STEADY-STATE RESPONSES AND THEIR STABILITY OF NONLINEAR VIBRATION OF AN AXIALLY ACCELERATING STRING

The steady-state transverse vibration of an axially moving string with geometric nonlinearity was investigated. The transport speed was assumed to be a constant mean speed with small harmonic variations. The nonlinear partial-differential equation that governs the transverse vibration of the string was derived by use of the Hamilton principle. The method of multiple scales was applied directly to the equation. The solvability condition of eliminating the secular terms was established. Closed form solutions for the amplitude and the existence conditions of nontrivial steady-state response of the two-to-one parametric resonance were obtained. Some numerical examples showing effects of the mean transport speed, the amplitude and the frequency of speed variation were presented. The Liapunov linearized stability theory was employed to derive the instability conditions of the trivial solution and the nontrivial solutions for the two-to-one parametric resonance. Some numerical examples highlighting influences of the related parameters on the instability conditions were presented.

THE EXISTENCE OF PERIODIC SOLUTION OF THE FOURTH ORDINARY NONLINEAR DIFFERENTIAL EQUATION CAUSED BY FLOW－IN

The vortex-induced vibration is a well-known problem in mechanics,In this paper,with the help fixed point principle, we study. this problem and find the existencecondition of the periodic solution as well as the region of parameters.

ON GALERKIN DISCRETIZATION OF AXIALLY MOVING NONLINEAR STRINGS

A computational technique is proposed for the Galerkin discretization of axially moving strings with geometric nonlinearity. The Galerkin discretization is based on the eigenfunctions of stationary strings. The discretized equations are simplified by regrouping nonlinear terms to reduce the computation work. The scheme can be easily implemented in the practical programming. Numerical results show the effectiveness of the technique. The results also highlight the feature of Galerkin＇s discretization of gyroscopic continua that the term number in Galerkin＇s discretization should be even. The technique is generalized from elastic strings to viscoelastic strings.

Solving Nonlinear Differential Equation Governing on the Rigid Beams on Viscoelastic Foundation by AGM

在现在的纸，在粘弹性的基础上在一根僵硬横梁上管理的一个震动的微分方程被调查了。在这个颤动的系统上管理的非线性的微分方程被一条简单、创新的途径解决，它被称为 Akbari-Ganjis 方法(AGM ) 。AGM 是一个很合适的计算过程并且为解决各种各样的非线性的微分方程是可用的。而且，用哪个的 AGM 解决一套代数学的方程，没有任何数学操作，复杂非线性的方程能容易被解决。另外，为三个周期每周期失去的抑制比率和精力被调查了。而且，比较被数字方法(Runk45 ) 和 AGM 在获得的结果之间做了。结果显示出 AGM 的高精确性。结果也证明由增加颤动(A) 的起始的振幅的数量，抑制比率的价值将被增加，并且精力由增加周期的数字每周期减少输了。AGM 是为解决微分方程的一条可靠、精确的途径，这被结束。在另一方面，说 AGM 能处于大多数状况直接解决线性、非线性的微分方程更好。这意味着没有任何无尺寸的过程，最后的答案能被获得。因此， AGM 能在非线性的科学被看作重要进步。

An Approximate Method for the Surge Response of the Tension Leg Platform

The solution for the Duffing equation in a nonlinear vibration problem is studied in this paper. Clearly, in the case of the perturb parameter being a larger value, the traditional perturbation method is no longer valid but the Homotopy Perturbation Method(HPM) is applicable usually.HPM is used to solve the weak and strong nonlinear differential equations for finding the perturbed frequency of the response. The obtained frequencies via HPM and the approximate method have good accordance for weak and strong nonlinear differential equations. Additionally, the calculated responses by use of the approximate method are compared with the responses obtained from the Numerical method in the time history of the response and phase plane.The results represent good accordance between them.

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.

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green＇s function. To solve the integral-differential equations, expansion method is used to obtain Green＇s function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green＇s function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells .

The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique

This paper outlines the vibrational motion of a nonlinear system with a spring of linear stiffness. Homotopy perturbation technique (HPT) is used to obtain the asymptotic solution of the governing equation of motion. The numerical solution of this equation is obtained using the fourth order Runge-Kutta method (RKM). The comparison between both solutions reveals high consistency between them which confirms that, the accuracy of the obtained solution using aforementioned perturbation technique. The time history of the attained solution is represented through some plots to reveal the good effect of the different parameters of the considered system on the motion at any instant. The conditions of the stability of the attained solution are presented and discussed.

Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators

In this study, the homotopy analysis method (HAM) is used to solve the generalized Duffing equation. Both the frequencies and periodic solutions of the nonlinear Duffing equation can be explicitly and analytically formulated. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HAM and numerical integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations which achieved between the results. Besides, the optimal HAM approach is introduced to accelerate the convergence of solutions.

一类非线性系统的广义伴随线性方程分析研究

非线性输出频率响应函数是线性系统理论中频率响应函数在非线性系统中的一种推广,越来越多的学者已将其应用在结构损伤检测及故障诊断中.基于Volterra级数理论与由非线性微分方程描述的单输入单输出非线性系统的广义频率响应函数递归计算公式,利用多重积分性质和多维傅里叶变换,将广义频率响应函数映射到了一维频域中,推导出了一类非线性系统的广义伴随线性方程计算公式.研究表明,这类系统的第n阶非线性输出响应是以系统输入激励和前n-1阶非线性输出响应的组合函数作为广义激励作用到系统各阶广义伴随线性方程中的输出响应,最后通过求解一系列线性微分方程可得到这类非线性系统的任意阶非线性输出响应,其结果弥补了伴随线性方程无法求解这类非线性系统的不足.同时,针对广义伴随线性方程的数值计算问题,论文提出了一种耦合计算法,提高了计算非线性输出响应的精度,为非线性输出频率响应函数的计算提供了一种新思路.最后利用广义伴随线性方程与线性算子理论研究了两种典型非线性系统中非线性现象产生的原因,研究结果为非线性系统的分析与设计提供了一种有效途径.

含表面缺陷双列调心滚子轴承动力学特性研究

为了分析双列调心滚子轴承故障位置,基于动量定理建立了缺陷处冲击载荷计算公式,引入缺陷位置分段函数和承载区分段函数,建立了双列调心滚子轴承外圈和内圈含缺陷的非线性动力学方程,并利用Matlab/Simulink进行数值仿真,得到了不同转速下缺陷轴承的振动加速度,由仿真结果可知:轴承转速越高,缺陷冲击载荷越大,且冲击载荷在轴承底部最大,沿两侧逐级递减。最后对该动力学方程仿真结果进行了试验验证,试验测得振动加速度与仿真结果基本一致,验证了冲击载荷计算公式和动力学模型的正确性,根据冲击载荷类型可清晰判断缺陷所在位置。

Nonlinear combination parametric resonance of axially accelerating viscoelastic strings constituted by the standard linear solid model

Nonlinear combination parametric resonance is investigated for an axially accelerating viscoelastic string.The governing equation of in-planar motion of the string is established by introducing a coordinate transform in the Eulerian equation of a string with moving boundaries.The string under investigation is constituted by the standard linear solid model in which the material,not partial,time derivative was used.The governing equation leads to the Mote model for transverse vibration by omitting the longitudinal component and higher order terms.The Kirchhoff model is derived from the Mote model by replacing the tension with the averaged tension over the string.The two models are respectively analyzed via the method of multiple scales for principal parametric resonance.The amplitudes and the existence conditions of steady-state response and its stability can be numerically determined.Numerical calculations demonstrate the effects of the string material parameters,the initial tension,and the axial speed fluctuation amplitude.The outcomes of the two models are qualitatively and quantitatively compared.

Equilibrium equations for nonlinear buckling analysis of drill-strings in 3D curved well-bores

With the development of drilling technology, the oil/gas well has evolved from its early vertical straight form to the inclined, horizontal, plane curved, or even 3D curved well-bore. Understanding of the buck- ling behavior of a drill-string in a well-bore is crucial for the success of a drilling operation. Therefore, equilibrium equations for analyzing the buckling behavior of a drill-string in a 3D curved well-bore are required. Based on Love’s equilibrium equations for a curved and twisted rod in space, a set of equi- librium equations for the nonlinear buckling analysis of a drill-string in a 3D curved well-bore are de- rived by introducing a radial constraint of the well-bore. The proposed formulae can account for the well curvature and tortuosity. Thus, it can be used to analyze the buckling behaviors of a drill-string constrained in a well-bore and subjected to axial compression, torsion at its upper end, and gravity simultaneously. It is worth noting that the existing equations in the literature for a drill-string in a straight and plane curved well-bore with a constant curvature are a special case of the proposed model. Thus, the present model can provide a theoretical basis for the nonlinear buckling analysis of a drill-string constrained in a 3D curved well-bore.

Distribution of Vibrational Energy Levels of Protein Molecular Chains

The distributions of the quantum vibrational energy levels of the protein molecular chain are found by the discretely nonlinear Schrodinger equation appropriate to protein obtained from the Davydov theory. The results calculated by this method are basically consistent with the experimental values. Furthermore, the energy spectra at high excited states have also been obtained for the molecular chain which is helpful in researching the properties of infrared absorption and Raman scattering of the protein molecules.

Vibration Analysis of Warp on Weaving

Warp is a nonlinear viscoelastic material. The loom efficiency was reduced considerably by the increase of the frequency of weak places in the warp. Warp should vibrate with low-frequency to reduce the friction times. The objective of this research was to establish the transverse/longitudinal vibration differential equations of warp and analyze the fluctuation process of warp movement by nonlinear method. Based on variable separation method, the time variable was separated from space variable, and the numerical solutions of vibration equations were obtained by the fourth-order Runge-Kutta method. Finally, the influencing factors and variable trends on warp vibration were discussed. The methods on control warp vibration were introduced, which could guide the engineering practice. The most important way to reduce warp vibration is quick adjustment of warp tension.

LARGE RANGE ANALYSIS FOR NONLINEAR DYNAMIC SYSTEMS——ELEMENT MAPPING METHOD

This paper presents a new method for global analysis of nonlinear system. By means of transforming the nonlinear dynamic problems into point mapping forms which are single-valued and continuous, the state space can be regularly divided into a certain number of finitely small triangle elements on which the non-linear mapping can be approximately substituted by the linear mapping given by definition. Hence, the large range distributed problem of the mapping fixed points will be simplified as a process for solving a set of linear equations. Still further, the exact position of the fixed points can be found by the iterative technique. It is convenient to judge the stability of fixed points and the shrinkage zone in the state space by using the deformation matrix of linear mapping. In this paper, the attractive kernel for the stationary fixed point is defined, which makes great advantage for describing the attractive domains of the fixed points. The new method is more convenient and effective than the cell mapping method. And an example for two-dimensional mapping is given.

Offshore Structural Control Considering Fluid Structure Interaction

Tuned Mass Damper （TMD） was applied to an offshore structure to control ocean wave-induced vibration, In the analysis of the dynamic response of the offshore structure, fluid-structure interaction is considered and the errors, which occur in the linearization of the interaction, are investigated. For the investigation of the performance of TMD in controlling the vibration, both regular waves with different periods and irregular waves with different significant wave heights are used. Based on the numerical analysis it is concluded that the fluid-structure interaction should be considered in the evaluation of the capability of TMD in vibration control of offshore structures.

非线性振动系统的多项式向量方法

对于常微分方程描述的非线性振动系统,当采用摄动方法求近似解时,先是给出满足各阶近似解的二阶常微分方程组,继而依次对每一个常微分方程进行求解,以致多自由度非线性振动系统的求解过程相当繁琐.文章针对常微分方程表示的非线性振动系统,提出了一种求解非线性振动系统近似解的多项式向量方法,该方法将二阶常微分方程组表示成一阶状态方程组,将非线性部分写成常数矩阵和多项式向量之积的形式.然后,采用直接摄动方法,获得每个幂次近似解所满足的一组状态方程,此时状态方程的非线性部分成为常数矩阵和前一幂次近似解作为元素组成的多项式向量的乘积.进一步,借助Toeplitz矩阵将多项式向量之乘法表示成矩阵形式,以解决多项式相乘带来的幂次方系数的确定问题,再根据一阶非齐次方程组的求解方法,获得状态方程组的全部近似解析解.多项式向量方法将二阶常微分描述的非线性振动求解过程转换为一阶非齐次状态方程组的求解问题,计算过程主要是矩阵和向量之间乘法运算,提高了计算效率和程序化水平.

高温高压高产斜井油管柱摩擦磨损失效机理研究

针对高温高压高产(三高)气井油管柱磨损失效问题,采用微元法、能量法结合哈密顿原理建立了油管柱非线性流致振动模型,模型考虑了井眼轨迹变化、井筒温度压力变化和套管接触碰撞等因素,通过开展油管柱振动模拟试验验证了模型的正确性和有效性。基于White-Fleisher磨损理论,提出了三高气井油管柱磨损量和磨损深度计算方法,试验测定了油管柱摩擦因数和磨损效率,在此基础上,系统分析了产量、井斜角、井段长度、封隔器位置和扶正器位置对油管磨损特性的影响规律,提出了现场油管柱防磨、减磨方法,即:随产量的增加,管柱磨损寿命呈现降低的趋势,且存在一个突变产量,现场配产设计时,采用提出的分析方法确定突变产量,使现场配产远离突变产量;井眼轨迹设计时,尽量减小直井段长度,增加造斜段长度和稳斜段井斜角,提高油管寿命;封隔器和扶正器设置都存在一个最优位置,此位置与井身结构、管柱结构的因素有关,需采用提出的分析方法确定最优位置,指导井下工具的设计。研究成果可有效提高三高气井油管柱的使用寿命。

纸张振动中的整数与分数振动

本文使用微扰论方法对纸张振动建立非线性动力学方程模型,通过该模型预测纸张产生的低音与高音现象以及它们之间的联系,并通过实验验证.该模型的非线性动力学方程中存在与低音解相关的分数倍频解,这与实验中存在的分数倍频振动现象相吻合.该模型解释了整数倍频高音产生的非线性本征性以及分数频低音与分数倍频高音产生的衍生性,实验结果中的整数振动与分数振动现象满足理论预测.