轴向可伸缩悬臂复合材料层合板的建模及数值分析

高艳红，张伟; Gao Yanhong; Zhang Wei

引 en

轴向可伸缩悬臂复合材料层合板的建模及数值分析^*

高艳红
机构：
北京工业大学材料与制造学部,北京 100124
×
，张伟
机构：
北京工业大学材料与制造学部,北京 100124
×

北京工业大学材料与制造学部,北京 100124；

Modeling and Numerical Analysis of Telescopic Cantilever Composite Plate^*

Gao Yanhong
Affiliation：
College of Mechanical Engineering, Beijing University of Technology, Beijing 100124 , China
×
，Zhang Wei
Affiliation：
College of Mechanical Engineering, Beijing University of Technology, Beijing 100124 , China
×

College of Mechanical Engineering, Beijing University of Technology, Beijing 100124 , China；

通讯作者:

高艳红,E-mail∶gydxgyh@163.com

中图分类号:O322

文献标识码:A

文章编号:1672-6553-2023-21(2)-041-008

DOI:10.6052/1672-6553-2021-077

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参考文献
出版信息

参考文献 1

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参考文献 2

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参考文献 3

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参考文献 4

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参考文献 5

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参考文献 6

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参考文献 8

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摘要

论文研究了受面内激励和三阶气动力联合作用下的可伸缩悬臂复合材料层合板的非线性振动问题.根据经典层合板理论和Hamilton原理建立可伸缩悬臂复合材料层合板在匀速轴向外伸和回收过程中的非线性动力学偏微分方程,然后采用Galerkin方法将偏微分方程离散成带有时变系数的常微分方程,通过数值方法得到频率变化图、时间历程图和相图,讨论轴向移动速度、宽厚比对可伸缩悬臂复合材料层合板的动力学特性的影响.结果表明,可伸缩悬臂板匀速外伸时,轴向速度越大可能越易出现振幅发散,回收过程未发现振幅发散现象.

Abstract

The nonlinear vibration of the telescopic cantilever composite plate subjected to the in-plane excitation and the third-order aerodynamic force is studied when it is in the deployment and retraction. Based on classical laminated plate theory and Hamilton principle, the nonlinear partial differential equation of the telescopic cantilever plate in the process of deployment and retraction is established. Then, the Galerkin method is used to discrete the nonlinear partial differential equation into the ordinary differential equation with time-varying coefficients. Frequency variation diagrams,time history diagrams and phase diagrams are obtained by numerical methods. The influence of axial velocity, width-to-thickness ratio on the nonlinear dynamic characteristics of the telescopic cantilever composite plate is discussed. The results show that the larger the axial moving speed is, the more likely it is to cause the amplitude to diverge when the telescopic cantilever plate is deploying at a uniform speed. However, amplitude divergence does not occur in the retracting process.

关键词

可伸缩悬臂板；经典层合板理论； Hamilton原理；非线性动力学

Keywords

telescopic cantilever plate ； classical laminated plate theory ； Hamilton principle ； nonlinear dynamics

引言
随着航空航天技术的快速发展，研究者对飞行器等高空作业的结构的关注度普遍提升，如太阳能翻板、空间大型可展天线、可变翼飞行器等.但是复杂的工作环境会导致很多工程问题，轴向可伸缩悬臂结构的非线性振动问题就是其中之一.由于这类结构在外载荷的作用下，轴向伸缩运动可能会导致结构失稳、破坏等，因此对可伸缩悬臂结构的动力学特性研究具有非常重要的工程价值.本文主要研究可伸缩机翼在匀速外伸和回收过程中的动力学问题.
近几年来，很多学者对轴向外伸悬臂梁和轴向移动板的动力学特性的研究产生了浓厚的兴趣.Park^[1]等采用欧拉和拉格朗日方法研究轴向运动梁的动力学特性，相比于拉格朗日描述轴向运动梁的方法，欧拉描述具有简洁高效的特点.Sahoo^[2]等研究了带有时变速度的轴向移动梁的参数共振和内共振特性.Zhu^[3]等研究了具有自旋运动的欧拉-伯努利梁的非线性横向振动，发现拍现象是由自旋轴向移动梁的第一阶和第二阶固有频率相互影响而产生的.Yan^[4]等研究了轴向运动铁木辛柯梁在参数激励和外激励联合作用下的非线性动力特性，并通过数值方法研究了双频激励之间的准周期运动特性.刘燕^[5]等采用数值方法研究了轴向移动速度对可伸缩悬臂矩形梁的非线性动力学特性的影响.段应昌^[6]等通过实验和数值方法对可伸缩悬臂梁的动力学特性进行了研究.Ghayesh等^[7]通过Von Karman板理论和Lagrange方法对轴向受迫简支移动板进行建模，通过数值方法研究了轴向移动速度和面外激励对板的振动特性的影响.吕书锋^[8]等研究了外伸悬臂板在三阶气动力和面内激励联合作用下的非线性动力学特点，研究表明外伸速度和面内激励对板的跳跃行为影响显著.Zhou^[9]等基于Kelvin-Voigt黏弹性模型，应用归一化幂级数法对轴向运动黏弹性板在两种边界条件下的横向振动和动力学稳定性进行了分析，发现外伸时间、轴向运动速度及长宽比对板的动力学行为和稳定性有很大影响.Zhang^[10]等通过Routh-Hurwitz标准以及多尺度方法对带有内共振的轴向移动板的稳定性进行了研究，并通过微分求积法验证解析结果的准确性.Zhang^[11]等将外伸机翼简化为悬臂壳的形式研究机翼在亚音速气流中的非线性动力学行为，讨论了外伸速度对悬臂壳一阶模态和二阶模态的影响.Wang等^[12]采用经典薄板理论、附加虚质量增量因子法和Rayleigh-Ritz法研究浸在无限域的轴向移动矩形板的振动特性.
本文主要研究可伸缩悬臂板在三阶气动力和面内激励联合作用下的非线性时变动力学问题.采用经典层合板理论和Hamilton原理对可伸缩悬臂板进行非线性建模，综合采用Galerkin和数值方法研究轴向移动速度和材料参数对可伸缩悬臂板动力学特性的影响.
1 基本方程
将可伸缩机翼简化为可伸缩悬臂复合材料层合板，其由三层等厚度的石墨/环氧（HT3/QY8911）树脂材料组成，面内激励的表达式为 $f = f_{y} + f_{1} c o s （ Ω t ）$ ，三阶气动力表达式为 $f_{z} ， f_{1} c o s （ Ω t ）$ 是扰动项.板长为 $l （ t ） = l_{0} + l_{1}$ ，板轴向移动速度为 $l_{1} = \int_{0}^{t} v d t$ ，板宽为b，伸缩板厚为h，如图1所示.
根据经典层合板理论^[13]，可伸缩悬臂复合材料层合板的位移场可以写为

u (x, y, t) = u_{0} (x (t), y, t) - z \frac{\partial w_{0}}{\partial x}

(1a)

v (x, y, t) = v_{0} (x (t), y, t) - z \frac{\partial w_{0}}{\partial y}

(1b)

w (x, y, t) = w_{0} (x (t), y, t)

(1c)

其中u₀，v₀，w₀分别表示可伸缩悬臂板中面上任意一点的位移.
图1 可伸缩悬臂复合板材料层合板动力学模型
Fig.1 The model of telescopic cantilever composite plate
位移-应变关系可以写为
$\begin{matrix} ε_{x x} = \frac{\partial u}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}, ε_{y y} = \frac{\partial v}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2}, \\ ε_{x y} = \frac{1}{2} (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} + \frac{\partial w}{\partial x \partial y}), \end{matrix}$

\begin{matrix} ε_{x z} = \frac{1}{2} (\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}), \\ ε_{y z} = \frac{1}{2} (\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}), ε_{z z} = \frac{\partial w}{\partial z} \end{matrix}

(2)

根据Hamilton原理可得非线性动力学方程

N_{x x, x} + N_{x y, y} = I_{0} \frac{d^{2} x}{d t^{2}} + I_{0} {\ddot{u}}_{0} - I_{1} (\frac{\partial {\ddot{w}}_{0}}{\partial x})

(3a)

N_{y y, y} + N_{x y, x} = I_{0} {\ddot{v}}_{0} - I_{1} (\frac{\partial {\ddot{w}}_{0}}{\partial y})

(3b)

\begin{matrix} \frac{\partial}{\partial x} (N_{x x} \frac{\partial w_{0}}{\partial x} + N_{x y} \frac{\partial w_{0}}{\partial y}) + \\ \frac{\partial}{\partial y} (N_{y y} \frac{\partial w_{0}}{\partial y} + N_{x y} \frac{\partial w_{0}}{\partial x}) + \frac{\partial^{2} M_{x x}}{\partial x^{2}} + \\ \frac{\partial^{2} M_{y y}}{\partial y^{2}} + 2 \frac{\partial^{2} M_{x y}}{\partial x \partial y} - γ {\dot{w}}_{0} + f_{z} = \\ I_{0} {\ddot{w}}_{0} + I_{1} (\frac{\partial {\ddot{u}}_{0}}{\partial x} + \frac{\partial {\ddot{v}}_{0}}{\partial y}) - I_{2} (\frac{\partial^{2} {\ddot{w}}_{0}}{\partial x^{2}} + \frac{\partial^{2} {\ddot{w}}_{0}}{\partial y^{2}}) \end{matrix}

(3c)

其中

(4)

根据文献^[8]，三阶气动力的表达式为

f_{z} = - \frac{4 q_{d} λ}{M_{\infty}} (\frac{1}{v} \frac{\partial w_{0}}{\partial x} \frac{d x}{d t} + \frac{\partial w_{0}}{\partial y} + \frac{1}{v} \frac{\partial w_{0}}{\partial t}) + p_{1}

(5a)

p_{1} = \frac{κ + 1}{3} q_{d} λ^{3} M_{\infty} {(\frac{1}{v} \frac{\partial w_{0}}{\partial x} \frac{d x}{d t} + \frac{\partial w_{0}}{\partial y} + \frac{1}{v} \frac{\partial w_{0}}{\partial t})}^{3}

(5b)

其中，q_d=1/2ρ_aυ²代表动压，κ为比热比，ρ_a表示空气密度，υ是来流速度， $M_{\infty}$ 和λ表示马赫数和动力修正因子， $λ = M_{\infty} / \sqrt{M_{\infty}^{2} - 1} .$
可伸缩悬臂板在固定和自由端的边界条件为

当 x = 0, u_{0} = v_{0} = w_{0} = 0

(6a)

当 x = l (t), N_{x x} = N_{x y} = M_{x x} = M_{x y} = 0

(6b)

$当 y = 和 b ， N_{x x} = N_{x y} = M_{x x} = M_{x y} = 0$

{\int_{0}^{h} N_{y y}|}_{y = 0} d z = - \int_{0}^{h} (f_{0} + f_{1} c o s (Ω t)) d z

(6c)

正交铺设复合材料层合板的应变-位移关系为

\{\begin{matrix} ε_{x x} \\ ε_{y y} \\ γ_{x y} \end{matrix}\} = \{\begin{matrix} ε_{x x}^{(0)} \\ ε_{y y}^{(0)} \\ γ_{x y}^{(0)} \end{matrix}\} + z \{\begin{matrix} ε_{x x}^{(1)} \\ ε_{y y}^{(1)} \\ γ_{x y}^{(1)} \end{matrix}\}

(7)

其中

\begin{matrix} \{\begin{matrix} ε_{x x}^{(0)} \\ ε_{y y}^{(0)} \\ γ_{x y}^{(0)} \end{matrix}\} = \{\begin{matrix} \frac{\partial u_{0}}{\partial x} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial x})}^{2} \\ \frac{\partial v_{0}}{\partial y} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial y})}^{2} \\ \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} + \frac{\partial w_{0}}{\partial x} \frac{\partial w_{0}}{\partial y} \end{matrix}\}, \\ \{\begin{matrix} ε_{x x}^{(1)} \\ ε_{y y}^{(1)} \\ γ_{x y}^{(1)} \end{matrix}\} = \{\begin{matrix} - \frac{\partial^{2} w_{0}}{\partial x^{2}} \\ - \frac{\partial^{2} w_{0}}{\partial y^{2}} \\ - 2 \frac{\partial^{2} w_{0}}{\partial x \partial y} \end{matrix}\} . \end{matrix}

(8)

正交各向异性层合板的内力和力矩与应变的关系为

\{\begin{matrix} N_{x} \\ N_{y} \\ N_{x y} \end{matrix}\} = \{\begin{matrix} A_{11} & A_{12} & 0 \\ A_{21} & A_{22} & 0 \\ 0 & 0 & A_{66} \end{matrix}\} \{\begin{matrix} ε_{x}^{(0)} \\ ε_{y}^{(0)} \\ γ_{x y}^{(0)} \end{matrix}\}

(9a)

\{\begin{matrix} M_{x} \\ M_{y} \\ M_{x y} \end{matrix}\} = \{\begin{matrix} D_{11} & D_{12} & 0 \\ D_{21} & D_{22} & 0 \\ 0 & 0 & D_{66} \end{matrix}\} \{\begin{matrix} ε_{x}^{(1)} \\ ε_{y}^{(1)} \\ γ_{x y}^{(1)} \end{matrix}\}

(9b)

层合板的刚度矩阵为

(A_{i j}, D_{i j}) = \int_{- h / 2}^{h / 2} Q_{i j} (1, z^{2}) d z, (i, j = 1,2, 6)

(10)

将方程（9）代入方程（3），由广义位移关系表示的可伸缩悬臂复合材料层合板的非线性动力学偏微分方程为

\begin{matrix} A_{11} \frac{\partial u_{0}^{2}}{\partial x^{2}} + (A_{12} + A_{66}) \frac{\partial^{2} v_{0}}{\partial x \partial y} + A_{66} \frac{\partial^{2} u_{0}}{\partial y^{2}} + \\ (A_{12} + A_{66}) \frac{\partial w_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial x \partial y} + A_{11} \frac{\partial w_{0}}{\partial x} \frac{\partial^{2} w_{0}}{\partial x^{2}} + \\ A_{66} \frac{\partial w_{0}}{\partial x} \frac{\partial^{2} w_{0}}{\partial y^{2}} = I_{0} \ddot{x} (t) + I_{0} {\ddot{u}}_{0} - I_{1} \frac{\partial {\ddot{w}}_{0}}{\partial x} \end{matrix}

(11a)

\begin{matrix} A_{22} \frac{\partial^{2} v_{0}}{\partial y^{2}} + (A_{12} + A_{66}) \frac{\partial^{2} u_{0}}{\partial x \partial y} + A_{66} \frac{\partial^{2} v_{0}}{\partial x^{2}} \\ (A_{12} + A_{66}) \frac{\partial w_{0}}{\partial x} \frac{\partial^{2} w_{0}}{\partial x \partial y} + A_{22} \frac{\partial w_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial y^{2}} + \\ A_{66} \frac{\partial w_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial x^{2}} = I_{0} {\ddot{v}}_{0} - I_{1} \frac{\partial {\ddot{w}}_{0}}{\partial y} \end{matrix}

(11b)

$\begin{matrix} - D_{11} \frac{\partial^{4} w_{0}}{\partial x^{4}} - (D_{12} + D_{21} + 4 D_{66}) \frac{\partial^{4} w_{0}}{\partial x^{2} \partial y^{2}} - \\ D_{22} \frac{\partial^{2} w_{0}}{\partial y^{2}} + A_{11} \frac{\partial u_{0}}{\partial x} \frac{\partial^{2} w_{0}}{\partial x^{2}} + A_{21} \frac{\partial u_{0}}{\partial x} \frac{\partial^{2} w_{0}}{\partial y^{2}} + \\ \frac{3}{2} A_{22} {(\frac{\partial w_{0}}{\partial y})}^{2} + 2 A_{66} \frac{\partial u_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial x \partial y} + p_{1} + \end{matrix}$

\begin{matrix} (A_{66} + \frac{1}{2} A_{21}) {(\frac{\partial w_{0}}{\partial y})}^{2} \frac{\partial^{2} w_{0}}{\partial x^{2}} + 2 A_{66} \frac{\partial u_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial x \partial y} + \\ A_{11} \frac{\partial^{2} u_{0}}{\partial x^{2}} \frac{\partial w_{0}}{\partial x} + (A_{66} + \frac{1}{2} A_{21}) {(\frac{\partial w_{0}}{\partial x})}^{2} \frac{\partial^{2} w_{0}}{\partial y^{2}} + \\ (A_{21} + A_{66}) \frac{\partial^{2} v_{0}}{\partial x \partial y} \frac{\partial w_{0}}{\partial x} + (A_{21} + A_{66}) \frac{\partial^{2} u_{0}}{\partial x \partial y} \frac{\partial w_{0}}{\partial y} + \\ A_{12} \frac{\partial v_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial x^{2}} + A_{22} \frac{\partial v_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial y^{2}} + A_{66} \frac{\partial^{2} u_{0}}{\partial y^{2}} \frac{\partial w_{0}}{\partial x} + \\ A_{66} \frac{\partial^{2} v_{0}}{\partial x^{2}} \frac{\partial w_{0}}{\partial y} + \frac{3}{2} A_{11} {(\frac{\partial w_{0}}{\partial x})}^{2} \frac{\partial^{2} w_{0}}{\partial x^{2}} + \\ (2 A_{66} + A_{21} + A_{12}) \frac{\partial w_{0}}{\partial x} \frac{\partial w_{0}}{\partial y} \frac{\partial^{2} w_{0}}{\partial x \partial y} + f_{y} \frac{\partial^{2} w_{0}}{\partial y^{2}} \\ A_{22} \frac{\partial^{2} v_{0}}{\partial y^{2}} \frac{\partial w_{0}}{\partial y} - \frac{4 q_{d} λ}{M_{\infty} ν} \frac{\partial w_{0}}{\partial x} \frac{d x}{d t} - \frac{4 q_{d} λ}{M_{\infty}} \frac{\partial w_{0}}{\partial y} + \\ A_{66} \frac{\partial^{2} u_{0}}{\partial y^{2}} \frac{\partial w_{0}}{\partial x} = I_{0} {\ddot{w}}_{0} + (\frac{4 q_{d} λ}{M_{\infty} ν} + γ) {\dot{w}}_{0} + \\ I_{2} (\frac{\partial^{2} {\ddot{w}}_{0}}{\partial x^{2}} + \frac{\partial^{2} {\ddot{w}}_{0}}{\partial y^{2}}) + I_{1} (\frac{\partial {\ddot{u}}_{0}}{\partial x} + \frac{\partial {\ddot{u}}_{0}}{\partial y}) \end{matrix}

(11c)

选取满足可伸缩悬臂板的位移边界条件的模态函数如下所示:

u (x, y, t) = \sum_{i = 1}^{N} u_{i} (t) s i n \frac{p π x}{2 l} c o s \frac{q π y}{b}

(12a)

\begin{matrix} v (x, y, t) = \sum_{j = 1}^{N} v_{j} (t) s i n \frac{p π x}{2 l} s i n \frac{q π y}{b}, \\ (p, q = 1,2, \dots) \end{matrix}

(12b)

\begin{matrix} w (x, y, t) = \sum_{i = 1}^{n} w_{i} (t) X_{i} (x) Y_{j} (y) \\ (i = 1,2, \dots, n) \end{matrix}

(12c)

\begin{matrix} X_{i} (x, t) = s i n \frac{k_{i} x}{l (t)} - s i n h \frac{k_{i} x}{l (t)} + \\ ξ_{i} (c o s h \frac{k_{i} x}{l (t)} - c o s \frac{k_{i} x}{l (t)}) \end{matrix}

(12d)

\begin{matrix} Y_{j} (y) = s i n \frac{u_{j} y}{b} + s i n h \frac{u_{j} y}{b} - \\ ζ_{j} (c o s h \frac{u_{j} y}{b} + c o s \frac{u_{j} y}{b}) \end{matrix}

(12e)

对方程（11）进行无量纲化，然后将时变模态函数（12）代入无量纲动力学方程，进行两阶伽辽金离散，可得非线性常微分方程，如方程（13）所示.
$\begin{matrix} β_{11} {\ddot{w}}_{2} + β_{12} {\dot{w}}_{2} + β_{13} {\dot{w}}_{1} + β_{14} w_{2} + \\ β_{15} (f_{0} + f_{1} c o s (Ω_{1} t)) w_{2} + β_{16} w_{1} + \\ β_{17} w_{1}^{3} + β_{18} w_{1}^{2} w_{2} + β_{19} w_{1} w_{2}^{2} + \\ β_{110} w_{2}^{3} + β_{111} w_{1} w_{2} {\dot{w}}_{2} + β_{112} w_{1} {\dot{w}}_{2}^{2} + \\ β_{113} w_{2}^{2} {\dot{w}}_{2} + β_{114} w_{1}^{2} {\dot{w}}_{2} + \end{matrix}$

β_{115} w_{2} {\dot{w}}_{2}^{2} + β_{116} {\dot{w}}_{2}^{3} = 0

(13a)

\begin{matrix} β_{21} {\ddot{w}}_{2} + β_{22} {\dot{w}}_{2} + β_{23} {\dot{w}}_{1} + β_{24} w_{2} + \\ β_{25} (f_{0} + f_{1} c o s (Ω_{1} t)) w_{2} + β_{26} w_{1} + \\ β_{27} w_{1}^{3} + β_{28} w_{1}^{2} w_{2} + β_{29} w_{1} w_{2}^{2} + \\ β_{210} w_{2}^{3} + β_{211} w_{1} w_{2} {\dot{w}}_{2} + β_{212} w_{1} {\dot{w}}_{2}^{2} + \\ β_{213} w_{2}^{2} {\dot{w}}_{2} + β_{214} w_{1}^{2} {\dot{w}}_{2} + \\ β_{215} w_{2} {\dot{w}}_{2}^{2} + β_{216} {\dot{w}}_{2}^{3} = 0 \end{matrix}

(13b)

其中β_ijk是关于时间的函数.
2 数值分析
为了研究可伸缩悬臂复合材料层板的动力学特性，选取的结构尺寸和材料参数具体表示如下:l₀=2m，b=1.5m，γ=300N.s/m，Ω=10， $M_{\infty}$ =3.0，f₀=2000N/m²，ρ_a=0.65kg/m³，υ=600m/s，E₁=125.0GPa，E₂=7.2GPa，G₂₃=1.43GPa，G₁₂=G13=4.1GPa，v₁₂=0.33，v₂₁=v₁₂E₂/E₁，ρ=1570kg/m³，根据非线性方程（13），通过数值方法研究材料参数和速度对可伸缩悬臂板在匀速外伸和回收过程中的非线性动力学特性的影响.
图2表示宽厚比对板前两阶无量纲固有频率的影响，其中，图（a）（b）和图（c）（d）分别表示可伸缩悬臂板匀速外伸和回收过程.可以看出，悬臂板匀速外伸时，板的固有频率逐渐减小; 匀速回收时，悬臂板的固有频率逐渐增大.由图2可以看出，随着板的宽厚比逐渐增加，可伸缩悬臂板的振动频率逐渐增大，这主要是由板的宽度增加，悬臂板的刚度增加所导致的.
图3给出了可伸缩悬臂板以两种轴向运动速度在匀速外伸和回收过程中的前两阶时域振动曲线及与之相对应的相图.其中，轴向速度分别为v₁=0.5m/s，v₂=0.4m/s.由图3可以看出，板在外伸过程中，振幅随着外伸时间的增加而减小，频率增大; 回收过程与之相反.外伸时，随着轴向运动速度增大，振幅和频率的趋势加快; 回收时，轴向运动速度越大，振幅和频率变化的速度增大，越快收敛.图3（e）-（h）分别表示可伸缩悬臂板匀速外伸和回收过程的前两阶模态相图.箭头表示相图起点，图3（e）、（f）表明可伸缩悬臂板匀速外伸时，横向速度减小，横向位移增大，轴向速度增大，振幅增加得快; 图3（g）、（h）表明可伸缩悬臂板匀速回收时，横向速度增大，横向位移减小，轴向速度越大，振幅减小得越快.
图2 可伸缩悬臂板匀速外伸和回收时宽厚比对前两阶固有频率的影响∶（a）（b）外伸过程的前两阶固有频率;（c）（d）回收过程的前两阶固有频率
Fig.2 The influence of ratio of width-to-thickness on natural frequency when the plate is deploying and retracting at a uniform speed∶ (a) (b) the first two-order natural frequency in deploying process; (c) (d) the first-two-order natural frequency in retracting process
图3 当可伸缩悬臂板以两种轴向速度匀速外伸和回收时的前两阶模态时间历程图和所对应的相图∶（a）（b）（e）（f）板外伸过程的前两阶模态时间历程图和相图;（c）（d）（g）（h）板回收过程的前两阶模态时间历程图和相图
Fig.3 The time history diagrams and corresponding phase diagrams of the first-two-order modes when the telescopic cantilever plate is deploying and retracting at two axially moving speeds∶ (a) (b) (e) (f) time history diagrams and phase diagrams of the first-two-orde rmodes in the deploying process; (c) (d) (g) (h) time history diagrams and phase diagrams of the first-two-order modes in the retracting process
图4 当可伸缩悬臂板以四种轴向速度匀速外伸时的前两阶模态时间历程图∶（a）（b）（e）（f）板外伸过程的前第一阶模态时间历程图;（c）（d）（g）（h）板外伸过程的前第二阶模态时间历程图
Fig.4 The time history diagrams of the first-two-order modes aregiven when the telescopic cantilever plate is deploying and retracting at four axially moving speeds (a) (b) (e) (f) time history diagram of the first-order modes in the deploying process, (c) (d) (g) (h) time history diagram of the second-order modes in the retracting process
图4（a）-（h）表示可伸缩悬臂板分别以v₃=0.002m/s，v₄=0.003m/s，v₅=1m/s，v₆=2m/s匀速外伸时的前两阶模态对应的时间历程图，其中图（f）、（h）分别为图（e）、（g）的局部放大图.由图5（a）~（d）可以看出，当可伸缩悬臂板以较小的轴向速度v₃和v₄轴向匀速外伸时，悬臂板的振幅先增大后减小，最后达到收敛.由图4（e）~（h）可以看出当可伸缩悬臂板以较大的轴向速度v₅和v₆匀速外伸时，板的横向振动位移是发散的，表示此刻板已经发生失稳; 除此之外，板的轴向外伸速度越大，振幅和频率的变化速率越大，这与图4的结论相同.
3 结论
本文以三阶气动力和面内激励联合作用下的可伸缩悬臂复合材料层合板为模型，通过经典层合板理论和Hamilton原理建立板的非线性时变动力学偏微分方程，采用Galerkin将其离散成常微分方程，通过数值方法研究了轴向移动速度和材料参数对板在匀速外伸和回收过程的动力学特性的影响，结论总结如下:随着板的宽厚比增大，板的固有频率增大.轴向移动速度对可伸缩悬臂板的动力学特性影响较大，当板匀速外伸较慢时，频率减小，振幅先增大后减小; 当板匀速外伸较快时，随着轴向速度的增大，板的振幅逐渐增大直至发散.回收时，振幅减小，频率增大，没有出现发散行为.无论外伸还是回收过程，轴向移动速度增大，板的频率和振幅的变化率增大.
参考文献
- [1] PARK S,YOO H H,CHUNG J T.Eulerian and Lagrangian descriptions for the vibration analysis of a deploying beam [J].Journal of Mechanical Science and Technology,2013,27∶ 2637-2643.
- [2] SAHOO B,PANDA L N,POHIT G.Parametric and internal resonances of an axially moving beam with time-dependent velocity [J].Modeling and Simulation in Engineering,2013,2013∶ 1-18.
- [3] ZHU K F,CHUNG J T.Nonlinear lateral vibrations of a deploying euler-bernoulli beam with a spinning motion [J].International Journal of Mechanical Science,2014,90∶ 200-212.
- [4] YAN Q Y,DING H,CHEN L Q.Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations [J].Applied Mathematics and Mechanics,2015,36∶971-984.
- [5] 刘燕,张伟,龚涛涛,轴向可伸缩复合材料悬臂梁的非线性振动研究 [J].动力学与控制学报,2020,18(4)∶19-26.LIU Y,ZHANG W,GONG T T.Nonlinear dynamics of an axially telecopic composite cantilever beam [J].Journal of Dynamics and Control,2020,18(4):19-26.(in Chinese)
- [6] 段应昌,王建平,刘虎,等.轴向运动悬臂梁横向振动理论及实验研究 [J].兵器装备工程学报,2021,42(4)∶138-144.DUAN Y C,WANG J P,LIU H,et al.Theoretical and experimental study on transverse vibration of an axially moving cantilever beam [J].Journal of Ordnance Equipment Engineering,2021,42(4)∶138-144.(in Chinese)
- [7] GHAYESH M H,AMABILI M,PADOUSSIS M P.Nonlinear dynamics of axially moving plates [J].Journal of Sound and Vibration,2013,332∶ 391-406.
- [8] 吕书锋,张伟.复合材料悬臂外伸板的非线性动力学建模及数值研究 [J].动力学与控制学报,2015,13(4)∶ 288-292.LV S F,ZHANG W.Nonlinear dynamic modeling and numerical study of composite cantilever overhanging plate [J].Journal of Dynamics and Control,2015,13(4)∶ 288-292.(in Chinese)
- [9] ZHOU Y F,WANG Z M.Dynamic instability of axially moving viscoelastic plate [J].European Journal of Mechanics A∶ Solids,2019,73∶ 1-10.
- [10] ZHANG D B,TANG Y Q,CHEN L Q.Internal resonance in parametric vibrations of axially accelerating viscoelastic plates [J].European Journal of Mechanics A∶ Solids,2019,75∶ 142-155.
- [11] ZHANG W,CHEN L L,GUO X Y,et al.Nonlinear dynamical behaviors of deploying wings in subsonic air flow [J].Journal of Fluids and Structures,2017,74∶ 340-355.
- [12] WANG Y Q,ZU J W.Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid [J].Composite Structures,2017,164∶ 130-144.
- [13] ASHLEY H,ZARTARIAN G.Piston theory-a new aerodynamic tool for the aeroelastician [J].Journal of Aeronautical Science,1956,23∶ 1109-1118.

基本信息

中图分类号: O322
文献标识码: A
DOI: 10.6052/1672-6553-2021-077
文章编号: 1672-6553-2023-21(2)-041-008

基金信息

国家自然科学基金资助项目(11832002)；
National Natural Science Foundation of China (11832002)；

引用信息

稿件历史

收稿日期: 2021-09-29

参考文献

[1] PARK S,YOO H H,CHUNG J T.Eulerian and Lagrangian descriptions for the vibration analysis of a deploying beam [J].Journal of Mechanical Science and Technology,2013,27∶ 2637-2643.
[2] SAHOO B,PANDA L N,POHIT G.Parametric and internal resonances of an axially moving beam with time-dependent velocity [J].Modeling and Simulation in Engineering,2013,2013∶ 1-18.
[3] ZHU K F,CHUNG J T.Nonlinear lateral vibrations of a deploying euler-bernoulli beam with a spinning motion [J].International Journal of Mechanical Science,2014,90∶ 200-212.
[4] YAN Q Y,DING H,CHEN L Q.Nonlinear dynamics of axially moving viscoelastic Timoshenko beam under parametric and external excitations [J].Applied Mathematics and Mechanics,2015,36∶971-984.
[5] 刘燕,张伟,龚涛涛,轴向可伸缩复合材料悬臂梁的非线性振动研究 [J].动力学与控制学报,2020,18(4)∶19-26.LIU Y,ZHANG W,GONG T T.Nonlinear dynamics of an axially telecopic composite cantilever beam [J].Journal of Dynamics and Control,2020,18(4):19-26.(in Chinese)
[6] 段应昌,王建平,刘虎,等.轴向运动悬臂梁横向振动理论及实验研究 [J].兵器装备工程学报,2021,42(4)∶138-144.DUAN Y C,WANG J P,LIU H,et al.Theoretical and experimental study on transverse vibration of an axially moving cantilever beam [J].Journal of Ordnance Equipment Engineering,2021,42(4)∶138-144.(in Chinese)
[7] GHAYESH M H,AMABILI M,PADOUSSIS M P.Nonlinear dynamics of axially moving plates [J].Journal of Sound and Vibration,2013,332∶ 391-406.
[8] 吕书锋,张伟.复合材料悬臂外伸板的非线性动力学建模及数值研究 [J].动力学与控制学报,2015,13(4)∶ 288-292.LV S F,ZHANG W.Nonlinear dynamic modeling and numerical study of composite cantilever overhanging plate [J].Journal of Dynamics and Control,2015,13(4)∶ 288-292.(in Chinese)
[9] ZHOU Y F,WANG Z M.Dynamic instability of axially moving viscoelastic plate [J].European Journal of Mechanics A∶ Solids,2019,73∶ 1-10.
[10] ZHANG D B,TANG Y Q,CHEN L Q.Internal resonance in parametric vibrations of axially accelerating viscoelastic plates [J].European Journal of Mechanics A∶ Solids,2019,75∶ 142-155.
[11] ZHANG W,CHEN L L,GUO X Y,et al.Nonlinear dynamical behaviors of deploying wings in subsonic air flow [J].Journal of Fluids and Structures,2017,74∶ 340-355.
[12] WANG Y Q,ZU J W.Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid [J].Composite Structures,2017,164∶ 130-144.
[13] ASHLEY H,ZARTARIAN G.Piston theory-a new aerodynamic tool for the aeroelastician [J].Journal of Aeronautical Science,1956,23∶ 1109-1118.

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轴向可伸缩悬臂复合材料层合板的建模及数值分析*

Modeling and Numerical Analysis of Telescopic Cantilever Composite Plate*

摘要

Abstract

关键词

Keywords

1 基本方程

(1a)

(1b)

(1c)

(2)

(3a)

(3b)

(3c)

(4)

(5a)

(5b)

(6a)

(6b)

(6c)

(7)

(8)

(9a)

(9b)

(10)

(11a)

(11b)

(11c)

(12a)

(12b)

(12c)

(12d)

(12e)

(13a)

(13b)

2 数值分析

3 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

微信公众号二维码

手机版网站二维码

轴向可伸缩悬臂复合材料层合板的建模及数值分析^*

Modeling and Numerical Analysis of Telescopic Cantilever Composite Plate^*