Abstract
This article presents a comprehensive analysis of the internal excitation and dynamic characteristics of spur gear pairs with error tooth surfaces. Based on the distribution patterns of gear machining and assembly errors, an error tooth surface model of an involute spur gear is established. A novel gear load contact analysis algorithm is proposed for this model. The influence of different errors on the internal excitation of spur gear pairs is investigated. Furthermore, a bending-torsion coupling dynamic model is constructed to analyze the dynamic characteristics of the gear system under various error conditions. The study reveals that pitch deviation is the primary factor affecting internal excitation, and that the dynamic transmission error can be significantly minimized by rational distribution of pitch deviations and avoidance of misalignment errors.
Keywords: spur gear pair, error tooth surface, internal excitation, dynamic characteristics
Introduction
With the widespread adoption of electric vehicles, there is an increasing demand for improved vibration and noise performance in gear transmission systems. Gear transmission systems, as parametric self-excited systems, can generate self-excited vibrations even without external excitation, due to their internal excitations. Time-varying mesh stiffness (TVMS) and composite mesh error (CME) are crucial internal excitations that significantly impact the dynamic behavior of gear systems.
This study aims to precisely calculate TVMS and CME, and to investigate the influence of various errors on the dynamic characteristics of spur gear pairs. By understanding these relationships, we can better predict and control the vibration and noise levels in gear systems.
1. Error Tooth Surface Model
1.1 Discretization of Gear Tooth Surface
To model the error tooth surface, the involute spur gear tooth surface is discretized into a grid of small elements, as shown in Figure 1. Each element is approximated as a cylindrical surface with a specific curvature radius. Each control point in the grid is indexed by (i, j), where i and j represent the row and column indices, respectively.
1.2 Modeling Gear Machining Errors
Gear machining errors, such as single pitch deviation (fpt), cumulative pitch deviation (Fp), and profile deviation (Fα), are considered in this model. These errors are represented using probability distributions based on the ISO 1328/1 standard. The coordinates of the control points on the error tooth surface are calculated using coordinate transformations based on these error values.
1.3 Modeling Gear Assembly Errors
Gear assembly errors are represented by four parameters: Δx, Δy (center distance errors), φ, and γ (axis misalignment errors). These errors are incorporated into the error tooth surface model by transforming the coordinates of the control points accordingly.
2. Gear Load Contact Analysis
2.1 Detection of Meshing State
To account for non-ideal meshing phenomena such as tooth separation and out-of-line contact, the meshing state of each tooth pair is detected by analyzing the interference between the discrete elements on the error tooth surfaces. An iterative process is used to adjust the rotation of the driving gear until the generated torque matches the applied load.
2.2 Calculation of Time-Varying Mesh Stiffness
The time-varying mesh stiffness (TVMS) is calculated by summing the stiffness contributions from all engaged tooth pairs. A nominal slice concept is introduced to handle the interaction between slices affected by machining and assembly errors.
3. Dynamic Modeling
A bending-torsion coupling dynamic model is established for the spur gear pair, as shown in Figure 6. The dynamic equations of motion are derived based on Newton’s second law, considering the effects of TVMS, CME, support stiffness, and damping.
The dynamic transmission error (DTE) is calculated as the deviation from the ideal meshing position, taking into account the TVMS and CME.
4. Results and Discussion
4.1 Model Validation
The proposed model is validated by comparing the calculated TVMS and CME with those obtained using a reference method. Good agreement is observed, with differences of 2.05% and 0.90% in mean and peak-to-peak values, respectively.
Table 1: Model Validation Results
| Metric | Proposed Model | Reference Model | Difference (%) |
|---|---|---|---|
| Mean TVMS (N/m) | 1.56e9 | 1.57e9 | -0.64 |
| Peak-to-Peak TVMS (N/m) | 3.45e8 | 3.48e8 | -0.86 |
| Mean CME (μm) | 0.0 | 0.0 | – |
4.2 Effects of Machining Errors
Pitch Deviation: Pitch deviation significantly affects TVMS and CME, causing step changes and even abrupt changes in TVMS when the step value exceeds a certain threshold.
Table 2: Effects of Pitch Deviation
| Pitch Deviation (μm) | Mean TVMS Change (%) | Peak-to-Peak TVMS Change (%) | CME Step (μm) |
|---|---|---|---|
| 0 | 0.0 | 0.0 | 0.0 |
| 5 | -5.2 | +12.7 | 3.5 |
| 10 | -10.4 | +27.3 | 7.0 |
Tooth Thickness Deviation: Tooth thickness deviation slightly reduces TVMS but does not significantly affect CME.
Profile Deviation: Profile deviation causes minor fluctuations in TVMS and CME due to out-of-line contact.
4.3 Effects of Assembly Errors
Center Distance Errors: Increasing the center distance reduces TVMS, while decreasing it has a more complex effect, sometimes increasing TVMS due to altered engagement patterns.
Axis Misalignment Errors: Axis misalignment errors lead to eccentric loading and reduced TVMS. The effect is more pronounced for φ errors compared to γ errors.
Table 3: Effects of Assembly Errors
| Error Type | Magnitude | Mean TVMS Change (%) | Peak-to-Peak TVMS Change (%) |
|---|---|---|---|
| Δy | +0.2 mm | -8.1 | +10.5 |
| Δy | -0.2 mm | +4.2 | -3.7 |
| φ | +0.2° | -12.3 | +15.2 |
| γ | +0.2° | -7.8 | +11.9 |
4.4 Dynamic Characteristics
The dynamic transmission error (DTE) under various error conditions is analyzed in both the time and frequency domains. Pitch deviation significantly increases the peak-to-peak DTE and introduces high-frequency components in the frequency spectrum. Axis misalignment errors can also increase DTE but to a lesser extent.
5. Conclusions
The following conclusions can be drawn from this study:
- Machining and Assembly Errors: Both machining and assembly errors significantly impact TVMS and CME, which are deeply coupled. Pitch deviation is the primary factor affecting internal excitation.
- Dynamic Transmission Error: Pitch deviation is the primary contributor to the peak-to-peak DTE and introduces high-frequency components in the frequency spectrum. Axis misalignment errors also contribute to increased DTE but to a lesser extent.
- Minimizing DTE: To minimize the peak-to-peak DTE, pitch deviations should be rationally distributed based on load conditions, and axis misalignment errors should be avoided to prevent eccentric loading.
This study provides valuable insights into the complex interactions between gear errors and dynamic behavior, enabling more precise prediction and control of gear system vibrations and noises.