Experimental‐based study of gear vibration characteristics incorporating the fractal topography of tooth surface

Microscopic roughness is inevitable on the gear meshing surface, which is also a key parameter affecting the dynamic response. The surface roughness exhibits self‐affine characteristics across multiscales. To explore the influence of surface fractal topography on the vibration amplitude of the gear system under different rotational speeds and loads, an experimental setup of spur gear transmission is devised. The fractal dimension and fractal roughness of the meshing surface are calculated by the power spectral density method. The relationships between gear response and fractal parameters are revealed experimentally. Results indicate that a rougher tooth surface, that is, a smaller fractal dimension or larger fractal roughness, corresponds to an intense vibration amplitude. The sensitivity of dynamic response to the tooth surface topography varies at different rotational speeds and loads. Under low speed and light load conditions, the fractal dimension and fractal roughness have a more obvious influence on the dynamic response of the gear transmission system. With the increase of speed and load, the macroworking conditions gradually become the main factor attributed to vibration amplitude.


| INTRODUCTION
Gear is an important component of the mechanical transmission system because of its simple structure and high transmission accuracy. The research on dynamic response has meanings for the low-vibration design of gears 1  The dynamic response of gears is also influenced by microsurface morphology. The contact between the tooth surface contains the macroform of a curved interface and microscopic rough features, which affects the dynamic response of gears. Kim and Singh 11 proposed a noise source model for gear surface roughness considering sliding contact between meshing teeth and verified that the noise level increases with the increase of speed and surface roughness.
Wang et al. 12 proposed a new model for calculating the power loss of planetary gear meshing considering tooth surface roughness and investigated the relationship between tooth surface roughness and the dynamic response. Zhou et al. 13 developed a finite element model of a megawatt wind turbine gear and used the developed model to comparatively study the effects of microstructure, inclusions, and surface roughness on the contact fatigue behavior of the gear. Xiao et al. 14 proposed a six-degree-of-freedom spur gear transmission system with coated teeth and used the finite element method to obtain the time-varying meshing stiffness of the coated gear pair during the meshing process. Huang et al. 15 proposed a new method to analyze the dynamics of high-contact-ratio gears by directly introducing the surface roughness into the multi-degree-offreedom model as well as establishing an indirect relationship between the dynamic response and the surface roughness.
Yu et al. 16 and Liu et al. 17  Some existing works have established the relationship between the gear surface morphology and vibration characteristics. It has been known that significant random components exist whose instantaneous power varies cyclically with the gear meshing, 25 and a widely accepted hypothesis attributes a portion of these CS2 components to friction and asperity contacts. However, the relationship between CS2 and surface roughness is very complex. On this basis, the carrier frequency f s was further investigated and the relationship between the surface morphology spatial frequency f v and the sliding vibration frequency f s was established 26 as f s ∝ v s · f v . The study showed that the frequency range characterizing vibration grew with sliding velocity and spatial frequency of the roughness of the mating surfaces. Considering the various spatial frequencies can affect the spectral content of the resulting gearmesh-cyclic second-order cyclostationary (CS2) components, Feng et al. 26

| EXPERIMENTAL SETUP AND DESIGN
In this section, an experimental setup is designed to collect the vibration signals of the gear systems. Figure 1 shows that the complete experimental setup is installed on a 1000 mm × 350 mm × 10 mm aluminum alloy ground bedplate. The ground bedplate is machined by finish-milling with a machining error of less than 0.05 mm. The test rig is mainly composed of a mechanical transmission system and a signal acquisition system. The main part of the mechanical transmission system is a single-stage spur gear transmission structure. The motor equipped with a frequency modulator provides variable power, the magnetic particle brake equipped with a controller provides variable load, and the speed and torque sensors collecting vibration signals. On this basis, the experimental study is designed. Three coupling conditions including rotational speed, load, and topography of tooth surfaces are considered. Figure 2 shows the configuration of the mechanical transmission system, which includes the power source, gear transmission mechanism, and loading device.

| Parameters of power source and load
A three-phase asynchronous motor is used as the power source to keep the input velocity of the system constant and take advantages of its simple structure, reliable operation, lightweight, and low price.
The frequency modulator is selected to change the velocity. It uses the encoder feedback speed control mode to form a closed-loop control, which makes the speed error less than 0.01%. The frequency modulator is selected to change the velocity because of its high accuracy of speed regulation. In addition, compared with other speed regulation modes, frequency conversion speed regulation has good performance in smooth speed regulation and high efficiency.
The magnetic particle brake is chosen to produce the load on the output. To simulate different working conditions, the current controller is used to adjust the load of the magnetic particle brake.
The tension controller can adjust the load by 0.01 N·m. It uses a proportion integration differentiation controller to make the control error less than 0.1%. The parameters of the setup mentioned above are listed in Table 1.

| Fractal parameters of spur gears
Generally, the friction coefficient is the key quantitative indicator to measure the surface morphology. 35 The friction coefficient can be increased by changing the surface morphology. 36 Nevertheless, this surface tribological parameter could not represent the surface treatment method. 37 Wear scar feature could analyze the morphology of the wear surface to determine the wear forms and characteristics of spherical contact friction pairs. 38 Besides, surface roughness S a was selected as the feature of surface morphology at the microlevel to represent the degradation process of the gear surface quantitatively. 39 However, this statistical parameter of roughness is limited by the instrument resolution and sampling length. 40,41 In fact, the profiles of machined surfaces are random, multiscale, disordered, and self-affine in rough structures. 42 The mathematical properties of the profiles are continuous everywhere but nondifferentiable at all points. Accordingly, the fractal dimension and fractal roughness are used to characterize the microscopic roughness of the tooth surface.
To make the tooth surface fractal topography of experimental gear meet the requirements of experimental design, three kinds of   Table 2.
The tooth profile height h(x) is required to acquire the corresponding fractal dimension and fractal roughness of the tooth surface. Using the Nano-μScan, the profile of the tooth surface by the grinding machining method is shown in Figure 3.
After Fourier-transforming, the power spectral density function P(ω) of the tooth profile height can be expressed as where ω is the spatial frequency of the tooth profile and x is the sampling point coordinates of the profile.
The relationship between the fractal parameters and the power spectrum can be expressed as 43 where D is the fractal dimension that reflects the complexity of the space, G is the fractal roughness that describes the amplitude of the surface profile at the microscopic scale, and γ is the frequency parameter that determines the spectral density and the relative phase differences between the spectral modes. To avoid the coincidence of the frequency modes, γ is generally taken as 1.5. 44 To calculate the fractal parameters, the logarithm of spatial frequency, lg(ω), is taken as abscissa and the logarithm of power spectral density, lg P(ω), is taken as ordinate. Figure 4 shows the logarithmic graph of the power spectral density function.  The least squares method is applied to fit the logarithmic graph.
The intercept and slope of the fitted function can be derived as The relationship between the fractal parameters and the intercept and slope can be given as

| Signal acquisition and processing system
To simplify the transmission system, the speed and torque sensors are used to collect the speed and load signals of the system. The error of the speed and torque sensors is 0.1%, which means the collected speed and load are accurate to ±5 r/min and ±0.02 N·m. The signal acquisition systems include speed and torque sensors and acceleration sensors. Relevant equipment parameters are shown in Table 4.
Since the horizontal and longitudinal vibrations of the single-stage spur gear transmission are both sensitive, the acceleration-sensor horizontal arrangement is selected.
To explore the laws of the relationships between surface topography and gear vibration response under different working conditions, three levels of rotational speed and load are selected. This experiment is based on dry friction, and the gears are cleaned carefully to ensure that the tooth surfaces are free from impurities, such as coolant left during machining.

| Time-frequency domain analysis
To eliminate the interference and noise in the experiment, it is necessary to filter the collected original data to reduce the influence of noise on the experimental results.
The smooth algorithm is used in this experiment, and its algorithm formula is as follows: where x t is the original data, y t is the processed data, i is the sampled signal, and N is the sampling length.

| Effect of rotational speed
The frequency spectrum analysis was carried out on the vibration acceleration signals. The load is 0.9 N·m. It can be seen from

| Effect of load
The frequency spectrum analysis was carried out on the vibration acceleration signals, which are collected under three types of load conditions, while the speed is fixed at 200 r/min.
As can be seen in Figure 6, the load increase leads to an increase in vibration acceleration amplitude at the first gear meshing frequency, but the vibration amplitude at the high frequency does not change significantly.
Taking rotational speed as the horizontal axis, the load as the vertical axis, and RMS as the third axis, a three-dimensional spatial coordinate system is established, and the points are marked in the coordinate system and the fitting surface is shown as Figure 7, which illustrates the effects of load and rotational speed on vibration.
T A B L E 5 RMS of the gear vibration under different rotational speeds and loads.

| Relationship between microscopic surface topography and macroscopic dynamic response
The analysis of the vibration signals of gears with four different surface topographies was carried out. Tables 6 and 7 give the experimental parameters and data.
It can be seen from Tables 6 and 7 that the dynamic response of the rough tooth surface still follows the rule mentioned in Section 3.1, which is increasing the rotational speed and load leads to an RMS increase. Meanwhile, the rough tooth surface topography intensifies the vibration. The rougher tooth surface corresponds to the more violent vibration.
To explore the influence of fractal dimension D and fractal roughness G on the vibration of the system, taking fractal dimension D as the horizontal axis, the fractal roughness G as the vertical axis, and RMS as the third axis, a three-dimensional spatial coordinate system was established, and the coordinate points corresponding to the four different tooth surface topographies were marked in the coordinate system. Figure 8 shows   smoother after driving. However, the partial area of a relatively rough surface tends to be rougher. According to the existing research, wear originates from the dedendum of the gear tooth. 27 The rougher surface always corresponds to intense vibration, which makes the tooth surface more prone to wear. Thus, the wear of the tooth surface appears in the area circled in red, as shown in Figure 9B. The results obtained in this experiment apply to the dynamic response analysis of the gear transmission system under dry friction conditions. Nevertheless, lubrication is essential in the gear transmission system. In our future work, lubrication conditions will be taken into account. In addition, the test rig will be further developed to provide better experimental conditions.