Design and simulation verification of differential spiral bevel gear transmission based on baja off-road vehicle

2.1. Determination of relevant parameters

Compared with the conventional differential design, this paper mainly analyzes the complex working conditions such as short-term impact and frequent start-stop in Baja race, optimizes the gear parameters, and focuses on ensuring the instantaneous overload capacity under muddy and steep slopes. For this design objective, 20CrMnTi is selected as the material for the spiral bevel gear. Its elastic modulus is 207 GPa, Poisson’s ratio is 0.25, and the material density is 7800 kg/m³. After carburizing and quenching treatment, the surface hardness and core toughness of this material can be enhanced, thus meeting the load characteristics of the Baja off-road vehicle, which are “short-term impact and frequent start-stop”. According to AGMA 2101 standard, the allowable contact stress of the material after heat treatment is [${\sigma }_H$] = 550 MPa. The engine of the Baja off-road vehicle is the Zongshen GB460V specified by the China Society of Automotive Engineers for the Baja competition, with a rated power of 12 kW. The competition rules require that the maximum engine speed must not exceed 3780 r/min. The power transmission from the Baja off-road vehicle to the differential is composed of a rubber belt CVT, a two-stage reduction gearbox, a universal joint and a drive shaft [8]. The composition and structure are shown in Fig. 1. Jin et al. [9] verified the speed regulation characteristics of rubber belt CVT used in Baja off-road vehicle through experiments. When the driving wheel speed is above 3000 r/min, it can maintain a high working efficiency, and its transmission efficiency is about 80 %. Pass the test, the speed ratio of rubber belt CVT can reach 0.5 when the soil is flat.

Fig. 1Transmission system of Baja off-road vehicle: 1 – rubber belt type CVT; 2 – two stage reduction gear; 3 – drive shaft length 760 mm (the display has been shortened to 250 mm); 4 – transfer case; 5 – gimbal; 6 – differential

The design speed of the racing car is 65 km/h, and the power of the transfer case comes from the intermediate shaft of the two-stage reducer and is input to the differential spiral bevel gear pair through the universal joint and transmission shaft. To ensure the same speed of the front and rear wheels when driving, the transmission ratio of its spiral bevel gear should be consistent with that of the low-speed stage of the second reducer. The transmission ratio of the two-stage reducer is 12.04, in which the transmission ratio of the low-speed stage is 3.09. Based on the selected parameters, materials, load-bearing capacity, and the complex driving conditions such as mud, steep slopes, and ditches that need to be faced [10], it combines the small size and high load-bearing requirements of the Baja off-road vehicle, and tries to minimize the radial size of the differential. Therefore, the number of teeth of the driving gear is chosen to be 10, and the number of teeth of the driven gear is 31. Based on the preliminary calculations of bending strength and contact strength, and considering the overall size of the differential, the gear module is selected as 4mm, and the tooth width is 20 mm. After calculation, the main parameters of the Grisson toothed bevel gear pair are shown in Table 1. Referring to the experimental results, the relevant input torque of the differential bevel gear pair's driving gear to be 52.03 N·m.

2.2. Establishment of 3D model

The formation principle of spherical involute [11] is shown in Fig. 2. By simulating the pure rolling motion of the base circular cone and the plane, an accurate tooth profile surface is generated. This method serves as the theoretical basis for precisely constructing the three-dimensional model of involute conical gears. The cone Angle ${\delta }_q$ of the moving point $Q$ and the deflection Angle ${\beta }_q$ after rotation are the basis for the construction of the size end curve of the tooth profile. The relationship between them can be expressed as:

Compared with Olcon’s system, Gleason's face milling method is chosen because it can produce higher meshing overlap under the configuration of small tooth difference and large helix angle (${\beta }_m=$ 35°). The tooth system meets the core requirements of the differential of Baja off-road vehicle for high stability and high bearing capacity. Therefore, according to Gleason's idea of machining spiral bevel gears by face milling, the tooth surface geometry is enveloped by the relative motion between the disc milling cutter and the workpiece. A three-dimensional model of spiral bevel gear pair is established based on the crown wheel forming. The tooth surface of the bevel gear can be regarded as the crown wheel and tooth surface formed through the meshing envelope.

The tooth surface equation of the Gleason face milling method is based on the principle of envelope of a single-parameter surface family. It couples the geometric parameters of the cutter head with the motion of the machining center machine tool, and the diameter of the cutter head needs to satisfy $D=2R\mathrm{c}\mathrm{o}\mathrm{s}\beta$, where $R$ is the radius of the crown gear and $\beta$ is the helix angle. Therefore, the cutter head coordinate system $S_c$ and the crown wheel coordinate system $S_g$ are defined, and the relative motion of the two is represented by the homogeneous transformation matrix $M_{gc}$:

where $\gamma =$ 35°, $R_c$ is the cone distance, and $\alpha$ is the rotation Angle of the cutter disk.

The parametric equation of the cutter tooth profile in the cutter coordinate system $S_c$ is:

The tooth surface equation of the crown wheel is obtained according to the coordinate transformation, and the tooth profile surface equation of the bevel gear is obtained through the engagement envelope of the crown wheel and the tooth surface, and the three-dimensional model of the bevel gear is established by defining the motion constraints of the envelope condition. The tooth surface equation of the crown wheel is and the motion constraint condition is:

Based on Gleason’s tooth surface equation and coordinate transformation theory [12]-[13], MATLAB is used to solve the equation and obtain the data point set of gear geometric shape. The data points are imported into UG software, and the tooth profile curve and tooth surface sheet are generated by parametric modeling method. Finally, the three-dimensional modeling of spiral bevel gear is completed through entity operation. Assemble and check the solid model of spiral bevel gear obtained by modeling, as shown in Fig. 3.

Fig. 2Spherical involute principle

Fig. 3Solid model of spiral bevel gear

3.1. Establishment of finite element mesh model

In CAE analysis, the size and quality of the mesh will have a great impact on the analysis, such as the solution speed and accuracy. Tetrahedral mesh [14] has high flexibility and can adapt to complex geometry, but its solution accuracy is not high, which may lead to instability of numerical solutions and poor shape quality. Aiming at the complex surface geometry of spiral bevel gear and the high stress gradient in the contact area of tooth surface, this paper chooses hexahedron-led grid division strategy. Compared with tetrahedral mesh, hexahedral mesh has higher calculation accuracy and numerical stability under the same mesh density, and can more accurately capture the distribution of tooth root bending stress and tooth surface contact stress. In this paper, the mesh division is a method of “dividing a single tooth into regular hexahedral mapping mesh, and then rotating and copying to generate a complete gear”, which ensures the high regularity and consistency of meshing tooth surface mesh, thus effectively improving the reliability of contact force and stress calculation and enhancing the convergence ability of nonlinear solution process. Aiming at the structure and working condition characteristics of the spiral bevel gear of Baja off-road vehicle differential, which has obvious complex surface characteristics and highly concentrated stress gradient, Hypermesh is used to divide the solid model of spiral bevel gear into hexahedral meshes. Solid edit is used to divide the solid model into simple and regular structures, 2D_automesh is used to mesh the single tooth surface, and then the general module of 3D_solid map is used to establish the single-tooth hexahedron mesh model. Finally, by using the rotate operation to replicate the rotation and the edges merge operation on the single-tooth hexahedral mesh model, the mesh model of the involute bevel gear can be obtained. As shown in Fig. 4. In finite element analysis, the quality of grid elements directly affects the accuracy and stability of solution. Jacobian ratio is a key index to evaluate the distortion degree of element shape. The closer it is to 1, the higher the calculation accuracy is. Referring to the general finite element analysis practice and related research, Jacobian ratio is usually required to be greater than 0.7 to ensure reliable calculation results in nonlinear contact problems and promote the convergence of solutions. In the grid created by this model, the proportion of cells with Jacobian > 0.7 reaches 95 %, which meets this recognized quality standard. And the remaining 5 % cells are in non-critical areas, which meets the simulation requirements and the calculation results are reliable.

Fig. 4Spiral bevel gear mesh model

3.2. Static contact stress analysis of spiral bevel gears

Tooth surface contact analysis of spiral bevel gear was carried out through finite element analysis software [15]-[17]. Create the contact surface of the master and slave gear teeth, define the contact pair contact type as S2S (face to face contact), set TRACK as FINITE (finite slip contact). Because Mobilube ^TM LS 85W-90 gear oil is used, the tooth surface has good lubrication conditions, so the friction coefficient between teeth is 0.06, in which the meshing tooth surface of gear teeth is the convex surface of driving gear and the concave surface of driven gear. The boundary conditions are defined as follows: the degrees of freedom of the inner ring of the driving gear are set as constraints except the degrees of freedom of the axial rotation, and the degrees of freedom of the cylindrical surface of the inner ring of the driven gear are all set as constraints. When the load is applied, the safety factor is taken to be 1.5, so the torque of 78.05 N·m is applied to the driving gear. The nonlinear analysis card NLPARM is created, and the PARAM card is loaded to improve the convergence of nonlinear analysis. The non-linear static module of Loadsteps analysis step is created, and OptiStruct is loaded to analyze the stress distribution of spiral bevel gear under this working condition. Element Stress [2D&3D] ($t$) and Contact Force ($v$) of Hyperview post-processing software were used to view the contact stress distribution cloud map and contact force of spiral bevel gear pair.

Fig. 5 shows the nephogram of contact stress distribution of spiral bevel gear pair. As can be seen from the figure, the maximum stress of the tooth surface at this meshing position is 411.4 MPa, which is less than the allowable contact stress of the material after heat treatment of 550 MPa, which is lower than the fatigue limit of the material, theoretically indicates that the gear has a good contact fatigue life, and the safety margin of contact strength is 1.34, which is enough to cover factors such as uncalculated dynamic load. The contact area is distributed in an elliptical step, and the maximum stress distribution area conforms to the actual meshing state of the gear. Edge contact occurred at the meshing position of the end face, which made the contact stress relatively high, and the highest stress was 168.2 MPa.

Fig. 5Stress cloud diagram of master and slave driving gear

a) Driving gear contact stress cloud diagram

b) Contact stress cloud diagram of driven gear

As shown in Fig. 6, the maximum contact force of the master and slave driving gears appears on the tooth surface, with a value of 536.3 N. According to the distribution area of the force, the main contact position and contact trajectory distribution of the tooth surface can be obtained, which are consistent with the actual tooth meshing position. The maximum edge contact force obtained by loading the edge contact position is 16.13 N. For the case of edge contact, the contact mark can be improved by tiny tooth surface modification, which makes it more concentrated in the middle of the tooth surface, thus further improving reliability and service life.

Fig. 6Contact force of master and slave gears

a) Driving gear contact force

b) Driven gear contact force