Global analysis of crises in shape memory thin plate system

Yue Xiao-Le; Xiang Yi-Lin; Zhang Ying

doi:10.7498/aps.68.20190155

Abstract

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1. 引　言
2. SMA薄板动力系统
3. 振幅对全局特性的影响
3.1 边界激变
3.3 域边界突变
4. 温度对全局特性的影响
4.1 边界激变
4.2 合并激变
4.3 域边界突变
5. 结　论

References

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:

PACS:

05.45.-a	(Nonlinear dynamics and chaos)
05.45.Ac	(Low-dimensional chaos)
05.45.Pq	(Numerical simulations of chaotic systems)

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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1. 引　言

形状记忆合金(shape memory alloy, SMA)表现出的独特力学性能来源于马氏体相变及其逆变化, 相变的驱动力可以由温度变化和机械载荷提供, 温度诱发的相变引起形状记忆效应, 外加应力诱导的相变产生超弹性. 利用这两个性质, SMA被制成各种参数可控的智能元件, 在机翼结构的变形、精密机械系统的高速驱动及机器人仿生等方面得到广泛应用^[1—5]. 目前关于SMA元件的力学性能研究主要集中在温度和载荷变化下SMA梁、振子及支架系统的平衡点稳定性、分岔行为及混沌现象等非线性动力学特征方面^[6—10], 对于SMA薄板系统的全局动力学研究较少, 相较于局部分析, 全局分析能够揭示更多的动力学信息^[11], 有利于从力学理论角度突破SMA薄板系统在机械驱动和振动控制等领域的局限性.

激变^[12,13]作为混沌系统中较为常见的全局分岔现象, 主要刻画了混沌吸引子和混沌鞍的不连续变化. 常见的激变包括边界激变、内部激变和合并激变. 边界激变是指混沌吸引子与吸引域边界上的不稳定周期轨道碰撞, 导致吸引子的突然消失. 内部激变是混沌吸引子与其所在吸引域内的不稳定周期轨道发生碰撞引起的. 合并激变是指两个及以上的吸引子同时与吸引域边界上的不稳定周期轨道发生碰撞, 合并形成新的吸引子. 尽管通过分岔图可以观测到激变现象, 但具体类型并不清楚, 需进一步绘制全局图进行判断. 近年来, 不少系统的激变现象均有所研究, 例如分数阶和单边碰撞系统^[14—17].

当系统中存在多个吸引子时, 吸引域边界可以是光滑的, 也可以呈现出“你中有我, 我中有你”的分形结构. 如果吸引子的个数为三个及以上, 且边界上任一点的任意小领域, 覆盖三个及以上吸引域, 则称此域边界具有Wada特性. 随着系统参数变化, 域边界结构会突然发生改变, 称为域边界突变, 包括光滑-Wada域边界突变、分形-Wada域边界突变和Wada-Wada域边界突变等. 域边界突变作为混沌动力学的一个研究重点, 对于确定系统的整体结构具有重要的物理意义, 在非光滑系统和准周期强迫系统^[18,19]中都有所研究, 但分形性使得动力系统的力学行为很难预测, 随着胞映射技术的发展, 域边界突变和激变现象的研究得到了突破. 其中经典的广义胞映射、图胞映射、插值胞映射等方法均可获得系统的全局信息^[20—22], 但在刻画域边界时有一定的局限性. 复合胞坐标系方法通过对连续相空间的多级分割构造一个复合胞空间, 基于点映射的原理, 不仅可以获取动力系统的吸引子、吸引域和鞍等信息, 还能细化任意小区域, 在全局图中优化对域边界的刻画.

本文以SMA薄板动力系统为研究对象, 温度和外部激励的振幅为分岔参数, 在全局分岔图的基础上, 通过复合胞坐标系方法进一步分析系统在演变过程中出现的激变类型和域边界突变现象, 并通过对指定区域的细化, 展示域边界的分形特征. 研究结果在工程领域中对控制系统的动态响应有重要意义.

2. SMA薄板动力系统

在SMA薄板中, 相对于奥氏体, 马氏体更具延展性, 当外加应力低于马氏体的屈服强度时, SMA薄板可表现出超弹性. 马氏体的转变与温度相关, 低温相称为马氏体, 高温相称为奥氏体, 降温时SMA薄板发生马氏体相变, 呈现形状记忆效应. 定义T_M为马氏体相变临界温度, 该温度之下, 马氏体稳定; T_A为奥氏体相变临界温度, 该温度之上, 奥氏体稳定, a, b, c均为材料常数, 满足:

${T_{\rm{A}}} = {T_{\rm{M}}} + \frac{{{b^2}}}{{4ac}}.$

(1)

根据Falk ^[23]的研究, SMA薄板的本构模型可以由一个五次多项式表示, 该模型给出了应变(ε)-应力(σ)的关系:

$\sigma = a(T - {T_{\rm{M}}})\varepsilon - b{\varepsilon ^3} + c{\varepsilon ^5}.$

(2)

Savi等^[10,24]给出了很多关于SMA弹簧振子的数学模型, 文献[25]在此基础上演变出SMA薄板横向振动的非线性动力学模型. 本文考虑密度ρ, 厚度h, 长宽分别为l₁, l₂的SMA薄板, 在环境黏性阻尼e下, 受到横向简谐激振力F₀ = FsinΩt作用, 如图1所示.

$图 1 SMA薄板的横向振动\r\nFig. 1. Transverse vibration of SMA thin plate.$

图 1 SMA薄板的横向振动

Fig. 1. Transverse vibration of SMA thin plate.

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薄板的横向位移为w (x, y, t), 单位长度上的内力矩分别为U_x(x, y, t), U_y(x, y, t), U_xy(x, y, t), 其横向振动方程为^[25]

$\frac{{{\partial ^2}{U_x}}}{{\partial {x^2}}} + 2\frac{{{\partial ^2}{U_{xy}}}}{{\partial x\partial y}} + \frac{{{\partial ^2}{U_y}}}{{\partial {y^2}}} + {F_0} - e\dot w - \rho h\frac{{{\partial ^2}w}}{{\partial {t^2}}} = 0.$

(3)

薄板面内的3个应力分别为σ_x, σ_y, τ_xy, 内力矩为^[25]

$\begin{array}{l} {U_x} = \displaystyle\int_{{{ - h} / 2}}^{{h / 2}} {{\sigma _x}z{\rm{d}}z},\; {U_y} = \displaystyle\int_{{{ - h} / 2}}^{{h / 2}} {{\sigma _y}z{\rm{d}}z},\\ {U_{xy}} = \displaystyle\int_{{{ - h} / 2}}^{{h / 2}} {{\tau _{xy}}z{\rm{d}}z} . \end{array}$

(4)

在满足四周简支边界条件下取位移模式^[25]:

$w(x,y,t) = W(t)\sin \frac{{{\text{π}}x}}{{{l_1}}}\sin \frac{{{\text{π}}y}}{{{l_2}}}.$

(5)

通过Galerkin原理和应变-应力本构关系, 并对系统参数量纲归一化, 得到SMA薄板的横向振动非线性动力学方程^[25]:

$\ddot W + \xi \dot W + \alpha (\theta - 1)W - \beta {W^3} + \gamma {W^5} = g{\rm{sin}}\varOmega t.$

(6)

方程(6)可化为以下形式的一阶微分方程组:

$\left\{ \begin{array}{l} \dot x = y, \\ \dot y = - \xi y - \alpha (\theta - 1)x + \beta {x^3} - \gamma {x^5} + g{\rm sin}\varOmega t, \\ \end{array} \right.$

(7)

其中α, ξ, β, γ均为正常数; θ, g分别是无量纲化的温度和简谐激振力振幅. 参数选取同文献[24]所示, ξ = 0.2, α = 1, β = 1300, γ = 470000, Ω = 1. 接下来基于复合胞坐标系方法, 分析温度θ和激励振幅g两个参数变化时, SMA薄板系统的全局动力学特性变化过程.

3. 振幅对全局特性的影响

当温度θ = 0.7, 此时SMA薄板位于马氏体临界温度下, 含有两个稳定的马氏体相. 选取不同的初值点(–0.0553, –0.09), (–0.031, 0.045), (0.083, –0.032), (0.047, 0.059)构造Poincare映射, 保留稳态映射点, 得到对应的多值分岔图, 如图2所示. 随着幅值g值的变化, 系统呈现明显的多吸引子共存特征, 并伴随有丰富的激变和域边界突变等现象.

$图 2 系统(7)随振幅g变化的多值分岔图\r\nFig. 2. Multivalued bifurcation diagram of the system (7) with the variation of amplitude g.$

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.

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在区域D = {(x, y)|–0.1 ≤ x ≤ 0.1, –0.12 ≤ y ≤ 0.12}上, 均匀划分1000 × 1200个胞, 利用复合胞坐标系方法获取系统的吸引子、吸引域和边界鞍等全局信息, 阐述g在0.0460到0.0727的范围内激变现象出现的机理. 为更直观理解全局图, 对以下图形做以下说明: 用△表示吸引子A, 不同颜色代表不同的吸引子; 吸引域用B表示, 不同吸引域颜色不同; S表示边界鞍; IS表示吸引域内部鞍; 下标表示吸引子、吸引域和内部鞍的个数, 例如A₁, A₂表示系统两个吸引子共存, B₁, B₂为其对应的吸引域.

3.1 边界激变

当g从0.0471增大到0.0472时, 发生两次逆的边界激变, 如图3所示. A₅和A₆表示周期为2的吸引子, A₅与嵌在吸引域B₁, B₅边界的鞍S碰撞, A₆和嵌入吸引域B₂, B₆边界的鞍S碰撞, 使得A₅, A₆和部分边界鞍S突然消失, 成为内部周期鞍IS₁, IS₂, 同时吸引域B₅, B₆消失, 吸引域B₁, B₂变大.

$图 3 系统(7)的全局图　(a) g = 0.0471; (b) g = 0.0472\r\nFig. 3. Global diagram of the system (7): (a) g = 0.0471; (b) g = 0.0472.$

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.

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当g从0.0482变化到0.0483时, 系统再次发生两次边界激变, 两个周期1吸引子A₁和A₂与边界上的鞍S发生碰撞, 变为吸引域B₅, B₆的内部鞍IS₅, IS₆, 同时吸引域B₁和B₂消失, 如图4所示.

$图 4 系统(7)的全局图　(a) g = 0.0482; (b) g = 0.0483\r\nFig. 4. Global diagram of the system (7): (a) g = 0.0482; (b) g = 0.0483.$

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.

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当g从0.0491增大到0.0492时, 混沌吸引子A₂与吸引域边界上的混沌鞍S发生碰撞, 发生边界激变, 变成新的更大的边界混沌鞍, 吸引域B₂随之消失, 状态空间中为四个周期吸引子和混沌边界鞍共存, 如图5所示.

$图 5 系统(7)的全局图　(a) g = 0.0491; (b) g = 0.0492\r\nFig. 5. Global diagram of the system (7): (a) g = 0.0491; (b) g = 0.0492.$

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.

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3.2 合并激变

当g从0.0490变化到0.0491时, 混沌吸引子A₂和A₄不断接近吸引域边界上的混沌鞍S, 发生合并激变, 成为新的混沌吸引子A₂, 与此同时吸引域B₂, B₄合并为新的吸引域B₂, 如图6所示.

$图 6 系统(7)的全局图　(a) g = 0.0490; (b) g = 0.0491\r\nFig. 6. Global diagram of the system (7): (a) g = 0.0490; (b) g = 0.0491.$

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.

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当g从0.0596变化到0.0597时, 混沌吸引子A₁, A₂与边界上的混沌鞍S碰撞, 发生合并激变, 变为新的混沌吸引子A₁, 如图7(a)和图7(b)所示. 当g从0.0726增大到0.0727时, 发生逆合并激变, 混沌吸引子A₁消失, 出现两个新的周期1吸引子A₁, A₂及混沌边界鞍S, 如图7(c)和图7(d)所示.

$图 7 系统(7)的全局图　(a) g = 0.0596; (b) g = 0.0597; (c) g = 0.0726; (d) g = 0.0727\r\nFig. 7. Global diagram of the system (7): (a) g = 0.0596; (b) g = 0.0597; (c) g = 0.0726; (d) g = 0.0727.$

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.

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3.3 域边界突变

当g = 0.0460时, 系统存在三个周期1吸引子A₁, A₂, A₃和嵌在吸引域B₁, B₂, B₃边界上的混沌鞍S, 此时的域边界呈现出Wada特性. 当g增大到0.0461时, 系统新出现一个周期3吸引子A₄, 此时的域边界由4个吸引域构成, 仍具有Wada特性, 系统发生Wada-Wada域边界突变, 如图8所示.

$图 8 系统(7)的全局图　(a) g = 0.0460; (b) g = 0.0461\r\nFig. 8. Global diagram of the system (7): (a) g = 0.0460; (b) g = 0.0461.$

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.

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当g从0.0478增大到0.0479时, 吸引子的个数从6个变为7个, 为判断吸引子变化过程, 对区域v ={(x, y)|–0.075 ≤ x ≤ –0.055, 0.068 ≤ y ≤ 0.092}进行细化. 可以发现, 原周期3吸引子A₄消失, 并在其附近出现两个新的周期3吸引子A₄和A₇, 如图9(c)和图9(d). 原吸引域B₄分裂为新吸引域B₄和B₇, 且参数变化前后域边界均呈现Wada特性, 域边界结构更加复杂, 系统再次发生Wada-Wada域边界突变.

$图 9 系统(7)的全局图　(a) g = 0.0478; (b) g = 0.0479; (c), (d) 分别对应于(a), (b)图的区域细化\r\nFig. 9. Global diagram of the system (7): (a) g = 0.0478; (b) g = 0.0479; (c), (d) the region refinement of panels (a) and (b).$

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).

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当g = 0.0533时, 状态空间中有4个周期1吸引子共存, 此时域边界仍具有Wada特性. 当g增大到0.0534时, 周期吸引子A₃和A₄消失, 状态空间中仅剩2个周期1吸引子A₁, A₂, 此时域边界的Wada特性消失, 只呈现分形特性, 系统发生Wada-分形域边界突变, 如图10所示.

$图 10 系统(7)的全局图　(a) g = 0.0533; (b) g = 0.0534\r\nFig. 10. Global diagram of the system (7): (a) g = 0.0533; (b) g = 0.0534.$

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.

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4. 温度对全局特性的影响

温度作为SMA薄板的一个可控参数, 利用温度改变SMA薄板的力学特性在工程领域中有重要运用. 取g为0.06, 分析温度θ变化对系统的影响, 选取不同的初值点(–0.0553, –0.09), (–0.031, 0.045), (0.083, –0.032), (0.047, 0.059)构造Poincare映射, 得到对应的分岔图, 如图11所示. 可以发现, 随温度θ的变化, 系统中吸引子个数、类型及大小会发生改变, 并出现激变现象. 为从全局角度分析温度变化对系统激变的影响, 本节基于复合胞坐标系方法, 将区域D = {(x, y)| –0.1 ≤ x ≤ 0.1, –0.12 ≤ y ≤ 0.12}, 均匀划分为1000 × 1200个胞, 获得系统的吸引子、吸引域、鞍和域边界等全局特性.

$图 11 系统(7)随温度θ变化的多值分岔图\r\nFig. 11. Multivalued bifurcation diagram of the system (7) with the variation of temperature θ.$

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.

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4.1 边界激变

当θ = 0.8379时, 状态空间中两个周期1吸引子A₁和A₂共存, S为嵌入在分形域边界上的混沌鞍, IS₁, IS₂是吸引域B₁, B₂内部的混沌鞍. 当θ增大为0.8380时, 系统出现两个新的吸引子A₃和A₄, 内部鞍IS₁, IS₂消失, 系统发生两次逆边界激变, 如图12所示.

$图 12 系统(7)的全局图　(a) θ = 0.8379; (b) θ = 0.8380\r\nFig. 12. Global diagram of the system (7): (a) θ = 0.8379; (b) θ = 0.08380.$

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.

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4.2 合并激变

当θ从0.4088变化到0.4089时, 周期2吸引子A₁和A₂同时与域边界上的混沌鞍S发生碰撞, 合并为一个新的混沌吸引子A₁, 吸引域B₁和B₂合并成为新吸引域B₁, 系统发生合并激变, 如图13(a)和图13(b)所示. 当θ从0.7182增大到0.7183时, 混沌吸引子A₁消失, 出现两个新的混沌吸引子A₁和A₂, 以及嵌在域边界上的混沌鞍S, 系统发生逆的合并激变, 如图13(c)和图13(d)所示.

$图 13 系统(7)的全局图　(a) θ = 0.4088; (b) θ = 0.4089; (c) θ = 0.7182; (d) θ = 0.7183\r\nFig. 13. Global diagram of the system (7): (a) θ = 0.4088; (b) θ = 0.4089; (c) θ = 0.7182; (d) θ = 0.7183.$

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.

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4.3 域边界突变

当g = 0.0950时, 状态空间中两个周期2吸引子A₁, A₂, 和两个周期1吸引子A₃, A₄共存, 吸引域边界呈现Wada特性. 当g增大为0.0951时, 两个周期1吸引子A₃和A₄消失, 吸引域边界变为由B₁和B₂构成的分形边界, 系统发生Wada-分形域边界突变, 如图14所示.

$图 14 系统(7)的全局图　(a) θ = 0.0950; (b) θ = 0.0951\r\nFig. 14. Global diagram of the system (7): (a) θ = 0.0950; (b) θ = 0.0951.$

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.

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5. 结　论

考虑实际工程中温度和应力对于SMA的力学特性的影响, 本文选取SMA薄板为研究对象, 以温度θ和激振力振幅g作为分岔参数, 采用复合胞坐标系方法分析其全局分岔特性, 探究在参数的连续变化下, 系统激变现象及域边界突变的演化过程.

在一定的参数变化范围内, 系统呈现丰富的激变现象, 如周期或混沌吸引子与域边界上的周期鞍或混沌鞍发生碰撞的边界激变, 周期吸引子或混沌吸引子同时与边界上的混沌鞍发生碰撞的合并激变等. 当多吸引子共存时, 域边界会呈现分形结构, 并随着参数的变化, 发生Wada-Wada, Wada-分形和分形-Wada等域边界突变现象. 本文的研究结果对于通过控制温度和激励强度等参数, 调控SMA薄板系统的动态响应, 优化机械设备的变形及振动控制等问题上提供有效的分析手段.

References

[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

图 1 SMA薄板的横向振动

Figure 1. Transverse vibration of SMA thin plate.

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图 2 系统(7)随振幅g变化的多值分岔图

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

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

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

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

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

DownLoad: Full-Size Img PowerPoint

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

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

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

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

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

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

Figure 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).

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

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

DownLoad: Full-Size Img PowerPoint

图 11 系统(7)随温度θ变化的多值分岔图

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

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

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

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

DownLoad: Full-Size Img PowerPoint

[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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Vol. 68, Issue 18 - 2019-09-20

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