受垂直激励和水平约束的单摆系统亚谐共振分岔与混沌  

Subharmonic resonance bifurcation and chaos of simple pendulum system with vertical excitation and horizontal constraint

作　　者：赵武[1] 张鸿斌 孙超凡 黄丹[2] 范俊锴[1] Zhao Wu;Zhang Hong-Bin;Sun Chao-Fan;Huang Dan;Fan Jun-Kai(School of Mechanical and Power Engineering,Henan Polytechnic University,Jiaozuo 454003,China;School of Materials Science and Engineering,Henan Polytechnic University,Jiaozuo 454003,China)

机构地区：[1]河南理工大学机械与动力工程学院,焦作454003 [2]河南理工大学材料科学与工程学院,焦作454003

出　　处：《物理学报》2021年第24期2-17,共16页Acta Physica Sinica

基　　金：国家自然科学基金联合基金(批准号:U1604140);河南省科技攻关计划(批准号:172102210269,192102210052,212102210108,212102210004);河南省重大成果培育项目(批准号:NSFRF170503);河南理工大学创新团队基金(批准号:T2019-5)资助的课题.

摘　　要：为解决一类典型工程摆的工作性能参数优选,抽象该类系统为“受垂直激励和水平约束”的物理单摆模型.运用多尺度法解析系统的亚谐共振响应,明确了系统参数对幅值共振带宽、多值性的作用规律.利用Melnikov函数法,求解得到系统的同宿轨和Smale意义上混沌的阈值条件.通过数值法解析系统单参分岔、最大Lyapunov指数、双参分岔及吸引域流形转迁等动力特性,揭示了这类摆系统的亚谐共振分岔、周期吸引子倍增、周期与混沌吸引子共存等全局特性的运动规律,进一步明确了相关参数改变对系统运动形态转化、能量分布与演变规律的作用机理,得到了相关参数对工程系统工作性能的影响和作用机制.研究结果对工程中该类典型物理系统的工作性能参数调整,及其对实际工况中系统的减振抑振提供了理论依据.In order to improve the working performance and optimize the working parameters of the typical engineering pendulum of a typical system that it is abstracted as a physical simple pendulum model with vertical excitation and horizontal constraint. The dynamical equation of the system with vertical excitation and horizontal constraint is established by using Lagrange equation. The multiple-scale method is used to analyze the subharmonic response characteristics of the system. The amplitude-frequency response equation and the phase-frequency response equation are obtained through calculation. The effects of the system parameters on the amplitude resonance bandwidth and variability are clarified. According to the singularity theory and the universal unfolding theory, the bifurcation topology structure of the subharmonic resonance of the system is obtained. The Melnikov function is applied to the study of the critical conditions for the chaotic motion of the system. The parameter equation of homoclinic orbit motion is obtained through calculation. The threshold conditions of chaos in the sense of Smale are analyzed by solving the Melnikov function of the homoclinic motion orbit. The dynamic characteristics of the system, including single-parameter bifurcation, maximum Lyapunov exponent, bi-parameter bifurcation, and manifold transition in the attraction basin, are analyzed numerically. The results show that the main path of the system entering into the chaos is an almost period doubling bifurcation. Complex dynamical behaviors such as periodic motion, period doubling bifurcation and chaos are found. The bi-parameter matching areas of the subharmonic resonance bifurcation and chaos of the system are clarified. The results reveal the global characteristics of the system with vertical excitation and horizontal constraint, such as subharmonic resonance bifurcation, periodic attractor multiplication, and the coexistence of periodic and chaotic attractors. The results further clarify the mechanism of the influence of system

关 键 词：单摆 亚谐共振 分岔 混沌 

分 类 号：O321[理学—一般力学与力学基础]