固支弹簧联接的旋转摆的非线性动力学行为研究
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摘要
本文提出并建立了一类全新的摆类模型,并依据模型的力学结构命名为固支弹簧联接的旋转摆。它是一类典型的柱面摆,通过对参数的控制,系统表现出光滑与不连续的动力学行为。应用非线性动力学理论方法与数值方法,对固支弹簧联接的旋转摆的非线性动力学行为从定性和定量两个方面进行了分析和计算。
本文具体安排如下:
第一章,介绍了摆类的发展史和非线性动力学相关问题的研究历史与现状以及本文的主要创新点。
第二章,建立固支弹簧联接的旋转摆模型,给出动力学运动微分方程,利用Matlab软件得到了系统的平衡点分岔曲线图,恢复力函数分析,势能函数分析、相图分析与吸引盆分析,研究结果表明系统不但具有光滑的动力学行为,而且还包含了不连续的动力学行为。
第三章,通过构造新的柱面近似系统对光滑固支弹簧联接的旋转摆非线性动力学行为进行了定量分析,用Melnikov方法分别对同宿轨道、柱面同宿轨道以及两类同宿轨道并存系统进行了混沌解析预测,得到了阈值曲线。利用Dynamics软件进行了的数值模拟,验证了周期解和混沌解存在。
第四章,本章对不连续固支弹簧联接的旋转摆的非线性动力学行为进行了定性分析,包括恢复力分析、势能函数分析与相图分析,可以得到系统具有柱面类同宿轨道、类同宿轨道的复杂动力学行为。利用Dynamics软件进行了的数值模拟,验证了周期解和混沌解存在。
第五章,柱面近似系统在周期扰动下呈现,Hopf分岔,二次闭轨分岔,柱面同宿轨道分岔以及同宿轨道分岔。应用Melnikov函数和广义Melnikov函数对系统的分岔曲线进行求解,利用Matlab软件得到全局分岔图,并用Dynamics软件进行数值验证。
第六章,本章主要考虑不连续系统,在不连续系统中,首先,巧妙的求得系统的解;其次,在黏性阻尼和周期外激励作用下,我们用Melnikov函数得到类异宿轨道横截相交的阈值曲线;最后,研究了不连续系统在周期函数下的分岔。
第七章,总结本文工作,并对固支弹簧联接的旋转摆今后的工作进行了展望,可以在理论分析和工程应用等方面进行下一步研究。
附录
1.求解近似系统的同宿轨道和柱面同宿轨道的解;
2.求解不连续系统柱面类同宿轨道的解;
3.求解类异宿轨道的解;
4.用Matlab软件编写实物演示程序,形象生动的展现周期解和混沌解的可视动画。本程序精髓在于运用了余弦定理及符号函数。
In this article,a kind of novel pendulum modle is proposed and established. Accoroding to the mechanics stucture of the modle, it is named Rotating Pendulum Linked a Clamped and Supported Spring. This kind of pendulum is a typical cylinder pendulum, and the system displays smooth and discontinuous dynamics behaviour depending on the value of a system parameter. The nonlinear dynamics behaviour is analized and computed from the pualitative and the puota which used the nonlinear dynamics theory method and the numerical method.
The concrete arrangement of this article as follows:
In the first chapter, the development of the pendulum, the research history and the present situation of the non-linear dynamics, and innovation spots are introduced.
In the second chapter, the modle of the Rotating Pendulum Linked a Clamped and Supported Spring is established, at the same time, the dynamic motion differential equation is given, then using the Matlab, the diagram of the system’s balance points, the analy of the resume strength fuction, the potential fuction, the phase diagram and the attractor basins are obtained. The conclution indicates that the system not only has the smooth dynamics behaviour but also contains the discontinual dynamics behaviour.
In the third chapter, the puantitative analysis of the smooth Rotating Pendulum Linked a Clamped and Supported Spring is given by investigating the approximate system. The threshold value curve is obtained by the Melnikov function analysis forecast to the system of homoclinic orbits, the system of cylinder homoclinic orbits and the coexistence of two kinds of homoclinic orbits. In the end, the nrmerical simrlation is done by using the Dynamics software, then the existence of the cycle solution and the chaos solution is cinfirmed.
In the forth chapter, the puantitative analysis of the discontinual Rotating Pendulum Linked a Clamped and Supported Spring is given. It contains the analysis of restoring nonlinear force, the potential function and the phase portraits. The conclution of the analysis indicates that the system has the complex dynamics behaviour of cylinder homoclinic-like orbits and homoclinic-like orbits. In the end, the nrmerical simrlation is done using the Dynamics software, then the existence of the cycle solution and the chaos solution is cinfirmed.
In the fifth chapter, the cylinder approximate system under the periodic disturbances present Hopf bifurcation, two closed orbit bifurcations, homoclinic orbit bifurcation and cylindrical homoclinic orbit bifurcation.The bifurcation of the system is solved using the Melnikov function and Submelnikov function ,then we use the Matlab software to obtain global bifurcation diagram and Dynamics software for numerical validation.
In the sixth chapter, this chapter is mainly focused on the study of discontinuous limit case, In the discontinuous limit regime, firstly, we obtained the solution of special orbits by using smart measurd; secondly, we make the Melnikov analysis to detect the discontinuous heteroclinic-orbits tangling under the perturbation of damping and driving.at last, the bifurcation of discontinuous system can be investigated under the periodic function.
In the seventh chapter, the main work of this article is summarized, and then predicts the next step work of the rotating pendulum linked the clamped and supported spring. It can be researched on the theoretical analysis and the project application.
Appendix
1. The solutions of cylinder homoclinic orbits and homoclinic orbits;
2. The solutions of cylinder homoclinic-like orbits;
3. The solutions of cylinder heteroclinic-like orbits;
4. The visual illustration procedure is compiled using the Matlab, the visible animation of the cycle solution and the chaos solution are vividly. The essence of the procedure lays in using the law of cosines and the corner formula.
本文具体安排如下:
第一章,介绍了摆类的发展史和非线性动力学相关问题的研究历史与现状以及本文的主要创新点。
第二章,建立固支弹簧联接的旋转摆模型,给出动力学运动微分方程,利用Matlab软件得到了系统的平衡点分岔曲线图,恢复力函数分析,势能函数分析、相图分析与吸引盆分析,研究结果表明系统不但具有光滑的动力学行为,而且还包含了不连续的动力学行为。
第三章,通过构造新的柱面近似系统对光滑固支弹簧联接的旋转摆非线性动力学行为进行了定量分析,用Melnikov方法分别对同宿轨道、柱面同宿轨道以及两类同宿轨道并存系统进行了混沌解析预测,得到了阈值曲线。利用Dynamics软件进行了的数值模拟,验证了周期解和混沌解存在。
第四章,本章对不连续固支弹簧联接的旋转摆的非线性动力学行为进行了定性分析,包括恢复力分析、势能函数分析与相图分析,可以得到系统具有柱面类同宿轨道、类同宿轨道的复杂动力学行为。利用Dynamics软件进行了的数值模拟,验证了周期解和混沌解存在。
第五章,柱面近似系统在周期扰动下呈现,Hopf分岔,二次闭轨分岔,柱面同宿轨道分岔以及同宿轨道分岔。应用Melnikov函数和广义Melnikov函数对系统的分岔曲线进行求解,利用Matlab软件得到全局分岔图,并用Dynamics软件进行数值验证。
第六章,本章主要考虑不连续系统,在不连续系统中,首先,巧妙的求得系统的解;其次,在黏性阻尼和周期外激励作用下,我们用Melnikov函数得到类异宿轨道横截相交的阈值曲线;最后,研究了不连续系统在周期函数下的分岔。
第七章,总结本文工作,并对固支弹簧联接的旋转摆今后的工作进行了展望,可以在理论分析和工程应用等方面进行下一步研究。
附录
1.求解近似系统的同宿轨道和柱面同宿轨道的解;
2.求解不连续系统柱面类同宿轨道的解;
3.求解类异宿轨道的解;
4.用Matlab软件编写实物演示程序,形象生动的展现周期解和混沌解的可视动画。本程序精髓在于运用了余弦定理及符号函数。
In this article,a kind of novel pendulum modle is proposed and established. Accoroding to the mechanics stucture of the modle, it is named Rotating Pendulum Linked a Clamped and Supported Spring. This kind of pendulum is a typical cylinder pendulum, and the system displays smooth and discontinuous dynamics behaviour depending on the value of a system parameter. The nonlinear dynamics behaviour is analized and computed from the pualitative and the puota which used the nonlinear dynamics theory method and the numerical method.
The concrete arrangement of this article as follows:
In the first chapter, the development of the pendulum, the research history and the present situation of the non-linear dynamics, and innovation spots are introduced.
In the second chapter, the modle of the Rotating Pendulum Linked a Clamped and Supported Spring is established, at the same time, the dynamic motion differential equation is given, then using the Matlab, the diagram of the system’s balance points, the analy of the resume strength fuction, the potential fuction, the phase diagram and the attractor basins are obtained. The conclution indicates that the system not only has the smooth dynamics behaviour but also contains the discontinual dynamics behaviour.
In the third chapter, the puantitative analysis of the smooth Rotating Pendulum Linked a Clamped and Supported Spring is given by investigating the approximate system. The threshold value curve is obtained by the Melnikov function analysis forecast to the system of homoclinic orbits, the system of cylinder homoclinic orbits and the coexistence of two kinds of homoclinic orbits. In the end, the nrmerical simrlation is done by using the Dynamics software, then the existence of the cycle solution and the chaos solution is cinfirmed.
In the forth chapter, the puantitative analysis of the discontinual Rotating Pendulum Linked a Clamped and Supported Spring is given. It contains the analysis of restoring nonlinear force, the potential function and the phase portraits. The conclution of the analysis indicates that the system has the complex dynamics behaviour of cylinder homoclinic-like orbits and homoclinic-like orbits. In the end, the nrmerical simrlation is done using the Dynamics software, then the existence of the cycle solution and the chaos solution is cinfirmed.
In the fifth chapter, the cylinder approximate system under the periodic disturbances present Hopf bifurcation, two closed orbit bifurcations, homoclinic orbit bifurcation and cylindrical homoclinic orbit bifurcation.The bifurcation of the system is solved using the Melnikov function and Submelnikov function ,then we use the Matlab software to obtain global bifurcation diagram and Dynamics software for numerical validation.
In the sixth chapter, this chapter is mainly focused on the study of discontinuous limit case, In the discontinuous limit regime, firstly, we obtained the solution of special orbits by using smart measurd; secondly, we make the Melnikov analysis to detect the discontinuous heteroclinic-orbits tangling under the perturbation of damping and driving.at last, the bifurcation of discontinuous system can be investigated under the periodic function.
In the seventh chapter, the main work of this article is summarized, and then predicts the next step work of the rotating pendulum linked the clamped and supported spring. It can be researched on the theoretical analysis and the project application.
Appendix
1. The solutions of cylinder homoclinic orbits and homoclinic orbits;
2. The solutions of cylinder homoclinic-like orbits;
3. The solutions of cylinder heteroclinic-like orbits;
4. The visual illustration procedure is compiled using the Matlab, the visible animation of the cycle solution and the chaos solution are vividly. The essence of the procedure lays in using the law of cosines and the corner formula.
引文
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[2] GALILEO G. De motu (on motion) [M], 1592.
[3] CHRISTIAN HUYGENS. Pendulum clock [M], 1658.
[4] HATVANI L. Stability Problems for the Mathematical pendulum[J].Periodica Mathematica Hungarica, 2008 56(1), 71-82.
[5] BEVILACQUA F, The Pendulum: From Constrained Fall to the Concept of Potential[J], Science & Education ,2006(15):553–575.
[6] JEONG S, TAKAHASHI T,Wheeled inverted pendulum type assistant robot: design concept and mobile control[J], Intel Serv Robotics ,2008(1):313–320.
[7] KIM Y H, KIM S Hand KWAK Y K,Dynamic Analysis of a Nonholonomic Two-Wheeled Inverted Pendulum Robot[J], Journal of Intelligent and Robotic Systems 2005 (44): 25–46
[8] LIU T K, CHEN C H, LI Z S, CHOU J H, Method of inequalities-based multiobjective genetic algorithm for optimizing a Card-double-pendulum system[J],International Journal of Automation and Computing,2009:29-37.
[9] PAUL A and RICHER P H, Application of Greene's method and the MacKay residue criterion to the double pendulum[J],Z.Phys. B,1994(93):515-520.
[10] LEE W. and PARK H. Chaotic dynamics of a harmonically exited spring pendulum system with internal resonance[J]. Nonlinear Dynamics,1997(14):211-229.
[11] OBUSEK J P, HOLT K G, ROSENSTEIN R M, The hybrid mass-spring pendulum modle of human leg swinging: stiffness in the control of cycle period[J],Biological Cybernetics,1995(73):139-147.
[12] BAYLY P V and VIRGIN L N, An empirical study of the stability of periodic motion in the forced spring-pendulum[J]. Proceedings of Royal Society of London A ,1993(443):391-408.
[13] BREITENBERGER E. and MUELLER R.D., The elastic pendulum: A nonlinear paradigm, Journal of mathematical Physics[J],1981(22):1196-1210.
[14] RANDALL D.PETERS. Soup-can pendulum[J]. Science&Education,2004(13):641-652.
[15] MILES W.J., Resonant motion of a spherical pendulum[J], Physica D, 11,1984:309-323.
[16] KANA D.D., Distinguishing the transition to chaos in a spherical pendulum,Chaos[J], 1995(5):298-310.
[17] Zou M., Kolmogorov's condiction for the square potential pendulum[J], Physical Letters A,1992(166):321-329.
[18] BRYANT P.J., Breakdown to chaotic motion of a forced damped spherical pendulum[J],Physical D,1993(64):324-339.
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