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研究了含分数阶导数阻尼的一类线性系统在不同周期信号激励下系统的响应问题. 首先在简谐信号的激励下, 利用谐波平衡法得到了系统响应的近似解, 这一结果和已有文献(申永军, 杨绍普, 邢海军 2012 61 110505)的结果完全相同, 但本文的求解过程大为简化, 而且本文进一步扩展了分数阶导数阻尼微分阶数的取值范围. 接着, 利用傅里叶级数展开法和线性系统的叠加原理, 求得了一般周期信号激励下系统响应的近似解, 并以周期方波信号和周期全波正弦信号为例进行了说明. 本文的结果表明, 分数阶导数阻尼的微分阶数影响系统响应中各阶谐波的共振频率和共振振幅. 系统响应的幅值与分数阶导数阻尼的微分阶数之间的单调关系主要受外激信号频率的影响. 除解析分析外, 本文还用数值模拟对相关结论进行了验证, 两种结果符合良好, 表明本文的分析方法是可行的.Under excitations of different periodical signals, the response of a fractional linear system is investigated. First, by the harmonic balance method, the approximate solutions of the fractional-order linear system excited by harmonica signals are obtained. The results in this paper are idenified with the existing results obtained by the average method (Shen Y J, Yang S P, Xing H 2012 Acta Phys. Sin. 61 110505). However, the solving process here is much simpler. Further, the value of the fractional-order is extended in this paper. Then, according to the Fourier expansion and the method of linear superposition, the response of the system to a general periodical signal is obtained, and two examples are given for the case of periodical square wave and modulus of sine wave respectively. The results in this paper show that the value of the factional-order influences the resonance frequency and resonance amplitude of each order harmonic. The monotonicity between the response amplitude and the value of the fractional-order is influenced mainly by the frequency of the external excitation. Besides the analytical analysis, the numerical simulations are also performed, and the approximate solutions are in good agreement with the numerical ones. Hence, the process of the analysis of this paper is feasible.
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Keywords:
- fractional-order damping /
- periodical signal /
- harmonic balance method
[1] Torvik P J, Bagley R L 1984 ASME J. Appl. Mech. 51 294
[2] Yang F, Zhu K Q 2011 Theor. Appl. Mech. Lett. 1 012007
[3] Mainardi F 2010 Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (London: Imperial College Press)
[4] Oldham K B 2010 Adv. Eng. Software 41 9
[5] Magin R L 2010 Comput. Math. Appl. 59 1586
[6] Carpinteri A, Mainardi F 1997 Fractals and Fractional Calculus in Continuum Mechanics, (Wien and New York: Springer)
[7] Agrawal O P 2004 Nonlinear Dyn. 38 323
[8] Das S, Pan I 2011 Fractional Order Signal Processing: Introductory Concepts and Applications (Berlin: Springer)
[9] Kusnezov D, Bulgac A, Dang G D 1999 Phys. Rev. Lett. 82 1136
[10] Ortigueira M D 2008 IEEE Circuits Syst. Mag. 8 19
[11] Ge Z M, Ou C Y 2007 Chaos Soliton. Fract. 34 262
[12] Cao J, Ma C, Jiang Z 2010 ASME J. Comput. Nonlinear Dyn. 5 041012
[13] Cao J, Ma C, Jiang Z, S Liu 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 1443
[14] Yang J H, Zhu H 2012 Chaos 22 013112
[15] Shen Y, Yang S, Xing H, Gao G 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3092
[16] Shen Y J, Yang S P, Xing H 2012 Acta Phys. Sin. 61 110505 [申永军, 杨绍普, 邢海军 2012 61 110505]
[17] Monje C A, Chen Y, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer)
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[1] Torvik P J, Bagley R L 1984 ASME J. Appl. Mech. 51 294
[2] Yang F, Zhu K Q 2011 Theor. Appl. Mech. Lett. 1 012007
[3] Mainardi F 2010 Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (London: Imperial College Press)
[4] Oldham K B 2010 Adv. Eng. Software 41 9
[5] Magin R L 2010 Comput. Math. Appl. 59 1586
[6] Carpinteri A, Mainardi F 1997 Fractals and Fractional Calculus in Continuum Mechanics, (Wien and New York: Springer)
[7] Agrawal O P 2004 Nonlinear Dyn. 38 323
[8] Das S, Pan I 2011 Fractional Order Signal Processing: Introductory Concepts and Applications (Berlin: Springer)
[9] Kusnezov D, Bulgac A, Dang G D 1999 Phys. Rev. Lett. 82 1136
[10] Ortigueira M D 2008 IEEE Circuits Syst. Mag. 8 19
[11] Ge Z M, Ou C Y 2007 Chaos Soliton. Fract. 34 262
[12] Cao J, Ma C, Jiang Z 2010 ASME J. Comput. Nonlinear Dyn. 5 041012
[13] Cao J, Ma C, Jiang Z, S Liu 2011 Commun. Nonlinear Sci. Numer. Simulat. 16 1443
[14] Yang J H, Zhu H 2012 Chaos 22 013112
[15] Shen Y, Yang S, Xing H, Gao G 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3092
[16] Shen Y J, Yang S P, Xing H 2012 Acta Phys. Sin. 61 110505 [申永军, 杨绍普, 邢海军 2012 61 110505]
[17] Monje C A, Chen Y, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer)
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