As I carried out this research, I focused on the helical gear pair used in the gearbox of a high-speed electric multiple unit. The operating environment of modern gear systems is becoming increasingly severe, especially in rail transit, aerospace, and large marine applications. When a mechanical device suffers a serious failure, it inevitably causes economic loss and may even lead to a safety accident. The equivalent stress between gear teeth, its distribution, and the temperature field distribution all cause deformation of the helical gear. To improve the working state of the gear and ensure smooth operation, I needed to modify the helical gear. In this work, I chose the helical gear in the CRH380A train gearbox as the research object. I first used Creo software to perform parametric modeling, combined with the meshing principle of the gear pair, to ensure that the gear runs without interference. Then, based on the Workbench module, I carried out static analysis of the gear, compared the analysis results with the calculation results of Hertz contact theory to ensure the correctness of the simulation results, and analyzed the stress magnitude and distribution of the gear meshing state as well as the deformation of the gear teeth under stress. Next, I analyzed the temperature field of the gear, calculated the friction heat flow and convective heat transfer coefficient of each part of the gear, and simulated the maximum temperature and temperature distribution of the gear. After that, I coupled the static analysis and temperature field analysis of the gear, compared the coupling analysis results with the analysis results under single load, and determined the deformation of the helical gear. Finally, according to the gear modification theory, I determined the modification amount, modified the gear, and compared the analysis results before and after modification. After modification, the maximum temperature of the helical gear dropped from 140°C to 116°C, the position of the maximum temperature shifted from the tooth tip to the middle position of the gear, and the maximum equivalent stress in the coupled state dropped from 1305 MPa to 1100 MPa. This reduced the probability of gear scuffing and improved the service performance of the helical gear. Therefore, the modification is effective.
The helical gear transmission system is a main working component of high-speed EMUs. Compared with belt drives and chain drives, helical gear transmissions have higher reliability and a relatively compact structure. With the healthy and rapid development of national high-speed railways, high-speed trains are becoming closer to people’s daily lives. Therefore, ensuring the safe and stable operation of high-speed trains is of great significance. The gearbox is a key component of high-speed train operation. Only the stable operation of the helical gear in the gearbox can ensure the safety and stability of train operation. If a fault occurs, it will produce serious safety hazards, affect the stable operation of the entire line, and inevitably cause large economic losses. The scuffing problem of helical gears has always been an important point for many mechanical researchers. From ordinary agricultural machinery to large heavy industrial machinery, from small buses to high-speed train gears, and from original small-displacement diesel engines to large marine power machinery, the study of helical gear scuffing cannot be separated from the analysis of contact stress and temperature. Scuffing of helical gears is often caused by contact stress between teeth. Therefore, analyzing the contact stress of helical gears plays a vital role in improving the service life and transmission stability.
During torque transmission, the temperature field also has a great influence on the scuffing of helical gears. In the transmission process of the gear pair, friction exists on the tooth surface and generates heat, which is also transferred to the lubricating oil and air, thus forming a temperature field. The temperature of the helical gear has an important influence on its transmission performance. The higher the rotational speed of the helical gear, the more times the gear meshes in the same time, the more times the tooth surfaces rub against each other, and the more heat is generated. Therefore, high-speed gears often suffer from scuffing failure caused by excessively high gear temperature, and deformation of gear teeth caused by excessively high gear temperature also occurs from time to time. When the gear teeth deform, it leads to meshing impact of the helical gear. In severe cases, vibration intensifies, which aggravates the heat generation of the gear. At present, there are many temperature measuring instruments, such as FLUK thermal imagers, but for high-speed moving gears such as those in EMUs, measurement is difficult and accuracy is low. There are also high-precision instruments, such as high-speed gear temperature infrared measuring instruments, but such highly specialized instruments are quite expensive and have high requirements for the experimental site. However, finite element theory has gradually matured, so using the finite element method for temperature field analysis of helical gears has certain value.
In this research, I took the helical gear of a high-speed train gearbox as the research object and analyzed the contact stress and temperature field of the gear pair separately. Because the contact stress and temperature field between gear teeth are both important causes of helical gear scuffing, the analysis of the gear needs to combine the temperature field and stress field to analyze the deformation of the gear under this coupling state. However, improving the scuffing phenomenon of the helical gear requires modification of the gear. The determination of the modification amount is based on the deformation of the gear analyzed under the previous coupling state. Through modification of the gear teeth, the position of stress concentration and the position of the temperature field are changed. In this way, stable operation of the helical gear is achieved, and a foundation is provided for the design, optimization, use, and maintenance of helical gear transmission. The analysis of tooth surface stress and temperature field of helical gear meshing conforms to the upgrading and reform of national industry, gradually changing the production mode that relies solely on resource consumption and productivity at the cost of technology as the core. Mastering the influencing factors of the temperature field generated by helical gear meshing and changing these factors enables the helical gear to adapt to high-speed and heavy-load working environments.
For the contact stress analysis of helical gear meshing, there are roughly three research methods at home and abroad: the experimental method, the formula method, and the finite element method. The experimental method means building a test bench, completing the contact fatigue experiment of the helical gear on the test bench, simulating the actual working condition of the gear operation, allowing the gear to run on the test bench, and obtaining the ultimate stress of the gear working state through testing. The formula method refers to obtaining the contact stress during helical gear meshing through theoretical calculation. However, this theoretical calculation method has relatively large errors. The actual working condition of the helical gear is highly nonlinear contact, and many tooth surfaces are processed by modification, so the meshing surface is relatively complex. Nevertheless, the results obtained by this method still have good reference value. Compared with the Hertz theoretical calculation method, the finite element method can more intuitively show the stress situation of the helical gear, and the change of contact stress can be intuitively seen through the cloud diagram. Compared with the experimental method, it has lower cost and is easier to obtain results. Therefore, from the perspective of cost control, the finite element algorithm is a good method for calculating contact algorithms. However, the finite element method is not perfect because mesh division and condition settings also have a great influence on the solution results. Therefore, it is still necessary to combine the Hertz theoretical calculation method with the experimental method to obtain more accurate results.
In the study of contact stress of helical gears, many domestic and foreign scholars have done a lot of research on these three methods. Some researchers conducted six groups of roller fatigue experiments under different working conditions and analyzed the relationship among contact stress, nominal contact area, and actual contact area of the helical gear. On this basis, they also studied the correlation between the contact stress of the gear teeth and the friction force of the tooth surface. Other researchers, based on tribology principles, used spur gears to build a dynamic analysis model. Their analysis method considered two aspects: lubrication analysis and transverse-torsional dynamics. These two aspects analyzed the relationship between friction force and motion with meshing force and tooth surface line. The two were used as analysis conditions for each other, thereby realizing iterative calculation of coupled dynamics and tribology of the gear pair, making the model closer to actual working conditions and the calculation results relatively more accurate. Some researchers took a certain type of locomotive traction helical gear as the research object and, based on gear tribology, took gear speed and load as influencing factors to deeply study the mechanical properties of locomotive traction gears. Through experiments and simulation analysis, they conducted contact stress analysis on the gear. Other researchers used ANSYS software to first perform adaptive mesh division of the model and then conduct contact stress analysis, giving the variation law of contact stress during the contact meshing process of the gear. Some researchers used the method of refining the meshing area of the gear to solve contact stress and also proposed a modular calculation system for calculating the contact stress amplification of spur gears and helical gears. This system was also verified by a pair of aviation mechanism helical gears. Other researchers used the APDL module in ANSYS software to parametrically model the gear and used programming language to conduct contact stress analysis on the model. Comparing the analysis results with theoretical calculation results verified that the analysis results are relatively accurate and the method is correct and effective. Some researchers simplified the gear teeth of the helical gear into a truncated cone, calculated the equivalent stress between gear teeth based on Hertz contact theory and the meshing principle of the gear, and analyzed the variation trend of equivalent stress within one meshing cycle. Other researchers started from the two perspectives of statics and dynamics, used MATLAB software as the development platform, and studied the static uniform load and dynamic uniform load of the gear system respectively. Some researchers first combined Newton-Raphson iterative solution and the finite element mixed method to solve the boundary problem of contacting objects and defined mechanical problems on the boundary, establishing a finite element array of elastic and plastic coupling. The above studies are based on different analysis objectives and adopt different analysis methods, all analyzing and calculating the equivalent contact stress of helical gears. However, some methods are difficult to calculate and the process is cumbersome. With the development of computer technology, many calculations have gradually become simple, and the accuracy of calculation has also been improved. Learning from previous experience and simplifying the gear model can more conveniently obtain calculation results.
The temperature field of helical gears and its influencing factors have also been widely studied. One researcher proposed a one-dimensional simplified equation for friction heat generation between objects. A similar flash temperature can be expressed by this equation, and the anti-scuffing ability of gear teeth can also be evaluated by this equation. However, the accuracy of this method needs to be improved. Other researchers used the finite element analysis method to study the temperature field of helical gears under different influencing factors such as load, rotational speed, and gear oil immersion depth. Some researchers developed formulas for calculating the convective heat transfer coefficients of gear end faces and meshing surfaces, and based on verification from multiple aspects, this calculation formula is relatively accurate. On this basis, a finite element model for gear temperature analysis was established, and the gear temperature field was obtained. One researcher studied the temperature field of spiral gears through two different research methods: experimental methods and theoretical calculation methods. In the analysis of gear friction factors, gear load distribution, and convective heat transfer coefficients of gear tooth surfaces, the boundary conditions for temperature field analysis were determined, and analysis results related to the gear temperature field were obtained. Some researchers realized the finite element calculation of three-dimensional bulk temperature field and studied the load distribution and friction heat flow between gear teeth during non-single-tooth meshing. They also calculated the thermal deformation of the gear under temperature load, laying the foundation for gear modification. One researcher took the planetary gear reducer of a certain helicopter as the research object, studied the relationship between heat generation and heat transfer of the gear and the heat source, determined the maximum temperature at the position where the most heat is generated in the gear, and calculated the thermal deformation, providing a basis for future research on thermal deformation and thermal stress of gears. Other researchers established an instantaneous temperature model of gear contact. Based on the finite element contact method and Hertz contact theory, they analyzed the tooth surface equivalent stress of the gear before and after modification. They also analyzed the calculation methods of gear friction coefficient and heat flux and calculated the bulk temperature of the gear before and after modification. Some researchers analyzed the temperature field of helical gears through transient thermoelastic coupling methods. In this experiment, the research object was the lubricating oil film, and the main research content was the influence of lubricating oil film thickness, pressure distribution, and lubrication coefficient on the gear temperature field. Other researchers used finite element software to simulate the bulk temperature of high-speed gears and compared the gear bulk temperature field under two different lubrication conditions: oil injection lubrication and oil mist lubrication. The analysis results showed that the effect of oil injection lubrication is better than that of oil mist lubrication and can better reduce the temperature during gear pair meshing. Some researchers studied the tooth surface temperature field of spiral bevel gears based on friction theory and thermoelastic flow theory and studied the influence of modification on the temperature field. Other researchers studied a heliostat planetary gear reducer, studied the variation law of relative sliding speed and friction coefficient between gears, and calculated the equivalent stress and friction heat flow of the gear at different meshing positions. Some researchers studied the fatigue characteristics of gearboxes using road conditions as influencing factors. To study the accuracy of this method, they combined simulation and experiment, established a gearbox working condition model in MATLAB based on real road driving conditions, and then combined it with the equivalent stress calculation formula to obtain the load spectrum of the gearbox. Other researchers studied the improvement of tooth surface stress in aero-engine gears during transmission, reducing gear meshing impact caused by deformation caused by high speed and heavy load through modification, reducing vibration and noise, and improving the load-carrying capacity of the transmission system. Some researchers took the meshing gears of a car gearbox as a prototype, used Pro/engineer and ANSYS interfaces to establish a heat transfer model of the gearbox meshing gears, established the heat balance equation of the gear steady-state bulk temperature field, combined theoretical analysis and finite element analysis methods, and carried out thermal analysis and calculation through ANSYS. The calculation results provided a theoretical basis for the study of gear scuffing and also provided a basis for the design of gear load-carrying capacity, gear modification, and lubrication systems. The previous research on temperature fields has gradually matured, mostly by studying the relative sliding speed and friction coefficient of gears, calculating the heat flux of the tooth surface, and comprehensively considering the influence of gear specific heat, rotational speed, density, thermal conductivity of lubricating oil, kinematic viscosity, density, etc. These are all studies on a single temperature field, which do not quite match the actual working condition of the helical gear because, in practical applications, the working state of the helical gear has both temperature field and torque simultaneously. Therefore, the analysis of the helical gear temperature field needs to be combined with the actual load of the gear. Therefore, in this paper, I considered the coupling effect of equivalent stress and temperature field of the helical gear and analyzed the deformation and stress distribution of the gear.
Regarding the modification of helical gears, research is still in the exploratory stage. The modification methods adopted by various countries and enterprises are inconsistent and are in a confidential stage, and there is no complete set of modification theory. However, helical gears that have been modified often run more smoothly and have improved reliability. In 1940, researchers began to study changing the profile shape of gears to improve the meshing state. After that, many experts and scholars also began to study improving the running performance of gears by modifying the profile. They concluded that tooth profile modification is indeed an effective method to reduce the meshing impact of gear teeth caused by the change of gear meshing stiffness in the transmission system. Therefore, tooth profile modification of helical gears is considered an effective method to reduce vibration and noise in the transmission system. Regarding the influence of tooth profile modification on static transmission error, time-varying meshing stiffness, gear tooth bending stress, and gear tooth contact stress, many scholars at home and abroad have done a lot of experimental analysis and theoretical analysis. Some researchers took spur gears as the research object and studied the relationship between load and equivalent stress of spur gears before and after modification. Other researchers used parametric modeling methods to change the profile of the gear and combined the finite element method to compare the comprehensive meshing stiffness curve of the gear before and after modification. Some researchers modified the same gear with different methods, compared the relationship between different modification methods in terms of equivalent stress, transmission error, and contact pressure of the gear, and compared experimental results with theoretical calculations to ensure the accuracy of the analysis. Finally, they concluded that the modification amount corresponding to different modification shapes is also different. Other researchers established a calculation model for gear transmission error and, based on tooth tip modification, analyzed the influence of modification on torque, load distribution, and transmission error. Based on the above research methods, some researchers established a nonlinear semi-definite spur gear rotor model. This model is a nonlinear semi-definite spur gear model containing six degrees of freedom. It considers factors such as time-varying meshing stiffness, tooth profile modification, and transmission error. Other researchers used ANSYS software to establish a spur gear model considering tooth tip modification, modified these gears with different tooth tip modifications, analyzed the transmission error and meshing stiffness after modification, input data streams, and imported these models into the dynamic model to analyze the vibration response before and after modification. Some researchers, based on the finite element method under static contact, analyzed the influence of gear modification amount on gear transmission error and meshing stiffness, as well as the difference between long modification and short modification. They also proposed that the determination of modification amount needs to be based on the amplitude of gear vibration and the amount of dynamic factor.
Based on previous research experience, I used the dimensional parameters of high-speed train gearbox gears from previous studies to perform finite element analysis of statics and temperature fields for this parameterized helical gear. I analyzed the deformation amount and deformation characteristics of the gear under the influence of these two single loads. Then I coupled these two fields to analyze the deformation of the gear teeth under the coupling of these two fields. Taking the deformation of the gear teeth as a reference, I modified the tooth profile of the gear teeth. The main content of this paper is as follows. First, I searched for gear parameters of the CRH380A EMU gearbox, used these parameters to perform parametric modeling in Creo software, meshed the gears, and checked for meshing interference based on the meshing principle of the gear to ensure that the helical gear meshes without interference. Second, based on Hertz contact theory, I calculated the theoretical contact stress of the two gears in the meshing state. I imported this model into ANSYS/Workbench for static analysis, obtained the stress cloud diagram of the two gears meshing, compared the theoretical calculation results with the simulation results, and determined the accuracy of the simulation results. I also analyzed the stress distribution of the helical gear under different rotational speeds and different meshing angles and analyzed the deformation generated when the stress is maximum. Third, I analyzed the temperature field of gear meshing. Based on tribology theory, I calculated the friction heat flow and convective heat transfer coefficient of the working face and non-working face of the gear. Then I used the Static Thermal (steady-state temperature field) module in ANSYS/Workbench for analysis and analyzed the temperature magnitude and distribution of the helical gear under different friction coefficients. Fourth, I indirectly coupled the static field and steady-state temperature field, selected the working condition with the maximum contact stress in the static analysis, coupled it with the temperature field, and obtained the equivalent stress and deformation of the gear teeth from the coupling results. Fifth, I used the deformation under coupling analysis as the modification amount of the helical gear, calculated the modification length, modified the gear, and compared the contact stress distribution of the gear pair under the coupled state after modification with that before modification to verify the effect of modification.
For the establishment of the three-dimensional solid model of the helical gear, I followed the features of Creo software. Since Creo software cannot recognize Greek letters, I needed to use English letters to replace the gear parameter relationships expressed by Greek letters and input the relationships in order to define these parameters. The basic parameters of the gear pair I used are listed in Table 1.
| Parameter | Symbol | Unit | Driving gear (pinion) | Driven gear (wheel) |
|---|---|---|---|---|
| Normal module | MN | mm | 7 | 7 |
| Number of teeth | ZZ | — | 29 | 69 |
| Height modification coefficient | XX | — | 0 | 0 |
| Helix angle | BT | ° | 20 | 20 |
| Gear width | BB | mm | 75 | 70 |
| Helix direction | — | — | right-hand | left-hand |
| Elastic modulus | E | MPa | 210000 | 210000 |
The new relationships used for the helical gear model are given in Eq. (1):
$$
\begin{cases}
DD = MN \cdot ZZ / \cos(BT) \\
AF = \arctan(\tan(20^\circ)/\cos(BT)) \\
DA = DD + 2 \cdot MN \cdot (1 + XX) \\
DF = DD – 2 \cdot MN \cdot (1.25 – XX) \\
DK = 0.3 \cdot DD \\
RC = 0.25 \cdot MN \\
C_0 = BB \cdot \tan(BT) / 180 \cdot \pi
\end{cases}
\tag{1}
$$
For the involute of the helical gear, I used the parametric equation. The involute is formed when a straight line rolls along the circumference of a base circle. The point on the line traces a curve, which is the involute. The shape of the involute is determined by the size of the base circle. The normal at any point on the involute is tangent to the base circle. The radius of curvature at any point on the involute increases with the increase of the base circle. If the radius of the base circle approaches infinity, the involute will resemble a straight line. The point on the base circle is the starting point of the involute. The involute exists outside the base circle. There is no involute inside the base circle. The length rolled by the generating line around the base circle is equal to the length of the involute. The curvature radius of the point on the base circle, which is the starting point of the involute, is zero. The longer the generating line rolls, the larger the curvature radius at that point. The involute equation of the helical gear is shown in Eq. (2):
$$
\begin{cases}
R = (DD/2) \cdot \cos(AF) / \cos(t \cdot AF) \\
\theta = 40 \cdot \tan(AF) – 40 \cdot AF + 90 \cdot (XX – 1) \cdot \tan(AF) / (ZZ \cdot \pi) \\
Z = 0
\end{cases}
\tag{2}
$$
For the helical line of the helical gear, I used Eq. (3):
$$
\begin{cases}
X = (DD/2) \cdot \cos(C_0 \cdot t) \\
Y = (DD/2) \cdot \sin(C_0 \cdot t) \\
Z = BB \cdot t
\end{cases}
\tag{3}
$$
The tooth space surrounding line is the section of the part to be removed from the gear tooth. After establishing the tooth space surrounding line, I performed three-dimensional processing. I mirrored the involute with the TOP plane as the middle reference and created a reference axis using the TOP plane and RIGHT plane. Then I drew the spiral tooth space end face surrounding line in the RIGHT plane, using two involutes and two arcs. The relationships are shown in Eq. (4):
$$
\begin{cases}
R_1 = DF/2 \\
R_2 = DA/2 \\
R_3 = RC
\end{cases}
\tag{4}
$$
I copied the sketched surrounding line to the other end of the helical line, selectively pasted it, translated and rotated it, and established the relationships shown in Eq. (5):
$$
\begin{cases}
L = BB \\
\Delta r = RC
\end{cases}
\tag{5}
$$
For the gear solid modeling, I extruded the gear blank. I sketched two circles in the FRONT plane and extruded the feature backward to establish the dimensional relationships shown in Eq. (6):
$$
\begin{cases}
R_w = DA \\
R_n = DK \\
B = BB
\end{cases}
\tag{6}
$$
I created a swept blend feature, sheared the spiral tooth space, took the helical line as the pulling direction, and used the tooth space surrounding line and its copied surrounding line as the section. I patterned the swept blend feature, arrayed it around the axis within 360°, established the relationship, and used the number of teeth as the array quantity to obtain the model of the helical gear. For the involute helical gear, the geometric dimensions need to be calculated according to the parameters of the end face. The parameters provided above are the normal parameters of the gear, and the end face parameters need to be calculated from the normal parameters. The end face module, end face pressure angle, pitch circle diameter, addendum circle diameter, dedendum circle diameter, pitch circle radius, and base circle radius are calculated as shown in Eq. (7):
$$
\begin{aligned}
m_t &= m_n / \cos\beta \\
\alpha_t &= \arctan(\tan\alpha_n / \cos\beta) \\
d &= z m_t \\
d_a &= d + 2 m_n h_a^* \\
d_f &= d – 2 m_n (h_a^* + c^*) \\
r &= d/2 \\
r_b &= r \cos\alpha_t
\end{aligned}
\tag{7}
$$
After the helical gear pair was assembled, I performed interference analysis. When a pair of helical gears is not meshed properly during the meshing transmission process, tooth interference occurs. If the gear teeth collide with each other, causing the gear teeth to embed into each other, the gear pair will not converge in subsequent analysis, resulting in no output. Therefore, ensuring the correct meshing of the gear is a very important step, which will lead to the accuracy of the output results. Interference checking is an evaluation of the accuracy of the gear pair. In Creo 3D modeling software, there are two meshing methods: one is by constraining the features of the gear, and the other is by using the simulation module in the software. Using the motion simulation module, by defining the power mechanism of the gear, the motion simulation of the gear is realized. In this scheme, the rotation angle of the driving gear can be controlled, or the gear pair can be rotated by manual dragging, to detect whether there is interference during the movement of the gear pair. If there is no interference, the meshing of the gear pair is successful. In Creo, the mechanism command in the application can be used to detect whether there is interference in the gear pair. After detection, this gear pair has no interference.
For the contact analysis of the helical gear pair meshing transmission, the helical gear tooth surface is in line contact during meshing. However, there is load on the gear and there is elastic modulus, so the line contact of the helical gear is regarded as a continuous contact surface composed of segments. Therefore, the contact pressure between the gear teeth can be calculated by the Hertz contact theory. I compared the calculated value with the simulation result to ensure the reliability of the simulation. Hertz contact theory is applied to the deformation and contact problems between elastic bodies of quadratic surfaces. Before using Hertz contact theory, two objects in contact need to satisfy the following two assumptions: first, both objects in contact need to be elastic half-space bodies; second, when the two objects are in contact, the size of the contact area needs to be much smaller than the size of the contacting objects themselves. For the model in this paper, these two necessary conditions are fully satisfied. When the helical gear meshes with each other, the contact between the tooth surfaces is line contact. The interaction force between the two gears is distributed on the contact line and diffuses to both sides of the contact line, and the diffused area forms the contact surface. I established a three-dimensional coordinate system OXYZ with the contact point O of the tooth surface as the origin. The direction of the Z axis is the common normal direction of the contact point O, and the plane OXY is the tangent plane at the contact point O of the two tooth surfaces during meshing.
When the gear runs under a certain working condition for a period of time, the contact line initially formed will deform as the meshing position changes. According to the assumption of Hertz theory, it is assumed that this deformation is regular and continuous on the contact surface of the gear teeth. Therefore, it can be determined that the contact surface after deformation forms a contact patch, and the curved surface of the contact patch can be expressed by Eq. (8):
$$
Z = \mu_1 X^2 + \mu_2 Y^2 + \mu_3 XY + \cdots
\tag{8}
$$
For the convenience of calculation, the higher-order terms in Eq. (8) can be omitted, and the xy term can be eliminated through the orientation of the XY axes, so it can be rewritten as Eq. (9):
$$
Z_1 = \frac{1}{2R_1′} x^2 + \frac{1}{2R_1”} y^2
\tag{9}
$$
where \(R_1’\) and \(R_1”\) are the principal curvature radii of the contact surface of the helical gear pair at point O. Similarly, the principal curvature radii of the other tooth surface are shown in Eq. (10):
$$
Z_2 = \frac{1}{2R_2′} x^2 + \frac{1}{2R_2”} y^2
\tag{10}
$$
The tooth surface clearance \(h\) can be calculated from the above two equations, \(h = Z_1 – Z_2\). Combining Eq. (8), (9), and (10), we obtain Eq. (11):
$$
h = \frac{1}{2R’} x^2 + \frac{1}{2R”} y^2 = \mu_1 x^2 + \mu_2 y^2
\tag{11}
$$
where \(\mu\) is a positive constant used to define the relative principal curvature radii of \(R’\) and \(R”\). The axes of the two gears are parallel to each other. According to the characteristics of the contact surface, the proof of geometric characteristics is shown in Eq. (12):
$$
\begin{aligned}
\frac{1}{2}\left(\frac{1}{R_1′} + \frac{1}{R_1”} + \frac{1}{R_2′} + \frac{1}{R_2”}\right) &= \mu_1 + \mu_2 \\
\frac{1}{2}\left(\frac{1}{R_1′} – \frac{1}{R_1”} + \frac{1}{R_2′} – \frac{1}{R_2”}\right) &= \mu_1 – \mu_2
\end{aligned}
\tag{12}
$$
The interaction force between the meshed gears can be regarded as an applied normal pressure. During the meshing process of the two gears, the contact area diffuses to both sides of the contact line, and the distances from the contact point O after diffusion are \(\delta_1\) and \(\delta_2\). The tooth surface deforms during contact. Assuming that under the action of the normal force, that is, the contact pressure, the tooth surface displacements parallel to the OZ direction are \(u_{z1}\) and \(u_{z2}\), the relationship is shown in Eq. (13):
$$
u_{z1} + u_{z2} + h = \delta_1 + \delta_2
\tag{13}
$$
Assuming \(\delta = \delta_1 + \delta_2\), then combining with Eq. (11), Eq. (13) can be rewritten as Eq. (14):
$$
u_{z1} + u_{z2} = \delta – A x^2 – B y^2
\tag{14}
$$
According to Hertz contact theory, it is generally assumed that the shape of the surface forming the contact line is an ellipse, and its semi-axes are \(a\) and \(b\), respectively. The semi-axes can be calculated by Eq. (15) and Eq. (16):
$$
a = \alpha \sqrt[3]{\frac{3 k_1 F_n}{E’}}
\tag{15}
$$
$$
b = \beta \sqrt[3]{\frac{3 k_2 F_n}{E’}}
\tag{16}
$$
where \(E’\) is the equivalent elastic modulus, \(R_0\) is the comprehensive curvature radius, and \(k_1\) and \(k_2\) are constants determined by the geometric parameter \(k_0\) of the helical gear. The calculation method of \(k_0\) is shown in Eq. (17):
$$
R_0 = \frac{1}{k_0} = \frac{1}{\cos^2\gamma} \left[ \left( \frac{1}{R_1′} – \frac{1}{R_1”} \right)^2 + \left( \frac{1}{R_2′} – \frac{1}{R_2”} \right)^2 + 2 \left( \frac{1}{R_1′} – \frac{1}{R_1”} \right) \left( \frac{1}{R_2′} – \frac{1}{R_2”} \right) \cos 2\gamma \right]^{1/2}
\tag{17}
$$
where \(\gamma\) is the angle between the principal curvatures of the contact surfaces of the two gear teeth. Based on Hertz contact theory, in the contact area, the contact pressure of the gear contact surface changes with the longitudinal coordinate of the ellipsoid. The contact pressure distribution is shown in Eq. (18):
$$
p = p_0 \sqrt{1 – (x/a)^2 – (y/b)^2}
\tag{18}
$$
where \(p_0\) is the maximum contact pressure in the contact area, and \(p_0\) is expressed as Eq. (19):
$$
p_0 = \frac{3 F_n}{2 \pi a b}
\tag{19}
$$
where \(F_n\) is the normal contact force of the tooth surface. To calculate the normal contact stress during gear meshing, considering the friction between the tooth surfaces increases the difficulty of calculation. Due to the particularity of structural steel, the friction between the tooth surfaces is relatively small, so the influence of friction between tooth surfaces can be ignored. The final calculation result is not much different from the actual result. The specific process for calculating the normal force of the tooth surface is as follows. The forces during the meshing transmission of the helical gear are decomposed. The three mutually perpendicular forces constitute the normal force \(F_n\): the tangential force \(F_t\) in the rotational direction of the gear, the radial force \(F_r\), and the axial force \(F_a\). The calculation formulas for each force are shown in Eq. (20):
$$
\begin{aligned}
F_t &= 2M/d \\
F_r &= F_t \tan\alpha_n / \cos\beta \\
F_a &= F_t \tan\beta \\
F_n &= F_t / (\cos\alpha_n \cos\beta)
\end{aligned}
\tag{20}
$$
From the meshing theory of helical gears, both the maximum curvature and the minimum curvature exist on the tooth surface. The two curvatures at the meshing point of the helical gear pair can be calculated by Eq. (21):
$$
\begin{aligned}
R_1′ &= \infty \\
R_1” &= \frac{d_1 \sin\alpha_t}{2 \cos\beta \cos^2\alpha_n} \\
R_2′ &= \infty \\
R_2” &= \frac{d_2 \sin\alpha_t}{2 \cos\beta \cos^2\alpha_n}
\end{aligned}
\tag{21}
$$
where \(d_1\) and \(d_2\) are the diameters of the two helical gears at the meshing point, \(\beta_1\) and \(\beta_2\) are the helix angles of the two helical gears at the meshing point, \(\alpha_{t1}\) and \(\alpha_{t2}\) are the end face pressure angles of the two helical gears, and \(\alpha_n\) is the normal pressure angle of the helical gear. The angle between the principal curvatures during helical gear meshing can be calculated by Eq. (22):
$$
\gamma = \frac{\sin\alpha_n \cos\beta}{\sin^2\alpha_n + \cos^2\alpha_n \cos^2\beta} \left( \frac{\tan\beta}{\cos\alpha_n} – \frac{\tan\beta_1}{\cos\alpha_n} \right)
\tag{22}
$$
For the equivalent elastic modulus, according to the principle of bending stiffness equivalence, the non-uniform distribution of the tooth surface caused by different temperatures can be converted into a uniform modulus \(E’\). The calculation formula for the equivalent elastic modulus is shown in Eq. (23):
$$
\frac{1}{E’} = \frac{1}{2} \left( \frac{1 – \nu_1^2}{E_1} + \frac{1 – \nu_2^2}{E_2} \right)
\tag{23}
$$
where \(E_1\) and \(E_2\) are the elastic moduli of the two gears, and \(\nu_1\) and \(\nu_2\) are the Poisson’s ratios of the two gears. According to the actual operating conditions of the CRH380 model, when the vehicle starts, the input torque is 2900 N·m, and the torque during continuous operation under stable conditions is about 1100 N·m. The materials of the two gears are the same, both using 20CrNiMo7-6, with an elastic modulus of 210 GPa and a Poisson’s ratio of \(\nu = 0.31\). According to the mechanical design manual, the tooth surface contact fatigue strength check formula for gear transmission is shown in Eq. (24):
$$
\sigma_H = Z_H Z_E Z_\varepsilon \sqrt{\frac{F_t}{b d_1} \frac{u+1}{u} K_A K_V K_{H\beta} K_{H\alpha}}
\tag{24}
$$
where the gear ratio \(u = 2.379\), the application factor \(K_A = 1.35\), the dynamic load factor \(K_V = 1.05\), the face load distribution factor \(K_{H\beta} = 1.15\), and the transverse load distribution factor \(K_{H\alpha} = 1\). By consulting the mechanical manual, the node region coefficient \(Z_H = 2.36\), \(Z_\varepsilon = 0.767\), and the elasticity coefficient \(Z_E\) can be calculated by Eq. (25):
$$
Z_E = \sqrt{\frac{1}{\pi \left( \frac{1 – \nu_1^2}{E_1} + \frac{1 – \nu_2^2}{E_2} \right)}}
\tag{25}
$$
The allowable stress is shown in Eq. (26):
$$
\sigma_{HP} = \frac{\sigma_{H\lim}}{S_{H\min}} Z_{NT} Z_{LVR} Z_W Z_X
\tag{26}
$$
where \(Z_{NT} = 0.98\), \(Z_{LVR} = 0.96\), \(Z_W = 1\), \(Z_X = 1\), and according to the table, \(\sigma_{H\lim} = 1600\) MPa. According to the calculation, the elasticity coefficient is about \(191.6\) (N/mm²)\(^{0.5}\), and the allowable contact stress of the gear is about 1132 MPa. The contact stress of the helical gear is about 1020 MPa. The contact stress of the gear is less than the allowable stress, so the strength in static analysis is satisfied.
During the finite element analysis, I defined the material properties of the gear. The material of the gear pair is 20CrNiMo7-6, and the tooth surface is carburized and quenched. According to the mechanical design manual, the material properties are Poisson’s ratio \(\nu = 0.31\), elastic modulus \(E = 210000\) MPa, and density \(\rho = 7851\) kg/m³. In the ANSYS Workbench module, there are three methods to solve the contact algorithm: the Lagrange multiplier method, the penalty function method, and the augmented Lagrangian method. The augmented Lagrangian method is a combination of the Lagrange multiplier method and the penalty function method in iteration. The contact condition is determined by penalty stiffness. When the penalty stiffness reaches equilibrium, the penetration amount is checked. If the penetration amount is found to be large and has exceeded the maximum penetration value during iteration, the contact stiffness of each contact element plus the contact force multiplied by the multiplier value is used to continue the iteration. Therefore, the augmented Lagrangian method is used to correct the penalty function contact stiffness for accurate positioning. On this basis, the iterative calculation is repeated until the penetration value is less than the maximum allowable value. Therefore, compared with the Lagrange multiplier method and the penalty function method, the augmented Lagrangian method has higher accuracy and easier-to-obtain results. It is not easy to cause unrealistic conditions, has low sensitivity to contact stiffness, and does not produce the phenomenon of zero diagonal elements of the sub-matrix of the stiffness matrix.
In Workbench, two contact types are provided: one is rigid-to-flexible contact, used when the stiffness of the contact surfaces of the two contacting objects differs greatly. The other is flexible-to-flexible contact, which is more common and suitable when the hardness of the two contact bodies is similar and both are deformable. In the research situation of this paper, both gears use the same material, so the elastic modulus and structural rigidity are the same, so the flexible-to-flexible contact type is adopted. ANSYS Workbench has three gear contact methods: point-to-point contact, point-to-face contact, and face-to-face contact. Face-to-face contact is not limited by the shape of the contact surface. I used the Mechanical meshing method and the hexahedral-dominated meshing method, combined with the Element Sizing method for contact surface refinement. I set the Relevance to 0 and the mesh size of the contact area to 1 mm. To reduce the difficulty of analysis and solution, the mesh size of the non-contact area can be slightly increased. The Size Function refinement method was set to default. For structural analysis, the smoothness was selected as low, and the Span Angle Center was selected as Medium. Because the size of the gear pair used in this paper is too large, if the entire gear is analyzed, the amount of analysis data is too large. According to the analysis content of this paper, intercepting part of the gear pair for analysis does not affect the analysis results.
I applied loads and constraints according to the design parameters of the high-speed train gearbox. The input torque of the driving gear, which is coaxial with the motor in the gearbox, is 1100 N·m. According to mechanical theory, the dynamic equation of the object is shown in Eq. (27):
$$
[M]\{\ddot{x}\} + [C]\{\dot{x}\} + [K]\{x\} = \{F(t)\}
\tag{27}
$$
where \([M]\) is the mass matrix, \([C]\) is the damping matrix, \([K]\) is the stiffness coefficient matrix, \(\{x\}\) is the displacement vector, \(\{F\}\) is the force vector, \(\{\dot{x}\}\) is the velocity vector, and \(\{\ddot{x}\}\) is the acceleration vector. This chapter is a static analysis of the helical gear structure, so all parameters related to time can be ignored. Therefore, Eq. (27) can be modified as Eq. (28):
$$
[K]\{x\} = \{F\}
\tag{28}
$$
In Mechanical, loads mainly include thermal loads, structural loads, and inertial loads. Among them, inertial loads need to be applied to the entire model for analysis. In this chapter, structural loads are mainly used. Because the pressure on the gear teeth changes when the contact line changes. This chapter will select the changes in equivalent stress and deformation between the gear teeth at different rotation angles under different rotational speeds during the gear rotation process. After the finite element model is established, not only loads but also constraints need to be added. The following are five common constraint types in static analysis: displacement constraint, fixed constraint, simple constraint, compression-only constraint, and rotational constraint. In this chapter, when analyzing the equivalent stress and deformation between gear teeth, I applied a fixed constraint to the inner hole surface of the gear to prevent translation and applied a rotational constraint to the inner hole surface so that it can only rotate around the z-axis.
Under the above constraint conditions, I performed static analysis on the gear at a rotational speed of 376 r/min and a torque of 1100 N·m. The contact stress of the gear was 1100.2 MPa, and the error compared with the calculated result was about 7.2%, which is within the allowable range, indicating the correctness of the simulation value. The maximum equivalent stress appears at the tooth tip of the driving gear and the tooth root of the driven gear, which are also the areas where the helical gear is most prone to scuffing. After confirming the correctness of the simulation results, I analyzed the contact stress of the helical gear at different meshing positions at driving gear speeds of 376 r/min, 675 r/min, 750 r/min, 875 r/min, and 960 r/min. The equivalent stresses at different speeds and rotation angles are summarized in Table 2.
| Speed (r/min) | Rotation angle 2° (MPa) | Rotation angle 5° (MPa) | Rotation angle 10° (MPa) | Rotation angle 15° (MPa) |
|---|---|---|---|---|
| 376 | 1102.2 | 978.46 | 835.67 | 898.89 |
| 675 | 972.81 | 908.94 | 752.31 | 810.52 |
| 750 | 902.06 | 875.47 | 718.94 | 776.83 |
| 875 | 898.52 | 821.39 | 642.83 | 690.45 |
| 960 | 884.37 | 850.42 | 617.11 | 677.42 |
From Table 2, it can be seen that at any rotation angle and speed of the gear, the equivalent stress of the gear satisfies the maximum allowable stress of the gear. Combined with the calculation results of Hertz contact theory, the simulation value is very close to the theoretical calculation value, indicating that the simulation results are reliable. From the curve diagram, it can be seen that at the same rotational speed, the contact stress of the gear will show a trend of first increasing, then decreasing, and then increasing. The increase is because when the gear meshes, it changes from three-tooth meshing to two-tooth meshing, so the equivalent stress of the gear increases. The decrease is because when the gear teeth mesh, it rotates from three-tooth meshing to two-tooth meshing. In this way, one meshing cycle of a gear tooth is formed. From the same rotation angle at different speeds, the equivalent stress of the gear teeth will show a decreasing trend. Because the output torque of the motor is constant, the initial rotation speed of the gear is small, so the extrusion between the gear teeth is more obvious. Therefore, as the rotational speed increases, the equivalent stress between the gear teeth gradually decreases. The variation trend of the equivalent stress of the gear teeth at different rotation angles between different speeds is similar. From the perspective of analyzing equivalent stress, the equivalent stress is the largest at a speed of 376 r/min and a rotation angle of 2°. Therefore, the deformation of the gear in this state is the largest. For the analysis results of the gear operating conditions, the maximum stress is 1100.5 MPa, which is smaller than the traditional calculation result and less than the allowable stress. Compared with the calculation result, the error is about 7.2%, which is within a reasonable range. The deformation of the gear is 0.0057 mm, which is also within a reasonable range. Because the traditional calculation method has certain errors in parameter selection and model simplification during finite element analysis, the analysis results also verify the feasibility of the model for this working condition and the correctness of the results. The main position where the gear teeth deform is the tooth tip of the driving gear and the tooth root of the driven gear.
For the temperature field analysis of the helical gear pair meshing transmission, heat transfer is a discipline that specifically studies the laws and processes of heat transfer. In daily life, knowledge of heat transfer exists everywhere. Under the normal operating conditions of high-speed trains, at thermal equilibrium, the friction between the driving gear teeth and the driven gear teeth causes a temperature field in the gear body. The temperature field of the entire gear pair includes the bulk temperature of the gear and the flash temperature. The bulk temperature refers to the temperature value of the gear that does not change with time when the gear operates under a certain working condition. The flash temperature of the tooth surface refers to the maximum temperature generated at the moment when the driving and driven gears contact. The bulk temperature of the gear is the main cause of thermal deformation of the gear. Therefore, it changes the involute characteristics of the gear, thereby causing comprehensive performance such as vibration and noise of the gear. The analysis and research of the temperature field can provide a basis for the tooth profile modification of the helical gear and the subsequent thermoelastic coupling.
Generally, the heat transfer process is composed of three forms: heat conduction, heat convection, and thermal radiation. Heat conduction, also known as thermal conduction, refers to the phenomenon of heat transfer relying on the thermal motion of microscopic particles when different objects are in contact without relative motion or relative displacement. Heat conduction is an inherent physical property of an object, and this conduction exists in objects in any state. According to the thermal analysis example, the formula for heat conduction can be obtained as Eq. (29) and Eq. (30):
$$
\phi = \lambda \frac{\Delta t}{\delta} A
\tag{29}
$$
$$
q = \lambda \frac{\Delta t}{\delta}
\tag{30}
$$
where \(\delta\) is the wall thickness, \(A\) is the wall area, \(\lambda\) is the proportionality coefficient, \(\Delta t\) is the temperature difference between the two sides of the wall, \(\phi\) is the heat conduction amount, and \(q\) is the heat flux density. The process in which a fluid transfers heat from one object to another during motion is called heat convection. Heat convection is a basic mode of heat transfer. When a fluid flows through two objects with different temperatures, when the temperatures at these two places are \(t_1\) and \(t_2\), and the specific heat capacity of the fluid is \(c_p\), the heat flux density and heat conduction amount can be calculated according to these conditions, as shown in Eq. (31) and Eq. (32):
$$
\phi = h A (t_w – t_f) = h A \Delta t
\tag{31}
$$
$$
q = M C_p (t_2 – t_1)
\tag{32}
$$
where \(t_w\) is the temperature of the gear wall surface, \(t_f\) is the fluid temperature, \(\Delta t\) is the temperature difference between the tooth surface and the fluid, and \(h\) is the convective heat transfer coefficient of the surface. Thermal radiation is different from heat conduction and heat convection. Heat conduction and thermal radiation both transfer heat through temperature differences, but thermal radiation transfers heat through invisible rays emitted from the surface of an object. The transferred energy is called radiant force, denoted by \(E\). However, this radiant energy is very small and can be ignored relative to the heat dissipation of the gear pair.
Thermal analysis is generally divided into two types: steady-state thermal analysis and transient thermal analysis. In the temperature field analysis process, the analysis of the temperature field that does not change with time is called steady-state temperature field analysis. The analysis method that separates the entire temperature field and divides it into heat conduction, heat convection, and thermal radiation that change with time is called transient thermal analysis. The heat balance equation of the steady-state temperature field is shown in Eq. (33):
$$
[K]\{I\} = \{Q\}
\tag{33}
$$
where \(\{I\}\) is the node temperature vector, \([K]\) contains the heat conduction matrix and convection coefficient, and \(\{Q\}\) is the node heat flow vector. The heat balance equation of the transient temperature field is shown in Eq. (34):
$$
[C]\{\dot{T}\} + [K]\{T\} = \{Q\}
\tag{34}
$$
where \(\{T\}\) is the node temperature vector, \([C]\) is the specific heat matrix, \(\{\dot{T}\}\) is the derivative of node temperature with respect to time, and \(\{Q\}\) is the node heat flow vector.
For the analysis of the temperature field of the helical gear, the basis is thermal analysis theory. From thermal analysis theory, it can be known that during heat conduction and heat convection, the heat generated by friction of the helical gear cannot be directly transferred between the gears but is transferred through the lubricating oil in the gearbox. A portion of the gear oil is immersed in the lubricating oil, and the heat on the gear is taken away by the lubricating oil. When the temperature field of the gear pair reaches a balanced state, the temperature of the gear pair is actually still in a state of continuous change, but the amount of change is negligible for the entire temperature field because the temperature change is too small. Therefore, the temperature field in this state can be treated as a steady-state temperature field. The analysis process of the temperature field is similar to the static field, and a simplified model can also be made to shorten the analysis time. The specific steps are as follows. According to the parameters of the helical gear, I performed parametric modeling to obtain the solid model of the gear set. I conducted meshing contact analysis on the gear pair to check whether there is interference. I used finite element software to mesh the simplified model. I calculated the convective heat transfer coefficient of the gear surface. I calculated the friction heat flow in the meshing area of the gear. I obtained the distribution law of the gear bulk temperature field.
The flash temperature and bulk temperature are the manifestations of the friction heat flow generated during the meshing process of the gear teeth in different time periods. The heat conduction differential equation of the gear transmission system temperature field with time established by the law of conservation of energy and Fourier’s law is shown in Eq. (35):
$$
\lambda \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) = \rho c \frac{\partial T}{\partial \tau}
\tag{35}
$$
The bulk temperature is the form of friction heat flow existing on the contact surface of the gear teeth. During the operation of the helical gear, heat is generated due to mutual friction of the contact surfaces. A portion of the heat enters the interior of the gear through heat conduction, and another portion is taken away by heat transfer from various surfaces of the gear under the action of the lubricating oil. Therefore, the heat of the gear pair exists simultaneously in the input and output processes. After a period of operation, the temperature of the gear teeth of the gear pair will reach a state of thermal equilibrium, and the temperature of the gear teeth does not change with time and space. Therefore, the heat balance equation of the system can be simplified as Eq. (36):
$$
\int \lambda \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) d\tau = 0
\tag{36}
$$
For the gear pair, the tooth tip and both sides of the gear are non-meshing surfaces, and the gear teeth exchange heat by convection with the lubricating oil. According to Fourier’s law and Newton’s cooling formula, as shown in Eq. (37):
$$
-\lambda \left( \frac{\partial t}{\partial n} \right) = h (t_w – t_f)
\tag{37}
$$
Under normal operating conditions of the helical gear, the meshing surface of the gear teeth both generates heat and takes away a portion of the heat due to the action of the lubricating oil, so heat generation and heat dissipation exist simultaneously. The heat balance condition is shown in Eq. (38):
$$
-\lambda \left( \frac{\partial t}{\partial n} \right) = h (t_w – t_f) – q
\tag{38}
$$
The part where the gear and the shaft are connected is generally an interference fit or key connection, so there is no lubricating oil between the gear and the shaft, so there is no heat conduction, and the gear and the shaft are considered adiabatic. After meshing the gear, in the finite element software, the matrix form of the heat balance equation of the system is shown in Eq. (39):
$$
[D][T] + [C][\partial T / \partial \tau] = [P]
\tag{39}
$$
where \([D]\) is the heat conduction matrix, \([C]\) is the total heat capacity matrix, and \([P]\) is the node flow column matrix. When the gearbox operation is in a stable state, that is, when the temperature of the gear system does not change with time, that is, when the temperature field of the gear is in a steady-state temperature field, Eq. (39) can be simplified to a linear equation system as shown in Eq. (40):
$$
[D][T] = [P]
\tag{40}
$$
When the gear operation reaches a thermal equilibrium state, the bulk temperature field of the gear does not change much, and the temperature distribution of each gear tooth on the gear is basically similar. Therefore, I studied a single gear tooth. The temperature field of a single gear tooth can be expressed as \(T = T(x,y,z)\). The boundary problem can be expressed as Eq. (41):
$$
\nabla^2 T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = 0
\tag{41}
$$
I divided a single gear tooth into several regions. For the meshing working tooth surface, region \(m\), the friction heat flow generated during gear meshing is \(q\). During the contact process, heat is both generated and dissipated to the surroundings, so the boundary condition is Eq. (42):
$$
-\lambda \left( \frac{\partial T}{\partial n} \right) = h_m (T – T_0) – q
\tag{42}
$$
For the non-working surface: tooth tip, tooth root, and non-meshing area, region \(t\), the boundary condition is Eq. (43):
$$
-\lambda \left( \frac{\partial T}{\partial n} \right) = h_t (T – T_0)
\tag{43}
$$
For the gear end face, region \(s\), the boundary condition is Eq. (44):
$$
-\lambda \left( \frac{\partial T}{\partial n} \right) = h_s (T – T_0)
\tag{44}
$$
For the axial center part of the gear, region \(d\), this position is relatively far from the gear teeth, and there is almost no heat conduction with the gear teeth, so it can be treated as an adiabatic surface:
$$
\frac{\partial T}{\partial n} = 0
\tag{45}
$$
For the gear segmented regions, regions \(p\) and \(q\), heat conduction exists in the gear segmented regions, and the heat conduction on these two surfaces is equal.
$$
T_p = T_q
\tag{46}
$$
$$
\lambda \frac{\partial T}{\partial n_p} = \lambda \frac{\partial T}{\partial n_q}
\tag{47}
$$
where \(T_0\) is the ambient temperature, \(n\) is the outer normal direction of the heat exchange surface, \(\lambda\) is the thermal conductivity of the gear material, and \(h_m\), \(h_t\), and \(h_s\) are the heat transfer coefficients of each region of the gear. The finite element method can be used to express this steady-state temperature field problem with surface heat sources using a finite number of node temperatures. The solution process can be converted into a problem of solving the limit value of the functional. The solution \(T(x,y,z)\) satisfying the conditions makes the functional as shown in Eq. (48):
$$
J[T] = \int \int \int_\Omega \left[ \left( \frac{\partial T}{\partial x} \right)^2 + \left( \frac{\partial T}{\partial y} \right)^2 + \left( \frac{\partial T}{\partial z} \right)^2 \right] d\Omega + \int \int_{\Sigma_m} h_m \left( \frac{1}{2} T^2 – T_0 T \right) dS + \int \int_{\Sigma_s} h_s \left( \frac{1}{2} T^2 – T_0 T \right) dS + \int \int_{\Sigma_t} h_t \left( \frac{1}{2} T^2 – T_0 T \right) dS – \int \int_{\Sigma_m} q T dS
\tag{48}
$$
Taking its minimum value, \(\Sigma_m\), \(\Sigma_s\), and \(\Sigma_t\) correspond to the boundaries of the gear regions. The calculation region is discretized into \(E\) elements, and the functional is decomposed into a series of element functionals. Assuming that the temperature field in each element can be represented by the node temperature interpolation function, as shown in Eq. (49):
$$
T = [N] \{T\}^e
\tag{49}
$$
where \(\{T\}^e\) and \([N]^e\) are the element node temperature vector and the element shape function matrix, respectively. In this way, the functional can become a function of the temperature of each node, and the problem of finding the extreme value of the functional can be transformed into a problem of finding the extreme value of the temperature of each node. The element temperature stiffness matrix is shown in Eq. (50):
$$
k_{ij}^e = \int \int \int_\Omega \left( \frac{\partial N_i}{\partial x} \frac{\partial N_j}{\partial x} + \frac{\partial N_i}{\partial y} \frac{\partial N_j}{\partial y} + \frac{\partial N_i}{\partial z} \frac{\partial N_j}{\partial z} \right) d\Omega + \int \int_{\Sigma_m} h_m N_i N_j dS + \int \int_{\Sigma_s} h_s N_i N_j dS + \int \int_{\Sigma_t} h_t N_i N_j dS
\tag{50}
$$
The element heat load vector is shown in Eq. (51):
$$
P_i^e = \int \int_{\Sigma_m} q N_i dS + \int \int_{\Sigma_m} h_m T_0 N_i dS + \int \int_{\Sigma_s} h_s T_0 N_i dS + \int \int_{\Sigma_t} h_t T_0 N_i dS
\tag{51}
$$
The overall temperature stiffness equation can be obtained as Eq. (52):
$$
[K]\{T\} = \{P\}
\tag{52}
$$
where \(\{P\}\) is a vector related to the lubricating medium, \(\{T\}\) is the temperature vector at any time, and \([K]\) is a matrix related to the medium, gear material, and node coordinates. According to the flow of lubricating oil along the tooth surface and end face, it can be divided into turbulent flow and laminar flow. Turbulent flow and laminar flow correspond to different Reynolds number ranges to determine the convective heat transfer coefficient. The calculation method of the Reynolds number is shown in Eq. (53):
$$
R_e = \frac{\omega r_c}{v_{oil}}
\tag{53}
$$
where \(r_c\) is any radius on the gear, \(v_{oil}\) is the kinematic viscosity of the lubricating oil, and \(\omega\) is the angular velocity of the gear. The convective heat transfer coefficient of the gear tooth end face is shown in Eq. (54): when \(R_e < 5 \times 10^5\), the flow mode of the lubricating oil is laminar, and the convective heat transfer coefficient \(h_s\) is expressed as Eq. (54):
$$
h_s = 0.308 \lambda_{oil} (m + 2)^{0.5} \left( \frac{\omega}{v_{oil}} \right)^{0.5}
\tag{54}
$$
where \(\rho_{oil}\) is the density of the lubricating oil, \(\lambda_{oil}\) is the thermal conductivity of the lubricating oil, \(c_{oil}\) is the specific heat of the lubricating oil, and \(m\) is generally set to a constant to define the radial distribution of the temperature on the disk surface, generally taken as 2. When \(R_e > 5 \times 10^5\), the flow mode of the lubricating oil on the gear surface is turbulent, and the convective heat transfer coefficient \(h_s\) is shown in Eq. (55):
$$
h_s = 0.0197 \lambda_{oil} (m + 2.6)^{0.2} \left( \frac{\omega}{v_{oil}} \right)^{0.8} r_c^{0.6} P_r^{0.6}
\tag{55}
$$
The convective heat transfer coefficient of the gear tooth meshing surface is shown in Eq. (56):
$$
h_s = 0.228 \left( \frac{\rho_{oil} c_{oil} \lambda_{oil}}{L} \right)^{0.333} \left( \frac{v}{v_{oil}} \right)^{0.731} \lambda_{oil} / L
\tag{56}
$$
where \(v\) is the linear velocity on the pitch circle, \(m_L\) is the average tooth height, and \(L\) is the pitch circle diameter. The friction heat flow \(Q\) at the meshing point of the gear is expressed as Eq. (57):
$$
Q = \frac{\mu_c}{J} F_{NC} v_c
\tag{57}
$$
where \(J\) is the mechanical equivalent of heat, \(v_c\) is the relative sliding speed of the two gears at the contact point, \(\mu_c\) is the friction coefficient at the contact point, and \(F_{NC}\) is the average contact pressure at the contact point. Because the material properties and sliding speeds of the driving gear and the driven gear are different, the friction heat flow is distributed between the teeth of the meshing gears. The friction heat flows obtained by the driving gear and the driven gear are \(Q_1\) and \(Q_2\), respectively, and their calculation formulas are shown in Eq. (58):
$$
\begin{aligned}
Q_1 &= \xi \frac{\mu_c}{J} F_{NC} v_c \\
Q_2 &= (1 – \xi) \frac{\mu_c}{J} F_{NC} v_c \\
\xi &= \frac{\sqrt{\lambda_1 \rho_1 c_1 v_1}}{\sqrt{\lambda_1 \rho_1 c_1 v_1} + \sqrt{\lambda_2 \rho_2 c_2 v_2}}
\end{aligned}
\tag{58}
$$
where \(\lambda_1\) and \(\lambda_2\) are the thermal conductivities of the driving and driven gear materials, \(\rho_1\) and \(\rho_2\) are the densities of the two gear materials, \(c_1\) and \(c_2\) are the specific heats of the two gears, and \(v_1\) and \(v_2\) are the tangential velocities at the meshing point of the two gears. Because the materials used for the driving and driven gears are the same, the thermal conductivity, material density, and specific heat are all the same, so only the respective tangential velocities need to be calculated. The heat generated by the helical gear rotating one revolution can be expressed by the formula shown in Eq. (59):
$$
\begin{aligned}
q_1 &= \frac{\omega_1}{2\pi} \frac{e}{v_1} Q_1 \\
q_2 &= \frac{\omega_2}{2\pi} \frac{e}{v_2} Q_2
\end{aligned}
\tag{59}
$$
where \(e\) is the contact half-width at the meshing point, \(\omega_1\) and \(\omega_2\) are the rotational angular velocities of the driving and driven gears, \(R_1\) and \(R_2\) are the equivalent curvature radii of the driving and driven gears, \(\mu_1\) and \(\mu_2\) are the Poisson’s ratios of the driving and driven gears, and \(E_1\) and \(E_2\) are the elastic moduli of the driving and driven gears. When the driving gear speed is 960 r/min, the end face heat transfer coefficient of the driving gear is 2576.2 W/(m·K), the end face heat transfer coefficient of the driven gear is 1538.2 W/(m·K), the heat transfer coefficient of the meshing surface of the driving gear is about 429 W/(m·K), the heat transfer coefficient of the meshing surface of the driven gear is about 275 W/(m·K), the friction heat flow of the driving gear is 4470 W/m², and the friction heat flow of the driven gear is about 2571.6 W/m². Substituting these parameters into the temperature field solution of the helical gear, I analyzed the distribution law of the gear temperature field and the temperature value of the gear under normal working conditions.
Substituting the convective heat transfer coefficient of the gear end face and the friction heat flow of the meshing surface calculated above into the ANSYS Workbench module for temperature analysis, I obtained the temperature field distribution. Through the simulation results, it can be seen that the temperature field at the meshing line of the driving and driven gears is relatively high. Because this part generates the most friction, the heat flow at this part is the largest, and the heat dissipation is poor, so it is easy to cause heat accumulation. Therefore, the temperature at the meshing part is relatively high. When the driving gear speed is fixed, the temperature at the meshing line of the driven gear is lower than that at the meshing line of the driving gear. Because during the meshing process of the gear pair, the small gear must rotate more times than the large gear, so the heat flow at the meshing line of the driving gear is higher than that at the meshing line of the driven gear. When changing the friction coefficient of the helical gear, I analyzed the general temperature field distribution of the gear. I briefly analyzed the influence of the lubrication condition of the gear on the temperature field. There are three types of gear lubrication: liquid lubrication, boundary lubrication, and no lubrication. The difference between these three lubrication methods lies in the friction coefficient. I analyzed the temperature field of the same gear according to different friction coefficients. The friction coefficients were taken as 0.01, 0.05, 0.08, and 0.15. Because the span of the friction coefficient is larger, the difference in the temperature field of the gear can be seen more easily. When the friction coefficient is 0.01, the high-temperature region is mainly distributed at the tooth tip of the driving gear, and the maximum temperature is 131°C. The temperature of the driven gear is also slightly lower than that of the driving gear. When the friction coefficient is 0.05, the high-temperature region is distributed at the tooth tip of the driving gear, and the maximum temperature is 133°C. The maximum temperature of the driven gear is lower than that of the driving gear. Compared with the gear pair with a friction coefficient of 0.01, the temperature has increased. When the friction coefficient is 0.08, the maximum temperature of the helical gear is 138°C. When the friction coefficient is 0.15, the maximum temperature of the helical gear is 140°C. From the analysis, it can be seen that the larger the friction coefficient between the gears, the higher the temperature of the helical gear. When the lubrication condition is poor, the friction coefficient becomes larger. When the friction coefficient becomes larger, the friction heat flow of the helical gear increases, and the temperature generated by the helical gear becomes higher. The maximum temperature is distributed at the tooth tip of the driving gear, and the maximum temperature of the driving gear is higher than that of the driven gear. The influence of friction coefficient on the maximum temperature of the helical gear is summarized in Table 3.
| Friction coefficient | Maximum temperature (°C) | Location of maximum temperature |
|---|---|---|
| 0.01 | 131 | Driving gear tooth tip |
| 0.05 | 133 | Driving gear tooth tip |
| 0.08 | 138 | Driving gear tooth tip |
| 0.15 | 140 | Driving gear tooth tip |
For the thermoelastic coupling contact analysis of the helical gear system, during the operation of the helical gear, the contact stress, bending stress, and deformation of the gear teeth are studied through the contact analysis of the gear. However, the operating state of the helical gear is also affected by the bulk temperature field of the gear. Therefore, coupling analysis is required in the analysis of the contact process of the gear. For the helical gear used in the high-speed train gearbox, in the thermal equilibrium state, the gear pair is simultaneously affected by the temperature field and the load. Therefore, the influence of the temperature field needs to be combined in the analysis of the contact process. In this chapter, the contact state of the helical gear is a thermoelastic coupling state. During the operation of the helical gear, it is simultaneously affected by force and temperature field, both of which cause deformation of the gear. I used the static structural module in ANSYS Workbench software to study the equivalent stress and deformation of the helical gear under static conditions during contact. I used the transient thermal module to analyze the temperature field of the helical gear. Based on the contact analysis and temperature field analysis of the helical gear in the previous chapters, I conducted a thermoelastic coupling analysis of the helical gear. Through the above analysis combined with the content of this chapter, I can obtain the respective differences in equivalent stress and deformation of the gear teeth.
Because the helical gear will have a temperature field during operation, the gear will have thermal expansion. Therefore, the gear teeth of the helical gear will become larger and thicker in the direction of the tooth profile, resulting in transmission error in the process of torque transmission of the gear transmission system, and even interference. These problems will cause the equivalent stress of the helical gear transmission system to increase. Therefore, the influence of the temperature field needs to be considered in the analysis of the gear transmission system. In the fourth chapter of this paper, I used the finite element method to analyze the components of the gear temperature field and the heat convection coefficients of each part. On this basis, I applied torque to the helical gear, constrained all degrees of freedom of the gear shaft hole, so that the gear can only perform circular motion around the axis. During the temperature change of the helical gear, the thermal expansion coefficient of the gear does not basically change with temperature, about \(1.14 \times 10^{-5}\). I obtained the thermal deformation results of the gear pair. From the analysis results, it can be seen that from the temperature point of view, the temperature of the driven gear is slightly lower than that of the driving gear. Because the small gear is the driving gear, the number of revolutions of the driving gear during meshing is more than that of the driven gear, so the number of frictions experienced by the driving gear is greater than that of the driving gear. From the perspective of deformation, the thermal deformation of the driving gear is also slightly larger than that of the driven gear, and the reason for this phenomenon is the same as the temperature. From the temperature distribution trend, along the helical line direction, the deformation in the middle of the tooth profile is greater than the deformation on both sides. Due to heat dissipation, both sides are easier to dissipate heat, so it presents a trend of bulging in the middle and flat on both sides, and the distribution trend of the driving gear is similar to that of the driven gear.
Under the combined influence of torque and thermal load, because the elastic deformation caused by torque and the thermal deformation caused by the temperature field act together, it is necessary to analyze the thermoelastic coupling state when analyzing the meshing state of the helical gear. The thermoelastic coupling contact state is different from the contact state under pure torque. To study the difference between the contact stress and deformation of the helical gear under a single load and under the coupling of two fields, based on the static contact analysis of the helical gear under a single load in the third chapter, I conducted a thermoelastic coupling analysis of the helical gear in this section. In the process of thermoelastic coupling analysis, I applied the temperature field of the helical gear as a load to the gear, coupled it with the torque load for analysis, and compared the analysis results to study the influence of the gear temperature field on the contact behavior, in order to determine the modification amount of the gear teeth later.
When the helical gear deforms due to the thermal load and torque load, the tooth profile of the gear becomes thicker, causing the gear to mesh earlier, and the time from entering meshing to leaving meshing becomes longer, and the backlash also becomes smaller. The teeth of the driving gear and the teeth of the driven gear squeeze each other, and the contact pressure in the meshing area also increases. The actual contact state of the helical gear pair is an inclined curve along the tooth width direction. In the simulation cloud diagram, the sliding area is the meshing area of the gear, which is similar to the actual meshing state, thus verifying the correctness of the simulation and ensuring the accuracy of the subsequent work. According to the equivalent stress cloud diagram in the coupled state, it can be seen that the maximum equivalent stress in the coupled state is 1305.8 MPa. In this meshing state, the maximum equivalent stress appears at the tooth tip of the driving gear and the tooth root of the driven gear. The overlap ratio of the helical gear pair in this paper is about 2.4. During the operation of the gear, there will be two-tooth meshing and three-tooth meshing. During the meshing process of the helical gear, because the number of loaded teeth changes, the equivalent stress of the helical gear changes with the number of loaded teeth. In this chapter, I conducted thermoelastic coupling analysis on the mesh-in position and mesh-out position of the helical gear respectively and compared the coupling analysis results with the analysis results of the gear pair under a single torque load. To improve the calculation efficiency and speed up the calculation while also considering the accuracy of the analysis, I intercepted the part of the gear analysis for mesh division and locally refined the contact part of the gear teeth. The remaining part of the gear does not need too high mesh density. The contact of the helical gear needs to be manually established as a contact pair. The small gear acts as the driving gear, so the tooth surface of the small gear acts as the contact surface, and the tooth surface of the large gear acts as the target surface. The thermal load is to apply the bulk temperature field to the gear teeth of the helical gear, and to define the thermal expansion coefficient and reference temperature of the gear. The deformation of the gear teeth in the coupled state is 13 μm, mainly occurring at the tooth tip of the driving gear and the tooth root of the driven gear, and the contact state is two-tooth contact. The comparison of contact pressure, equivalent stress, and deformation before and after coupling is summarized in Table 4.
| Item | Structural analysis | Coupled analysis |
|---|---|---|
| Maximum contact pressure (MPa) | 884.37 | 1100.2 |
| Maximum equivalent stress (MPa) | 1100.2 | 1305.8 |
| Maximum deformation (μm) | 5 | 13 |
From Table 4, it can be seen that the contact pressure, equivalent stress, and deformation in the coupled state are all larger than those in the single load state. Therefore, in the subsequent modification, the determination of the modification amount needs to be based on the deformation in the coupled state. With the rapid development of the high-speed railway industry, the quality test of helical gears is becoming more and more strict, and the use environment is becoming more and more severe. It is necessary to meet the requirements of high speed and heavy load. If only traditional methods such as improving machining accuracy and reducing manufacturing and installation errors are relied upon, a large amount of manpower, material resources, and financial resources need to be invested, but the improvement in effect is very small. Therefore, it can be considered to modify the tooth profile of the helical gear by using the deformation after coupling analysis as a reference for tooth modification, so as to improve the load-carrying capacity of the helical gear.
During the actual meshing of the helical gear, manufacturing errors, installation errors, and deformation will cause deviation from the theoretical meshing line. Because of the difference in instantaneous speed between the driving gear and the driven gear during the meshing process, impact and interference will exist between the gear teeth, resulting in an increase in equivalent stress during gear operation. When the tooth pair just enters the mesh-in point, the gear fully bears the load on the gear. Tooth pair I deforms due to the load, making the actual base pitch of the driving gear smaller and the actual base pitch of the driven gear larger. The actual meshing point of tooth pair II is point A1 on the involute, which deviates from the meshing point B1 on the theoretical meshing line. The common normal of the axis line and the actual meshing point does not intersect at point P but actually intersects at point P’. The actual instantaneous transmission ratio of the gear pair is shown in Eq. (60):
$$
i_{12}’ = \frac{r_2 – \Delta r}{r_1 + \Delta r} < \frac{r_2}{r_1}
\tag{60}
$$
where \(r_1\) and \(r_2\) represent the pitch circle radii of the driving gear and the driven gear, respectively, and \(\Delta r\) represents the change in base pitch. At the critical point of gear tooth mesh-out, tooth II deforms elastically due to the load, and the actual base pitch of the driven gear is smaller, making the actual base pitch of the driving gear larger. The intersection point of the axis line and the common normal of the contact point of gear tooth I also shifts, not at the theoretical point P, but actually at point P’. Therefore, in terms of instantaneous transmission ratio, the actual instantaneous transmission ratio is greater than the theoretical instantaneous transmission ratio, resulting in meshing impact. The actual instantaneous transmission ratio is expressed as Eq. (61):
$$
i_{12}’ = \frac{r_2 + \Delta r}{r_1 – \Delta r} > \frac{r_2}{r_1}
\tag{61}
$$
During the meshing process, in order to avoid sudden changes in the load on the gear teeth caused by gear tooth deformation, the tooth profile of the standard involute helical gear is modified. According to the deformation, a part of the material on the tooth surface is removed, so that the interference and impact caused by the deformation of the gear teeth during operation are reduced, making the helical gear run more smoothly and reducing the noise and vibration of the system. The meshing cycle of the gear teeth refers to a process from mesh-in to mesh-out along the gear meshing line during the meshing process of the helical gear. In the helical gear of this paper, the gear teeth will experience from two-tooth meshing to three-tooth meshing and then to two-tooth meshing. During the meshing process of the helical gear, the gear teeth will deform due to contact, which also causes the load change during gear tooth meshing. There are generally two modification methods for the tooth profile of the gear teeth. One is to modify the tooth tip and tooth root of one gear at the same time; the other is to modify the tooth tip of both the driving gear and the driven gear at the same time. Theoretically, the modification amount of the tooth profile is consistent with the deformation of the gear teeth. However, in the actual determination of the modification amount, it is also necessary to compensate for the influence of gear manufacturing errors on meshing according to the accuracy of the gear. In actual manufacturing, many companies often combine production practice experience and have their own empirical calculation formulas and standards. According to the technical data of the MAAG gear company and related literature, the simplified calculation formula for the tooth profile modification amount of helical gears is given in Eq. (62) and Eq. (63). The profile modification amount at the meshing starting point is shown in Eq. (62) and Eq. (63):
$$
\Delta_{1u} = 0.00511 W + 0.000407 \times 10^3
\tag{62}
$$
$$
\Delta_{1o} = 0.0131 W + 0.000407 \times 10^3
\tag{63}
$$
The tooth profile modification amount at the meshing end point is shown in Eq. (64) and Eq. (65):
$$
\Delta_{2u} = 0.000406 \times 10^3 W
\tag{64}
$$
$$
\Delta_{2o} = 0.0762 W + 0.000406 \times 10^3
\tag{65}
$$
where \(\Delta\) is the modification amount, \(W\) is the unit tooth width load, \(W = F_t / B\), \(B\) is the tooth width, \(F_t\) is the tangential force, \(\Delta_u\) is the upper tolerance limit, and \(\Delta_o\) is the lower tolerance limit. According to the gear modification theory in the literature, the modification height of the helical gear is calculated as Eq. (66):
$$
R_g = \pi m_t \varepsilon_\alpha \cos\alpha_t
\tag{66}
$$
where \(R_g\) is the modification height of the helical gear, \(\varepsilon_\alpha\) is the overlap ratio of the helical gear, \(m_t\) is the end face module, and \(\alpha_t\) is the end face pressure angle. According to Eq. (66), the modification length is 4.2 mm. During normal operation, the helical gear will simultaneously produce thermal deformation and elastic deformation. These two deformations will reduce the clearance between the gear teeth. Therefore, the modification amount of the helical gear needs to be determined according to the deformation results of the thermoelastic coupling analysis. If only elastic deformation is considered, the modification amount of the helical gear will be too small. In actual operation, the clearance between the gear teeth is still too small, and the equivalent stress will also be too large. According to the deformation of the coupling analysis in Section 5.3.3, the modification amount is 13 μm.
The tooth profile of the helical gear is mainly composed of two parts: one is the involute equation of the helical gear, and the other is the transition curve of the helical gear. For the involute equation, in the standard involute, the coordinate equation of any point K is shown in Eq. (67):
$$
\begin{aligned}
x_k &= r_k \cos\phi_k \\
y_k &= r_k \sin\phi_k \\
\phi_k &= \frac{\pi}{2} – \frac{4m \tan\alpha}{\pi r} + \text{inv}\alpha – \text{inv}\alpha_k
\end{aligned}
\tag{67}
$$
where \(r_k\) is the radius vector of any point K, \(\phi_k\) is the angle between the y-axis and the radius vector of any point K, \(\alpha_k\) is the pressure angle of any point K on the involute, \(m\) is the module, \(x\) is the modification coefficient, and \(\alpha\) is the pressure angle. When the basic parameters of the helical gear are fixed, the variation of the involute equation is \(r_k\), and its range is between the radius of the addendum circle and the radius of the dedendum circle. The transition curve can be regarded as the curve formed by the tooth corner arc when the pitch line and pitch circle make pure rolling. Its equation is shown in Eq. (68):
$$
\begin{aligned}
d &= m h_a^* + r_0 \cos\alpha \\
a &= m h_a^* + c^* – r_0 \sin\alpha \\
r_0 &= \frac{m c^*}{1 – \sin\alpha}
\end{aligned}
\tag{68}
$$
where \(\alpha\) is the distance from the tool fillet center to the tool tooth profile centerline, \(r_0\) is the tool fillet radius, and \(d\) is the distance from the tool fillet center to the tool tooth slot centerline. The base circle centers of the helical gear before and after modification are coincident, but the base circle radius of the modified thickness is smaller. Point H is the starting point of modification, L is the modification length, and \(r_b’\) is the base circle radius after modification. The base circle radius after modification is shown in Eq. (69):
$$
r_b’ = \frac{r_a \cos\alpha_a – C_h \tan\alpha_h}{r_h \tan\alpha_h}
\tag{69}
$$
where \(r_a\) is the addendum circle radius, \(\alpha_a\) is the addendum pressure angle, \(r_h\) is the radius at the starting point of tooth profile modification, and \(\alpha_h\) is the pressure angle at the starting point of tooth profile modification. After modification, the gear teeth can avoid sharp corner contact during mesh-in and mesh-out. The method I adopted is to simultaneously modify the tooth tip of the driving gear and the driven gear. In the literature, the involute is segmented, and the helical gear is parametrically modeled. First, the starting position of the modification and the modification amount are determined. The curve between the starting point and the end point of the modification is taken as a new involute. The center of the base circle of the involute after modification coincides with the previous center, but the radius of the base circle is reduced. The equation at the tooth profile modification is shown in Eq. (70):
$$
\begin{aligned}
x_k’ &= r_k’ \cos\phi_k’ \\
y_k’ &= r_k’ \sin\phi_k’ \\
\phi_k’ &= \frac{\pi}{2} – \frac{4m \tan\alpha’}{\pi r’} + \text{inv}\alpha’ – \text{inv}\alpha_k’
\end{aligned}
\tag{70}
$$
where \(\alpha_k’\) is the pressure angle of point K on the involute after modification, and \(\alpha’\) is the pressure angle of the pitch circle on the involute after modification. I used the basic parameters of the helical gear in Table 1. After modifying the tooth profile of the helical gear, I obtained the modified involute and transition curve.
The purpose of tooth profile modification is to change the tooth surface shape and thus change the meshing position of the helical gear, thereby changing the load distribution of the tooth surface and then changing the temperature field distribution of the tooth surface. During the meshing process of the helical gear, the places where relative sliding is more obvious are the tooth tip of the driving gear and the tooth root of the driven gear, so more friction heat flow will be generated, and scuffing is also likely to occur during the meshing process of the helical gear. Therefore, it is necessary to analyze the temperature field distribution before and after tooth profile modification. From the analysis, it can be seen that after modification, the temperature field distribution of the helical gear has changed. The maximum temperature of the gear teeth has dropped from 138°C to 116°C, and the area of the maximum temperature has also changed, gradually moving from the previous tooth tip to the middle of the gear teeth. Because scuffing of the gear teeth is most likely to occur at the tooth tip of the driving gear and the tooth root of the driven gear, the maximum temperature has avoided these two areas, so the probability of scuffing is reduced. The comparison of the temperature field before and after modification is summarized in Table 5.
| Item | Before modification | After modification |
|---|---|---|
| Maximum temperature (°C) | 138 | 116 |
| Location of maximum temperature | Driving gear tooth tip | Middle of gear tooth |
In the simulation analysis of the fifth chapter and the third chapter, it is known that the maximum equivalent stress of the helical gear, whether under a single torque load or under a coupled load, appears at the tooth tip of the driving gear. After modification of the helical gear, the maximum equivalent stress in the coupled state is 1100.2 MPa. Compared with the equivalent stress in the coupled state of the unmodified helical gear, it has decreased by about 200 MPa. The position where the maximum equivalent stress appears has also changed, moving from the tooth tip of the driving gear and the tooth root of the driven gear to the middle of the gear teeth, reducing the probability of scuffing of the helical gear. This indicates that this modification method of the helical gear is effective. The comparison of the equivalent stress before and after modification is summarized in Table 6.
| Item | Before modification | After modification |
|---|---|---|
| Maximum equivalent stress (MPa) | 1305.8 | 1100.2 |
| Location of maximum stress | Driving gear tooth tip and driven gear tooth root | Middle of gear tooth |
In conclusion, I took the helical gear in the gearbox of the CRH380A EMU as the research object and studied its stress deformation and temperature field under stable operating conditions. I used Creo modeling software to create a three-dimensional model of the helical gear. Because the equivalent stress and temperature field between the gear teeth are the two most important factors that cause scuffing of the helical gear and affect the smooth operation of the helical gear, I conducted static analysis and steady-state temperature field analysis of the helical gear and compared its stress distribution and temperature field distribution under different rotational speeds and meshing angles. Because the equivalent stress and temperature exist simultaneously in the actual operation of the helical gear, the coupling analysis of the equivalent stress and temperature between the gear teeth is more consistent with the actual operating environment of the helical gear. I compared the differences among elastic deformation, thermal deformation, and thermoelastic coupling deformation, selected the appropriate deformation amount as the modification amount of the helical gear, and then compared the equivalent stress in the thermoelastic coupling state before and after modification to evaluate the effect of modification. The conclusions are as follows. Using ANSYS Workbench, through the static analysis of the helical gear, I studied the equivalent stress of the helical gear under different rotational speeds and different rotation angles. The maximum equivalent stress of the helical gear was 1100.2 MPa, and the maximum deformation was 5 μm. The location of the maximum equivalent stress distribution was at the tooth tip of the driving gear and the tooth root of the driven gear, and the direction of deformation was mainly the circumferential direction of the gear. Under the analysis of the temperature field of gear meshing, it can be concluded that the higher the friction coefficient, the higher the temperature generated by the helical gear. The maximum temperature is about 140°C and is concentrated at the tooth tip. Under the action of the temperature field, the helical gear will undergo thermal deformation, and the deformation trend is that along the tooth width direction, the deformation in the middle is relatively large, and the deformation at both ends is relatively small. The deformation of the driving gear is greater than that of the driven gear, but the deformation trend of the driving gear is similar to that of the driven gear. By coupling the structural field and the temperature field, it can be seen from the comparison results that the equivalent stress under coupling analysis is about 1300 MPa, which is greater than the equivalent stress under structural analysis. The maximum equivalent stress still appears at the tooth tip of the driving gear and the tooth root of the driven gear, similar to the stress distribution trend under static analysis. From the perspective of deformation, under the analysis of the coupled state, the axial, radial, and circumferential directions of the gear teeth all deform, and the deformation amount is about 13 μm. Therefore, compared with the previous single temperature field and structural field, it can be concluded that the circumferential deformation of the helical gear is mainly affected by torque, and the axial and radial deformation of the helical gear are mainly affected by the temperature field. In terms of deformation trend, the results under coupling analysis are similar to those under single load. According to the deformation results of the helical gear under coupling analysis, I modified the tooth profile of the helical gear with a modification amount of 13 μm and a modification length of 4.6 mm. Comparing before and after modification, the maximum temperature of the tooth surface decreased, and the high-temperature area moved from the tooth tip to the middle of the tooth surface, avoiding the area prone to scuffing. The maximum equivalent stress after modification also decreased, and the stress concentration phenomenon was also improved.
Gear contact is a quite complex process. Due to insufficient experience and limited time, there are still some shortcomings. The analysis in this paper did not use a complete gear model. In future research, if hardware conditions permit, a complete helical gear pair can be analyzed, and the results will be more consistent with the actual meshing situation. The working conditions selected in this paper are relatively few. I only selected several different rotational speeds under a fixed torque, and the results obtained can only show the general trend of stress variation. The research on modification is also not perfect. I only performed short modification in tooth profile modification on the helical gear and did not use other methods to modify the helical gear and compare the effects of the two modification methods.