Research on the Calculation Method of Time-Varying Mesh Stiffness for Micro-Textured Gears

Abstract: To investigate the influence of surface micro-textures on the mesh stiffness of gears, this paper proposes a calculation method for the time-varying mesh stiffness (TVMS) of micro-textured gears based on the potential energy method. By establishing a geometric model of the micro-textured gear teeth, the reduction in effective contact area and the change in the moment of inertia of the tooth section caused by the micro-texture are analyzed. The mesh stiffness components, including Hertzian contact stiffness, bending stiffness, shear stiffness, and axial compressive stiffness, are calculated by integrating the micro-texture parameters. Furthermore, a bionic tooth surface texture is designed by simulating the surface morphology of shark skin. The TVMS model is verified using the finite element method. The results indicate that the presence of textures reduces the gear TVMS; specifically, the TVMS decreases as texture width and depth increase, but increases as texture spacing increases. This study provides a theoretical basis for the vibration and noise control of high-performance gear systems.

Keywords: micro-texture; gear; time-varying mesh stiffness; potential energy method; bionic texture

1 Introduction

Gear systems are critical components in mechanical power transmission, and their dynamic performance directly affects the reliability and noise levels of industrial machinery. Micro-texturing has emerged as a promising surface engineering technique to enhance the tribological properties of contacting surfaces. While the benefits of micro-textures in reducing friction and wear are well-documented, their impact on the structural integrity and fundamental dynamic parameters of gears—most notably the time-varying mesh stiffness (TVMS)—requires rigorous quantification.

The TVMS is a primary internal excitation source for gear vibration and noise. In micro-textured gears, the removal of material to create bionic patterns, such as those inspired by shark skin, alters the effective contact area and the local compliance of the gear teeth. This paper aims to bridge the gap between surface texture design and gear dynamics by developing an analytical model to calculate the TVMS of gears featuring bionic micro-textures.

2 Modeling of Micro-Textured Gear Mesh Stiffness

2.1 Bionic Texture Design

In this study, the tooth surface texture is designed by simulating the riblet structures found on shark skin, which are known for their drag-reduction and anti-friction properties. These bionic textures are characterized by specific geometric parameters, including width, depth, and spacing.

2.2 TVMS Calculation Method

The calculation of TVMS typically involves the potential energy method, considering the contributions of Hertzian contact stiffness, bending stiffness, shear stiffness, and axial compressive stiffness. For micro-textured gears, the conventional model must be corrected to account for the reduction in the effective contact length caused by the texture grooves.

The effective contact length $L_{eff}$ is modified as follows:
$$\begin{aligned} L_{eff} = L_{total} - \sum w_i \end{aligned}$$
where $L_{total}$ is the total contact length and $w_i$ represents the width of the textures intersecting the contact line.

The total mesh stiffness $k_t$ for a single pair of teeth in mesh can be expressed as:
$$k_t = \frac{1}{\frac{1}{k_h} + \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_{f2}}}$$
where $k_h$, $k_b$, $k_s$, $k_a$, and $k_f$ represent the Hertzian contact, bending, shear, axial compressive, and foundation stiffness, respectively.

2.3 Geometric Constraints and Texture Parameters

The spacing of the textured grooves in both the horizontal and vertical directions is denoted by $L_x$ and $L_y$, respectively. These parameters are critical in determining the overall density and distribution of the surface features.

[FIGURE:1]

As illustrated in [FIGURE:1], the geometric arrangement of these grooves follows a periodic pattern. The precise control of $L_x$ and $L_y$ allows for the optimization of lubricant retention and the reduction of frictional resistance. The modification of the area moment of inertia $I(x)$ and the cross-sectional area $A(x)$ due to the presence of textures is calculated as:
$$A(x) = 2h(x)L - \Delta A$$
$$I(x) = \frac{1}{12}(2h(x))^3 L - \Delta I$$
where $\Delta A$ and $\Delta I$ represent the reductions in area and moment of inertia caused by the micro-texture grooves of depth $d$ and width $w$.

3 Finite Element Verification

The geometric modeling and meshing process for the gear components were conducted using HyperMesh and Abaqus to ensure computational accuracy. Given the complex involute profile, precise discretization is essential for capturing stress gradients.

[FIGURE:4]

As illustrated in [FIGURE:4], the axonometric and sectional drawings provide a detailed view of the individual tooth geometry. A hexahedral-dominant mesh was generated to evaluate the gear's performance under operational loads.

[TABLE:1]

The study utilized three sets of micro-textured gears with varying parameters. Based on the TVMS results presented in [FIGURE:6] and [TABLE:2], it is observed that the stiffness values obtained via the analytical method and the finite element method are highly consistent, with a relative error of approximately 4%.

[TABLE:2]

4 Results and Discussion

The introduction of micro-textures leads to a periodic reduction in the time-varying mesh stiffness. The magnitude of this reduction is closely related to the texture density and geometric dimensions. Specifically:
1. Texture Width and Depth: As the width ($W$) and depth ($D$) of the bionic textures increase, the effective contact area and the structural rigidity of the tooth decrease, leading to a lower TVMS.
2. Texture Spacing: Increasing the spacing ($S$) between textures results in a higher TVMS, as the tooth surface approaches a smooth state with less material removal.

Since textures can reduce the friction coefficient and mitigate gear tooth surface wear, it is essential to account for the simultaneous effects of texture on both the friction coefficient and TVMS when designing high-performance gear systems.

5 Conclusion

This paper established a TVMS model for micro-textured gears using an improved potential energy method. The model accounts for the reduction in effective contact length and the change in sectional properties due to bionic shark-skin textures. The analytical results were verified against finite element simulations, showing good agreement. These findings provide a theoretical basis for the dynamic analysis and optimization of micro-textured gear systems, allowing for a balance between improved tribological performance and structural vibration control.