初始几何缺陷对含裂纹的双向功能梯度梁振动特性研究

史迎辉; 唐怀平; 赵英治

doi:10.21656/1000-0887.450213

初始几何缺陷对含裂纹的双向功能梯度梁振动特性研究

doi: 10.21656/1000-0887.450213

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 51778548

详细信息

作者简介:
史迎辉(1999—)，男，硕士生(E-mail: syhui@my.swjtu.edu.cn)

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: 36
- HTML全文浏览量: 12
- PDF下载量: 3
- 被引次数: 0
出版历程
- 收稿日期: 2024-07-15
- 修回日期: 2025-06-07
- 刊出日期: 2025-09-01

Study on Natural Vibration Characteristics of Bidirectional Functionally Graded Material Beams With Cracks and Geometric Imperfections

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

摘要

摘要:
基于Timoshenko梁理论，考虑初始几何缺陷与裂纹，利用Hamilton原理推导了含初始几何缺陷与裂纹的双向功能梯度梁的振动控制方程，使用无质量扭转弹簧模型模拟裂纹，采用微分求积法对结构控制方程求解. 探究了初始几何缺陷类型、几何缺陷的无量纲振幅、裂纹深度等因素对梁结构自由振动无量纲频率的影响. 结果表明：在一定轴向功能梯度指标下，无量纲基频随初始几何缺陷无量纲振幅增大而增大，随裂纹深度的增加而减小，且全局缺陷对无量纲基频的影响要大于正弦缺陷.
- 双向功能梯度梁 /
- 初始几何缺陷 /
- 裂纹 /
- 无质量扭转弹簧 /
- 微分求积法
Abstract:
Based on the Timoshenko beam theory, in view of geometric imperfections and cracks, the governing equations for bi-directional functionally graded beams were derived under the Hamiltonian principle. The cracked section was modeled with a massless elastic rotational spring, and the governing equations were solved with the differential quadrature method. The effects of geometric imperfection types, geometric imperfection dimensionless vibration amplitudes, crack depths, and other factors on the dimensionless frequencies were explored. The results show that, for a certain axial functional gradient index value, the dimensionless fundamental frequency increases with the dimensionless amplitude of the geometric imperfection, and decreases with the of crack depth. Moreover, the influence of the global imperfection on the dimensionless fundamental frequency is greater than the sine imperfection.
- bi-directional functionally graded beam /
- geometric imperfection /
- crack /
- massless elastic rotational spring /
- differential quadrature method

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图 1 双向功能梯度梁几何尺寸及坐标系

Figure 1. The geometry and coordinates of the bi-directional functionally graded beam

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图 2 无质量弹性旋转弹簧模拟FGM裂纹梁的裂纹

Figure 2. The crack modeled with a massless elastic rotational spring

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图 3 弯曲缺陷梁与不同缺陷类型

Figure 3. The bending imperfect beam and different imperfection modes

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图 4 几何缺陷无量纲振幅A_c与轴向功能梯度指标β共同对双向FGM裂纹梁无量纲基频的影响

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 4. The effects of A_c and β on the dimensionless 1st order frequency of the bi-directional FGM cracked beam

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图 5 裂纹深度a对双向FGM裂纹梁无量纲基频的影响

Figure 5. The effects of crack depth a on the dimensionless 1st order frequency of the bi-directional FGM cracked beam

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图 6 初始几何缺陷的无量纲振幅A_c对双向FGM裂纹梁无量纲基频的影响

Figure 6. The effects of dimensionless amplitude A_c of the geometric imperfection on the dimensionless 1st order frequency of the bi-directional FGM cracked beam

下载: 全尺寸图片幻灯片

表 1 含初始几何缺陷双向FGM裂纹梁无量纲基频收敛性分析

Table 1. Convergence verification of dimensionless the 1st order frequency of the bi-directional FGM cracked beam with geometric imperfection

	N
	9	11	13	15	17	19	21	23
Ω	9.932 7	9.974 0	9.969 3	9.969 6	9.969 6	9.969 6	9.969 6	9.969 6

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表 2 含初始几何缺陷无裂纹均质梁无量纲基频理论计算与仿真结果对比

Table 2. Comparison of dimensionless 1st order frequencies between theoretical calculations and simulation results of the crackless homogeneous material beam with geometric imperfection

imperfection	method	A_c
imperfection	method	0	0.2	0.4	0.6	0.8	1
sine	DQM	6.454 1	6.599 3	7.016 7	7.661 8	8.482 6	9.432 9
sine	ANSYS	6.454 2	6.599 4	7.015 7	7.660 0	8.479 5	9.427 4
global	DQM	6.454 1	7.154 9	8.872 7	11.004 4	13.169 1	15.162 4
global	ANSYS	6.454 2	7.154 1	8.868 3	10.994 4	13.150 6	15.132 8

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表 3 无初始几何缺陷双向FGM裂纹梁无量纲基频与文献对比

Table 3. Comparison of dimensionless 1st order frequencies of the bi-directional FGM cracked beam without geometric imperfection

E₂/E₁	this paper	ref. [22]	ref. [23]
0.2	0.369 4	0.361 0	0.360 2
1	0.392 2	0.384 8	0.384 0
5	0.369 9	0.364 9	0.364 1

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表 4 不同裂纹位置与不同缺陷无量纲振幅对双向FGM梁无量纲基频影响

Table 4. The influences of different crack positions and imperfection degrees on the dimensionless 1st order frequency of the bi-directional FGM beam

crack location	imperfection	A_c
crack location	imperfection	0.2	0.4	0.6	0.8	1
n=0.25	sine	6.196 8	6.580 2	7.173 7	7.927 1	8.794 1
n=0.25	global	6.666 1	8.139 7	9.932 3	11.684 0	13.209 3
n=0.5	sine	6.176 2	6.558 7	7.151 3	7.903 3	8.768 1
n=0.5	global	6.650 6	8.139 5	9.952 5	11.725 7	13.271 5
n=0.75	sine	6.206 8	6.593 4	7.189 0	7.943 1	8.809 7
n=0.75	global	6.682 2	8.165 5	9.966 5	11.725 8	13.258 9

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