Optimal Design of Super Reduction Ratio Hypoid Gear Based on Slip Rate

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1. Introduction

  • 1.1 Research Background
    • 1.1.1 Project Source
      • The super reduction ratio hypoid gear (SRH) is developed from the ordinary hypoid gear. It can gradually replace the traditional worm gear device due to its lower processing cost and similar structure, and has strong practicability. This project is derived from the National Natural Science Foundation of China Youth Project (No. 51805555) and the Hunan Provincial Natural Science Foundation Project (No. 2020JJ5989).
    • 1.1.2 Research Background and Significance
      • With the development of mechanical manufacturing technology, large ratio transmission gears have received more and more attention. Hypoid gears have the characteristics of large transmission ratio, high coincidence degree, large bearing capacity, and high transmission stability. However, compared with spiral bevel gears, the large and small wheels of hypoid gears have sliding during movement, which will accelerate mechanical wear and cause tooth surface gluing, reducing the service life of the gears. In recent years, the hypoid gears with large speed ratio transmission have been studied. SRH is a special form of hypoid gear, which can achieve single-stage large transmission ratio transmission with higher transmission efficiency. The optimization research of SRH can provide reference for the design and manufacture of SRH gears.
  • 1.2 Research Status at Home and Abroad
    • 1.2.1 Research and Development of Hypoid Gears
      • Gleason Company began to study hypoid gears in the 1920s and designed a hypoid gear processing machine tool in 1925. Scholars in China have also studied the design and manufacturing technology of hypoid gears. The research on hypoid gears mainly focuses on design theory, processing technology, finite element analysis, and optimization design.
    • 1.2.2 Research Status of Super Reduction Ratio Hypoid Gears
      • Gleason Company has the design and manufacturing technology of SRH gears, and some scholars have also studied the design and geometric characteristics of SRH gears. However, due to the lack of relevant research on the sliding rate of SRH gears, this paper studies the tooth surface sliding of SRH gears and optimizes the design.
    • 1.2.3 Research Status of Gear Sliding Rate
      • Although there is no relevant research on the sliding rate of SRH gears in China, some scholars have studied the sliding rate of other gears, which provides a reference for the research on the sliding rate of SRH gears.
  • 1.3 Main Research Contents of the Project
    • This paper studies the tooth blank design formula of SRH gears, analyzes the tooth surface sliding, and optimizes the design. The main research contents include establishing the spatial position relationship of the pitch cones of SRH gears, deriving the calculation method of the tooth blank geometric parameters, establishing the tooth surface mathematical model, deriving the sliding rate formula, optimizing the geometric parameters of the gears, and verifying the effectiveness of the optimization method.

2. Tooth Blank Design of Super Reduction Ratio Hypoid Gear

  • 2.1 Determination of the Pitch Cone of the Super Reduction Ratio Hypoid Gear
    • The pitch cone of the SRH gear can be determined by the node P. When the node P is determined, there is a unique straight line passing through P and perpendicular to the pitch plane that can intersect with the pitch cone axes of the large and small wheels. The pitch cone of the hypoid gear is formed by rotating the pitch cone distance around the axis of the gear. The angle between the two pitch cone axes is the shaft angle, and the angles between the pitch cone generatrix and the pitch cone axis are the pitch cone angles of the large and small wheels.
  • 2.2 Relationship between the Geometric Parameters of the Gear Pitch Cone
    • The geometric parameters of the pitch cone of the SRH gear are related to each other. The formulas for calculating the pitch cone angles, pitch cone distances, and other parameters are derived based on the spatial geometric relationship. The motion parameters at the node are also analyzed, and the relationship between the gear speed vectors is discussed.
  • 2.3 Calculation Example of the Tooth Blank
    • Taking a 3:74 gear pair as an example, the tooth blank parameters are calculated according to the proposed method, and the gears are processed and inspected. The results show that the new tooth blank derivation formula can be used for the design of SRH gears.

3. Establishment of the Tooth Surface Equation of the Super Reduction Ratio Hypoid Gear

  • 3.1 Mathematical Model of the Large Wheel Tooth Surface
    • The cutter coordinate system and the machining machine coordinate system of the SRH gear are constructed. The equations of the tool production surfaces in the tool coordinate system for the large wheel of the SRH gear under different machining methods are derived by space vector operations. The tooth surface equation of the large wheel is obtained by combining the meshing principle and the space transformation between the tool coordinate system and the machine tool coordinate system.
  • 3.2 Mathematical Model of the Small Wheel Tooth Surface
    • The cutter coordinate system and the machining machine coordinate system of the small wheel of the SRH gear are established. The equations of the tool production surfaces in the tool coordinate system for the small wheel of the SRH gear under different machining methods are derived. The tooth surface equation of the small wheel is obtained by combining the meshing principle and the space transformation between the tool coordinate system and the machine tool coordinate system.
  • 3.3 Grid Division and Discrete Point Solution of the Tooth Surface
    • The grid division of the tooth surface and the solution of the discrete points are analyzed. The tooth surface grid points are represented in the coordinate system, and the relationships between the discrete points and the gear intersection points are obtained. The tooth surface equation is used to calculate the coordinates of the tooth surface points, and the gear three-dimensional model is established by importing the discrete points into the three-dimensional modeling software.

4. Derivation and Verification of the Sliding Rate of the Super Reduction Ratio Hypoid Gear

  • 4.1 Spatial Vector Motion Analysis
    • The spatial vector motion analysis is the basis of gear meshing analysis. The relationship between the change of any spatial vector in the stationary coordinate system and its change in the moving coordinate system is derived, which is important for analyzing the spatial motion speed of the gear.
  • 4.2 Analysis of the Relative Motion Speed of the Two Gears
    • The relative motion speed of the two gears is analyzed. The speed of the meshing point of the gear is calculated, and the relationship between the relative motion speed of the gears and the speed of the meshing point is obtained. The spatial motion relationship of the gear meshing point is derived, which is the basis for analyzing the sliding of the tooth profile.
  • 4.3 Causes of Sliding between Tooth Profiles and the Basic Expression of the Sliding Rate
    • The causes of sliding between tooth profiles are analyzed. The sliding rate formula of the SRH gear is derived based on the meshing principle and the spatial position relationship of the gears. The sliding rate formula is used to calculate the sliding rate of the gears with different tooth numbers, and the results are compared with the actual situation to verify the correctness of the formula.

5. Optimization Design of the Geometric Parameters of the Super Reduction Ratio Hypoid Gear

  • 5.1 Optimization Algorithm
    • The genetic algorithm is introduced. The basic principle and operation steps of the genetic algorithm are described. The genetic algorithm is a kind of evolutionary algorithm based on the theory of biological evolution, which has strong search ability and fast optimization speed. It is widely used in engineering optimization problems.
  • 5.2 Optimization of the Tooth Surface Sliding Rate
    • The optimization design parameters of the SRH gear are selected, and the objective function and constraint conditions are established. The genetic algorithm is used to optimize the geometric parameters of the SRH gear with the minimum sliding rate as the objective function. The optimized tooth blank parameters are obtained by substituting the optimal solution into the iterative calculation formula of the SRH gear tooth blank.
  • 5.3 Gear Parameters before and after Optimization
    • Taking a 2:60 gear pair as an example, the optimization calculation is carried out. The optimized tooth blank parameters are compared with the original parameters, and the results show that the optimization method can effectively reduce the sliding rate of the gear.

6. Tooth Surface Contact Analysis of the Super Reduction Ratio Hypoid Gear

  • 6.1 Finite Element Overview
    • The finite element method is introduced. The development history, basic principles, and characteristics of the finite element method are described. The finite element method is a powerful tool for analyzing engineering problems, which can handle complex structures and nonlinear problems.
  • 6.2 Main Contents of ANSYS Simulation
    • The ANSYS simulation process is introduced. The preprocessing, solution, and postprocessing of ANSYS are described. The contact methods and types of ANSYS are introduced, and the contact algorithms of ANSYS are analyzed. The selection of the appropriate contact algorithm can improve the accuracy and convergence of the simulation results.
  • 6.3 Simulation Settings of the Super Reduction Ratio Hypoid Gear
    • The simulation settings of the SRH gear are described. The gear models are imported into ANSYS, and the material properties and boundary conditions are set. The simulation results of the gear before and after optimization are compared, and the effectiveness of the optimization method is verified.

7. Conclusion and Outlook

  • 7.1 Research Summary
    • In this paper, the tooth blank design, sliding rate analysis, and geometric parameter optimization of the SRH gear are studied. The tooth surface mathematical model of the SRH gear is established, and the sliding rate formula is derived. The genetic algorithm is used to optimize the geometric parameters of the SRH gear, and the effectiveness of the optimization method is verified by simulation.
  • 7.2 Research Outlook
    • Future research can focus on analyzing the influence of the geometric parameters of the SRH gear on the gear sliding, calculating the transmission efficiency of the SRH gear, and studying the meshing performance of the SRH gear. This will help to further improve the performance of the SRH gear and promote its application in various fields.
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