含初始变形的多孔功能梯度矩形板自由振动研究

王煜程; 高芳清; 邓鹤轩; 古定翱

doi:10.21656/1000-0887.450227

含初始变形的多孔功能梯度矩形板自由振动研究

doi: 10.21656/1000-0887.450227

西南交通大学力学与航空航天学院，成都 610031

详细信息

作者简介:
王煜程(2000—), 男，硕士生(E-mail: 997757358@qq.com)

通讯作者:
高芳清(1969—), 男，副教授(通讯作者. E-mail: gfq_vib@163.com)

中图分类号: O327
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- HTML全文浏览量: 16
- PDF下载量: 12
- 被引次数: 0
出版历程
- 收稿日期: 2024-08-06
- 修回日期: 2024-12-02
- 刊出日期: 2025-11-01

Study of Free Vibration of Porous Functionally Graded Rectangular Plates With Initial Deformation

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

摘要

摘要: 面向多参数动力影响的功能梯度板定型前高效设计优化需求，建立了统一考虑初始变形、孔隙分布和梯度效应耦合影响的功能梯度板动力分析理论框架与模型. 采用一阶剪切变形理论下的位移场，描述了梯度材料厚度方向变化所致横向剪切效应. 使用Chebyshev-Ritz法对位移进行离散，推导出了含初始变形的多孔功能梯度板的能量泛函表达式. 根据最小能量原理得出振动系统的特征方程，并通过数值求解分析了多功能梯度板的自由振动特性. 通过与文献结果和有限元分析的对比，验证了该方法的收敛性和准确性. 结果表明：初始变形的引入会显著影响多孔功能梯度矩形板的自振频率，此外，孔隙、梯度指数等参数也会影响板的自振频率.
- 初始变形 /
- 自由振动 /
- 功能梯度 /
- 孔隙率 /
- Chebyshev-Ritz法
Abstract: To meet the efficient design optimization requirements of functionally graded plates under multi-parameter dynamic effects, a unified theoretical framework and model for dynamic analysis was established, under the coupled influences of initial deformation, pore distribution, and gradient effects. The displacement field was described with the 1st-order shear deformation theory to account for the transverse shear effects caused by the graded material variation in the thickness direction. The displacement was discretized with the Chebyshev-Ritz method, and the energy functional expression of the porous functionally graded plate with initial deformation was derived. The characteristic equation of the vibration system was derived based on the principle of minimum energy, and the free vibration properties of the multi-functional graded plate were analyzed through numerical solutions. The convergence and accuracy of this method were validated by comparison with results from previous literatures and finite element analysis. The results show that, the introduction of initial deformation will significantly affect the natural vibration frequencies of porous functional gradient rectangular plates, in addition to the porosity, gradient indexes and other parameters will also affect the natural vibration frequency of the plate.
- initial deformation /
- free vibration /
- functional gradient /
- porosity /
- Chebyshev-Ritz method

HTML全文

图 1 多孔FGM矩形板

Figure 1. The porous FGM rectangular plate

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图 2 孔隙分布模式

Figure 2. Pore distribution patterns

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图 3 FEM法与Chebyshev-Ritz法结果对比

Figure 3. Comparison of results between the FEM method and the Chebyshev-Ritz method

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图 4 S-S-S-S边界下梯度指数和孔隙对无量纲基频的影响

Figure 4. The influence of gradient indexes and porosities on dimensionless fundamental frequency under the S-S-S-S boundary

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图 5 C-C-C-C边界下梯度指数和孔隙对无量纲基频的影响

Figure 5. The influence of porosities and gradient indexes on dimensionless fundamental frequencies under the C-C-C-C boundary

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图 6 S-S-S-S边界下变形指标对无量纲基频的影响

Figure 6. The influence of deformation indexes under S-S-S-S boundary on dimensionless fundamental frequencies

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图 7 S-S-S-S边界下变形阶次对无量纲基频的影响

Figure 7. The influence of deformation orders on dimensionless fundamental frequencies under the S-S-S-S boundary

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表 1 不同情况下的辅助函数系数

Table 1. Auxiliary function coefficients in different situations

pu, qu	ru, su	pv, qv	rv, sv	pw, qw	rw, sw	px, qx	rx, sx	py, qy	ry, sy
C-C-C-C	uniaxial buckling, biaxial buckling and shear buckling analysis
0	0	0	0	1	1	1	1	1	1
C-C-C-C	bending, free vibration analysis
1	1	1	1	1	1	1	1	1	1
S-S-S-S	uniaxial buckling, biaxial buckling, bending and free vibration analysis
0	1	1	0	1	1	1	0	1	0
S-S-S-S	shear buckling analysis
0	0	0	0	1	1	0	0	0	0

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表 2 FGM的组成特性

Table 2. Constituent properties of FGM

material	constituent	property
material	constituent	E/(N/m²)	μ	ρ/(kg/m³)
metal	aluminum (Al)	7×10¹⁰	0.30	2 707
ceramic	alumina (Al₂O3)	3.8×10¹¹	0.30	3 800

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表 3 无初始变形时无量纲频率收敛情况

Table 3. Convergence of dimensionless frequencies without initial deformation

boundary	C-C-C-C			S-S-S-S
N=M	mode 1	mode 2	mode 4	mode 1	mode 2	mode 4
6	9.844 3	18.778 4	26.317 7	5.769 3	13.798 3	19.483 3
7	9.842 6	18.775 6	26.317 4	5.769 3	13.763 8	19.483 3
8	9.842 6	18.774 2	26.312 7	5.769 3	13.763 8	19.483 3
9	9.842 3	18.773 9	26.312 2	5.769 3	13.763 7	19.483 3
10	9.842 3	18.773 8	26.312 0	5.769 3	13.763 7	19.483 3
FEM	9.855 2	18.840 8	26.430 2	5.733 1	13.704 9	19.482 3
ref. [25]	9.871 0	18.881 4	26.406 2	5.761 9	13.798 0	19.494 4

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表 4 梯度指数对无量纲频率的影响(h/a=0.1, θ=0)

Table 4. The impact of gradient indexes on dimensionless frequencies (h/a=0.1, θ=0)

	BC	C-C-C-C			S-S-S-S
	n	mode 1	mode 2	mode 4	mode 1	mode 2	mode 4
present	0	9.842 3	18.773 9	26.312 6	5.769 3	13.798 3	19.483 3
	1	7.603 5	14.595 3	20.531 8	4.417 5	10.612 3	16.195 2
	2	6.901 3	13.229 8	18.593 8	4.012 0	9.626 0	14.624 3
	5	6.448 3	12.263 4	17.153 9	3.785 3	9.027 1	12.630 9
	10	6.182 3	11.700 3	16.322 0	3.655 0	8.685 1	11.516 6
ref. [25]	0	9.871 0	18.881 4	26.406 2	5.761 9	13.798 0	19.494 4
	1	7.628 0	14.686 2	20.613 9	4.410 6	10.613 0	16.204 4
	2	6.923 3	13.311 7	18.667 3	4.005 9	9.626 6	14.632 6
	5	6.466 7	12.332 6	17.213 5	3.780 6	9.026 7	12.638 1
	10	6.198 8	11.762 5	16.374 3	3.651 0	8.684 3	11.523 2

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表 5 长厚比和孔隙对多孔FGM无量纲基频ω^*的影响(n=0)

Table 5. The influence of aspect ratios and porosities on dimensionless fundamental frequency ω^* of the porous FGM (n=0)

λ	θ	boundary
		S-S-S-S		C-C-C-C
		uniform	linear	uniform	linear
10	0	0.113 5	0.113 5	0.193 5	0.193 5
	0.1	0.114 1	0.114 8	0.195 2	0.195 9
	0.2	0.115 2	0.116 3	0.197 5	0.198 4
	0.3	0.116 8	0.117 9	0.200 6	0.201 2
12.5	0	0.073 5	0.073 5	0.128 0	0.128 0
	0.1	0.073 9	0.074 4	0.129 0	0.129 6
	0.2	0.074 6	0.075 4	0.130 4	0.131 3
	0.3	0.075 6	0.076 4	0.132 3	0.133 1
15	0	0.051 4	0.051 4	0.090 7	0.090 7
	0.1	0.051 7	0.052 0	0.091 3	0.091 8
	0.2	0.052 1	0.052 7	0.092 2	0.093 0
	0.3	0.052 8	0.053 4	0.093 5	0.094 3

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