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1、关于非线性振动第一页，讲稿共六十五页哦2机械振动的形成机械振动的形成惯性惯性+恢复力恢复力惯惯 性：维持系统的性：维持系统的运动运动状态状态恢复力：维持系统的恢复力：维持系统的平衡平衡状态，恒指向平衡位置状态，恒指向平衡位置线性与非线性系统遵循同样的物理原理线性与非线性系统遵循同样的物理原理 第二页，讲稿共六十五页哦3运动微分方程的建立运动微分方程的建立线性系统具有简单的线性系统具有简单的特征特征简谐运动简谐运动第三页，讲稿共六十五页哦4第四页，讲稿共六十五页哦5特征方程特征方程特征根特征根纯虚根纯虚根上述方程有非零解，要求系数矩阵的行列式为零上述方程有非零解，要求系数矩阵的行列式为零第五页，
2、讲稿共六十五页哦6满足上述方程的满足上述方程的特征向量特征向量第六页，讲稿共六十五页哦7振型：振型：第一阶振型第一阶振型第二阶振型第二阶振型方程的解方程的解第七页，讲稿共六十五页哦8线性系统具有线性系统具有特征特征第八页，讲稿共六十五页哦9例：例：已知质量已知质量m,杆长杆长l,求系统运动方程求系统运动方程系统的动能和势能系统的动能和势能非线性振动仍然可以用周期、振幅、非线性振动仍然可以用周期、振幅、相位等来刻画相位等来刻画非线性运动形式通常无法用非线性运动形式通常无法用初等函数表示初等函数表示第九页，讲稿共六十五页哦10怎样判断其路径？摆动周期的变化？怎样判断其路径？摆动周期的变化？定性分析
3、定性分析第十页，讲稿共六十五页哦11摆动周期的变化摆动周期的变化第十一页，讲稿共六十五页哦12线性情况线性情况相点沿相轨迹相点沿相轨迹匀速圆周运动匀速圆周运动量纲看物理概念量纲看物理概念无量纲概括一般规律无量纲概括一般规律第十二页，讲稿共六十五页哦13例：例：已知质量已知质量m,杆长杆长l,求系统运动方程求系统运动方程解：系统的动能和势能解：系统的动能和势能线性化线性化线性与非线性的联系线性与非线性的联系第十三页，讲稿共六十五页哦14例：当例：当基座周期运动时，求系统运动方程基座周期运动时，求系统运动方程解：系统的动能和势能解：系统的动能和势能线性化线性化？周期系数非线性周期系数非线性第十四页
4、，讲稿共六十五页哦The motion equation Where P is the elastic potential energy-a piece wisely differentiable 分段线性分段线性第十五页，讲稿共六十五页哦Three Elements Amplitude,Frequency and Phase(difference)Vibrating SystemDriving System第十六页，讲稿共六十五页哦Phase difference and vibration energy第十七页，讲稿共六十五页哦Stiffness increase第十八页，讲稿共六十五页哦
5、Phase modulationstiffness increase:Amplitude-frequency 第十九页，讲稿共六十五页哦Phase modulationstiffness increase:Amplitude-frequency 第二十页，讲稿共六十五页哦Phase modulationstiffness increase:Phase-frequency 第二十一页，讲稿共六十五页哦Phase modulationstiffness increase:Phase-frequency 第二十二页，讲稿共六十五页哦Mechanism on stiffness increaseMec
6、hanism on stiffness increasePhase difference 0-Phase difference 0-/2/2 第二十三页，讲稿共六十五页哦Mechanism on stiffness increaseMechanism on stiffness increasePhase difference 0-Phase difference 0-/2/2 第二十四页，讲稿共六十五页哦Phase modulation/stiffness increase第二十五页，讲稿共六十五页哦Phase modulationstiffness increase第二十六页，讲稿共六十五页
7、哦 混沌，指一种貌似无规则的运动，但支配它的规律混沌，指一种貌似无规则的运动，但支配它的规律却是用确定型方程来描述的。上面提到的庞加莱在总结天却是用确定型方程来描述的。上面提到的庞加莱在总结天体力学中的问题时，已经对这种现象有了认识。到体力学中的问题时，已经对这种现象有了认识。到2020世纪世纪5050年代，有些物理学家年代，有些物理学家(如玻恩如玻恩(M.Born)(M.Born)也已明确知道经也已明确知道经典力学中会有长期动态的不可预测性。但混沌现象和理论开典力学中会有长期动态的不可预测性。但混沌现象和理论开始受到重视，一般认为始于始受到重视，一般认为始于6060年代两件事。一是罗仑兹年代
8、两件事。一是罗仑兹(E.Lorenz)(E.Lorenz)在天气预报方程的研究中发现，尽管描述用的在天气预报方程的研究中发现，尽管描述用的方程是确定性的，天气长期动态却是不可预测的。另一个是，方程是确定性的，天气长期动态却是不可预测的。另一个是，几位数学家证明了有关经典力学动态的一个定理，即现在按几位数学家证明了有关经典力学动态的一个定理，即现在按他们的姓称谓的卡姆他们的姓称谓的卡姆(KAM)(KAM)理论。理论。第二十七页，讲稿共六十五页哦 这两件事也分别代表混沌理论两类对象和两种方法：这两件事也分别代表混沌理论两类对象和两种方法：罗仑兹的对象是罗仑兹的对象是耗散系统耗散系统(这类系统和周围
9、环境有联系，这类系统和周围环境有联系，在自然和工程中广泛存在在自然和工程中广泛存在)，而卡姆的对象是保守系统，而卡姆的对象是保守系统(当当作是孤立的、封闭的，在天体研究和统计物理中常见作是孤立的、封闭的，在天体研究和统计物理中常见)。罗。罗仑兹依靠的是数值计算，卡姆用的是严格数学推理，这两仑兹依靠的是数值计算，卡姆用的是严格数学推理，这两种方法在混沌理论研究里都是必不可少的。种方法在混沌理论研究里都是必不可少的。第二十八页，讲稿共六十五页哦Poincars Note on Chaos“If we knew exactly the laws of nature and the situation
10、 of the universe at the initial moment,we could predict exactly the situation of that same universe at a succeeding moment.But even if it were the case that the natural laws had no longer any secret for us,we could still only know the initial situation approximately.If that enabled us to predict the
11、 succeeding situation with the same approximation,that is all we require,and we should say that the phenomenon had been predicted,that it is governed by laws.第二十九页，讲稿共六十五页哦Poincars Note on ChaosBut it is not always so;it may happen that small differences in the initial conditions produce very great
12、ones in the final phenomena.A small error in the former will produce an enormous error in the latter.Prediction becomes impossible,and we have the fortuitous phenomenon.”（in a 1903 essay“Science and Method”by Poincar）第三十页，讲稿共六十五页哦Non-linearDynamic Phenomena第三十一页，讲稿共六十五页哦Linear EquationDamped linear
13、oscillatorNonlinear EquationDamped Duffing oscillatorDamped pendulumNonlinear systemsNonlinear dynamic systems contain products or functions of the dependent variable.第三十二页，讲稿共六十五页哦Non-linear Dynamic PhenomenaLinear System Fixed natural frequency Responds at excitation frequency One solution or“attr
14、actor”Non-linear System Natural frequency depends on amplitude May respond at frequencies other than excitation frequency Possibility of multiple solutions or“attractors”第三十三页，讲稿共六十五页哦Linear-SmallAmplitude PendulumNon-linear Phenomena VariableNatural FrequencyNon-linear-LargeAmplitude Pendulum第三十四页，
15、讲稿共六十五页哦Non-linear Phenomena-Jump phenomena for harmonic Excitation第三十五页，讲稿共六十五页哦Non-linear Phenomena Effects ofJump Phenomenon Sudden changes in amplitude of vibration can occur for small changes in frequency.It is possible to have more than one stable solution at a particular frequency.The region
16、E-B is unstable.Initial conditions determine which of the two solutions is attained,e.g.a large initial velocity may jump the system to the upper solution.第三十六页，讲稿共六十五页哦Sub-harmonic motionSuper-harmonic motionQuasi-periodic motionChaotic motion(random like)Non-linear Phenomena-Types of Response to H
17、armonic ExcitationHarmonicInputLinear SystemNonlinear SystemHarmonic motion第三十七页，讲稿共六十五页哦Examples of Systems ExhibitingChaos-Biological Systems Described initially by Robert May.Human physiology Brain-normal brain activity is thought to be chaotic.Heart-normal heart activity is more or less periodic
18、 but has variability thought to be chaotic.Fibrillation(loss of stability of the heart muscle)is thought to be chaotic.第三十八页，讲稿共六十五页哦Examples of Systems ExhibitingChaos-Fluid Systems Weather systems Models of the weather including convection,viscous effects and temperature can produce chaotic result
19、s.First shown by Edward Lorenz in 1963.Long term prediction isimpossible since the initial state is not known exactly.Turbulence Experiments and modelling show that turbulence in fluidsystems is a chaotic phenomenon.第三十九页，讲稿共六十五页哦Examples of Systems ExhibitingChaos-Mechanical Systems Systems with cl
20、earance Gear systems-gears can“rattle”against each other in a chaotic manner Rotor systems-clearance in bearings can induce chaos which can be used to diagnose bearing faults Two potential well system If a pendulum or the tip of a cantilever beam is set up between two strong magnets the pendulum or
21、cantilever will be attract to one or other magnet.The final solution of which attractor is achieved is chaotic.第四十页，讲稿共六十五页哦第四十一页，讲稿共六十五页哦Two Potential WellDivergence on Phase Plane第四十二页，讲稿共六十五页哦Forced Two Potential WellDivergence on Phase Plane第四十三页，讲稿共六十五页哦第四十四页，讲稿共六十五页哦Poincare Map Representation
22、 Sampled Data第四十五页，讲稿共六十五页哦Poincare Map第四十六页，讲稿共六十五页哦Examples of Systems ExhibitingChaos Logic Map第四十七页，讲稿共六十五页哦第四十八页，讲稿共六十五页哦x-Rotation of the eddy y-Horizontal temperature distribution z-Vertical temperature profile a=10,b=28,c=8/3Chaos in the Lorenzs EquationsAn Atmospheric Model第四十九页，讲稿共六十五页哦第五十
23、页，讲稿共六十五页哦第五十一页，讲稿共六十五页哦第五十二页，讲稿共六十五页哦Divergence of Close SolutionsLyapunov Exponent Divergence between close trajectories is measured by the Lyapunov exponent The Lyapunov exponent is calculated by propagating two initially close trajectories and measuring the divergence in each dimension with time
24、.One positive Lyapunov exponent for a system implies chaotic motion.第五十三页，讲稿共六十五页哦Lyapunov Exponents第五十四页，讲稿共六十五页哦Lyapunov A.M.(1857-1918)Alexander Lyapunov was born 6 June 1857 in Yaroslavl,Russia in the family of the famous astronomer M.V.Lypunov,who played a great role in the education of Alexand
25、er and Sergey.Alexander Lyapunov was a school friend of Markov and later a student of Chebyshev at Physics&Mathematics department of Petersburg University which he entered in 1876.第五十五页，讲稿共六十五页哦In 1885 he brilliantly defends his MSc diploma“On the equilibrium shape of rotating liquids”,which attract
26、ed the attention of physicists,mathematicians and astronomers of the world.The same year he starts to work in Kharkov University at the Department of Mechanics.He gives lectures on Theoretical Mechanics,ODE,Probability.In 1892 defends PhD.In 1902 was elected to Science Academy.After wifes death 31.1
27、0.1918 committed suicide and died 3.11.1918 第五十六页，讲稿共六十五页哦What is“chaos”?Chaos is a aperiodic long-time behavior arising in a deterministic dynamical system that exhibits a sensitive dependence on initial conditions.Trajectories which do not settle down to fixed points,periodic orbits or quasiperiod
28、ic orbits as t The system has no random or noisy inputs or parameters the irregular behavior arises from systems nonliniarityThe nearby trajectories separate exponentially fast Lyapunov Exponent 0第五十七页，讲稿共六十五页哦The Lyapunov ExponentA quantitative measure of the sensitive dependence on the initial con
29、ditions is the Lyapunov exponent .It is the averaged rate of divergence(or convergence)of two neighboring trajectories in the phase space.Actually there is a whole spectrum of Lyapunov exponents.Their number is equal to the dimension of the phase space.If one speaks about the Lyapunov exponent,the largest one is meant.第五十八页，讲稿共六十五页哦Simple Chaotic Systems and Circuits第五十九页，讲稿共六十五页哦Operational Amplifiers第六十页，讲稿共六十五页哦Lorenz第六十一页，讲稿共六十五页哦Poincar第六十二页，讲稿共六十五页哦Poincar Section第六十三页，讲稿共六十五页哦Poincar Section:Examples第六十四页，讲稿共六十五页哦感谢大家观看第六十五页，讲稿共六十五页哦