Duffing 系统的主-亚谐联合共振

李航; 申永军; 李向红; 韩彦军; 彭孟菲

doi:10.6052/0459-1879-19-349

Duffing 系统的主-亚谐联合共振

李航^*†,,
申永军^*†2),,
李向红^***,
韩彦军^†,
彭孟菲^†

* 石家庄铁道大学交通工程结构力学行为与系统安全国家重点实验室,石家庄 050043
† 石家庄铁道大学机械工程学院,石家庄 050043
** 石家庄铁道大学数理系,石家庄 050043

基金项目: 1)国家自然科学基金资助项目(U1934201);国家自然科学基金资助项目(11772206)

详细信息

通讯作者:
申永军

中图分类号: O322,O313
计量
- 文章访问数: 02047
- HTML全文浏览量: 0390
- PDF下载量: 0309
出版历程
- 收稿日期: 2019-12-08
- 刊出日期: 2020-04-09

PRIMARY AND SUBHARMONIC SIMULTANEOUS RESONANCE OF DUFFING OSCILLATOR

Li Hang^*†,,
Shen Yongjun^*†2),,
Li Xianghong^***,
Han Yanjun^†,
Peng Mengfei^†

* State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University,Shijiazhuang 050043,China
† Department of Mechanical Engineering,Shijiazhuang Tiedao University,Shijiazhuang 050043,China
** Department of Mathematics and Physics,Shijiazhuang Tiedao University,Shijiazhuang 050043,China

摘要

摘要: 以Duffing系统为研究对象,研究在多频激励下同时发生主共振和1/3次亚谐共振的动力学行为与稳定性.首先,通过多尺度法得到系统的近似解析解,利用数值方法检验近似程度,结果吻合良好,证明了求解过程和解析解的正确性.然后,从解析解中导出稳态响应的幅频方程和相频方程,从幅频曲线以及相频曲线中发现系统最多存在7个不同的周期解,这种多解现象可用于对系统状态进行切换.基于Lyapunov稳定性理论,得到联合共振定常解的稳定条件,利用该条件分析了系统的稳定性,并与Duffing系统的主共振和1/3次亚谐共振单独存在时比较.最后,通过数值方法分析了非线性项和外激励对系统动力学行为与稳定性的影响,发现了联合共振特有的现象：刚度软化时,非线性项不仅影响系统的响应幅值,同时还影响系统的多值性和稳定性;刚度硬化时,非线性项对系统的影响与单一频率下主共振和1/3次亚谐共振类似,仅影响系统的响应幅值.这些结果对Duffing系统动力学特性的研究具有重要意义.
- 非线性振动 /
- 非线性微分方程 /
- Duffing系统 /
- 联合共振
Abstract: In this paper, the dynamics and stability of the Duffing oscillator subjected to the primary resonance together with the 1/3 subharmonic resonance are studied. At first, the approximate analytical solution and amplitude-frequency equation are obtained through the method of multiple scales, and the correctness and satisfactory precision of the approximate solution are verified by simulation. Then, the amplitude-frequency equation and phase-frequency equation of steady-state response are derived from the approximate analytical solution, and it can be found there are at most seven different periodic solutions, which are called multi-value characteristics and can be used to switch the state of the system. Moreover, the stability condition of steady-state response is derived based on Lyapunov theory, and the amplitude-frequency curves of steady-state response are compared with the cases where the primary or 1/3 subharmonic resonance exists alone, and it is found that the system contains both resonance characteristics. At last, the effects of nonlinear factor and excitations on the system response are analyzed by simulation. The particular phenomena in this system are revealed, i.e., the nonlinear factor affects the response amplitude, multi-value characteristics and stability of the system with stiffness softening. However, for the stiffness hardening system, the nonlinear factor only affects the response amplitude, which is similar to the cases of single-frequency excitation. These results are important for the study on the Duffing system or other similar systems.
- nonlinear vibration /
- nonlinear differential equation /
- Duffing oscillator /
- simultaneous resonance

HTML全文

参考文献(1)

[1]	孟光, 薜中擎 . 带挤压油膜阻尼器的柔性转子非线性响应的Duffing特性分析. 航空动力学报, 1989(2):173-178, 197
[1]	( Meng Guang, Xue Zhongqing . Study on nonlinear Duffing characteristics of flexible rotor with SFDB. Journal of Aerospace Power, 1989(2):173-178, 197 (in Chinese))
[2]	Ertas A, Chew EK . Non-linear dynamic response of a rotating machine. International Journal of Non-linear Mechanics, 1990,25(2-3):241-251
[3]	吴敬东 . 转子系统碰摩的若干非线性动力学问题研究. [博士论文]. 沈阳: 东北大学, 2006: 38-42
[3]	( Wu Jingdong . Study on some nonlinear dynamics problems of rotor system with impact-rubbing. [PhD Thesis]. Shenyang: Northeastern University, 2006: 38-42 (in Chinese))
[4]	刘辉, 浦金云, 陈晓洪等. 基于胞映射法的破损进水船非线性横摇运动. 华中科技大学学报(自然科学版), 2009,37(8):116-119
[4]	( Liu Hui, Pu Jinyun, Chen Xiaohong , et al. Study on non-linear rolling motion of a forced flooded ship using cell mapping method. Journal of Huazhong University of Science and Technology (Natural Science Edition), 2009,37(8):116-119 (in Chinese))
[5]	Pan R, Davies HG . Responses of a non-linearly coupled pitch-roll ship model under harmonic excitation. Nonlinear Dynamics, 1996,9(4):349-368
[6]	束立红, 周炜, 吕志强等. 钢丝绳隔振器在大型机械设备的振动冲击隔离设计中的应用. 振动与冲击, 2006(4): 78-81, 177-178
[6]	( Shu Lihong, Zhou Wei, Lü Zhiqiang , et al. Stainless steel wire-rope isolator used in vibration and impact isolation design for large machine equipment. Journal of Vibration and Shock, 2006(4): 78-81, 177-178 (in Chinese))
[7]	韩祥临, 林万涛, 许永红等. 广义Duffing扰动振子随机共振机理的渐近解. 物理学报, 2014,63(17):35-39
[7]	( Han Xianglin, Lin Wantao, Xu Yonghong , et al. Asymptotic solution to the generalized Duffing equation for disturbed oscillator in stochastic resonance. Acta Physica Sinica, 2014,63(17):35-39 (in Chinese))
[8]	李瑞红, 徐伟, 李爽 . 含三次耦合项的2自由度Duffing系统的共振及混沌行为. 应用力学学报, 2007, ( 2):200-203, 337
[8]	( Li Ruihong, Xu Wei, Li Shuang . Resonance and chaos behavior in a two-degree-of-freedom Duffing system with cubic coupled terms. Chinese Journal of Applied Mechanics, 2007, ( 2):200-203, 337 (in Chinese))
[9]	Shen Y, Yang S, Xing H , et al. Primary resonance of Duffing oscillator with fractional-order derivative. Communications in Nonlinear Science and Numerical Simulation, 2012,17(7):3092-3100
[10]	申永军, 杨绍普, 邢海军 . 分数阶Duffng振子的超谐共振. 力学学报, 2012,44(4):762-768
[10]	( Shen Yongjun, Yang Shaopu, Xing Haijun . Super-harmonic resonance of fractional-order Duffing oscillator. Chinese Journal of Theoretical and Applied Mechanics, 2012,44(4):762-768 (in Chinese))
[11]	Holmes C, Holmes P . Second order averaging and bifurcations to subharmonics in Duffing's equation. Journal of Sound and Vibration, 1981,78(2):161-174
[12]	张毅, 韩修静, 毕勤胜 . 串联式叉型滞后簇发振荡及其动力学机制. 力学学报, 2019,51(1):228-236
[12]	( Zhang Yi, Han Xiujing, Bi Qinsheng . Series-mode pitchfork-hysteresis bursting oscillations and their dynamical mechanisms. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(1):228-236 (in Chinese))
[13]	曲子芳, 张正娣, 彭淼等. 双频激励下Filippov系统的非光滑簇发振荡机理. 力学学报, 2018,50(5):1145-1155
[13]	( Qu Zifang, Zhang Zhengdi, Peng Miao , et al. Non-smooth bursting oscillation mechanisms in a Filippov-type system with multiple periodic excitations. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(5):1145-1155 (in Chinese))
[14]	吕小红, 罗冠炜 . 冲击渐进振动系统相邻基本振动的转迁规律. 力学学报, 2017,49(5):1091-1102
[14]	( Lü Xiaohong, Luo Guanwei . Transition law of adjacent fundamental motions in vibro-impact system with progression. Chinese Journal of Theoretical and Applied Mechanics, 2017,49(5):1091-1102 (in Chinese))
[15]	毕勤胜, 陈予恕 . 强Duffing系统的周期共振解及其转迁集. 振动工程学报, 1997,10(1):29-34
[15]	( Bi Qinsheng, Chen Yushu . The periodic resonant solutions and the transition boundaries of strongly nonlinear Duffing's equation. Journal of Vibration Engineering, 1997,10(1):29-34 (in Chinese))
[16]	毕勤胜, 陈予恕 . Duffing系统解的转迁集的解析表达式. 力学学报, 1997(5):55-63
[16]	( Bi Qinsheng, Chen Yushu . Analytical expression of transition boundaries of the solution of Duffing systems. Chinese Journal of Theoretical and Applied Mechanics, 1997(5):55-63 (in Chinese))
[17]	Kimiaeifar A, Saidi AR, Bagheri GH , et al. Analytical solution for van der Pol-Duffing oscillators. Chaos, Solitons & Fractals, 2009,42(5):2660-2666
[18]	Jin Y, Hu H . Dynamics of a Duffing oscillator with two time delays in feedback control under narrow-band random excitation. Journal of Computational and Nonlinear Dynamics, 2008,3(2):021205
[19]	Jin Y, Hu H . Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation. Nonlinear Dynamics, 2007,50(1):213-227
[20]	戎海武, 徐伟, 方同 . 谐和与窄带随机噪声联合作用下Duffing系统的参数主共振. 力学学报, 1998,30(2):50-57
[20]	( Rong Haiwu, Xu Wei, Fang Tong . Principal response of Duffing oscillator to combined deterministic and narrow-band random parameteric excitation. Chinese Journal of Theoretical and Applied Mechanics, 1998,30(2):50-57 (in Chinese))
[21]	Hosseini SAA . Some considerations on higher order approximation of Duffing equation in the case of primary resonance. Scientia Iranica, 2013,20(5):1464-1473
[22]	徐伟, 方同, 戎海武 . 有界窄带激励下具有黏弹项的Duffing振子. 力学学报, 2002(5):764-771
[22]	( Xu Wei, Fang Tong, Rong Haiwu . Duffing oscillator with visco-elastic term under narrow-band random excitation. Chinese Journal of Theoretical and Applied Mechanics, 2002(5):764-771 (in Chinese))
[23]	Nayfeh AH, Mook DT . Nonlinear Oscillations. New York: John Wiley & Sons, 2008: 162-192
[24]	姜源, 申永军, 温少芳等. 分数阶达芬振子的超谐与亚谐联合共振. 力学学报, 2017,49(5):1008-1019
[24]	( Jiang Yuan, Shen Yongjun, Wen Shaofang , et al. Super-harmonic and sub-harmonic simultaneous resonances of fractional-order Duffing oscillator. Chinese Journal of Theoretical and Applied Mechanics, 2017,49(5):1008-1019 (in Chinese))
[25]	姜源, 申永军, 温少芳 . 分数阶van der Pol振子的超谐与亚谐联合共振. 振动工程学报, 2019,32(5):863-873
[25]	( Jiang Yuan, Shen Yongjun, Wen Shaofang . Super-harmonic and sub-harmonic simultaneous resonances of fractional-order van der Pol oscillator. Journal of Vibration Engineering, 2019,32(5):863-873 (in Chinese))
[26]	安木金 . 车辆多相关子系统激励下响应的分析研究. [硕士论文]. 南京: 南京航空航天大学, 2011: 1-2
[26]	( An Mujin . Research on response of the vehicle under multi-correlative subsystems excitations. [Master Thesis]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2011: 1-2 (in Chinese))
[27]	Riganti R . Subharmonic solutions of the duffing equation with large non-linearity. International Journal of Non-Linear Mechanics, 1978,13(1):21-31
[28]	韦鹏, 申永军, 杨绍普 . 分数阶Duffing振子的亚谐共振. 振动工程学报, 2014,27(6):811-818
[28]	( Wei Peng, Shen Yongjun, Yang Shaopu . Sub-harmonic resonance of Duffing oscillator with fractional-order derivative. Journal of Vibration Engineering, 2014,27(6):811-818 (in Chinese))
[29]	Van Khang N, Chien T . Subharmonic resonance of Duffing oscillator with fractional-order derivative. Journal of Computational and Nonlinear Dynamics, 2016,11(5):051018
[30]	Hassan A . On the third superharmonic resonance in the Duffing oscillator. Journal of Sound and Vibration, 1994,172(4):513-526
[31]	Rahman Z, Burton TD . Large amplitude primary and superharmonic resonances in the Duffing oscillator. Journal of Sound and Vibration, 1986,110(3):363-380
[32]	杨绍普, 申永军 . 滞后非线性系统的分岔与奇异性. 北京: 科学出版社, 2003: 35-40
[32]	( Yang Shaopu, Shen Yongjun. Singularity and Bifurcation of Hysteretic Nonlinear System. Beijing: Science Press, 2003: 35-40(in Chinese))
[33]	胡海岩 . 应用非线性动力学. 北京: 航空工业出版社, 2000: 64-73
[33]	( Hu Haiyan . Applied Nonlinear Dynamics. Beijing: Aviation Industry Press, 2000: 64-73(in Chinese))