带裂纹石墨烯增强金属泡沫梁的振动特性研究

石兴东，程鹏，张宇飞; Shi Xingdong; Cheng Peng; Zhang Yufei

引 en

带裂纹石墨烯增强金属泡沫梁的振动特性研究^*

石兴东¹
机构：
1. 沈阳航空航天大学航空宇航学院,沈阳 110136
×
，程鹏¹
机构：
1. 沈阳航空航天大学航空宇航学院,沈阳 110136
×
，张宇飞^1,2
机构：
1. 沈阳航空航天大学航空宇航学院,沈阳 110136
2. 广西大学土木建筑工程学院,南宁 530004
×

1. 沈阳航空航天大学航空宇航学院,沈阳 110136；
2. 广西大学土木建筑工程学院,南宁 530004；

Study on Vibration Characteristics of Cracked Metal Foam Beams Reinforced with Graphene^*

Shi Xingdong¹
Affiliation：
1. College of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136 , China
×
，Cheng Peng¹
Affiliation：
1. College of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136 , China
×
，Zhang Yufei^1,2
Affiliation：
1. College of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136 , China
2. School of Civil Engineering and Architecture, Guangxi University, Nanning 530004 , China
×

1. College of Aerospace Engineering, Shenyang Aerospace University, Shenyang 110136 , China；
2. School of Civil Engineering and Architecture, Guangxi University, Nanning 530004 , China；

通讯作者:

张宇飞,E-mail:yufeizhang73@163.com

中图分类号:V21;O39

文献标识码:A

文章编号:1672-6553-2024-22(3)-048-008

DOI:10.6052/1672-6553-2023-025

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参考文献
出版信息

参考文献 1

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目录contents

摘要 Abstract
关键词 Keywords
1 材料特性与旋转弹簧模型
1.1 材料特性
1.2 旋转弹簧模型
2 运动方程与微分变换法
2.1 运动方程
2.2 微分变换法
3 结果和讨论
4 结论
参考文献

摘要

分析了带裂纹功能梯度石墨烯增强金属泡沫梁的自由振动.采用Timoshenko梁理论进行建模,裂纹由无质量扭转弹簧模拟,利用Halpin-Tsai微观力学模型预测材料的有效性能.通过哈密顿原理,得到了带裂纹功能梯度石墨烯增强金属泡沫(FG-GPLRMF)梁的运动方程及其边界条件.采用微分变换法分析带裂纹FG-GPLRMF梁的自由振动.结果表明,带裂纹FG-GPLRMF梁的振动特性受到石墨烯几何尺寸、孔隙类型和石墨烯分布的影响显著.

Abstract

The free vibration of a cracked functionally graded graphene reinforced metal foam beams is analyzed in this paper. Modeling was carried out with Timoshenko beam theory, cracks were simulated by mass-free elastic torsion springs, and material effectiveness was predicted with Halpin-Tsai micro-mechanical model. The motion equation and boundary conditions of a cracked functionally graded graphene reinforced metal foam (FG-GPLRMF) beams are obtained by Hamilton's principle. Free vibration of FG-GPLRMF beams is analyzed by differential transformation method. The results show that the vibration characteristics of FG-GPLRMF beams are influenced by the geometrical size of graphene, the pore type and the distribution of graphene platelet.

关键词

金属泡沫梁；石墨烯增强；自由振动；裂纹；微分变换法

Keywords

metal foam beam ； graphene reinforced ； free vibration ； crack ； differential transformation method

引言
金属泡沫材料由于内部孔隙的存在，具有轻质、隔热、减震的特性，通过定制内部孔隙的密度和尺度，可以获得所需的材料力学性能，因此，金属泡沫材料在能量吸收、轻质结构和热控制^[1]等方面得到了广泛的应用.然而，多孔材料的内部孔隙降低了结构的强度和刚度，这限制了其在某些工程领域的应用.相关研究^[2]表明，多孔材料可以填充低体积分数的碳材料，如石墨烯薄片（GPL）和碳纳米管，以弥补其强度和刚度的降低，特别是GPL增强复合材料^[3，4]具有轻质、良好的导电和导热性能，因此，关于GPL增强复合材料结构的研究受到了广泛的关注.
Kitipornchai等^[5]研究了GPL增强多孔梁的自由振动特性.Ye和Wang^[6]研究了FG-GPLRMF薄壁圆柱壳的非线性强迫振动特性.Yang等^[7]基于切比雪夫-里兹方法分析了GPL增强多孔纳米复合板的屈曲和自由振动.Dong等^[8，9]研究了GPL增强多孔纳米复合圆柱壳的屈曲和线性振动行为.Wang等^[10]分析了GPLRMF圆柱壳的非线性自由振动特性.Yang和Chen^[11]分析了裂纹对不同边界条件下功能梯度材料Euler-Bernoulli梁振动和屈曲的影响.刘涛等^[12]分析了轴向运动功能梯度梁的横向振动频率特性.周磊^[13]以含裂纹石墨烯增强功能梯度层合梁为研究对象，进行了动力稳定性行为分析.刘金建等^[14]基于欧拉梁理论，研究了轴向运动功能梯度粘弹性梁横向振动的稳定性问题.
虽然一些学者已经对石墨烯增强多孔结构开展了研究，但关于带裂纹石墨烯增强金属泡沫梁振动问题的研究尚未开展.本论文基于Timoshenko梁理论，研究带裂纹功能梯度石墨烯增强金属泡沫梁（FG-GPLRMF）的自由振动问题，包括控制方程推导和微分变换法的应用，研究了裂纹深度、裂纹位置、GPL几何尺寸、孔隙类型、GPL分布和长细比对带裂纹FG-GPLRMF梁自由振动特性的影响.
1 材料特性与旋转弹簧模型
1.1 材料特性
图1（a）是长度为L、厚度为h的两端固定的Timoshenko梁模型，在距左端L₁处包含一个深度为a的边缘裂纹.
沿梁高度方向考虑了三种孔隙分布类型^[10]，即孔隙-a、孔隙-b和孔隙-c.孔隙-a和孔隙-b在梁高方向上呈对称分布.孔隙-a和孔隙-b的差异在于前者的中平面孔隙最大，后者的中平面孔隙最小.孔隙-c的孔隙大小均匀分布.
图1 裂纹梁结构
Fig.1 Configuration of cracked beam
考虑了三种不同GPL分布方式^[10]，对于GPL-Ⅰ和GPL-Ⅱ，GPL相对梁中平面对称分布.对于GPL-I，最大的体积分数发生在梁的上表面和下表面；对于GPL-Ⅱ，最大体积分数发生在中面.对于GPL-Ⅲ，GPL均匀分布.
FG-GPLRMF梁的杨氏模量E_GM（z）和质量密度ρ_GM（z）分别为^[7，15]

E_{G M} (z) = \{\begin{matrix} E_{g} [1 - P_{1} c o s (z π / h)] 孔 隙 - a \\ E_{g} \{1 - P_{2} [1 - c o s (z π / h)]\} 孔 隙 - b \\ E_{g} P_{3} 孔 隙 - c \end{matrix}

(1)

ρ_{G M} (z) = \{\begin{matrix} ρ_{g} [1 - P_{m 1} c o s (z π / h)] 孔 隙 - a \\ ρ_{g} \{1 - P_{m 2} [1 - c o s (z π / h)]\} 孔 隙 - b \\ ρ_{g} P_{m 3} 孔 隙 - c \end{matrix}

(2)

其中E_g和ρ_g表示无孔GPL增强金属的杨氏模量和质量密度；P₁、P₂和P₃表示孔隙系数；P_m1、P_m2和P_m3表示质量密度系数.
1.2 旋转弹簧模型
假定裂纹垂直于梁表面，且始终保持开放状态.Timoshenko梁的边缘裂纹造成了弯曲斜率的不连续以及开裂截面的横向位移.已有研究^[16，17]表明，与弯曲斜率中的不连续性（I型断裂）相比，横向位移中的不连续性（Ⅱ型断裂）对系统总应变能的贡献要小得多，因此本分析中忽略Ⅱ型断裂的影响.将裂纹截面建模为如图1b所示的无质量弹性弹簧，可将开裂梁视为在开裂截面上由转动弹簧连接的两个子梁，其弯曲刚度为：

K_{T} = 1 / G

(3)

其中G是裂纹引起的柔度，并且^[18]

\frac{(1 - ν^{2}) K_{1}^{2}}{E (a)} = \frac{M^{2}}{2} \frac{d G}{d a}

(4)

其中M为裂纹截面弯矩，K₁为I型弯曲荷载作用下的应力强度因子，E（a）为裂纹尖端的杨氏模量，ν为泊松比.
通过拉格朗日插值技术，可以从Erdogan和Wu^[19]给出的数据中得到应力强度因子的大小，即

K_{1} = \frac{6 M \sqrt{π h ζ}}{h^{2}} F (ζ), ζ = \frac{a}{h} (ζ ⩽ 0.7)

(5)

其中ζ≤0.7意味着仅考虑0至0.7的裂纹深度比.根据式（3）~式（5），可以确定裂纹部分的弯曲刚度.
2 运动方程与微分变换法
2.1 运动方程
基于Timoshenko梁理论，梁任意点沿x轴和z轴的位移，用 $\tilde{U} （ x ， z ， t ）$ 和 $\tilde{V} （ x ， z ， t ）$ 表示，分别为

\begin{matrix} \tilde{U} (x, z, t) = U (x, t) + z Θ (x, t) \\ \tilde{V} (x, z, t) = V (x, t) \end{matrix}

(6)

其中U（x，t）和V（x，t）是中平面（z=0）上的位移分量，Θ是梁截面的转动角位移，t是时间.von Kármán型非线性应变-位移关系由下式给出^[20]：

ε_{x} = \frac{\partial U}{\partial x} + z \frac{\partial Θ}{\partial x}, γ_{x z} = \frac{\partial V}{\partial x} + Θ

(7)

正应力和剪切应力分别为

\begin{matrix} σ_{x x} = Q_{11} (z) (\frac{\partial U}{\partial x} + z \frac{\partial Θ}{\partial x}) \\ σ_{x z} = Q_{55} (z) (\frac{\partial V}{\partial x} + Θ) \end{matrix}

(8)

这里

Q_{11} (z) = \frac{E_{G M} (z)}{1 - ν^{2}}, Q_{55} (z) = \frac{E_{G M} (z)}{2 (1 + ν)}

(9)

裂纹FG-GPLRMF梁的动能T和势能W为^[20]

T = \frac{1}{2} \int_{0}^{L_{1}} \int_{- h / 2}^{h / 2} ρ_{G M} [{(\frac{\partial U_{1}}{\partial t} + z \frac{\partial Θ_{1}}{\partial t})}^{2} + {(\frac{\partial V_{1}}{\partial t})}^{2}] d z d x + \frac{1}{2} \int_{L_{1}}^{L} \int_{- h / 2}^{h / 2} ρ_{G M} [{(\frac{\partial U_{2}}{\partial t} + z \frac{\partial Θ_{2}}{\partial t})}^{2} + {(\frac{\partial V_{2}}{\partial t})}^{2}] d z d x

(10)

\begin{matrix} W = \frac{1}{2} \int_{0}^{L_{1}} \int_{- h / 2}^{h / 2} [Q_{11} {(\frac{\partial U_{1}}{\partial x} + z \frac{\partial Θ_{1}}{\partial x})}^{2} + Q_{55} {(\frac{\partial V_{1}}{\partial x} + Θ_{1})}^{2}] d z d x + \\ \frac{1}{2} \int_{L_{1}}^{L} \int_{- h / 2}^{h / 2} [Q_{11} {(\frac{\partial U_{2}}{\partial x} + z \frac{\partial Θ_{2}}{\partial x})}^{2} + Q_{55} {(\frac{\partial V_{2}}{\partial x} + Θ_{2})}^{2}] d z d x + \frac{1}{2} K_{T} (Δ Θ) 2 \end{matrix}

(11)

其中ΔΘ为Θ₂（L₁）-Θ₁（L₁）；U_i、Θ_i、V_i中的下标i等于1，2，表示被裂纹分割的左子梁和右子梁.最后一项表示旋转弹簧产生的势能.
将式（10）、式（11）中含Q₁₁、Q₅₅和ρ_GM（z）项定义为：

\begin{matrix} (A_{11}, B_{11}, D_{11}) = \int_{- h / 2}^{h / 2} Q_{11} (z) \{1, z, z^{2}\} d z, \\ A_{55} = \int_{- h / 2}^{h / 2} κ Q_{55} (z) d z \\ \{I_{1}, I_{2}, I_{3}\} = \int_{- h / 2}^{h / 2} ρ_{G M} (z) \{1, z, z^{2}\} d z \end{matrix}

(12)

其中κ=5/6是剪切修正系数.
哈密顿原理由下式给出

δ \int_{t_{1}}^{t_{2}} (T - W) d t = 0

(13)

其中t₁和t₂分别表示初始时间和最终时间.
引入如下无量纲变量
$\bar{x} = \frac{x}{L}, α_{1} = \frac{L_{1}}{L}, (u, v) = \frac{(U, V)}{h},$

\begin{matrix} (I_{11}, I_{22}, I_{33}) = (\frac{I_{1}}{I_{10}}, \frac{I_{2}}{I_{10} h}, \frac{I_{3}}{I_{10} h^{2}}), \\ Θ = Θ, α_{2} = \frac{L}{h}, (a_{11}, a_{55}, b_{11}, d_{11}) \\ = (\frac{A_{11}}{A_{110}}, \frac{A_{55}}{A_{110}}, \frac{B_{11}}{A_{110} h}, \frac{D_{11}}{A_{110} h^{2}}) \end{matrix}

(14)

其中A₁₁₀和I₁₀是无孔梁的A₁₁和I₁的值.
梁连续性条件

\begin{matrix} u_{1} = u_{2}, v_{1} = v_{2}, \frac{\partial u_{1}}{\partial x} = \frac{\partial u_{2}}{\partial x} \\ \frac{\partial v_{1}}{\partial x} - Θ_{1} = \frac{\partial v_{2}}{\partial x} - Θ_{2}, \\ K_{T} (Θ_{2} - Θ_{1}) = D_{11} \frac{\partial Θ_{1}}{\partial x} = D_{11} \frac{\partial Θ_{2}}{\partial x} \end{matrix}

(15)

梁的位移函数设为^[21]：

\begin{matrix} V (\bar{x}, t) = \bar{V} (\bar{x}) e^{i λ t}, U (\bar{x}, t) = \bar{U} (\bar{x}) e^{i γ t}, \\ Θ (\bar{x}, t) = \bar{Θ} (\bar{x}) e^{i γ t} \end{matrix}

(16)

这里λ是径向固有频率，γ是弦向固有频率.
将式（10）~式（12），式（14）~式（16）代入式（13），即可得到简化后的运动方程：
$\begin{matrix} I_{11} α_{1}^{2} (γ^{2} + λ^{2}) {\bar{U}}_{1} + a_{11} \frac{\partial^{2} {\bar{U}}_{1}}{\partial {\bar{x}}^{2}} = 0 \\ I_{11} α_{1}^{2} λ^{2} {\bar{V}}_{1} - \frac{1}{2} I_{11} α_{1}^{2} γ^{2} \frac{\partial {\bar{V}}_{1}}{\partial \bar{x}} - \frac{1}{2} (2 a_{55} + \\ I_{11} γ^{2}) \frac{\partial^{2} {\bar{V}}_{1}}{\partial {\bar{x}}^{2}} - a_{55} α_{1} α_{2} \frac{\partial {\bar{Θ}}_{1}}{\partial \bar{x}} = 0 \\ a_{55} α_{1} α_{2} \frac{\partial {\bar{V}}_{1}}{\partial \bar{x}} - α_{1}^{2} [a_{55} α_{2}^{2} - I_{33} (γ^{2} + λ^{2})] {\bar{Θ}}_{1} - \\ d_{11} \frac{\partial^{2} {\bar{Θ}}_{1}}{\partial {\bar{x}}^{2}} = 0 \end{matrix}$

\begin{matrix} I_{11} {(1 - α_{1})}^{2} (γ^{2} + λ^{2}) {\bar{U}}_{2} + a_{11} \frac{\partial^{2} {\bar{U}}_{2}}{\partial {\bar{x}}^{2}} = 0 \\ I_{11} {(1 - α_{1})}^{2} λ^{2} {\bar{V}}_{2} - \frac{1}{2} I_{11} {(1 - α_{1})}^{2} γ^{2} \frac{\partial {\bar{V}}_{2}}{\partial \bar{x}} - \\ \frac{1}{2} [2 a_{55} + I_{11} (1 - α_{1}^{2}) γ^{2}] \frac{\partial^{2} {\bar{V}}_{2}}{\partial {\bar{x}}^{2}} - \\ a_{55} (1 - α_{1}) α_{2} \frac{\partial {\bar{Θ}}_{2}}{\partial \bar{x}} = 0 \\ a_{55} α_{2} (1 - α_{1}) \frac{\partial {\bar{V}}_{2}}{\partial \bar{x}} - {(1 - α_{1})}^{2} [a_{55} α_{2}^{2} - I_{33} (γ^{2} + \\ λ^{2})] {\bar{Θ}}_{2} - d_{11} \frac{\partial^{2} {\bar{Θ}}_{2}}{\partial {\bar{x}}^{2}} = 0 \end{matrix}

(17)

2.2 微分变换法
本文将变换函数用大写字母表示，将原函数用小写字母表示.在定义域D中定义一个解析函数g（x），设x=x₀表示D中的任意点.函数g（x）的微分变换定义为^[15]

g (x) = \sum_{k = 0}^{m} {(x - x_{0})}^{k} G (k)

(18)

其中，m是截断数.
将Xu^[15]中的表1和表2公式代入方程（17）以及两端固定边界条件，可以得到变换函数和对应边界条件.
相容方程：

\begin{matrix} U_{2} (0) = \sum_{k = 0}^{\infty} U_{1} (k) \\ U_{2} (1) = \sum_{k = 0}^{\infty} \frac{1 - α_{1}}{α_{1}} k U_{1} (k) \end{matrix}

(19)

$V_{2} (0) = \sum_{k = 0}^{\infty} V_{1} (k)$

\begin{matrix} V_{2} (1) = \sum_{k = 0}^{\infty} \frac{1 - α_{1}}{α_{1}} k V_{1} (k) + \\ \sum_{k = 0}^{\infty} \frac{d_{11} (1 - α_{1})}{α_{1} k_{T}} k_{1} Θ_{1} (k) \end{matrix}

(20)

\begin{matrix} Θ_{2} (0) = \sum_{k = 0}^{\infty} (\frac{d_{11} k}{α_{1} k_{c}} + 1) Θ_{1} (k) \\ Θ_{2} (1) = \sum_{k = 0}^{\infty} \frac{1 - α_{1}}{α_{1}} k Θ_{1} (k) \end{matrix}

(21)

在变换函数、边界条件和相容方程中：U_i（k），V_i（k）和Θ_i（k）分别是 ${\bar{u}}_{i}$ ， ${\bar{v}}_{i}$ 和 ${\bar{θ}}_{i}$ 的微分变换.
设U₁（0）= C₁，V₁（0）= C₂，Θ₁（0）= C₃，代入变换函数方程和对应边界条件，可以计算U_i（k），V_i（k）和Θ_i（k）的值.
将U_i（k），V_i（k）和Θ_i（k）代入对应边界条件方程，可以解出λ.
3 结果和讨论
下面对图1中所示的FG-GPLRMF梁进行研究，参数如表1所示.
FG-GPLRMF梁弯曲振动的固有频率收敛结果见表2，其中以P₁ =0.5、W_GPL =1.0%、GPL-I和孔隙-a为例.结果表明，在m=16时，固有频率趋于收敛，但考虑到精确性，在下面的计算中采用了m= 60.
表1 FG-GPLRMF梁参数^[20]
Table1 Parameters of FG-GPLRMF beams^[20]
表2 FG-GPLRMF梁弯曲振动固有频率（rad/s）的收敛性
Table2 Convergence of natural frequency of bending vibration (rad/s) of FG-GPLRMF beams
在图2中，研究了孔隙系数P₁对固有频率的作用，其中考虑了不同的GPL分布方式和孔隙分布.随着P₁的增加，孔隙-b和孔隙-c梁的固有频率减小.然而，对于孔隙-a梁，其固有频率先减小后增加.造成这种现象的原因是，当梁的几何尺寸保持不变时，固有频率与弯曲刚度（E_Iy）呈正相关，与单位长度的质量（ρT）呈负相关.进一步计算表明，当P₁大于0.4时，弯曲刚度的相对变化小于质量的相对变化.此外，这三种孔隙分布的固有频率中，GPL-Ⅰ分布的固有频率最高，而GPL-Ⅱ分布的固有频率最低.因此，GPL-Ⅰ分布是提高结构刚度最有效的分布方式.以上结果表明，孔隙系数和GPL分布对FG-GPLRMF的自由振动有显著影响.
图2 孔隙系数P₁对梁的固有频率的影响（W_GPL=0.01；L/h=16；a/h=0.2；L₁/L=0.5）
Fig.2 Natural frequencies of FG-GPLRMF beams for different porosity coefficient (W_GPL=0.01; L/h=16; a/h=0.2; L₁/L=0.5)
图3给出了GPL重量分数W_GPL对不同GPL分布方式和孔隙分布的FG-GPLRMF固有频率的影响.对于不同的GPL分布方式，固有频率随W_GPL的增加而增加，其原因就在于增加石墨烯含量可增强梁刚度.通过将极少量的GPL分散到基质中，即可提高多孔梁的有效刚度.与其他分布模式相比，孔隙-a和GPL-Ⅰ分布方式的梁具有最大的固有频率.
图3 GPL重量分数对FG-GPLRMF梁的固有频率的影响（L/h=16；a/h=0.2；L1/L=0.5；P1=0.50）
Fig.3 Effect of GPL weight fraction on natural frequencies of FG-GPLRMF beams (L/h=16; a/h=0.2; L₁/L=0.5; P₁=0.50)
图4给出了不同GPL尺寸对FG-GPLRMF梁固有频率的影响.图4（a）为FG-GPLRMF梁的固有频率随GPL长厚比l_GPL/t_GPL的变化情况.随着l_GPL/t_GPL的增加，固有频率增加.结果表明，当l_GPL和w_GPL保持固定时，单个GPL越薄，其刚度强化效果越好.图4（b）为固有频率随GPL长宽比l_GPL/w_GPL的变化规律.随着l_GPL/w_GPL的增加，固有频率减小.这表明，石墨烯的表面积越大，刚度增强效果越好.同时，由图4可以看出，当l_GPL/t_GPL≥102或l_GPL/w_GPL≥10时，强化效应随这些比值的变化不大.
图4 GPL尺寸对FG-GPLRMF梁固有频率的影响（孔隙-a；P₁=0.5；W_GPL=1%；L/h=16；a/h=0.2；L₁/L=0.5）
Fig.4 Effect of geometrical size of GPL on natural frequency of FG-GPLRMF beams (Porosity-a; P₁=0.5; W_GPL=1%; L/h=16; a/h=0.2; L₁/L=0.5; )
图5研究了裂纹位置对FG-GPLRMF梁的固有频率的影响.结果表明，固有频率对位于梁中心的裂纹最为敏感.当裂纹靠近梁端时，裂纹的影响较小，裂纹位置由梁中心向梁端变化时，固有频率逐渐增加.
图6研究了裂纹深度对FG-GPLRMF梁固有频率的影响.结果表明，随着裂纹深度增加，梁的固有频率降低，对于不同GPL分布方式，固有频率降低的速度大致相同，不同孔隙下裂纹梁固有频率都逐渐变小，这说明裂纹位置对频率的影响远超其他因素.
图5 裂纹位置对FG-GPLRMF梁固有频率的影响（孔隙-a；W_GPL=0.01；L/h=6；h=0.1；P₁=0.5；L₁/L=0.5）
Fig.5 Effect of crack location on natural frequency of FG-GPLRMF beam (Porosity-a; W_GPL=0.01; L/h=6; h=0.1; P₁=0.5; L₁/L=0.5)
图6 裂纹深度对梁固有频率的影响（W_GPL=0.01；L/h=6；P₁=0.5；L₁/L=0.5；h=0.1m）
Fig.6 Effect of crack depth on natural frequency of beam (W_GPL=0.01; L/h=6; P₁=0.5; L₁/L=0.5; h=0.1m)
4 结论
本文研究了带裂纹FG-GPLRMF梁的自由振动特性.利用哈密顿原理，得到了带裂纹FG-GPLRMF梁的运动方程，采用微分变换法分析了梁的固有频率.研究了裂纹深度、裂纹位置、GPL几何尺寸、孔隙类型、GPL分布和长细比对裂纹FG-GPLRMF梁自由振动特性的影响.主要结论如下：
对于孔隙-b和孔隙-c梁，弯曲振动的固有频率随孔隙系数P₁的增加而减小，对于孔隙-a梁，其固有频率先减小后增大.GPL-Ⅰ分布是提高结构刚度的最有效的分散方法.GPL的增强效应受到GPL分布和孔隙分布的影响.
弯曲振动的固有频率随W_GPL值的增加而增加，孔隙-a和GPL-Ⅰ分布方式的FG-GPLRMF梁具有最大固有频率.裂纹位置对于梁固有频率的影响是呈现对称性的，距离梁中心位置越近，固有频率受到影响越大.随着裂纹深度的增加，FG-GPLRMF梁的固有频率降低.
参考文献
- [1] WANG Y Q,ZHAO H L.Free vibration analysis of metal foam core sandwich beams on elastic foundation using Chebyshev collocation method [J].Archive of Applied Mechanics,2019,89(11):2335-2349.
- [2] ZHANG Z,DING J,XIA X C,et al.Fabrication and characterization of closed-cell aluminum foams with different contents of multi-walled carbon nanotubes [J].Materials & Design,2015,88:359-365.
- [3] GARCIA-MACIAS E,RODRIGUEZ-TEMBLEQUE L,SAEZ A.Bending and free vibration analysis of functionally graded graphene vs.carbon nanotube reinforced composite plates [J].Composite Structures,2018,186:123-138.
- [4] RAFIEE M A,RAFIEE J,WANG Z,et al.Enhanced mechanical properties of nanocomposites at low graphene content [J].ACS Nano,2009,3(12):3884-3890.
- [5] KITIPORNCHAI S,CHEN D,YANG J.Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets [J].Materials & Design,2017,116:656-665.
- [6] YE C,WANG Y Q.Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells:internal resonances [J].Nonlinear Dynamics,2021,104(3):2051-2069.
- [7] YANG J,CHEN D,KITIPORNCHAI S.Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method [J].Composite Structures,2018,193:281-294.
- [8] DONG Y H,HE L W,WANG L,et al.Buckling of spinning functionally graded graphene reinforced porous nanocomposite cylindrical shells:an analytical study [J].Aerospace Science and Technology,2018,82/83:466-478.
- [9] DONG Y H,LI Y H,CHEN D,et al.Vibration characteristics of functionally graded graphene reinforced porous nanocomposite cylindrical shells with spinning motion [J].Composites Part B:Engineering,2018,145:1-13.
- [10] WANG Y Q,YE C,ZU J W.Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets [J].Aerospace Science and Technology,2019,85:359-370.
- [11] YANG J,CHEN Y.Free vibration and buckling analyses of functionally graded beams with edge cracks [J].Composite Structures,2008,83(1):48-60.
- [12] 刘涛,周洋忻,胡伟鹏.轴向运动功能梯度梁横向振动问题的保结构分析[J].动力学与控制学报,2022,20(6):101-105.LIU T,ZHOU Y X,HU W P.Structure-preserving analysis on transverse vibration of functionally graded beam with an axial velocity [J].Journal of Dynamics and Control,2022,20(6):101-105.(in Chinese)
- [13] 周磊.含裂纹石墨烯增强功能梯度结构的动力学行为分析[D].镇江:江苏大学,2022.ZHOU L.Dynamic analysis of cracked functionally graded graphene nanoplatelet-reinforced structures [D].Zhenjiang:Jiangsu University,2022.(in Chinese)
- [14] 刘金建,蔡改改,谢锋,等.轴向运动功能梯度粘弹性梁横向振动的稳定性分析[J].动力学与控制学报,2016,14(6):533-541.LIU J J,CAI G G,XIE F,et al.Stability analysis on transverse vibration of axially moving functionally graded viscoelastic beams [J].Journal of Dynamics and Control,2016,14(6):533-541.(in Chinese)
- [15] XU H,WANG Y Q,ZHANG Y F.Free vibration of functionally graded graphene platelet-reinforced porous beams with spinning movement via differential transformation method [J].Archive of Applied Mechanics,2021,91(12):4817-4834.
- [16] DIMAROGONAS A D.Vibration of cracked structures:a state of the art review [J].Engineering Fracture Mechanics,1996,55(5):831-857.
- [17] CHONDROS T G,DIMAROGONAS A D,YAO J.A continuous cracked beam vibration theory [J].Journal of Sound and Vibration,1998,215(1):17-34.
- [18] BROEK D.Elementary engineering fracture mechanics [M].Dordrecht,Netherland:Martinus Nijhoff Publishers,1986.
- [19] ERDOGAN F,WU B H.The surface crack problem for a plate with functionally graded properties [J].Journal of Applied Mechanics,1997,64(3):449.
- [20] KITIPORNCHAI S,KE L L,YANG J,et al.Nonlinear vibration of edge cracked functionally graded Timoshenko beams [J].Journal of Sound and Vibration,2009,324(3/4/5):962-982.
- [21] GORMAN D J.Free Vibration Analysis of Beams and Shafts [M].Research Supported by the National Research Council of Canada.New York:Wiley-Interscience,1975.

基本信息

中图分类号: V21;O39
文献标识码: A
DOI: 10.6052/1672-6553-2023-025
文章编号: 1672-6553-2024-22(3)-048-008

基金信息

国家自然科学基金资助项目 (12072201)；
National Natural Science Foundation of China (12072201)；

引用信息

稿件历史

收稿日期: 2023-02-22

参考文献

[1] WANG Y Q,ZHAO H L.Free vibration analysis of metal foam core sandwich beams on elastic foundation using Chebyshev collocation method [J].Archive of Applied Mechanics,2019,89(11):2335-2349.
[2] ZHANG Z,DING J,XIA X C,et al.Fabrication and characterization of closed-cell aluminum foams with different contents of multi-walled carbon nanotubes [J].Materials & Design,2015,88:359-365.
[3] GARCIA-MACIAS E,RODRIGUEZ-TEMBLEQUE L,SAEZ A.Bending and free vibration analysis of functionally graded graphene vs.carbon nanotube reinforced composite plates [J].Composite Structures,2018,186:123-138.
[4] RAFIEE M A,RAFIEE J,WANG Z,et al.Enhanced mechanical properties of nanocomposites at low graphene content [J].ACS Nano,2009,3(12):3884-3890.
[5] KITIPORNCHAI S,CHEN D,YANG J.Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets [J].Materials & Design,2017,116:656-665.
[6] YE C,WANG Y Q.Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells:internal resonances [J].Nonlinear Dynamics,2021,104(3):2051-2069.
[7] YANG J,CHEN D,KITIPORNCHAI S.Buckling and free vibration analyses of functionally graded graphene reinforced porous nanocomposite plates based on Chebyshev-Ritz method [J].Composite Structures,2018,193:281-294.
[8] DONG Y H,HE L W,WANG L,et al.Buckling of spinning functionally graded graphene reinforced porous nanocomposite cylindrical shells:an analytical study [J].Aerospace Science and Technology,2018,82/83:466-478.
[9] DONG Y H,LI Y H,CHEN D,et al.Vibration characteristics of functionally graded graphene reinforced porous nanocomposite cylindrical shells with spinning motion [J].Composites Part B:Engineering,2018,145:1-13.
[10] WANG Y Q,YE C,ZU J W.Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets [J].Aerospace Science and Technology,2019,85:359-370.
[11] YANG J,CHEN Y.Free vibration and buckling analyses of functionally graded beams with edge cracks [J].Composite Structures,2008,83(1):48-60.
[12] 刘涛,周洋忻,胡伟鹏.轴向运动功能梯度梁横向振动问题的保结构分析[J].动力学与控制学报,2022,20(6):101-105.LIU T,ZHOU Y X,HU W P.Structure-preserving analysis on transverse vibration of functionally graded beam with an axial velocity [J].Journal of Dynamics and Control,2022,20(6):101-105.(in Chinese)
[13] 周磊.含裂纹石墨烯增强功能梯度结构的动力学行为分析[D].镇江:江苏大学,2022.ZHOU L.Dynamic analysis of cracked functionally graded graphene nanoplatelet-reinforced structures [D].Zhenjiang:Jiangsu University,2022.(in Chinese)
[14] 刘金建,蔡改改,谢锋,等.轴向运动功能梯度粘弹性梁横向振动的稳定性分析[J].动力学与控制学报,2016,14(6):533-541.LIU J J,CAI G G,XIE F,et al.Stability analysis on transverse vibration of axially moving functionally graded viscoelastic beams [J].Journal of Dynamics and Control,2016,14(6):533-541.(in Chinese)
[15] XU H,WANG Y Q,ZHANG Y F.Free vibration of functionally graded graphene platelet-reinforced porous beams with spinning movement via differential transformation method [J].Archive of Applied Mechanics,2021,91(12):4817-4834.
[16] DIMAROGONAS A D.Vibration of cracked structures:a state of the art review [J].Engineering Fracture Mechanics,1996,55(5):831-857.
[17] CHONDROS T G,DIMAROGONAS A D,YAO J.A continuous cracked beam vibration theory [J].Journal of Sound and Vibration,1998,215(1):17-34.
[18] BROEK D.Elementary engineering fracture mechanics [M].Dordrecht,Netherland:Martinus Nijhoff Publishers,1986.
[19] ERDOGAN F,WU B H.The surface crack problem for a plate with functionally graded properties [J].Journal of Applied Mechanics,1997,64(3):449.
[20] KITIPORNCHAI S,KE L L,YANG J,et al.Nonlinear vibration of edge cracked functionally graded Timoshenko beams [J].Journal of Sound and Vibration,2009,324(3/4/5):962-982.
[21] GORMAN D J.Free Vibration Analysis of Beams and Shafts [M].Research Supported by the National Research Council of Canada.New York:Wiley-Interscience,1975.

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带裂纹石墨烯增强金属泡沫梁的振动特性研究*

Study on Vibration Characteristics of Cracked Metal Foam Beams Reinforced with Graphene*

摘要

Abstract

关键词

Keywords

1 材料特性与旋转弹簧模型

1.1 材料特性

(1)

(2)

1.2 旋转弹簧模型

(3)

(4)

(5)

2 运动方程与微分变换法

2.1 运动方程

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

2.2 微分变换法

(18)

(19)

(20)

(21)

3 结果和讨论

4 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

带裂纹石墨烯增强金属泡沫梁的振动特性研究^*

Study on Vibration Characteristics of Cracked Metal Foam Beams Reinforced with Graphene^*