1. Hypoid Gear Tooth Surface Measurement and Modeling
Measurement Instrument
Gleason 350GMM
Sampling Points on Tooth Surface
45 points (9 columns in tooth length direction, 5 rows in tooth height direction)
Fitting Method for Tooth Surface
NURBS surface fitting
The tooth surface coordinates of the hypoid gear are measured by Gleason 350GMM, and then the NURBS surface is used for fitting to construct the 3D model of the tooth surface. The NURBS curve is defined as
, And the NURBS surface is .
2. Hypoid Gear Installation Error and Transmission Error
Installation Error
Definition
Direction Convention
Axial Installation Error of Driving Gear (ΔH)
Deviation in the axial direction of the driving gear
Positive when gears approach each other, negative when gears move away
Axial Installation Error of Driven Gear (ΔG)
Deviation in the axial direction of the driven gear
Positive when gears approach each other, negative when gears move away
Offset Distance Error (ΔV)
Deviation in the offset distance between gears
Positive when the pinion axis moves downwards, negative when it moves upwards
Shaft Angle Error (Δ∑)
Deviation in the shaft angle between gears
Positive when the angle increases, negative when it decreases
The transmission error (ET) is calculated as ,
which reflects the performance of hypoid gear meshing.
3. Finite Element Modeling and Simulation Analysis
Software
Function
Hypermesh
Finite element mesh generation for the hypoid gear model, ensuring fine mesh in the meshing area for accuracy and speed
Abaqus
Simulation analysis of the hypoid gear transmission error
The material properties of the gears are set as follows: 20CrMnTi, with elastic modulus E, Poisson’s ratio ε=0.3, and density ρ=7.8✖10^3kg/mm^3. Boundary conditions are applied, including coupling constraints and loading steps.
4. Results and Analysis
4.1 Comparison of Transmission Errors between Two Tooth Surfaces
Tooth Surface
Constant Part of Transmission Error (rad)
Shape of Transmission Error Fluctuation
Theoretical Tooth Surface
Around 0
Smooth parabolic curve
Real Tooth Surface
Around 2.5✖10^-3
Parabolic curve with jaggedness
The real tooth surface has larger transmission error amplitude and verifies the existence of tooth surface error.
4.2 Influence of Installation Errors on Transmission Error
4.2.1 Offset Distance Error (ΔV)
ΔV (mm)
Effect on Transmission Error
Increase
Overall positive shift of transmission error
Decrease
Slight change in peak position, especially when ΔV is positive
4.2.2 Pinion Installation Distance Error (ΔH)
ΔH (mm)
Effect on Transmission Error
Increase
Overall negative shift of transmission error
Decrease
Larger amplitude change and different peak positions, more sensitive in negative direction
4.2.3 Large Gear Installation Distance Error (ΔE)
ΔE (mm)
Effect on Transmission Error
Increase
Overall negative shift of transmission error
Decrease
Similar amplitude and peak positions, mainly affects meshing backlash
4.2.4 Shaft Angle Error (Δ∑)
Δ∑ (°)
Effect on Transmission Error
Increase
Overall positive shift of transmission error
Decrease
Smoother fluctuation curve when negative, significant increase in amplitude in both directions
4.2.5 Sensitivity Analysis of Transmission Error
Installation Error
Sensitivity
Shaft Angle Error (Δ∑) and Pinion Installation Distance Error (ΔH)
Highest
Large Gear Axial Installation Error (ΔE)
Lowest, close to 0
Offset Distance Error (ΔV)
More sensitive in positive direction
The sensitivity of transmission error to installation errors is ranked as Δ∑=ΔH>ΔV>ΔE.
5.1 Comparison of Transmission Errors at Standard Installation Distance
Tooth Surface Type
Transmission Error Amplitude (rad)
Theoretical Tooth Surface
3.6800
Real Tooth Surface
6.4200
Actual Gear
6.6286
The simulation with the real tooth surface has higher accuracy.
5.2 Influence of Installation Errors on Transmission Error
Installation Error
Transmission Error Amplitude (rad) in Different Conditions
ΔV = ±0.04mm
Values for theoretical, real tooth surfaces, and actual gear are compared
ΔH = ±0.04 mm
Values for theoretical, real tooth surfaces, and actual gear are compared
Δ∑ = ±4°
Values for theoretical, real tooth surfaces, and actual gear are compared
The real tooth surface model is verified to be reasonable.
The experimental results show that the transmission error amplitudes obtained from the actual gear rolling inspection are closer to those of the real tooth surface simulation compared to the theoretical tooth surface simulation. This indicates that the evaluation benchmark with the real tooth surface simulation and actual gear rolling inspection has higher accuracy, further validating the rationality of the real tooth surface model.
In conclusion, this study reconstructs the real tooth surface of the hypoid gear using the NURBS surface fitting method and analyzes its transmission error characteristics. The existence of tooth surface error and its impact on the transmission error are verified. The influence of different installation errors on the transmission error is studied, and the sensitivity of the transmission error to installation errors is determined. The rolling tester experiment further validates the rationality of the real tooth surface simulation model, which is of great significance for controlling the vibration and noise of the gear transmission system and guiding the assembly of the rear axle main reducer.
