## 1 Introduction
The roughness parameters in these studies were directly measured in experiments, and mathematical models for tooth surface roughness were not considered when establishing the gear dynamic model. [PERSON] et al. [17] studied the surface morphology of gears and believed that fractal theory could describe surface waviness and roughness. [PERSON] et al. [18] constructed the W m fractal function to characterize the nonlinear random clearances of gears using fractal theory. They compared and studied the impact of nonlinear random clearances on gear dynamics. [PERSON] et al. [19] pointed out that the engineering surface is a nonstationary random surface. Due to different gear manufacturing processes, such as hobbing and grinding, gears' surfaces exhibit varying roughness. This paper proposes an eight-degrees-of-freedom dynamic analysis method that takes into account the diverse microgeometries of tooth surfaces. The method is employed to analyze the influence of tooth surface microgeometries on time-varying mesh stiffness and vibration characteristics. In the realm of gear dynamic research considering errors, [PERSON] and [PERSON] [20] established a gear dynamic model incorporating random manufacturing errors and analyzed the impact of randomly distributed profile errors on dynamics. [PERSON] et al. [21] studied gear dynamic characteristics mainly driven by long-period quasi-static transmission errors induced by pitch errors. [PERSON] and [PERSON] [22] decomposed transmission errors into harmonic components and random components, proposing a dynamic analysis model for gear systems considering random errors. [PERSON] et al. [23] investigated the calculation method of comprehensive meshing errors in gears and their impact on system vibration. [PERSON] et al. [24] introduced the concept and acquisition method of overall gear pair errors and studied the gear dynamic behavior considering overall errors. [PERSON] et al. [25] conducted research on the nonlinear dynamics of gears under the influence of roughness based on fractal theory. However, the simplified pure torsional model is currently the most commonly used. FEA methods provide adequate means to achieve a higher degree of engineering component safety and accurate service life prediction [26]. Based on the aforementioned research, a dynamic parameter solution model for spiral level gears, considering the microtopography of tooth surfaces, has been proposed in the paper. Models for spiral level gears with different fractal dimensions have been established, and finite element analysis has been employed to address time-varying mesh stiffness under various fractal dimensions. Initially, models for spiral level gears with different surface roughness have been developed, with surface roughness characterized by the profile arithmetic mean deviation, an early parameter proposed for roughness evaluation, still widely adopted internationally. The relationship between roughness, fractal dimension, and characteristic scale factor has been provided by [PERSON] et al. [27; 28] based on experimental determinations. Through model simulation and numerical calculations, parameters such as vibration velocity, vibration acceleration, and time-varying mesh stiffness of the spiral level gears have been obtained. The vibration characteristics of spiral level gears under different surface roughness have been deduced through comparative analysis of results, highlighting the significance of this method in examining the dynamic characteristics of spiral level gears with diverse surface microtopographies.