Nodal-diameter vibration modes identification and wide-speed-range resonance-avoidance optimization of gears based on Campbell diagram
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摘要:
针对某航空发动机锥齿轮在较宽的工作速域内易发生节径型行波共振破坏及设计优化中发现的振型跳变问题，提出一种基于Campbell图的齿轮危险节径振型识别与宽域避振优化方法，将事后的、人工的、基于经验的传统判断，转变为事前的、自动化的、基于数学特征的优化导向，实现齿轮避振优化中对节径振型的分离与控制。首先，基于模态置信准则区分各转速下各模态固有频率数据，拟合Campbell图并提取图中各阶固有频率直线斜率，根据斜率特征识别各阶次是否为节径振型，基于振型识别结果预测节径型行波共振频率并存储节径信息。然后，基于提出的危险节径振型识别方法，以齿轮质量最小为优化目标，以应力及节径型行波共振频率为约束参数，建立基于危险节径振型识别的齿轮宽域避振优化数学模型，并基于赋大值法和Pointer优化策略构建齿轮避振优化流程。优化后，齿轮在75%～107%的工作速域内成功避免了节径型行波共振，且齿轮质量降低了6.566%，证实了提出的齿轮节径振型识别与避振优化方法在工程应用中的有效性，也为某型航空发动机锥齿轮结构优化提供了重要支撑。
Abstract:Gears are crucial mechanical components to transmit power and torque, playing an irreplaceable role in aero-engines while continuously evolving towards higher speeds, heavier loads, and lighter weights. However, their dynamic performance faces substantial challenges. The issues of nodal-diameter resonance failure and modal jumping that occur within a wide operational speed range in the design optimization of an aero-engine bevel gear was addressed. A method based on the Campbell diagram was proposed to identify dangerous nodal-diameter vibration modes and optimize the structure of the gear to avoid resonance across a wide speed range. The method transformed traditional post-event, manual, experience-based judgments into a preemptive, automated, mathematically feature-driven optimization indicator, enabling separation and control of nodal-diameter vibration modes in gear vibration optimization. First, modal frequencies were distinguished by the Modal Assurance Criterion, ensuring an accurate fit to the Campbell diagram. The slopes of the frequency lines in the Campbell diagram were extracted, and based on these slopes, the presence of nodal-diameter vibration modes was identified. Their resonance frequencies were then predicted and recorded. Next, an optimization model was developed based on the dangerous nodal-diameter vibration mode identification method, to avoid resonance over a wide speed range. This aimed to minimize the gear mass while constraining the stress and the nodal-diameter resonance frequencies. The resonance-avoidance optimization process was constructed by assigning large values and the Pointer strategy. After optimization, the gear successfully avoided nodal-diameter resonance within the 75%—107% operational speed range, and its mass was reduced by 6.566%. This confirmed the effectiveness of the proposed gear nodal-diameter identification and resonance avoidance optimization method in engineering applications, offering significant support for the structural enhancement of aero-engine bevel gears.
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图 5 基于Campbell图的齿轮危险节径振型识别及宽域避振优化流程
Figure 5. Process of nodal-diameter vibration modes identification and wide-speed-range resonance-avoidance optimization of the gear based on Campbell diagram
图 10 仅含节径振型的Campbell图
Figure 10. Campbell diagram with only nodal-diameter vibration modes
图 12 齿轮宽域避振优化流程
Figure 12. Process of gear wide-speed-range resonance-avoidance optimization
图 16 优化前后只含节径振型的Campbell图对比
Figure 16. Comparison of Campbell diagrams before and after optimization with only nodal diameter vibration modes
图 17 优化前后节径型行波共振频率对比
Figure 17. Comparison of resonance frequency of nodal-diameter travelling wave before and after optimization
图 18 优化前后齿轮剖面形状对比
Figure 18. Comparison of gear cross-section shape before and after optimization
图 19 优化前后齿轮辐板轮廓线对比
Figure 19. Comparison of gear web contour line before and after optimization
图 20 优化前后各设计参数对比
Figure 20. Comparison of design parameters before and after optimization
图 22 额定工况下最大等效应力位置示意
Figure 22. Schematic of maximum equivalent stress location under rated operating conditions
表 1 齿轮振型图及振型类型
Table 1. Gear modal-shape diagrams and types
阶次 1阶 2阶 3阶 4阶 5阶 振型图 
振型类型 1节径 1节径 节圆 2节径 2节径 阶次 6阶 7阶 8阶 9阶 10阶 振型图 
振型类型 扭转 3节径 3节径 4节径 4节径 阶次 11阶 12阶 13阶 14阶 15阶 振型图 
振型类型 摆动 摆动 5节径 5节径 耦合 
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表 2 节径型行波共振转速及频率
Table 2. Nodal diameter resonance speed and frequency
节径共振点
编号节径数 转速/
104 （r/min）频率/
103 Hz归一化 1 1节径 1.081 6.486 0.358 2 1节径 1.140 6.848 0.378 3 2节径 1.310 7.863 0.434 4 2节径 1.453 8.715 0.481 5 3节径 2.072 1.243 0.686 6 3节径 2.394 1.437 0.793 7 4节径 3.182 1.910 1.054 8 4节径 3.804 2.283 1.260 9 5节径 4.470 2.681 1.480 注：加粗部分表示在齿轮75%～107%的工作速域内，存在3节径前行波共振与4节径后行波共振两个节径型行波共振点。
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表 3 振型跳变示例之振型类型对比
Table 3. Comparison of modal-shape types for example of modal jumping
阶次 原振型图 原振型类型 原归一化频率/% 改后振型图 改后振型类型 改后归一化频率/% 9 
4节径 105.381 
摆动 107.756 10 
4节径 125.972 
摆动 107.967 11 
摆动 126.998 
4节径 108.907 12 
摆动 127.027 
4节径 129.456 注：加粗部分表示原模型的4节径振动发生在第9、10阶次，且4节径前行波归一化共振频率为105.381%，对应共振点落在75%～107%的工作转速范围内。
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表 4 齿轮各阶次对应固有频率直线斜率及振动类型
Table 4. Slopes of natural frequency lines and modal-shape types of the gear’s each mode
阶次 1阶 2阶 3阶 4阶 5阶 斜率 −1.502×10−2 1.698×10−2 −2.660×10−6 −2.772×10−2 3.349×10−2 振型 1节径 1节径 节圆 2节径 2节径 阶次 6阶 7阶 8阶 9阶 10阶 斜率 −6.183×10−8 −3.949×10−2 4.658×10−2 −4.956×10−2 5.649×10−2 振型 扭转 3节径 3节径 4节径 4节径 阶次 11阶 12阶 13阶 14阶 15阶 斜率 −1.568×10−6 2.632×10−6 −5.771×10−2 6.415×10−2 1.902×10−7 振型 摆动 摆动 5节径 5节径 耦合 注：加粗部分表示节径振型。
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表 5 齿轮各设计参数变化范围
Table 5. Variation range of design parameters of the gear
mm 设计参数 下限 上限 初始值 H1 5.000 8.800 7.600 H2 8.600 9.300 8.747 H3 9.500 14.100 14.130 H4 14.000 15.800 15.137 L1 2.000 5.500 2.740 L2 3.500 8.000 5.870 L3 0.900 2.700 0.900 L4 5.000 13.000 11.300 T1 1.200 3.700 1.200 T2 1.200 3.700 3.170
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表 6 齿轮优化前后质量、节径共振点数量及应力对比
Table 6. Comparison of the gear’s mass, number of nodal diameter resonance points in the working speed domain and stress before and after optimization
参数 优化前 优化后 变化率/% 质量/102 g 2.315 2.163 −6.566 共振点数 2 0 100 $ {{\sigma }}_{{1},\rm{max}} $/102 MPa 1.113 1.422 27.763 $ {{\sigma }}_{{2},\rm{max}} $/103 MPa 1.226 1.106 −9.788
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表 7 优化前后节径型行波共振频率对比
Table 7. Comparison of resonance frequency of nodal-diameter travelling wave before and after optimization
节径共振点编号 节径数 优化前频率/% 优化后频率/% 1 1 35.789 20.980 2 1 37.757 22.081 3 2 43.396 29.777 4 2 48.103 32.896 5 3 68.613 63.309 6 3 79.296 72.857 7 4 105.381 107.013 8 4 125.972 127.274 9 5 148.031 注：加粗部分表示原本存在的两个节径共振点。
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表 8 齿轮优化前后各设计参数对比
Table 8. Comparison of the gear’s each design parameter before and after optimization
设计变量 初始值/mm 优化值/mm 变化率/% H1 7.600 6.469 −14.882 H2 8.747 8.600 −1.681 H3 14.130 9.528 −32.569 H4 15.137 15.800 4.380 L1 2.740 3.041 10.985 L2 5.870 7.496 27.700 L3 0.900 2.583 187.000 L4 11.300 9.092 −19.540 T1 1.200 1.945 62.083 T2 3.170 1.944 −38.675
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