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A locally resonant stiffened plate is constructed by attaching a two-dimensional periodic array of spring-mass resonators to a traditional periodic stiffened plate. A method based on the finite element method and Bloch theorem is presented for calculating the flexural wave dispersion relation and forced vibration response of the proposed locally resonant stiffened plate. The method is validated by comparing the predictions with simulations by FEM software COMSOL. The effects of the spring-stiffness and mass ratio of local resonators on the flexural wave band gap and vibration reduction performance are analysed, which can facilitate the design of the locally resonant stiffened plate for vibration-reduction applications in engineering. The main findings of this work are as follows. 1) The local resonator can have a significant effect on the propagation of flexural wave in stiffened plate. On the one hand, the local resonator is able to create a low-frequency local resonance band gap; on the other hand, it can enhance the high-frequency Bragg band gap. Within the band gap frequency range, the vibration of the locally resonant stiffened plate can be reduced remarkably. 2) The spring-stiffness of local resonators shows a notable influence on the band gap and vibration reduction performance of the locally resonant stiffened plate. As the spring-stiffness gradually increases, the nature frequency of local resonator is gradually tuned to higher frequency, and the phenomenon of band-gap transition and band-gap near-coupling may arise. Under the near-coupling condition, the pass band between two band gaps turns narrow, and it seems that these two band gaps form a super-wide pseudo-gap (within which only a very narrow pass band exists). This behaviour is of great interest for the broad band vibration reduction applications. Moreover, the complete band gap will disappear if the nature frequency of local resonator is tuned to a higher value than a threshold frequency, which is dependent on the geometrical and material parameters. 3) The influence of the additional mass ratio of local resonator on the band gap behavior is highly relevant to the nature frequency of local resonator. If the nature frequency of resonator is lower than the band-gap near-coupling frequency, both the local resonance band gap and Bragg band gap are broadened with increasing the additional mass ratio of resonator. When the nature frequency of resonator is close to the band-gap near-coupling frequency, the phenomenon of band-gap near coupling and band-gap transition may arise or disappear as the additional mass ratio of resonator gradually changes. When the nature frequency of resonator is higher than the band-gap near-coupling frequency, on the one hand, the lower frequency band gap will disappear rapidly with increasing the mass ratio of resonator. However, it will be present again if the mass ratio of resonator increases up to a large enough value. On the other hand, the higher frequency band gap is broadened with increasing the mass ratio, but if the mass ratio is tuned to a larger value than a specific value, this band gap will transform from local resonance band gap to Bragg band gap, and the normalized gap width of this band gap will be narrowed with increasing the mass ratio.
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Keywords:
- local resonance /
- stiffened plate /
- flexural wave band gap /
- vibration reduction
[1] Xiao Y 2012 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese) [肖勇 2012 博士学位论文 (长沙: 国防科技大学)]
[2] Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T, Sheng P 2000 Science 289 1734
[3] Wen J H, Yu D L, Zhao H G, Cai L, Xiao Y, Wang G, Yin J F 2015 Propagation of Elastic Waves in Artificial Periodic Structures: Vibrational and Acoustical Properties (Beijing: Science Press) pp5-200 (in Chinese) [温激鸿, 郁殿龙, 赵宏刚, 蔡力, 肖勇, 王刚, 尹剑飞 2015 人工周期结构中弹性波的传播-振动与声学特性 (北京:科学出版社) 第5-200页]
[4] Wen X S, Wen J H, Yu D L, Wang G, Liu Y Z, Han X Y 2009 Phononic Crystals (Beijing: National Defense Industry Press) pp53-195 (in Chinese) [温熙森, 温激鸿, 郁殿龙, 王刚, 刘耀宗, 韩小云 2009 声子晶体 (北京: 国防工业出版社) 第53-195页]
[5] Kushwaha M S, Halevi P, Dobrzynski L, Djafari-Rouhani B 1993 Phys. Rev. Lett. 71 2022
[6] Brillouin L 1946 Wave Propagation in Periodic Structures (New York: Dover) pp115-117
[7] Xiao Y, Wen J H, Yu D L, Wen X S 2013 J. Sound Vib. 332 867
[8] Xiao Y, Wen J H, Huang L 2014 J. Phys. D: Appl. Phys. 47 045307
[9] Wang G 2005 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese) [王刚 2005 博士学位论文 (长沙:国防科技大学)]
[10] Hsu J C 2011 J. Phys. D: Appl. Phys. 44 055401
[11] Oudich M, Li Y, Assouar B M, Hou Z 2010 New J. Phys. 12 083049
[12] Pennec Y, Djafari-Rouhani B, Larabi H, Vasseur J O, Hladky-Hennion A C 2008 Phys. Rev. B 78 104105
[13] Zhang S, Wu J, Hu Z 2013 J. Appl. Phys. 113 163511
[14] Xiao Y, Wen J, Wen X 2012 J. Phys. D: Appl. Phys. 19 195401
[15] Xiao Y, Wen J, Wen X 2012 J. Sound Vib. 331 5408
[16] Assouar B M, Sun J, Lin F, Hsu J 2014 Ultrasonics 54 2159
[17] Wu J, Bai X, Xiao Y, Geng X, Yu D, Wen J 2016 Acta Phys. Sin. 65 064602 (in Chinese) [吴健, 白晓春, 肖勇, 耿明昕, 郁殿龙, 温激鸿 2016 物理学报 65 064602]
[18] Liu J, Hou Z L, Fu X J 2015 Acta Phys. Sin. 64 154302 (in Chinese) [刘娇, 侯志林, 傅秀军 2015 物理学报 64 154302]
[19] Wang X Z, Pang F Z, Yao X L, Su N 2012 Acta Mech. Solida Sin. 33 583 (in Chinese) [王献忠, 庞福振, 姚熊亮, 苏楠 2012 固体力学学报 33 583]
[20] Jin Y Q, Pang F Z, Yao X L, Wang X Z 2012 J. Vib. Eng. 25 579 (in Chinese) [金叶青, 庞福振, 姚熊亮, 王献忠 2012 振动工程学报 25 579]
[21] Yao X L, Wang X Z, Pang F Z, Kang P H 2012 J. Huazhong Univ. of Sci. Tech. (Natural Science Edition) 40 119 (in Chinese) [姚熊亮, 王献忠, 庞福振, 康蓬辉 2012 华中科技大学学报 40 119]
[22] Li S, Zhao D Y 2001 Acta Acustica. 26 174 (in Chinese) [黎胜, 赵德有 2001 声学学报 26 174]
[23] Li S, Zhao D Y 2004 Acta Acustica. 29 200 (in Chinese) [黎胜, 赵德有 2004 声学学报 29 200]
[24] Lu T J, Xin F X 2012 Vibroacoustics of Lightweight Sandwich Structures (Beijing: Science Press) pp80-157 (in Chinese) [卢天健, 辛锋先 2012 轻质板壳结构设计的振动和声学基础 (北京:科学出版社) 第80-157页]
[25] Xin F X, Lu T J, Chen C Q 2010 J. Vib. Acoust. 132 011008
[26] Maurice P 1990 Introduction to finite element vibration analysis (New York: Cambridge University Press) pp229-313
[27] Xu S Y, Huang X C, Hua H X 2013 J. ShangHai JiaoTong Univ. 47 167 (in Chinese) [徐时吟, 黄修长, 华宏星 2013 上海交通大学学报 47 167]
[28] Song Y, Feng L, Wen J, Yu D, Wen X 2015 Compos. Struct. 128 428
[29] Langley R S 1993 J. Sound Vib. 169 377
[30] Farzbod F, Leamy M J 2009 J. Sound Vib. 325 545
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[1] Xiao Y 2012 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese) [肖勇 2012 博士学位论文 (长沙: 国防科技大学)]
[2] Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T, Sheng P 2000 Science 289 1734
[3] Wen J H, Yu D L, Zhao H G, Cai L, Xiao Y, Wang G, Yin J F 2015 Propagation of Elastic Waves in Artificial Periodic Structures: Vibrational and Acoustical Properties (Beijing: Science Press) pp5-200 (in Chinese) [温激鸿, 郁殿龙, 赵宏刚, 蔡力, 肖勇, 王刚, 尹剑飞 2015 人工周期结构中弹性波的传播-振动与声学特性 (北京:科学出版社) 第5-200页]
[4] Wen X S, Wen J H, Yu D L, Wang G, Liu Y Z, Han X Y 2009 Phononic Crystals (Beijing: National Defense Industry Press) pp53-195 (in Chinese) [温熙森, 温激鸿, 郁殿龙, 王刚, 刘耀宗, 韩小云 2009 声子晶体 (北京: 国防工业出版社) 第53-195页]
[5] Kushwaha M S, Halevi P, Dobrzynski L, Djafari-Rouhani B 1993 Phys. Rev. Lett. 71 2022
[6] Brillouin L 1946 Wave Propagation in Periodic Structures (New York: Dover) pp115-117
[7] Xiao Y, Wen J H, Yu D L, Wen X S 2013 J. Sound Vib. 332 867
[8] Xiao Y, Wen J H, Huang L 2014 J. Phys. D: Appl. Phys. 47 045307
[9] Wang G 2005 Ph. D. Dissertation (Changsha: National University of Defense Technology) (in Chinese) [王刚 2005 博士学位论文 (长沙:国防科技大学)]
[10] Hsu J C 2011 J. Phys. D: Appl. Phys. 44 055401
[11] Oudich M, Li Y, Assouar B M, Hou Z 2010 New J. Phys. 12 083049
[12] Pennec Y, Djafari-Rouhani B, Larabi H, Vasseur J O, Hladky-Hennion A C 2008 Phys. Rev. B 78 104105
[13] Zhang S, Wu J, Hu Z 2013 J. Appl. Phys. 113 163511
[14] Xiao Y, Wen J, Wen X 2012 J. Phys. D: Appl. Phys. 19 195401
[15] Xiao Y, Wen J, Wen X 2012 J. Sound Vib. 331 5408
[16] Assouar B M, Sun J, Lin F, Hsu J 2014 Ultrasonics 54 2159
[17] Wu J, Bai X, Xiao Y, Geng X, Yu D, Wen J 2016 Acta Phys. Sin. 65 064602 (in Chinese) [吴健, 白晓春, 肖勇, 耿明昕, 郁殿龙, 温激鸿 2016 物理学报 65 064602]
[18] Liu J, Hou Z L, Fu X J 2015 Acta Phys. Sin. 64 154302 (in Chinese) [刘娇, 侯志林, 傅秀军 2015 物理学报 64 154302]
[19] Wang X Z, Pang F Z, Yao X L, Su N 2012 Acta Mech. Solida Sin. 33 583 (in Chinese) [王献忠, 庞福振, 姚熊亮, 苏楠 2012 固体力学学报 33 583]
[20] Jin Y Q, Pang F Z, Yao X L, Wang X Z 2012 J. Vib. Eng. 25 579 (in Chinese) [金叶青, 庞福振, 姚熊亮, 王献忠 2012 振动工程学报 25 579]
[21] Yao X L, Wang X Z, Pang F Z, Kang P H 2012 J. Huazhong Univ. of Sci. Tech. (Natural Science Edition) 40 119 (in Chinese) [姚熊亮, 王献忠, 庞福振, 康蓬辉 2012 华中科技大学学报 40 119]
[22] Li S, Zhao D Y 2001 Acta Acustica. 26 174 (in Chinese) [黎胜, 赵德有 2001 声学学报 26 174]
[23] Li S, Zhao D Y 2004 Acta Acustica. 29 200 (in Chinese) [黎胜, 赵德有 2004 声学学报 29 200]
[24] Lu T J, Xin F X 2012 Vibroacoustics of Lightweight Sandwich Structures (Beijing: Science Press) pp80-157 (in Chinese) [卢天健, 辛锋先 2012 轻质板壳结构设计的振动和声学基础 (北京:科学出版社) 第80-157页]
[25] Xin F X, Lu T J, Chen C Q 2010 J. Vib. Acoust. 132 011008
[26] Maurice P 1990 Introduction to finite element vibration analysis (New York: Cambridge University Press) pp229-313
[27] Xu S Y, Huang X C, Hua H X 2013 J. ShangHai JiaoTong Univ. 47 167 (in Chinese) [徐时吟, 黄修长, 华宏星 2013 上海交通大学学报 47 167]
[28] Song Y, Feng L, Wen J, Yu D, Wen X 2015 Compos. Struct. 128 428
[29] Langley R S 1993 J. Sound Vib. 169 377
[30] Farzbod F, Leamy M J 2009 J. Sound Vib. 325 545
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