SUPER-HARMONIC AND SUB-HARMONIC SIMULTANEOUS RESONANCES OF FRACTIONAL-ORDER DUFFING OSCILLATOR

doi:10.6052/0459-1879-17-105

Abstract

Abstract:

The super-harmonic and sub-harmonic simultaneous resonance of Duffing oscillator with fractional-order derivative is studied in this paper. The first-order approximate analytical solution is obtained by averaging method. The definitions of equivalent linear damping coefficient and equivalent linear stiffness efficient for super-harmonic and subharmonic simultaneous resonance are presented. The analytical amplitude-frequency equation for steady-state solution of simultaneous resonance is established. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness and higher-order precision of the approximately analytical results. Then, a further comparison between the fractional-order and traditional integer-order Duffing oscillator is fulfilled through the definitions of equivalent linear damping coefficient and equivalent linear stiffness coefficient, and the results prove that the fractional-order parameters has the effects of both damping and stiffness, which is similar in other fractionalorder system. At last the numerical simulation is used to analyze the effects of different fractional-order parameters on multi-value characteristics and jumping phenomena of the amplitude-frequency curve under simultaneous resonance, and the differences between the super-harmonic and sub-harmonic resonances under single-frequency excitation are analyzed in detail. It could be found that the fractional-order parameters not only affect the response amplitude and resonant frequency of the system, but also has significant influence on the number, existing area and the occurrence order of periodic solutions. Moreover, single super-harmonic resonance, single sub-harmonic resonance and both existing of these two resonances could be respectively found under different basic parameters, which is important to study the dynamic characteristics of the similar system.

Key words:

fractional-order derivative|Duffing oscillator|simultaneous resonance|averaging method

CLC Number:

O313
O322

Jiang Yuan, Shen Yongjun, Wen Shaofang, Yang Shaopu. SUPER-HARMONIC AND SUB-HARMONIC SIMULTANEOUS RESONANCES OF FRACTIONAL-ORDER DUFFING OSCILLATOR[J]. Chinese Journal of Theoretical and Applied Mechani, 2017, 49(5): 1008-1019.

References

1 Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam:Elsevier, 2006
2 Petras I. Fractional-order Nonlinear Systems:Modeling, Analysis and Simulation. Beijing:Higher Education Press, 2011:18-19
3 Rossikhin YA, Shitikova MV. Application of fractional calculus for dynamic problems of solid mechanics:novel trends and recent results. Applied Mechanics Reviews, 2010, 63:010801-1-52
4 Machado JAT, Kiryakova V, Mainardi F. Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 2011, 16:1140-1153
5 陈章耀, 王亚茗, 张春. 双状态切换下 BVP 振子的复杂行为分析. 力学学报, 2016, 48(4):953-962 (Chen Zhangyao, Wang Yaming, Zhang Chun. Complicated behaviors as well as the mechanism in BVP oscillator with switches related to two states. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(4):953-962 (in Chinese))
6 苏二龙, 罗建军. 高超声速飞行器横侧向失稳非线性分岔分析. 力学学报, 2016, 48(5):1192-1201 (Su Erlong, Luo Jianjun. Nonlinear bifurcation analysis of lateral loss of stability for hypersonic vehicle. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(5):1192-1201 (in Chinese))
7 高雪, 陈前, 刘先斌. 一类分段光滑隔振系统的非线性动力学设计方法. 力学学报, 2016, 48(1):192-200 (Gao Xue, Chen Qian, Liu Xianbin. Nonlinear dynamics design for piecewise smooth vibration isolation system. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(1):192-200 (in Chinese))
8 王浩宇, 吴勇军. 1:1 内共振对随机振动系统可靠性的影响. 力学学报, 2015, 47(5):807-813 (Wang Haoyu, Wu Yongjun. The influence of one-to-one internal resonance on reliability of random vibration system. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(5):807-813 (in Chinese))
9 李占龙, 孙大刚, 韩斌慧. 基于分数阶导数的黏弹性减振系统时频特性. 应用基础与工程科学学报, 2017, 25(1):187-198 (Li Zhanlong, Sun Dagang, Han Binhui. Time and frequency features of viscoelastic vibration damping system based on fractional derivative. Journal of Basic Science and Engineering, 2017, 25(1):187-198 (in Chinese))
10 蔡伟, 陈文. 复杂介质中任意阶频率依赖耗散声波的分数阶导数模型. 力学学报, 2016, 48(6):1265-1280 (Cai Wei, Chen Wen. Fractional derivative modeling of frequency-dependent dissipative mechanism for wave propagation in complex media. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6):1265-1280 (in Chinese))
11 韦鹏, 申永军, 杨绍普. 分数阶 van der Pol 振子的超谐共振. 物理学报, 2014, 63(1):47-58 (Wei Peng, Shen Yongjun, Yang Shaopu. Super-harmonic resonance of fractional-order van der Pol oscillator. Acta Phys. Sin, 2014, 63(1):47-58 (in Chinese))
12 林世敏, 许传炬. 分数阶微分方程的理论和数值方法研究. 计算数学, 2016, 38(1):1-24 (Lin Shimin, Xu Chuanju. Theoretical and numerical investigation of fractional differential equations. Mathematica Numerica Sinica, 2016, 38(1):1-24 (in Chinese))
13 谭健, 周洲, 祝小平. 飞翼布局无人机分数阶积分滑模姿态控制. 控制理论与应用, 2015, 32(5):607-614 (Tan Jian, Zhou Zhou, Zhu Xiaoping. Attitude control for flying wing unmanned aerial vehicles based on fractional order integral sliding-mode. Control Theory and Applications, 2015, 32(5):607-614 (in Chinese))
14 文家燕, 高远, 刘传国. 永磁同步电动机的双闭环分数阶控制研究. 微特电机, 2016, 44(1):34-38 (Wen Jiayan, Gao Yuan, Liu Chuanguo. Control of PMSM via a double closed-loop fractionalorder control strategy. Small and Special Electrical Machines, 2016, 44(1):34-38 (in Chinese))
15 牛江川, 申永军, 杨绍普. 基于速度反馈分数阶 PID 控制的达芬振子的主共振. 力学学报, 2016, 48(2):422-429 (Niu Jiangchuang, Shen Yongjun, Yang Shaopu. Primary resonance of Duffing oscillator with fractional-order PID controller based on velocity feedback. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(2):422-429 (in Chinese))
16 张碧陶, 高福荣, 姚科. 集成神经网络与自适应算法的分数阶滑模控制. 控制理论与应用, 2016, 33(10):1373-1377 (Zhang Bitao, Gao Furong, Yao Ke. Neural network and adaptive algorithm-based fractional order sliding mode controller. Control Theory and Applications, 2016, 33(10):1373-1377 (in Chinese))
17 孙会来, 金纯, 张文明. 基于分数阶微积分的油气悬架建模与试验分. 振动与冲击, 2014, 34(17):167-172 (Sun Huilai, Jin Chun, Zhang Wenming. Modeling and tests for a hydro-peneumatic suspension based on fractional calculus. Journal of Vibration and Shocck, 2014, 34(17):167-172 (in Chinese))
18 吴光强, 黄焕军, 叶光湖. 基于分数阶微积分的汽车空气悬架半主动控制. 农业机械学报, 2014, 45(7):19-25 (Wu Guangqiang, Huang Huanjun, Ye Guanghu. Semi-active control of automotive air suspension based on fractional calculus. Transactions of The Chinese Society of Agricultural Machinery, 2014, 45(7):19-25 (in Chinese))
19 陈丙三, 曾寿金, 江吉彬. 磁流变阻尼器的分数阶模型及其减振系统分析. 机械设计与制造, 2012 (7):219-221 (Chen Bingsan, Zeng Shoujin, Jiang Jibin. Fractional calculus modeling of the megnetorheological damper and the analysis of its damping system. Machinery Design and Manufacture, 2012 (7):219-221 (in Chinese))
20 马冬冬, 王志强, 王进君. 单相逆变器分数阶建模及分析. 电测与仪表, 2017, 54(6):106-112 (Ma Dongdong, Wang Zhiqiang, Wang Jinjun. Fractional order modeling and analysis of single phase inverter. Electrical Measurement & Instrumentation, 2017, 54(6):106-112 (in Chinese))
21 何志磊, 朱珍德, 朱明礼. 基于分数阶导数的非定常蠕变本构模型研究. 岩土力学, 2016, 37(3):737-744 (He Zhilei, Zhu Zhende, Zhu Mingli. An unsteady creep constitutive model based on fractional order derivatives. Rock and Soil Mechanics, 2016, 37(3):737-744 (in Chinese))
22 段晓梦, 殷德顺, 安丽媛. 基于分数阶微积分的黏弹性材料变形研究. 中国科学:物理学 · 力学 · 天文学, 2013, 43(8):971-977 (Duan Xiaomeng, Yin Deshun, An Liyuan. The deformation study in viscoelastic materials based on fractional order calculus. Scientia Sinica Physica, Mechanica & Astronomica, 2013, 43(8):971-977 (in Chinese))
23 陈林聪, 朱位秋. 谐和与宽带噪声联合激励下含分数导数型阻尼的达芬振子的平稳响应. 应用力学学报, 2010, 27(1):517-521 (Chen Lincong, Zhu Weiqiu. Stationary response of Duffing oscillator with fractionalderivative damping under combined harmonic and wide band noise excitations. Journal of Applied Mechanics, 2010, 27(1):517-521 (in Chinese))
24 申永军, 杨绍普, 邢海军. 含分数阶微分的线性单自由度振子的动力学分析. 物理学报, 2012, 61(11):158-163 (Shen Yongjun, Yang Shaopu, Xing Haijun. Dynamical analysis of linear single degreeof-freedom oscillator with fractional-order derivative. Acta Phys Sin, 2012, 61(11):158-163 (in Chinese))
25 申永军, 杨绍普, 邢海军. 含分数阶微分的线性单自由度振子的动力学分析 (Ⅱ). 物理学报, 2012, 61(15):55-63 (Shen Yongjun, Yang Shaopu, Xing Haijun. Dynamical analysis of linear SDOF oscillator with fractional-order derivative(Ⅱ). Acta Phys Sin, 2012, 61(15):55-63 (in Chinese))
26 Shen YJ, Yang SP, Xing HJ, et al. Primary resonance of Duffing oscillator with fractional-order derivative. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(7):3092-3100
27 Shen Yongjun, Yang Shaopu, Xing Haijun, et al. Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives. International Journal of Non-Linear Mechanics, 2012, 47(9):975-983
28 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 (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))
29 温少芳, 申永军, 杨绍普. 分数阶时滞反馈对达芬振子动力学特性的影响. 物理学报, 2016, 65(9):166-175 (Wen Shaofang, Shen Yongjun, Yang Shaopu. Dynamical analysis of Duffing oscillator with fractional-order feedback with time delay. Acta Phys. Sin, 2016, 65(9):166-175 (in Chinese))
30 韦鹏, 申永军, 杨绍普. 分数阶达芬振子的亚谐共振. 振动工程学报, 2014, 28(6):811-818 (Wei Peng, Shen Yongjun, Yang Shaopu, Sub-harmonic resonance of Duffing oscillator with fractional-order derivative. Journal of Vibration Engineering, 2014, 28(6):811-818 (in Chinese))
31 Atanackovic TM, Konjik S, Pilipovic S. Variational problems with fractional derivatives:Euler-Lagrange equations. Journal of Physics A Mathematical & Theoretical, 2011, 41(9):1937-1940
32 王学彬. 拉普拉斯变换方法解分数阶微分方程. 西南师范大学学报 (自然科学版), 2016, 41(7):7-12 (Wang Xuebin. On laplace transform method for solving fractional differential equations. Journal of Southwest China Normal University(Natural Science Edition), 2016, 41(7):7-12 (in Chinese))
33 陈明杰. 分数阶傅里叶变换的数值实现. 重庆大学学报 (自然科学版), 2003, 26(5):129-132 (Chen Mingjie. A numerical algorithms of fractional fourier transform. Journal of Chongqing University (Natural Science Edition), 2003, 26(5):129-132 (in Chinese))
34 奈弗 AH, 穆克 DT. 非线性振动. 北京:高等教育出版社, 1980 (Nayfeh AH, Mook DT. Nonlinear oscillations. Beijing:High Education Press, 1980 (in Chinese))

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