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本文研究了基础激励下含分数阶阻尼的线性系统的响应特性. 当基础激励为简谐激励时, 通过待定系数方法求得系统的动力传递系数; 当基础激励为非简谐周期激励时, 首先将激励展开成傅里叶级数, 然后根据线性系统的叠加原理求得激励中各阶频率成分所引起的动力传递系数, 并根据展开的傅里叶级数解决了数值运算中的不可导问题. 用数值仿真的方法对解析结果进行了验证, 两者符合良好, 证明了解析分析的正确性. 研究表明, 基础激励引起的动力传递系数依赖于分数阶阻尼阶数的值, 通过调节阻尼阶数可以控制动力传递系数的大小. 对于基础激励为非简谐的周期激励情况, 当激励频率一定时, 激励中的高阶频率成分引起的动力传递系数可能大于激励中的低阶频率成分引起的动力传递系数. 因此, 激励中的高阶频率成分所起的作用是不可忽略的.We investigate the response property of a linear system that is excited by the base excitation. The linear system contains the ordinary damping or the fractional-order damping. In our studies, the base excitation is in the harmonic form or in the general periodic form. When the base excitation is in the harmonic form, we obtain the dynamic transfer coefficient by the undetermined coefficient method. When the base excitation is in the general periodic form, we first expand the excitation into the Fourier series, then, according to the linear superposition principle, we obtain the dynamic transfer coefficient that is induced by each harmonic component in the excitation. By expanding the general periodic excitation into the Fourier series, we can solve the non-differentiable problem that is induced by the periodic base excitation for the numerical calculations. Based on the Grnwald-Letnikov definition, the discretization formula for the fractional-order system is obtained explicitly. The analytical results are in good agreement with the numerical simulations, which verifies the validity of the analytical results. Both the analytical and the numerical results show that the dynamic transfer coefficient depends on the fractional-order of the damping closely. The dynamic transfer coefficient can be controlled by tuning the value of the fractional-order. For the general periodic excitation, when the frequency is fixed, the dynamic transfer coefficient that is induced by the high-order harmonic component may be stronger than that induced by the low-order harmonic component in the base excitation. Hence, the effect of the high-order harmonic component in the excitation cannot be ignored although its amplitude is small. Further, when the base excitation is in the full sine form, or the square form, or the triangular form, the response property of the system can be described by center frequency, resonance peak, cutoff frequency, and the filter bandwidth. For a fixed fractional-order, the center frequencies of each order corresponding to the response, obtained by the three kinds of the periodic base excitations mentioned above, are identical. However, the corresponding resonance peaks are different. The resonance peak and the filter bandwidth are both maximal when the base excitation is in the square form. The resonance peak and the filter bandwidth are both minimal when the base excitation is in the triangular form. We believe that our results are useful for solving the vibration problem in the engineering field such as the vibration isolation and the vibration control.
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Keywords:
- fractional calculus /
- base excitation /
- dynamic transfer coefficient
[1] Ortigueira M D 2008 IEEE Circuits Syst. Mag. 8 19
[2] Achar B N, Hanneken J, Clarke T 2002 Physica A 309 275
[3] Deng W H, Li C P 2008 Phys. Lett. A 372 401
[4] Deng W H 2007 J. Comput. Phys. 227 1510
[5] Shen Y, Yang S, Xing H, Ma H 2012 Int. J. NonLin. Mech. 47 975
[6] Shen Y, Yang S, Xing H, Gao G 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3092
[7] Shen Y, Yang S, Sui C 2014 Chaos Soliton Fract. 67 94
[8] Rostami M, Haeri M 2014 Signal Process. 107 361
[9] Litak G, Borowiec M 2014 Nonlinear Dynam. 77 681
[10] Yang J H, Zhu H 2012 Chaos 22 149
[11] Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[12] Yang J H, Sanjuan Miguel A F, Tian F, Yang H F 2015 Int. J. Bifurcat. Chaos 25 1550023
[13] Yang J H, Sanjuan Miguel A F, Xiang W, Zhu H 2013 Pramana 81 943
[14] Yang J H, Zhu H 2012 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2012 物理学报 62 024501]
[15] Chen L, Li H, Li Z, Zhu W 2013 Sci. China: Phys. Mech. Astron. 43 670
[16] Chen L C, Zhu W Q 2009 Nonlinear Dynam. 56 231
[17] Chen L C, Zhu W Q 2009 Acta Mech. 207 109
[18] Xu Y, Li Y, Liu D 2014 J Comput. Nonlin. Dyn. 9 031015
[19] Baleanu D, Magin R L, Bhalekar S, Daftardar-Gejji V 2015 Commun. Nonlinear Sci. Numer. Simulat. 25 41
[20] Monje C A, Chen Y Q, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer) p11
[21] Chen W, Sun H G, Li X C 2010 Fractional Derivative Modeling in Mechanics and Engineering (Beijing: Science Press) pp12-14 (in Chinese) [陈文, 孙洪广, 李西成 2010 力学与工程问题中的分数阶导数建模 (北京: 科学出版社) pp12-14]
[22] Zhou J L, Pu Y F, Liao K 2010 Principle of Fractional Calculus and its Applications to Advanced Signal Processing (Beijing: Science Press) p54 (in Chinese) [周激流, 蒲亦非, 廖科 2010 分数阶微积分原理及其在现代信号分析与处理中的应用 (北京: 科学出版社) p54]
[23] Yu Y J, Wang Z H 2015 Acta Phys. Sin. 64 238401 (in Chinese) [俞亚娟, 王在华 2015 物理学报 64 238401]
[24] Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) p88
[25] Li C, Zhang F, Kurths J, Zeng F 2013 Philos. T. Roy. Soc. A 371 20120156
[26] Cao J, Ma C, Xie H, Jiang Z 2010 ASME J. Comput. Nonlin. Dyn. 5 041012
[27] Balachandran B, Magrab E B 2008 Vibrations (Australia: Cengage Learning) pp210-212
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[1] Ortigueira M D 2008 IEEE Circuits Syst. Mag. 8 19
[2] Achar B N, Hanneken J, Clarke T 2002 Physica A 309 275
[3] Deng W H, Li C P 2008 Phys. Lett. A 372 401
[4] Deng W H 2007 J. Comput. Phys. 227 1510
[5] Shen Y, Yang S, Xing H, Ma H 2012 Int. J. NonLin. Mech. 47 975
[6] Shen Y, Yang S, Xing H, Gao G 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 3092
[7] Shen Y, Yang S, Sui C 2014 Chaos Soliton Fract. 67 94
[8] Rostami M, Haeri M 2014 Signal Process. 107 361
[9] Litak G, Borowiec M 2014 Nonlinear Dynam. 77 681
[10] Yang J H, Zhu H 2012 Chaos 22 149
[11] Yang J H, Zhu H 2013 Commun. Nonlinear Sci. Numer. Simulat. 18 1316
[12] Yang J H, Sanjuan Miguel A F, Tian F, Yang H F 2015 Int. J. Bifurcat. Chaos 25 1550023
[13] Yang J H, Sanjuan Miguel A F, Xiang W, Zhu H 2013 Pramana 81 943
[14] Yang J H, Zhu H 2012 Acta Phys. Sin. 62 024501 (in Chinese) [杨建华, 朱华 2012 物理学报 62 024501]
[15] Chen L, Li H, Li Z, Zhu W 2013 Sci. China: Phys. Mech. Astron. 43 670
[16] Chen L C, Zhu W Q 2009 Nonlinear Dynam. 56 231
[17] Chen L C, Zhu W Q 2009 Acta Mech. 207 109
[18] Xu Y, Li Y, Liu D 2014 J Comput. Nonlin. Dyn. 9 031015
[19] Baleanu D, Magin R L, Bhalekar S, Daftardar-Gejji V 2015 Commun. Nonlinear Sci. Numer. Simulat. 25 41
[20] Monje C A, Chen Y Q, Vinagre B M, Xue D, Feliu V 2010 Fractional-order Systems and Controls (London: Springer) p11
[21] Chen W, Sun H G, Li X C 2010 Fractional Derivative Modeling in Mechanics and Engineering (Beijing: Science Press) pp12-14 (in Chinese) [陈文, 孙洪广, 李西成 2010 力学与工程问题中的分数阶导数建模 (北京: 科学出版社) pp12-14]
[22] Zhou J L, Pu Y F, Liao K 2010 Principle of Fractional Calculus and its Applications to Advanced Signal Processing (Beijing: Science Press) p54 (in Chinese) [周激流, 蒲亦非, 廖科 2010 分数阶微积分原理及其在现代信号分析与处理中的应用 (北京: 科学出版社) p54]
[23] Yu Y J, Wang Z H 2015 Acta Phys. Sin. 64 238401 (in Chinese) [俞亚娟, 王在华 2015 物理学报 64 238401]
[24] Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) p88
[25] Li C, Zhang F, Kurths J, Zeng F 2013 Philos. T. Roy. Soc. A 371 20120156
[26] Cao J, Ma C, Xie H, Jiang Z 2010 ASME J. Comput. Nonlin. Dyn. 5 041012
[27] Balachandran B, Magrab E B 2008 Vibrations (Australia: Cengage Learning) pp210-212
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