曲壁板在超音速气流中的分岔特性

采用Galerkin方法建立了超音速气流中二维曲壁板的非线性热气动弹性运动方程。用von Karman大变形理论来考虑曲壁板的大变形。用准定常的一阶活塞理论模拟曲壁板上表面受到的气动力。在不同来流速压和温升条件下,基于分岔理论研究了具有不同初始几何曲率的曲壁板系统对应的定常状态方程（组）的解的个数、性态和动态稳定性,并对方程（组）进行了解曲线的跟踪分析。研究表明,不同条件下,方程组的解特性不同,并且随着初始几何曲率和温升条件的变化,系统的失稳机理发生变化。超音速气流中的二维曲壁板系统存在动态Hopf分岔和静态鞍－结点分岔两种失稳现象,但不会发生热屈曲失稳。      

Bifurcation of the Curved Panel in Supersonic Air Flow

采用Galerkin方法建立了超音速气流中二维曲壁板的非线性热气动弹性运动方程。用von Karman大变形理论来考虑曲壁板的大变形。用准定常的一阶活塞理论模拟曲壁板上表面受到的气动力。在不同来流速压和温升条件下,基于分岔理论研究了具有不同初始几何曲率的曲壁板系统对应的定常状态方程（组）的解的个数、性态和动态稳定性,并对方程（组）进行了解曲线的跟踪分析。研究表明,不同条件下,方程组的解特性不同,并且随着初始几何曲率和温升条件的变化,系统的失稳机理发生变化。超音速气流中的二维曲壁板系统存在动态Hopf分岔和静态鞍－结点分岔两种失稳现象,但不会发生热屈曲失稳。     

跳跃振子平衡及其分岔的能量分析

用能量方法研究了跳跃振子的平衡与分岔.用势能驻值条件确定了平衡位置所满足的方程,通过势能极值判断平衡的稳定性.在不同的弹簧构型下,数值计算了平衡随系统弹性刚度和质量比变化的分岔图.结果表明,弹性刚度和质量比较小时,系统只有一个稳定平衡点和一个不稳定平衡点;刚度和质量比充分大时,系统分岔出一个新的稳定平衡点和一个新的不稳定平衡点.     

Duffing系统的主--亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.     

Duffing 系统的主-亚谐联合共振

以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象:刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.      

压电复合材料层合梁的分岔、混沌动力学与控制

研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分岔和混沌动力学响应. 基于vonKarman理论和Reddy高阶剪切变形理论,推导出了压电复合层合梁的动力学方程. 利用Galerkin法离散偏微分方程,得到两个自由度非线性控制方程,并且利用多尺度法得到了平均方程. 基于平均方程,研究了压电层合梁系统的动态分岔,分析了系统各种参数对倍周期分岔的影响及变化规律. 结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分岔解,从而阻止系统通过倍周期分岔进入混沌运动,并给出了控制分岔图.      

封闭水平矩形腔内流体自然对流第一次逆叉形分岔的数值研究

本文采用二阶全展开ETG(Euler-Taylor-Galerkin)分裂步有限元方法,对长宽比为3.5(L/B=3.5,如图 1所示)的封闭矩形腔体内,三种Pr数条件下,定常层流范围内,流体自然对流叉形分岔随Rayleigh数的演化过程进行了数值模拟。研究结果表明,该矩形腔内对应三种Pr数条件下,流体的叉形分岔的演化过程中,在第二次模态Ⅱ型叉形分岔之后,均会出现两个较小尺度涡旋合并,突变为一个较大尺度涡旋的全新叉形分岔模态。即在某临界Ra数两侧,存在定常四涡结构和定常三涡结构两个定常解支,当系统控制参数Ra越过临界值,前者被后者突发性取代,这是完全不同于传统叉形分岔的逆叉形分岔。其数值预报,则采用分半法结合流动拓扑结构及典型截面处速度扩线上鞍点的变化来确定。计算结果表明,在计算的Pr数条件下,随Pr数的增加逆叉形分岔对应临界Ra数的取值也会提高。     

进动圆筒内液体流动不稳定的非线性特征

在建立进动充液圆筒内液体偏差流动方程的基础上，结合液体惯性波和轴向二次流动线性解，通过对定常二次流动的线性稳定性分析，提出了函数空间表达的流动不稳定性非线性分岔分析方程. 对非惯性坐标系下液体流动的Navier-Stokes方程进行了数值求解，并对惯性波发生破裂(实验提供的3种主模态下得出的共振破裂现象)时的压力时间序列进行分析，得出了液体流动不稳定的基本非线性特征.     

外激励作用下弹性支撑浅拱的非线性动力学分析

研究了外激励下两端采用转动弹簧约束的铰支浅拱在发生1:1内共振时的非线性动力学行为。通过引入基本假定和无量纲化变量得到浅拱的动力学控制方程, 将阻尼项、外荷载项和非线性项去掉后,所得线性方程及对应边界条件即可确定考虑转动弹簧影响的频率和模态, 发现转动约束取不同刚度值时系统存在模态交叉与模态转向两种内共振形式。对动力方程进行Galerkin全离散, 并采用多尺度法对内共振进行了摄动分析, 得到了极坐标和直角坐标两种形式的平均方程, 其中平均方程系数与转动弹簧刚度一一对应。最低两阶模态之间1:1内共振的数值研究结果表明: 外激励能激发内共振模态的非线性相互作用, 参数处于某一范围时系统存在周期解、准周期解和混沌解窗口, 且通过(逆)倍周期分岔方式进入混沌。     

船舶甲板上浪的非线性横摇响应

考虑甲板上浪引起的横倾力矩、非线性阻尼和非线性恢复力矩,建立了规则横浪中船舶横摇运动方程;基于伯努利方程,推导了船舶甲板上浪水质量的计算公式.以66.01米长的拖网渔船为例,计算了不同波浪扰动力矩作用下的横摇响应,并构造了响应的分岔图和庞加莱截面.结果表明:甲板上浪后船舶横摇运动包括混沌运动和周期二、周期三、周期四等多周期运动;随着波浪扰动力矩幅值的增大,横摇响应发生从周期运动到混沌运动或从混沌运动到周期运动的阵发性分岔、正向及反向倍周期分岔.     

Stability and Hamiltonian Hopf bifurcation for a nonlinear symmetric bladed rotor

The modal interaction which leads to Hamiltonian Hopf bifurcation is studied for a nonlinear rotating bladed-disk system. The model, which is discussed in the paper, is a Jeffcott rotor carrying a number of planar blades which bend in the plane of the motion. The rigid rotating disk is supported on nonlinear bearings. It is supposed that this dynamical system is a Hamiltonian system which is perturbed by small dissipative and nonlinear forces. Krein’s theorem is employed for obtaining a stability criterion. The nonlinear eigenvalue equations on the stability boundary are turned into ordinary differential equations (ODEs) by differentiating them over the rotating speed. By solving these ODEs, the eigenmodes and the eigenvalues on the stability boundary are obtained. The bifurcation analysis is performed by applying multiple scales method around the boundary. The rotor nonlinear behavior and damping effects are studied for different conditions on the rotating speed and nonlinearity type by the bifurcation equation. It is shown that the damping distribution between the blades and bearings may shift the unstable mode. Depending on the nonlinearity type, subcritical and supercritical Hopf bifurcation are possible.     

Stability analysis of a nonlinear rotating asymmetrical shaft near the resonances

An asymmetrical rotating shaft with unequal mass moments of inertia and flexural rigidities in the direction of principal axes is considered. In this system, there are two excitation sources, including a harmonic excitation due to the dynamic imbalances and a parametric excitation due to shaft asymmetry. Nonlinearities are due to the in-extensionality of the shaft and large amplitude. In this study, harmonic and parametric resonances due to the mentioned effects are considered. The influences of inequality of mass moments of inertia and flexural rigidities in the direction of principal axes, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation of steady state response of the rotating asymmetrical shaft are investigated. In addition, the characteristic of stable stationary points and loci of bifurcation points as function of damping coefficient are determined. In order to analyze the resonances of the system the multiple scales method is applied to the complex form of partial differential equations of motion. The achieved results show a good agreement with those of numerical computation.     

Resonances of an in-extensional asymmetrical spinning shaft with speed fluctuations

In this study, main and parametric resonances of an asymmetrical spinning shaft with in-extensional nonlinearity and large amplitude are simultaneously investigated. The main resonance is due to inhomogeneous part of the equations of motion, which is due to dynamic imbalances of shaft whereas the parametric resonances are due to parametric excitations due to speed fluctuations and a shaft asymmetry. The shaft is simply supported with unequal mass moments of inertia and flexural rigidities in the direction of principal axes. The equations of motion are derived by the extended Hamilton principle. The stability and bifurcations are obtained by multiple scales method, which is applied to both partial and ordinary differential equations of motion. The influences of asymmetry of shaft, speed fluctuations, inequality between two eccentricities corresponding to the principal axes and external damping on the stability and bifurcation are studied. To investigate the effect of speed fluctuations on the bifurcations and stability the loci of bifurcation points are plotted as function of damping coefficient. The numerical solutions are used to verify the results of multiple scales method. The results of multiple scales method show a good agreement with those of numerical solutions.     

Bifurcation of a shaft with hysteretic-type internal friction force of material

IntroductionItwasfoundlongtimeagothattheinternalfrictionofmaterialcancauseinstabilityofrotatingshaft.Soitisalwaysoneoftheimportantsubjectsinrotordynamics[1].Earlyinvestigationswerefocusedonthedynamicalstabilityproblemofrotorinfluencedbythelinearinternalfrictionofmaterial,aimingtoobtainthecriterionofstability[2～4 ].Asthedevelopmentofnonlineardynamics,moreandmoreattentionswerepaidtothestudyoftheself_excitedmotionofrotatingshaft,thatisthebifurcation .Thestabilityregionsandbifurcationsofbothanau…     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

复合材料层合板1:1参数共振的分岔研究

针对复合材料对称铺设各向异性矩形层合板的物理模型,在同时考虑了材料、阻尼和几何等非线性因素后,建立了二自由度非线性参数振动系统动力学控制方程,并应用多尺度法求得基本参数共振下的近似解析解,利用数值模拟分析了系统的分岔和混沌运动.指出了伽辽金截断对系统动力学分析的影响,以及系统进入混沌的途径.     

具不对称粘弹性支承弹性盘轴转子系统的非线性动力稳定性分析

考虑轴和盘的质量及轴的几何非线性，建立了在周期轴向载荷作用下具不对称粘弹性支承弹性盘轴转子系统的非线性动力学方程．基于Lyapunov运动稳定性理论和Floquet判据，对系统的线性和非线性动力稳定性进行了分析．     

Two-mode combination resonances of an in-extensional rotating shaft with large amplitude

In this paper, two-mode combination resonances of a simply supported rotating shaft are investigated. The shaft is modeled as an in-extensional spinning beam with large amplitude. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. The equations of motion are derived with the aid of the Hamilton principle and then transformed to the complex form. The method of harmonic balance is applied to obtain analytical solutions. Frequency-response curves are plotted for the combination resonances of the first and the second modes. The effects of eccentricity and external damping are investigated on the steady state response of the rotating shaft. The loci of saddle node bifurcation points are plotted as functions of external damping and eccentricity. The results are validated with numerical simulations.     

Dynamic Response and Stability of a Rotating Asymmetric Shaft Mounted on a Flexible Base

This paper presents the dynamic response and stability of an asymmetric rotating shaft supported by a flexible base near the major critical speed and the secondary critical speed. In this system, the base is movable only in a direction transversal to the shaft. In the theoretical analysis, taking into account the effects of damping, the unstable vibrations near the major critical speed are mainly considered, and also the behavior of the forced oscillations near the major and secondary critical speeds is investigated. From the theoretical analysis, the unstable region is found to be divided into at most six subregions which depend on the mass of the base, the stiffness of the base, and the asymmetry of the shaft. In addition, the resonance curves near unstable subregions are calculated. It is found that there exist two shapes of resonance curves. In experiments, five types of response curves, which contained n unstable subregion (n = 1, 2, ¨, 5) near the major critical speed, were obtained by changing the mass of the base. It was ascertained that the theoretical results for the behavior near the major critical speed agreed quantitatively with the experimental results.     

具有曲率和惯性非线性以及材料内阻的旋转复合材料轴的主共振

旋转复合材料轴作为一类典型的转子动力学系统,在先进直升机和汽车动力驱动系统中有着广阔的应用前景.研究旋转复合材料轴的非线性振动特性具有重要的理论与实用价值.然而,目前有关旋转轴的非线性振动研究仅限于各向同性金属材料轴,很少考虑材料内阻的影响.本文研究具有材料内阻的旋转非线性复合材料轴的主共振.非线性来源于不可伸长复合材料轴的大变形引起的非线性曲率和非线性惯性,材料内阻来源于复合材料的黏弹性.动力学建模计入转动惯量和陀螺效应.基于扩展的Hamilton原理,导出具有偏心激励的旋转复合材料轴的弯-弯耦合非线性振动偏微分方程组.采用Galerkin法将偏微分方程离散化为常微分方程,采用多尺度法对常微分方程进行摄动分析,导出主共振响应的解析表达式.对内阻、外阻、铺层角、长径比、铺层方式和偏心距进行数值分析,研究上述参数对旋转非线性复合材料轴的稳态受迫振动响应行为的影响.研究发现,角铺设复合材料轴的内阻系数随着铺层角的增大而增大;内阻对主共振响应特性的影响主要体现在对抑制振幅和改变频率响应的稳定性方面;发生在正进动固有频率附近的主共振响应具有典型的硬弹簧非线性特性.本文提出的模型能够用于描述旋转复合材料轴的主共振特性,是对不可伸长旋转金属轴非线性动力学模型的重要推广.