非均质物理双摆的混沌特性研究  

Study on chaotic characteristics of heterogeneous physical double pendulum

作　　者：孔令辉 刘丁杨 蹇开林[1,2] KONG Linghui;LIU Dingyang;JIAN Kailin(School of Aeronautics and Astronautics,Chongqing University,Chongqing 400044,P.R.China;Chongqing Key Laobratory of Heterogenous Materisal Mechanics,Chongqing University,Chongqing 400044,P.R.China;CSSC Haizhuang Windpower Co.,Ltd.,Chongqing 401122,P.R.China)

机构地区：[1]重庆大学航空航天学院,重庆400044 [2]重庆大学非均质材料力学重庆市重点实验室,重庆400044 [3]中国船舶集团海装风电股份有限公司,重庆401122

出　　处：《重庆大学学报》2024年第2期106-118,共13页Journal of Chongqing University

摘　　要：为了解决工程实际中材料质量不均匀分布对双摆系统运动的影响,在均质物理双摆模型的基础上,将摆的质心位置和摆的转动惯量提取为变量,建立非均质双摆模型。将非均质双摆系统由Hamilton系统近似为拟Hamilton系统,运用双自由度的Melnikov法,得到拟Hamilton系统存在Smale马蹄意义下混沌的能量阈值,以此作为Hamilton系统的混沌条件。利用最大Lyapunov指数图、分岔图、Poincaré截面图等数值方法验证混沌条件的正确性,并详细分析了各参数对系统运动状态的影响和作用机制。结果表明,非均质双摆的混沌阈值有较高复杂性,而且摆长、摆重、第一摆的质心位置同时影响着系统的能量与混沌阈值,解释了质心位置和转动惯量等参数发生变化时,系统在混沌和拟周期之间交替变换的原因。进一步研究了参数取值与Melnikov法适用性之间的关系,通过数值仿真分类讨论了Melnikov法不适用时的参数取值情况。To address the impact of materials with uneven mass distribution on the motion of double pendulum systems in engineering practice,a heterogeneous double pendulum model was established based on the homogeneous physical double pendulum model.This model incorporated variables,such as the position of the center of mass and the moment of inertia of the pendulum.To further explore the chaotic characteristics of the system,the heterogeneous double pendulum system was approximated from a Hamilton system to a quasi-Hamilton system.The dynamics equation of the double pendulum system was obtained using the Euler-Lagrange equation of the second kind.The energy threshold for Smale horseshoe chaos in the quasi-Hamiltonian system was determined using Melnikov method with two degrees of freedom,serving as the chaos condition for the Hamiltonian system.Through programming in Matlab,the correctness of the chaos condition was verified using numerical methods such as maximum Lyapunov exponential diagram,bifurcation diagram and Poincare section diagram. The influence of each parameter on the system’ s motion state and action mechanism was analyzed in detail. The results show that the chaos threshold of heterogeneous double pendulums depends on various factors, including the initial energy of the double pendulums, the position of the center of mass of the first pendulums and the ratio of the moment of inertia of the two pendulums. With different values for each parameter, the system changes from a regular motion state with a chaos threshold to an irregular and complex motion state without a chaos threshold. The reasons for the system alternating between chaotic and quasi-periodic states were explained, and theoretical predictions were validated. The differences between the theoretical threshold and actual numerical simulation results were explained when the center of mass of the first pendulum, the mass ratio of the two pendulums, and the ratio of moment of inertia of the two pendulums were set to their limit values. On this basis,

关 键 词：非均质物理双摆 混沌 拟周期 Melnikov法 

分 类 号：O322[理学—一般力学与力学基础]