一类刚-梁耦合系统定态运动的稳定性

采用带有一悬臂梁的刚体模型, 研究了一类自由的刚-弹耦合系统定态运动的稳定 性. 直接从原始系统出发(未做离散化处理)，综合考虑了系统的平动与姿态运动的 耦合，在非完整坐标的Lagrange力学体系下选取状态变量，结合Lyapunov直接方 法和Chetaev的从运动方程的首次积分构造Lyapunov泛函的 方法. 引入的变量使得Lyapunov泛函形式简单，给运动稳定性分析带来了很大的方便. 最终给出了系统的定态运动按尺度稳定的充分条件.     

摩擦约束广义变分不等原理的临界变分现象及分析

对于具有摩擦约束的弹塑性接触问题,由于边界接触面上的摩擦力由不等式表示,导致得到包含摩擦约束的广义变分原理为广义变分不等原理.广义变分不等原理通过将摩擦力纳入问题的能量泛涵,可避免考虑摩擦力变化的具体过程,便于数值方法如有限元等在弹性接触问题上的应用.但是,通过对广义变分不等原理的研究,发现在弹性力学广义变分不等原理中,势能型和余能型广义变分不等原理,均存在临界变分现象,即变分时拉格朗日乘子为零,变分失败;或者得到的能量泛函变分后得不到问题的欧拉方程.在对弹性力学广义变分不等原理临界变分现象进行分析后,提出了避免发生临界变分现象的方法.实际应用证明了方法的有效性.通过避免临界变分现象的发生,可以保证拉格朗日乘子方法的有效使用.     

关于可变区域问题的变分法以及广义数学规划问题

本文具体研究了定义在可变区域上或可变边界表面上的泛函的一阶、二阶变分问题，得到了与经典变分法相对应的关于可变区域问题的变分法。并用该变分法讨论了具有可变区域的弹性系统的势能原理。另一方面，与传统的数学规划相对应，研究了可变区域上泛函的约束极值问题——广义数学规划问题，给出了相应的广义Ｋｕｈｎ－Ｔｕｃｋｅｒ条件。     

关于可变区域问题的变化法以及广义数学规划问题

本文具体研究了定义在可主域上或可变边界表面上的泛函的一阶，二阶变分问题，得到了与经典变分法相对应的关于可变区域问题的变分法，并用该变分法讨论了具有可变区域的弹性系统的势能原理。另一方面，与传统的数学模型相对应，研究了可变区域上泛函的约束极值──广义数学规划问题，给出相应的广义Ｋｕｈｎ－Ｔｕｃｋｅｒ条件。     

一类刚-梁耦合系统平面稳态运动的稳定性

研究了中心力场中的一类刚-弹耦合系统的平面运动动力学，模型是带有一悬臂 梁的刚体. 综合考虑了系统轨道运动与姿态运动，在Lagrange力学体系下给出了系统的运 动方程，在保守系统和考虑梁的材料黏滞阻尼两种情况下，利用能量-动量方法给出了一类 相对平衡点稳定性的充分条件.     

正交各向异性圆板大挠度问题的一种解法

本文基于Berger方法研究了正交各向异性圆板的大挠度问题。在所讨论问题的总势能泛函中引入中面应变不变量并应用欧拉变分方程,导得了非耦联的控制方程。最后由加权积分法给出均布载荷作用下周边固定和周边不动简支圆板的解析数值结果。     

旋转运动导电圆板电磁弹性耦合振动方程

针对磁场环境中旋转运动导电圆板的电磁弹性耦合振动理论建模问题进行研究。在考虑几何非线性效应下,给出了旋转运动圆板的形变势能、动能及变分表达式。应用哈密顿变分原理,推得磁场中旋转运动导电圆板的磁弹性耦合非线性振动方程。根据麦克斯威尔电磁场方程及相应的电磁本构关系,并基于磁弹性基本假设,推得磁场环境中旋转运动圆板所受的电磁力表达式和磁弹性二维电动力学方程。通过算例,分析了横向磁场中旋转运动圆板的轴对称振动问题,得到了圆板的固有振动频率随转速、磁感应强度的变化规律,并对结果进行了分析。     

大气运动不稳定的变分原理(Ⅰ)

本文利用非线性基本方程组和变分原理研究旋转大气运动的不稳定性问题。文中所用方法是普遍适用的,对各种可能的基流模型都可获得其不稳定性判据。例如:适合于正压或斜压、分层或连续模型以及准地转模式或原始方程组;基流可以是带状或非带状的(即平行流或非平行流),定常的或非定常的。 虽然基流具有多样性,但它们在由相应的不变泛函所决定的空间内都是驻点(临界点)。如果在相应的泛函中将角动量守恒包括进去,那么基流可以是非定常流。 无论线性或非线性二阶变分都给出不稳定性判据。但尤其值得一提的是,本文第一次得到关于非定常基流,地形扰动流和非地转流的不稳定性判据。 同时本文也指出了线性理论和我们发展的变分原理所得到的不稳定性判据之间的差别,该差别说明用非线性基本方程组的重要性。 在最后一节,将该理论推广到流体不具有有限能量,因而变分原理不能直接应用的情况,如卢平面,然而广义的Liapounoff函数仍然可以在变分考虑下得到。      

结构动力分析的随机变分原理及随机有限元法

将结构动力系统的参数及激励的随机性直接引入结构的动力泛函变分表达式中，基于瞬时最小势能原理，应用小参数摄动法，建立了随机结构动力分析的随机变分列式及相应的确机有限元法。算例表明，应用此法分析随机结构动力响应，具有程序实施简便，计算效率高的优点。     

有限位移弹性理论变分泛函的代入构成法

提出了变分泛函的一种新的数学构成法—代入法.将此法应用于弹性有限位移理论中,除了得到前人得到的变分泛函外,还得到了一对新泛函—狭义泛函.并通过狭义泛函,给出应力应变关系是目前所得到的各种变分泛函的Euler方程的结论.     

A preliminary study of 3D difference scheme with energy dynamic equilibrium

In this paper, a difference scheme with energy dynamic equilibrium (DS-EDE) is presented, which can be used for the simulation of long-term atmosphere and sea motion. Based on three dimensional nonlinear evolution equations for atmosphere and sea motion, a three dimensional compact upwind scheme (CUWS) is constructed, as the basis of the DS-EDE. The DS-EDE satisfies the following condition of energy dynamic equilibrium (EDE): the total work of external forces on the region boundary is equal to the sum of the total effective variation of the kinetic energy and the energy dissipation in the average flow motion and the effective variation of the potential energy per unit time within the region of interest. It really reflects the basic mechanism of the action of external forces and dissipation in atmosphere and sea movement. Therefore, the DS-EDE developed in this paper is a suitable model for simulating long-term atmosphere and sea movement with forcing and dissipation.     

Self-synchronization theory of dual motor driven vibration system with two-stage vibration isolation frame

This paper studies self-synchronization and stability of a dual-motor driven vibration system with a two-stage vibration isolation frame. Oscillation amplitude of the material box large enough can be ensured on the vibration system in order to screen materials. Reduction of the dynamic load transmitted to the foundation can also be achieved for the vibration system. A Lagrange equation is used to set up the motion differential equations of the system, and a dimensionless coupled equation of the eccentric rotors is obtained using a method of modified average small parameter. According to the existence condition of zero solution in the dimensionless coupled equation of the eccentric rotors, the precondition for commencing self-synchronization motion is achieved.The stability condition of self-synchronization is obtained based on the Routh-Hurwitz criterion. The theoretical analysis is validated by simulations and experiments.     

Stability of motion of two rigid bodies connected by a cable in the atmosphere

The spatial motion of two rigid bodies connected by a weightless inextensible cable in the atmosphere is considered. They are assumed to be bodies of revolution with static and dynamic symmetry. The condition of static stability of the system angular motion with respect to the direction of the incident airflow velocity vector is written out and analyzed. The influence of gyroscopic terms and damping moments on the stability condition is studied. An example of analysis of motion in the atmosphere of two connected bodies that are cones with a spherical tip is given. It is shown that the stable motion in the atmosphere can always be ensured by an appropriate matched choice of the parameters of the entire system on the basis of the obtained stability conditions. A numerical example of estimating the cable tension forces arising as the system descends on a ballistic trajectory in the atmosphere is presented.     

功能梯度旋转圆板的热弹性固有振动

在考虑热因素及旋转运动条件下，针对金属－陶瓷功能梯度圆板的固有振动问题进行研究．给出随温度变化且材料组分沿厚度方向按幂律分布的材料物性参数，依据热弹性理论得到圆板的能量关系式．基于哈密顿原理建立旋转金属－陶瓷功能梯度圆板热弹性动力学方程．采用伽辽金法得到边界约束下圆板的自由振动方程，确定了静挠度及固有振动频率．基于数值计算，得到系统固有频率值随体积分数指数、转速和温度等参量的变化曲线，讨论了静挠度变化规律及动力系统的奇点稳定性问题．结果表明，固有频率随体积分数指数、材料表面温度以及转速的增加而减小．     

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.     

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electro- magnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period- doubling motion and quasi-period motion are discussed.     

Stability analysis of the active control system with time delay using IHB method

In this paper, the incremental harmonic balance method is employed to solve the periodic solution that a vibration active control system with double time delays generates, and the stability analysis of which is achieved by the Poincare theorem. The system stability regions can be obtained in view of time delay and feedback gain, the variation of which is also studied. It turns out that along with the increase of time delay, the active control system is not always from stable to unstable, and the system can be from stable to unstable state, whereas the system can be from unstable to stable state. The extent that the two times delays impact on the system stability region is mainly related to the relative magnitude of the two feedback gains. The system can maintain the stable state under the condition of the well-matched feedback gains. The results can provide evidence to design the control strategy of time-delayed feedback.     

Hopf分岔的代数判据及其在车辆动力学中的应用

利用Hurwitz行列式,给出平衡点失稳而发生Hopf分岔的代数判定准则和计算方法,这一方法将Hopf分岔点的求解转化为一个非线性方程的求解问题,从而克服了以前方法在计算Hopf分岔点时,对于参数的每一次变化通过求特征根并判定特征根的实部是否为零的庞大工作量。应用这一方法,我们进行了非线性车辆系统蛇行运动稳定性的研究,得到了轮对系统发生蛇行运动的临界速度的解析表达式。     

Bursting-like motion induced by time-varying delay in an internet congestion control model

Time delay is an important parameter in the problem of internet congestion control. According to some researches, time delay is not always constant and can be viewed as a periodic function of time for some cases. In this work, an internet congestion control model is considered to study the time-varying delay induced bursting-like motion, which consists of a rapid oscillation burst and quiescent steady state. Then, for the system with periodic delay of small amplitude and low frequency, the method of multiple scales is employed to obtain the amplitude of the oscillation. Based on the expression of the asymptotic solution, it can be found that the relative length of the steady state increases with amplitude of the variation of time delay and decreases with frequency of the variation of time delay. Finally, an effective method to control the bursting-like motion is proposed by introducing a periodic gain parameter with appropriate amplitude. Theoretical results are in agreement with that from numerical method.