Differential helical planetary gear transmission simulation and fatigue life analysis

The differential system is a key component in the transmission system of automobiles, agricultural machinery, tanks, etc. It is mainly used to distribute the power generated by the engine to the wheels. The use of differentials allows the left and right drive wheels to rotate at different speeds, which is essential for mechanical equipment to remain stable in corners and reduce tyre wear. The differential allows the left and right wheels to rotate at the same RPM in normal straight-line driving conditions; and when cornering, the outer wheels need to rotate faster to cover a greater distance, which is when the differential allows the inner and outer wheels to fall out of RPM.A differential locking device locks the differentials in certain situations, allowing the left and right wheels to rotate at the same speed to improve vehicle traction on low traction surfaces.

As shown in Figure 1, a model of differential of a new tracked agricultural farm machinery. Typically, the differential system consists of a planetary gear mechanism that includes parts such as planetary gears, half shaft gears, differential housing, and a locking mechanism.

Figure 1.

Three-dimensional model of tracked agricultural machinery differentials

This paper adopts the helical bevel gear to replace the traditional straight bevel gear transmission for analysis and research, the helical bevel gear has a high load carrying capacity, smooth movement, transmission efficiency and other characteristics, can better adapt to the complex working environment, can be applied to high-speed and heavy-duty transmission occasions.

When modelling the geometry of a bevel gear pair, it is first necessary to determine the basic parameters of the bevel gear, including the number of teeth, module, compression angle, helix angle, etc. in order to decide the basic size and shape of the gear. In addition an involute tooth profile is usually used in modelling because it has a constant ratio and good meshing properties. Finally, determine the size of the tooth top, tooth root circle, and indexing circle. The size and position of these circles have a direct impact on the strength and meshing performance of the gear. The geometric theory of bevel gears is based on space geometry, gear meshing theory, and material mechanics. Through accurate geometric modeling, bevel gears can be designed to meet specific requirements for efficient and smooth transmission.

According to the above design principle of helical bevel gears, this paper uses the expression tool to define the key geometric parameters of helical bevel gears and the related size relationship, and combined with the secondary development technology, based on the UG three-dimensional modelling software to establish a parametric three-dimensional model of helical bevel gears, as shown in Figure 2.

Figure 2.

Parametric 3D model of bevel gears

The parameters of the helical bevel gear are obtained from the involute equation, The equations is [12]: 1{ x=Rb(sinδbcosφcosψ+sinφsinψ)y=Rb(sinδbsinφcosψ−cosφsinψ)z=−Rbcosδbcosψ

Where R_b is the base circle conical taper distance (in terms of the radius of the circular plane); δ_b is the reference angle; and the spreading angles of the base circle and the circular plane are φ and ψ, respectively [13]. 2ψ=ϕsinδb

As shown in Table 1, the pitch cone angle of the helical bevel gear is 45°, the helix angle is 20°, the top height coefficient is 1, the top clearance coefficient is 0.2, and the root fillet radius is 0.2.

Compared to conventional straight bevel gears, helical angle and module of helical bevel wheels are smaller than that of straight bevels, but because they are larger than straight bevels, they are more prone to breakage than straight bevels. Therefore, it is necessary to check the strength of the helical bevel gears, and improve the strength of the helical bevel gears by reducing the bevel of the helical teeth, so as to make them meet the working requirements of the actual use.

In order to analyse the individual gears in the differential gear can not truly reflect the meshing state of the gears, therefore, when analysing the contact stress of the helical bevel gears in operation and carrying out strength checking of the gear train, it is necessary to establish an assembly model of the helical bevel gear train. The model of the driving and driven wheels are built up with UG software, and then assembled and suppressed with the “Gear Mesh” command. The helical bevel gear subassembly model is finalised by the gear meshing command of the UG modelling software, as shown in Figure 3, which avoids problems such as model interference and parametric update failure due to the deviation of the meshing position of the two gear entities.

Figure 3.

Helical Bevel Gear Sub-Mesh Assembly Model

In 1882 physicist Heinrich Rudolf Hertz proposed the Hertz contact theory, which essentially describes the mechanical behaviour of two elastic objects when they are in contact with each other.Hertz contact theory treats the contacting body as an ideal elastic object, where the contact surface is smooth and the contact area is relatively small enough to ignore the effects of gravity and other external forces.When two spheres or cylinders are in contact with each other, the contact area is an ellipse, and the shape of the contact area varies according to the shape of the contacting bodies.In the contact area, the stress distribution is not uniform, the maximum stress usually occurs in the centre of the contact area and decreases along the depth direction of the contact surface. Based on the Hertz contact theory it can be concluded that. 3σHmax=Fπb⋅1ρ1+1ρ21−u12E1+1−u22E2

It is known that the bending and contact stress equations for helical bevel gears are [14], respectively: 4σV=YFS4KT1FΨR(1−0.5ψR)2Z12Me3μ2+1 5σH=4.98ZE1−0.5ψRKT1HψRd13μ

Where K is the load factor, ZE is the elasticity coefficient, and YFS is the tooth complex factor. By comparing the experimental data, it can be concluded that there exists a certain relationship between the tooth surface meshing force and the speed of the gear, and at the same position when the gear speed is higher the tooth surface meshing force is also higher.This is due to the fact that when the tooth speed increases, the speed of the tooth contact area is also greater, and the tooth contact time becomes shorter, which leads to an increase in the tooth meshing force. If the rotational speed is too high, it causes an increase in heat on the gear surface, which in turn affects the working life of the gear.

Hertz contact theory has a wide range of applications in engineering, materials science, and mechanical design, such as bearing design, gear contact analysis, and biomechanics. Through the theory, engineers can predict and calculate the stress distribution and deformation of the contact body when it is subjected to force, so as to optimise the design and improve the performance and life of mechanical components.

Gear models are generally categorized into: standard gears and gears with larger modules. For standard gears, the adjustment of the gear profile can be realized by modifying the number of teeth or the length of the base section; while for gears with larger module, the adjustment of the profile can be realized by changing the center distance. Helical bevel gears are then standard gears that need to be adjusted by modifying the number of teeth or the length of the base knuckle to adjust the gear profile.

This paper focuses on the simulation and strength testing of helical bevel gears in differential applications. Since the gear contact stress occurs in the involute tooth profile curve area, the mesh of the gear hub area and tooth root area is coarsened, and the mesh of the tooth contact meshing area is encrypted and refined. The meshing is done using a tetrahedral swept mesh. Q235 was selected as the material for the helical bevel gear sub-master and driven wheels. For regions like the helical bevel gear system, where the topology is particularly complex, tetrahedral meshes tend to be of better quality than hexahedral meshes and can be adapted to a variety of complex structures.

According to the principle of gear meshing, it can be seen that during the meshing process, the maximum contact stress generated by the gear teeth gradually increases from the root to the top of the teeth. Since there are 4-node tetrahedral cells in the finite element model, it is only necessary to assign them to the whole gear model with the boundary conditions set as shown in Table 2:

The use of the geometric solid method to divide the nodes during meshing can improve the clarity of reflection of local features within each cell. The tooth profile part of the gear is divided by tetrahedral mesh cells, i.e., the number of mesh nodes of tetrahedral cells in the direction of tooth thickness of the 1st gear is 65535 and the number of mesh nodes of tetrahedral cells in the direction of tooth thickness of the 2nd gear is 51077. The results of the meshing are shown in Figure 4.

Figure 4.

Helical Bevel Gear Sub-Mesh Division

As shown in Figure 5 is a comparison of the stress cloud of the helical bevel gear at different moments in the transmission process.

Figure 5.

Stress cloud at the meshing of helical bevel gears at different moments

Figure 5(a) shows the stress cloud of helical bevel gear meshing at the moment of 0.25s, and its maximum Mises stress value is 293.36Mpa; Figure 5(b) shows the stress cloud of helical bevel gear meshing at the moment of 0.5s, and its maximum Mises stress value is 280.38Mpa; Figure 5(c) shows the stress cloud of helical bevel gear meshing at the moment of 0.755s, and its maximum Mises stress value is 216.65Mpa; Figure 5(d) shows the meshing stress cloud of the helical bevel gear at the moment of 1s, and its maximum Mises stress value is 88.913Mpa.

The finite element analysis of the helical bevel gear pair reveals that the maximum contact stress δ max [15] is obtained at the moments of meshing in and out of the gears with a value of 307.11 MPa, which is in accordance with the required stresses in the material.

Separately for different speeds and different moments under the maximum equivalent force comparison, as shown in Figure 6, through the line graph can be more intuitive to see the helical bevel gear meshing region of the equivalent force change rule, the same moment under the helical bevel gear speed is greater, the meshing equivalent force is also greater.

Figure 6.

Comparison of gear meshing forces: (a) different speeds; (b) different moments

According to the equation (3) to solve the helical gear tooth surface maximum contact stress, helical gear tooth surface maximum contact should be solved theoretical calculation value of 313.10MPa, tooth surface maximum contact stress of the finite element simulation value and the theoretical calculation of the value of the difference between the difference of 1.95%, to verify the reasonableness of the finite element analysis results.

A large difference in tooth contact force between the teeth as they mesh in and out will have an impact on the smoothness of the transmission and the load carrying capacity of the gear during operation. As shown in Figure 7 for the equivalent force graphs when gear teeth are meshing in and out respectively, the equivalent force gradually increases when the gear teeth are meshing in. As shown in Figure 7(a), the equivalent force reaches a maximum value of 243.25 MPa when the two teeth are fully meshed. The equivalent force of the gear teeth decreases as they mesh out. As shown in Figure 7(b), the equivalent force reaches a minimum value of 88.97 MPa at engagement out. The difference between the equivalent forces at the time of meshing in and meshing out satisfies the operational requirements of the gear train, and it can be seen that the helical bevel gear meets the strength requirements for differential operation.

Figure 7.

Gear tooth stresses: (a) meshing in; (b) meshing out

In order to be able to further analyze and derive the influence of transmission speed on the dynamic meshing effect in the process of helical bevel gear transmission, the dynamic meshing performance of the gear mechanism is simulated under the conditions of 0.1rad/S and 0.2rad/S, respectively, under the condition of other working conditions remaining unchanged. As shown in Figure 8, the degree of fluctuation of the dynamic engagement force gradually increases with the gradual increase of the gearing speed, which indicates that the lower the gearing speed, the degree of oscillation of the dynamic load decreases. The dynamic meshing force of the gears always fluctuates up and down in the range of 3.2 × 10⁴N when the rotational speed is varied, which indicates that the variation of the gear rotational speed does not have much effect on the average meshing force. At the same time, the impact effect produced by the gear pair during the dynamic meshing process causes periodic changes in the gear speed [16], which results in non-uniformity of the meshing contact area.

Figure 8.

Force convergence curves at different speeds

From Figure 8(b), it can be seen that there is a convergence of the helical bevel gear meshing force at a speed of 0.2 rad/s compared to 0.1 rad/s. And it can be seen by comparing the service life curves of helical bevel gears under different rotational speeds, when the rotational speed ratio is 1.0, the rotational cycle of the gears under 0.1 rad/s rotational speed is close to infinity, and as the rotational speed ratio rises, the usable life (cycle) of the gears under different rotational speeds decreases accordingly, but on the whole, it is still that the smaller the rotational speed is, the bigger the usable life (cycle) is, so that, in the case of a lower speed Therefore, the lower the rotational speed, the tooth cone in the actual application of the differential, should try to avoid due to the speed change is too fast, so that the gear surface wear and damage to the differential and the occurrence of dangerous accidents, which will cause rapid and sudden changes in the gear sub gear force. A comparison of the service life of helical bevel gears at different speeds is shown in Figure 9.

Figure 9.

Curve of usable life (cycles) of gears at different rotational speeds

As shown in Figure 10 (a) to (c) is the variation curve of the helical bevel gear tooth flank wear with the pinion rotation angle under 1×10¹⁰ meshing cycles. Because of the helix angle and the right-hand rotation of the main wheel, the helical bevel gear engages from the rear face and disengages from the front face during movement. Figures 10 (a) to (c) show the distribution of gear tooth wear on the front face, intermediate face, and rear face of the helical bevel gear teeth, respectively. It can be seen that the greatest wear of the helical bevel gears occurs at the roots of the teeth of the helical bevel gears, while the wear at the top of the teeth is also greater, due to the fact that the root and top positions, which have a greater relative sliding distance, cause more severe wear at this position. The relative sliding at the position of the pitch circle is approximated to be 0, so the wear near the pitch circle is the smallest, and the wear distribution law of the front face, intermediate face and rear face is basically the same. The wear amount shows non-uniform distribution in the direction of tooth width, and the maximum wear depth is distributed as follows: front face < intermediate face < rear face, and the maximum wear depth from the front face to the rear face increases gradually along the direction of tooth width.

Figure 10.

Hertz contact half-widths: (a) front face; (b) middle face; (c) back face; (d) fitted surface

In the process of gear meshing, it is difficult to accurately represent the relationship between the gear contact stress and the amount of gear wear, so an approximate fitting surface is introduced to analyze the wear situation. The role of the gear wear fitting surface is mainly to accurately describe and analyze the wear of gears during use. By fitting the surface, a three-dimensional model of the gear surface wear can be obtained, so as to better assess the maximum wear of the helical bevel gear, predict its performance changes, and provide a scientific basis for the maintenance and replacement of gears. In addition, by analyzing the wear pattern and improving the gear material and processing technology, it is possible to reduce tooth surface wear and extend the service life of the gear. The approximate fitted surface of the maximum wear of the helical bevel gear face is shown in Figure 10(d).

As a key component in chassis transmission, differentials are widely used in heavy vehicles. Due to the complex working environment, the wear rate of helical bevel gears will be aggravated, the gear gap will increase, and the oil leakage phenomenon will occur. In the process of oil loss, the lubrication between the gear teeth deteriorates, the friction heat between the teeth increases and the oil film between the teeth gradually decreases until it ruptures, resulting in the gears appearing to be glued, and other conditions. In serious cases, the transmission system may fail, resulting in major accidents.

In order to ensure that the car can still return safely after the oil loss, the transmission system needs to have at least thirty minutes of dry running ability, which puts forward very high requirements for the dry running ability of the bevel gear located in the differential system.

The time-varying friction coefficient of the helical bevel gear can be calculated according to the formula, and then the time-varying friction heat source in the oil loss process of the differential system can be obtained according to the formula, as shown in Figure 11.

Figure 11.

Heat source of gear meshing friction during oil loss

Applying the helical bevel gear sub-model, the simulation calculation of the gear system is carried out by ANASYS software, and the temperature variation rule of the front face, middle face and rear face of the gear under the oil loss condition with time is shown in Figure 12 [18].

Figure 12.

Gear face temperature change under oil loss condition

As can be seen from the figure, the temperature rise of the front face, middle face and rear face of the helical bevel gear shows a sharp rise first, then a slow rise, and finally a gradual stabilization of the trend. Among them, the temperature rise of the middle surface of the gear teeth is significantly larger than that of the front and rear surfaces, which also means that the middle surface will be the first to wear and lead to the failure of the helical bevel gear.

As can be seen from Table 3, the helical bevel gears in the actual use of the differential into the state of complete oil loss of about 2.2 hours, the lubrication of the middle surface of the gears first disappeared into the state of complete dry friction, waiting with the front surface and the rear surface is also completely into the state of dry friction, the helical bevel gears will be rapid wear, gluing until failure.