Foreword Spiral-tooth hypoid gears are widely used transmission components in vehicle power transmission. However, hypoid gears are the most complex type of gears designed and manufactured. The design content includes the overall geometric parameter design, tooth surface parameter design and Gear cutting machine adjustment parameter design. The traditional design method of the geometric parameters of the curved tooth hypoid gear is to apply 150 calculation steps in the Gleason calculation card, but since the calculation number is used extensively in the 150 calculation steps, the designer does not really understand The entire design process. Although some design manuals replace the calculation numbers with calculation formulas and simplify the calculation steps, it is still difficult for designers to understand the design principles of geometric parameters from these numerous formulas. Understanding the design principles of mastering the geometric parameters of hypoid gears is the key to improving their design. The geometric parameter design of the hypoid gear includes the design of the pitch cone geometry and the design of the geometric parameters.

1 hypoid gear sub-section cone geometry The hypoid gear pair's pitch cone is a pair of staggered tangent contact cones, which are also the split cones of the wheel blank geometry parameters design, two cones tangent At node P, r1 is the pitch circle radius of the small wheel, r2 is the pitch circle radius of the big wheel; C1C2 is the shortest distance line between the axes a1 and a2 of the two pitch cones, and E=C1C2 is called the hypoid gear pair The offset angle, the angle between the axes a1 and a2 is called the angle of intersection of the hypoid gear pair; the pitch plane T passes through the node P and is tangent to the two cones, Î´1 is the pitch angle of the small wheel, Î´2 Is the pitch angle of the large wheel, Îµ' is the offset angle of the hypoid gear pair; the intersection of the line passing through the node P and perpendicular to the pitch plane T and the small wheel axis a1 is K1, and the intersection with the large wheel axis a2 is K2, plane Î¦1 is a small axle section defined by K1K2 and axis a1, and plane Î¦2 is a large axle section determined by K1K2 and axis a2.

The plane Î¦ is perpendicular to the shortest distance line C1C2 through the parallel line C1H of the small wheel axis a1 and the large wheel axis a2; the plane Î¦' is perpendicular to the shortest distance line C1C2 through the parallel line W2K2 of the large wheel axis a2 and the small wheel axis a1; The Î· angle is the angle between the small axle section Î¦1 and the plane Î¦, and the Îµ angle is the angle between the large axle section Î¦2 and the plane Î¦.

, Î²1 is the helix angle of the small wheel, and Î²2 is the helix angle of the big wheel. According to the geometrical relationship of the pitch cone [5]: sinÎµ=cosÎ´1sinÎµâ€²/sinÎ£(1)sinÎ·=cosÎ´2sinÎµâ€²/sinÎ£(2)E=(r1cosÎ´2 r2cosÎ´1)sinÎµâ€²/sinÎ£cosÎ£=cosÎ´1cosÎ´2coscosâ€²â€²-sinÎ´1sinÎ´2Î²1=Î²2 Îµâ€²z2z1= R2cosÎ²2r1cosÎ²1(3) Equation (3) is called the basic equations of the hypoid gear sub-cone parameters.

In addition, according to the pitch cone geometry, the following formula can also be obtained. Q2=EtanÎ·sinÎ£(4)G2=r2sinÎ´2coscos2-EtanÎ·sinÎ£(5)G1=r1sinÎ´1cosÎ´1-EtanÎµsinÎ£(6)EcosÎ´1cosÎ´2sinÎµâ€²sinÎ£=G1sinÎ´1 G2sinÎ´2(7) where: G1 is small The distance from the apex O1 of the wheel cone to the intersection point C1; G2 is the distance from the apex O2 of the large wheel cone to the intersection point C2.

After the distance G2 from the apex of the large wheel cone to the staggered point has been determined, the distance G1 from the apex of the small wheel cone to the staggered point can also be determined by equation (7).

Two-section cone geometry The pitch-cone geometry of the hypoid gear pair includes the pitch circle diameters r1 and r2, the pitch cone angles Î´1 and Î´2, and the midpoint helix angles Î²1 and Î²2. In the basic equations of hypoid gear subsection cone parameters, the offset distance E, the angle of intersection Î£, the number of teeth z1 and z2 are the known conditions of the design, and the remaining r1, r2, Î´1, Î´2, Î²1, Î²2 and Îµâ€² There are 7 variables, but only 4 equations, so there are 3 free variables. In theory, any three of these seven variables can be taken as free variables, but the methods for determining the remaining four variables corresponding to different combinations of free variables and their difficulty levels will be different. Take r2, Î´2 and Î²1 as free variables, and Îµ' as the intermediate variable, Îµ'.

3 Geometric parameters The geometric parameters of the hypoid gear pair include the tooth width, the distance from the apex of the pitch cone to the staggered point, the tip cone angle, the root cone angle, the distance from the top of the cone to the point of intersection, and the apex of the root cone to the point of intersection. Distance, outer diameter, crown top distance, crown distance, installation distance, etc. The tooth width is the basic design variable. The installation distance is determined according to the structural design. The design calculation of the outer diameter and the crown distance is relatively simple. The determination of these parameters is omitted here.

The distance from the apex of the pitch cone to the staggered point is determined by the above equations (5) and (6). The following is the design calculation method for other geometric parameters.

The top cone angle Î´a2 of the 311 large wheel and the distance Ga2 from the apex of the top cone to the staggered point, the top cone angle Î´a2 of the large wheel is: Î´a2=Î´2 Î¸a2 (8) where: Î¸a2 is the large gear apex angle.

The distance Ga2 from the apex of the top cone of the big wheel to the interlaced point C2 (ha2 is the height of the midpoint of the midpoint of the large tooth width; R2 is the pitch of the midpoint of the large wheel, R2=r2/sinÎ´2(10) 312 The root cone angle Î´f2 and the distance Gf2 from the apex of the root cone to the staggered point, the root cone angle Î´f2 of the large wheel is Î´f2=Î´2-Î¸f2 (11) where: Î¸f2 is the large root angle.

Big wheel to the root cone and the apex O'2 the cross point C2 is Gf2 and Gf2 = G2 O'2O2 O'2O2 = R2sinÎ´2-hf2cosÎ´2tanÎ´f2-R2cosÎ´2-hf2sinÎ´2 = R2 (sinÎ´2cosÎ´f2-cosÎ´2sinÎ´f2) sinÎ´f2-hf2 (cosÎ´2cosÎ´f2 sinÎ´2sinÎ´f2) sinÎ´f2 = R2sin Î¸f2 - hf2cos Î¸f2sin Î´f2 Therefore Gf2 = G2 R2sin Î¸f2 - hf2cos Î¸f2sin Î´f2 (12) where: hf2 is the root height of the midpoint of the large tooth width.

The top cone angle Î´a1 of the 313 small wheel and the distance Ga1 from the apex of the top cone to the staggered point determine the basic principle of the top cone and the root cone of the small wheel: in order to make the gear pair do not interfere when meshing, without considering the head gap The top cone of the small wheel is tangent to the root cone of the large wheel, and the root cone of the small wheel is tangent to the top cone of the large wheel. Therefore, the top cone of the small wheel and the root cone of the big wheel, the root cone of the small wheel and the top cone of the big wheel can be regarded as two pairs of pitch cones of the hypoid gear pair, and the pitch cone geometric relationship is used to determine the two pairs of knots. The geometric parameters of the cone. When directly studying the two pairs of pitch cones, the position of the nodes is not well determined. For this purpose, we study the equidistant cones with the same cone angle, and assume that the nodes of the big wheel are unchanged, and the radius is still equal to r2, but around the big wheel. The axis is rotated through an angle such that the formed large wheel cone angle is equal to the large wheel cone angle or the large wheel tip cone angle.

G2 Qf2sin(90Â°-Î¸f2)=R2sin(90Â°-Î´f2), ie Qf2=cosÎ¸f2cosÎ´f2R2-G2(13) is obtained from equation (4): tanÎ·f=EQf2sinÎ£(14) is obtained from equation (2): sinÎµâ€²f = sin Î· fsin Î£ / cos Î´ f2 (15) According to the second formula of the formula (3), cos Î£ = cos Î´ a1 cos Î´ f2 cos Îµ 'f - sin Î´ a1 sin Î´ f 2 (16) The small wheel tip cone angle Î´ a1 can be obtained from the equation (16).

According to formula (7), EcosÎ´a1cosÎ´f2sinÎµ'fsinÎ£=Ga1sinÎ´a1 Gf2sinÎ´f2 is obtained. Therefore, Ga1=1sinÎ´a1 (EcosÎ´a1cosÎ´f2sinÎµ'fsinÎ£-Gf2sinÎ´f2) When considering the headspace, Ga1 should also reduce c/sinÎ´a1, that is, the top of the small round cone to the intersection point C1. The distance Ga1 is Ga1=1sinÎ´a1(EcosÎ´a1cosÎ´f2sinÎµ'fsinÎ£-Gf2sinÎ´f2-c) (17) where: c is the headspace. The root cone angle Î´f1 of the 314 small wheel and the distance Gf1 from the apex of the root cone to the staggered point are similarly obtained by G2 Qa2sin(90Â° Î¸a2)=R2sin(90Â°-Î´a2), ie Qa2=cosÎ¸a2cosÎ´a2R2-G2(18) according to the formula ( 4) can obtain tanÎ·a=EQa2sinÎ£(19) according to formula (2), sinÎµâ€²a=sinÎ·asinÎ£/cosÎ´a2(20) can be obtained according to the second formula of formula (3): cosÎ£=cosÎ´f1cosÎ´a2cosÎµâ€²a-sinÎ´f1sinÎ´a2(21) (21) The small cone root angle Î´f1 can be solved.

According to the formula (7), EcosÎ´f1cosÎ´a2sinÎµ'asinÎ£=Gf1sinÎ´f1 Ga2sinÎ´a2 is thus obtained, so Gf1=1sinÎ´f1(EcosÎ´f1cosÎ´a2sinÎµ'asinÎ£-Ga2sinÎ´a2) When considering the headspace, Gf1 should also decrease c/sinÎ´f1, that is, the peak of the small round root cone to the intersection point C1. The distance Gf1 is Gf1=1sinÎ´f1(EcosÎ´f1cosÎ´a2sinÎµ'asinÎ£-Ga2sinÎ´a2-c) (22)315 The distance from the crown to the staggered point of the large wheel Za2 The distance from the crown to the staggered point of the large wheel is Za2=Oâ€²â€²2Mâ€²â€²2cos2a2- Ga2 is Za2=Re2-(G2-Ga2)cosÎ´2cos Î¸a2cosÎ´a2-Ga2(23) where Re2=R2 b2/2(24) where: Re2 is the outer wheel pitch of the large wheel; b2 is the large wheel tooth width.

The distance from the crown of the 316 small wheel to the staggered point Za1 is similarly the distance from the crown of the small wheel to the staggered point Za1 is Za1=Re1-(G1-Ga1)cosÎ´1cosÎ¸a1cosÎ´a1-Ga1(25)Re1=R1 b1/2( 26) R1=r1/sinÎ´1(27) where: Re1 is the small wheel outer taper; R1 is the small wheel midpoint taper; b1 is the small wheel tooth width. 4 Example analysis An aligned hyperboloid gear: the number of small teeth z1=7, the number of large teeth z2=38, the offset distance E=35mm, the angle of intersection Î£=90Â°. It has been determined that the radius of the midpoint of the small wheel is r1=3319231mm, the radius of the midpoint of the large wheel is r2=16515893mm, the cone angle of the small wheel is Î´1=1213758333Â°, the cone angle of the big wheel is Î´2=7713591667Â°, the helix angle of the small wheel midpoint Î’1=45Â°, the midpoint helix angle Î²2=3310593469Â°, the offset angle Îµâ€²=1119406531Â°, the tip clearance c=21021mm, the large gear apex angle Î¸a2=016636146Â°, the large gear root angle Î¸f2=414413744Â° , the large tooth width midpoint height h2=11708531mm, large wheel tooth width midpoint root height hf2=131455399mm, large wheel tooth width b2=45, small wheel tooth width b1=50.

Geometric parameter calculation parameter calculation examples described Remarks Î· / of formula (2) 215950900G2 / mm of formula (5) 312 492 528 Large gear pitch cone apex of the cross point of the wheel -715706545 small distance Îµ / formula (1) 1116592423G1 / mm of formula (6) The distance from the apex of the pitch cone to the staggered point Î´a2/ (8) 7810227813 The top cone angle of the large wheel R2/mm (10)16917027159 The calculation of the midpoint cone distance parameter of the big wheel Gauge case Ga2/mm type (9) 219864511 round nose cone vertex to the cross point distance Î´f2 / formula (11) root angle 7219177923 large wheel Gf2 / mm of formula (12) 219 632 504 large wheel root cone vertex to the cross point distance Qf2 / mm of formula (13) 57217400426Î·f / formula (14)314969820Îµâ€²f/(Â°) Formula (15)1119846965Î´a1/ Formula (16)1617308875 Small wheel top cone angle Ga1/mm Type (17)-917577835 Small wheel top cone apex to the intersection point Qa2/mm type (18)81414506366Î·a/ Formula (19)214607006Îµ'a/ Formula (20)1119400880Î´f1/ Formula (21)1117253356 Root cone angle Gf1/mm Equation (22)-1710804749 Small wheel root cone apex to the intersection point Re2 /mm type (24)19212027159 big wheel outer cone distance Za2/mm type (23) 3618907355 big wheel crown to the intersection The distance of the wrong point Re1/mm type (26) 18112799638 small wheel outer taper Za1/mm type (25) 18317756172 small round wheel crown to the intersection point distance 5 Conclusion (1) hypoid gear pair due to the axis offset Making the geometric relationship complex, understanding the simple expression of this geometric relationship and characterizing this relationship is the key to flexible design of hypoid gear transmission and improve the quality of its design.

(2) The geometric design of the hypoid gear includes the geometric parameters of the gear sub-segment cone and the geometric parameters of the gear pair. The pitch-to-cone geometry of the gear pair is the basis of the geometric design of the hypoid gear. Through the basic equations of the hypoid curve of the hypoid gear in the pitch-cone geometry, the three parameters of the pitch angle and the pitch radius of the large wheel and the helix angle of the small wheel are used as free variables to the hypoid gear pair. The offset angle is an intermediate variable, and the numerical solution method can determine the pitch cone geometry parameters very conveniently and reliably.

(3) Without considering the headspace, the top cone of the small wheel is tangent to the root cone of the big wheel, the root cone of the small wheel is tangent to the top cone of the big wheel, and the top cone and the big wheel of the small wheel The root cone, the root cone of the small wheel and the top cone of the big wheel are regarded as two pairs of pitch cones of the hypoid gear pair. The pitch cone geometry can be used to easily determine the tip cone angle, the root cone angle and the top cone apex of the pinion. The geometrical parameters such as the distance from the apex of the root cone to the point of intersection of the gear's secondary axis, the formula provided in the paper is a simple expression formula for designing and calculating these geometric parameters.

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