弹性地基上多孔二维功能梯度材料微梁自由振动研究

赵英治; 唐怀平; 赖泽东; 章家杰

doi:10.21656/1000-0887.440050

弹性地基上多孔二维功能梯度材料微梁自由振动研究

doi: 10.21656/1000-0887.440050

西南交通大学力学与航空航天学院, 成都 610031

基金项目:

国家自然科学基金项目 51778548

详细信息

作者简介:
赵英治(1999—)，男，硕士生(E-mail: 964003289@qq.com)

通讯作者:
唐怀平(1967—)，男，副教授，博士(通讯作者. E-mail: thp-vib@163.com)

中图分类号: O342
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- 文章访问数: 386
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- PDF下载量: 52
- 被引次数: 0
出版历程
- 收稿日期: 2023-02-27
- 修回日期: 2023-04-03
- 刊出日期: 2023-11-01

Free Vibration Analysis of Porous 2D Functionally Graded Material Microbeams on Winkler's Foundation

School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu 610031, P.R.China

摘要

摘要: 基于修正偶应力和Timoshenko梁理论，利用Hamilton变分原理推导了Winkler弹性地基上多孔二维功能梯度材料(2D-FGM)微梁的振动控制方程, 采用微分求积法获得固支-固支(C-C)、简支-简支(S-S)边界条件下微梁的振动频率和基本振型, 对刚度矩阵进行数学处理后极大地提高了计算效率, 将该文模型退化为宏观和微观二维功能梯度模型且与已有文献对比验证其正确性.算例结果表明：该文数学模型适用于不同类型的二维材料分布；微梁的无量纲振动频率随着Winkler弹性地基模量的增大而增大; 在一定Winkler弹性地基模量下, 微梁的无量纲振动频率随着功能梯度指数、轴向功能梯度指数、孔隙率的增大而减小.材料变化对振动模态的影响随着振动模态阶数的增加而增加.同样参数下, 孔隙均匀分布时梁频率略小于孔隙线性分布的情况.
- 修正偶应力理论 /
- 孔隙率 /
- Winkler弹性地基 /
- 二维功能梯度材料 /
- 微分求积法
Abstract: Based on the modified couple stress theory and the Timoshenko beam theory, the governing equations for free vibration of porous 2D functional graded material (FGM) on Winkler's foundation were derived under Hamilton's principle. The differential quadrature method was used to obtain the numerical solutions of the vibration frequencies and fundamental mode shapes of microbeams with both ends clamped (C-C) and simply supported (S-S). The improved stiffness matrix was used to greatly improve the calculation efficiency. The proposed model was degenerated to the macro and micro 2D-FGM models, which were compared with those in previous literatures for validation. The results show that, the present mathematical model is suitable for different types of 2D material distributions. The dimensionless frequencies increase with the dimensionless elastic modulus of Winkler's foundation. Under a certain dimensionless elastic foundation modulus, the dimensionless frequencies decrease with the functionally graded index, the axial functionally graded index and the porosity. The effect of the material variation on the mode shape increases with the mode number. For the same parameter, the dimensionless frequencies of the beam with uniform porosity distribution are slightly lower than those with linear porosity distribution.
- modified couple stress theory /
- porosity /
- Winkler's foundation /
- 2D functionally graded material /
- differential quadrature method

HTML全文

图 1 Winkler弹性地基上多孔2D-FGM微梁几何尺寸及坐标系

Figure 1. The geometry and coordinates of a porous 2D functionally graded microbeam on Winkler's foundation

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图 2 孔隙分布模式

Figure 2. Porosity distribution patterns

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图 3 孔隙率对前三阶无量纲频率的影响(P_z=2, L/h=5, λ=3×10^-6)

Figure 3. Effects of the porosity on the 1st 3 dimensionless frequencies (P_z=2, L/h=5, λ=3×10^-6)

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图 4 不同理论下功能梯度指数对前二阶无量纲频率的影响(b=2h, L=5h, θ=0.2, λ=3×10^-6)

Figure 4. Effects of the functionally graded index on the 1st 2 dimensionless frequencies under different theories (b=2h, L=5h, θ=0.2, λ=3×10^-6)

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图 5 功能梯度指数P_z和P_x共同对弹性地基多孔2D-FGM微梁一阶无量纲频率的影响(θ=0.2, λ=3×10^-6)

Figure 5. The effects of P_z and P_x on the dimensionless fundamental frequency of porous 2D-FGM microbeams(θ=0.2, λ=3×10^-6)

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图 6 跨厚比和功能梯度指数共同对弹性地基多孔2D-FGM梁一阶无量纲频率的影响(孔隙均匀分布, θ=0.2, λ=3×10^-6)

Figure 6. The effects of the span-to-depth ratio and the functionally graded index on the dimensionless fundamental frequencies of porous 2D-FGM microbeams(uniform porosity distribution, θ=0.2, λ=3×10^-6)

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图 7 弹性地基上多孔2D-FGM微梁的前三阶振动模态(C-C边界)

Figure 7. The 1st 3 mode shape of the porous 2D-FGM microbeam on Winkler's foundation (C-C boundary condition)

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图 8 弹性地基上多孔2D-FGM微梁的前三阶振动模态(S-S边界)

Figure 8. The 1st 3 mode shapes of the porous 2D-FGM microbeam on Winkler's foundation (S-S boundary condition)

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表 1 弹性地基多孔2D-FGM微梁无量纲频率收敛性分析(S-S边界, P_x=1, P_z=1, θ=0.1, L/h=5, h=2l, λ=3×10^-6)

Table 1. Convergence verification of dimensionless frequencies of the porous 2D-FGM microbeam on Winkler's foundation (S-S boundary condition, P_x=1, P_z=1, θ=0.1, L/h=5, h=2l, λ=3×10^-6)

	N
	7	9	11	13	15	17	19	21
Ω₁	5.429 7	5.428 5	5.428 3	5.428 3	5.428 3	5.428 3	5.428 3	5.428 3
Ω₂	19.390 2	18.200 1	18.193 4	18.193 2	18.193 2	18.193 2	18.193 2	18.193 2
Ω₃	38.905 4	35.116 2	34.682 9	34.672 5	34.672 4	34.672 4	34.672 4	34.672 4

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表 2 C-C边界条件下2D-FGM微梁一阶振动频率

Table 2. Dimensionless frequencies of the 2D functionally graded microbeam under the C-C boundary condition

model	P_z=0, P_x=0	P_z=1, P_x=0	P_z=0, P_x=1	P_z=1, P_x=1
this paper	28.585 4	23.883 1	22.223 6	19.534 0
ref. [22]	28.577 9	23.677 7	22.427 6	19.546 4

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表 3 基于本文模型的宏观2D-FGM梁无量纲频率与文献中结果的对比(C-C边界)

Table 3. Comparison of the dimensionless fundamental frequencies of the C-C bi-directional functionally graded beam

	model	P_z=0	P_z=2	P_z=4	P_z=6	P_z=8
P_x=0	ref. [31]	6.454 1	5.872 9	4.664 3	3.557 0	2.766 1
P_x=0	this papaer	6.455 2	5.873 7	4.664 7	3.557 1	2.766 1
P_x=2	ref. [31]	6.616 8	6.021 0	4.782 0	3.646 7	2.835 9
P_x=2	this papaer	6.617 9	6.021 8	4.782 3	3.646 9	2.835 9
P_x=4	ref. [31]	7.150 6	6.506 8	5.167 9	3.941 1	3.064 8
P_x=4	this papaer	7.152 3	6.508 1	5.168 6	3.941 4	3.065 0
P_x=6	ref. [31]	8.162 0	7.427 3	5.899 0	4.498 7	3.498 5
P_x=6	this papaer	8.168 3	7.432 7	5.903 0	4.501 6	3.500 6
P_x=8	ref. [31]	9.753 2	8.875 3	7.049 3	5.376 1	4.180 8
P_x=8	this papaer	9.802 4	8.919 9	7.084 5	5.402 8	4.201 5

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表 4 无量纲弹性地基模量对前三阶无量纲频率的影响(P_x=1, P_z=1, θ=0.1, L/h=5, h=2l, 孔隙均匀分布)

Table 4. Effects of the dimensionless elastic foundation modulus on the 1st 3 orders of dimensionless frequencies (P_x=1, P_z=1, θ=0.1, L/h=5, h=2l, uniform porosity distribution)

λ	C-C			S-S
λ	Ω₁	Ω₂	Ω₃	Ω₁	Ω₂	Ω₃
0	10.2652	23.536 3	40.299 4	5.581 7	18.666 0	35.487 9
1×10^-5	10.272 0	23.539 2	40.301 1	5.593 5	18.669 5	35.489 8
2×10^-5	10.278 7	23.542 2	40.302 8	5.605 3	18.673 0	35.491 6
3×10^-5	10.285 5	23.545 1	40.304 4	5.617 2	18.676 5	35.493 5
4×10^-5	10.292 2	23.548 0	40.306 1	5.628 9	18.680 0	35.495 3
5×10^-5	10.299 0	23.550 9	40.307 8	5.640 7	18.683 5	35.497 2
6×10^-5	10.305 7	23.553 8	40.309 5	5.652 4	18.687 0	35.499 1
7×10^-5	10.312 4	23.556 7	40.311 2	5.664 1	18.690 5	35.500 9
8×10^-5	10.319 2	23.559 6	40.312 9	5.675 8	18.694 0	35.502 8
9×10^-5	10.325 9	23.562 5	40.314 6	5.687 5	18.697 5	35.504 6
1×10^-4	10.332 6	23.565 5	40.316 3	5.699 1	18.701 0	35.506 5

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