形状记忆合金薄板系统全局激变现象分析

岳晓乐; 向以琳; 张莹

doi:10.7498/aps.68.20190155

摘要

形状记忆合金在工程应用中的难点主要来自于系统在温度和外载荷作用下产生的复杂全局动力学行为. 本文以形状记忆合金薄板动力系统为研究对象, 分析在温度和激励振幅两个控制参数作用下系统的全局动力学. 通过全局分岔图, 可以观测到系统会发生复杂的激变现象, 然后利用复合胞坐标系方法, 获取系统的吸引子、吸引域、鞍和域边界等信息, 展现系统的全局演变过程. 研究发现, 系统随着振幅和温度变化会呈现复杂的全局结构, 并发生一系列的边界激变、合并激变现象, 同时多次发生分形-Wada, Wada-Wada, Wada-分形等域边界突变. 通过对指定区域细化, 可以清晰地显示域边界的分形特征. 研究结果对于如何通过调控温度与外载荷强度, 使形状记忆合金薄板在系统中发挥最佳性能具有理论指导意义.

关键词:

Abstract

The unique global properties of shape memory alloy are mainly derived from the martensite phase transition and its inverse, which result from the change of temperature and external load. In this paper, the global characteristics of shape memory alloy thin plate system are analyzed with the temperature and harmonic excitation amplitude as control parameters. Based on the method of Poincare map, the complex crisis phenomenon of the system including the sudden change in number, size and type of attractors can be observed through the global multivalued bifurcation diagram. However, the specific crisis type is not clear, it is necessary to be analyzed from the global viewpoint. By computing the global diagram with the composite cell coordinate system method which constructs a composite cell state space by multistage division of the continuous phase space, the attractors, saddles and basins of attraction of the system can be obtained more accurately. The vivid evolutionary processes of the crisis phenomena of the system are illustrated, and it can be found that the system presents a complex global structure with amplitude and temperature changing. There exist two kinds of crises: one is the boundary crisis resulting from the collision between a chaotic/periodic attractor and a chaotic saddle within the basin boundary, which causes the attractor to vanish, and the other is the merging crisis caused by the collision of two or more attractors with the chaotic saddle within the basin boundary where a new chaotic attractor appears. When multiple attractors coexist in the system, the basin boundary may be smooth or fractal, and for any point at boundary, its small open neighborhood always has a nonempty intersection with three or more basins, which is known as Wada basin boundary. It is difficult to predict the dynamic behavior of the system accurately due to the fractal, the Wada-Wada, Wada-fractal and fractal-Wada basin boundary metamorphoses which can be observed along with the variation of temperature and amplitude through the composite cell coordinate system method, which owns a unique advantage in depicting basin boundary. Furthermore, the Wada property is displayed more clearly by refining specified region. The results of this paper provide a theoretical analysis tool for adjusting the dynamic response of shape memory alloy thin plate system and optimizing the deformation and vibration control of mechanical equipment through controlling temperature and excitation intensity.

Keywords:

作者及机构信息

西北工业大学应用数学系, 西安　710072

通信作者: 岳晓乐, xiaoleyue@nwpu.edu.cn

基金项目: 国家自然科学基金(批准号: 11672230, 11672232)和陕西省自然科学基础研究计划(批准号: 2017JM1029)资助的课题.

Authors and contacts

Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China

Corresponding author: Yue Xiao-Le, xiaoleyue@nwpu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11672230, 11672232) and the Natural Science Basic Research Plan in Shaanxi Province, China (Grant No. 2017JM1029).

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参考文献

[1]	Yuan B, Zhu M, Chung C 2018 Materials 11 1716 Google Scholar
[2]	Hartl D J, Lagoudas D C 2007 Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 221 535
[3]	Lee J, Jin M, Ahn K K 2013 Mechatronics 23 310 Google Scholar
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施引文献

图 1 SMA薄板的横向振动

Fig. 1. Transverse vibration of SMA thin plate.

下载: 全尺寸图片幻灯片

图 2 系统(7)随振幅g变化的多值分岔图

Fig. 2. Multivalued bifurcation diagram of the system (7) with the variation of amplitude g.

下载: 全尺寸图片幻灯片

图 3 系统(7)的全局图　(a) g = 0.0471; (b) g = 0.0472

Fig. 3. Global diagram of the system (7): (a) g = 0.0471; (b) g = 0.0472.

下载: 全尺寸图片幻灯片

图 4 系统(7)的全局图　(a) g = 0.0482; (b) g = 0.0483

Fig. 4. Global diagram of the system (7): (a) g = 0.0482; (b) g = 0.0483.

下载: 全尺寸图片幻灯片

图 5 系统(7)的全局图　(a) g = 0.0491; (b) g = 0.0492

Fig. 5. Global diagram of the system (7): (a) g = 0.0491; (b) g = 0.0492.

下载: 全尺寸图片幻灯片

图 6 系统(7)的全局图　(a) g = 0.0490; (b) g = 0.0491

Fig. 6. Global diagram of the system (7): (a) g = 0.0490; (b) g = 0.0491.

下载: 全尺寸图片幻灯片

图 7 系统(7)的全局图　(a) g = 0.0596; (b) g = 0.0597; (c) g = 0.0726; (d) g = 0.0727

Fig. 7. Global diagram of the system (7): (a) g = 0.0596; (b) g = 0.0597; (c) g = 0.0726; (d) g = 0.0727.

下载: 全尺寸图片幻灯片

图 8 系统(7)的全局图　(a) g = 0.0460; (b) g = 0.0461

Fig. 8. Global diagram of the system (7): (a) g = 0.0460; (b) g = 0.0461.

下载: 全尺寸图片幻灯片

图 9 系统(7)的全局图　(a) g = 0.0478; (b) g = 0.0479; (c), (d) 分别对应于(a), (b)图的区域细化

Fig. 9. Global diagram of the system (7): (a) g = 0.0478; (b) g = 0.0479; (c), (d) the region refinement of panels (a) and (b).

下载: 全尺寸图片幻灯片

图 10 系统(7)的全局图　(a) g = 0.0533; (b) g = 0.0534

Fig. 10. Global diagram of the system (7): (a) g = 0.0533; (b) g = 0.0534.

下载: 全尺寸图片幻灯片

图 11 系统(7)随温度θ变化的多值分岔图

Fig. 11. Multivalued bifurcation diagram of the system (7) with the variation of temperature θ.

下载: 全尺寸图片幻灯片

图 12 系统(7)的全局图　(a) θ = 0.8379; (b) θ = 0.8380

Fig. 12. Global diagram of the system (7): (a) θ = 0.8379; (b) θ = 0.08380.

下载: 全尺寸图片幻灯片

图 13 系统(7)的全局图　(a) θ = 0.4088; (b) θ = 0.4089; (c) θ = 0.7182; (d) θ = 0.7183

Fig. 13. Global diagram of the system (7): (a) θ = 0.4088; (b) θ = 0.4089; (c) θ = 0.7182; (d) θ = 0.7183.

下载: 全尺寸图片幻灯片

图 14 系统(7)的全局图　(a) θ = 0.0950; (b) θ = 0.0951

Fig. 14. Global diagram of the system (7): (a) θ = 0.0950; (b) θ = 0.0951.

下载: 全尺寸图片幻灯片

[1]	Yuan B, Zhu M, Chung C 2018 Materials 11 1716 Google Scholar
[2]	Hartl D J, Lagoudas D C 2007 Proc. Inst. Mech. Eng. Part G: J. Aerosp. Eng. 221 535
[3]	Lee J, Jin M, Ahn K K 2013 Mechatronics 23 310 Google Scholar
[4]	Jani J M, Leary M, Subic A, Gibson M A 2014 Mater. Des. 56 1078 Google Scholar
[5]	Song G, Ma N, Li H N 2006 Eng. Struct. 28 1266 Google Scholar
[6]	Bernardini D, Rega G 2011 Int. J. Bifurcation Chaos 21 2769 Google Scholar
[7]	Paula A S, Savi M A, Lagoudas D C 2012 J. Braz. Soc. Mech. Sci. Eng. 34 401 Google Scholar
[8]	Sado D, Pietrzakowski M 2010 Int. J. Non-Linear Mech. 45 859 Google Scholar
[9]	Hashemi S M T, Khadem S E 2006 Int. J. Mech. Sci. 48 44 Google Scholar
[10]	Savi M A 2015 Int. J. Non-Linear Mech. 70 2 Google Scholar
[11]	Han Q, Xu W, Yue X 2014 Int. J. Bifurcation Chaos 24 1450051 Google Scholar
[12]	Grebogi C, Ott E, Yorke J A 1982 Phys. Rev. Lett. 48 1507 Google Scholar
[13]	Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181 Google Scholar
[14]	Chian A C L, Borotto F A, Rempel E L, Rogers C 2005 Chaos Solitons Fractals 24 869 Google Scholar
[15]	Yue X, Xu W, Zhang Y 2012 Nonlinear Dyn. 69 437 Google Scholar
[16]	刘莉, 徐伟, 岳晓乐, 韩群 2013 物理学报 62 200501 Google Scholar Liu L, Xu W, Yue X L, Han Q 2013 Acta Phys. Sin. 62 200501 Google Scholar
[17]	刘晓君, 洪灵, 江俊 2016 物理学报 65 180502 Google Scholar Liu X J, Hong L, Jiang J 2016 Acta Phys. Sin. 65 180502 Google Scholar
[18]	Yue X, Xu W, Wang L 2013 Commun. Nonlinear Sci. Numer. Simul. 18 3567 Google Scholar
[19]	Zhang Y 2013 Phys. Lett. A 377 1269 Google Scholar
[20]	Hsu C S 1992 Int. J. Bifurcation Chaos 2 727 Google Scholar
[21]	Hsu C S 1995 Int. J. Bifurcation Chaos 5 1085 Google Scholar
[22]	Tongue B H 1987 Physica D 28 401 Google Scholar
[23]	Falk F 1980 Acta Metall. Sin. 28 1773 Google Scholar
[24]	Machado L G, Savi M A, Pacheco P M C L 2004 Shock Vib. 11 67 Google Scholar
[25]	黄志华, 刘平, 杜长城, 李映辉 2009 力学季刊 30 71 Google Scholar Huang Z H, Liu P, Du C C, Li Y H 2009 Chin. Quarterly Mech. 30 71 Google Scholar

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形状记忆合金薄板系统全局激变现象分析