Spiral bevel gears play a crucial role in the transmission systems of various fields, such as aerospace and automotive. They have strict requirements in terms of load-carrying capacity, contact performance, dynamic characteristics, and reliability. With the continuous development of modern machinery towards a high power-to-weight ratio, the load transmitted in the system increases, and the problems of vibration and noise become more prominent. During the operation of gears, the actual contact ratio and transmission error are important factors that cause vibration and noise. Improving the design contact ratio directly affects the transmission error and dynamic behavior of the gear pair, and the transmission error serves as an excitation factor for the transmission system. Therefore, a tooth surface design with a high contact ratio and a low transmission error amplitude will significantly improve the smoothness of gear transmission, the load distribution between teeth, and achieve good dynamic, meshing, and strength performance.
Basic Concepts
- Design Contact Ratio and Actual Contact Ratio
- In the case of ignoring the elastic deformation of the gear teeth (assuming the gears are completely rigid), the tooth surfaces of the pinion and the gear form a pair of locally conjugate surfaces, that is, point contact surfaces. To ensure the normal operation of the gear pair, it is required to maintain continuous contact during the meshing process. Therefore, this point contact is characterized by a continuous series of points on the tooth surface, namely the contact path.
- In actual working conditions, the gears are non-rigid, and the tooth loading deformation is inevitable. According to the theory of elastic contact, the contact point will sink under load at a certain moment, forming an instantaneous contact ellipse, and the contact path will eventually form the tooth surface contact imprint. The design (theoretical) contact ratio of the gear pair is defined by Equation (1): , where and are shown in the transmission error curve as in Figure 1. The actual contact ratio is defined as the average number of pairs of teeth in simultaneous meshing under load conditions.
- Design Transmission Error and Load Distribution Coefficient
- The main factors causing the transmission error of spiral bevel gears include the quasi-conjugate characteristics of the tooth surfaces, the deformation caused by loading, installation errors, and processing errors. The definition is as follows: when the pinion rotates at a constant speed, the difference between the actual meshing angle of the gear and the theoretical meshing angle can be obtained by the second-order Taylor expansion to get a parabolic transmission error curve, as shown in Figure 1. The specific expression is , where and are the number of teeth of the pinion and the gear, respectively; and are the current meshing angles of the pinion and the gear, respectively; and are the initial meshing angles of the pinion and the gear, respectively; is the derivative of the first-order transmission ratio.
- The relationship between the transmission error amplitude at the meshing transition point and the derivative of the first-order transmission ratio is . When the gear pair enters meshing at point A and exits at point B, point M represents the design reference point of the tooth surface; , , and represent the meshing angles of the pinion at the corresponding moments. By combining Equations (2) and (3), the expressions for the meshing angles of the pinion and the gear can be obtained as .
- Based on this, the transmission error of the spiral bevel gear can be pre-controlled to meet the required meshing performance. When the actual contact ratio of the gear pair is greater than one, multiple pairs of teeth will participate in meshing simultaneously and jointly bear the load. The load distribution coefficient between teeth is defined as , where is the maximum load on the tooth at the current meshing instant, and is the total load transmitted by the gear pair.
- Tooth Geometric Contact Analysis and Loaded Contact Analysis
- To more intuitively reflect the meshing characteristics of the spiral bevel gear pair, the Tooth Contact Analysis (TCA) technique is usually used to simulate the meshing process of the gear pair. Firstly, based on the local synthesis method, the cutting adjustment parameters of the pinion are deduced from the cutting adjustment parameters of the gear, thereby obtaining the tooth surface equations of the pinion and the gear. To ensure that the tooth surfaces are always in continuous contact at each meshing instant, the basic equation of TCA is , where the subscript “h” indicates that each vector is located in the meshing coordinate system ; , and , are the surface coordinate parameters of the pinion and the gear tooth surfaces, respectively; and are the meshing angles of the pinion and the gear, respectively.
- Since the TCA technique assumes that the gear teeth are completely rigid and does not consider the effect of the load, it describes the meshing process of the gear pair from a purely mathematical perspective. Therefore, the analysis results are generally only used to evaluate the quality of the tooth surface corresponding to the initial design parameters, which is quite different from the actual situation during the actual load-bearing operation. The Load Tooth Contact Analysis (LTCA) technique considers the loading deformation and the influence of the load distribution on the gear meshing characteristics during the actual working of the gear pair. Based on the TCA technique, the finite element method, the flexibility matrix method, and the mathematical programming method, the basic equation of LTCA is , where is the flexibility matrix, is the total number of discrete points that may be in contact at the current meshing instant; is the -dimensional load vector; is the -dimensional initial clearance vector; is the rotation angle of the gear caused by elastic deformation; and are the -dimensional rotation radius vectors of the contact points of the pinion and the gear, respectively; is the -dimensional clearance vector after elastic deformation; is the input torque.
Optimization Strategy
- Multi-objective Optimization Problem
- For multi-objective optimization problems (MOPs), it is necessary to simultaneously optimize multiple objectives, and these objectives may be contradictory. Therefore, it is difficult or even impossible to find a solution that makes each objective function reach the optimal value, that is, any two solutions may not be able to compare their superiority or inferiority. Such solutions are called non-inferior solutions (Pareto optimal solutions), and all the Pareto optimal solutions combined together form the Pareto optimal solution set. Therefore, according to the design requirements and practical experience of the specific multi-objective problem, selecting a relatively reasonable solution from the Pareto optimal solution set as the final optimization result is an important goal to solve the multi-objective optimization problem.
- NSGA-II Algorithm
- There are many genetic algorithms for solving multi-objective optimization problems, such as the Niched Pareto Genetic Algorithm (NPGA) and the Non-dominated Sorting Genetic Algorithms (NSGA). Although the NSGA algorithm is widely used in many engineering practical problems, it still has some shortcomings. The NSGA-II algorithm refines the NSGA algorithm, making it more effective in practical engineering problems. Therefore, this algorithm is proposed to solve the multi-objective optimization problem of the tooth surface in this paper. The main process of the algorithm is shown in Figure 2: Non-dominated Sorting, Density Distance Sorting, F, 1, Discard if not meeting the conditions, R.
- Mathematical Model for the Tooth Surface Optimization Design of High Speed and Heavy Load Spiral Bevel Gears
- Since the contact ratio and the transmission error amplitude at the transition point of the spiral bevel gear are mainly affected by the second-order contact parameters, and the three second-order contact parameters are independent of each other, based on the local synthesis method, TCA, and LTCA techniques, a tooth surface with a high contact ratio and a low amplitude at the meshing transition point is designed, and the following multi-objective optimization design model is established:
- Optimization Variables: .
- Constraint Conditions: .
- Objective Function: , where , , and are the three second-order contact parameters of the tooth surface.
Numerical Example and Analysis
- Numerical Example
- A pair of spiral bevel gears is taken as an example for the multi-objective optimization design of the tooth surface. The basic parameters of the gear blank are shown in Table 1. Some parameter settings related to the NSGA-II algorithm are shown in Table 2. After 500 generations of evolution, the Pareto optimal solution set is obtained, and through uniform sampling, the Pareto approximate frontier of the multi-objective optimization problem of the spiral bevel gear tooth surface is finally obtained, as shown in Figure 3. The two objectives are dimensionless processed, and according to the importance of the optimization objectives, the weight factor of the contact ratio is set to 0.7, and the weight factor of the transmission error amplitude at the meshing transition point is set to 0.3, to obtain the final optimization result. The second-order contact parameters before and after optimization are listed in Table 3, and the calculation results of the objective functions before and after optimization are listed in Table 4.
ParameterPinionGearNumber of Teeth2873Module at the Large End / mm3.853.85Tooth Width / mm4040Pressure Angle / (°)2020Midpoint Helix Angle / (°)3030Shaft Angle / (°)7070Hand of RotationLeft-handedRight-handed
ParameterValuePopulation Size200Mutation Probability0.1Crossover Probability0.9
ParameterBefore OptimizationAfter OptimizationContact Trace and Root Cone Angle (°)6030.377Derivative of Transmission Ratio at First Order -0.008-0.001Ratio of Major Axis of Contact Ellipse to Tooth Width 0.180.15016
ParameterBefore OptimizationAfter OptimizationContact Ratio1.4581.75Transmission Error Amplitude at Meshing Transition Point / (“)11.1301.410 - Meshing Performance Analysis
- Based on the TCA technique, the tooth surface imprints and transmission error curves of the pinion and the gear before and after optimization are obtained, as shown in Figures 4 and 5.
- By comparing the tooth surface contact imprints before and after optimization in Figures 4 and 5 and Table 4, it can be found that the area of the tooth surface meshing region, the number of meshing points, and the inclination degree of the contact trace increase significantly, and the contact trace tends to be straight. The contact ratio is very sensitive to the change of the angle between the convex surface contact trace of the gear and the root cone of the gear. Reducing the size of this angle significantly increases the contact ratio. The contact ratio after optimization increases by 16.69% compared to that before optimization. The transmission error amplitude at the meshing transition point is very sensitive to the change of the derivative of the first-order transmission ratio. The amplitude after optimization decreases by 87.33% compared to that before optimization. The ratio of the major axis radius of the contact ellipse on the convex surface of the gear to the tooth width has a relatively small influence on the contact ratio and the transmission error amplitude at the meshing transition point.
- To further compare and verify the meshing characteristics after optimization, based on the LTCA technique, the loaded transmission error curves, tooth surface contact stress distribution curves, load distribution coefficient curves, and tooth root bending stress curves of the pinion and the gear before and after optimization are calculated.
- By comparing Figures 6c and 7c, due to the increase in the design contact ratio of the tooth surface after optimization, the meshing period of a single tooth increases significantly, and the proportion of a single tooth bearing the full load within one meshing period is greatly reduced. For the convenience of subsequent explanation, the specific numerical information involved in Figures 6 and 7 is listed in Table 5. According to Figures 6, 7, and Table 5, it can be concluded that after optimization, the overall loaded transmission error decreases, and the number of 突变次数 decreases within one meshing period, which is beneficial to the stable operation of the gear pair and improves its dynamic characteristics. The mean values of the tooth surface contact stress and the tooth root bending stress of the pinion and the gear after optimization are reduced to different degrees, and the reduction in the bending stress is the largest. The tooth surface after optimization has good meshing characteristics, and the contact strength and bending strength are both improved.
ParameterBefore OptimizationAfter OptimizationMean Loaded Transmission Error / (“)-83.2911-70.4445Mean Tooth Surface Contact Stress / MPa1411.3221407.748Mean Tooth Root Bending Stress of the Gear / MPa328.440287.963Mean Tooth Root Bending Stress of the Pinion / MPa321.891278.068
Conclusion
By using the local synthesis method, TCA technique, etc., two objective functions and three design variables are proposed, and the constraint conditions are reasonably set. The multi-objective optimization of the spiral bevel gear is realized based on the NSGA-II algorithm, and the actual tooth surface contact process is simulated by combining the LTCA technique. The following conclusions are drawn:
- By comparing the tooth surface contact imprints before and after optimization, the area of the tooth surface meshing region, the number of meshing points, and the inclination degree of the contact trace all increase, and the contact trace approaches a straight line.
- By comparing the contact ratio of 1.458 and the transmission error amplitude of 11.130″ before optimization, the contact ratio after optimization increases by 16.69%, and the amplitude decreases by 87.33%, with specific values of 1.75 and 1.410″, respectively.
- By comparing the responses of various contact characteristics before and after optimization, the mean values of the loaded transmission error, the tooth surface contact stress, and the tooth root bending stress of the pinion and the gear after optimization decrease by 11.82%, 0.25%, 12.32%, and 13.61%, respectively. The proportion of a single tooth bearing the full load within one meshing period is greatly reduced, and the contact performance and strength of the tooth surface are overall improved.
In summary, the multi-objective optimization design of the tooth surface of spiral bevel gears can effectively improve the performance and reliability of the gear transmission system, which has important practical significance for the development of modern machinery. Further research can focus on the optimization of more objective functions, the application of more advanced algorithms, and the consideration of more complex working conditions to further improve the performance of spiral bevel gears.