1. Introduction
Gear transmission systems are widely used in various industries, and their performance directly affects the stability and reliability of the entire equipment. Cylindrical gears, as a crucial component, often encounter issues such as wear during operation. Among the factors influencing gear wear, cumulative pitch error, a common manufacturing error, plays a significant role. This error can change the tooth – contact state, affect the load distribution between teeth, and ultimately influence the wear behavior of gears. Understanding the relationship between cumulative pitch error and gear dynamic wear is essential for improving gear design, predicting gear lifespan, and ensuring the normal operation of gear – based machinery.
1.1 Research Background
In industrial applications, the accurate operation of gear systems is of great importance. Manufacturing and installation errors are inevitable in the production process of gears, and these errors can cause changes in the contact state of the tooth surface. Cumulative pitch error can lead to uneven load distribution among teeth, which in turn affects the fatigue life, vibration, and wear of gears. For example, in automotive transmissions, gear wear due to cumulative pitch error can lead to increased noise, reduced transmission efficiency, and even premature failure of the transmission system. Therefore, it is necessary to study the influence of cumulative pitch error on gear dynamic wear to improve the performance and reliability of gear systems.
1.2 Significance of the Research
This research is of great significance for several reasons. Firstly, it can provide a theoretical basis for gear design. By understanding the impact of cumulative pitch error on gear wear, designers can optimize gear parameters to reduce wear and extend the service life of gears. Secondly, it helps in the prediction of gear failure. Through the establishment of a dynamic wear model considering cumulative pitch error, it is possible to predict the wear state of gears in advance, enabling timely maintenance and replacement, and reducing the risk of equipment failure. Finally, this research can also contribute to the improvement of gear manufacturing processes. By understanding the relationship between manufacturing errors and gear wear, manufacturers can take measures to reduce cumulative pitch error and improve the quality of gears.
2. Gear Meshing and Wear – Related Theories
2.1 Gear Meshing Principles
Gears work based on the principle of meshing. When two gears are in operation, the teeth of the driving gear engage with those of the driven gear, transferring torque and motion. The meshing process involves the contact and relative motion between the tooth surfaces. In an ideal situation, the meshing of gears should be smooth and stable, with the load evenly distributed among the teeth. However, in reality, manufacturing errors such as cumulative pitch error can disrupt this ideal meshing state.
2.1.1 Ideal Gear Meshing
In ideal gear meshing, the tooth profiles of the gears are designed according to specific geometric principles, usually involute curves. The meshing process follows the law of conjugate motion, ensuring a constant transmission ratio and smooth power transfer. The contact between the teeth occurs along the line of action, and the load is evenly distributed across the engaged teeth. Table 1 shows the characteristics of ideal gear meshing.
| Characteristics of Ideal Gear Meshing | Details |
|---|---|
| Tooth Profile | Involute curve |
| Transmission Ratio | Constant |
| Contact Location | Along the line of action |
| Load Distribution | Even among engaged teeth |
2.1.2 Impact of Cumulative Pitch Error on Meshing
Cumulative pitch error refers to the deviation of the actual pitch from the theoretical pitch along the circumference of the gear. This error can cause the teeth to engage prematurely or delay, disrupting the normal meshing process. As a result, the load distribution between the teeth becomes uneven. Teeth with larger pitch errors may bear more load, while those with smaller errors may bear less. This uneven load distribution can lead to increased stress on some teeth, accelerating wear and affecting the overall performance of the gear system. Figure 1 shows a schematic diagram of the impact of cumulative pitch error on gear meshing.
2.2 Wear Theories
Wear is a complex phenomenon that occurs on the surface of gears during operation. There are several wear theories, among which the Archard wear theory is widely used.
2.2.1 Archard Wear Theory
The Archard wear theory is based on the idea that the wear volume is proportional to the load, sliding distance, and inversely proportional to the hardness of the material. The formula for calculating the wear depth is given by \(\Delta h = k\frac{F_s d}{H}\), where \(\Delta h\) is the wear depth, k is the wear coefficient, \(F_s\) is the normal load, d is the sliding distance, and H is the hardness of the material. In the case of gears, this theory is used to predict the wear of the tooth surface. Different lubrication states can affect the value of the wear coefficient k. For example, in boundary lubrication, k has a certain value, while in mixed lubrication and elastohydrodynamic lubrication, the value of k changes accordingly. Table 2 shows the values of the wear coefficient k under different lubrication states.
| Lubrication State | Wear Coefficient k |
|---|---|
| Boundary Lubrication | \(k_{w0}\) |
| Mixed Lubrication | \(2k_{w0}(4 – \lambda)/7\) (\(0.5\leq\lambda\leq4\), \(\lambda\) is the film – thickness ratio) |
| Elastohydrodynamic Lubrication | 0 (\(\lambda>4\)) |
2.2.2 Other Wear – Related Factors
In addition to the Archard wear theory, there are other factors that affect gear wear. The relative sliding velocity between the teeth, surface roughness, and temperature all play important roles. High relative sliding velocity can increase the wear rate, while rough surfaces are more prone to wear. Temperature can also affect the material properties and lubrication state, thereby influencing the wear process. For example, high – temperature environments can reduce the viscosity of the lubricant, making it less effective in reducing wear.
3. Establishment of the Dynamic Wear Model Considering Cumulative Pitch Error
3.1 Load Distribution Algorithm Considering Cumulative Pitch Error
The load distribution between the teeth of a gear pair is a key factor in determining the wear behavior. To accurately analyze the load distribution considering cumulative pitch error, the Load Tooth Contact Analysis (LTCA) method is used.
3.1.1 LTCA Method
The LTCA method separates the overall deformation and contact deformation during the gear meshing process. The contact deformation is calculated using analytical formulas to improve calculation efficiency, while the overall deformation is calculated by the finite – element method. In this study, the finite – element calculation is implemented in the MATLAB environment. Figure 2 shows the finite – element model used in the LTCA method.
3.1.2 Conversion of Cumulative Pitch Error
The cumulative pitch error needs to be converted to the normal direction of the tooth surface. The conversion formula is \(E_{pr}=f_{pr}\cos\alpha_{t}\cos\beta_{b}\), where \(E_{pr}\) is the converted error, \(f_{pr}\) is the pitch error, \(\alpha_{t}\) is the transverse pressure angle, and \(\beta_{b}\) is the base – helix angle. This conversion ensures that the cumulative pitch error can be correctly considered in the load – distribution calculation.
3.1.3 Load – Distribution Calculation
The load – distribution calculation in the LTCA method involves solving a set of equations. The iterative equation is \(\begin{bmatrix}-(\lambda_{c}+\lambda_{b})&I_{n\times1}\\I_{1\times n}&0\end{bmatrix}\begin{bmatrix}F_{n}\\\delta_{s}\end{bmatrix}=\begin{bmatrix}\varepsilon\\F_{s}\end{bmatrix}\), where \(F_{n}\) is the normal contact force vector, \(\delta_{s}\) is the static transmission error, \(F_{s}\) is the static meshing force, \(\varepsilon\) is the tooth – profile deviation vector, and I is a vector with all elements equal to 1. By solving this equation, the load distribution on the tooth surface and the static transmission error can be obtained.
3.2 Gear Rotor System Dynamics Model Considering Cumulative Pitch Error
To analyze the dynamic behavior of the gear system considering cumulative pitch error, a gear rotor system dynamics model is established.
3.2.1 Modeling of the Gear Meshing Unit
The gear meshing unit is used to simulate the meshing relationship between the input and output shafts. Considering the degrees of freedom of bending, torsion, and axial direction, the generalized coordinates of the gear meshing unit are defined as \(x_{m}=[x_{p},y_{p},z_{p},\theta_{xp},\theta_{yp},\theta_{zp},x_{g},y_{g},z_{g},\theta_{xg},\theta_{yg},\theta_{zg}]\). The displacement of the gear is projected onto the meshing line using the projection vector \(V_{m}\), and the unit stiffness matrix of the gear meshing unit is \(K_{m}(t)=k(t)V_{m}^{T}V_{m}\), where \(k(t)\) is the meshing stiffness.
3.2.2 Simulation of Shaft and Bearing
The flexibility of the shaft is simulated using the Timoshenko beam unit, and the bearing support at both ends of the transmission shaft is simulated using linear springs. By assembling the meshing unit, beam unit, and bearing unit, the overall stiffness matrix of the gear rotor system can be formed. The overall dynamics equation of the gear rotor system is \(M\ddot{X}+(C + G)\dot{X}+KX = F\), where M is the mass matrix, C is the damping matrix, G is the gyroscopic matrix, K is the stiffness matrix, F is the external load vector, and X is the displacement column matrix.
3.2.3 Influence of Cumulative Pitch Error on Dynamic Response
Due to the presence of cumulative pitch error, the vibration response of the gear system changes. The traditional gear rotor dynamics analysis usually takes the meshing period as the cycle of vibration response change. However, considering the cumulative pitch error, a hunting – tooth period needs to be considered for dynamics analysis. The hunting – tooth period \(T_{ht}=\frac{1}{f_{ht}} = lcm(z_{1},z_{2})\cdot T_{m}=\frac{z_{1}z_{2}}{N_{ap}}\cdot T_{m}\), where \(T_{m}\) is the gear meshing period, lcm is the least – common – multiple function, \(z_{1}\) and \(z_{2}\) are the numbers of teeth of the driving and driven gears respectively, \(N_{ap}\) is the combination – state coefficient, and \(f_{ht}\) is the hunting – tooth frequency. The hunting – tooth frequency represents the frequency at which two specific teeth on the driving and driven gears meet once. In addition, the non – uniformity of tooth meshing can also generate combination – state frequencies. Table 3 shows the calculation formulas of relevant frequencies.
| Frequency | Calculation Formula |
|---|---|
| Hunting – Tooth Frequency \(f_{ht}\) | \(f_{ht}=\frac{f_{m}N_{ap}}{z_{1}z_{2}}\) (\(f_{m}\) is the meshing frequency) |
| Combination – State Frequency \(f_{sp}\) | \(f_{sp}=\frac{f_{m}}{N_{ap}}\) |
3.3 Wear Prediction Model Based on Archard Wear Theory
Based on the Archard wear theory, a wear prediction model for gears is established.
3.3.1 Calculation of Wear Depth
The wear depth caused by a single wear is calculated using the Archard wear model. For the driving and driven wheels, the wear depth formulas are \(\Delta h_{p}=2k_{w}\overline{\sigma}_{H}a_{H}\left|1 – v_{g}/v_{p}\right|\) and \(\Delta h_{g}=2k_{w}\overline{\sigma}_{H}a_{H}\left|1 – v_{p}/v_{g}\right|\) respectively, where \(v_{p}\) and \(v_{g}\) are the sliding velocities of the meshing points on the driving and driven wheels, \(a_{H}\) is the semi – width of the Hertz contact area, \(\overline{\sigma}_{H}\) is the average stress in the Hertz contact area, and \(k_{w}\) is the wear coefficient considering different lubrication states. The semi – width of the Hertz contact area \(a_{H}=\sqrt{\frac{8F_{d}\cdot lsr_{s}\rho_{e}}{L\pi E_{e}}}\), and the average stress \(\overline{\sigma}_{H}=\frac{3F_{d}\cdot lsr_{s}}{2\pi La_{H}}\), where \(F_{d}\) is the dynamic load, \(lsr_{s}\) is the static load – distribution coefficient obtained from the LTCA method, \(\rho_{e}\) and \(E_{e}\) are the equivalent curvature radius and equivalent elastic modulus of the gear pair, and L is the tooth width.
3.3.2 Iterative Process of Dynamic Wear Simulation
To consider the coupling relationship between dynamics and wear, an iterative approach is used for dynamic wear simulation. Instead of solving the load – distribution and dynamics models for each wear cycle, the tooth profile is updated only when the cumulative wear depth is greater than the wear – update threshold \(\varepsilon_{n}\) (in this study, \(\varepsilon_{n} = 2\ \mu m\)). This method divides the full – life – cycle wear process into several stages, and within each stage, the load distribution and steady – state vibration response are considered to be unchanged. Figure 3 shows the flow chart of the dynamic wear simulation process.
4. Model Verification
To verify the effectiveness of the established model, a comparison is made with the experimental results in the literature.
4.1 Experimental Data Selection
The dynamic transmission error experimental results of a helical gear pair with cumulative pitch error in the literature [18] are selected for verification. The gear pair parameters, tooth – profile deviation parameters, shaft – segment parameters, and bearing stiffness parameters used in the experiment are detailed in the literature [18]. To facilitate comparison with the experimental results, the same tooth – profile deviation parameters as those in the experiment are adopted in the simulation, including tooth – profile modification, tooth – direction modification, cumulative pitch error, and geometric eccentricity.
4.2 Comparison of Simulation Results and Experimental Results
The frequency spectrum of the dynamic transmission error obtained from the simulation and the experimental results are compared. Figure 4 shows the comparison of the frequency spectra. It can be seen that the rotation – frequency components caused by the cumulative pitch error are very obvious, and there are obvious side – frequency bands near the meshing frequency. The frequency – spectrum characteristics obtained from the simulation are very similar to those of the experimental results, indicating the effectiveness of the established simulation model.
5. Results Analysis and Discussion
5.1 Meshing Characteristics Analysis
5.1.1 Gear Parameters and Error Settings
The gear pair parameters used in the dynamic wear simulation are shown in Table 4. The diameters of the driving and driven shafts are 60 mm and 100 mm respectively, and the shaft length is 300 mm. The bearing stiffness is shown in Table 5. The cumulative pitch errors of the driving and driven wheels are based on the measured results in the literature [19], as shown in Figure 5.
| Gear Pair Parameters | Values |
|---|---|
| Number of Teeth (Driving/Driven) | 28/56 |
| Elastic Modulus (GPa) | 210 |
| Poisson’s Ratio | 0.3 |
| Inner Hole Radius (mm) | 30/50 |
| Normal Module (mm) | 4 |
| Helix Angle (\(^{\circ}\)) | 0 |
| Pressure Angle (\(^{\circ}\)) | 20 |
| Tooth Width (mm) | 40 |
| Torque (\(N\cdot m\)) | 500 |
| Rotational Speed (\(r\cdot min^{-1}\)) | 2865 |
| Tooth – Tip Height Coefficient | 1 |
| Profile Shift Coefficient | 0 |
| Bearing Stiffness | Value |
|---|---|
| Radial Stiffness (\(N/m\)) | \(7.6\times10^{7}\) |
| Axial Stiffness (\(N/m\)) | \(1\times10^{6}\) |
5.1.2 Wear Distribution and Meshing Stiffness
Figure 6 shows the cumulative wear depth on the middle surface of the tooth width of the driving wheel at different operating times. The wear amount at the pitch point is zero because the relative sliding velocity is zero. There is a large amount of wear at the tooth root and tooth tip. As the number of operating times increases, due to severe wear at the tooth root and tooth tip, the contact stress in this area decreases, and the wear rate slows down. Figure 7 shows the time – varying meshing stiffness under different conditions. For a gear pair without cumulative pitch error, the meshing stiffness in each meshing cycle is the same. The introduction of cumulative pitch error makes the meshing stiffness in each meshing cycle different, showing an overall fluctuating trend within the hunting – tooth period. The cumulative pitch error only affects the meshing stiffness in the double – tooth region, and the meshing stiffness in the single – tooth region is independent of the cumulative pitch error. After \(400\times10^{6}\) operating times, the stiffness between the single – and double – tooth meshing regions becomes more stable, and the length of the double – tooth region is reduced, that is, the contact ratio of the gear pair is decreased.
5.2 Vibration Characteristics Analysis
5.2.1 Frequency Components in the Spectrum
Figure 8 shows the dynamic transmission error spectrum after \(400\times10^{6}\) operating times. If the meshing non – uniformity caused by cumulative pitch error is not considered, the spectrum of the worn gear only contains the meshing frequency component. When cumulative pitch error is considered, the spectrum of the worn gear contains richer frequency components. In the low – frequency range, obvious hunting – tooth frequency and combination – state frequency components can be observed, and there is also side – frequency modulation near the meshing frequency and its harmonics. In this example, due to the special number of teeth, the hunting – tooth frequency \(f_{ht}\) is equal to the rotational frequency \(f_{s}\) of the driven shaft, and the combination – state frequency \(f_{sp}\) is equal to the rotational frequency \(f_{a1}\) of the driving shaft.
5.2.2 Influence of Wear on the Dynamic Load Coefficient
The dynamic load coefficient is defined as the ratio of the maximum dynamic meshing force to the static meshing force during the steady – state vibration period. Figure 9 shows the dynamic load coefficient at different wear degradation stages. In the early wear stage (running – in stage), wear can improve the meshing stability. The main reason is that mild wear can relieve the tooth interference during the single – tooth to double – tooth transition process, playing a role in running – in and “passive profile modification”. As the number of operating times increases, the meshing clearance (no – load transmission error) caused by wear becomes larger, and the gear system shows an obvious performance degradation trend, with the vibration level increasing linearly. Although both the proposed model and the traditional model (without considering pitch error) can predict this performance degradation process, the traditional model underestimates the dynamic load coefficient by about 10%.
5.3 Wear Non – uniformity Analysis
5.3.1 Comparison between Traditional and Proposed Models
Traditional wear simulation analysis is based on the assumption of uniform load distribution between teeth, so the wear amount on each tooth is the same. However, in reality, due to the existence of cumulative pitch error, the wear morphology on the surface of each tooth is often different. The proposed model can effectively simulate the wear non – uniformity caused by uneven load distribution between teeth by considering cumulative pitch error. Figure 10 shows the wear distribution obtained by the proposed model, where the color of the contour plot represents the wear depth at the tooth root of each tooth.
5.3.2 Wear Non – uniformity Coefficient and Its Variation
The wear non – uniformity coefficient is defined as the ratio of the standard deviation of the wear amount of each tooth to the average value, \(c_{weur}=\frac{std(h_{root})}{\overline{h}_{root}}\), where \(h_{root}\) is the vector of the wear amount at the tooth root of each gear tooth. Figure 11 shows the wear depth at the tooth root and the wear non – uniformity coefficient at different wear stages. The wear depth at the tooth root obtained by the traditional model is only close to that of the proposed model in the initial wear stage. As the wear degradation process progresses, the difference between the two models becomes larger. The difference between the two models is closely related to the wear non – uniformity between teeth. The variation trend of the wear non – uniformity coefficient predicted by the proposed model is similar to that of the dynamic load coefficient. Early wear helps to relieve the uneven load distribution between teeth caused by pitch error, but severe wear will deteriorate the load – sharing characteristics and contact state, leading to increased vibration and even failure.
5.3.3 Influence of Tooth Number Combination on Wear Distribution
To study the wear distribution characteristics under different tooth number combinations, the wear uniformity of five groups of tooth number combinations shown in Table 6 is evaluated. The hunting – tooth coefficient is defined as \(c_{ht}=\frac{N_{ap}}{max(z_{1},z_{2})}\). Figure 12 shows the change rule of the wear non – uniformity coefficient of the driving wheel with the hunting – tooth coefficient. As the hunting – tooth coefficient increases, the wear non – uniformity coefficient increases, that is, the difference in wear amount between teeth increases. For gear pair A, the number of teeth of the two gears is relatively prime, and its wear non – uniformity coefficient is the smallest. This is because when the number of teeth is relatively prime, the period (hunting – tooth period) for two specific teeth on the driving and driven wheels to meet once is the longest. During the hunting – tooth period, the teeth of the driving wheel will mesh with each tooth of the driven wheel once. As the meshing process progresses, the wear between teeth gradually becomes more uniform. The design with relatively prime numbers of teeth is called the hunting – tooth design. Compared with the original design (gear pair D), the hunting – tooth design (gear pair A) can reduce the wear non – uniformity coefficient by about 30%. For gear pair E, the teeth of the driving wheel will always mesh with a certain tooth of the driven wheel, which will naturally cause severe wear of certain specific teeth. In the gear design stage, the hunting – tooth problem should be fully considered to obtain a more uniform wear form and reduce the probability of local gear failure.
5.3.4 Influence of Cumulative Pitch Error on Vibration and Hunting – Tooth Design
Figure 13 shows the dynamic load coefficient under different cumulative pitch errors. For different pitch error conditions, the hunting – tooth design can effectively reduce vibration. When the gear machining accuracy is high (such as 20% cumulative pitch error in Figure 13), the difference in vibration response between the hunting – tooth design and the original design is small. Due to the limitations of transmission ratio and structural size, not all gears in engineering can meet the requirements of the hunting – tooth design. When the cumulative pitch error of the gear is small, the requirement of the hunting – tooth design can be appropriately relaxed.
6. Conclusions
Based on the Load Tooth Contact Analysis method, gear rotor dynamics model, and Archard wear model, a dynamic wear prediction model for gears considering cumulative pitch error is proposed. The effectiveness of the model is verified by comparing the simulation spectrum with the experimental results in the literature. The following conclusions are obtained through analyzing the influence of cumulative pitch error on response characteristics and wear distribution:
(1) Cumulative pitch error introduces rich frequency components such as shaft rotation frequency, hunting – tooth frequency, and combination – state frequency into the meshing stiffness, transmission error, and vibration response. In contrast, the spectrum obtained by the traditional dynamics model usually only contains the meshing frequency component.
(2) The proposed model can evaluate the wear non – uniformity between teeth caused by cumulative pitch error, improving the idealized assumption in the traditional wear model that the wear amount of each gear tooth is the same. Mild wear helps to relieve the uneven load distribution between teeth, while severe wear deteriorates the load – sharing characteristics and leads to increased vibration.
(3) The proposed dynamic wear model can establish the mapping relationship between the hunting – tooth coefficient and the wear non – uniformity coefficient. The hunting – tooth design can reduce the wear non – uniformity coefficient by about 30%. When the cumulative pitch error is small, the requirement of the hunting – tooth design can be appropriately relaxed.
This research provides a theoretical basis for gear wear mechanism research and gear parameter design. Future research can be further extended to consider the influence of more complex factors, such as variable operating conditions and multi – gear – system interactions, on gear dynamic wear.
