This article delves into the grazing collision characteristics of non – orthogonal face gear transmission systems. By building a dynamic model considering multiple nonlinear factors, establishing Poincaré maps, and analyzing bifurcation conditions, the impact of grazing bifurcation on the system’s dynamics is explored. The study findings can offer a theoretical basis for enhancing the reliability and lifespan of face gear transmission systems in engineering applications.
1. Introduction
Face gear transmissions have significant application value in helicopter main reducers due to their unique structure, which can reduce the mass and volume of the transmission system while providing a high transmission ratio. However, research on grazing collision phenomena in gear systems remains limited. Grazing collision, a special form of tooth – to – tooth collision, can lead to vibrations, noise, and affect the normal operation of the transmission system. Therefore, it is crucial to study the grazing collision characteristics of non – orthogonal face gear transmission systems.
2. Non – orthogonal Face Gear Transmission System Dynamics Model
2.1 System Composition and Model Simplification
The non – orthogonal face gear transmission system consists of an involute spur pinion and a conjugate non – orthogonal face gear. As shown in Figure 1, the axes of the face gear and the spur gear have an angle \(\gamma_{n}\). By using the lumped parameter method, considering nonlinear factors such as tooth side clearance, time – varying meshing stiffness, and comprehensive transmission error, and neglecting internal friction and support shaft deformation during motion, the system is simplified into a 6 – degree – of – freedom bending – torsion vibration dynamics model, as depicted in Figure 2.
| Parameter | Value | Parameter | Value | Parameter | Value |
|---|---|---|---|---|---|
| Non – orthogonal face gear teeth number \(Z_{f}\) | 120 | Face gear bearing support stiffness in x – direction \(k_{1x}\) (\(N\cdot m^{-1}\)) | \(4.0\times 10^{8}\) | Face gear bearing support damping in y – direction \(c_{1y}\) (\(N\cdot s\cdot m^{-1}\)) | \(1.7\times 10^{4}\) |
| Cylindrical gear teeth number \(Z_{p}\) | 40 | Face gear bearing support stiffness in y – direction \(k_{1y}\) (\(N\cdot m^{-1}\)) | \(4.0\times 10^{8}\) | Face gear bearing support damping in z – direction \(c_{1z}\) (\(N\cdot s\cdot m^{-1}\)) | \(1.7\times 10^{4}\) |
| Module m (mm) | 3 | Face gear bearing support stiffness in z – direction \(k_{1z}\) (\(N\cdot m^{-1}\)) | \(5.2\times 10^{8}\) | Cylindrical gear bearing support damping in x – direction \(c_{2x}\) (\(N\cdot s\cdot m^{-1}\)) | \(0.3\times 10^{4}\) |
| Meshing pressure angle \(\alpha_{n}\) (°) | 25 | Cylindrical gear bearing support stiffness in x – direction \(k_{2x}\) (\(N\cdot m^{-1}\)) | \(3.2\times 10^{8}\) | Cylindrical gear bearing support damping in y – direction \(c_{2y}\) (\(N\cdot s\cdot m^{-1}\)) | \(0.3\times 10^{4}\) |
| Gear axis angle \(\gamma_{n}\) (°) | 72 | Cylindrical gear bearing support stiffness in y – direction \(k_{2y}\) (\(N\cdot m^{-1}\)) | \(3.4\times 10^{8}\) | Meshing damping \(c_{m}\) (\(N\cdot s\cdot m^{-1}\)) | \(0.7\times 10^{4}\) |
| Non – orthogonal face gear mass \(m_{1}\) (kg) | 35 | Time – varying meshing stiffness average value \(k_{m}\) (\(N\cdot m^{-1}\)) | \(3.8\times 10^{8}\) | Tooth side clearance b (mm) | 0.1000 |
| Cylindrical gear mass \(m_{2}\) (kg) | 4 | Face gear bearing support damping in x – direction \(c_{1x}\) (\(N\cdot s\cdot m^{-1}\)) | \(1.7\times 10^{4}\) | Face gear output torque \(T_{1}\) (\(N\cdot m\)) | 720 |
| Non – orthogonal face gear tooth width \(h_{1}\) (mm) | 35 | Cylindrical gear input torque \(T_{2}\) (\(N\cdot m\)) | 240 | ||
| Cylindrical gear tooth width \(h_{2}\) (mm) | 40 |
2.2 Relative Displacement and Meshing Force
During the meshing process of the gear pair, there is a relative displacement \(X_{n}\) along the meshing line direction. The expression for \(X_{n}\) is: \(X_{n}=(y_{2}+r_{2}\theta_{2}-y_{1}-r_{1}\theta_{1})\cos\alpha_{n}+(x_{2}-x_{1}\sin\gamma – z_{1}\cos\gamma)\sin\alpha_{n}-e(t)\) where \(r_{1}\), \(r_{2}\) are the distances from the mid – point of the face gear tooth width to the rotation axis and the pitch circle radius of the cylindrical gear respectively; \(\alpha_{n}\) is the meshing pressure angle; \(\gamma\) is the complementary angle of the coordinate system rotation angle; \(e(t)\) is the comprehensive transmission error of the gear pair, and its expression is \(e(t)=e_{0}+A_{er}\cos(\omega_{0}t + \varphi_{cr})\).
The meshing force \(F_{n}\) between the teeth is given by: \(F_{n}=k(t)f_{0}(X_{n})+c_{m}\dot{X}_{n}\) where \(k(t)\) is the time – varying meshing stiffness of the gear pair, \(f_{0}(X_{n})\) is the clearance function substituting displacement, and \(c_{m}\) is the meshing damping.
2.3 Vibration Differential Equation
The vibration differential equations of the non – orthogonal face gear transmission system are as follows: \(\begin{cases}m_{1}\ddot{x}_{1}+c_{1x}\dot{x}_{1}+k_{1x}x_{1}=F_{1x}\\m_{1}\ddot{y}_{1}+c_{1y}\dot{y}_{1}+k_{1y}y_{1}=F_{1y}\\m_{1}\ddot{z}_{1}+c_{1z}\dot{z}_{1}+k_{1z}z_{1}=F_{1z}\\I_{1}\ddot{\theta}_{1}=F_{n1}l_{1}-T_{1}\\m_{2}\ddot{x}_{2}+c_{2x}\dot{x}_{2}+k_{2x}x_{2}=F_{2x}\\m_{2}\ddot{y}_{2}+c_{2y}\dot{y}_{2}+k_{2y}y_{2}=F_{2y}\\I_{2}\ddot{\theta}_{2}=T_{2}-F_{n2}l_{2}\end{cases}\) where \(F_{1x}\), \(F_{1y}\), \(F_{1z}\) are the components of the meshing force \(F_{n}\) along the fixed coordinate system of the face gear; \(F_{2x}\), \(F_{2y}\) are the components of the meshing force \(F_{n}\) along the fixed coordinate system of the cylindrical gear; \(l_{1}\), \(l_{2}\) are the meshing force arms of the face gear and the cylindrical gear respectively; \(I_{1}\), \(I_{2}\) are the concentrated moments of inertia of the face gear and the cylindrical gear; \(T_{1}\) is the output torque of the face gear; \(T_{2}\) is the input torque of the cylindrical gear.
3. Poincaré Map
3.1 Grazing Bifurcation Conditions
Only the grazing bifurcation conditions at the tooth – face interface are discussed in this article. By transforming the non – autonomous system of the face gear transmission system into an autonomous system and translating the tooth – face interface, the system equations are changed. The grazing bifurcation at the grazing point of the face gear transmission system trajectory needs to meet four conditions. Through calculation and analysis, it can be determined whether the system undergoes grazing bifurcation at a certain point. For example, in the selected parameters of this article, the trajectory takes a maximum value at the grazing point, and the phase trajectory undergoes grazing bifurcation from the left side of the tooth – face interface to the right.
3.2 Poincaré Sections
According to the tooth – to – tooth collision conditions, the corresponding phase space is divided into three regions: the tooth – separation region \(\sum_{0}\), the tooth – face collision region \(\sum_{1}\), and the tooth – back collision region \(\sum_{2}\).
3.2.1 Time Poincaré Section
The time Poincaré section samples every period \(2\pi/\omega\). It can obtain the periodic solutions of the face gear transmission system under different external excitations, and its expression is: \(\sigma_{0}=\{(x_{f},\dot{x}_{f},y_{f},\dot{y}_{f},z_{f},\dot{z}_{f},x_{p},\dot{x}_{p},y_{p},\dot{y}_{p},x_{n},\dot{x}_{n},\theta)\in R^{12}\cdot S^{1},\theta = \omega\tau=0\ mod(2\pi)\}\)
3.2.2 Tooth – face Collision Surface Poincaré Section
The tooth – face collision surface Poincaré section takes the interface with non – dimensional relative displacement \(x_{n}=1\) as the section. Through this section, the number of tooth – face collisions can be judged and the meshing force at this moment can be calculated. The section is recorded as: \(\sigma_{1}=\{(x_{f},\dot{x}_{f},y_{f},\dot{y}_{f},z_{f},\dot{z}_{f},x_{p},\dot{x}_{p},y_{p},\dot{y}_{p},x_{n},\dot{x}_{n},F_{1},\theta)\in R^{13}\cdot S^{1},x_{n}=1\}\)
3.2.3 Tooth – back Collision Surface Poincaré Section
The tooth – back collision surface Poincaré section takes the interface with non – dimensional relative displacement \(x_{n}=-1\) as the section. Through this section, the number of tooth – back collisions can be judged and the meshing force at this moment can be calculated. The section is recorded as: \(\sigma_{2}=\{(x_{f},\dot{x}_{f},y_{f},\dot{y}_{f},z_{f},\dot{z}_{f},x_{p},\dot{x}_{p},y_{p},\dot{y}_{p},x_{n},\dot{x}_{n},F_{1},\theta)\in R^{13}\cdot S^{1},x_{n}=-1\}\)
4. Grazing Bifurcation Characteristics Analysis
4.1 Variation of Non – dimensional Meshing Frequency \(\omega\)
Fix the non – dimensional tooth side clearance \(b_{m}=1.0000\) (corresponding to a tooth side clearance of \(0.0500\) mm), and take the non – dimensional meshing frequency \(\omega\in[1.6000,2.3000]\). Analyze the dynamic characteristics of the face gear transmission system with a rotational speed in the range of \([1.10\times 10^{4},1.50\times 10^{4}]\) r/min.
As \(\omega\) increases, the system changes from chaotic motion to periodic motion through inverse period – doubling bifurcation. When \(\omega = 1.6661\), the system shows period – 6 motion with a period – 5 tooth – face meshing force. When \(\omega = 1.6686\), the system undergoes grazing bifurcation and inverse period – doubling bifurcation (codimension – two grazing bifurcation). The system’s motion state changes from period – 6 to period – 3, and the meshing behavior changes from \(6 – 5 – 0\) to \(3 – 3 – 0\).
| \(\omega\) | \(\lambda_{max}\) | Bifurcation Type |
|---|---|---|
| 1.6684 | 1.0685 | – |
| 1.6686 | 1.0230 | Inverse period – doubling bifurcation |
| 1.6688 | 0.9808 | – |
As \(\omega\) continues to increase, the system’s motion state and meshing force change continuously. For example, when \(\omega = 2.1241\) and \(\omega = 2.1791\), the system undergoes grazing bifurcation again, and the meshing force period changes.
4.2 Variation of Non – dimensional Tooth Side Clearance \(b_{m}\)
Select the non – dimensional meshing frequency \(\omega = 2.1500\) (corresponding to a face gear rotational speed of \(1.37\times 10^{4}\) r/min), and take the non – dimensional tooth side clearance \(b_{m}\in[0,1.0000]\) (corresponding to a tooth side clearance in the range of \([0,0.0500]\) mm). Analyze the dynamic characteristics of the non – orthogonal face gear transmission system.
When \(b_{m}\) is small, the system dynamics behavior is unstable, and tooth – back collisions occur. As \(b_{m}\) increases, the system may undergo grazing bifurcation. For example, when \(b_{m}=0.1355\), \(b_{m}=0.1565\), and \(b_{m}=0.2046\), the system undergoes grazing bifurcation, and the impact state and meshing behavior of the system change. When \(b_{m}>0.2046\), the system dynamics behavior does not fluctuate significantly with the change of \(b_{m}\).
5. Simulation Verification
Due to the complexity of actual working conditions and the limitations of existing measurement techniques, it is difficult to directly test the grazing collision phenomenon. In this article, a non – orthogonal face gear transmission system model is established based on the gear shaping processing principle and meshing conditions. The model is simulated in Adams software.
When \(\omega = 1.7500\), the time – response diagram of the meshing force \(F_{1}\) shows that the system is in period – 1 motion. The simulation results of the contact force between the two gears are consistent with the theoretical analysis. When \(\omega = 2.2500\), the system is in period – 2 motion, and the simulation results also verify the theoretical analysis. Although there may be some deviations between the theoretical analysis and the simulation results due to factors such as 重合度 and tooth – face friction ignored in the theoretical analysis, the rationality and reliability of the theoretical analysis can still be verified.
6. Conclusions
(1) When grazing collision occurs between teeth, the face gear transmission system will experience grazing bifurcation. Usually, it only causes a mutation in the meshing force period diagram, changing the number of impacts or the magnitude of the impact force between teeth, without changing the motion state of the face gear transmission system. (2) When grazing bifurcation and period – doubling bifurcation occur simultaneously, the face gear transmission system will experience codimension – two grazing bifurcation. At this time, grazing collision not only causes changes in the impact state but also leads to a mutation in the motion state. In actual working conditions, efforts should be made to avoid the occurrence of codimension – two grazing bifurcation to maintain the stability of the face gear transmission system. (3) In actual engineering, it is difficult to directly observe the grazing collision phenomenon. However, abnormal vibrations and noises in the gear system may be related to it. The theoretical results of this study can provide a reference for actual engineering problems. Appropriately adjusting system design parameters (such as tooth side clearance, rated rotational speed, etc.) can prevent or reduce grazing collision phenomena, helping to maintain the relative stability of the gear system’s motion state and impact state, and enhancing the reliability and lifespan of the gear transmission system.
