ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR FREE VIBRATION DISTURBANCE OF CIRCULARLY CURVED BEAMS WITH MULTIPLE CRACKS
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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。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.
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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) $\omega _k^{\rm{h}}$ $\omega _k^{\rm{h}}$[29] 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 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) $\omega _{0.0,k}^{\rm{h}}$ $\omega _{0.0,k}^{\rm{h}}$[14] $\omega _{0.0,k}^{\rm{h}}$[6] $\omega _{0.16,k}^{\rm{h}}$ $\omega _{0.16,k}^{\rm{h}}$[14] $\omega _{0.16,k}^{\rm{h}}$[6] $\omega _{0.5,k}^{\rm{h}}$ $\omega _{0.5,k}^{\rm{h}}$[14] $\omega _{0.5,k}^{\rm{h}}$[6] 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 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 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 $\omega _k^{\rm{h}}$,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 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 $\omega _{{\rm{u}}k}^h$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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