Spiral bevel gears are widely used in various mechanical systems due to their excellent transmission performance, such as smooth operation, low vibration and noise, and high load-carrying capacity. However, in order to meet the requirements of high-speed and heavy-duty transmissions, it is necessary to improve the meshing performance of spiral bevel gears by increasing the contact ratio. This article will discuss the design and meshing performance comparison of high contact ratio spiral bevel gears.
Introduction
Spiral bevel gears have many advantages, making them an important component in power transmission systems. However, early designs of spiral bevel gears had some problems, such as low contact ratio, poor tooth strength, and high vibration and noise, which could not meet the current transmission requirements. To address these issues, scholars have proposed various methods to increase the contact ratio of spiral bevel gears, such as the “inner diagonal design” by reducing the angle between the contact path and the root cone.
Presetting Meshing Performance
The meshing performance of spiral bevel gears is mainly evaluated by several indicators, including transmission error, tooth surface imprint, and design contact ratio. The design parameters of the transmission error mainly include the amplitude at the meshing transition point and the symmetry of the transmission error curve. The amplitude of the meshing transition point determines the impact force and stability when the meshing teeth change, and the symmetrical transmission error can avoid premature edge contact under light and medium loads. To make the gear pair transmission more stable, it is required to ensure that at least two pairs of teeth are in meshing simultaneously under working loads. By actively designing the contact path of the spiral bevel gear as a straight line, the sensitivity of the meshing performance to installation errors can be reduced, and the installation accuracy requirements can be lowered.
The processing parameter design flow chart for presetting the meshing performance is shown in Figure 2. The flow chart presets the design contact ratio εr, the transmission error amplitude at the meshing transition point, and the symmetry of the curve shape. The specific steps are as follows:
- Given the local synthesis parameters of the design reference point M0 (η2, Δx, Δy, m’21), the processing parameters of the large and small wheels can be obtained through the active design of the spiral bevel gear. The amplitude of the meshing transition point δφ2 is determined by the first-order derivative m’21 of the reciprocal of the transmission ratio, and its value is generally determined based on conditions such as no edge contact occurring under the working load and the smallest load transmission error.
- The meshing position T0 of the design reference point M0 is directly calculated by the local synthesis method. According to the geometric conditions of the meshing of the tooth top of the large wheel and the tooth root of the small wheel, the meshing-in position Tin can be calculated. Similarly, by the geometric conditions of the meshing of the tooth top of the small wheel and the tooth root of the large wheel, the meshing-out position Tout can be calculated.
- The symmetry ς is defined as the ratio of the meshing-in time |Tin – T0| to the meshing-out time |T0 – Tout|. It can be seen that when the symmetric ς is closer to 1, it means that Tin is closer to Tout. Since the selection of the design reference point M0 is arbitrary, it is generally difficult to determine ς = 1. Therefore, in order to achieve the required position symmetry, we usually adjust the tooth height position Δy of the design reference point M0 for the design calculation.
- In order to design a satisfactory contact ratio εr, when the meshing type is single-tooth meshing of the driving wheel, the contact ratio εr is the ratio of the total meshing time |Tin – Tout| from the meshing-in to the meshing-out of the driving wheel to the meshing cycle Tmesh. It can be calculated that when the εr obtained by the TCA is greater (less) than the predetermined design contact ratio ε0, the direction η2 of the contact trace can be increased (decreased) to obtain the required design contact ratio.
High Contact Ratio Design
There are different design contact ratios for the five contact paths between the root cone and the large wheel convex surface shown in Figure 4. These contact paths can be roughly divided into two categories: one is the contact paths ①, ②, and ③, which can be classified as inner diagonal contact. The contact mode is that the contact path of the convex surface of the large wheel starts from the tooth top of the large end to the tooth root of the small end, and the design contact ratio εr is ① < ② < ③. The starting and ending points of the contact path are on the top cone of the large and small wheels. The other is ④ and ⑤, where the contact path runs through the entire tooth surface along the tooth length direction, and the starting and ending points are on the inner and outer cones of the gear. The contact path ④ is in the middle of the tooth width and is basically parallel to the pitch cone line, which is suitable for the case where the midpoint spiral angle is greater than 30°. Its design contact ratio is only related to the tooth width. The contact path ⑤ has a certain inclination with the pitch cone line, and the design contact ratio εr is ④ < ⑤, which is suitable for the case where the midpoint spiral angle is less than 30°. The purpose is to increase the design contact ratio and the area of the tooth surface imprint. When the midpoint spiral angle is small, the contact path along the tooth length will cause the angle between the meshing line and the contact path to be very small, thereby reducing the area of the tooth surface imprint and making it difficult to form a lubricating oil film, which will lead to premature gear failure.
From the above analysis, it can be seen that the design contact ratios obtained by the contact paths ① and ② are too small to meet the requirements of high contact ratio design. The contact path ③ is the main high contact ratio design method at present, but its relative sliding speed is relatively large, mostly occurring at the tooth root or tooth top, so it is prone to tooth surface scratches and gluing, and it is more sensitive to installation errors, and it is easy to form edge contact, which is not universally applicable to some transmission occasions with high speed and heavy load. The contact path ④ is similar to the contact path of helical gears, and the contact path is basically coincident with the pitch cone line, and there is no friction. However, since the bevel gears generally adopt the tooth height modification design, if the contact path is designed to coincide with the pitch cone line, it will inevitably lead to the tooth surface imprint being too close to the tooth top of the large wheel or the tooth root of the small wheel, which is easy to cause premature edge contact and reduce the load-carrying capacity. Therefore, the contact path is generally designed in the middle of the tooth width. The relative sliding speed of the tooth surface of the contact path ⑤ is larger than that of the contact path ④, thereby increasing the probability of tooth surface scratches and gluing.
Example Calculation
In this section, a pair of spiral bevel gear pairs (where the large wheel is the convex surface and the small wheel is the concave surface) is used as an example to illustrate the two high contact ratio design methods proposed in this article. Table 1 shows the geometric parameters of the example gear pair, and Table 2 shows the processing parameters of the working surface of the example gear pair, including the design along the tooth length and the inner diagonal design. The preset meshing performance parameters are the design contact ratio εr = 2.5, the transmission error amplitude of the meshing transition point of the spiral bevel gear pair is δφ(tr)2 = 20”, the symmetry is ς = 1.0, and the type is required to be a parabolic transmission error.
Figures 5 and 6 show the tooth surface imprint and transmission error of the contact path inner diagonal design and the along-tooth length design, respectively. It can be seen from Figure 5 that when the inner diagonal design is adopted, the tooth surface imprint starts from the tooth top near the large end of the large wheel and gradually moves to the tooth root near the small end, showing an obvious inner diagonal shape. It can be seen from Figure 6 that when the along-tooth length design is adopted, the tooth surface imprint spans the entire tooth surface from the large end to the small end of the large wheel. Comparing the transmission error curves in Figures 5(b) and 6(b), it can be seen that the transmission error amplitude at the meshing transition point is basically the same in these two designs. Similarly, in the parabolic transmission error with a symmetry of ς = 1.0, if the design contact ratio εr is the same as the transmission error amplitude δφ(tr)2 at the meshing transition point, then the transmission error amplitudes at the meshing-in point and the meshing-out point are also basically equal.
According to the relevant regulations of the GB/T 11365 – 2019 standard, the specific values of the accuracy of bevel gears and hypoid gears can be determined. Four installation errors are specified, namely, the shaft angle deviation ΔΣ is 0.005 rad, the shaft spacing deviation ΔE is 0.5 mm, the axial displacement of the small wheel Δ4 is 0.3 mm, and the axial displacement of the large wheel Δ4 is 0.5 mm. The sensitivity of the tooth surface imprint to the installation error is analyzed. Figures 7(a) – 7(d) show the influence of various installation errors on the tooth surface imprint under the inner diagonal design. When there is an installation error, the tooth surface imprint will move to the small end or the large end, and the inner diagonal design is prone to edge contact at the tooth top or tooth root. Especially when moving to the small end, because the tooth height of the tapered bevel gear at the small end gradually decreases, it is more prone to edge contact. The sensitivity relationship of each individual error is as follows: the axial displacement of the large wheel and the axial displacement of the small wheel are roughly the same, the shaft spacing deviation is greater than the axial displacement, and the shaft angle deviation is the most sensitive.
Taking the same installation error values, the tooth surface imprint of the along-tooth length design is shown in Figures 8(a) – 8(d). It can be seen that this design is not sensitive to the axial displacement of the small wheel and the axial displacement of the large wheel, and the tooth surface imprint moves up and down, but it is more sensitive to the shaft spacing deviation. This is because the tooth surface imprint is along the tooth length direction, and the shaft spacing deviation acts on the movement of the tooth surface imprint almost in a 1:1 relationship. And its shaft angle deviation will cause the tooth surface imprint to tilt to a certain extent. Analyzing its sensitivity, it can be seen that the axial displacement of the large wheel and the axial displacement of the small wheel are roughly the same, followed by the shaft angle deviation, and the shaft spacing deviation is the most sensitive. It can be seen that the along-tooth length design can reduce the error sensitivity of the spiral bevel gear to a greater extent.
With the help of the Loaded Tooth Contact Analysis (LTCA) technology, when the load torque of the large wheel is 1000 N·m, the tooth root bending strength and tooth surface contact strength under the two high contact ratio designs are analyzed, and the results are shown in Figure 9. It can be seen from Figure 9(a) that the maximum tensile stress at the tooth root of the small wheel in the inner diagonal design is 110.2367 MPa. It can be seen from Figures 9(b) – 9(c) that the maximum tensile stress at the tooth root of the large wheel can reach 127.7034 MPa, and the maximum contact stress on the tooth surface is 1033.3867 MPa. Therefore, it can be obtained that the maximum tensile stress at the tooth root of the small wheel in the along-tooth length design is 95.2387 MPa, the maximum tensile stress at the tooth root of the large wheel is 114.8642 MPa, and the maximum contact stress on the tooth surface is 973.7385 MPa. Figure 10 shows the loaded transmission errors of the two high contact ratio designs, and the amplitudes of the loaded transmission errors of the inner diagonal design and the along-tooth length design are -13.53” and -9.292”, respectively.
Conclusion
Through the above analysis, the following conclusions can be drawn:
- By optimizing the local synthesis parameters, the high contact ratio design of the spiral bevel gear contact path as the inner diagonal and along the tooth length can be realized to meet the preset meshing performance.
- Compared with the inner diagonal design, the high contact ratio design with the contact path along the tooth length has lower error sensitivity. When there is an installation error, the tooth surface imprint will shift along the tooth height direction, but it is more sensitive to the shaft spacing deviation. Therefore, higher installation accuracy requirements are put forward for the center distance of the two axes.
- The tooth surface imprint of the contact path along the tooth length covers the entire tooth surface, which is conducive to improving the bending strength and contact strength of the gear teeth and reducing the fluctuation of the loaded transmission error of the gear pair.
| Design Method | Contact Path | Design Contact Ratio | Features |
|---|---|---|---|
| Inner Diagonal Design | From the large end tooth top to the small end tooth root | εr is ① < ② < ③ (depending on the specific path) | Relatively large relative sliding speed, prone to tooth surface scratches and gluing, sensitive to installation errors, easy to form edge contact, not suitable for high-speed and heavy-duty transmissions |
| Along-Tooth Length Design | Across the entire tooth surface from the large end to the small end | εr is ④ < ⑤ (depending on the specific path) | Lower error sensitivity, but more sensitive to shaft spacing deviation, requires higher installation accuracy for the center distance of the two axes, the tooth surface imprint covers the entire tooth surface, which is beneficial to improve the strength and reduce the transmission error fluctuation |
In future research, we can further explore the optimization of the high contact ratio design of spiral bevel gears, such as considering more factors that affect the meshing performance, optimizing the processing parameters, and improving the manufacturing accuracy. Additionally, the development of more accurate and efficient analysis methods to evaluate the performance of spiral bevel gears under different operating conditions is also an important direction for future research.