Numerical Modeling and Nonlinear Mechanism Analysis of Spline Coupling Stiffness Considering Contact Surface Friction

doi:10.3969/j.issn.1004-132X.2026.02.008

Abstract

Abstract:

Taking the flexible spline coupling structure of an engine as the research object， a contact-friction model was introduced at the meshing points of all shaft segment in the spline coupling. Combined with the geometric relationships of the structure， force equilibrium conditions， and numerical iteration methods， a modeling and calculation approach for the nonlinear stiffness of the spline couplings was proposed. The high solving efficiency and accuracy of the proposed calculation method were validated through comparisons with ANSYS finite element model results and experimental results. Using the proposed method， detailed analyses were conducted on the nonlinear stiffness characteristics of the spline couplings and the influence patterns of key parameters. The intrinsic mechanism of the nonlinear stiffness variation was revealed by combining the changing patterns of contact states in the meshing tooth pairs. Results indicate that the stiffness of the spline coupling structures gradually decreases with the increasing linear displacement， while the decreasing rate progressively diminishes. The reduction in contact area of meshing teeth during this process was identified as the fundamental cause of the stiffness degradation. The stiffness of the spline couplings shows improvement with the increase of the friction coefficient， the meshing stiffness， the spline width， and the applied torque. Among these parameters， the torque exhibits the most significant influence.

Key words: spline connection, stiffness characteristic, contact-friction model, numerical modeling

摘要：

以发动机柔性套齿连接结构为对象，通过在套齿连接结构轴段微元各啮合点处引入接触-摩擦模型，并结合整体结构的几何关系、受力平衡关系和数值迭代法提出了一种套齿非线性刚度的建模与计算方法，通过与ANSYS有限元模型结果进行对比，验证了所提计算方法的高求解效率和求解准确性。利用所提求解方法详细分析了套齿连接结构刚度非线性特征及关键参数的影响规律，并结合啮合齿对接触状态的变化规律揭示了其刚度非线性变化的内在机理。结果表明：套齿连接结构刚度随线位移的增加逐渐下降，但下降速率逐渐减小，此过程中套齿连接结构啮合齿接触面积减小是刚度降低的根本原因。接触面摩擦因数、啮合刚度、套齿齿宽以及扭矩增加时，套齿连接结构刚度均有所提高，其中扭矩的影响最为明显。

关键词: 套齿连接结构, 刚度特性, 接触-摩擦模型, 数值建模

CLC Number:

V231.96

JIANG Ke, YU Pingchao, YAN Xunjin, ZHENG Huaqiang, JIANG Zihan, TAO Xuanjun. Numerical Modeling and Nonlinear Mechanism Analysis of Spline Coupling Stiffness Considering Contact Surface Friction[J]. China Mechanical Engineering, 2026, 37(2): 332-341.

蒋科, 于平超, 严循金, 郑华强, 蒋紫菡, 陶玄君. 考虑接触面摩擦的柔性套齿连接结构刚度数值建模与非线性机理分析[J]. 中国机械工程, 2026, 37(2): 332-341.

Figures/Tables 21

Fig.1 A splined low-pressure rotor system of an aeroengine

Fig.2 Deformation of spline coupling under external load

Fig.3 Change of meshing distance

Fig.4 Schematic diagram of the penetration and slide caused by the deflection and torsion of the turbine shaft

Fig.5 Definition of parameters of meshing teeth

Fig.6 Contact-friction model

Fig.7 The solution flow chart of the proposed model

Tab.1 The structural parameters of spline coupling

参数名称	涡轮轴	风扇轴	参数名称	涡轮轴	风扇轴
齿数	22	22	模数/mm	3	3
压力角/（°）	20	20	齿宽L/mm	30	30
内轮廓半径r₁/ mm	21		外轮廓半径r₂/mm		44
泊松比	0.25	0.25	摩擦因数	0.15	0.15
弹性模量/GPa	210	210

Fig.8 Number and position of teeth

Fig.9 The solid finite element model of spline coupling

Fig.10 Comparison of results obtained by the proposed model in this paper and ANSYS model

Fig.11 Schematic diagram of the stiffness testing device of the spline coupling

Tab.2 The structure parameter of the spline test piece

参数名称	涡轮轴	风扇轴
模数/mm	2	2
齿数	30	30
压力角/（°）	30	30
基圆直径/mm	51.6915	51.6915
分度圆直径/mm	60	60
花键大径/mm	63	62
花键小径/mm	58.15	57

Fig.12 Comparison of stiffness results between numerical calculation and experiment

Fig.13 The effect of the load amplitude on the proportion of contact area and the torsion angle

Fig.14 The influence of load on the contact area of each teeth

Fig.15 Influence of load on the penetration of each teeth

Fig.16 The influence of the friction coefficient on the stiffness of the spline coupling

Fig.17 The influence of the meshing stiffness on the stiffness of the spline coupling

Fig.18 The influence of the torque on the stiffness of the spline coupling

Fig.19 The influence of the tooth width on the stiffness of the spline coupling

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