Abstract
The dynamic characteristics of a highcontactratio (HCR) spurgear system having rough surfaces generated by shot peening (SP) were studied, with specific emphasis on characterisation of the gearsurface topography as well as modelling of the gear backlash and static transmission error. Accordingly, a fourdegreeoffreedom dynamic model was established. Simulation experiments were then conducted using surface roughness, rotational velocity, input torque, and shaftbearing stiffness as the variables. The results show that the dynamic characteristics of the gear system tend toward instability with increasing surface roughness. The models developed in this study outline a method for building indirect relationships between the vibration, dynamics, and toothsurface microscopic features. This research thus provides a theoretical basis for designing the toothsurface topography of HCR gears in the future.
1. Introduction
The rapid growth and development of the manufacturing industry has promoted the need for gear systems in various domains, such as aerospace, robots, and vehicles, with demand for high bearing capacities and lengthy service lifetimes. Highcontactratio (HCR) spur gears are defined as gears whose contact ratios are greater than two [1]. The mesh stiffness of the HCR spur gear is much larger than that of the lowcontactratio (LCR) spur gear [2] when all the other parameters are equal; therefore, HCR spur gears are widely used in the abovementioned fields [3]. In addition, highperformance gears must demonstrate low vibration and noise, especially for Noise, Vibration and Harshness (NVH) performance optimisation in batterypowered electric vehicles. The dynamic characteristics of gears influence their vibrations considerably. Research on gear dynamics have been reported in literature extensively [411]. Yang [12] adopted a multiscale method to study gearsystem stability. Xiang [13] researched the effects of oilfilm stiffness on spurgear dynamics. Ouyang [14] studied the lubricating and dynamic performances of highspeed spur gears. Pan [15] investigated the gear dynamic behaviours under the effects of toothcontact temperature and random excitations. However, most of these studies involved LCR gear systems.
The meshing surfaces of gears constitute a type of sportive joint surface, where the surface contact pressures are limited within a certain range. There may also be geometric shape errors and microscopic irregularities on the contact surfaces and in various media. Different machining techniques often produce surfacefeature differences. Shot peening (SP) can improve the fatigue properties of metallic components exposed to cyclic loadings [16], so this method is widely used in gearsurface treatments [1719]. During the SP process, numerous highspeed shots impact a given surface to modify its morphology, refine the grain [20], and promote phase transformation [21]. Surface morphology changes often have positive effects on the mechanical properties of the treated gears; however, they may also produce negative effects that make the gears more prone to unexpected faults. Previous studies on gear SP primarily investigated the relationships among the process parameters, residual stresses, roughnesses, corrosion behaviours, and fatigue properties [2225]. However, none of the reported studies focused on the dynamic behaviours of HCR gear systems by considering the rough surfaces generated by SP. Many researchers have focused on characterising machined rough surfaces using fractal theory [2627]. In the study of gear dynamics, the gear teeth are often regarded as cantilever beams. In terms of beam dynamics, Pankaj Charan Jena et al. considered the effects of cracks, vibration characteristics, bidirectional fibreangle changes, and other factors [2831]. In addition, the Timoshenko beam theory, higherorder sheardeformation theory, and higherorder stratification theory are widely used in beam vibration and dynamics analyses [3236]. However, these studies have some disadvantages as follows. First, the static transfer error calculation is based on a smooth surface that neglects the effects of surface roughness. Second, the dynamic characteristics of HCR spur gears having rough surfaces are seldom considered. Solving the above deficiencies is the main contribution of this study.
Therefore, the dynamic behaviours of HCR gear systems with rough surfaces generated by SP are discussed in this work in terms of backlash and transmission errors. Fig. 1 depicts a flow chart of the simulations performed in this study. The remainder of this paper is organised as follows: Section 2 presents the gearsurface topography characterisation. Section 3 illustrates the proposed dynamic model. Section 4 presents analyses of the dynamic behaviours of the system under multiple nonlinear parameters. Section 5 presents the main conclusions of this study.
Fig. 1Flow chart of the simulations in this study
2. Characterisation of gearsurface topography
2.1. Roughness measurement experiment
SP is typically used as the final step in the gear manufacturing process, before which machining trace lines are produced on the gear surfaces by the preceding procedures. During SP, many highspeed shots impact the gear surfaces randomly to produce regular craters. With increasing coverage from these shots, the craters overlap and influence each other, thereby changing the micromorphology of the gear surfaces completely. After SP, the original machining traces on the surface disappear. Different SP processes produce dissimilar surface topographies; however, improper process parameters may cause surface cracking, folding, broken projectile embedment on the surface, and fatigue resistance reduction of the metal parts [37]. 39NiCrMo_{3} is a type of lowalloy steel that is widely used to produce gears owing to its resistance to thermal cracking [38]. In the present study, 39NiCrMo_{3} was chosen to manufacture HCR gears by wire cutting, and the gears were subsequently subjected to SP treatment.
Aroughsurfaceiscomposedofharmoniccomponentswithseveralfrequenciesandmaybedescribedbyitscorrespondingroughnessparameters.Thesurfacetopographyofanobject is described by its surface profile; herein, the surface profiles and corresponding roughness parameters were directly tested using a threedimensional measurement system (Alicona Infinite focus G5), as shown in Fig. 2. Among the parameters used to quantify the surface roughness ${R}_{a}$, the arithmetic mean deviation of the assessed profile defined by Eq. (1) is the most widely used parameter [39]:
where $y=f\left(x\right)$ represents the peak heights within a sampling length $l$. The test results in this study indicate that the roughness parameter ${R}_{a}$ of the tooth surfaces of the SPtreated gears changed from 0.59 μm to 5.13 μm.
Fig. 2Threedimensional measurement equipment
2.2. Roughsurface simulation
Based on previous reports, a machined surface has fractal characteristics that can be described using a formula, and the Weierstrass Mandelbrot (WM) function is suitable for characterising the rough surface [4041]. The height of the random asperity of the geartooth profile based on the WM function is expressed as follows:
where, $D$ denotes the fractal dimension, ${\gamma}^{n}$ is the discrete frequency spectrum of the surface roughness, ${n}_{l}$ is the lower cutoff frequency of the profile, and $G$ is the characteristic scale coefficient [1]. Therefore, the measured surface profile of a gear can be approximately simulated by selecting appropriate fractal parameters, as shown in Fig. 3.
Fig. 3Twodimensional gearsurface profile with Ra= 1.13 μm
3. Dynamic model
3.1. Gear backlash and static transmission error
A meshing gear pair with rough surfaces is shown in Fig. 4, where $P$ is the highest point among the tooth asperities in the driving gear, and $Q$ is the highest point among the tooth asperities in the driven gear; ${b}_{0}$ is the initial backlash; ${R}_{a}$ is the real surface roughness; ${R}_{ac}\left(D\right)$ is a function to obtain the corresponding ${R}_{a}$ with a definite $D$, which can be obtained from an extant work [27]. According to another report [42], the new height of the surface asperities can be defined as:
The actual tooth topography and toothprofile deviation can be simulated and calculated using Eq. (3). Considering the differences between the roughnesses of two teeth and their fractal dimensions, gear backlash can be redefined as follows [42]:
where, $b\left(t\right)$ is the timevarying backlash of the gear pair; ${R}_{a1}$ and ${R}_{a2}$ denote the measured surface roughness values of the driving and driven gears, respectively; $\lambda $ is the characteristic scale coefficient (set as $\lambda =$ 1.5 according to [43]); ${D}_{1}$ and ${D}_{2}$ denote the measured fractal dimensions of the driving and driven gears, respectively.
Fig. 4Illustration of gear backlash with toothsurface microscopic features
The deviation between the real profile of the gear surface and a perfect surface is the main cause of the static transmission error. As noted above, the tooth profile may be uneven after SP. Here, $e\left(t\right)$ is expressed as the static transmission error, which may be regarded as the synthesised toothprofile microdeviation of each tooth given a teeth pair, ignoring the effects of spacing and runout errors. Therefore, according to Eq. (3), the static transmission error considering rough surfaces can be expressed as follows:
3.2. Mesh stiffness
The numbers of teeth engaged during the meshing process may vary. An analytical method, a finiteelement method, and an analyticalfiniteelement approach are commonly used to solve the mesh stiffness [44]. As shown in Fig. 5, Ma [45] proposed an improved potential energy method to calculate the mesh stiffness, which was validated by the finiteelement method. Therefore, the potentialenergy method proposed by Ma is adopted in the present work and expressed as follows:
where the subscripts 1 and 2 represent the driving and driven gears of the tooth pair, respectively. Based on previous research [46], the singletoothpair mesh stiffness is defined as follows:
Fig. 5Geometric model of a gear profile [45]
The stiffness for a single pair of teeth making and breaking contact can be solved using a computer program and then fitted to the form of the Fourier function as follows:
where ${k}_{m}$ is the average mesh stiffness; ${k}_{aj}$ and ${k}_{bj}$ are the amplitudes of the $j$th order harmonics; ${w}_{f}$ is a fitting frequency of the Fourier series with the fitting period ${T}_{f}=2\pi /{w}_{f}$ being equal to the mesh period. The stiffness excitation period $T$ can be deduced as $T={T}_{f}/\epsilon $. Generally, a good fit can be achieved with the first six orders of the fourier series. Based on results calculated using the potentialenergy method, the fitting parameters are presented in Table 1.
Table 1Coefficients of the harmonic components (N/m)
$j$  ${k}_{aj}$  ${k}_{bj}$ 
1  –1.003e+07  1.233e+08 
2  1.757e+07  2.923e+06 
3  2.393e+06  –9.463e+06 
4  –3.053e+06  –1.055e+06 
5  –3.234e+05  7.274e+05 
6  9.327e+04  5.15e+04 
k_{m}  3.8925e+08 
Then, the timevarying mesh stiffness can be calculated by defining three teeth pairs in the mesh. Ignoring the effects between the teeth, the piecewise meshstiffness functions of the three teeth pairs are expressed as follows:
At last, the timevarying mesh stiffness of the HCR gear system can be calculated using Eqs. (7)(12), whose results are displayed in Fig. 6:
3.3. Dynamic equation
The HCR gear system is mainly composed of gears, shafts, and bearings. The nonlinear dynamic model of this gear system is shown in Fig. 7, where ${T}_{1}$ and ${T}_{2}$ are the torques impacting the driving and driven gears, and ${F}_{1}$ and ${F}_{2}$ are the external radial preloads sustained by the corresponding bearings, respectively. Further, ${I}_{i}$ is the mass moment of inertia, ${m}_{i}$ is the mass, ${c}_{i}$ is the equivalent support damping, ${c}_{h}$ is the damping coefficient of the gear mesh, ${r}_{i}$ is the radius of the base circle, ${k}_{i}$ is the equivalent support stiffness of the bearing, and $Y$ is the mesh direction. The essential dynamics of the gear system can be described as a fourdegreesoffreedom system with coordinates $x\left(t\right)=\left\{{\theta}_{1},{\theta}_{2},{y}_{1},{y}_{2}\right\}$, where ${\theta}_{1}$ and ${\theta}_{2}$ are the dynamic angular displacements of gears 1 and 2, and ${y}_{1}$ and ${y}_{2}$ are the displacements of the corresponding bearings, respectively. The quantities $2{b}_{i}$ and $2b$ refer to the bearing clearance and gear meshing backlash, respectively. As a new relative coordinate, y is defined as transmission error (TE), as follows:
Fig. 6Curves of mesh stiffness with respect to mesh cycle
Fig. 7Dynamic model of the HCR geartransmission system
The dynamic differential equation of the gear system can now be derived using Newton’s second law of motion as follows:
where $f\left({y}_{i},{b}_{i}\right)$ ($i=$ 1, 2) is the radial clearance displacement function of the bearing along the $Y$ direction, which can be expressed as:
and $f\left(y,b\right)$ is a piecewise backlash function described as follows:
Then, the dynamic differential equations can be transformed as follows:
where ${m}_{e}$ is the equivalent mass of the HCR gear system, and $F$ is the average force related to the mean torque. Additionally, the dynamic meshing force (DMF) can be deduced as follows:
4. Results and discussion
Eq. (17) illustrates a series of dynamic equations. In this study, the solutions of these coupled nonlinear equations are solved using the RungeKutta numerical method. The main parameters of the HCR gear system are listed in Table 2. As a gear pair rotates in the mesh, the TE is the main excitation that dictates the lineofaction (LOA) vibrational motion [47]. It is therefore necessary to choose suitable parameters for assessing the motion state of the HCR gear system to reduce vibrations and avoid chaos. To understand the dynamic features of the system comprehensively, the surface roughness, rotating speed, input torque, and shaftbearing stiffness were selected as the control parameters.
Table 2Main parameters of the HCR gear system
Parameters  Pinion/Gear  Parameters  Pinion/Gear 
Material  39NiCrMo_{3}  Elasticity modulus (GPa)  190 
Number of teeth ${z}_{1}$/${z}_{2}$  45  Addendum coefficient  1.3 
Transverse modulus (mm)  3  Poisson ratio  0.3 
Mass (kg) m_{1}/m_{2}  1.914  Pressure angle (°)  20 
Moment of inertia (kg·m^{2}) ${J}_{1}$/${J}_{2}$  0.003  Initial backlash (μm)  10 
Bearing stiffness (N/m) ${k}_{1}$/${k}_{2}$  1e8  Initial mesh damping ratio  0.043 
Bearing clearance (μm) ${b}_{1}$/${b}_{2}$  10  ${F}_{1}$/${F}_{2}$_{}(N)  0 
Tooth width (mm)  25  Hubcore radius (mm)  13.5 
4.1. Effects of surface roughness
To study the surface roughness effects on the system responses, the parameter ${R}_{a}$ was selected as the variable in the corresponding timehistory chart, phase diagram, Poincare map, and time response of the DMF. The other system parameters used in the numerical simulations are as follows: $n=$ 1500 r/min; ${D}_{1}$, ${D}_{2}=$ 1.1; ${T}_{1}=$ 500 N·m. Thus, Fig. 8 illustrates the dynamic responses for different surface roughness values.
Fig. 8System responses of the transmission errors and dynamic meshing forces for varying surface roughnesses: a) timehistory chart; b) phase diagram; c) Poincare map; d) DMF curve
Based on the simulation results, the system dynamic responses are generally divided into simple harmonic, nonharmonic single periodic, subharmonic, quasiperiodic, and chaotic responses. From Fig. 8, it is clear that the HCR gear system is under the nonharmonic single periodic response state when ${R}_{a}=$ 0.59 μm. As shown in Fig. 8(b), the phase graph is a noncircular and nonelliptical curve, and the Poincare map is a dot. Here, AMP represents the amplitude of displacement or force. From the experimental results, the changes in the gear dynamic responses with respect to ${R}_{a}$ are tracked. As the magnitude of ${R}_{a}$ increases from 0.59 μm to 5.13 μm, the vibration and DMF amplitudes change slightly, except for the curve widths in the phase diagram and point distribution on the Poincare map. As the surface roughness increases, the phasediagram curve widths increase gradually, and the point distribution on the Poincare map becomes more disorderly, as can be seen in Figs. 8(b) and 8(c).
This phenomenon indicates that the system state changes from singleperiod to quasiperiodic motions with increasing roughness. More broadly, as shown in Fig. 9, the variations in the bifurcation diagram show that the dynamic responses of the HCR gear system tend from stable toward unstable motions. Therefore, it is concluded that the dynamic characteristics of the HCR gear system are influenced by SP, where the increase in ${R}_{a}$ produced by SP causes the gear dynamics to tend toward instability.
Fig. 9Bifurcation diagram with respect to surface roughness
4.2. Effects of rotating speed
The rotating speed of the pinion is commonly used as a regulating parameter to study the nonlinear dynamic characteristics of a gear transmission system. For enhanced practical engineering value, different rotating speeds are simulated to investigate their influences on the dynamic responses. The HCR spur gears are generally used under heavy loading conditions. Therefore, the load torque is set to 1500 N·m, and the other system parameters are set as follows: ${D}_{1}$, ${D}_{2}=$1.1; ${R}_{a}=$1.51 μm. Fig. 10 depicts the simulated system responses at low, medium, and high rotating speeds. The HCR gear system is under the nonharmonic single periodic response state for all three speed conditions. The response periods of the timehistory charts are equal to the excitation periods. With increase in rotating speed, the amplitude of the transmission error increases first and then decreases. Simultaneously, the DMF amplitude varies similarly; the degree of fluctuation of the DMF is significantly enhanced with increasing rotating speeds and decreases later.
4.3. Effects of input torque
The input torque plays a critical role in the dynamic behaviour, and the system parameters used in the numerical simulations are as follows: $n=$ 2000 r/min; ${D}_{1}$, ${D}_{2}=$ 1.1; ${R}_{a}=$1.51 μm. As seen in Fig. 11, all systems demonstrate nonharmonic single periodic responses, and the phase diagrams are nonelliptical curves. The shapes of all phase diagrams are similar, but the results indicate that increases in the input torques, transmission errors, and DMFs are accompanied by increasing amplitudes. In addition, the ranges of the phase curves are expanded.
Fig. 10System responses of the transmission errors and dynamic meshing forces for varying rotating speeds: a) timehistory chart; b) phase diagram and Poincare map; c) DMF curve
4.4. Effects of shaftbearing stiffness
To understand the effects of the shaftbearing stiffness, ${k}_{1}$ is selected as the variable in the corresponding dynamic system response. The other system parameters used in the numerical simulations are as follows: ${T}_{1}=$ 500 N·m; $n=$ 3000 r/min; ${D}_{1}$, ${D}_{2}=$ 1.1; ${R}_{a}=$ 1.51 μm. The shaftbearing stiffness value is varied from 1e7 N/m to 1e11 N/m, and the results indicate that the bearing support state changes from elastic to rigid supports. It is obvious that the shaftbearing stiffness influences the dynamic characteristics. As shown in Fig. 12, the HCR gear system is under the nonharmonic single periodic response state for different shaftbearing stiffness conditions, and the phase diagram is a noncircular and nonelliptical curve. Moreover, each Poincare map shows a set of centralised points. The amplitudes of the transmission errors and DMF curves increase first and then decrease as the shaftbearing stiffness improves. However, the minimum DMF amplitude is always greater than zero in all the numerical experiments. Thus, the nonimpact state of the HCR gear system can be further verified.
Fig. 11System responses of the transmission errors and dynamic meshing forces for varying input torques: a) timehistory chart; b) phase diagram and Poincare map; c) DMF curve
Fig. 12System responses of the transmission errors and dynamic meshing forces for varying shaftbearing stiffness values: a) timehistory chart; b) phase diagram and Poincare map; c) DMF curve
5. Conclusions
This study focuses mainly on the dynamic responses of the HCR gear system by considering the effects of rough surfaces produced through SP. The main conclusions of the study are summarised as follows:
1) As the surface roughness increases, the dynamic characteristics of the gear system tend toward instability.
2) The minimum DMF is always greater than zero, indicating that the gear system is always in the nonimpact state.
The next step in our research is to try to study the effects of the shot peening parameters on gear dynamics.
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About this article
This study was supported by the Open Fund of Anhui Undergrowth Crop Intelligent Equipment Engineering Research Centre (AUCIEERC202212), the Fundamental Research Funds for the Central Universities of China (PA2023GDSK0064), Highlevel talent research startup funding project (WGKQ2023006).
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
Zhenbang Cheng: writingoriginal draft, writing review and editing, methodology. Yu Zhou: software, supervision. Zhengyu Liu: formal analysis.
The authors declare that they have no conflict of interest.