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A linear single degree-of-freedom oscillator with fractional-order derivative is researched by the averaging method, and the approximately analytical solution is obtained. The effects of the parameters on the dynamical property, including the fractional coefficient and the fractional order, are characterized by the equivalent linear damping coefficient and the equivalent linear stiffness, and this conclusion is entirely different from the published results. The comparison of the analytical solution with the numerical results verifies the correctness of the approximately analytical results. The following analysis on the effects of the fractional parameters on the amplitude-frequency is fulfilled, and it is found that the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient, but also the resonance frequency by the equivalent linear stiffness.
[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 Press) p10
[3] Petras I 2011 Fractional-Order Nonlinear System (Beijing: Higher Education Press) p19
[4] Rossikhin Y A, Shitikova M V 2010 Appl. Mech. Rev. 63 010801
[5] Riewe F 1997 Phys. Rev. E 53 3581
[6] Wang Z Z, Hu H Y 2010 Sci. China Phys. Mech. 53 345
[7] Wang Z Z, Du M L 2011 Shock Vib. 18 257
[8] Rossikhin Y A, Shitikova M V 1997 Acta Mech. 120 109
[9] Li G G, Zhu Z Y, Cheng C J 2011 Appl. Math. Mech. 22 294
[10] Cao J Y, Ma C B, Xie H, Jiang Z D 2010 J. Comput. Nonlin. Dyn. 5 041012
[11] Wu X J, Lu H T, Shen S L 2009 Phys. Lett. A 373 2329
[12] Chen J H, Chen W C 2008 Chaos Soliton Fract. 35 188
[13] Lu J G 2006 Phys. Lett. A 354 305
[14] Wahi P, Chatterjee A 2004 Nonlinear Dynam. 38 3
[15] Chen L C, Zhu W Q 2009 J. Vib. Control 15 1247
[16] Huang Z L, Jin X L 2009 J. Sound Vib. 319 1121
[17] 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]
[18] Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 56 6865]
[19] 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]
[20] 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]
[21] Hu J B, Zhang G A, Zhao L D, Zeng J Q 2011 Acta Phys. Sin. 60 060504 (in Chinese) [胡建兵, 章国安, 赵灵冬, 曾金全 2011 60 060504]
[22] Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502
[23] Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295
[24] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[25] Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901
[26] Deng W H, Li C P 2008 Phys. Lett. A 372 401
[27] Deng W H 2007 J. Comput. Phys. 227 1510
[28] Sanders J A, Verhulst F, Murdock J 2007 Averaging Methods in Nonlinear Dynamical Systems (New York: Springer) p150
[29] Ni Z H 1988 Vibration Mechanics (Xi'an: Xi'an Jiaotong University Press) p79 (in Chinese) [倪振华 1988 振动力学 (西安: 西安交通大学出版社) 第79页]
[30] Liu Y Z, Chen W L, Chen L Q 1998 Vibration Mechanics (Beijing: Higher Education Press) p36 (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 Press) p10
[3] Petras I 2011 Fractional-Order Nonlinear System (Beijing: Higher Education Press) p19
[4] Rossikhin Y A, Shitikova M V 2010 Appl. Mech. Rev. 63 010801
[5] Riewe F 1997 Phys. Rev. E 53 3581
[6] Wang Z Z, Hu H Y 2010 Sci. China Phys. Mech. 53 345
[7] Wang Z Z, Du M L 2011 Shock Vib. 18 257
[8] Rossikhin Y A, Shitikova M V 1997 Acta Mech. 120 109
[9] Li G G, Zhu Z Y, Cheng C J 2011 Appl. Math. Mech. 22 294
[10] Cao J Y, Ma C B, Xie H, Jiang Z D 2010 J. Comput. Nonlin. Dyn. 5 041012
[11] Wu X J, Lu H T, Shen S L 2009 Phys. Lett. A 373 2329
[12] Chen J H, Chen W C 2008 Chaos Soliton Fract. 35 188
[13] Lu J G 2006 Phys. Lett. A 354 305
[14] Wahi P, Chatterjee A 2004 Nonlinear Dynam. 38 3
[15] Chen L C, Zhu W Q 2009 J. Vib. Control 15 1247
[16] Huang Z L, Jin X L 2009 J. Sound Vib. 319 1121
[17] 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]
[18] Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese) [刘崇新 2007 56 6865]
[19] 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]
[20] 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]
[21] Hu J B, Zhang G A, Zhao L D, Zeng J Q 2011 Acta Phys. Sin. 60 060504 (in Chinese) [胡建兵, 章国安, 赵灵冬, 曾金全 2011 60 060504]
[22] Li Q D, Chen S, Zhou P 2011 Chin. Phys. B 20 010502
[23] Zhang R X, Yang S P 2009 Chin. Phys. B 18 3295
[24] Qi D L, Wang Q, Yang J 2011 Chin. Phys. B 20 100505
[25] Wu Z M, Xie J Y 2007 Chin. Phys. 16 1901
[26] Deng W H, Li C P 2008 Phys. Lett. A 372 401
[27] Deng W H 2007 J. Comput. Phys. 227 1510
[28] Sanders J A, Verhulst F, Murdock J 2007 Averaging Methods in Nonlinear Dynamical Systems (New York: Springer) p150
[29] Ni Z H 1988 Vibration Mechanics (Xi'an: Xi'an Jiaotong University Press) p79 (in Chinese) [倪振华 1988 振动力学 (西安: 西安交通大学出版社) 第79页]
[30] Liu Y Z, Chen W L, Chen L Q 1998 Vibration Mechanics (Beijing: Higher Education Press) p36 (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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