基于几何积分方法的压电岛-桥结构动力行为分析

鲁康，王博，毕皓皓，师岩，邓子辰; Lu Kang; Wang Bo; Bi Haohao; Shi Yan; Deng Zichen

引 en

基于几何积分方法的压电岛-桥结构动力行为分析*

鲁康¹
机构：
1. 西北工业大学工程力学系,西安 710072
×
，王博^1,4
机构：
1. 西北工业大学工程力学系,西安 710072
4. 大连理工大学工业装备结构分析国家重点实验室,大连 116024
×
，毕皓皓²
机构：
2. 西北工业大学应用数学系,西安 710072
×
，师岩³
机构：
3. 南京航空航天大学机械结构力学及控制国家重点实验室,南京 210016
×
，邓子辰^1,5
机构：
1. 西北工业大学工程力学系,西安 710072
5. 西北工业大学复杂系统动力学与控制工信部重点实验室,西安 710072
×

1. 西北工业大学工程力学系,西安 710072；
2. 西北工业大学应用数学系,西安 710072；
3. 南京航空航天大学机械结构力学及控制国家重点实验室,南京 210016；
4. 大连理工大学工业装备结构分析国家重点实验室,大连 116024；
5. 西北工业大学复杂系统动力学与控制工信部重点实验室,西安 710072；

DYNAMIC ANALYSIS OF PIEZOELECTRIC ISLAND-BRIDGE STRUCTURE BASED ON GEOMETRIC INTEGRAL METHOD*

Lu Kang¹
Affiliation：
1. Department of Engineering Mechanics,Northwestern Polytechnical University,Xi’an 710072 ,China
×
，Wang Bo^1,4
Affiliation：
1. Department of Engineering Mechanics,Northwestern Polytechnical University,Xi’an 710072 ,China
4. State key Laboratory of Industrial equipment structure Analysis,Dalian University of Technology, Dalian 116024 ,China
×
，Bi Haohao²
Affiliation：
2. Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an 710072 ,China
×
，Shi Yan³
Affiliation：
3. State Key Laboratory of Mechanics and Control for Mechanical Structures,Nanjing University,of Aeronautics and Astronautics,Nanjing 210016 ,China
×
，Deng Zichen^1,5
Affiliation：
1. Department of Engineering Mechanics,Northwestern Polytechnical University,Xi’an 710072 ,China
5. Ministry of Industry and Information Technology Key Laboratory of Dynamics and Control of Complex Systems,Northwestern Polytechnical University,Xi’an 710072 ,China
×

1. Department of Engineering Mechanics,Northwestern Polytechnical University,Xi’an 710072 ,China；
2. Department of Applied Mathematics,Northwestern Polytechnical University,Xi’an 710072 ,China；
3. State Key Laboratory of Mechanics and Control for Mechanical Structures,Nanjing University,of Aeronautics and Astronautics,Nanjing 210016 ,China；
4. State key Laboratory of Industrial equipment structure Analysis,Dalian University of Technology, Dalian 116024 ,China；
5. Ministry of Industry and Information Technology Key Laboratory of Dynamics and Control of Complex Systems,Northwestern Polytechnical University,Xi’an 710072 ,China；

通讯作者:

王博,E-mail:bowang@nwpu.edu.cn;

邓子辰,dweifan@nwpu.edu.cn

中图分类号:O175;T/V384

文献标识码:A

文章编号:1672-6553-2022-20(3)-015-10

DOI:10.6052/1672-6553-2021-034

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目录contents

摘要 Abstract
关键词 Keywords
1 压电屈曲薄膜的动力学模型
2 近似求解
3 Magnus几何积分方法
4 数值算例
4.1 几何积分方法的有效性验证
4.2 岛-桥结构动力学响应
5 结论
参考文献

摘要

基于岛-桥结构的柔性电子器件因其优越的延展性,在可穿戴皮肤电子和航空航天等领域展现出巨大的应用潜力.压电材料因其良好的电学和力学性能成为制造岛-桥结构蛇形互联导线的理想材料.然而,岛-桥结构的电子器件在工作过程中极易受温度激励等复杂物理场影响,产生非线性振动,会严重影响电子器件的稳定性和灵敏度.故而本文研究压电岛-桥结构的非线性动力学响应.首先,基于Timoshenko梁理论,建立两端固支的压电岛-桥结构偏微分振动控制方程;其次,利用Galerkin截断法,得到形式较为简单的常微分控制方程,并利用几何积分方法数值求解该常微分方程;最后,通过数值实验验证几何积分方法的有效性和优越性,讨论温度变化量和电压等参数对岛-桥结构动力学响应的影响规律.本文研究结果将为基于压电岛-桥结构的柔性电子器件的设计提供理论参考.

Abstract

Due to its excellent ductility, the flexible electronic device, which is based on the island-bridge structure, has wide applications in wearable skin electronics and aerospace. Lead zirconate titanate (PZT) is an ideal material to fabricate the island-bridge structure for flexible electronic devices, because of its excellent electrical and mechanical properties. However, those kinds of devices would work in a complex environment, and their stabilities would be influenced by the complex environment. Therefore, dynamic behaviors of the buckled island-bridge structure will be investigated in this paper. Firstly, based on the theory of Timoshenko beam, the governing equation of the buckled piezoelectric island-bridge structure with fixed ends is derived. Secondly, the Galerkin method is utilized to transform the partial differential equation into an ordinary differential equation. Thirdly, the geometric integral method is used to solve the corresponding dynamic equations. Finally, through several numerical examples, effectiveness and advantages of the geometric integral method are verified, and influences of temperature change and voltage on the dynamic response of the buckled island-bridge structure are discussed. The results of this paper would be useful to guide design of piezoelectric island-bridge-based flexible electronics.

关键词

岛-桥结构； Timoshenko梁；几何积分方法；动态力学响应

Keywords

island-bridge structure ； timoshenko beam ； geometric integration method ； dynamic response

引言
柔性电子器件以其独特的延展性在生物医疗、电子设备和航空航天等领域展现出巨大的应用潜力^[1].近年来,柔性电子技术发展迅速,衍生出的基于力学设计的延展柔性结构设计技术^[2,3],既能保持无机电子器件“硬而脆”的电学性能,又使其具备一定的延展性.在可延展柔性结构设计方法中,一种基于岛-桥结构的设计方法能够提升柔性电子器件的延展性至少100%,因而得到学术和工业界的广泛关注^[4,5].
岛-桥结构又称非共面网格结构^[6],由刚性功能部件(“岛”)和互联导线(“桥”)共同组成.通常,“岛”和“桥”都粘接在柔性基底上,通过释放基底的预拉伸,“桥”发生面外屈曲,使得刚性功能模块保持在低应变状态,从而实现岛-桥结构的可延展性,图1给出了扫描电子显微镜拍摄的岛-桥阵列和结构示意图^[7].针对屈曲岛-桥结构,Zhang等^[8]对其屈曲行为展开了研究,他们给出了岛-桥结构屈曲的临界应变解析表达式,并对互联导线的设计进行了优化,提升了其拉伸延展率.压电材料具有优良的力学和电学性能,是作为岛-桥结构互联导线的理想材料^[9].Huang^[10]团队基于压电材料的蛇形/自相似岛-桥结构,制备出高拉伸性、高稳定性、高灵敏度的类人工皮肤.基于压电薄膜结构,Lin^[11]等设计出屈曲梁阵列结构的能量采集器.然而,对于屈曲岛-桥结构的柔性电子器件,无论是在制备过程中^[12],还是在工作过程中^[13],都不可避免地受到外界复杂的温度场、电场等物理场的作用,影响其灵敏性^[14].近年来,该类问题得到国内外学者们的广泛关注,郭伟国等^[15]研究了温度变化对压电薄膜的压电系数的影响,理论和实验表明:压电薄膜对温度非常敏感.申胜平等^[16]研究了温度对压电材料中挠曲电效应的影响,实验表明:温度对压电材料挠曲系数产生非线性影响.李世荣等^[17]考虑了温度及电场的耦合作用,研究了压电层合梁的屈曲及后屈曲行为,他们发现:在屈曲和后屈曲阶段,温度对压电梁的低阶固有频率有着显著而不同的影响.Shariyat^[18]研究了压电层合结构在温度场、电场以及变形场耦合作用下的动态屈曲行为,结果表明:温度梯度对结构的动态屈曲有显著的影响.
图1 岛-桥结构示意图
Fig.1 Diagram of island-bridge structure
对于岛-桥结构的力学行为研究,一般将作为互联导线的压电薄膜等效成Euler-Bernoulli梁来研究.彭剑等^[19]针对轴向周期载荷激励下的压电Euler梁,采用多尺度分析方法给出了压电梁的振动稳定域.Nayfeh等^[20]基于Euler-Bernoulli梁理论研究了各种边界条件下梁的非线性后屈曲问题,并得到了相应的临界屈曲载荷.Wang和Feng等^[21]将屈曲薄膜等效为Euler梁,研究了表面残余应力对屈曲薄膜静态和动态行为的影响.当梁是“短粗梁”时,梁的横截面剪切变形和转动惯量会对其动力学特性产生一定的影响.盛冬发等^[22]采用Timoshenko梁理论,分析了具有非线性损伤的粘弹性梁的动力学响应和混沌现象.杨晓东等^[23]针对两端铰支的Timoshenko梁,对其横向振动问题展开了研究,给出了其固有频率表达式.周强等^[24]基于Timoshenko梁理论,针对压电纳米梁,研究了表面效应对梁振动响应的影响.
本文针对屈曲压电岛-桥结构,采用Timoshenko梁理论建立其振动控制方程.并利用几何积分方法求解相应的动力学方程.分析了温度变化和工作电压对屈曲压电岛-桥结构动力学特性的影响.
1 压电屈曲薄膜的动力学模型
在岛-桥结构柔性电子器件中,屈曲的压电梁结构被用来连接功能模块,如图1(b)所示.两端固支的压电Timoshenko梁如图2(a)所示,其中L为屈曲后长度,b为宽度,h为厚度,ρ为密度,I为面积惯性矩,A为横截面面积.
图2 岛-桥结构力学分析模型
Fig.2 Mechanical analysis model of island-bridge structure
鉴于屈曲压电梁结构将发生大变形,梁的横截面剪切变形和转动惯量将会对其力学行为产生一定的影响^[26].基于Timoshenko梁理论,压电梁的本构方程为^[27,28],
$σ_{x} = c_{11} ε_{x} - e_{31} E_{z} - c_{11} α Δ T$ (1a)
$σ_{x z} = c_{55} γ_{x z}$ (1b)
$D_{z} = e_{31} ε_{x} + k_{33} E_{z} + p_{3} Δ T$ (1c)
其中,σ_x,σ_xz和D_z分别为轴向应力、剪切应力和z方向上的电位移;c₁₁,e₃₁,k₃₃和p₃分别表示弹性模量、压电常数、介电常数以及热电常数;c₅₅=κG为修正剪切模量,E_z为z方向上的电场强度,α为轴向热膨胀系数,ΔT为温度变化量.ε_x为轴向应变,γ_xz为剪切应变,其表达式如下:
$ε_{x} = \frac{\partial u}{\partial x} = - z \frac{\partial φ}{\partial x}$ (2a)
$γ_{x z} = \frac{\partial w}{\partial x} + \frac{\partial u}{\partial z} = \frac{\partial w}{\partial x} - φ$ (2b)
其中,u,w和φ分别表示梁的径向挠度,横向挠度和横截面转角.
由于电场的作用只在z方向上产生应变,故假设只在z方向存在电场,则电场E_z可以由电势Φ确定^[28]

E_{z} = - \frac{\partial Φ}{\partial z}

(3)

不考虑自由电荷,z方向电位移D_z满足高斯定律

\frac{\partial D_{z}}{\partial z} = 0

(4)

将几何方程(2a)和电势(3)代入本构方程(1c),将结果再代入式(4),得到

Φ = - \frac{1}{2} z^{2} \frac{e_{31}}{k_{33}} \frac{\partial φ}{\partial x} + z f_{1} (x, y) + f_{2} (x, y)

(5)

假设在梁的上表面施加电势为V,下表面电势为零,即电势的边界条件为

Φ (\frac{h}{2}) = V, Φ (- \frac{h}{2}) = 0

(6)

对于方程(5),利用边界条件(6),电场可以表示为

Φ = \frac{1}{2} \frac{e_{31}}{k_{33}} (- z^{2} + \frac{h^{2}}{4}) \frac{\partial φ}{\partial x} + z \frac{V}{h} + \frac{V}{2}

(7)

再将电场表达式(7)代入式(3),可以得到电势表达式为

E_{z} = z \frac{e_{31}}{k_{33}} \frac{\partial φ}{\partial x} - \frac{V}{h}

(8)

通过公式(1a)和式(8)可以得到

σ_{x} = - z (c_{11} + \frac{e_{31}^{2}}{k_{33}}) \frac{\partial φ}{\partial x} + e_{31} \frac{V}{h} - c_{11} α Δ T

(9)

对于Timoshenko梁,定义其弯矩M、剪力Q和轴力N分别为,
$M = \int_{A} σ_{x} z d A$ (10a)
$Q = \int_{A} σ_{x z} d A$ (10b)
$N = \int_{A} σ_{x} d A$ (10c)
将公式(9)代入公式(10)可以得到
$M = (E I)^{*} \frac{\partial φ}{\partial x}$ (11a)
$Q = b h_{κ} G (\frac{\partial w}{\partial x} - φ)$ (11b)
$N = b (V e_{31} - h c_{11} α Δ T)$ (11c)
其中, $(E I)^{*} = b h^{3} (c_{11} + p_{31}) / 12$ 为等效刚度,N为含有压电参数的膜力.
根据力平衡方法,如图2(b)所示,可以得到压电Timoshenko梁的非线性振动控制方程为
$κ G A \frac{\partial}{\partial x} (\frac{\partial w}{\partial x} - φ) + (N + H) \frac{\partial φ}{\partial x} = ρ A \frac{\partial^{2} w}{\partial t^{2}}$ (12a)
$(E I) * \frac{\partial^{2} φ}{\partial x^{2}} + κ G A (\frac{\partial w}{\partial x} - φ) = ρ I \frac{\partial^{2} φ}{\partial t^{2}}$ (12b)
其中, $H = E A \int_{0}^{L} (\partial w / \partial x)^{2} d x / 2 L$ 是由中性层拉伸引起的附加轴向力^[29],它是产生几何非线性的重要因素.
根据文献^[24],对于等截面压电梁,由方程(12a)求出 $\partial φ / \partial x$ 后,将其代入方程(12b),则可以得到一个含高阶项的单变量控制方程

\begin{matrix} \frac{\partial^{4} w}{\partial x^{4}} - \frac{N + H}{(E I)^{*}} \frac{\partial^{2} w}{\partial x^{2}} + \frac{ρ A}{(E I)^{*}} \frac{\partial^{2} w}{\partial t^{2}} - \\ [\frac{ρ I}{(E I)^{*}} + \frac{ρ}{κ G}] \frac{\partial^{4} w}{\partial x^{2} \partial t^{2}} + \frac{ρ^{2} I}{κ G (E I)^{*}} \frac{\partial^{4} w}{\partial t^{4}} = 0 \end{matrix}

(13)

2 近似求解
对于偏微分方程(13)无法得到其解析解,Galerkin截断法作为一种简便有效的近似方法,可以将偏微分方程离散为常微分方程.本文采用Galerkin方法^[23]分析屈曲压电岛-桥结构的动力学行为.
对于图2描述的压电梁,两端固支边界条件表述为

w (0) = \frac{\partial w}{\partial x} (0) = w (L) = \frac{\partial w}{\partial x} (L) = 0

(14)

如实验发现^[30],对于岛-桥结构的柔性电子器件,其模态函数为一阶余弦函数形式,故而压电梁的位移可以定义为

w (x, t) = W (t) c o s (\frac{2 π x}{L})

(15)

其中, $W (t)$ 为横振幅.
将方程(15)代入方程(13),并给所得结果乘以cos(2πx/L),最后再在[0,L]上积分;于是压电Timoshenko梁的常微分振动控制方程可以描述为

W^{''''} + a_{1} \ddot{W} + a_{2} W + a_{3} W^{3} = 0

(16)

其中, $W^{''''}$ 表示 $W (t)$ 对时间的四阶导数, $\ddot{W}$ 表示 $W (t)$ 对时间的二阶导数,各个系数表达式如下,
$a_{1} = \frac{κ G A}{ρ I} + {(\frac{2 π}{L})}^{2} [\frac{κ G}{ρ} + \frac{(E I)^{*}}{ρ I}]$ (17a)
$a_{2} = \frac{κ G (E I)^{*}}{ρ^{2} I} {(\frac{2 π}{L})}^{4} + \frac{N_{κ} G}{ρ^{2} I} {(\frac{2 π}{L})}^{2} = ω_{0}^{2}$ (17b)
$a_{3} = \frac{κ G E A}{4 ρ^{2} I} {(\frac{2 π}{L})}^{4}$ (17c)
为方便起见,引入如下形式的无量纲参数

\begin{matrix} \bar{t} = \frac{t}{t_{0}}, ξ = \frac{x}{L}, \hat{W} = \frac{W}{L}, β = \frac{ω_{0}^{2}}{a_{1}^{2}} \\ γ = \frac{a_{3} L^{2}}{ω_{0}^{2}}, t_{0} = \frac{\sqrt{a_{1}}}{ω_{0}} \end{matrix}

(18)

将式(18)代入方程(16),可以得到无量纲方程

{\hat{W}}^{''''} + \frac{1}{β} \ddot{\hat{W}} + \frac{1}{β} \hat{W} + \frac{γ}{β} {\hat{W}}^{3} = 0

(19)

为了便于分析,取状态变量 $x_{1} = \hat{W}, x_{2} = \dot{\hat{W}}, x_{3} = \ddot{\hat{W}}, x_{4} = {\hat{W}}^{''''}$ ,则四阶常微分方程(19)可等效成一阶常微分方程组,

\{\begin{matrix} \dot{x_{1}} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = x_{4} \\ {\dot{x}}_{4} = - \frac{1}{β} x_{3} - \frac{1}{β} x_{1} - \frac{γ}{β} x_{1}^{3} \end{matrix}

(20)

令 $x = [x_{1}, x_{2}, x_{3}, x_{4}]$ ,则方程(20)可以表示为如下的矩阵微分方程形式

\dot{x} = D (x) x

(21)

其中, $D (x) = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - \frac{1}{β} - \frac{γ}{β} x_{1}^{2} & 0 & - \frac{1}{β} & 0 \end{matrix}]$ .
3 Magnus几何积分方法
对于非线性方程(21),无法得到解析解,只能通过数值方法进行求解.本文考虑应用Magnus几何积分方法求解非线性动力学控制方程(21).Magnus方法几何积分方法在数值计算方面具有显著的优点:即使适当截断,数值结果依然可以保持原动力学问题的几何性质^[31].
方程(21)的解可以局部地表示为

x (t) = e x p (Ω (t)) x_{0}

(22)

其中, $Ω (t) = \sum_{k = 1}^{\infty} Ω_{k} (t)$ ,称为Magnus级数.它的前两项表示为^[31]

\begin{matrix} Ω_{1} = \int_{t_{0}}^{t} D (x, s) d s \\ Ω_{2} = \frac{1}{2} \int_{t_{0}}^{t} [D (x, s_{1}), D (x, s_{2})] d s_{1} \end{matrix}

(23)

取步长为τ,则 $t_{k + 1} = t_{k} + τ$ ,在一个时间步长[ $t_{k}$ , $t_{k + 1}$ ]内讨论问题,采用四阶Guass-Legendre积分公式计算上式,可以得到方程(21)的一个Magnus四阶近似格式^[32]

\{\begin{matrix} D_{1} = D (t_{k} + c_{1} τ, x (t_{k} + c_{1} τ)) \\ D_{2} = D (t_{k} + c_{2} τ, x (t_{k} + c_{2} τ)) \\ \hat{Ω} = \frac{τ}{2} (D_{1} + D_{2}) + \frac{\sqrt{3}}{12} τ^{2} (D_{1} D_{2} - D_{2} D_{1}) \\ x_{k + 1} = e x p (\hat{Ω}) x_{k} \end{matrix}

(24)

其中, $c_{1} = \frac{1}{2} - \frac{\sqrt{3}}{6}, c_{2} = \frac{1}{2} + \frac{\sqrt{3}}{6} .$
4 数值算例
4.1 几何积分方法的有效性验证
对于压电岛-桥结构,由方程(13)不难发现其将做无阻尼的自由振动,故而压电岛-桥结构的总能量应该守恒.本文将分别采用四阶Magnus方法和四阶Runge-Kutta方法计算岛-桥结构Hamilton函数.压电岛-桥结构的材料参数和几何参数在表1中给出.
表1 材料参数和几何参数^[33]
Table1 Material parameters and geometric parameters^[33]
由无量纲控制方程(20),压电岛-桥结构的Hamilton函数可以表示为

\begin{matrix} H (\bar{t}) = T + V = \\ \frac{1}{2} x_{2}^{2} + \frac{1}{β} (\frac{1}{2} x_{1}^{2} + \frac{1}{2} x_{3}^{2} + \frac{1}{4} γ x_{1}^{4}) \end{matrix}

(25)

定义Hamilton函数相对误差为

Δ H (\bar{t}) = \frac{|H (\bar{t}) - E_{0}|}{E_{0}}

(26)

其中,E₀为系统初始能量.
图3分别采用4阶Runge-Kutta方法和4阶Magnus几何积分方法计算了压电岛-桥结构Hamilton函数的相对误差图,其初始条件为x₁=0,x₂=0.1,x₃=0,x₄=0,步长取0.5.由图3结果不难发现:采用4阶Magnus几何积分方法得到的Hamilton函数相对误差一直在零附近震荡;而利用4阶Runge-Kutta方法得到Hamilton函数相对误差随着积分时间的增加而不断增长.对于保守系统(21),其能量应该守恒,图3的结果验证了几何积分方法保能量的优越性.
为了进一步验证几何积分方法的有效性,图4给出了岛-桥结构的时间历程曲线.由图4数值结果可以发现:利用几何积分方法得到的位移响应幅值x₁能够长时间保持解的稳定性,可以认为该方法在较大积分范围内不会产生算法阻尼和数值耗散^[31];然而,经典的四阶Runge-Kutta方法自身存在算法阻尼,在迭代过程中,所得位移响应幅值x₁经过数值耗散不断减小.图3和图4显示了Magnus几何积分方法在保守系统非线性数值计算方面的优越性.通过该方法得到的Hamilton函数相对误差一直在零附近震荡,而通过Runge-Kutta方法得到Hamilton函数相对误差随着积分时间的增加而不断增长,在算法阻尼的作用下,误差持续积累,最终必然会超过数值计算的误差限.
图3 Hamilton函数相对误差
Fig.3 The relative error of Hamilton's function
图4 位移时间历程图
Fig.4 Displacement time history diagram of the system
对于保守系统(21)的另一个重要性质是:在时间演化过程中,轨线在相空间中围成的体积不变.为了验证几何积分方法的保结构特性,分别采用四阶几何积分方法和经典的四阶Runge-Kutta方法给出了系统(21)的相图,如图5所示.其中,时间步长分别取τ=0.1s和τ=0.5s.
观察图5,从中可以发现:当步长增加时,采用几何积分方法得到的系统相空间面积保持不变,而采用Rung-Kutta方法得到的相空间会因为步长的增加,导致相空间面积增加.
由图3-图5可以说明几何积分方法能够保持保守系统的能量和结构.因此,在下一节的动力学分析中,考虑采用几何积分方法对方程(21)进行求解.
图5 不同时间步长下的相图
Fig.5 Phase diagram at different time steps
4.2 岛-桥结构动力学响应
对于岛-桥结构,由振动理论可知,其无量纲固有频率为

ω_{n} = \sqrt{\frac{1}{β}}

(27)

取压电薄膜的材料为PTZ-5H,仍取表1中的材料参数和几何参数.取初值x₁=0,x₂=0.1,x₃=0,x₄=0.
为了讨论温度场和电压对压电岛-桥结构动态力学行为的影响,图6和图7给出了其对应的频谱曲线.通过数值仿真,图6和图7给出了基于压电薄膜的岛-桥结构互联导线自由振动时固有频率与温度场和膜间工作电压之间的变化曲线.
图6的数值结果讨论了温度变化量对屈曲压电结构固有频率的影响规律.从图6可以发现:当取定电压,随着温度变化量的增加,屈曲压电结构的固有频率随之增加,同时还发现其增长趋势呈线性增加趋势.对于这一现象,可以由方程(11c)进行解释:原因在于压电参数和温度参数控制的膜力N呈现线性关系.
图6 温度对压电薄膜固有频率的影响
Fig.6 Effect of temperature on natural frequency of piezoelectric films
图7讨论了电压改变量对屈曲压电结构固有频率的影响规律.由图7的数值结果可以发现:随着电压的增加,系统固有频率呈线性增加,这一点仍然可以由式(11c)来解释.
图7 电压对压电薄膜固有频率的影响
Fig.7 Effect of voltage on natural frequency of piezoelectric films
由图6和图7的结果,从中可以得到这样的结论:随着温度变化量和电压的增加,屈曲压电岛-桥结构的固有频率会随之增加,即说明了温度变化量和电压会使结构变“硬”.如果让岛-桥结构工作于过高的温度场和电场中,会导致该类型电子器件失稳的风险增加.
图8和图9给出了温度变化量和电压对屈曲压电岛-桥结构振动特性的影响.由图8结果可以发现:当工作电压均匀增加时,岛-桥结构振幅的峰值呈现递增趋势,而振动周期几乎没有变化,说明岛-桥结构电压对结构振动呈现正激励作用.观察图8(b),可以进一步发现:当工作电压固定时,随着温度改变量的增加,岛-桥结构的振幅会随之增加;当系统温度变化量较少时,岛-桥结构的振动幅值呈现“波动”式变化,原因在于温度场对该岛-桥结构的影响被等效成了轴向力.由方程(11)可以知道:温度变化量较小时,轴向力对原系统的影响较小,系统将会按其原固有频率振动.
图8 电压对压电薄膜振幅的影响
Fig.8 The effect of voltage on the amplitude of piezoelectric film
图9 温度对压电薄膜振幅的影响
Fig.9 The effect of temperature on the amplitude of piezoelectric film
从图9(a)可以发现:随着温度变化量的增加,屈曲岛-桥结构振幅会随之增加;同时,岛-桥结构振动周期也发生了变化,其规律为:当温度变化量增加300K,结构振动周期将减小大约1/8,这进一步验证了图8(b)中所展示的现象—温度场能够改变岛-桥结构振动频率.由图9(b)的数值结果可以发现:当固定温度变化量时,岛-桥结构振幅会随着工作电压的升高而增加.通过对比图8(b)和图9(b),可以得到这样的规律:相比于工作电压,温度变化量对岛-桥结构振动特性的影响更大;另外,屈曲压电岛-桥结构工作于较低温度场中时,其振动呈现低幅波动振荡,而对于较高温度场作用时,结构的振动明显快速增加.因此,对于通过调整“硬”薄膜结构的振幅来实现延展性的岛-桥式柔性电子器件来说,可以通过改变工作温度和电压来提高该类电子器件的延展率及稳定性.
5 结论
针对屈曲压电岛-桥结构,本文将其等效成两端固支的梁结构.鉴于岛-桥结构的柔性电子器件会发生大变形,本文基于Timoshenko梁理论,建立屈曲压电岛-桥结构的动力学控制方程;采用Galerkin方法对原方程进行近似求解,再通过引入状态变量,将四阶常微分方程降阶为一阶常微分方程组,然后采用几何积分方法进行求解,通过数值实验,讨论了温度变化量和工作电压对屈曲岛-桥结构振动特性的影响,得到如下结论:
(1)对于保守系统,几何积分方法展现出优异的保结构,保能量特性;
(2)通过改变温度变化量和工作电压,能够改变岛-桥结构的刚度,同时,随着温度变化量和电压的增加,屈曲结构固有频率随之增加,即温度和工作电压的升高会使岛-桥结构变“硬”;
(3)在相同工作电压下,随着温度变化量的增加,屈曲结构振幅随之增加,当取定工作温度,岛-桥结构的振幅也会随电压的升高而增加.另外,相比于工作电压,屈曲压电岛-桥结构更容易受温度场的影响.
本文所得结论将会为基于压电岛-桥结构的柔性电子器件设计提供理论支撑,同时,通过改变温度场和电场,能够在一定程度上改变基于压电岛-桥结构的柔性电子器件的延展性和稳定性.
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- [16] 卢建锋,梁旭,申胜平.温度对PVDF材料中挠曲电效应的影响研究.中国力学大会论文集(CCTAM 2019),中国力学大会(CCTAM 2019),杭州,2019年8月25日.杭州:浙江大学出版社,2019:13;(Lu J F,Liang X,Shen S P.Influence of temperature on flexoelectric effect in PVDF.In:Proceedings of China Mechanics Conference(CCTAM 2019),China Mechanics Conference(CCTAM 2019),Hangzhou,2019-8-25.Hangzhou:Zhejiang University press,2019:13(in Chinese))
- [17] 李世荣,苏厚德,程昌钧.热环境中粘贴压电层功能梯度材料梁的自由振动.应用数学和力学,2009,30(8):907~918;(Li S R,Su H D,Cheng C J.Free vibration of functionally graded material beams bonded with piezoelectric layers in thermal environment.Applied Mathematics and Mechanics,2009,30(8):907~918(in Chinese))
- [18] Shariyat M.Dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical shells with temperature-dependent material properties under thermo-electro-mechanical loads.International Journal of Mechanical Sciences,2008,50(12):1561~1571
- [19] 彭剑,李禄欣,马建军.时滞反馈作用下压电梁的参数共振分析.动力学与控制学报,2016,14(5):412~416;(Peng J,Li L X,Ma J J.Parametric resonance of piezoelectric beams with time-delayed feedback.Journal of Dynamics and Control,2016,14(5):412~416(in Chinese))
- [20] Nayfeh A H,Emam S A.Exact solution and stability of postbuckling configurations of beams.Nonlinear Dynamics,2008,54(4):395~408
- [21] Wang Y,Feng X,Lu B,et al.Surface effects on the mechanical behavior of buckled thin film.Journal of Applied Mechanics,2013,82(2):021002
- [22] 盛冬发,程昌钧.几何非线性的损伤粘弹性Timoshenko梁的动力学行为.动力学与控制学报,2004,2(4):77~83;(Sheng D F,Cheng C J.Dynamical behaviors of nonlinear viscoelastic timoshenko beams with damage.Journal of Dynamics and Control,2004,2(4):77~83(in Chinese))
- [23] 杨晓东,唐有绮,戈新生.轴向运动Timoshenko梁固有频率的求解方法研究.机械强度,2009,31(2):208~210;(Yang X D,Tang Y Q,Ge X S.Study on the solution method of natural frequency of Timoshenko beam with axial motion.Journal of Mechanical Strength,2009,31(2):208~210(in Chiniese))
- [24] 周强,张志纯,龙志林,等.考虑表面效应的压电纳米梁的振动研究.应用数学和力学,2020,41(8):853~865;(Zhou Q,Zhang Z C,Long Z L,et al.Vibration of piezoelectric nanobeam considering surface effect.Applied Mathematics and Mechanics,2020,41(8):853~865(in Chinese))
- [25] Kim D H,Ahn J H,Choi W M,et al.Stretchable and foldable silicon integrated circuits.Science,2008,320(5875):507~511
- [26] Komijani M,Kiani Y,Esfahani S E,et al.Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams.Composite Structures,2013,98:143~152
- [27] Qian D.Bandgap properties of a piezoelectric phononic crystal nanobeam based on nonlocal theory.Jourmal of Materials Science,2019,54(5):4038~4048
- [28] 尚福林,王子昆,李中华.压电层合板热屈曲问题的精确分析.固体力学学报,1997,18(1):1~10;(Shang F L,Wang Z K,Li Z H.Accurate analysis of thermal buckling of piezoelectric laminates.Chinese Journal of Solid Mechanics,1997,18(1):1~10(in Chinese))
- [29] Wang Y,Feng X.Dynamic behaviors of controllably buckled thin films.Applied Physics Letters,2009,95(23):231915
- [30] Dagdeviren C,Joe P,Tuzman O L,et al.Recent progress in flexible and stretchable piezoelectric devices for mechanical energy harvesting,sensing and actuation.Extreme Mechanics Letters,2016,9:269~281
- [31] 张素英,邓子辰.非线性动力学系统的几何积分理论及应用.西安:西北工业大学出版社,2004;(Zhang S Y,Deng Z C.Geometric integral theory of nonlinear dynamic system and its application.Xi’an:Northwestern Polytechnical University Press,2004(in Chiniese))
- [32] 王博,邓子辰,李文成,等.Hill’s方程的Magnus积分.西北工业大学学报,2011,29(6):988~991;(Wang B,Deng Z C,Li W C,et al.Magnus integral of Hill's equation.Journal of Northwestern Polytechnical University,2011,29(6):988~991(in Chinese))
- [33] Yan Z,Jiang L Y.The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects.Nanotechnology,2011,22(24):245703

基本信息

中图分类号: O175;T/V384
文献标识码: A
DOI: 10.6052/1672-6553-2021-034
文章编号: 1672-6553-2022-20(3)-015-10

基金信息

国家自然科学基金资助项目(11802319)；
机械结构力学及控制国家重点实验室开放课题资助项目(MCMS-E-0221K01)；

引用信息

稿件历史

收稿日期: 2021-04-20

参考文献

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[12] Im M,Lee W H,Kweon S H,et al.New lead-free piezoelectric thin film fabricated using metal-oxide nanosheets at low temperature.Ceramics International,2019,45(17):21773~21780
[13] Kim J H,Kweon S H,Nahm S.Low-temperature crystalline lead-free piezoelectric thin films grown on 2D perovskite nanosheet for flexible electronic device applications.Nano Research,2019,12(10):2559~2567
[14] Wang B L,Noda N,Han J C,et al.Surface thermal shock fracture of a semi-infinite piezoelectric medium(poling axis parallel to the crack plane).Mechanics of Materials,2002,34(3):135~144
[15] 郭伟国,褚世永.不同力和温度对PVDF压电薄膜压电系数的影响.科学技术与工程,2008,8(2):347~352;(Guo W G,Chu S Y.Influence of different force and temperature on piezoelectric coefficient of PVDF piezoelectric film.Science Technology and Engineering,2008,8(2):347~352(in Chinese))
[16] 卢建锋,梁旭,申胜平.温度对PVDF材料中挠曲电效应的影响研究.中国力学大会论文集(CCTAM 2019),中国力学大会(CCTAM 2019),杭州,2019年8月25日.杭州:浙江大学出版社,2019:13;(Lu J F,Liang X,Shen S P.Influence of temperature on flexoelectric effect in PVDF.In:Proceedings of China Mechanics Conference(CCTAM 2019),China Mechanics Conference(CCTAM 2019),Hangzhou,2019-8-25.Hangzhou:Zhejiang University press,2019:13(in Chinese))
[17] 李世荣,苏厚德,程昌钧.热环境中粘贴压电层功能梯度材料梁的自由振动.应用数学和力学,2009,30(8):907~918;(Li S R,Su H D,Cheng C J.Free vibration of functionally graded material beams bonded with piezoelectric layers in thermal environment.Applied Mathematics and Mechanics,2009,30(8):907~918(in Chinese))
[18] Shariyat M.Dynamic buckling of suddenly loaded imperfect hybrid FGM cylindrical shells with temperature-dependent material properties under thermo-electro-mechanical loads.International Journal of Mechanical Sciences,2008,50(12):1561~1571
[19] 彭剑,李禄欣,马建军.时滞反馈作用下压电梁的参数共振分析.动力学与控制学报,2016,14(5):412~416;(Peng J,Li L X,Ma J J.Parametric resonance of piezoelectric beams with time-delayed feedback.Journal of Dynamics and Control,2016,14(5):412~416(in Chinese))
[20] Nayfeh A H,Emam S A.Exact solution and stability of postbuckling configurations of beams.Nonlinear Dynamics,2008,54(4):395~408
[21] Wang Y,Feng X,Lu B,et al.Surface effects on the mechanical behavior of buckled thin film.Journal of Applied Mechanics,2013,82(2):021002
[22] 盛冬发,程昌钧.几何非线性的损伤粘弹性Timoshenko梁的动力学行为.动力学与控制学报,2004,2(4):77~83;(Sheng D F,Cheng C J.Dynamical behaviors of nonlinear viscoelastic timoshenko beams with damage.Journal of Dynamics and Control,2004,2(4):77~83(in Chinese))
[23] 杨晓东,唐有绮,戈新生.轴向运动Timoshenko梁固有频率的求解方法研究.机械强度,2009,31(2):208~210;(Yang X D,Tang Y Q,Ge X S.Study on the solution method of natural frequency of Timoshenko beam with axial motion.Journal of Mechanical Strength,2009,31(2):208~210(in Chiniese))
[24] 周强,张志纯,龙志林,等.考虑表面效应的压电纳米梁的振动研究.应用数学和力学,2020,41(8):853~865;(Zhou Q,Zhang Z C,Long Z L,et al.Vibration of piezoelectric nanobeam considering surface effect.Applied Mathematics and Mechanics,2020,41(8):853~865(in Chinese))
[25] Kim D H,Ahn J H,Choi W M,et al.Stretchable and foldable silicon integrated circuits.Science,2008,320(5875):507~511
[26] Komijani M,Kiani Y,Esfahani S E,et al.Vibration of thermo-electrically post-buckled rectangular functionally graded piezoelectric beams.Composite Structures,2013,98:143~152
[27] Qian D.Bandgap properties of a piezoelectric phononic crystal nanobeam based on nonlocal theory.Jourmal of Materials Science,2019,54(5):4038~4048
[28] 尚福林,王子昆,李中华.压电层合板热屈曲问题的精确分析.固体力学学报,1997,18(1):1~10;(Shang F L,Wang Z K,Li Z H.Accurate analysis of thermal buckling of piezoelectric laminates.Chinese Journal of Solid Mechanics,1997,18(1):1~10(in Chinese))
[29] Wang Y,Feng X.Dynamic behaviors of controllably buckled thin films.Applied Physics Letters,2009,95(23):231915
[30] Dagdeviren C,Joe P,Tuzman O L,et al.Recent progress in flexible and stretchable piezoelectric devices for mechanical energy harvesting,sensing and actuation.Extreme Mechanics Letters,2016,9:269~281
[31] 张素英,邓子辰.非线性动力学系统的几何积分理论及应用.西安:西北工业大学出版社,2004;(Zhang S Y,Deng Z C.Geometric integral theory of nonlinear dynamic system and its application.Xi’an:Northwestern Polytechnical University Press,2004(in Chiniese))
[32] 王博,邓子辰,李文成,等.Hill’s方程的Magnus积分.西北工业大学学报,2011,29(6):988~991;(Wang B,Deng Z C,Li W C,et al.Magnus integral of Hill's equation.Journal of Northwestern Polytechnical University,2011,29(6):988~991(in Chinese))
[33] Yan Z,Jiang L Y.The vibrational and buckling behaviors of piezoelectric nanobeams with surface effects.Nanotechnology,2011,22(24):245703

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基于几何积分方法的压电岛-桥结构动力行为分析*

DYNAMIC ANALYSIS OF PIEZOELECTRIC ISLAND-BRIDGE STRUCTURE BASED ON GEOMETRIC INTEGRAL METHOD*

摘要

Abstract

关键词

Keywords

1 压电屈曲薄膜的动力学模型

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(13)

2 近似求解

(14)

(15)

(16)

(18)

(19)

(20)

(21)

3 Magnus几何积分方法

(22)

(23)

(24)

4 数值算例

4.1 几何积分方法的有效性验证

(25)

(26)

4.2 岛-桥结构动力学响应

(27)

5 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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