三线摆周期的旋转矢量求法

由能量方程出发，引入新变量，用旋转矢量法讨论了三线摆大角度摆动时的周期。     

Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

The Sumcient and Necessary Condition of Lagrange Stability of Quasi-periodic Pendulum Type Equations

The quasi-periodic pendulum type equations are considered. A sufficient and necessary condition of Lagrange stability for this kind of equations is obtained. The result obtained answers a problem proposed by Moser under the quasi-periodic case.     

悬摆法测量气体推进剂激光推进冲量耦合系数

 给出了悬摆法测量激光推进冲量耦合系数的原理，分别以空气、氩气、氮气和氦气为推进剂，用悬摆法测量了抛物形推力器在能量不同的单脉冲TEA CO₂激光辐照下的冲量耦合系数。实验结果表明：氩气的冲量耦合系数最高，氦气的冲量耦合系数最低；在实验测试的激光能量范围内，4种推进剂气体的激光推进冲量耦合系数基本上都随着激光能量的增加而线性增大，冲量耦合系数的相对误差为5.4%~6.4%。实验结果与国外相关实验结论一致。     

世界首例"平面运动二级倒立摆实物系统控制"试验成功

20 0 3年 3月 2 5日北京师范大学李洪兴教授领导的复杂系统智能控制实验室成功地实现了平面运动二级倒立摆实物系统控制。自从 2 0 0 2年 8月李洪兴教授领导的科研团体在世界上首次成功实现四级倒立摆实物系统以后 ,他们又将“平面运动倒立摆实物系统控制的实现”作为重要研究课题之一。此前他们已经实现的一级至四级倒立摆均为“直线运动倒立摆”。据李洪兴教授介绍 ,从控制理论和控制工程的意义上讲 ,平面运动倒立摆实物系统控制的实现要比直线运动倒立摆实物系统控制的实现困难得多 ;这不仅是因为这样的系统其变量、非线性程度及不稳定性…     

激光烧蚀硬铝产生等离子体温度和力学效应测量

 为了研究激光推进技术中激光与材料相互作用的机制，获取等离子体状态参数及力学参数，采用Nd:YAG被动调Q固体激光器烧蚀硬铝，通过激光诱导等离子体光谱技术测得等离子体光谱和温度，由冲量摆测得力学参数。实验结果显示：在激光功率密度0.534×10⁸ W/cm²时，靶材表面的等离子体温度在等离子体辐射过程中呈二次曲线衰减；改变靶材等离子体点燃阈值附近的激光功率密度时，随着功率密度的增加，等离子体温度、冲量耦合系数也随着增大，当功率密度达到靶材的等离子体点燃阈值时，各参数达到最大，此后随着功率密度增加，由于等离子体对能量的屏蔽作用，导致靶材表面的等离子体温度降低，等离子体获得的动能减少，靶材耦合的冲量降低。     

沙摆实验误差分析

从实验误差理论角度，分析了沙摆重心变化引起的实验误差，结果表明由重心变化引起的误差相对于其它误差而言在实验中占主要地位．     

干簧管"永动摆"

介绍了干簧管的特性。分析电流磁场做功过程．认识“永动摆”能量来源．