This article focuses on spiral bevel gears, which are crucial components in various mechanical systems. It delves into their dynamics under steady – state and variable – speed transient conditions. A detailed analysis of the off – line – of – action – contact phenomenon, caused by tooth loading deformation, is presented. The establishment of an 8 – degree – of – freedom torsional vibration model for spiral bevel gear pairs is described, and a numerical example based on helicopter gear parameters is used to illustrate the dynamic responses in different operating conditions. The research aims to enhance the understanding of spiral bevel gear dynamics and provide a basis for improving the performance and reliability of mechanical systems.
1. Introduction
Spiral bevel gears are widely used in numerous fields such as automotive, helicopter, and construction machinery industries due to their favorable characteristics. They possess a large overlap coefficient, which enables smooth power transmission. Their ability to change the direction of torque transfer makes them suitable for complex mechanical setups. Additionally, they have a high load – carrying capacity and operate with low noise levels, ensuring the stability and efficiency of the entire transmission system [1 – 6].
In the field of spiral bevel gear system dynamics, many scholars have made significant contributions. Wang Sanmin et al. established an equivalent 8 – degree – of – freedom dynamics model for spiral bevel gear transmission systems considering tooth flank clearance and time – varying mesh stiffness. Through numerical calculations, they obtained the system’s vibration characteristics and meshing laws. developed a multi – degree – of – freedom coupled dynamics model for spiral bevel gears, taking into account the elastic deformation of drive shafts and bearings, as well as time – varying mesh stiffness and meshing impact. Wang et al. studied the nonlinear characteristics of hypoid gears, such as the excitation effects of time – varying parameters like tooth side clearance and meshing stiffness, and conducted dynamics analysis using a nonlinear time – varying model. established a bending – torsion – axis coupling dynamics model and analyzed the influence of transmission error amplitude and overlap ratio on the vibration of spiral bevel gear pairs. investigated the impact of tooth surface friction and contact overlap ratio during the meshing process of spiral bevel gears on their vibration and noise. studied the effects of transmission error, time – varying mesh stiffness, support stiffness, and wheel body displacement under different working conditions on the vibration of gear pairs.
However, most previous studies mainly focused on the relationship between time – varying mesh stiffness and gear rotation angle during meshing transmission. The influence of early and late meshing of gear teeth, which is caused by installation errors and tooth deformation under load, on the time – varying mesh stiffness of spiral bevel gears has not been thoroughly considered. Moreover, regarding the influencing factors of dynamic meshing force, most studies only took into account the effects of load, damping, friction, and static working conditions, without detailed analysis of the system response under variable – speed conditions. This article aims to fill these research gaps.
2. Off – Line – of – Action – Contact Caused by Tooth Loading Deformation
2.1 Generalized Displacement Representation
When a gear pair is in meshing, the generalized displacement can be represented as a vector: \(q=\left[x_{p}, y_{p}, z_{p}, \theta_{p}, x_{g}, y_{p}, z_{g}, \theta_{g}\right]\) where \(x_{i}, y_{i}, z_{i}(i = p, g)\) are the translational displacements relative to the initial position, and \(\theta_{i}(i = p, g)\) are the angular displacements.
2.2 Phenomenon of Off – Line – of – Action – Contact
As shown in Figure 1 (not provided here, but referred to in the original paper), for a pair of spiral bevel gears in the normal view at the mid – point of the tooth width, A and B are the theoretical meshing – out and meshing – in points respectively. When the gear pair rotates under no – load conditions, the gear teeth do not undergo elastic deformation, and the length of the meshing line remains unchanged, with the transmission ratio being the theoretical value. However, when the gear is under load, the teeth deform due to the applied force. This leads to the phenomenon of early meshing – in and late meshing – out. The actual meshing area is longer than the theoretical one, which is called off – line – of – action – contact or extended meshing.
| Condition | Meshing Line Length | Transmission Ratio | Meshing Phenomenon |
|---|---|---|---|
| No – load | Unchanged | Theoretical value | No early or late meshing |
| Loaded | Extended | Different from theoretical (slight change) | Early meshing – in and late meshing – out |
Figure 1: Normal view of gear meshing at no – load (referred from the original paper)
The elastic deformation of gear teeth causes a pair of teeth to enter meshing earlier and exit later. As depicted in Figure 2 (not provided here, but referred to in the original paper), due to the elastic deformation of tooth pair Ⅱ, tooth pair Ⅰ enters meshing before the theoretical meshing – in point, and tooth pair Ⅲ exits meshing after the theoretical meshing – out point. This early meshing – in and late meshing – out increase the actual overlap ratio of the gear pair. Moreover, as the transmitted load increases, the increase in the overlap ratio becomes more significant. The meshing distance GTSD (Gear Teeth Separation Distance) is defined as the distance between a pair of teeth that are about to enter or just exit meshing when there is no elastic deformation. It is measured along the meshing line, such as \(\xi_{a}\) and \(\xi_{s}\) shown in the figure.
Figure 2: Normal view of gear meshing under load (referred from the original paper)
2.3 Calculation of Elastic Deformation
Based on the different positions of gear teeth on the actual meshing line, let S be the intersection point of the active gear tooth profile on BA. BS is a periodic function of the angular displacement \(\theta_{1}\) of the active gear, expressed as: \(|BS|=\text{mod}(r_{b1}\theta_{1}, |B’A’|)\) where \(\text{mod}(x, y)\) is the remainder function.
The actual meshing line \(B’A’\) is divided into three parts for calculating the elastic deformation:
- Before entering the theoretical meshing line (\(BB’\)): This is the early – meshing – in stage.
- Within the theoretical meshing area (BA): When S is located within the theoretical meshing line BA, the elastic deformation \(\lambda\) is calculated as: \(\lambda=\lambda_{0}=(x_{p}-x_{g})a_{1}+(y_{p}-y_{g})a_{2}+(z_{p}-z_{g})a_{3}+\theta_{p}r_{b1}-\theta_{g}r_{b2}\)
- After exiting the theoretical meshing area (\(AA’\)): This is the late – meshing – out stage. When S is outside the theoretical meshing line BA, the elastic deformation \(\lambda\) is calculated as: \(\lambda=\begin{cases}\max(\delta_{0}-\xi_{a,0}),&0 < |BS| < BB\\\max(\delta_{0}-\xi_{b,0}),&B’A < |BS| < AA’\end{cases}\)
3. Dynamics of Spiral Bevel Gear Pairs
3.1 Establishment of the 8 – Degree – of – Freedom Torsional Vibration Model
A concentrated – parameter method is employed to establish an 8 – degree – of – freedom torsional vibration model for spiral bevel gears, as shown in Figure 3 (not provided here, but referred to in the original paper). Taking the intersection point of the two axes as the coordinate origin, with the small gear shaft as the X – axis and the large gear shaft as the Z – axis, and without considering tooth surface friction and torsional – pendulum vibration, the entire system has eight degrees of freedom. These include lateral vibrations along the X, Y, and Z axes, axial vibrations, and torsional vibrations.
The projection vector of the meshing gears separated along the coordinate direction is: \(V=\left[\begin{array}{c}-\left(\sin\alpha\cos\delta_{\beta}+\cos\alpha\sin\beta\sin\delta_{p}\right)\\\left(\sin\alpha\sin\delta_{p}-\cos\alpha\sin\beta\cos\delta_{p}\right)\\\cos\alpha\cos\beta\\-r_{b1}\\-\left(\sin\alpha\cos\delta_{p}+\cos\alpha\sin\beta\sin\delta_{p}\right)\\\sin\alpha\sin\delta_{p}-\cos\alpha\sin\beta\cos\delta_{p}\\\sin\alpha\sin\delta_{p}-\cos\alpha\sin\beta\cos\delta_{p}\\\cos\alpha\cos\beta\\r_{b2}\end{array}\right]\) where \(\alpha\) is the normal pressure angle of the active gear, \(\delta_{p}\) is the pitch cone angle of the active gear, and \(\beta\) is the mid – point helix angle of the active gear.
The normal force \(F_{n}\) of the meshing gear pair and its component forces and torques along each coordinate direction can be expressed as: \(\begin{cases}F_{s}=k_{n}(t)\lambda + c_{n}\dot{\lambda}\\F_{x}=-F_{n}\left(\sin\alpha\cos\delta_{p}+\cos\alpha\sin\beta\sin\delta_{p}\right)\\F_{y}=F_{n}\left(\sin\alpha\sin\delta_{p}-\cos\alpha\sin\beta\cos\delta_{p}\right)\\F_{z}=F_{x}\cos\alpha\cos\beta\end{cases}\) where \(k_{n}(t)\) is the time – varying mesh stiffness and \(c_{n}\) is the meshing damping.
The differential equations of the concentrated – parameter dynamics model of spiral bevel gears are as follows: \(\begin{cases}m_{p}\ddot{x}_{p}+c_{xp}\dot{x}_{p}+k_{xp}x_{p}=-F_{x}\\m_{p}\ddot{y}_{p}+c_{xy}\dot{y}_{p}+k_{yp}y_{p}=-F_{y}\\m_{p}\ddot{z}_{p}+c_{xp}z_{p}+k_{zp}z_{p}=-F_{z}\\\ddot{I}_{p}\theta_{p}=T_{p}-F_{n}r_{b1}\\m_{g}\ddot{x}_{g}+c_{xy}\dot{x}_{g}+k_{xg}x_{g}=F_{x}\\m_{g}\ddot{y}_{g}+c_{xy}y_{g}+k_{yg}y_{g}=F_{y}\\m_{g}\ddot{z}_{g}+c_{xz}z_{g}+k_{gz}=F_{z}\\\dot{T}_{g}\theta_{g}=-T_{g}+F_{gx}r_{b2}\end{cases}\) where \(m_{p}\), \(m_{g}\) are the concentrated masses of the active and driven gears respectively; \(I_{p}\), \(I_{s}\) are the corresponding moments of inertia; \(k_{ij}\), \(c_{ij}(i = x, y, z; j = p, g)\) are the stiffness and damping coefficients of the active and driven wheels along the three coordinate axes respectively; \(T_{1}\), \(T_{2}\) are the input torque and load torque respectively; \(r_{b1}\), \(r_{b2}\) are the equivalent base – circle radii at the meshing points of the active and driven wheels respectively.
Figure 3: Concentrated – mass model of spiral bevel gears (referred from the original paper)
3.2 Matrix Representation and Equation Transformation
The above – mentioned dynamics equations can be represented in matrix form as: \(\begin{gathered}MX + CX+KX = F\\X=\left[X_{1}, Y_{1}, Z_{1}, \theta_{1}, X_{2}, Y_{2}, Z_{2}, \theta_{1}\right]\end{gathered}\) where X, \(\dot{X}\), \(\ddot{X}\) are the acceleration vector, velocity vector, and generalized displacement vector of the system respectively; M is the mass and moment – of – inertia matrix; C is the damping matrix; K is the support stiffness matrix; and F is the external load vector.
By introducing state variables \(U=[u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6}, u_{7}, u_{8}, u_{9}, u_{10}, u_{11}, u_{12}, u_{15}, u_{14}, u_{15}, u_{16}]=[y_{a}, z_{n}, \theta_{0}, \dot{x}_{p},\cdots, z_{g}, \theta_{g}]\), the system of equations is reduced in order and transformed into a 16 – degree – of – freedom first – order differential equation system.
4. Numerical Example
4.1 Gear Parameters
Taking a pair of spiral bevel gears in a helicopter’s first – stage reducer as an example, the specific gear parameters are shown in Table 1.
| Parameter | Active Gear | Driven Gear |
|---|---|---|
| Number of Teeth | 23 | 75 |
| Module | 3.25 | 3.25 |
| Normal Pressure Angle | 20° | 20° |
| Mid – point Helix Angle | 35° | 35° |
| Pitch Cone Angle | 17.05° | 72.95° |
4.2 Dynamic Response under Steady – State Conditions
4.2.1 Solution Method
Using MATLAB’s ode(45) function, with \(U(0)\) as the initial calculation value (where the displacement terms are determined by the initial position of the system and are all set to 0), the active gear speed set to 5000 r/min, and the driven gear load set to 300 N·m, the time – varying displacements, velocities, and normal dynamic transmission errors and dynamic meshing forces of the gears in the coordinate directions can be solved.
4.2.2 Analysis of Vibration Characteristics
From the obtained results, it can be observed that the vibration shock is relatively large at the beginning and end of the meshing cycle. As shown in Figure 4 (not provided here, but referred to in the original paper), the axial vibration displacement in the z – direction is the largest. The vibration displacements and velocities of the active and driven wheels show a high degree of consistency in their changing trends.
4.2.3 Analysis of Dynamic Meshing Force
Figure 6 (not provided here, but referred to in the original paper) shows the time – domain and frequency – domain simulation results of the dynamic meshing force of the spiral bevel gear pair at a speed of 5000 r/min. In the time – domain, the dynamic meshing force of the spiral bevel gear pair exhibits periodic fluctuations, and no tooth separation occurs, indicating a relatively stable transmission quality. In the frequency – domain, the main amplitudes of the dynamic meshing force are at the meshing frequency (1916 Hz) and its multiple frequencies, with the maximum amplitude appearing at 5 times the meshing frequency.
The dynamic load coefficient \(K_{v}\) is defined as: \(K_{v}=\frac{\max\left(DF_{w}(t)\right)}{\max\left(SF_{w}(t)\right)}\) where \(\max DF_{m}(t)\) is the maximum dynamic meshing force and \(\max SF_{m}(t)\) is the maximum static meshing force.
Figure 7 (not provided here, but referred to in the original paper) shows the dynamic load coefficient of the spiral bevel gear pair in the range of 0 – 5000 r/min. The \(K_{v}\) curve of the spiral bevel gear pair rises rapidly with the increase in discrete speed, then fluctuates, forming multiple obvious peak – valley structures.
4.3 Dynamic Response under Variable – Speed Transient Conditions
4.3.1 Simulation of Variable – Speed Process
When the load of the driven gear is 300 N·m, the speed of the active gear is uniformly increased from 0 r/min to 5000 r/min within 0.5 s. The angular velocity of the active gear is shown in Figure 8 (not provided here, but referred to in the original paper). From Figure 9 (not provided here, but referred to in the original paper), it can be seen that the average angular velocity of the active gear increases linearly, indicating that this model can be applied to the variable – speed process of spiral bevel gear transmission.
4.3.2 Analysis of Dynamic Meshing Force
By taking \(U(0)\) as the initial calculation value (with displacement terms set to 0 according to the system’s initial position), the dynamic meshing force of the system is simulated. As shown in Figure 9, under the uniform variable – speed transient condition, compared with the steady – state condition, the amplitude of the dynamic meshing force during the meshing process of the spiral bevel gear pair fluctuates significantly, but no tooth – disengaging phenomenon occurs.
5. Discussion
5.1 Influence of Off – Line – of – Action – Contact on Gear Dynamics
The off – line – of – action – contact phenomenon has a profound impact on the dynamics of spiral bevel gears. The extension of the actual meshing interval due to tooth loading deformation changes the time – varying mesh stiffness. This, in turn, affects the dynamic meshing force and vibration characteristics of the gear pair. When the teeth enter and exit meshing earlier or later, the distribution of forces and the vibration patterns are altered. For example, the increased overlap ratio resulting from off – line – of – action – contact can lead to a more continuous transfer of load, reducing the impact at the start and end of the meshing cycle to some extent. However, it also introduces new challenges, such as additional stress concentrations at the edges of the extended meshing areas.
5.2 Comparison between Steady – State and Transient Conditions
In steady – state conditions, the vibration displacement and dynamic meshing force of the spiral bevel gear pair exhibit periodic fluctuations. The well – defined frequency components, with the dominant ones at the meshing frequency and its multiples, provide a stable but still fluctuating pattern. In contrast, during variable – speed transient conditions, the system experiences larger amplitude and dynamic meshing force fluctuations. This is mainly because the change in rotational speed causes sudden changes in the inertial forces and the meshing conditions. The linear increase in the angular velocity of the active gear in the transient example results in a non – stationary dynamic response, highlighting the importance of considering transient states when analyzing gear systems, especially in applications where frequent speed changes occur, like in automotive transmissions.
5.3 Implications for Gear Design and System Optimization
The findings from this study have important implications for gear design and system optimization. Understanding the dynamic behavior under different conditions can help engineers select appropriate gear parameters. For example, when designing gears for applications with high – speed and heavy – load requirements, considering the off – line – of – action – contact can prevent premature failure. By optimizing the tooth profile and material properties, the negative effects of increased dynamic meshing forces during transient conditions can be mitigated. Additionally, in system – level design, the knowledge of vibration characteristics can be used to design better – tuned damping and support structures to reduce overall system vibrations.
6. Future Research Directions
6.1 Incorporating More Complex Factors
Future research could focus on incorporating more complex factors into the gear dynamics model. This includes considering the influence of manufacturing errors, such as tooth profile errors and pitch errors, on the off – line – of – action – contact and dynamic performance. Moreover, the impact of lubrication conditions on the meshing process, especially in terms of friction and wear, can be further investigated. The interaction between different factors, like the combined effect of lubrication and manufacturing errors on gear dynamics, remains an area for exploration.
6.2 Multi – Physics and System – Level Modeling
There is a need for more comprehensive multi – physics and system – level modeling. Spiral bevel gears operate in complex mechanical systems, and their performance is affected by factors from other components, such as the stiffness and damping of the shafts and bearings. By integrating gear dynamics with the dynamics of the entire mechanical system, a more accurate prediction of the overall system behavior can be achieved. Additionally, considering thermal effects, which can cause changes in material properties and gear dimensions, is crucial for high – performance applications.
6.3 Experimental Validation
Although numerical simulations provide valuable insights, experimental validation is essential. Future studies should aim to conduct more in – depth experimental investigations to verify the theoretical models. This can involve measuring the vibration displacement, velocity, and dynamic meshing forces of spiral bevel gears under various operating conditions using advanced sensors. The experimental data can then be used to refine the models, improving their accuracy and reliability.
7. Conclusion
This article has presented a comprehensive analysis of spiral bevel gears’ dynamics under steady – state and variable – speed transient conditions. By establishing a model that accounts for off – line – of – action – contact, the research has provided detailed insights into the vibration and dynamic meshing force characteristics of spiral bevel gear pairs. In steady – state, the periodic fluctuations and dominant frequency components were identified, while in variable – speed transient conditions, the increased amplitude and force fluctuations were observed. The findings have significant implications for gear design, system optimization, and future research directions. Further studies incorporating more complex factors, multi – physics modeling, and experimental validation are expected to enhance the understanding and performance of spiral bevel gear – based mechanical systems.