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The wall of elastic microtubules can be described as a membrane-type elastic structure. An oscillating system driven by ultrasound consists of liquid columns, a bubble and elastic wall of the tube. The nonlinear properties of this system are explored. Based on the successive approximation method, the nonlinear resonance frequencies, the amplitude response of fundamental and third oscillations to driving acoustic wave, and the mechanism of resonance response related to the driving wave whose frequency is lower than the resonant frequency are analyzed theoretically. The nonlinear system is oscillating in two directions: the axial and radial directions of the bubble in the microtubule. Numerical results show that the resonance responses cannot be present simultaneously. It has been found that the amplitudes of the fundamental and third harmonic oscillation are multivalued, which may lead to instable response. The third harmonic oscillation is stronger in the region of lower frequencies.
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Keywords:
- elastic microtubule /
- nonlinear oscillations /
- forced oscillations /
- acoustical response of bubble
[1] Hu Y T, Qin S P, Hu T, Ferrara K, Jiang Q 2005 Int. J. Nonlin. Mech. 40 341
[2] Qin S P, Hu Y T, Jiang Q 2006 IEEE. T. Ultrason. Ferr. 53 1322
[3] Freund J B 2008 J. Acoust. Soc. Am. 123 2867
[4] Cancelos S, Moraga F J, Lahey R T, Shain W, Parsons R H 2010 J. Acoust. Soc. Am. 128 2726
[5] Qin S P, Ferrara K W 2007 Ultrasound Med. Biol. 33 1140
[6] Miao H Y, Gracewski S M, Dalecki D 2009 J. Acoust. Soc. Am. 126 2374
[7] Martynov S, Stride E, Saffari N 2009 J. Acoust. Soc. Am. 126 2963
[8] Sassaroli E, and Hynynen K 2005 Phys. Med. Biol. 50 5293
[9] Gao F R, Hu Y T, Hu H P 2007 Int. J. Solids Struct. 44 7197
[10] Zhen H R, Dayton P A, Caskey C, Zhao S K, Qin S P, Ferrara K W 2007 Ultrasound Med. Biol. 33 1978
[11] Wang Z Y, Tong A Y 2008 Int. J. Therm. Sci. 47 221
[12] Wang C H, Lin S Y 2010 Sci China Phys. Mech. Astron. 53 496
[13] Leighton T G, White P R, Marsden M A 1995 Acta. Acust. 3 517
[14] Oguz H N, Prosperetti A 1998 J. Acoust. Soc. Am. 103 3301
[15] Sassaroli E, Hynynen K 2004 J. Acoust. Soc. Am. 115 3235
[16] Chen X M, Prosperetti A 1998 J. Acoust. Soc. Am. 104 1389
[17] Jang N W, Gracewski S M, Abrahamsen B, Buttaccio T, Halm Robert, Dalecki D 2009 J. Acoust. Soc. Am. 126 EL34
[18] Wang C H, Cheng J C. 2005 Acta. Phys. Sin. 61 194303 (in Chinese) [王成会, 程建春 2012 61 194303]
[19] Landau L D, Lifshitz E M 1976 Mechanics (Third Edition) (London: Pergamon Press) p58
[20] Du G H, Zhu Z M, Gong X F 2001 Fundamentals of Sound (Nanjing: Nanjing University Press) p502 (in Chinese) [杜功焕, 朱哲民, 龚秀芬 2001 声学基础. (南京: 南京大学出版社)第 502页]
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[1] Hu Y T, Qin S P, Hu T, Ferrara K, Jiang Q 2005 Int. J. Nonlin. Mech. 40 341
[2] Qin S P, Hu Y T, Jiang Q 2006 IEEE. T. Ultrason. Ferr. 53 1322
[3] Freund J B 2008 J. Acoust. Soc. Am. 123 2867
[4] Cancelos S, Moraga F J, Lahey R T, Shain W, Parsons R H 2010 J. Acoust. Soc. Am. 128 2726
[5] Qin S P, Ferrara K W 2007 Ultrasound Med. Biol. 33 1140
[6] Miao H Y, Gracewski S M, Dalecki D 2009 J. Acoust. Soc. Am. 126 2374
[7] Martynov S, Stride E, Saffari N 2009 J. Acoust. Soc. Am. 126 2963
[8] Sassaroli E, and Hynynen K 2005 Phys. Med. Biol. 50 5293
[9] Gao F R, Hu Y T, Hu H P 2007 Int. J. Solids Struct. 44 7197
[10] Zhen H R, Dayton P A, Caskey C, Zhao S K, Qin S P, Ferrara K W 2007 Ultrasound Med. Biol. 33 1978
[11] Wang Z Y, Tong A Y 2008 Int. J. Therm. Sci. 47 221
[12] Wang C H, Lin S Y 2010 Sci China Phys. Mech. Astron. 53 496
[13] Leighton T G, White P R, Marsden M A 1995 Acta. Acust. 3 517
[14] Oguz H N, Prosperetti A 1998 J. Acoust. Soc. Am. 103 3301
[15] Sassaroli E, Hynynen K 2004 J. Acoust. Soc. Am. 115 3235
[16] Chen X M, Prosperetti A 1998 J. Acoust. Soc. Am. 104 1389
[17] Jang N W, Gracewski S M, Abrahamsen B, Buttaccio T, Halm Robert, Dalecki D 2009 J. Acoust. Soc. Am. 126 EL34
[18] Wang C H, Cheng J C. 2005 Acta. Phys. Sin. 61 194303 (in Chinese) [王成会, 程建春 2012 61 194303]
[19] Landau L D, Lifshitz E M 1976 Mechanics (Third Edition) (London: Pergamon Press) p58
[20] Du G H, Zhu Z M, Gong X F 2001 Fundamentals of Sound (Nanjing: Nanjing University Press) p502 (in Chinese) [杜功焕, 朱哲民, 龚秀芬 2001 声学基础. (南京: 南京大学出版社)第 502页]
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