Study on the evolution of crack extension frequency characteristics under traveling wave resonance of bevel gears
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摘要:
针对某型航空发动机中央传动锥齿轮在实际工况下裂纹扩展断裂失效问题，采取实验与数值仿真相结合的方法，探究齿轮处于行波共振状态下裂纹扩展频率特性演化规律。基于行波共振理论与声测法，开展了正常齿轮和预制缺陷齿轮疲劳特性实验，实验结果表明：预制缺陷齿轮处于行波共振状态下裂纹迅速扩展，共振频率不断减小，裂纹扩展后期无法跟踪行波共振点，齿轮发生瞬断。基于模态分析方法，建立了不同裂纹尺度齿轮有限元模型，提出了裂纹扩展前期裂纹长度与模态频率比的函数关系，实验所测频率比与预测模态频率比相对误差在0.02%以内。仿真结果表明：随着裂纹程度的加深，齿轮结构不对称性加大，相同节径型振动的两个对称模态逐渐分化为两个不同的振动模态，模态频率比差值增大。综合实验与仿真结果，裂纹扩展前期行波共振导致裂纹尖端应力强度因子过大而促使裂纹发生扩展，裂纹扩展后期齿轮两对称模态交替出现，齿轮无法维持行波共振状态，由于裂尖应力强度因子接近或超过材料断裂韧性导致裂纹快速扩展，进而剩余强度不足而瞬断。
Abstract:Considering the problem of crack extension and fracture failure of central transmission bevel gear of a certain type of aero-engine under actual working conditions, a combination of experimental and numerical simulation was adopted to investigate the crack extension frequency characteristic evolution law of the gear in traveling wave resonance state. Based on the traveling wave resonance theory and acoustic measurement method, fatigue characteristic experiments of normal gears and prefabricated defective gears were carried out, and the acoustic signal tracking method was adopted to make the gears always in the state of traveling wave resonance. The experimental results showed that: the cracks of prefabricated defective gears in the state of traveling wave resonance were expanded rapidly, the resonance frequency decreased continuously, and the gears cannot be tracked to the point of traveling wave resonance at the late stage of the crack expansion, and the gears were instantaneously broken. Based on the modal analysis method, a finite element model of gears with different crack scales was established, and the functional relationship between crack length and modal frequency ratio in the pre-crack expansion stage was proposed, which can accurately predict the state of crack expansion in the pre-crack expansion stage with a relative error of 0.02% or less. The simulation results showed that with the deepening of the crack, the intrinsic frequency of the two symmetric modes of the same node diameters vibration of the gear decreased at the same time, but due to the increase of the asymmetry of the gear structure, it was gradually divided into two different vibration modes, and the difference of the intrinsic frequency of the two symmetric vibration modes became larger. Comprehensive experimental and simulation results showed that the traveling wave resonance in the early stage of crack expansion led to too large crack tip stress intensity factor, which caused crack expansion; then two symmetric modes appeared alternately, the gear can not maintain the traveling wave resonance state; as the crack tip stress intensity factor was close to or more than the fracture toughness of the material leading to rapid expansion of the crack, the residual strength was insufficient, leading to transient breakage.
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Key words:
- traveling wave resonance /
- crack extension /
- acoustic method /
- modal analysis /
- frequency ratio
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图 2 基于静坐标系的共振分析坎贝尔图
Figure 2. Campbell diagram for resonance analysis based on a static coordinate system
图 3 中央传动锥齿轮行波共振噪声监测实验系统框图
Figure 3. Block diagram of central drive bevel gear traveling wave resonance noise monitoring system
图 4 从动弧齿锥齿轮行波共振疲劳性能实验现场
Figure 4. Driven spiral bevel gear travelling wave resonance fatigue performance experiment site
图 6 三节径前行波共振时噪声信号最大点频谱
Figure 6. Spectrum of the maximum point of the noise signal at the resonance of the three-node diameter front traveling wave
图 7 四节径后行波共振时噪声信号最大点频谱
Figure 7. Spectrum of the maximum point of the noise signal at the resonance of the four-node diameter back traveling wave
图 9 弧齿锥齿轮实验初始预制裂纹（单位：mm）
Figure 9. Experimental initial prefabrication cracking of bevel gears （unit：mm）
图 10 齿轮三节径前行波共振裂纹扩展路径
Figure 10. Gear three-node diameter front traveling wave resonance crack extension paths
图 11 三节径前行波共振断裂前噪声频谱
Figure 11. Three-node diameter front traveling wave resonance pre-fracture noise spectrum
图 12 齿轮四节径后行波共振裂纹扩展路径
Figure 12. Gear four-node diameter back traveling wave resonance crack extension paths
图 13 四节径后行波共振断裂前噪声频谱
Figure 13. Four-node diameter back traveling wave resonance pre-fracture noise spectrum
图 14 弧齿锥齿轮仿真初始预制裂纹
Figure 14. Initial prefabrication cracking for bevel gear simulation
图 15 三节径裂纹扩展过程与端点定义
Figure 15. Three-node diameter crack extension process and endpoint definition
图 16 四节径裂纹扩展过程与端点定义
Figure 16. Four-node diameter crack extension process and endpoint definition
图 17 正常齿轮三节径和四节径振动对称振型云图
Figure 17. Symmetric mode shapes of normal gears with three and four-node diameter vibration
图 22 行波共振疲劳特性实验最后裂纹状态
Figure 22. Final crack state of traveling wave resonance fatigue characterization experiments
图 24 实验最后裂纹状态应力分布云图
Figure 24. Cloud view of stress distribution in the cracked state at the end of the experiment
表 1 由静频预测的行波共振转速和频率
Table 1. Traveling wave resonance speed and frequency predicted from static frequency
振动
模态f /Hz 前行波 后行波 n2/（r/min） ff /Hz n2/（r/min） fb /Hz 三节径 7904 11036 8645 9294 7280 四节径 13340 19227 15061 15283 11972 
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表 2 三节径裂纹长度与深度尺寸
Table 2. Three-node diameter crack length and depth dimensions
裂纹尺寸 阶段1 阶段2 阶段3 阶段4 阶段5 阶段6 阶段7 阶段8 阶段9 阶段10 阶段11 裂纹长度L/mm 1.84 5.43 9.02 12.58 16.19 19.18 23.43 27.37 31.31 35.25 39.2 ΔL/mm 1.84 3.59 3.59 3.56 3.61 2.99 4.25 3.94 3.94 3.94 3.95 裂纹深度H/mm 0.6 1.76 2.92 4.07 5.23 6.39 7.55 8.35 9.15 9.95 10.75 ΔH/mm 0.6 1.16 1.16 1.15 1.16 1.16 1.16 0.8 0.8 0.8 0.8 裂纹尺寸 阶段12 阶段13 阶段14 阶段15 阶段16 阶段17 阶段18 阶段19 阶段20 阶段21 阶段22 裂纹长度L/mm 43.15 47.12 51.09 55.05 59.03 63.01 68.23 75.49 80.38 93.85 122.02 ΔL/mm 3.95 3.97 3.97 3.96 3.98 3.98 5.22 7.26 4.89 13.47 28.17 裂纹深度H/mm 11.55 12.35 13.15 13.95 14.75 15.55 15.55 15.55 15.55 16.55 17.55 ΔH/mm 0.8 0.8 0.8 0.8 0.8 0.8 0 0 0 1 1
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表 3 四节径裂纹长度与深度尺寸
Table 3. Four-node diameter crack length and depth dimensions
裂纹尺寸 阶段1 阶段2 阶段3 阶段4 阶段5 阶段6 阶段7 阶段8 阶段9 阶段10 阶段11 裂纹长度L/mm 1.84 5.43 9.02 12.67 16.42 20.18 23.94 29.44 34.95 40.46 45.97 ΔL/mm 1.84 3.59 3.59 3.65 3.75 3.76 3.76 5.5 5.51 5.51 5.51 裂纹深度H/mm 0.6 1.76 2.92 4.08 5.24 6.4 7.55 8.35 9.15 9.95 10.75 ΔH/mm 0.6 1.16 1.16 1.16 1.16 1.16 1.15 0.8 0.8 0.8 0.8 裂纹尺寸 阶段12 阶段13 阶段14 阶段15 阶段16 阶段17 阶段18 阶段19 阶段20 阶段21 阶段22 裂纹长度L/mm 51.49 57.02 62.56 68.09 73.63 79.18 86.55 93.92 101.29 111.24 122.24 ΔL/mm 5.52 5.53 5.54 5.53 5.54 5.55 7.37 7.37 7.37 9.95 11 裂纹深度H/mm 11.55 12.35 13.15 13.95 14.75 15.55 15.55 15.55 15.55 16.86 18.17 ΔH/mm 0.8 0.8 0.8 0.8 0.8 0.8 0 0 0 1.31 1.31
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表 4 不同裂纹阶段下的节径振动模态频率
Table 4. Modal frequencies of diameter vibration at different crack stages
裂纹
阶段三节径前行波共振
裂纹扩展三节径
振动模态频率四节径后行波共振
裂纹扩展四节径
振动模态频率振型1 振型2 振型1 振型2 正常 8026.4 8053.7 13380 13380 阶段1 8027 8054.4 13380 13380 阶段2 8025.2 8052 13374 13380 阶段3 8020 8048.1 13359 13378 阶段4 8009.3 8043.3 13334 13374 阶段5 7989.7 8037.2 13298 13367 阶段6 7962.5 8029.8 13249 13355 阶段7 7927.6 8019.9 13193 13339 阶段8 7899.1 8011.2 13146 13324 阶段9 7866.2 8000.7 13095 13307 阶段10 7828.8 7988.1 13039 13286 阶段11 7788.6 7974.4 12978 13263 阶段12 7744.9 7958.5 12915 13234 阶段13 7699.8 7942 12848 13204 阶段14 7653.6 7924.5 12783 13174 阶段15 7608.1 7906 12707 13135 阶段16 7563 7885.8 12625 13094 阶段17 7517.3 7862.4 12455 13043 阶段18 7516.1 7860.5 12446 13040 阶段19 7511.3 7853.8 12444 13033 阶段20 7504.1 7844.3 12424 13019 阶段21 7426.9 7801.2 11901 12859 阶段22 7305.8 7636.7 10866 12517
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