This paper focuses on the study of helical cylindrical gears’ meshing dislocation and modification. By considering the influence of system stiffness and random factors, the meshing dislocation of gear pairs is analyzed in detail. Four different cases are set up for comparison and research, and the effectiveness of gear modification in compensating for meshing dislocation is verified through contact analysis and stress analysis. The research results can provide a reference for optimizing the design of gear transmission.
1. Introduction
1.1 Background and Significance
Gears are crucial components in mechanical transmission systems, and their performance directly affects the operation stability and efficiency of the entire system. In the actual working process of gears, meshing dislocation often occurs due to various factors such as thermal deformation and contact load during transmission. This not only affects the contact stress, meshing stiffness, transmission efficiency, but also leads to failure and damage of gears. For example, in the gear transmission system of a vehicle, meshing dislocation can cause abnormal noise, increased wear, and reduced service life. Therefore, studying how to improve the tooth surface bearing capacity of gears through reasonable modification methods is of great significance for enhancing the reliability and durability of mechanical systems.
1.2 Research Status
At present, many scholars at home and abroad have conducted in – depth research on gear meshing dislocation. Some studies mainly focus on the analysis of gear bearing contact, such as Wang Peng et al. who analyzed the system sources and influencing factors of gear meshing dislocation in electric vehicle reducers in detail and proposed that micro – modification design of gears can reduce meshing dislocation deviation. Others, like Yuan Bing et al., established relevant models to verify the vibration – reducing effect of different modification methods under a certain amount of meshing dislocation. However, most of the existing research is aimed at the tooth surface modification under ideal meshing conditions, while ignoring the influence of various complex factors in the actual gear transmission process.
2. Analysis Model and Examples
2.1 Gear Transmission System Model
The gear transmission system studied in this paper consists of a set of gear pairs, two pairs of bearings (both are tapered roller bearings), and two parallel rotating shafts, as shown in Figure 1. The power is transmitted from the left end of shaft 1 to the driving gear, and then output from the right end of shaft 2 through the driven gear. The maximum input torque is 450 N·m, and the highest speed is 7500 r/min. The macroscopic geometric parameters of the gear pair are shown in Table 1.
| Geometric Parameter | Driving Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 25 | 31 |
| Tooth Width/mm | 48 | 48 |
| Helical Direction | Right | Left |
| Normal Module/mm | 2.954 | 2.954 |
| Normal Pressure Angle/(°) | 23.45 | 23.45 |
| Helix Angle/(°) | 21.5 | 21.5 |
| Standard Center Distance/mm | 88.9 | 88.9 |
| Figure 1: Schematic diagram of gear transmission |
2.2 Analysis Examples
Four cases are designed for comparative research, as shown in Table 2. Case 1 represents a gear pair with misalignment caused by systematic errors but without modification. Case 2 is a gear pair with systematic meshing misalignment and gear modification. Case 3 introduces misalignment caused by random errors without gear modification, and Case 4 is a gear pair with both systematic and random misalignments and gear modification.
| Case | Meshing Dislocation Caused by Systematic Errors | Meshing Dislocation Caused by Random Errors | Micro – modification |
|---|---|---|---|
| 1 | Yes | No | No |
| 2 | Yes | No | Yes |
| 3 | Yes | Yes | No |
| 4 | Yes | Yes | Yes |
3. Gear Meshing Dislocation
3.1 Definition and Classification of Meshing Dislocation
According to the ISO 6336 standard, meshing dislocation is defined as the maximum separation between corresponding points of two gears along the meshing tooth width direction, denoted by the symbol \(F_{\beta x}\). Hourser et al. classified gear meshing dislocation into three types: parallel misalignment (center – distance error), misalignment parallel to the meshing plane \(M_{LOA}\), and misalignment perpendicular to the meshing plane MoLoA. In this study, since the center – distance error has a small impact on tooth – direction load distribution and the influence of MoLoA on tooth – direction load distribution is negligible, the focus is on \(M_{LOA}\), which can lead to stress concentration at one end of the meshing teeth and worsen the tooth – surface load – distribution situation.
3.2 Sources of Meshing Dislocation
The factors causing tooth – direction load distribution and meshing dislocation can be generally divided into two parts: manufacturing and installation errors of gears and the deformation and clearance of components in the transmission system. In ISO standards, the misalignment caused by manufacturing and installation errors is denoted as \(f_{man}\), the misalignment caused by gear and support – shaft deformation is denoted as \(f_{shì}\), the misalignment caused by the deformation of the gearbox is denoted as \(f_{car}\), and the misalignment caused by bearing clearance and deformation is denoted as \(f_{be}\). The total meshing dislocation \(F_{\beta x}\) can be expressed by the formula: \(F_{\beta x}=B_{1}\left(f_{s h 1}+f_{s h 2}+f_{c a}+f_{b e}\right)+B_{2} f_{m a}\) where \(B_{1}\), \(B_{2}\) are correction coefficients for deformation and manufacturing errors, and \(f_{sh1}\), \(f_{sh2}\) are the deformations of the driving and driven gear shafts respectively. The errors caused by load or temperature are systematic errors, such as shaft bending, torsional deformation, gear deformation, bearing deformation, and bearing clearance. The manufacturing and installation errors related to tolerances and accuracy are random errors. In this study, the influence of helix – line slope deviation \(\Delta f_{H \beta}\) and helix – line waviness deviation \(\Delta f_{f \beta}\) on misalignment parallel to the meshing plane is mainly considered.
3.3 Calculation of Meshing Dislocation
3.3.1 Calculation of System Meshing Dislocation
ISO 6336 – 1:2006 provides three methods for calculating meshing dislocation. In this paper, the calculation method based on computer simulation (Method B) is used. First, a global coordinate system is defined with the origin of the gearbox as the coordinate origin o, the axial direction as the Z – axis (right – hand positive), the center – distance direction of the gear pair as the Y – axis (positive from the driving gear to the driven gear), and the normal direction of the YOZ plane as the X – axis. The continuous meshing surface of the gear is discretized into meshing points, and the tooth width is divided into n equal – interval measuring points (\(n = 5\) in this case). The linear displacements of the discrete points of the gear support shaft in the X and Y axes are calculated and denoted as \(M_{i n X}\) and \(M_{i n Y}\). Then, the linear displacements of the measuring points of the gear pair are projected onto the meshing – line direction of the gear pair according to the projection relationship to obtain the equivalent displacements of the two gears along the meshing line at the measuring points. Finally, the minimum value of the difference between the displacements of the driving and driven gears at each measuring point is taken for normalization, and the maximum value of the meshing dislocation at 5 measuring points is taken as the meshing dislocation \(F_{\beta_{x}}\) of the gear pair along the meshing line. Based on the model of Case 1, without considering the meshing dislocation caused by random errors, the meshing – line – direction misalignment distribution of the gear pair under the maximum load torque is calculated as shown in Table 3.
| Offset Distance along Tooth Width Coincidence/mm | Displacement of Driving Gear/μm | Displacement of Driven Gear/μm | Normalized Meshing Dislocation/μm |
|---|---|---|---|
| 0 | – 14.39 | 14.08 | 3.65 |
| 12 | – 13.59 | 15.25 | 4.02 |
| 24 | – 12.79 | 16.42 | 4.40 |
| 36 | – 11.99 | 15.02 | 2.20 |
| 48 | – 11.19 | 13.62 | 0 |
| If the influence of the stiffness of each component and bearing clearance on gear transmission is considered, further deformation analysis is required. The shafts are divided into multiple shaft – segment units, and through static analysis and iterative calculation, the total meshing dislocation \(M_{s}=-3.4112\ \mu m/mm\) caused by system errors is obtained. The contribution of each component in the system to meshing dislocation is shown in Table 4. It can be seen that the components on the driving – gear shaft play a leading role in meshing dislocation, and the bearing on the driving – gear shaft has the largest contribution rate, reaching 47.44%. | |||
| Component | Dislocation/μm | Component | Dislocation/μm |
| — | — | — | — |
| Driving – Gear Shaft Assembly | – 3.00 | Driven – Gear Shaft Assembly | – 0.4124 |
| Gear | 0.9555 | Gear | – 0.3515 |
| Support Shaft | – 8.001×10⁻³ | Support Shaft | – 1.688×10⁻² |
| Bearing Inner Ring | – 3.95 | Bearing Inner Ring | – 4.398×10⁻² |
| Bearing Outer Ring | 0 | Bearing Outer Ring | 0 |
3.3.2 Calculation of Random Meshing Dislocation
Based on the ISO 7 – level accuracy of the gear pair in this study, referring to relevant quality standards, the helix – line slope deviation \(\Delta f_{H \beta}\) and helix – line waviness deviation \(\Delta f_{f \beta}\) of the two gears are both ±14μm, and the direction is random. The total random meshing dislocation \(M_{r}\) caused by these two errors can be calculated by the formula: \(M_{r}=2\left(\Delta f_{H \beta}+\Delta f_{f \beta}\right)/b\) where b is the working tooth width. The calculation result is \(M_{r}= \pm 1.1667\ \mu m/mm\). Since there is no relevant literature on converting this random meshing dislocation into an equivalent dislocation, a method used by Shehata et al. is referred to in the model of Case 3 to introduce this error.
4. Gear Modification
4.1 Purpose and Types of Gear Modification
The purpose of gear modification is to improve the tooth – surface contact performance of gear transmission under the influence of meshing dislocation and make the tooth – surface load distribution more reasonable. Common modification methods include tooth – direction modification, tooth – profile modification (two – dimensional modification), and for helical gears, diagonal modification (three – dimensional modification). Tooth – direction modification can effectively avoid edge contact problems; tooth – profile modification can avoid sudden changes in gear load; diagonal modification can reduce tooth – surface contact stress, bearing transmission error, and meshing – in and meshing – out impacts.
4.2 Tooth – Direction Modification
Tooth – direction modification is the main modification method considered in this paper to compensate for meshing dislocation. It includes tooth – direction end – face chamfering, tooth – direction inclination, and tooth – direction crowning, as shown in Figure 2. Figure 2: Tooth – direction modification
4.2.1 Tooth – Direction End – Face Chamfering
For the gear pair in Case 1 and Case 3, the right tooth surface of the driving gear is in a state of concentrated load at the II – end. Therefore, the modification direction is to chamfer from the I – end face to the II – end face. For quenched and tempered steel gears, the ISO provides empirical formulas to calculate the modification length and modification amount of the end – face chamfering. For Case 2, the modification amount is calculated as 0.4818μm according to the formula, and for Case 4, it is 1.3568μm considering the influence of 1.5 times the misalignment caused by random deviation. The modification lengths of both cases are 2.954 mm.
4.2.2 Tooth – Direction Inclination
The direction of tooth – direction inclination is the same as that of chamfering. It starts from the left tooth end and thins the entire tooth width. The acceptable modification amount is \(10\ \mu m\leq C_{H \beta}\leq25\ \mu m\). In Case 2, \(C_{H \beta}=f_{H \beta}=14\ \mu m\), and in Case 4, \(C_{H \beta}=f_{H \beta}+M_{r}=15.1667\ \mu m\).
4.2.3 Tooth – Direction Crowning
Tooth – direction crowning is aimed at the middle part of the tooth width. Only the crowning amount needs to be determined. For the model in Case 2, the crowning amount is calculated as 0.4818μm, and for Case 4, it is 2.2318μm considering the influence of random errors.
4.3 Tooth – Profile Modification and Diagonal Modification
For wide – helical gears in this model, although there is no sudden change in load due to the alternation of single – and double – tooth meshing, in order to improve the bearing capacity of the gears, it is necessary to shorten the length of the single – tooth meshing area under 100% load and make the load change more gentle during double – tooth meshing. Therefore, tooth – profile modification is required, which can include tooth – tip/tooth – root chamfering, end – face tooth – profile modification, and tooth – profile crowning. Since diagonal modification basically includes the effects of tooth – profile modification, diagonal modification is used to replace tooth – profile modification in this paper, which can be divided into tooth – tip meshing – in angle modification \(C_{E a}\) and tooth – root meshing – out angle modification \(C_{E f}\). Based on the tooth – surface load distribution results of Case 1, only the tooth – tip meshing – in angle of the gear pair in Case 2 is modified. The modification reference diameter \(d_{E a}=82.035\ mm\), the modification length \(b_{E a}=11.429\ mm\), and the modification amount \(C_{E a}\approx3.84\ \mu m\). For Case 3, since the tooth – surface load – bearing situation is similar to that of Case 1, the modification reference diameter and length remain unchanged, and the modification amount \(C_{E a}\approx5.73\ \mu m\). The modification amounts and methods of Case 2 and Case 4 are summarized in Table 5.
| Parameter | Case 2 | Case 4 |
|---|---|---|
| Tooth – Right – End – Face Chamfering Amount \(C_{\beta II}\)/μm | 0.4818 | 1.3568 |
| Tooth – Right – End – Face Chamfering Length \(L_{C II}\)/μm | 2.954 | 2.9574 |
| Starting Point of Tooth – Right – End – Face Chamfering/mm | 40.8 | 40.8 |
| Tooth – Direction Inclination Amount \(C_{H \beta}\)/μm | 14 | 15.1667 |
| Tooth – Direction Crowning Amount \(C_{\beta}\)/μm | 0.4818 | 2.2318 |
| Tooth – Tip Meshing – In Angle Modification Amount \(C_{Ea}\)/μm | 3.84 | 5.73 |
| Tooth – Tip Meshing – In Angle Modification Length \(b_{Ea}\)/mm | 11.429 | – |
| Tooth – Tip Meshing – In Angle Modification Reference Diameter \(d_{Ea}\)/mm | 82.035 | 82.035 |
5. Contact Analysis Comparison
5.1 Evaluation Indexes
In order to verify the effectiveness of gear micro – modification in compensating for meshing dislocation, tooth – surface contact analysis and tooth – root stress analysis are carried out on the driving gears of all cases. The evaluation indexes include transmission error, contact spots on the working tooth surface of the driving gear, and tooth – root stress of the driving gear.
5.2.2 Contact Spots and Tooth – Root Stress
The contact spots on the tooth surface and tooth – root stress are analyzed by different methods. The contact spots are analyzed using the tooth – surface bearing contact analysis technology (LTCA), and the tooth – root stress is analyzed by the finite – element method (FEA). The unit – length normal load distribution and tooth – root stress distribution of the driving gear in each case model are shown in Table 6.
| Case | Tooth – Surface Contact Spot (Unit – Length Load/N/mm) | Tooth – Root Stress (Von Mises Stress/MPa) |
|---|---|---|
| 1 | 429.509, 371.681, 313.852, 256.024, 140.367, 198.196, 82.539 | 275.514688, 229.655712, 183.796737, 137.937761, 92.078786, 46.219810, 0.360835 |
| 2 | 297.061, 247.551, 198.041, 148.530, 99.020, 49.510, 0.000 | 180.608352, 150.608723, 120.609094, 90.609465, 60.609836, 30.610207, 0.610578 |
| 3 | 431.948, 371.637, 311.326, 251.015, 190.704, 130.393, 70.082 | 281.049792, 234.267470, 187.485147, 140.702825, 93.920502, 47.138180, 0.355857 |
| 4 | 294.566, 245.472, 196.377, 147.283, 98.189, 49.094, 0.000 | 189.716368, 158.207772, 126.699176, 95.190580, 63.681984, 32.173388, 0.664792 |
Case 1 shows a typical uneven load – distribution situation, with the maximum load on the right – hand side of the tooth surface. The length of the contact spot is about 23.8% of the total tooth width, and the height is about 47.8% of the total tooth height. The tooth – root stress is concentrated at the meshing – in end of the gear. After the gear modification in Case 2, the tooth – surface load is significantly improved. The unit – length load is reduced from 429.509 N/mm to 297.061 N/mm, and the length and height distribution of the contact spot become more uniform. The tooth – root stress is reduced from 275.5 MPa to 180.6 MPa, and the stress distribution gradually moves towards the middle of the tooth width.
Case 3 has no obvious difference in stress – distribution form from Case 1, either on the tooth surface or at the tooth root, except for a slight increase in numerical values. After the modification in Case 4, the stress amplitude decreases, and the stresses on the tooth surface and at the tooth root are more evenly distributed. However, the modification effect of Case 4 is not as good as that of Case 2, indicating that the modification has little compensation effect on the meshing dislocation caused by random errors.
6. Conclusion
6.1 Influence of Various Factors on Meshing Dislocation
Among the systematic factors, the bearing deformation has the greatest influence on the misalignment amount. The influence effects of various factors may be superimposed or offset each other. Among the random factors, the helix – line deviation has the greatest influence on the meshing misalignment, and the influence effects of various factors are superimposed.
6.2 Dominant Factor Affecting Gear Transmission Performance
The meshing misalignment caused by systematic factors is the dominant factor affecting the transmission stability and bearing capacity of gears.
6.3 Compensation Effect of Gear Modification
The combined modification method of tooth – direction modification and diagonal modification has a good compensation effect on the influence of meshing misalignment caused by systematic factors, while it has almost no compensation effect on the load – unevenness caused by random meshing misalignment.
7. Future Research Directions
7.1 Consideration of More Complex Working Conditions
In this study, although some common influencing factors are considered, in actual engineering applications, gears may be in more complex working conditions, such as high – temperature, high – humidity, and variable – load environments. Future research can focus on analyzing the meshing dislocation and modification of gears under these complex conditions to further improve the accuracy of gear design.
7.2 Optimization of Modification Algorithms
The modification methods used in this paper are mainly based on empirical formulas. Future research can explore more advanced optimization algorithms, such as genetic algorithms and neural – network – based methods, to optimize the modification parameters more accurately and improve the performance of gears more effectively.
7.3 Experimental Verification
Most of the research in this paper is based on simulation analysis. In the future, it is necessary to conduct a large number of experiments to verify the accuracy of the simulation results. Through experimental research, the actual performance of gears under different modification conditions can be obtained, which can provide more reliable data support for gear design and modification.
In conclusion, the research on the modification of helical cylindrical gears based on meshing dislocation is of great significance for improving the performance of gear transmission systems. By continuously exploring and researching, more effective methods can be found to optimize gear design and enhance the reliability and durability of mechanical systems.
