轴向运动弦线横向振动的频域分析

轴向运动弦线是多种工程系统的模型。为明确轴向运动横向振动的频域特性，及探索频域方法的应用特点．本文用频域方法分析轴向运动弦线的横向振动。基于轴向运动弦线横向振动方程和边界条件．通过Laplace变换导出频率域中的控制方程，并将该控制方程和边界条件用状态变量表示。由状态空间中的控制方程导出特征方程，从而求出固有频率。由轴向运动弦线的矩阵函数计算得到系统的传递函数，然后用留数定理计算传递函数的Laplace逆变换．这样就可以得到时域响应。最后分析了轴向运动弦线的横向共振，若简谐外激励的频率与系统固有频率相同，系统响应将随时间无限增加。     

黏弹性轴向运动梁非线性受迫振动稳态幅频响应

本文研究了黏弹性轴向运动梁横向受迫振动稳态幅频响应问题.在控制方程的推导中,对黏弹性本构关系采用物质导数.把多尺度法直接应用于梁横向振动的非线性控制方程,利用可解性条件消除长期项,得到系统稳态的幅频响应曲线.运用Lyapunov一次近似理论分析幅频响应曲线的稳定性.通过算例研究了黏性系数,外部激励幅值以及非线性项系数对稳态幅频响应曲线及其稳定性的影响.运用数值方法对两端固定边界下黏弹性轴向运动梁的控制方程直接数值解,分析梁横向非线性振动的稳态幅频响应,通过数值算例验证直接多尺度法的结论.     

轴向运动弦线横向振动的线性反馈控制

研究轴向运动弦线和作动器组成的耦合系统的横向振动控制。此系统被作动器分成未控和受控两部分,通过作用在作动器上的控制力对受控部分的横向振动进行控制。采用能量方法获得反馈控制规律,得到两种控制力,并用半群理论证实受控弦线横向振动的渐近稳定性和指数稳定性。在初始扰动和激励力作用下,通过数值仿真证实控制规律的有效性。     

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.      

黏弹性土层中单桩纵向振动的对比分析

在频率域内研究了黏弹性土层中端承桩纵向振动的动力特性.将土骨架视为具有分数阶导数本构关系的黏弹性体,基于黏弹性理论,采用平面应变模型给出了分数阶导数黏弹性土层的动力阻抗.考虑桩纵向振动时的横向惯性效应,将桩等效为Rayleigh-Love杆,得到了桩头动力复刚度和导纳的解析表达式.通过数值计算,分析了不同模型土条件下桩头动刚度因子和阻尼随激励频率的动力响应.同时,研究了Rayleigh-Love和Euler-Bernoulli两种模型桩动力特性的差异.分析了桩-土界面连续性模型和相对滑移模型对黏弹性土层中桩纵向振动的影响.结果表明:1随着阶数和材料参数比的增加,桩头刚度因子和阻尼明显减小;2对于大直径桩,随着外荷载激励频率的增加,桩横向效应对刚度因子和阻尼有显著影响.3连续性模型条件下桩头的刚度因子和阻尼在共振时的振幅小于相对滑移模型条件.     

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.     

轴向变速运动粘弹性弦线的横向振动分岔

研究轴向运动弦线横向振动的分岔 .弦线轴向速度为常平均速度带有简谐涨落 ,其粘弹性材料由Kelvin模型描述 .建立系统的动力学方程并应用 2阶Galerkin截断进行简化 .计算了弦线中点的Poincar啨截面映射对平均轴向速度、轴向速度涨落幅值和弹性模量的分岔图 .     

作大范围运动弹性梁刚—柔耦合动力学建模

利用弹性梁的变形理论和 Hamilton力学原理对作大范围运动弹性梁的刚 -柔耦合动力学建模理论进行了研究。分析了大范围运动对弹性梁的横向振动和纵向振动的影响 ,得到了大范围运动与弹性梁的中线耦合变形之间的耦合作用对该系统动力学性质有显著的影响 ,从而提出了作大范围运动弹性梁的刚柔耦合动力学模型     

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。     

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.     

Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.     

Natural frequencies of nonlinear vibration of axially moving beams

Axially moving beam-typed structures are of technical importance and present in a wide class of engineering problem. In the present paper, natural frequencies of nonlinear planar vibration of axially moving beams are numerically investigated via the fast Fourier transform (FFT). The FFT is a computational tool for efficiently calculating the discrete Fourier transform of a series of data samples by means of digital computers. The governing equations of coupled planar of an axially moving beam are reduced to two nonlinear models of transverse vibration. Numerical schemes are respectively presented for the governing equations via the finite difference method under the simple support boundary condition. In this paper, time series of the discrete Fourier transform is defined as numerically solutions of three nonlinear governing equations, respectively. The standard FFT scheme is used to investigate the natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results are compared with the first two natural frequencies of linear free transverse vibration of an axially moving beam. And results indicate that the effect of the nonlinear coefficient on the first natural frequencies of nonlinear free transverse vibration of axially moving beams. The numerical results also illustrate the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters.     

轴向移动系统的参数振动问题研究进展

对近二十年来轴向移动系统(移动弦，移动梁和移动带等)的参数振动研究进展进行了详细的评述，特别关注了轴向张紧力和移动速度随时间改变时轴向移动系统的参数振动特性和稳定性等问题。文章首先讨论了所研究问题的控制方程。然后详细说明了目前研究中人们较为关注的几个重点问题，如参数激励的形式，求解方法和所研究的问题等。接着在其后的两节中，分别评述了在张紧力和移动速度随时间变化时，轴向移动弦和轴向移动梁的振动问题近年来的研究进展，详细、深入讨论了模型的类型、张紧力和轴向移动速度随时间变化的形式以及在研究中使用的解题方法和系统的振动特性(振动响应、固有频率和动态稳定性)等；最后给出了在此领域今后研究中应关注的问题。     

Transverse vibration characteristics of axially moving viscoelastic plate

The dynamic characteristics and stability of axially moving viscoelastic rect- angular thin plate are investigated.Based on the two dimensional viscoelastic differential constitutive relation,the differential equations of motion of the axially moving viscoelastic plate are established.Dimensionless complex frequencies of an axially moving viscoelastic plate with four edges simply supported,two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method.The effects of the aspect ratio,moving speed and dimensionless delay time of the material on the trans- verse vibration and stability of the axially moving viscoelastic plate are analyzed.     

基于积分本构黏弹性带的横向非线性动力学

研究了黏弹性传动带在1:1内共振时的横向非平面非线性动力学特性. 首先，利用 Hamilton原理建立了黏弹性传动带横向非平面非线性动力学方程. 然后综合应用多尺度法和 Galerkin离散法对偏微分形式的动力学方程进行摄动分析，得到了四维平均方程. 对平均方 程的稳定性进行了分析，从理论上讨论了动力系统解的稳定性变化情况. 最后数值模拟结果 表明黏弹性传动带系统存在混沌运动、概周期运动和周期运动.     

Supercritical vibration of nonlinear coupled moving beams based on discrete Fourier transform

Natural frequencies of nonlinear coupled planar vibration are investigated for axially moving beams in the supercritical transport speed ranges. The straight equilibrium configuration bifurcates in multiple equilibrium positions in the supercritical regime. The finite difference scheme is developed to calculate the non-trivial static equilibrium. The equations are cast in the standard form of continuous gyroscopic systems via introducing a coordinate transform for non-trivial equilibrium configuration. Under fixed boundary conditions, time series are calculated via the finite difference method. Based on the time series, the natural frequencies of nonlinear planar vibration, which are determined via discrete Fourier transform (DFT), are compared with the results of the Galerkin method for the corresponding governing equations without nonlinear parts. The effects of material parameters and vibration amplitude on the natural frequencies are investigated through parametric studies. The model of coupled planar vibration can reduce to two nonlinear models of transverse vibration. For the transverse integro-partial-differential equation, the equilibrium solutions are performed analytically under the fixed boundary conditions. Numerical examples indicate that the integro-partial-differential equation yields natural frequencies closer to those of the coupled planar equation.     

轴向运动导电薄板磁弹性耦合动力学理论模型

针对磁场环境中轴向运动导电薄板的动力学理论建模问题进行研究，得到较为完备的磁弹性耦合振动基本方程及相应的补充关系式。在考虑几何非线性效应下，给出薄板运动的动能、应变能以及外力虚功的表达式。应用哈密顿变分原理，推得磁场中轴向运动薄板的非线性磁弹性耦合振动方程，并得到力和位移满足的边界条件。基于麦克斯威尔电磁场方程，并考虑相应的电磁本构关系和电磁边界条件，推得任意磁场环境中轴向运动导电薄板满足的电动力学方程和所受电磁力表达式。分别针对纵向磁场环境、横向磁场环境、条形板等具体情形，给出了振动方程、电动力学方程和电磁力的简化形式。所得结果，可为此类问题的进一步求解和分析提供理论参考。     

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

The weakly forced vibration of an axially moving viscoelastic beam is investigated.The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved.The nonlinear equations governing the transverse vibration are derived from the dynamical,constitutive,and geometrical relations.The method of multiple scales is used to determine the steady-state response.The modulation equation is derived from the solvability condition of eliminating secular terms.Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation.The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.