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研究了含两类分数阶微分项的线性单自由度振子, 通过平均法得到了系统的近似解析解. 在近似解中, 两类分数阶微分项的系数和阶次均以等效线性阻尼和等效线性刚度的形式影响着系统的动力学特性, 这一点和现有文献中大多数直接将分数阶微分项归类为阻尼进行处理是完全不同的. 对近似解析解和数值解进行了比较, 二者符合精度很高, 证明了该结果的准确性. 然后分析了两类分数阶微分项的系数和阶次对系统响应特性的影响, 发现两类分数阶微分项的系数和阶次都既可以影响系统的共振振幅, 又可以影响系统的共振频率. 最后研究了第二类分数阶微分项对共振频率的影响, 指出了在振动控制工程中如何通过选取合适的第二类分数阶微分项的系数达到满意的控制效果.A linear single degree-of-freedom (SDOF) oscillator with two kinds of fractional-order derivatives is investigated by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical properties, including the fractional coefficients and the fractional orders in the two kinds of fractional-order derivatives, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and the results is entirely different from the results given in the existing literature. A comparison of the analytical solution with the numerical results is made, and their satisfactory agreement verifies the correctness of the approximately analytical results. The following analysis of the effects of the fractional parameters on the amplitude-frequency is presented, and it is found that the fractional coefficients and the fractional orders can affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness. Finally, the effects of the fractional coefficient in the second fractional-order derivative on resonance frequency are analyzed, and the design rule for the fractional coefficient in the second fractional-order derivative to meet the satisfactory vibration control performance is pointed out.
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Keywords:
- fractional-order derivative /
- averaging method /
- approximately analytical solution /
- vibration control
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[2] Podlubny I 1999 Fractional Differential Equations (London: Academic) p10
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[4] Rossikhin Y A, Shitikova M V 2010 Applied Mechanics Reviews 63 010801
[5] Yang S P, Shen Y J 2009 Chao, Solitons and Fractals 40 1808
[6] Wang Z Z, Hu H Y 2010 Science China: Physics, Mechanics & Astronomy 53 345
[7] Wang Z Z, Du M L 2011 Shock and Vibration 18 257
[8] Rossikhin Y A, Shitikova M V 1997 Acta Mechanica 120 109
[9] Li G G, Zhu Z Y, Cheng C J 2011 Applied Mathematics and Mechanics 22 294
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[11] Wu X J, Lu H T, Shen S L 2009 Physics Letters A 373 2329
[12] Chen J H, Chen W C 2008 Chaos, Solitons and Fractals 35 188
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[25] Deng W H. 2007 Journal of Computational Physics 227 1510
[26] Chen L C, Zhu W Q 2009 Journal of Vibration and Control 15 1247
[27] Wahi P, Chatterjee A 2004 Nonlinear Dynamics 38 3
[28] Huang Z L, Jin X L 2009 Journal of Sound and Vibration 319 1121
[29] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[30] Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nonlinear Sci. Numer Simulat 17 3092
[31] Sanders J A, Verhulst F, Murdock J 2007 Averaging methods in nonlinear dynamical systems (New York: Springer) p150
[32] Ni Z H 1988 Vibration Mechanics p79 (Xi'an: Xi'an Jiaotong University Press) (in Chinese) [倪振华 1988 振动力学 (西安: 西安交通大学出版社) 第79页]
[33] Liu Y Z, Chen W L, Chen L Q 1998 Vibration Mechanics p36 (Beijing: Higher Education Press) (in Chinese) [刘延柱, 陈文良, 陈立群 1998 振动力学 (北京: 高等教育出版社) 第36页]
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[1] Oldham K B, Spanier J 1974 The Fractional Calculus-Theory and Applications of Differentiation and Integration to Arbitrary Order (New York: Academic Press) p1
[2] Podlubny I 1999 Fractional Differential Equations (London: Academic) p10
[3] Petras I 2011 Fractional-Order Nonlinear System (China: Higher Education Press) p19
[4] Rossikhin Y A, Shitikova M V 2010 Applied Mechanics Reviews 63 010801
[5] Yang S P, Shen Y J 2009 Chao, Solitons and Fractals 40 1808
[6] Wang Z Z, Hu H Y 2010 Science China: Physics, Mechanics & Astronomy 53 345
[7] Wang Z Z, Du M L 2011 Shock and Vibration 18 257
[8] Rossikhin Y A, Shitikova M V 1997 Acta Mechanica 120 109
[9] Li G G, Zhu Z Y, Cheng C J 2011 Applied Mathematics and Mechanics 22 294
[10] Cao J Y, Ma C B, Xie H, Jiang Z D 2010 ASME Journal of Computational and Nonlinear Dynamics 5 041012
[11] Wu X J, Lu H T, Shen S L 2009 Physics Letters A 373 2329
[12] Chen J H, Chen W C 2008 Chaos, Solitons and Fractals 35 188
[13] Lu J G 2006 Physics Letters A 354 305
[14] Zhang C F, Gao J F, Xu L 2007 Acta Phys. Sin. 56 5124 (in Chinese) [张成芬, 高金峰, 徐磊 2007 56 5124]
[15] Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 56 6865]
[16] Chen X R, Liu C X, Wang F Q, Li Q X 2008 Acta Phys. Sin. 57 1416 (in Chinese) [陈向荣, 刘崇新, 王发强, 李永勋 2008 57 1416]
[17] Zhang R X, Yang Y, Yang S P 2009 Acta Phys. Sin. 58 6039 (in Chinese) [张若洵, 杨洋, 杨世平 2009 58 6039]
[18] Hu J B, Xiao J, Zhao L D 2011 Acta Phys. Sin. 60 110515 (in Chinese) [胡建兵, 肖建, 赵灵冬 2011 60 110515]
[19] Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502
[20] Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295
[21] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[22] Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901
[23] Zhou P 2007 Chin. Phys. 16 1263
[24] Deng W H, Li C P 2008 Phys. Lett. A 372 401
[25] Deng W H. 2007 Journal of Computational Physics 227 1510
[26] Chen L C, Zhu W Q 2009 Journal of Vibration and Control 15 1247
[27] Wahi P, Chatterjee A 2004 Nonlinear Dynamics 38 3
[28] Huang Z L, Jin X L 2009 Journal of Sound and Vibration 319 1121
[29] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[30] Shen Y J, Yang S P, Xing H J, Gao G S 2012 Commun. Nonlinear Sci. Numer Simulat 17 3092
[31] Sanders J A, Verhulst F, Murdock J 2007 Averaging methods in nonlinear dynamical systems (New York: Springer) p150
[32] Ni Z H 1988 Vibration Mechanics p79 (Xi'an: Xi'an Jiaotong University Press) (in Chinese) [倪振华 1988 振动力学 (西安: 西安交通大学出版社) 第79页]
[33] Liu Y Z, Chen W L, Chen L Q 1998 Vibration Mechanics p36 (Beijing: Higher Education Press) (in Chinese) [刘延柱, 陈文良, 陈立群 1998 振动力学 (北京: 高等教育出版社) 第36页]
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