This article offers an in – depth exploration of the spiral bevel gear transmission system. It commences with an overview of its significance and applications in diverse industries. Subsequently, it delves into the construction of mechanical models for key components, including 传动轴 (transmission shafts), gear pairs, and bearings, with a focus on their equivalent modeling methods. Static and dynamic characteristic analyses are carried out, presenting detailed calculation methods and model – building processes. The experimental verification section details the experimental setup, results, and optimization strategies. Finally, the article summarizes the research and looks ahead to potential future research directions, aiming to provide a comprehensive understanding of the spiral bevel gear transmission system for engineers and researchers.
1. Introduction
1.1 Background and Significance
Spiral bevel gear transmission systems are crucial in numerous fields such as aerospace, automotive, and marine industries. Their unique features like high transmission efficiency, strong bearing capacity, and compact structure make them irreplaceable in power transmission scenarios. For example, in the automotive industry, they are widely used in differential systems to ensure smooth power distribution between wheels. However, during operation, the load – induced deformation of the transmission system can cause misalignment in the meshing state of gear pairs, significantly affecting transmission performance. This deformation can lead to increased wear, noise, and reduced transmission efficiency. Therefore, accurate and efficient performance analysis of spiral bevel gear transmission systems is of great importance for ensuring the reliable operation of mechanical equipment.
1.2 Research Objectives
The main objective of this research is to develop a method that can achieve both high – accuracy and high – efficiency in analyzing the performance of spiral bevel gear transmission systems. Specifically, it aims to construct appropriate models to calculate system deformation and meshing misalignment accurately, study the static and dynamic characteristics of the system, and verify the effectiveness of the analysis methods through experiments. Additionally, it seeks to provide guidance for the design and optimization of spiral bevel gear transmission systems to improve their overall performance.
1.3 Research Significance
The research on spiral bevel gear transmission systems has far – reaching significance. In terms of industrial applications, it can help manufacturers improve the quality and reliability of products, reduce production costs, and enhance the competitiveness of products in the market. From a theoretical perspective, it enriches the research content of gear transmission theory, providing a basis for further research on more complex gear systems. Moreover, the research results can also be applied to related fields such as mechanical design, manufacturing, and automation, promoting the development of these disciplines.
2. Mechanical Model Construction of Key Components in Spiral Bevel Gear Transmission Systems
2.1 Transmission Shaft Model
2.1.1 Solid Finite Element Model of the Transmission Shaft
The transmission shaft is a critical component for power transmission in mechanical equipment. To ensure the accuracy of the model and computational efficiency, a solid finite element model of the transmission shaft is constructed. As shown in Figure 1, certain positions of the transmission shaft are appropriately simplified. During the meshing process, the forces acting on the transmission shaft are complex, including the meshing forces from the gear pair and the supporting forces from the bearings.
When dividing the grid of the transmission shaft, it is necessary to consider the installation positions of the bearings and gears. As presented in Table 1, the use of hexahedral elements can ensure computational accuracy while reducing computational resource consumption. The installation areas of the bearings and gears require higher grid density to accurately capture the stress and deformation distribution. For example, in the bearing installation area, the grid density should be sufficient to adapt to the contact stress distribution and deformation caused by the interaction between the bearing and the shaft.
| Model Type | Element Type | Consideration in Grid Division |
|---|---|---|
| Solid finite element model of the transmission shaft | Hexahedral elements | Bearing and gear installation positions |
2.1.2 Beam Element Model of the Transmission Shaft
Beam elements, including Euler beams and Timoshenko beams, can be used to construct the transmission shaft model. Given that the transmission shaft is often a short – thick beam with complex stress conditions, the Timoshenko beam element is selected. The stiffness matrix of the beam element is calculated based on parameters such as the cross – sectional area, shear modulus, moment of inertia, and polar moment of inertia. For a two – node Timoshenko beam element, its stiffness matrix is a 12×12 matrix, as shown in Equation (1).
K12=[A1112A21A12A2212]
where A1112 is the stiffness at node 1, A2212 is the stiffness at node 2, and A12, A21 are the coupling stiffnesses. Each element in K12 contains 6 degrees of freedom.
For a multi – node beam element model, the stiffness matrix of the entire transmission shaft can be obtained by superimposing the stiffness matrices of each two – node beam element. As the number of nodes increases, the model can more accurately represent the actual situation of the transmission shaft, but the computational complexity also increases.
2.2 Gear Pair Model
2.2.1 Equivalent Meshing Force of the Gear Pair
To simplify the calculation of tooth surface contact analysis, the gear contact is equivalent to a meshing force. The meshing position on the tooth surface is constantly changing during the meshing process. For simplicity, the mid – point of the tooth width is regarded as the meshing node. The distance from the mid – point of the tooth width to the axis is calculated by Equation (2).
LB=(R2−2B)sinδ2
where R2 is the outer cone distance of the large gear, B is the tooth width of the large gear, and δ2 is the pitch cone angle of the large gear.
Based on the known torque or power, the normal force at the meshing node can be calculated. Then, the meshing force at the tooth of the spiral bevel gear is decomposed into axial force, radial force, and tangential force, as shown in Table 2.
| Force Component | Calculation Formula |
|---|---|
| Tangential force Fmt | Fmt=LB1000Tg |
| Axial force Fax (concave surface) | Fax=cosβgFmt(tanαg1sinδg+sinβgcosδg) |
| Axial force Fax (convex surface) | Fax=cosβgFmt(tanαg2sinδg−sinβgcosδg) |
| Radial force Frd (concave surface) | Frd=cosβgFmt(tanαg1cosδg−sinβgsinδg) |
| Radial force Frd (convex surface) | Frd=cosβgFmt(tanαg2cosδg+sinβgsinδg) |
When the gear pair has no offset, the translation calculation of the meshing force is relatively straightforward. For the input shaft, the radial force translation does not generate an additional moment, while the axial and tangential forces generate additional moments when translated to the projection point of the small gear tooth width mid – point on the axis. When there is an offset, the calculation becomes more complex, especially for the output shaft, where the direction of the moment generated by the large gear axial force needs to be carefully considered.
2.2.2 Equivalent Rigid Body Model of the Gear Pair
When the gear is loaded, the contact between the gear pair produces elastic deformation, similar to the behavior of a spring. The meshing stiffness of the gear contact is equivalent to a spring model, and the energy loss during gear transmission is equivalent to a damper. Therefore, the gear meshing contact can be equivalent to a mechanical model containing a spring and a damper, as shown in Figure 2.
In the construction of the gear – train dynamics model, considering the high computational cost of the solid finite – element model, the gear is regarded as a rigid disk with a certain moment of inertia. The moment of inertia of the gear is calculated by considering it as a frustum of a cone. The calculation formulas for the mass and moment of inertia in three directions are shown in Equation (3).
m=ρπ3h(R2−r2)
Ix=Iy=121m(3(R2+r2)+h2)
Iz=21m(3(R2+r2+4h2(R2−r2)2))
where m is the mass of the frustum of the cone, ρ is the density, R is the radius of the lower base, r is the radius of the upper base, and h is the distance from the lower base to the centroid.
2.3 Bearing Model
2.3.1 Structure Characteristics and Force Analysis of the Bearing
Take the tapered roller bearing as an example. As shown in Figure 3, it consists of an inner and outer raceway, rolling elements, and a cage. The cage ensures that the rolling elements do not collide with each other, and the rolling elements make the stress evenly distributed on the raceways.
When analyzing the bearing force, the bearing is simplified into two parts: the inner raceway and the rolling elements as one part, and the outer raceway as another part. The contact load of each rolling element is decomposed and superimposed to achieve load balance with the external force, and an additional moment is added to ensure the overall balance of the bearing. The deformation of the rolling element under the action of a deflection angle θ is calculated by Equation (4).
δaθi=0.5dmθcosφi
where dm is the pitch diameter of the rolling element, θ is the deflection angle, and φi is the angle of the rolling element at different positions.
The total deformation of the rolling element in the contact normal direction is δni=δricosαe+δaisinαe, where δai=δa+δaθi, δa is the axial deformation without considering the deflection angle, δri is the radial deformation, and αe is the contact angle between the outer raceway and the rolling element.
2.3.2 Calculation of the Bearing Stiffness Matrix
The stiffness of the tapered roller bearing is an important performance index. Due to the complex internal structure and load – deformation relationship of the bearing, its stiffness shows nonlinear characteristics and coupling in the axial, radial, and tilting directions.
Based on the force analysis of the bearing, the relationship between the load and displacement of the tapered roller bearing under the action of axial force, radial force, and moment is expressed by Equation (5).
⎩⎨⎧Fr=Kn∑j=1z(n1∑i=1n(δni,jncosαecosφj))Fa=Kn∑j=1z(n1∑i=1n(δni,jnsinαe))M=Kn∑j=1z(n1∑i=1n(2dmsinαe+xi)δni,jncosφj)
where δni,j is the normal displacement of the j – th roller at the i – th slice position, n is the value corresponding to different contact states (10/9 for line contact and 1.5 for point contact), Kn is the comprehensive stiffness coefficient, z is the number of rolling elements.
After coordinate decomposition, the relationship between the force and displacement of the bearing is established. By taking the partial derivatives of the 6 – direction loads with respect to the 6 – direction displacements, the 6×6 stiffness matrix KB of the bearing can be obtained. The stiffness matrix can accurately describe the stiffness characteristics of the bearing in different directions, providing a basis for analyzing the dynamic performance of the transmission system.
3. Static Characteristic Analysis of Spiral Bevel Gear Transmission Systems
3.1 Meshing Misalignment Calculation of the Gear Pair
3.1.1 Definition and Calculation Principle of Meshing Misalignment
Meshing misalignment has a significant impact on the comprehensive performance of the gear – train. During the operation of the transmission system, the meshing force between the gear pair causes elastic or plastic deformation of the transmission shaft and bearings, resulting in a relative displacement of the initial installation position of the gear pair, which is defined as meshing misalignment, as shown in Figure 4.
The calculation of meshing misalignment involves several key steps. First, the offset vector of the shaft intersection point after deformation is obtained by subtracting the vectors from the origin to the shaft intersection points before and after deformation. Then, this offset vector is decomposed into the initial gear coordinate system to obtain the meshing misalignment in different directions, as shown in Equation (6).
⎩⎨⎧ΔP=(R′−ROQ)⋅iΔW=(R′−ROQ)⋅jΔE=(R′−ROQ)⋅kΔ∑=θ2−θ1
where ΔP is the relative displacement along the small gear axis, ΔW is the relative displacement along the large gear axis, ΔE is the relative displacement along the offset direction, Δ∑ is the relative angular change along the shaft intersection angle direction, R′ is the offset vector of the shaft intersection point after deformation, ROQ is the vector from the origin to the original shaft intersection point, and i, j, k are the unit vectors in the X, Y, Z directions of the coordinate system.
3.1.2 Calculation Process of Meshing Misalignment
In the actual calculation, as shown in Figure 5, after the transmission shaft and bearings are deformed under load, the positions of the projection points of the small and large gear tooth width mid – points on the axis change. The distances δP and δw represent the displacements of these projection points. To simplify the analysis, the deformed axis is approximated as a straight line, and the tangent vectors s1 and s2 of the small and large gear axes after deformation are obtained.
The normal vector n of the plane formed by s1 and s2 is calculated by n=s1×s2, and the angle θ2 between s1 and s2 is calculated by θ2=arccos(∣s1∣⋅∣s2∣s1⋅s2). The distance lE between the two tangents is calculated by lE=∣n∣s3⋅n, where s3 is the direction vector of the line connecting the projection points of the small and large gear tooth width mid – points after deformation.
By establishing equations and solving for the distances lP2 and lW2, the total meshing misalignment of the system can be obtained by superimposing the deformation parameters before and after loading, as shown in Equation (7).
⎩⎨⎧ΔP=lP2−lP1ΔW=lW2−lW1ΔE=lE−lΔ∑=θ2−θ1
3.2 Analysis Model of the Step – by – Step Spiral Bevel Gear Transmission System Based on Solid Finite Elements
3.2.1 Coupled Analysis Model of the Transmission Shaft Based on Solid Finite Elements
A coupled analysis model of the transmission shaft based on solid finite elements is constructed, as shown in Figure 6. This model clearly shows the complex coupling relationship between the transmission shaft, bearings, and gear pairs.
The analysis steps of the solid finite – element model are as follows:
- Define materials: Assign the same material properties to the input and output shafts, such as an elastic modulus of 2.06×105MPa, a Poisson’s ratio of 0.3, and a density of 7800kg/m3.
- Define analysis steps: Use the static and general method, set the analysis time length to 0.1.
