CN110966374A - Design method of hypoid gear with large reduction ratio for high-precision robot - Google Patents

Design method of hypoid gear with large reduction ratio for high-precision robot Download PDF

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CN110966374A
CN110966374A CN201911163709.2A CN201911163709A CN110966374A CN 110966374 A CN110966374 A CN 110966374A CN 201911163709 A CN201911163709 A CN 201911163709A CN 110966374 A CN110966374 A CN 110966374A
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gear
tooth
wheel
angle
reduction ratio
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吕传贵
余锡泉
罗建勤
汪慧君
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Jinhua Newsky Gear Co ltd
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Jinhua Newsky Gear Co ltd
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • F16H55/0846—Intersecting-shaft arrangement of the toothed members
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H1/00—Toothed gearings for conveying rotary motion
    • F16H1/02—Toothed gearings for conveying rotary motion without gears having orbital motion
    • F16H1/04—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members
    • F16H1/12—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members with non-parallel axes
    • F16H1/14—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members with non-parallel axes comprising conical gears only
    • F16H1/145—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members with non-parallel axes comprising conical gears only with offset axes, e.g. hypoïd gearings

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  • Mechanical Engineering (AREA)
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Abstract

The invention discloses a design method of a hypoid gear with a large reduction ratio for a high-precision robot, which comprises the following steps of 1) calculating gear design parameters, wherein the calculation is divided into the geometric limitation conditions of the design parameters and the calculation of tooth blank pitch cone parameters, and the geometric limitation of the design parameters mainly comprises the steps of offset E, a helical angle β, a tooth face width b and an addendum height coefficient faLimiting; 2) the tooth surface equation derivation and adjustment parameter calculation of the hypoid gear with less tooth number and large reduction ratio are carried out, the hypoid gear with large reduction ratio is processed by adopting a forming method because the speed ratio is larger than 15, and the hypoid gear with large reduction ratio adopts equal-height teethThe taper forms of the large and small wheels, i.e. the face cone angle and root cone angle are equal to their pitch cone angle, and the crest coefficient of the large wheel is equal to 0. According to the invention, through analysis of gear undercut, contact ratio and gear meshing strength, the small gear adopts a larger helical angle to obtain a larger equivalent gear tooth number, so that undercut caused by a small gear with a small tooth number is avoided.

Description

Design method of hypoid gear with large reduction ratio for high-precision robot
Technical Field
The invention belongs to the field of gear machining, and particularly relates to a design method of a hypoid gear with a large reduction ratio for a high-precision robot.
Background
With the perspective planning of China manufacturing 2025, the industry of domestic industrial robots develops rapidly, bevel gears are used in wrist joints of industrial robots, and in order to reduce the weight of the joints and meet the requirements of transmission ratios, high requirements are put forward for the transmission ratios of gears, such as hypoid gears with the transmission ratio larger than 15 and the transmission ratio of 1:45/2: 60.
The hypoid gear that reduces greatly of domestic robot at present generally adopts the import, and the price is high, the trade is long, and main import source is the big gear that reduces of Japan small original gear company's standard, but does not have the gear grinding high accuracy gear, is difficult to satisfy the installation requirement, and domestic present research to reducing greatly the gear still stays the colleges and universities, and no producer can process the big gear that reduces, also does not form the volume production, consequently seriously restricts the quality promotion and the development of domestic robot.
Therefore, it is necessary to implement a research project for designing a high-precision robot with a large reduction ratio of bevel gears and an advanced manufacturing technology.
Foreign research is earlier and has been adopted in robots, standard large-reduction-ratio gears are available at Japan small primary gear company, but the module is less than 1.5mm and cannot be ground, and the U.S. or Germany has mature technology, but when the device is sold to the outside, design software cannot calculate bevel gears with the tooth number less than 5, so that the bevel gears with the tooth number less than 5 cannot be processed, and the application of the technology in the gear processing in China is also restricted.
Disclosure of Invention
In order to solve the defects and shortcomings in the prior art, the invention provides a design method for obtaining a hypoid gear with a large reduction ratio for a high-precision robot by analyzing gear undercut, contact ratio and gear meshing strength and adopting a larger helical angle for a small gear to obtain a larger equivalent gear tooth number so as to avoid undercut caused by less tooth number of the small gear, and simultaneously providing corresponding limiting conditions for geometric parameters of the hypoid gear with a large reduction ratio.
The technical scheme of the invention is as follows: a design method of a hypoid gear with a large reduction ratio for a high-precision robot comprises the following steps:
1) calculating gear design parameters, namely calculating geometric limitation conditions of the design parameters and tooth blank pitch cone parameters, wherein the geometric limitation of the design parameters is mainly to offset E, helix angle β, tooth face width b and tooth crest height coefficient faLimiting; the gear blank pitch cones refer to the large and small wheel axes of the hypoid gear with large reduction ratio, and the pitch cones are two tangent cones formed by respectively rotating around the two staggered axes;
2) the tooth surface equation derivation and adjustment parameter calculation of the hypoid gear with less tooth number and large reduction ratio are carried out, the hypoid gear with large reduction ratio is processed by a forming method because the speed ratio is larger than 15, the hypoid gear with large reduction ratio adopts a contraction form of equal-height teeth, namely, the cone angle of the large and small gear surfaces and the root cone angle are equal to the pitch cone angle of the large and small gear surfaces, and the tooth crest height coefficient of the large gear is equal to 0.
Preferably, the offset distance is limited, the small wheel offset distance E and the large wheel outer diameter d are limited due to longitudinal sliding when the hypoid gear with high reduction ratio is in meshing transmission2The ratio should be within a proper range, and is generally taken
Figure BDA0002286855880000021
The selection of the spiral angle is related to the undercut of the gear, the contact ratio of the gear pair and the axial force during meshing transmission; the larger the helix angle is, the smaller the possibility of undercut is, and the larger the contact ratio is; coincidence degree epsilon0The number of the meshing teeth at the same moment is reflected, the larger the value of the meshing teeth is, the more stable the transmission of the gear pair is, and the continuous transmission of the gear pair can be ensured only when the contact ratio is more than 1; however, an excessively large helical angle causes an excessively large axial force to be applied to the gear pair during rotation;
according to the limitation condition of the minimum tooth number of the undercut, the equivalent tooth number of the small wheel should not be less than 20 teeth, and considering that the pitch cone angle of the small wheel is smaller than 10 degrees when the transmission ratio is large, the relationship between the equivalent tooth number of the small wheel and the spiral angle can be approximated as follows:
Figure BDA0002286855880000031
the equivalent tooth number of the small wheel can be increased by selecting a larger helical angle for the small wheel, so that the risk of undercut of the small wheel is avoided;
the contact ratio of the hypoid gear can be simplified into the contact ratio of the equivalent gear of the reference point, including the contact ratio epsilon of the end surface of the equivalent gearvαAnd equivalent gear longitudinal overlap ratio epsilonvβTaking the reference point of the bull wheel as the helix angle βm2Reference point reference circle radius r of the bull wheelm2Reference point modulus mnIs composed of
Figure BDA0002286855880000032
Equivalent gear longitudinal contact ratio epsilonvβIs composed of
Figure BDA0002286855880000033
Equivalent gear mesh angle α for Gleason hypoid gearsnNamely the actual pressure angle α of the tooth surface, and the equivalent gear base circle helix angle βvbIs composed of
sinβvb＝sinβm2cosα (5)
Involving equivalent gear face contact ratio epsilonvαIs composed of
Figure BDA0002286855880000041
In the formula, gvαnThe normal equivalent gear meshing line effective length is obtained,
the total contact ratio epsilon of the gear pairvγIs composed of
Figure BDA0002286855880000042
In practical design, the design requirement can be met only if the contact ratio is larger than a preset value, the range of the helical angle of the large wheel meeting the requirement can be deduced by using the formulas (3) to (7), but in order to avoid overlarge axial force, the helical angle of the large wheel is smaller than 40 degrees;
the limit to the tooth crest height coefficient is that in a pair of hypoid gear pairs, the undercut generally occurs at the inner end, the undercut is more likely to occur on a large small wheel, the height reduction ratio is smaller than that of a small wheel in the hypoid gear pair, and in order to avoid the undercut phenomenon at the inner end of the small wheel, the actual tooth crest height of the large wheel is smaller than the limit tooth crest height, so the tooth crest height coefficient of the large wheel is 0;
the tooth surface width is limited, the tooth surface width of the height-reduction ratio hypoid gear pair is related to the tooth bottom groove width and the normal tooth top width, the small gear is processed by a double-face method, the height-reduction ratio hypoid gear is processed by an equal-height tooth shrinkage mode, the tooth bottom normal groove widths of the large gear and the small gear are equal, the tooth normal tooth top width and the tooth shrinkage condition of the tooth are related to the cutter radius and the tooth surface width of the processed gear, and if the tooth surface width is too large, the tooth top width difference between the outer end of the tooth and the inner end normal tooth top width is too large, so that the meshing strength of the gear pair is reduced;
firstly, determining the radius of a cutter for machining a hypoid gear pair with a high reduction ratio, and avoiding abnormal contraction of gear teeth when a root angle of a large gear tooth meets the formula (8);
Figure BDA0002286855880000051
in the formula, s1The thickness of the arc teeth of the small wheel is set; r iscIs the radius of the bull wheel cutter;
because the high-reduction-ratio quasi-curved-surface gear adopts the equal-height teeth, the root angle of the large gear is 0 degree; the tool radius r is determined from equation (8)c＝R2sinβ2When the radius of the cutter meets the condition, the gear teeth can not shrink abnormally;
in order to ensure that the normal tooth crest widths of the inner end and the outer end of the gear tooth are not too different, the tooth surface width needs to be limited; firstly, get the arbitrary cone distance R of the bull wheelx2(Ri2≤Rx2≤Re2，Ri2Is the inner cone pitch of the bull wheel, Re2Large outer cone distance), then Rx2Helix angle β at any distance from the large wheelx2Is composed of
Rx2＝R2-μb2
Figure BDA0002286855880000052
Wherein mu is a coefficient and-0.5 is not less than mu and not more than 0.5; r2The distance between the big wheel pitch and the cone is large; b2The width of the big gear teeth;
normal modulus m of large wheel at any conic distancemnxIs composed of
mmnx＝mncosβx2(10)
The normal tooth top width S of the large wheel at any cone distancen2xIs composed of
Sn2x＝(0.5π-k2)mmnx-(ha1-ha2)tanα (11)
In the formula, ha1、ha2Respectively the top heights of the small wheel and the big wheel; k is a radical of2Coefficient of tooth width [2]]；
Rx2Taking two limit values of the inner cone distance and the outer cone distance to obtain corresponding inner and outer end normal tooth crest widths, and taking the inner end normal tooth crest width larger than 0.5 times of the outer end normal tooth crest width as a limiting condition to obtain a large gear tooth face width b2The value range of (a).
Preferably, the geometrical relationship between the pitch cone parameters is determined using the following equation,
Figure BDA0002286855880000061
Figure BDA0002286855880000062
Figure BDA0002286855880000063
sinδ1＝cosδ2cosεsinΣ-sinδ2cosΣ＝cosδ2cosε
Figure BDA0002286855880000064
Figure BDA0002286855880000065
Figure BDA0002286855880000066
epsilon' is the section of big and small wheel axles1、π2The included angle of (a) is called the offset angle of the hypoid gear;
β1＝β2+ε′
according to the gear meshing principle, when the hypoid gear pair is meshed at the node P, the equation v is satisfied12n is 0. This gives:
Figure BDA0002286855880000071
and (4) obtaining the basic geometric parameters of the hypoid gear with the high subtraction ratio by adopting an initial value and iteration method through the relational expression.
Preferably, the limit pressure angle α when the node P is a type two boundary point is obtained from the hypoid gear characteristics*Radius of curvature r of sum limit method*：
Figure BDA0002286855880000072
Figure BDA0002286855880000073
Radius r of the bull wheel cutter0＝r*；
When the big wheel is processed by a forming method, the relative positions of the cutter head and the big wheel and the pitch-cone distance of the productive wheel are as follows:
Figure BDA0002286855880000074
then the horizontal tool position H and the vertical tool position V for machining the large wheel are as follows:
Figure BDA0002286855880000075
V＝R02cosβf2sinβM+△rcosβM＝R02cosβ2sinβ2+△rcosβ2
wherein:
△r＝r0-R02sinβf2cos△α2+hf2sin△α2＝r0-R02sinβ2cos△α2+h2sin△α2
△α2when the gear is machined by a bull wheel, the main shaft of the cutter is inclined, and if the tooth profile angle of the cutter head is equal to the root cone pressure angle of the bull wheel, △α2＝0；
Thereby obtaining the radial cutter position S of the large wheel processed by the forming method2And angular tool position q2：
Figure BDA0002286855880000081
Figure BDA0002286855880000082
When the big wheel is processed by a forming method, the curvature of the tooth surface of the big wheel is completely the same as that of the tooth surface of the cutter head; namely, it is
Figure BDA0002286855880000083
The small wheel of the hypoid gear with large reduction ratio is processed by a double-faced method because the small wheel of the hypoid gear with large reduction ratio adopts a larger helical angle, and the processing parameters of the small wheel of the hypoid gear with large reduction ratio can be obtained by the method by using the processing parameters of the large wheel.
According to the design method, through analysis of gear undercut, contact ratio and gear meshing strength, a small gear adopts a larger helical angle to obtain a larger equivalent gear tooth number so as to avoid undercut caused by the small gear with less tooth number, and meanwhile, corresponding limiting conditions are provided for geometric parameters of a hypoid gear with a large reduction ratio, so that the hypoid gear with the large reduction ratio for the high-precision robot is obtained.
Drawings
FIG. 1 is a diagram of a method of determining a hypoid gear pitch cone in accordance with the present invention;
FIG. 2 is a graph of the geometrical relationship between pitch cone parameters in the present invention;
FIG. 3 is a mathematical relationship diagram of a wheel formed by the forming method of the present invention;
FIG. 4 is a schematic view of the convex contact area of the bull wheel according to the present invention;
FIG. 5 is a schematic view of the concave contact area of the bull wheel of the present invention;
fig. 6 is a schematic view of a 2:60 high reduction ratio hypoid gear made by an embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the attached drawings, but the present invention is not limited thereto.
1. Calculating gear design parameters:
1.1 geometric constraints of design parameters
Before designing the hypoid gear, the shaft angle and the number of teeth z of the small gear should be determined according to the transmission requirement1Number of teeth of large gear z2Offset distance E and external diameter d of bull wheel2British umbrella and hypoid gear precision standard (BS ISO 23509: 2016) alignment hypoid gear tooth number has the following stipulations that the tooth number of a small gear is more than 5, the sum of the tooth numbers of the small gear and a large gear is not less than 40, and the tooth number of a hypoid secondary small gear with high reduction ratio is less than 5 teeth, so that the offset distance E, the helical angle β, the tooth surface width b and the tooth top height coefficient f are respectively considered comprehensively from multiple aspects such as contact ratio, meshing strength, meshing interference and the likeaA restriction is made.
(1) And (4) offset distance. Due to high reduction ratio quasi-dualThe gear with curved surface has longitudinal sliding, small wheel offset E and large wheel outer diameter d2The ratio should be within a proper range, and is generally taken
Figure BDA0002286855880000101
(2) A helix angle. The selection of the spiral angle is related to the undercut of the gear, the contact ratio of the gear pair and the axial force during meshing transmission. The larger the helix angle, the less likely an undercut and the greater the overlap. Coincidence degree epsilon0The number of the meshing teeth at the same moment is reflected, the larger the value of the meshing teeth is, the more stable the transmission of the gear pair is, and the continuous transmission of the gear pair can be ensured only when the contact ratio is greater than 1. However, an excessive helix angle will result in an excessive axial force on the gear pair during rotation.
The equivalent tooth number of the small wheel should generally not be less than 20 teeth, subject to the undercut minimum tooth number constraint. Considering that the pitch cone angle of the small wheel is small (<10 °) when the transmission is large, the relationship of the equivalent number of teeth of the small wheel to the helix angle can be approximated by:
Figure BDA0002286855880000102
from the above formula, it can be known that the equivalent number of teeth of the small wheel can be increased by selecting a larger helix angle for the small wheel, thereby avoiding the risk of undercut of the small wheel.
The contact ratio of the hypoid gear can be simplified into the contact ratio of the equivalent gear of the reference point, including the contact ratio epsilon of the end surface of the equivalent gearvαAnd equivalent gear longitudinal overlap ratio epsilonvβTake the helix angle β of the bull wheel reference pointm2Reference point reference circle radius r of the bull wheelm2Reference point modulus mnIs composed of
Figure BDA0002286855880000103
Equivalent gear longitudinal contact ratio epsilonvβIs composed of
Figure BDA0002286855880000104
Equivalent gear mesh angle α for Gleason hypoid gearsnI.e. the actual pressure angle α of the tooth surface, the equivalent gear base circle helix angle βvbIs composed of
sinβvb＝sinβm2cosα (5)
Involving equivalent gear face contact ratio epsilonvαIs composed of
Figure BDA0002286855880000111
In the formula, gvαnThe normal equivalent gear meshing line effective length is obtained,
the total contact ratio epsilon of the gear pairvγIs composed of
Figure BDA0002286855880000112
In actual design, the design requirement can be satisfied only if the overlap ratio is larger than a predetermined value, and the range of the large wheel helix angle that satisfies the requirement can be derived by the expressions (3) to (7). But to avoid excessive axial forces, the large wheel helix angle should be less than 40 °.
(3) Crest coefficient of tooth. In a hypoid gear pair, undercutting typically occurs at the inner end, and undercutting is more likely to occur with smaller and larger wheels. The small wheel in the hypoid gear pair with high reduction ratio is very small, and in order to avoid the undercut phenomenon at the inner end of the small wheel, the actual tooth top height of the large wheel is smaller than the limit tooth top height, so the tooth top height coefficient of the large wheel is 0.
(4) The tooth surface is wide. The face width of hypoid gear pair with high reduction ratio is related to the width of tooth bottom slot and normal tooth top width. Because the small wheel is processed by a double-face method, the hypoid gear with high reduction ratio adopts an equal-height tooth contraction mode, and the normal groove widths of the tooth bottoms of the large wheel and the small wheel are equal. The size of the tooth normal tooth top width and the tooth shrinkage are related to the radius of a cutter for machining the gear and the tooth face width. If the tooth surface width is too large, the difference between the tooth top widths of the outer end and the inner end of the gear tooth in the normal direction is too large, and the meshing strength of the gear pair is reduced.
First, the radius of the cutter for machining the hypoid gear pair with high reduction ratio is determined. As can be seen from document [2], abnormal contraction of the gear teeth can be avoided when the root angle of the large gear teeth satisfies the formula (8).
Figure BDA0002286855880000121
In the formula, s1The thickness of the arc teeth of the small wheel is set; r iscIs the radius of the bull wheel cutter.
Because the high-reduction-ratio quasi-curved-surface gear adopts the equal-height teeth, the root angle of the large gear is 0 degree. The tool radius r is determined from equation (8)c＝R2sinβ2When the radius of the cutter meets the condition, the gear teeth can not shrink abnormally.
In order to ensure that the normal tooth crest widths of the inner end and the outer end of the gear tooth are not too different, the tooth flank width needs to be limited. Firstly, get the arbitrary cone distance R of the bull wheelx2(Ri2≤Rx2≤Re2，Ri2Is the inner cone pitch of the bull wheel, Re2Large outer cone distance), then Rx2Helix angle β at any distance from the large wheelx2Is composed of
Rx2＝R2-μb2
Figure BDA0002286855880000122
Wherein mu is a coefficient and-0.5 is not less than mu and not more than 0.5; r2The distance between the big wheel pitch and the cone is large; b2The width of the big gear teeth.
Normal modulus m of large wheel at any conic distancemnxIs composed of
mmnx＝mncosβx2(10)
The normal tooth top width S of the large wheel at any cone distancen2xIs composed of
Sn2x＝(0.5π-k2)mmnx-(ha1-ha2)tanα (11)
In the formula, ha1、ha2Respectively the top heights of the small wheel and the big wheel; k is a radical of2Coefficient of tooth width [2]]。
Rx2Taking two limit values of the inner cone distance and the outer cone distance to obtain corresponding inner and outer end normal tooth crest widths, and taking the inner end normal tooth crest width larger than 0.5 times of the outer end normal tooth crest width as a limiting condition to obtain a large gear tooth face width b2The value range of (a).
1.2 calculation of pitch cone parameters of tooth blank
The axes of the large wheel and the small wheel of the hypoid gear with large reduction ratio are staggered, and the pitch cones of the hypoid gear are two tangent cones respectively formed by rotating around the staggered axes (shown in figure 1). P is the tangent point of the two cones, namely the node of the gear pair.
To determine the geometric relationship between the nodal cone parameters, we convert FIG. 1 to the form of FIG. 2, and from this we can derive the following relationship.
Figure BDA0002286855880000131
Figure BDA0002286855880000132
Figure BDA0002286855880000133
sinδ1＝cosδ2cosεsinΣ-sinδ2cosΣ＝cosδ2cosε
Figure BDA0002286855880000134
Figure BDA0002286855880000135
Figure BDA0002286855880000141
Epsilon' is the section of big and small wheel axles1、π2Is referred to as the offset angle of the hypoid gear.
β1＝β2+ε′
According to the gear meshing principle, when the hypoid gear pair is meshed at the node P, the equation v is satisfied12n is 0. This gives:
Figure BDA0002286855880000142
the basic geometric parameters of the hypoid gear with the high reduction ratio can be obtained by the relation and by adopting an initial value and iteration method.
2. And (3) deducing a tooth surface equation of the hypoid gear with less tooth number and large reduction ratio and calculating adjustment parameters:
the hypoid gear with large reduction ratio is greater than 15, and the large wheel is machined by a forming method. According to our previous derivation, hypoid gears with a large reduction ratio take the form of a contraction of teeth of equal height, i.e. the face cone angle and root cone angle of the large and small wheels are equal to their pitch cone angle, and the crest height factor of the large wheel is equal to 0.
Reference [4] according to hypoid gear characteristics]The ultimate pressure angle α for the node P being the class II boundary can be obtained*Radius of curvature r of sum limit method*：
Figure BDA0002286855880000143
Figure BDA0002286855880000144
Radius r of the bull wheel cutter0＝r*。
When the large wheel is processed by the forming method, the relative positions of the cutter head and the large wheel are shown in figure 3, and the pitch-cone distance of the productive wheel is as follows:
Figure BDA0002286855880000151
then the horizontal tool position H and the vertical tool position V for machining the large wheel are as follows:
Figure BDA0002286855880000152
V＝R02cosβf2sinβM+△rcosβM＝R02cosβ2sinβ2+△rcosβ2
wherein:
△r＝r0-R02sinβf2cos△α2+hf2sin△α2＝r0-R02sinβ2cos△α2+h2sin△α2
△α2when the gear angle of the cutter head is equal to the root cone pressure angle of the bull wheel, △α2＝0。
Thereby obtaining the radial cutter position S of the large wheel processed by the forming method2And angular tool position q2：
Figure BDA0002286855880000153
Figure BDA0002286855880000154
When the big wheel is processed by the forming method, the curvature of the tooth surface of the big wheel is completely the same as that of the tooth surface of the cutter head. Namely, it is
Figure BDA0002286855880000155
Because the hypoid gear small wheel with large reduction ratio adopts a larger helical angle, the small wheel is processed by a double-faced method. In document [5], a method of machining hypoid gear small wheels by a helical double-face method is described in detail. By using the processing parameters of the large wheel, the small wheel processing parameters of the hypoid gear with large reduction ratio can be obtained by the method.
3. Gear design examples and machining experiments:
according to the requirements of customers, a pair of hypoid gears with high reduction ratio is designed, and the number of teeth Z1＝2，Z260, outside diameter D of bull wheel265, the small wheel offset E is 16. From the 1.1 section constraint and 1.2 section tooth blank pitch cone parameter calculations, the geometric parameters of a high reduction ratio hypoid gear pair with a gear ratio of 2:60 can be obtained, as shown in table 1.
TABLE 1 Gear geometry parameters
Figure BDA0002286855880000161
Figure BDA0002286855880000171
According to the adjustment parameter calculation of 2 sections, a large wheel machining adjustment card of the gear pair can be obtained and is shown in a table 2, and a small wheel machining adjustment card is shown in a table 3.
TABLE 2 big wheel processing adjusting clamp
Diameter D of cutter head2/mm 36.36
Outer cutter tooth form angle αo2/(°) 20
Inner cutter tooth form angle αi2/(°) 20
Width W of tool tip2 0.40
V/mm vertical tool position 14.102
Horizontal tool position H/mm 17.12
Radial tool position S2/mm 22.18
Angular tool position q2/(°) 39.479
Horizontal wheel position X2/mm 0.248
Machine tool mounting angle deltam2/(°) 80.81
TABLE 3 Small wheel processing and adjusting clamp
Figure BDA0002286855880000172
Figure BDA0002286855880000181
The contact area of the gear design is shown in fig. 4 and 5.
The hypoid gear with large reduction ratio has large speed ratio and small gear number (less than 5 teeth), and can not be designed and manufactured because the traditional bevel gear design method can generate undercut. According to the method, through analysis of gear undercut, contact ratio and gear meshing strength, a small gear adopts a larger helical angle to obtain a larger equivalent gear tooth number so as to avoid undercut caused by less tooth number of the small gear, and meanwhile, corresponding limiting conditions are provided for geometric parameters of the hypoid gear with a large reduction ratio, so that the design calculation method of the hypoid gear with the large reduction ratio is obtained. Using a pair of gear examples in a robot, the method is adopted to carry out design calculation, and the pair of hypoid gears with large reduction ratio is processed by using the calculation result (as shown in figure 6).
The references of the present invention are as follows:
[1] the technical overview of the manufacturing technology of the small module spiral bevel gear in China [ J ]. mechanical transmission.40 (12);
[2] wen Bingyang, high reduction ratio hypoid gear design and gear cutting experiment [ J ]. the university of Henan science and technology newspaper (Nature science edition). 2015: 6;
[3] zhengchang-start spiral bevel gear and hypoid gear [ M ]. Beijing, mechanical industry Press, 1988;
[4] ontao. helical bevel gear design and manufacture [ M ]. Harbin industry university Press, 1989;
[5] and the spiral double-faced method is used for processing the cutting teeth of the small-modulus hypoid bevel gear.

Claims (4)

1. A design method of a hypoid gear with a large reduction ratio for a high-precision robot is characterized by comprising the following steps: the method comprises the following steps:
1) calculating gear design parameters, namely calculating geometric limitation conditions of the design parameters and tooth blank pitch cone parameters, wherein the geometric limitation of the design parameters is mainly to offset E, helix angle β, tooth face width b and tooth crest height coefficient faLimiting; the gear blank pitch cones refer to the large and small wheel axes of the hypoid gear with large reduction ratio, and the pitch cones are two tangent cones formed by respectively rotating around the two staggered axes;
2) the tooth surface equation derivation and adjustment parameter calculation of the hypoid gear with less tooth number and large reduction ratio are carried out, the hypoid gear with large reduction ratio is processed by a forming method because the speed ratio is larger than 15, the hypoid gear with large reduction ratio adopts a contraction form of equal-height teeth, namely, the cone angle of the large and small gear surfaces and the root cone angle are equal to the pitch cone angle of the large and small gear surfaces, and the tooth crest height coefficient of the large gear is equal to 0.
2. The design method of the hypoid gear with the large reduction ratio for the high-precision robot as claimed in claim 1, wherein the method comprises the following steps:
wherein the limitation of offset distance, small wheel deviation due to longitudinal sliding in the meshing transmission of the hypoid gear with high reduction ratioDistance E and external diameter d of large wheel2The ratio should be within a proper range, and is generally taken
Figure FDA0002286855870000011
The selection of the spiral angle is related to the undercut of the gear, the contact ratio of the gear pair and the axial force during meshing transmission; the larger the helix angle is, the smaller the possibility of undercut is, and the larger the contact ratio is; coincidence degree epsilon0The number of the meshing teeth at the same moment is reflected, the larger the value of the meshing teeth is, the more stable the transmission of the gear pair is, and the continuous transmission of the gear pair can be ensured only when the contact ratio is more than 1; however, an excessively large helical angle causes an excessively large axial force to be applied to the gear pair during rotation;
according to the limitation condition of the minimum tooth number of the undercut, the equivalent tooth number of the small wheel should not be less than 20 teeth, and considering that the pitch cone angle of the small wheel is smaller than 10 degrees when the transmission ratio is large, the relationship between the equivalent tooth number of the small wheel and the spiral angle can be approximated as follows:
Figure FDA0002286855870000021
the equivalent tooth number of the small wheel can be increased by selecting a larger helical angle for the small wheel, so that the risk of undercut of the small wheel is avoided;
the contact ratio of the hypoid gear can be simplified into the contact ratio of the equivalent gear of the reference point, including the contact ratio epsilon of the end surface of the equivalent gearvαAnd equivalent gear longitudinal overlap ratio epsilonvβTaking the reference point of the bull wheel as the helix angle βm2Reference point reference circle radius r of the bull wheelm2Reference point modulus mnIs composed of
Figure FDA0002286855870000022
Equivalent gear longitudinal contact ratio epsilonvβIs composed of
Figure FDA0002286855870000023
Equivalent gear mesh angle α for Gleason hypoid gearsnNamely the actual pressure angle α of the tooth surface, and the equivalent gear base circle helix angle βvbIs composed of
sinβvb＝sinβm2cosα (5)
Involving equivalent gear face contact ratio epsilonvαIs composed of
Figure FDA0002286855870000031
In the formula, gvαnThe normal equivalent gear meshing line effective length is obtained,
the total contact ratio epsilon of the gear pairvγIs composed of
Figure FDA0002286855870000032
In practical design, the design requirement can be met only if the contact ratio is larger than a preset value, the range of the helical angle of the large wheel meeting the requirement can be deduced by using the formulas (3) to (7), but in order to avoid overlarge axial force, the helical angle of the large wheel is smaller than 40 degrees;
the limit to the tooth crest height coefficient is that in a pair of hypoid gear pairs, the undercut generally occurs at the inner end, the undercut is more likely to occur on a large small wheel, the height reduction ratio is smaller than that of a small wheel in the hypoid gear pair, and in order to avoid the undercut phenomenon at the inner end of the small wheel, the actual tooth crest height of the large wheel is smaller than the limit tooth crest height, so the tooth crest height coefficient of the large wheel is 0;
the tooth surface width is limited, the tooth surface width of the height-reduction ratio hypoid gear pair is related to the tooth bottom groove width and the normal tooth top width, the small gear is processed by a double-face method, the height-reduction ratio hypoid gear is processed by an equal-height tooth shrinkage mode, the tooth bottom normal groove widths of the large gear and the small gear are equal, the tooth normal tooth top width and the tooth shrinkage condition of the tooth are related to the cutter radius and the tooth surface width of the processed gear, and if the tooth surface width is too large, the tooth top width difference between the outer end of the tooth and the inner end normal tooth top width is too large, so that the meshing strength of the gear pair is reduced;
firstly, determining the radius of a cutter for machining a hypoid gear pair with a high reduction ratio, and avoiding abnormal contraction of gear teeth when a root angle of a large gear tooth meets the formula (8);
Figure FDA0002286855870000041
in the formula, s1The thickness of the arc teeth of the small wheel is set; r iscIs the radius of the bull wheel cutter;
because the high-reduction-ratio quasi-curved-surface gear adopts the equal-height teeth, the root angle of the large gear is 0 degree; the tool radius r is determined from equation (8)c＝R2sinβ2When the radius of the cutter meets the condition, the gear teeth can not shrink abnormally;
in order to ensure that the normal tooth crest widths of the inner end and the outer end of the gear tooth are not too different, the tooth surface width needs to be limited; firstly, get the arbitrary cone distance R of the bull wheelx2(Ri2≤Rx2≤Re2，Ri2Is the inner cone pitch of the bull wheel, Re2Large outer cone distance), then Rx2Helix angle β at any distance from the large wheelx2Is composed of
Rx2＝R2-μb2
Figure FDA0002286855870000042
Wherein mu is a coefficient and-0.5 is not less than mu and not more than 0.5; r2The distance between the big wheel pitch and the cone is large; b2The width of the big gear teeth;
normal modulus m of large wheel at any conic distancemnxIs composed of
mmnx＝mncosβx2(10)
The normal tooth top width S of the large wheel at any cone distancen2xIs composed of
Sn2x＝(0.5π-k2)mmnx-(ha1-ha2)tanα (11)
In the formula, ha1、ha2Respectively the top heights of the small wheel and the big wheel; k is a radical of2Coefficient of tooth width[2]；
Rx2Taking two limit values of the inner cone distance and the outer cone distance to obtain corresponding inner and outer end normal tooth crest widths, and taking the inner end normal tooth crest width larger than 0.5 times of the outer end normal tooth crest width as a limiting condition to obtain a large gear tooth face width b2The value range of (a).
3. The design method of the hypoid gear with the large reduction ratio for the high-precision robot as claimed in claim 1, wherein the method comprises the following steps: the geometrical relationship between the pitch cone parameters is determined using the following equation,
Figure FDA0002286855870000051
Figure FDA0002286855870000052
Figure FDA0002286855870000053
sinδ1＝cosδ2cosεsinΣ-sinδ2cosΣ＝cosδ2cosε
Figure FDA0002286855870000054
Figure FDA0002286855870000055
Figure FDA0002286855870000056
epsilon' is the section of big and small wheel axles1、π2The included angle of (a) is called the offset angle of the hypoid gear;
β1＝β2+ε′
according to the gear meshing principle, the hypoid gear pair is in the jointWhen the points P are engaged, the equation v is satisfied12n is 0. This gives:
Figure FDA0002286855870000061
and (4) obtaining the basic geometric parameters of the hypoid gear with the high subtraction ratio by adopting an initial value and iteration method through the relational expression.
4. The design method of hypoid gear with large reduction ratio for high-precision robot as claimed in claim 1, wherein the limit pressure angle α when the node P is a class II boundary point is obtained according to the hypoid gear characteristics*Radius of curvature r of sum limit method*：
Figure FDA0002286855870000062
Figure FDA0002286855870000063
Radius r of the bull wheel cutter0＝r*；
When the big wheel is processed by a forming method, the relative positions of the cutter head and the big wheel and the pitch-cone distance of the productive wheel are as follows:
Figure FDA0002286855870000064
then the horizontal tool position H and the vertical tool position V for machining the large wheel are as follows:
Figure FDA0002286855870000065
V＝R02cosβf2sinβM+△rcosβM＝R02cosβ2sinβ2+△rcosβ2
wherein:
△r＝r0-R02sinβf2cos△α2+hf2sin△α2＝r0-R02sinβ2cos△α2+h2sin△α2
△α2when the gear is machined by a bull wheel, the main shaft of the cutter is inclined, and if the tooth profile angle of the cutter head is equal to the root cone pressure angle of the bull wheel, △α2＝0；
Thereby obtaining the radial cutter position S of the large wheel processed by the forming method2And angular tool position q2：
Figure FDA0002286855870000071
Figure FDA0002286855870000072
When the big wheel is processed by a forming method, the curvature of the tooth surface of the big wheel is completely the same as that of the tooth surface of the cutter head; namely, it is
Figure FDA0002286855870000073
Bf2＝0 Cf2＝0
The small wheel of the hypoid gear with large reduction ratio is processed by a double-faced method because the small wheel of the hypoid gear with large reduction ratio adopts a larger helical angle, and the processing parameters of the small wheel of the hypoid gear with large reduction ratio can be obtained by the method by using the processing parameters of the large wheel.
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