含多裂纹损伤圆弧曲梁自由振动扰动的有限元网格自适应分析

王永亮; 王建辉; 张磊

doi:10.6052/j.issn.1000-4750.2020.10.0708

含多裂纹损伤圆弧曲梁自由振动扰动的有限元网格自适应分析

doi: 10.6052/j.issn.1000-4750.2020.10.0708

王永亮^{1, 2, ,},
王建辉^1,,
张磊^1,

1.
中国矿业大学(北京)，力学与建筑工程学院，北京 100083
2.
中国矿业大学(北京)，煤炭资源与安全开采国家重点实验室，北京 100083

基金项目: 国家自然科学基金项目(41877275，51608301)；中央高校基本科研业务费专项资金项目(2019QL02)；天津市重点实验室开放课题基金项目(2017SCEEKL003)；中国矿业大学(北京)越崎青年学者计划项目(2019QN14)；大学生创新创业训练计划项目(C201906327，C202006976)

详细信息

作者简介:
王建辉(1997−)，男，河北石家庄人，硕士生，主要从事矿山岩体计算力学的研究(E-mail: wangjianhui@students.cumtb.edu.cn)

张　磊(1998−)，男，河北廊坊人，本科生，主要从事矿山岩体计算力学的研究(E-mail: zhanglei@students.cumtb.edu.cn)

通讯作者:
王永亮(1985−)，男，河北唐山人，副教授，博士，硕导，主要从事矿山岩体计算力学的研究(E-mail: wangyl@cumtb.edu.cn)

中图分类号: O242.21
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出版历程
- 收稿日期: 2020-10-04
- 修回日期: 2020-12-17
- 网络出版日期: 2021-02-08
- 刊出日期: 2021-10-18

ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR FREE VIBRATION DISTURBANCE OF CIRCULARLY CURVED BEAMS WITH MULTIPLE CRACKS

WANG Yong-liang^{1, 2
, ,},
WANG Jian-hui^1
,,
ZHANG Lei^1
,

1.
School of Mechanical and Civil Engineering, China University of Mining and Technology (Beijing), Beijing 100083, China
2.
State Key Laboratory of Coal Resources and Safe Mining, China University of Mining and Technology (Beijing), Beijing 100083, China

摘要

摘要: 该文建立圆弧形曲梁裂纹的截面损伤缺陷比拟方案，实施微裂纹损伤诱发截面弱化，实现多裂纹深度、位置、数目的模拟。引入变截面Timoshenko梁的h型有限元网格自适应分析方法，求解含裂纹损伤圆弧曲梁自由振动问题，得到优化的网格和满足预设误差限的高精度自振频率和振型解答，研究多裂纹损伤对圆弧曲梁振型的扰动行为。数值算例表明，该算法中网格非均匀加密可适应裂纹损伤引起的振型变化，应用于各类曲梁夹角和裂纹损伤分布工况下的自由振动研究，定量分析了多裂纹损伤深度、数目、分布对圆弧曲梁自振频率和振型的扰动影响，检验了该文算法的精确性和实用性。
- 裂纹损伤 /
- 圆弧曲梁 /
- 自由振动 /
- 振型扰动 /
- 网格自适应划分 /
- 有限元法
Abstract: The scheme for the cross-section damage defects in a circularly curved beam is established to simulate the depth, the location and the number of multiple cracks, through implementing cross-section reduction induced by microcrack damage. The h-version adaptive finite element method of a variable cross-section Timoshenko beam is introduced to solve the free vibration of a circularly curved beam with cracks damage. Using the proposed method, the final optimized meshes and high-precision solution of natural frequency and mode shape satisfying the preset error tolerance can be obtained, and the disturbance behavior of multi crack damage on the vibration mode of a circularly curved beam is studied. Numerical examples show that the non-uniform mesh refinement can adapt to the change of mode of vibration induced by crack damage, which is applied to the free vibration research of various circularly curved beams with included angles and crack damage distribution conditions. Furthermore, the influences of crack damage depth, crack damage number, and crack damage distribution on the natural frequency and mode of vibration of a circularly curved beam are quantitatively analyzed, and the accuracy and practicability of the proposed algorithm are verified.
- crack damage /
- circularly curved beams /
- free vibration /
- mode disturbance /
- adaptive mesh refinement /
- finite element method

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图 1 含裂纹损伤曲梁坐标系和符号

Figure 1. Coordinate systems and symbols of cracked curved beam

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图 2 无裂纹损伤1/4圆弧曲梁模型

Figure 2. Model of a quarter of uncracked circularly curved beam

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图 3 裂纹损伤1/4圆弧曲梁振型

Figure 3. Vibration modes of a quarter of uncracked circularly curved beam

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图 4 单裂纹损伤曲梁不同裂纹深度(α=0.0、0.16、0.5，β=0. 6125)模型

Figure 4. Model of curved beam with single crack in different depth cases (α=0.0, 0.16, 0.5, β =0.6125)

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图 5 裂纹深度对自振频率扰动影响

Figure 5. Disturbance influence of crack depth on natural frequencies

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图 6 单裂纹损伤曲梁裂纹深度α=0.5下振型

Figure 6. Vibration modes of curved beam with single crack in depth case α=0.5

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图 7 单裂纹损伤曲梁裂纹深度α=0.16下振型扰动

Figure 7. Vibration modes disturbance of curved beam with single crack in depth case α=0.16

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图 8 单裂纹损伤曲梁裂纹深度α=0.5下振型扰动

Figure 8. Vibration modes disturbance of curved beam with single crack in depth case α=0.5

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图 9 多裂纹损伤曲梁不同裂纹数目(n=2、3、4)模型

Figure 9. Model of curved beam with multiple cracks in different number cases (n=2, 3, 4)

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图 10 裂纹数目对自振频率扰动影响

Figure 10. Disturbance influence of crack number on natural frequencies

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图 11 多裂纹损伤曲梁不同裂纹数目振型扰动

Figure 11. Vibration modes disturbance of curved beam with multiple cracks in different number cases

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图 12 多裂纹损伤曲梁不同裂纹分布模型

Figure 12. Model of curved beam with multiple cracks in different distribution cases

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图 13 裂纹分布对自振频率扰动影响

Figure 13. Disturbance influence of crack distribution on natural frequencies

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图 14 多裂纹损伤曲梁不同裂纹分布振型扰动

Figure 14. Vibration modes disturbance of curved beam with multiple cracks in different distribution cases

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表 1 无裂纹损伤1/4圆弧曲梁自振频率值

Table 1. Natural frequencies of a quarter of uncracked circularly curved beam

阶次k	频率值		单元数	误差/(×10⁻⁶)
阶次k	$\omega _k^{\rm{h}}$	$\omega _k^{\rm{h}}$^[29]	单元数	误差/(×10⁻⁶)
1	1.641 294 01	1.641 294 00	20	0.004
2	1.894 921 53	1.894 920 42	20	0.383
3	2.888 817 91	2.888 817 87	28	0.010
4	3.929 098 42	3.929 098 42	32	0.205
5	4.462 234 01	4.462 234 00	32	0.002

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表 2 单裂纹损伤曲梁不同裂纹深度自振频率值

Table 2. Natural frequencies of curved beam with single crack in different depth cases

阶次k	频率值(α=0.0)			误差/(×10⁻²)	频率值(α=0.16)			误差/(×10⁻²)	频率值(α=0.5)			误差/(×10⁻²)
阶次k	$\omega _{0.0,k}^{\rm{h}}$	$\omega _{0.0,k}^{\rm{h}}$^[14]	$\omega _{0.0,k}^{\rm{h}}$^[6]	误差/(×10⁻²)	$\omega _{0.16,k}^{\rm{h}}$	$\omega _{0.16,k}^{\rm{h}}$^[14]	$\omega _{0.16,k}^{\rm{h}}$^[6]	误差/(×10⁻²)	$\omega _{0.5,k}^{\rm{h}}$	$\omega _{0.5,k}^{\rm{h}}$^[14]	$\omega _{0.5,k}^{\rm{h}}$^[6]	误差/(×10⁻²)
1	24.44506	24.45	24.52	0.02	24.42725	24.44	24.48	0.05	24.21468	24.34	24.32	0.49
2	61.69788	61.73	61.78	0.05	61.67788	61.72	61.72	0.07	61.35527	61.68	61.60	0.52
3	119.07079	119.19	118.05	0.10	118.83333	119.16	117.76	0.27	116.41556	118.27	116.43	1.55
4	188.20437	188.51	184.11	0.16	188.20171	188.49	184.03	0.15	188.10660	188.09	183.81	0.88
5	277.18047	277.87	269.02	0.25	276.75348	277.85	268.24	0.39	272.57374	277.25	267.94	1.68

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表 3 多裂纹损伤数目和位置工况

Table 3. Number and location of multiple cracks damage

工况	裂纹数目n	裂纹位置β
Ⅰ	2	0.3333、0.6667
Ⅱ	3	0.2500、0.5000、0.7500
Ⅲ	4	0.2000、0.4000、0.6000、0.8000

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表 4 多裂纹损伤曲梁不同裂纹数目自振频率值

Table 4. Natural frequencies of curved beam with multiple cracks in different number cases

阶次k	频率值
阶次k	$\omega _k^{\rm{h}}$，n=2	$\omega _k^{\rm{h}}$，n=3	$\omega _k^{\rm{h}}$，n=4
1	48.535 332 13	47.902 077 58	47.343 023 95
2	153.660 750 2	146.651 275 1	144.728 906 6
3	299.120 913 1	310.041 195 0	292.354 186 0
4	489.649 892 8	482.873 507 0	505.824 971 0
5	751.539 627 1	719.674 070 5	712.041 973 3
10	2545.139 009 0	2515.668 639 0	2502.870 283 0
20	7706.363 308 0	7603.176 785 0	7520.409 513 0
30	16 025.943 200 0	16 158.552 250 0	16 094.761 810 0
40	26 133.888 910 0	26 108.549 720 0	26 050.598 210 0
50	38 202.443 860 0	37 858.824 340 0	37 728.935 150 0

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表 5 多裂纹损伤曲梁不同裂纹分布自振频率值

Table 5. Natural frequencies of curved beam with multiple cracks in different distribution cases

阶次k	频率值
阶次k	$\omega _{{\rm{u}}k}^h$多裂纹均匀分布	$\omega _{{\rm{c}}k}^h$多裂纹左侧集中分布	频率差 $\omega _{{\rm{c}}k}^h - \omega _{{\rm{u}}k}^h$
1	46.808 799 67	47.106 856 43	0.298 06
2	143.036 785 3	144.704 910 70	1.668 13
3	288.869 077 2	290.343 790 10	1.474 71
4	471.643 268 9	477.601 844 00	5.958 58
5	751.792 226 0	703.902 905 80	−47.889 32
10	2461.536 651 0	2513.568 991 00	52.032 34
20	7578.276 595 0	7536.887 673 00	−41.388 92
30	16 051.321 530 0	16 049.900 190 00	−1.421 34
40	26 044.720 800 0	25 984.653 450 00	−60.067 35
50	37 833.740 600 0	37 557.169 530 00	−276.571 07

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