Resonance-avoidance Optimization Method of Gears Based on Tooth-root Circumferential Displacement Features of Critical Modes

doi:10.3969/j.issn.1004-132X.2026.10.024

Abstract

Abstract:

To address the frequent nodal-diameter resonance failures of an aero-engine bevel gear during operation， the cylindrical-coordinate displacements of the nodes along the tooth-root circumference were extracted from the modal analysis results. The distinct circumferential periodicity of a mode shape with k （k ≥ 2） nodal diameters was quantified as an identical number of k cycles in the displacement waveforms of the tooth-root circumferential nodes in all cylindrical-coordinate directions. Furthermore， the difference between pure and coupled nodal-diameter modes was characterized by the relative ordering of their modal orders， thereby allowing the two types of modes to be distinguished and the critical modes of the gear to be identified. Subsequently， using the fast Fourier transform， the number of cycles in each tooth-root circumferential displacement waveform was converted into the frequency corresponding to the dominant peak in the frequency-domain spectrum. Accordingly， the tooth-root circumferential displacement features of the critical modes were automatically extracted. Finally， multi-platform collaborative resonance-avoidance optimization of the gear was performed， with the gear mass minimization taken as the optimization objective and the stress and frequency specified taken as constraints. With both the resonance-avoidance and strength requirements satisfied， the gear mass was reduced by 25.31%， thereby demonstrating the validity and engineering applicability of the proposed resonance-avoidance optimization method of gears.

Key words: bevel gear, nodal diameter vibration, resonance-avoidance optimization, Fourier transform

摘要：

针对某航空发动机锥齿轮运行时频发的节径共振失效问题，提取模态分析后齿根环线节点的柱坐标位移，将含k（k≥2）节径振型图像在环向具有的明显周期特征量化为齿根环线节点在柱坐标各方向位移波形图中相等的k个周期数，并将纯节径振型与耦合节径振型间的差异转化为模态阶次的先后差异，实现节径振型与耦合节径振型的区分，完成齿轮危险节径振型的判别；通过快速傅里叶变换，将齿根环线位移图中的周期个数转换为频域图中主要峰值对应的频率数值，实现危险节径振型齿根环线位移特征的自动提取。最后，以齿轮质量最小为优化目标，以应力及频率为约束参数，开展多平台联合的齿轮避振优化。在满足避振要求及强度要求的前提下，齿轮质量减小了25.31%，证实了提出的齿轮避振优化方法的合理性及工程适用性。

关键词: 锥齿轮, 节径振动, 避振优化, 傅里叶变换

CLC Number:

V233.1

XU Kehan, LIU Kaicheng, ZHU Haoyuan, LI Mei, ZUO Yanfei, YAN Cheng. Resonance-avoidance Optimization Method of Gears Based on Tooth-root Circumferential Displacement Features of Critical Modes[J]. China Mechanical Engineering, 2026, 37(10): 2610-2618.

许可晗, 柳恺骋, 朱浩元, 李梅, 左彦飞, 闫成. 基于危险振型齿根环线位移特征的齿轮避振优化方法[J]. 中国机械工程, 2026, 37(10): 2610-2618.

Figures/Tables 24

Fig.1 Bevel gear of an aero-engine

Fig.2 Vibration modes’ diagrams and types of the gear

Fig.3 Schematic diagram of gear’s nodal-diameter vibration

Tab.1 Normalized nodal-diameter natural frequency corresponding to the upper and lower limits of the velocity domain

节径共振点	节径数	归一化F_l/%	归一化F_u/%
1	2节径	32.10	31.79
2	2节径	33.56	33.86
3	3节径	48.62	48.37
4	3节径	49.78	50.02
5	4节径	78.33	78.17
6	4节径	79.05	79.21
7	5节径	110.92	110.85
8	5节径	111.26	111.34

Fig.4 Comparison of structures in modal jumping examples

Tab.2 Comparison of modal types in modal jumping examples

模态阶次	原振型类型	改后振型类型
8	4节径	摆动
9	4节径	摆动
13	扭转	4节径
14	耦合	4节径

Fig.5 Nodal-diameter vibration modes’ of the gear

Fig.6 Schema of the position of the tooth-root ring

Fig.7 Tooth-root circumferential column coordinate displacement periodic features of the 2th nodal-diameter vibration mode

Fig.8 Tooth-root circumferential column coordinate displacement periodic features of the 2th coupled nodal-diameter vibration mode

Fig.9 Tooth-root circumferential column coordinate displacement periodic features of the nodal-circle mode

Fig.10 Principle of differentiation between pure and coupled nodal-diameter vibration modes

Fig.11 Tooth-root circumferential column coordinate displacement frequency domain plots of the 2th nodal-diameter vibration mode

Fig.12 Flow of the gear’s resonance-avoidance optimization

Fig.13 Parametric model of the gear

Tab.3 Range of the gear’s design parameters

设计变量	下限	上限	初始值
r₁	1.4000	3.5000	3.0910
r₂	4.8000	9.0000	4.8000
r₃	1.5000	5.0000	4.3479
r₄	0.9000	3.5000	3.5000
r₅	1.0000	5.4609	5.4609
r₆	0.2000	3.0000	2.7452
h₁	8.6000	9.4000	9.0367
h₂	1.0000	2.5000	1.8995
h₃	0.3000	1.0000	0.8607
h₄	0.5000	2.0000	1.4463
h₅	0.5000	1.5000	1.5000
h₆	3.0000	7.0000	5.0000
h₇	1.0000	3.5000	3.5000
h₈	0.5000	9.9000	7.2481

Fig.14 Historical curves of the optimization

Tab.4 Comparison of gears before and after optimization

	速域内节径共振点数	辐板最大等效应力值/MPa	质量/g
优化前	2	323.53	438.38
优化后	0	462.72	327.44
变化率	100%	43.02%	-25.31%

Fig.15 Comparison of the stress contour before and after optimization

Tab.5 Comparison of normalized nodal-diameter natural frequency before and after optimization

节径数	优化前F_l/%	优化后F_l/%	优化前F_u/%	优化后F_u/%
2节径	32.10	15.61	31.79	15.10
2节径	33.56	18.11	33.86	18.64
3节径	48.62	42.37	48.37	41.99
3节径	49.78	43.72	50.02	43.97
4节径	78.33	74.06	78.17	73.88
4节径	79.05	74.75	79.21	74.87
5节径	110.92	107.27	110.85	107.20
5节径	111.26	107.59	111.34	107.65

Fig.16 Comparison of campbell diagrams before and after optimization

Fig.17 Comparison of gear’s web profile contour lines before and after optimization

Tab.6 Comparison of the design parameters before and after gear optimization

设计变量	初始值/mm	优化值/mm	变化率/%
r₁	3.0910	3.4072	10.23
r₂	4.8000	8.1048	68.85
r₃	4.3479	4.9186	13.13
r₄	3.5000	0.9586	-72.61
r₅	5.4609	1.0737	-80.34
r₆	2.7452	2.5075	-8.66
h₁	9.0367	8.9828	-0.60
h₂	1.8995	2.3025	21.22
h₃	0.8607	0.8206	-4.66
h₄	1.4463	1.4285	-1.23
h₅	1.5000	1.4266	-4.89
h₆	5.0000	6.8938	37.88
h₇	3.5000	1.6590	-52.60
h₈	7.2481	9.4137	29.88

Fig.18 Parameter correlation of F4bu and F4fu

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