两跨输电线非线性主共振响应分析

研究两跨输电线非线性共振响应问题,应用Hamilton变分原理推导了两跨输电线的振动微分方程以及对应的边界条件。利用Galerkin离散方法和多尺度法,得到了单模态主共振响应。研究结果表明:幅频响应曲线表现出软、硬弹簧性质,随着外激励幅值的增大,输电线系统响应由软弹簧性质向硬弹簧性质转换;系统阻尼减小或外激励幅值增大时,系统幅值个数也随之发生变化,表现出多值和跳跃现象。     

Runge–Kutta methods for a semi-analytical prediction of milling stability

On the basis of Runge–Kutta methods, this paper proposes two semi-analytical methods to predict the stability of milling processes taking a regenerative effect into account. The corresponding dynamics model is concluded as a coefficient-varying periodic differential equation with a single time delay. Floquet theory is adopted to predict the stability of machining operations by judging the eigenvalues of the state transition matrix. This paper firstly presents the classical fourth-order Runge–Kutta method (CRKM) to solve the differential equation. Through numerical tests and analysis, the convergence rate and the approximation order of the CRKM is not as high as expected, and only small discrete time step size could ensure high computation accuracy. In order to improve the performance of the CRKM, this paper then presents a generalized form of the Runge–Kutta method (GRKM) based on the Volterra integral equation of the second kind. The GRKM has higher convergence rate and computation accuracy, validated by comparisons with the semi-discretization method, etc. Stability lobes of a single degree of freedom (DOF) milling model and a two DOF milling model with the GRKM are provided in this paper.     

Primary resonance of fractional-order van der Pol oscillator

In this paper the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically. At first the approximately analytical solution is obtained by the averaging method, and it is found that the fractional-order derivative could affect the dynamical properties of VDP oscillator, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude–frequency equation for steady-state solution is established, and the corresponding stability condition is also presented based on Lyapunov theory. Then, the comparisons of several different amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two fractional parameters, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated for some typical excitation amplitudes, which are different from the traditional integer-order VDP oscillator.     

2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator

The frequency-locking area of 2:1 and 1:1 resonances in a fast harmonically excited van der Pol–Mathieu–Duffing oscillator is studied. An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator. A perturbation technique is then performed on the slow dynamic near the 2:1 and 1:1 resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. These equations are used to determine the steady state responses, bifurcations and frequency-response curves. Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows. Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the 1:1 resonance. Numerical simulations are performed to confirm the analytical results.     

Evolution of the double-jumping in pipes conveying fluid flowing at the supercritical speed

Non-linear vibration of viscoelastic pipes conveying fluid around curved equilibrium due to the supercritical flow is investigated with the emphasis on steady-state response in external and internal resonances. The governing equation, a non-linear integro-partial-differential equation, is truncated into a perturbed gyroscopic system via the Galerkin method. The method of multiple scales is applied to establish the solvability condition in the first primary resonance and the 2:1 internal resonance. The approximate analytical expressions are derived for the frequency–amplitude curves of the steady-state responses. The stabilities of the steady-state responses are determined. The generation and the vanishing of a double-jumping phenomenon on the frequency–amplitude curves are examined. The analytical results are supported by the numerical integration results.     

One-to-three resonant Hopf bifurcations of a maglev system

This paper studies the dynamics of a maglev system around 1:3 resonant Hopf–Hopf bifurcations. When two pairs of purely imaginary roots exist for the corresponding characteristic equation, the maglev system has an interaction of Hopf–Hopf bifurcations at the intersection of two bifurcation curves in the feedback control parameter and time delay space. The method of multiple time scales is employed to drive the bifurcation equations for the maglev system by expressing complex amplitudes in a combined polar-Cartesian representation. The dynamics behavior in the vicinity of 1:3 resonant Hopf–Hopf bifurcations is studied in terms of the controller’s parameters (time delay and two feedback control gains). Finally, numerical simulations are presented to support the analytical results and demonstrate some interesting phenomena for the maglev system.     

Dynamical analysis of Mathieu equation with two kinds of van der Pol fractional-order terms

In this paper the dynamics of Mathieu equation with two kinds of van der Pol (VDP) fractional-order terms is investigated. The approximately analytical solution is obtained by the averaging method. The steady-state solution, existence conditions and stability condition for the steady-state solution are presented, and it is found that the two kinds of VDP fractional coefficients and fractional orders remarkably affect the steady-state solution, which is characterized by the additional damping coefficient (ADC) and additional stiffness coefficient (ASC). The comparisons between the analytical and numerical solutions verify the correctness and satisfactory precision of the approximately analytical solution. The presented typical amplitude–frequency curves illustrate the important effects of two kinds of VDP fractional-order terms on system dynamics. The application of two VDP fractional-order terms in vibration control is discussed. At last, the detailed results are summarized and the conclusions are made.     

Large amplitude oscillations of thick hyperelastic cylindrical shells

We present numerical solutions to the problem of large amplitude oscillations of a thick-walled hyperelastic cylindrical shell employing the general theory of finite dynamic deformations of elastic bodies. The material of the shell is considered incompressible and of Mooney-Rivlin type rubbers.

We apply a fourth-order Runge-Kutta numerical technique to the governing equation which was originally derived by J.K. Knowles in 1960.

We consider the free as well as forced oscillations due to a Heaviside step load and display graphs for the variations of amplitude against time and frequencies for different thicknesses and material constants. Discussions are presented on the significances of the results obtained.     

Buckling of Columns with Movable Boundaries∗

The Euler buckling problem of a slender tubular column subject to its own weight, tension or compression exerted at its top, and internal and external variable static fluid pressure is studied. This problem finds many applications in drilling and production risers, mining risers, hydraulic columns, and legs of Tension Leg Platforms. The supports at the upper end of the column are considered movable to properly simulate drill ships or platforms that support such columns. The corresponding eigenvalue problem is comprised of a fourth-order differential equation with variable coefficients and four homogeneous boundary conditions. Two methods of solution derived in previous work are implemented numerically. The first solution is expressed in terms of Airy functions of the first and second kind and the second in terms of power series. The combined results of the two methods yield the critical buckling curves over the entire domain of practical interest. The critical curves are plotted in the plane of the two loading variables for the first six buckling modes for four different sets of boundary conditions. The results reveal the dependence of the asymptotic behavior of the critical curves for long columns on the boundary conditions.     

Parametric excitation of a thin ring under time-varying initial stress; theoretical and numerical analysis

Initial stress in rings is one of the destructive effects which is almost inevitable due to various reasons such as being subsystems of a shrink-fitted joint, imperfections in the manufacturing, assembly or misalignment of the supporting mounts, and unbalancing in rotating condition. So, in this paper we focus on the effect of the initial stress and its variation with time on the dynamics of the pre-stressed ring. For this purpose, the equation of motion for in-plane bending vibration of a thin ring is derived using Hamilton’s principle. It is assumed that the initial stress is due to the distributed radially time-varying pressure. By representing the dynamic initial stress in the coefficients of the equation of motion; the equation is converted to Mathieu’s equation. The strained parameters method has been used to obtain the stability regions of motion and transition curves. Furthermore, to validate the obtained stability regions, numerical solutions of the equation of motion and Floquet theorem are used in some selected values of the parameters of the initial stress (magnitude of static and dynamic components of the initial stress). The fourth-order Runge-Kutta algorithm is used for numerical analysis of the equation of motion. The results show that the parameters of initial stress have direct impact on the stability of dynamic response. The obtained results from theoretical and numerical methods which are notably consistent with each other demonstrate that the initial stress, which has been almost always neglected in dynamic models of the systems, has a significant effect on the dynamics of the system, and it may even lead to an unstable dynamic response, while the excitation frequency is far enough from resonance region. So this paper can present the other application of modal analysis to non-destructive measure of initial stress.     

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.      

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。     

Planar nonlinear dynamic analysis of cable-stayed bridge considering support stiffness

Support stiffness is one of important factors on structure dynamics. Considering the vertical support stiffness, a multi-cable-stayed shallow-arch model of the cable-stayed bridge is established. Its differential equation governing the planar motion of cables and the shallow arch and the boundary conditions are derived by Hamilton’s principle. Firstly, the in-plane free vibration of the system is explored in order to find the modal functions and the possible internal resonances of nonlinear dynamics. Then, the 1:2:2 internal resonance among the different modes of the shallow arch and two cables are investigated by the multiple time scale method and pseudo-arclength algorithm. Meanwhile, the frequency-/force–response curves are used to explore the nonlinear behaviors of the system, especially the influence of vertical support stiffness, excitation frequency and amplitude on the internal resonance of the system is considered. To a certain extent, the support stiffness can reduce the response amplitudes of members by absorbing some energy from excitation.

     

Mathematical modelling and control of a nonholonomic spherical robot on a variable-slope inclined plane using terminal sliding mode control

In this paper, the three degrees-of-freedom motion of a two-dimensional rectangular liquid tank under wave action is simulated by the boundary element method in time domain. The coupling effects between tank motion and internal sloshing flow are investigated in partially filled conditions. The fourth-order Runge–Kutta method is adopted to update the wave shape and velocity potential on the free surface. The fully nonlinear mutual dependence of the incident wave, tank motion and internal sloshing flow is decoupled through an auxiliary function method, by which the liquid tank acceleration can be obtained directly without knowing the pressure distribution. The corresponding validation of numerical model is carried out and indicates that the accuracy of the present method is satisfactory to evaluate the dynamic responses of tank and sloshing motion. The corresponding response amplitude operators of tank motions for various wave frequencies, amplitudes and filling conditions are obtained, and the nonlinear coupling effects of sloshing flow on the tank responses are analyzed. It is found that the coupling effects have significant influence on sway and roll motion while have little impact on heave motion. The most important coupling effects on roll motion are the split of peak. In addition, due to the nonlinearity of sloshing flow, the roll motion amplitude is not linearly proportional to wave amplitude.     

Multi-Field Coupled Dynamics for a Micro Beam

This paper proposes a multi-field coupled dynamics equation for a micro beam. The natural frequencies and the amplitude–frequency relationship of the micro beam in the coupled fields are investigated. Changes in the natural frequencies of the micro beam along with time, bias voltage, and dynamic viscosity of gas are discussed. The effects of the system parameters on the amplitude–frequency relationship are investigated. A number of useful results are obtained. These results are useful in the sensitivity design of resonant micro gas sensors excited by the electrostatic force.     

Non-linear dynamics of a soft magneto-elastic Cartesian manipulator

This paper studies the non-linear dynamics of a soft magneto-elastic Cartesian manipulator with large transverse deflection. The system has been subjected to a time varying magnetic field and a harmonic base excitation at the roller-supported end. Unlike elastic and viscoelastic manipulators, here the governing temporal equation of motion contains additional two frequency forced, and linear and non-linear parametric excitation terms. Method of multiple scales has been used to solve the temporal equation of motion. The influences of various system parameters such as amplitude and frequency of magnetic field strength, amplitude and frequency of support motion, and the payload on the frequency response curves have been investigated for three different resonance conditions. With the help of numerical results, it has been shown that by using suitable amplitude and frequency of magnetic field, the vibration of the manipulator can be significantly controlled. The developed results and expressions can find extensive applications in the feed-forward vibration control of the flexible Cartesian manipulator using magnetic field.     

Computation of wave fields in the ocean around an Isaland

A linear wave equation correct to first order in bed slope is used to calculate the wave field in the sea around an idealized island. This is of circular cylindrical shape and is situated on a paraboloidal shoal in an ocean of constant depth (Figure 1). The sides of the island are assumed fully reflecting. The incident waves are plane and periodic. Wave periods up to 30 min are investigated, and the Coriolis force is neglected. The solution of the wave equation is represented by a finite Fourier series, and a large number of very accurate numerical computations are carried through. The results appear partly in figures showing amplitude and phase angle curves (in some cases extending to the water area of constant depth outside the shoal), partly in figures showing amplitude vs wave period in fixed points. Comparison with solutions to the linearized long-wave equation is made, and the validity range of the corresponding shallow water theory is given. The influence of the shoal is studied by investigating the wave field around an island in an ocean of constant depth. New criteria are given for the applicability of a geometrical optics approach (i. e. refraction). Complete numerical refraction solutions for points at the shoreline (corresponding to many wave orthogonals ending at the point) for shallows water waves, as for the general case, demonstrate the inadequacy of this approach for long-period waves (seismic seawaves: tsunamis). All non-linear effects, including dissipation, are excluded.     

Using delay to quench undesirable vibrations

A van der Pol type system with delayed feedback is explored by employing the two variable expansion perturbation method. The perturbation scheme is based on choosing a critical value for the delay corresponding to a Hopf bifurcation in the unperturbed ε=0 system. The resulting amplitude–delay relation predicts two Hopf bifurcation curves, such that in the region between these two curves oscillations will be quenched. The perturbation results are verified by comparison with numerical integration.     

轴向运动正交各向异性板的超谐波共振及稳定性

研究了轴向运动正交各向异性条形薄板在线载荷作用下的超谐波共振问题。通过哈密顿原理导出了几何非线性下正交各向异性条形板的非线性振动方程。运用伽辽金积分法,推得了关于时间变量的量纲归一化非线性振动微分方程组。应用多尺度法求解三阶超谐波共振问题,得到了稳态运动下一阶、二阶、三阶共振形式的共振幅值响应方程。利用Liapunov方法推得不同共振形式稳态解的稳定性判据,并据此分析不同参数对系统稳定性的影响。绘制了振幅特性变化曲线图和与之对应的激发共振多解临界点曲线图,分析系统参数对共振的影响,并预测系统进入非线性共振区域的临界条件。得出激励在特定位置区间时可激发系统的超谐波共振,随着激励幅值的增加,上稳定解支减小,下稳定解支增加,且一阶模态振幅大于二阶、三阶振幅。