedestal amplitudes, is likely to be unreliable if the particle diameter is comparable to the diameter of the incident beams and if the probability of two particles simultaneously present in each beam is not negligible. A method for estimating the level of turbidity at which discrimination is no longer possible is described.List of symbols b beam diameter - d particle diameter - D amplitude of high passed Doppler signal (equation 7) - D ₁ depth of field of water tank - E particle extinction coefficient (= 2) - I _1,2 instantaneous light intensity at LDA measuring volume of each incident beam - n particle concentration (number per unit volume) - N number of particles in the incident beam - P maximum amplitude of pedestal component of Doppler signal - T Transmittance of beam {(attenuated/unattenuated) beam intensity} - V signal visibility (equation 8) - relative error in measured particle diameter {(truemeasured)/true)} - wavelength of laser beam (632.8 nm)     

On the effect of gravity on the bifurcation of rectangular closed-loop thermosyphon

Closed-loop thermosyphons are systems in which heat is transferred from a source to a sink by means of a natural convective flow, i.e. without the help of mechanical pumping. In fact, the dynamics of such systems strongly depend both on the thermal boundary conditions and on the gravitational field in which they operate. While the effect of variations of the boundary conditions has been extensively analysed in the last decades, the dependence on gravity has never been explicitly studied.The aim of this paper is to examine the effect of variations of gravity as well as that of thermal boundary conditions on the dynamics of natural circulation loops. Such an analysis might point out some useful applications for the cooling of a generic source in reduced gravity conditions.To this purpose an experimental campaign was performed on a natural circulation operating under a gravity field varying in the range between 10^–2 g and 1.8 g, with g = 9.81 ms^–2. The dynamical behaviour detected during the experiment was used for the validation of a mathematical model, previously validated under terrestrial gravity conditions. Model simulations were found to satisfactorily reproduce the dynamics of the system under variable gravity. This proved the possibility to use the model for the construction of bifurcation diagrams describing the behaviours of natural circulation loops under variations of both the gravitational field and the thermal boundaries.     

剪切流作用下层合梁非线性振动特性研究

针对剪切流中层合梁的大变形非线性振动问题, 采用高阶剪切变形锯齿理论和冯·卡门应变描述层合梁的变形模式和几何非线性效应, 构建了大变形层合梁非线性振动有限元数值模型; 采用基于任意拉格朗日?欧拉方法的有限体积法求解不可压缩黏性流体纳维-斯托克斯方程, 结合层合梁和流体的耦合界面条件建立了剪切流作用下层合梁流固耦合非线性动力学数值模型, 采用分区并行强耦合方法对层合梁的流致非线性振动响应进行了迭代计算. 研究了不同速度分布的剪切流作用下单层梁和多层复合材料梁的振动响应特性, 并验证了本文数值建模方法的有效性. 结果表明: 剪切流作用下单层梁的振动特性与均匀流作用下的情况不同, 梁的运动轨迹受剪切流影响向下偏斜, 随着速度分布系数增加, 尾部流场中的涡结构发生改变; 刚度比对剪切流作用下层合梁的振动特性有显著影响, 随着刚度比的增加, 层合梁振动的振幅增大, 主导频率下降, 运动轨迹由‘8’字形逐渐变得不对称; 发现了不同厚度比和铺层角度情况下, 层合梁存在定点稳定模式、周期极限环振动模式和非周期振动模式三种不同的振动模式, 改变层合梁铺层角度可实现层合梁周期极限环振动模式向非周期振动模式转变.      

Nonlinear oscillations of a charged drop levitated in gravity and electrostatic fields

Analytically, on the basis of asymptotic methods, the problem of the nonlinear oscillations of a charged ideal incompressible electroconductive fluid drop levitated at rest in gravity and homogeneous electrostatic fields is solved in the quadratic approximation in two small parameters: the initial drop shape deformation amplitude and the stationary eccentricity of the equilibrium drop shape in the electrostatic field. The calculations are performed in fractional powers of the nonlinear oscillation amplitude. The nonlinear corrections to the oscillation frequencies are always negative and already present in the second-order approximation due to the stationary deformation of the drop in the external fields rather than nonlinear interaction between the modes. In the case considered, in contrast to the nonlinear oscillations of a free charged drop, the expression for the generator of the nonlinearly oscillating drop shape contains terms proportional to the oscillation amplitude to the power 3/2.     

Experimental investigations of steady and oscillating annular liquid curtains with different surface tensions

Experiments were performed on laminar, vertical, annular, liquid curtains to study the dynamics of steady curtains, and the onset and frequency of oscillating curtains. The experiments were conducted to observe the effects of inertia and pressure on liquid curtains with different surface tensions. For steady curtains, convergence lengths were measured as functions of Froude number and pressure differential for three different surface tensions. The factors causing the onset of oscillations in a pressurized curtain were observed and the frequency of the internal pressure fluctuations were measured for various Froude numbers and two surface tensions.List of symbols b local thickness of curtain sheet - b ₀ initial thickness of curtain or nozzle gap thickness (0.5 mm) - C _P pressure coefficient - Fr Froude number (V ₀ ² /g R ₀) - g gravitational acceleration - g gravitational acceleration - L convergence length of curtain - L ^* dimensionless convergence length (L/R ₀) - N _c convergence number (g ² R ₀ ² b ₀ /2v ₀ ² ) - P _e pressure outside the curtain (ambient) - P _i pressure inside the curtain - P pressure differential (P _i– P _e) - P _cr pressure differential at which curtain begins to oscillate - R local radius of curvature in the horizontal plane - R ₀ initial curtain radius or radius of nozzle exit (50 mm) - r _v local radius of curvature in the vertical plane - V local liquid velocity - V ₀ initial liquid velocity - V ^* dimensionless local liquid velocity (V/V ₀) - z axial distance from the nozzle - z ^* dimensionless axial distance from the nozzel (z/R ₀) - s differential length of curtain - differential angle in the horizontal plane - angle between the direction of the surface tension force in the vertical plane and the direction of r _v - deangle between the direction of the surface tension force in the horizontal plane and the direction of R - angle between r _vand R in the vertical plane - ₀ nozzle exit angle (zero degrees) - surface tension of liquid - liquid density (1.0 gm/cm³)     

The nonlinear vibration analysis of a clamped-clamped beam by a reduced multibody method

The nonlinear response characteristics for a dynamic system with a geometric nonlinearity is examined using a multibody dynamics method. The planar system is an initially straight clamped-clamped beam subject to high frequency excitation in the vicinity of its third natural mode. The model includes a pre-applied static axial load, linear bending stiffness and a cubic in-plane stretching force. Constrained flexibility is applied to a multibody method that lumps the beam into N elements for three substructures subjected to the nonlinear partial differential equation of motion and N-1 linear modal constraints. This procedure is verified by d'Alembert's principle and leads to a discrete form of Galerkin's method. A finite difference scheme models the elastic forces. The beam is tuned by the axial force to obtain fourth order internal resonance that demonstrates bimodal and trimodal responses in agreement with low and moderate excitation test results. The continuous Galerkin method is shown to generate results conflicting with the test and multibody method. A new checking function based on Gauss' principle of least constraint is applied to the beam to minimize modal constraint error.     

非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.      

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

不同重力场下颗粒冲击过程的离散元分析

开展不同重力场下颗粒材料冲击动力学研究有助于加深对颗粒运动机制的理解和深空探测任务的实施。本文采用离散元模拟对颗粒介质受球形冲击物的冲击过程进行了数值模拟,并与地球重力下冲击的试验结果进行对比验证。在此基础上,进一步研究了重力加速度对冲击物动力学的影响规律。计算结果表明,在所有重力加速度下,冲击物的穿透深度d与冲击速度v₀的关系可以用Poncelet模型表达;d与冲击物下落的总高度H表现为d～Hⁿ的幂律关系,当H<10 m时,d与H的幂率标度为0.322,而H>10 m时,d与H的幂率标度下降到0.211。此外,穿透深度小于冲击物半径时,重力加速度对冲击物减速过程无影响。在所有的重力加速度下,当冲击速度大于5 m/s时,冲击物的持续碰撞时间t_c是恒定的,且与重力的-1/2次方呈正比。     

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique (GITT) is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations (ODEs) in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.