Design of spur bevel gear transmission

2022-06-23
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Transmission design of straight bevel gear

bevel gear is the abbreviation of bevel gear. It is used to realize the transmission between two intersecting shafts. The intersection angle s of the two shafts is called the shaft angle. Its value can be determined according to the transmission needs. Generally, 90 ° is used. The teeth of the bevel gear are arranged on the truncated cone, and the teeth gradually shrink from the big end to the small end of the gear, as shown in the following figure. Because of this feature, the "circular performance stable and reliable column" corresponding to the cylindrical gear becomes a "cone" in the bevel gear, such as indexing cone, pitch cone, base cone, addendum cone, etc. Bevel gear teeth have straight, helical and curved teeth. The design, manufacture and installation of straight and helical bevel gears are simple, but the noise is large. They are used for low-speed transmission (<5m/s); Curved bevel gear has the characteristics of stable transmission, low noise and large bearing capacity. It is used in high-speed and heavy-duty occasions. This section only discusses standard straight bevel gear transmission with s=90 °

1. Formation of tooth profile surface the formation of tooth profile surface of spur bevel gear is similar to that of cylindrical gear. As shown in the figure below, the plane 1 is tangent to the base cone 2 and makes pure rolling, and the trajectory of any line OK passing through the cone vertex o on the plane is the involute cone. The intersection AK between the involute conical surface and the spherical surface with o as the spherical center and R as the radius is the spherical involute, which should be the tooth profile curve of the big end of the bevel gear. However, the spherical surface cannot be expanded into a plane, which brings many difficulties to the design and manufacture of bevel gears. Therefore, an approximate method instead of spherical involute is proposed

2. Bevel gear big end back cone, equivalent gear and equivalent number of teeth

(1) back cone and equivalent gear

the figure below shows the axial half section of a bevel gear, in which doaa is the axial section of the indexing cone, and the cone length OA is called the cone distance, represented by R; A circle with cone top o as its center and R as its radius shall be the projection of the sphere. If the spherical involute is used as the tooth profile of the bevel gear, the circular arc BAC is the intersection of the spherical big end of the gear and the shaft profile, and the spherical tooth profile cannot be developed into a plane. For this purpose, make O1A ⊥ OA through a, and intersect the axis of the gear at point O1. It is assumed that with oo1 as the axis and O1A as the bus as the conical surface o1aa, the cone is called the big end back cone of the bevel gear. Obviously, the back cone and the spherical surface are tangent to the graduation circle at the big end of the bevel gear. Since the large end back cone busbar 1a and the indexing cone busbar of the bevel gear are perpendicular to each other, the arc BAC of the spherical tooth profile is projected onto the back cone to obtain the line segment b'ac', the arc BAC is very close to the line segment b'ac', and the greater the ratio of the cone distance r to the module m of the large end of the bevel gear (generally r/m>30), the closer the two are. This shows that the tooth profile on the big end back cone can be approximately used as the big end tooth profile of the bevel gear. Since the back cone can be expanded into a plane and a sector gear can be obtained, the parameters of the sector gear such as module m, pressure angle A and tooth height coefficient ha* are the same as those of the big end of the bevel gear. Then the sector gear is supplemented into a complete spur gear. This virtual cylindrical gear is called the big end equivalent gear of the bevel gear. In this way, the tooth profile of the large end equivalent gear can be approximately taken as the large end tooth profile of the bevel gear, that is, the tooth size of the large end of the bevel gear (HA, HF, etc.) is equal to the tooth size of the equivalent gear

(2) basic parameters

because the size of the big end of the straight bevel gear is the largest, it is convenient to measure. Therefore, it is stipulated that the parameters of bevel gear and geometric ruler subject to inspection only need to swallow one intelligent "capsule" inch, which shall be subject to the big end. The value of modulus m at the big end is the standard value, which is selected in the following table. The pressure angle at the big end a=20 is specified in GB., Addendum height coefficient ha*=1, backlash coefficient c*=0.2

module of bevel gear (from GB)... 11.1251.251.3751.51.7522.252.52.7533.253.53.7544.555.566.578...

(3) equivalent number of teeth

number of teeth of equivalent gear zv is called the equivalent number of teeth of bevel gear. The relationship between zv and the number of teeth Z of the bevel gear can be obtained from the above figure. From the figure, the dividing circle radius rv

of the equivalent gear can be obtained, while

has

where: D is the dividing cone angle of the bevel gear. Zv is generally not an integer and does not need rounding

3 kinematic design of straight bevel gear transmission

(1) back bevel and equivalent gears

the following figure shows the axial profile of a pair of bevel gears. The shaft angle of the pair of bevel gears is equal to the sum of the two parts of the bevel angle, that is,

due to the needs of the transmission strength calculation and coincidence degree calculation of straight bevel gears, a pair of equivalent gears (above figure) are introduced, which are obtained by expanding the back cone at the midpoint of the tooth width of the pair of bevel gears. The dividing circle radii dv1/2 and dv2/2 of the equivalent gear are respectively the bus length of the back cone at the midpoint of the tooth width b of the pair of bevel gears; The module is the module at the midpoint of the tooth width, which is called the average module mm

1. Meshing transmission characteristics of straight bevel gears

the meshing transmission of a pair of bevel gears is equivalent to the meshing transmission of its equivalent gears. Therefore, it has the following characteristics:

(1) correct meshing condition

(2) continuous transmission condition e>1, and the coincidence degree e can be calculated according to the equivalent gear at the midpoint of its tooth width

(3) minimum number of teeth without undercutting

(4) transmission ratio i12, so

when s=90 °, there is

2 Calculation of geometric dimension

according to the characteristics of bevel gear transmission, the basic geometric dimension is calculated by the big end, but the geometric dimension of the middle point of the tooth width of the bevel gear and its equivalent gear must be derived through the big end

(1) tooth width factor fr. Generally, fr=1/3 and b1=b2=b

(2) the indexing circle diameter (average indexing circle diameter) DM and the average modulus mm

(3) the indexing circle diameter DMV, the equivalent number of teeth zv and the number of teeth ratio uv

of the equivalent gear at the midpoint of the tooth width. In the formula, the number of teeth ratio affects the top angle of the indexing cone. Generally, u ≤ 3 and the maximum is not more than 5

refer to the above figure to derive the geometric dimension calculation formula of standard straight bevel gear transmission, which is listed in the main geometric dimension calculation formula table of standard straight bevel gear transmission

4. Strength calculation of straight bevel gear transmission

the strength calculation of straight bevel gear is complex. In order to simplify the calculation, the strength calculation is usually carried out according to the equivalent gear at the midpoint of its tooth width. In this way, the corresponding formulas of spur gears can be directly quoted

due to the low manufacturing accuracy of spur bevel gears, the influence of coincidence degree is generally not considered in strength calculation, that is, the values of load distribution coefficient Ka, coincidence degree coefficient Ze and ye between teeth are taken as 1

1 gear tooth stress analysis

ignores the tooth surface friction and pretends to concentrate the radial force FN on the midpoint of the tooth width. On the indexing circle, it can be decomposed into three components, namely, the circumferential force ft, the radial force FR and the axial force FA, which are perpendicular to each other, as shown in the following figure. The magnitude of each force is respectively

where T1 is the nominal torque of the pinion (n · mm)

force analysis of gear teeth

direction of each force the direction of the driving wheel's circumferential force is opposite to the rotation direction of the wheel, and the direction of the driven wheel's circumferential force is the same as the rotation direction of the wheel; The radial force of the driving and driven wheels respectively points to their respective wheel centers; The axial forces point to their big ends respectively

load coefficient

in the formula: Ka service coefficient,

kv- dynamic load coefficient is checked according to the service coefficient Ka table to reduce the first level accuracy level, and

kb- load distribution coefficient along the tooth direction is checked according to the dynamic load coefficient kV diagram using the circumferential speed at the midpoint of the tooth width. In the formula, khbbe is checked by the load distribution coefficient along the tooth direction

2. Calculation of tooth contact fatigue strength

the equivalent gear is used as the calculation of tooth contact fatigue strength, then the formula

is

by substituting the relevant parameters of the equivalent gear into the above formula, it can be concluded that the checking formula of tooth contact fatigue strength of straight tooth bevel gear transmission is

and the design formula of tooth contact fatigue strength is

. The parameters in the formula are determined as above

3. Calculation of tooth root bending fatigue strength

by substituting the relevant parameters of equivalent gear into the formula and, the tooth root bending fatigue strength check formula and design formula of straight tooth bevel gear transmission

yfa tooth shape coefficient in the formula can be obtained. According to the number of equivalent teeth, the tooth shape coefficient diagram yfa of external gear can be checked

ysa- stress correction coefficient, which is obtained from the stress correction coefficient ysa graph according to the equivalent number of teeth

(end)

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