Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model

Nonlinear parametric vibration of axially accelerating viscoelastic strings is investigated via an approximate analytical approach. The standard linear solid model using the material time derivative is employed to describe the string viscoelastic behaviors. A coordinate transformation is introduced to derive Mote’s model of transverse motion from the governing equation of the stationary string. Mote’s model leads to Kirchhoff’s model by replacing the tension with the averaged tension over the string. An asymptotic perturbation approach is proposed to study principal parametric resonance based on the two models. The amplitude and the existence conditions of the steady-state responses are determined by locating the nonzero fixed points in the modulation equations resulting from the solvability condition. Numerical results are presented to highlight the effects of the material parameters, the axial-speed fluctuation amplitude, and the initial stress on steady-state responses.     

Transverse vibrations of an axially accelerating viscoelastic string with geometric nonlinearity

Two-to-one parametric resonance in transverse vibration of an axially accelerating viscoelastic string with geometric nonlinearity is investigated. The transport speed is assumed to be a constant mean speed with small harmonic variations. The nonlinear partial differential equation that governs transverse vibration of the string is derived from Newton's second law. The method of multiple scales is applied directly to the equation, and the solvability condition of eliminating secular terms is established. Closed-form solutions for the amplitude of the vibration and the existence conditions of nontrivial steady-state response in two-to-one parametric resonance are obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation are presented. Lyapunov's linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions for two-to-one parametric resonance. Some numerical examples highlighting the effects of the related parameters on the stability conditions are presented.     

Asymptotic analysis of axially accelerating viscoelastic strings

An asymptotic approach is proposed to investigate nonlinear parametric vibration of axially accelerating viscoelastic strings. The string is constituted by the Kelvin model with the material time derivative. Both the Mote model and the Kirchhoff model of transverse motion are analyzed via the asymptotic approach in principal parametric resonance. The modulation equation is derived from the solvability condition. Closed-form expressions of the amplitudes and the existence conditions of steady-state responses are solved from the modulation equation. Numerical results are presented to highlight the effects the initial stress, the parameters in the Kelvin model, and the axial speed fluctuation amplitude on the amplitudes and the existence conditions of steady-state responses.     

Nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to mono-frequency excitation

In this paper, the nonlinear transversal vibration of an axially moving viscoelastic string on a viscoelastic guide subjected to a mono-frequency excitation is considered. The model of the viscoelastic guide is a parallel combination of springs and viscous dampers. The governing equation of motion is developed using Hamilton’s principle. Applying the method of multiple scales to the governing partial differential equation, the solvability condition and approximate solutions are derived. Three cases, namely primary, subharmonic and superharmonic resonances are studied and appropriate analytical solutions are obtained. The effect of mean value velocity, force amplitude, guide stiffness and viscosity coefficient of the string on the frequency-response and bifurcation points is investigated. Findings are in good agreement with results extracted from numerical modeling.     

Dynamic buckling of functionally graded cylindrical thin shells under non-periodic impulsive loading

Summary.  This paper is concerned with the study of the amplitude of the steady-state vibration in a right finite cylinder made of a mixture consisting of three components: an elastic solid, a viscous fluid and a gas. An exponential decay estimate of Saint-Venant type in terms of the distance from one end of the cylinder is obtained from a first-order differential inequality for a cross-sectional area measure associated with the amplitude of the steady-state vibration. The decay constant depends on the excitation frequency, the constitutive coefficients and the first positive eigenvalue for the fixed membrane problem for the cross-sectional geometry. The paper also indicates how to extend the results to a semi-infinite cylinder. Received October 24, 2002 Published online: April 17, 2003     

十字弦结构非线性自由振动的频率分析

该文研究了十字交叉弦几何大变形的非线性自由振动问题。首先根据哈密顿原理推导了控制其自由振动的运动方程,然后采用摄动方法求解了该弦的非线性耦合固有频率。通过将求得的非线性耦合固有频率解析解与各单弦的非线性频率解析解进行比较发现,非线性耦合频率的解析解除了具有非线性特性,还反应了各子结构对整体结构频率的影响,即存在耦合特性。并且,当一个子结构自身参数改变时,整体结构的频率也会发生变化,但是变化的幅度小于子结构的变化幅度,即耦合特性增加了十字弦系统的稳定性。     

轴向运动粘弹性弦线的横向非线性动力学行为

采用Poincaré映射和分岔图分析轴向运动黏弹性弦线横向振动的非线性动力学行为。考虑由积分型本构关系描述的黏弹性弦线,并计及微小但有限的变形导致的几何非线性,建立了系统的控制方程。应用Galerkin方法将系统控制方程截断,并通过引入辅助变量将截断后的方程转化为便于数值积分的形式。计算了弦线中点Poincaré映射对轴向张力涨落幅值、轴向运动速度、黏弹性系数和黏弹性指数的分岔图。     

轴向运动黏弹性梁横向非线性受迫振动

运用微分求积法数值研究不同边界条件下轴向运动黏弹性梁受到简谐外激励的横向受迫     

随机地震激励作用下自复位结构的平稳响应

自复位结构是一种震后不需修复或仅需少量修复即可继续投入使用的新型抗震结构。目前有关自复位结构地震响应的研究多是在确定性激励下进行,鲜有涉及随机激励环境。假定地震动加速度过程为金井清过滤白噪声模型,研究了随机地震激励下单自由度自复位体系的平稳响应。应用广义谐波平衡技术分解旗帜形的恢复力,建立原系统的等效非线性随机系统。通过Stratonovich-Khasminskii极限定理作随机平均,得到关于幅值的近似一维扩散过程。建立并求解对应的FPK方程,得到关于幅值的稳态概率密度函数并进行参数分析。数值结果表明,能量耗散系数与屈服位移的减小能降低系统响应,同时,随着这两个参数的变化,系统会出现随机P分岔现象。通过蒙特卡罗数值模拟法验证了解析解的有效性。     

横向风荷载作用下轴向运动弦线自激振动和稳定性分析

研究了横向定常风荷载作用下轴向运动弦线的非线性自激振动问题。将风荷载模型化为平均风速的非线性函数,建立动力学微分方程。采用Galerkin方法,将运动弦线简化为离散的二维系统并进行线性化,分析弦线平衡构型的稳定性,根据Routh-Hurtwitz判据确定了平衡点的稳定域。确定了多参数下Hopf分岔点及产生稳定极限环的条件。使用增量谐波平衡(IHB)法求解了自激振动的周期响应,按照Floquet理论确定了周期解的稳定性。最后,讨论了运动速度和平均风速稳定性的影响,并给出相应的稳定性条件。     

复合材料层合板的非线性组合共振特性及分岔

考虑几何非线性项和阻尼的影响, 给出了四边简支的正交各向异性矩形层合板在两项横向简谐激励作用下的非线性振动微分方程, 利用伽辽金法导出了相应的达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解, 得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论, 得到了解的稳定性判定条件。通过数值算例, 分析了不同参数对系统组合共振及其分岔特性的影响。结果表明, 随着调谐参数、板厚度、阻尼系数以及激励力等参数的改变, 系统存在多幅值现象、滞后现象和跳跃现象, 出现不稳定解, 且在某些参数点处具有运动性态发生变化的分岔特性, 表现出较为复杂的动力学特性。      

平行导线间轴向运动导电梁主-内联合共振EI北大核心CSCD

研究轴向运动导电梁在平行导线产生的磁场环境中的主-内联合共振问题。基于电磁场基本理论和哈密顿原理,导出轴向运动梁在外激励和磁场共同作用下的非线性振动方程。针对一端夹支一端铰支的导电梁,采用多尺度法求解方程,得到非线性方程的近似解析解和幅频响应方程,并对稳态解的稳定性进行了分析。通过算例,得到系统前两阶幅值随频率调谐参数、外激励力、轴向速度、电流强度等参数的变化规律。结果表明:系统发生主-内联合共振时一阶和二阶响应都被激发,且存在不同的多解区域;一阶和二阶幅值的稳态解个数在几个多解区域同步变化,其个数取决于外激励力、运动速度和电流强度值。     

Effect of transverse oscillations on the vibrational transport of a piezoelement along a string

We have studied conditions admitting the vibrational transport of a piezoelement along a string by means of low-frequency excitation of this piezoelement. Transportation of the piezoelement takes place only in certain regions of the string. The positions of these transport regions correspond to the nodes of standing transverse waves in the string and are determined by the frequencies of excitation and the natural transverse oscillations of the string. The position of the piezoelement on the string can be controlled by smoothly varying the frequency of the applied voltage.     

三交叉弦结构非线性自由振动频率特性分析

依据哈密顿原理获得了三交叉弦结构非线性自由振动的运动方程,并应用摄动法推导了自振频率下的一阶摄动解。相较于传统的单根弦线非线性振动运动方程多采用单三角级数,三交叉弦结构首次采用三重三角级数解法并成功获取一阶摄动解。通过分析表明,非线性自振频率的解析解除了具有典型的非线性特性,还体现了各个子结构参数变化对整体结构自振频率的影响,即存在子结构间的耦合特性。结果表明,整个结构与局部子结构在子结构自身因参数发生改变时,变化幅度之间不是线性关系,且整体结构小于子结构自身因参数改变的变化幅度。     

Forced periodic vibrations of laminated composite plates by a p-version,first order shear deformation,finite element

The subject of this study is the large amplitude, geometrically non-linear periodic vibrations of shear deformable composite laminated plates. A p-version, hierarchical finite element is employed to define the model, taking into account the effects of the rotary inertia, transverse shear and geometrical non-linearity. Harmonic forces are applied transversely to the plates and the steady-state periodic solutions are sought in the time domain by the shooting method. Fixing the amplitude of excitation and varying its frequency, response curves are derived. Several cases of modal coupling are found and the ensuing motions are analysed. The influences that the fibres orientations have on the forced vibrations are investigated. The efficiency and accuracy of the methods employed are discussed.     

电磁开关强非线性系统主共振分析

以电磁开关系统为研究对象,根据拉格朗日麦克斯韦方程建立三阶非线性动力学模型,应用MLP法得到电磁开关强非线性系统主共振的幅频响应方程及位移和电流分别随时间变化的振动规律,分析系统各参数对系统主共振幅频响应曲线的影响。结果表明,系统的幅频响应曲线存在跳跃现象;随着阻尼系数的增大,系统的振幅减小;随着激励电压幅值增大,系统的振幅和共振区域增大;随着电阻增大,系统的振幅和共振区域减小。     

窄带随机激励带集中质量梁的主参激共振

以受窄带随机激励带集中质量柔性梁为研究对象,采用多尺度法导出系统单模态主参激共振调谐方程组,结合FPK方程分析了力-幅特性下联合概率密度随激励幅值所出现的随机跳跃现象,结果显示:在三值响应范围内存在一外扇形峰及一内火山口峰,随着激励幅值减小,外扇形峰减弱而内火山口峰增强,表明响应逐渐从非平凡稳态响应向平凡稳态响应跳跃。     

受机械力作用金属／陶瓷功能梯度板主共振奇异性

研究机械力作用下金属，陶瓷功能梯度薄板主共振奇异性问题。按照功能梯度薄板的非线性动力学方程，得到金属，陶瓷功能梯度薄板受横向机械力作用的非线性振动方程。应用非线性振动的多尺度法得到系统主共振幅频响应分岔方程并进行奇异性分析，求得幅频响应分岔方程在开折参数平面的转迁集和分岔图。     

抽油杆柱纵向振动共振条件的质疑与力学模型的修正

在研究抽油杆柱纵向振动特性时，将抽油杆柱简化为上端固定于随悬点运动的基础之上、下端受液体负荷激励的力学模型。研究了单级抽油杆柱纵向自由振动的固有频率以及抽油杆柱对各种激励的稳态响应。建立了单级杆柱第i阶固有频率pi和基频p0之间的关系，即pi=(2i-1)p0，指出了单级抽油杆柱产生共振的条件是激励频率等于基频的奇数倍，证明了APIRPllL有杆抽油系统的设计计算推荐方法中关于激励频率等于偶数倍基频时，单级杆柱将产生共振的结论是错误的，并指出了导致这一错误结论的原因是所采用的力学模型是错误的。通过对抽油杆柱动态特性的计算机仿真，进一步证明了本文结论的正确性。     

矩形贮箱类液固耦合系统的平动响应研

采用与以往不同的方式,综合运用两种不同形式的Ｌａｇｒａｎｇｅ函数,以简洁的方式建立了常重力下受到水平激励的矩形贮箱类液固耦合系统的耦合动力学方程。该方法大大减少了公式推导,且所得的动力学方程是降阶的。通过数值仿真发现,在一定的激励幅值和频率下,该耦合系统会出现零点漂移等非线性动力学现象。