CN108533684B - Convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission - Google Patents

Convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission Download PDF

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CN108533684B
CN108533684B CN201810603089.9A CN201810603089A CN108533684B CN 108533684 B CN108533684 B CN 108533684B CN 201810603089 A CN201810603089 A CN 201810603089A CN 108533684 B CN108533684 B CN 108533684B
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wheel
spiral
meshing
small wheel
small
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CN108533684A (en
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陈祯
丁华锋
曾鸣
杨静
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China University of Geosciences
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China University of Geosciences
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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
    • 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
    • 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
    • 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
    • F16H55/17—Toothed wheels
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/30—Circuit design
    • G06F30/36—Circuit design at the analogue level
    • G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
    • 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
    • F16H2055/0866—Profiles for improving radial engagement of gears, e.g. chamfers on the tips of the teeth

Abstract

The invention provides a convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission, which comprises a pair of transmission pairs consisting of a small wheel and a large wheel, wherein the small wheel is fixedly connected with a driver through an input shaft, the large wheel is connected with an output shaft, the axis of the small wheel is crossed with the axis of the large wheel, and n is arranged on the outer surface of a cone of the small wheel1The convex spiral arc teeth are uniformly distributed, the central lines of all the spiral arc teeth are equal-lift-distance conical spiral lines, a transition fillet is arranged between each spiral arc tooth and the outer surface of the small wheel cone, and n is arranged on the outer surface of the large wheel cone2The spiral trapezoidal grooves are evenly distributed, the central lines of all the spiral trapezoidal grooves are equal-lift-distance conical spiral lines, transition fillets are arranged between the spiral trapezoidal grooves and the outer surface of the cone of the large wheel, and the spiral circular arc teeth of the small wheel and the spiral trapezoidal grooves of the large wheel are in point-contact pure rolling meshing transmission. The invention has the beneficial effects that: the transmission efficiency is high, the contact ratio is large, the bearing capacity is strong, and the lubricating oil can be widely applied to the fields of micro machines and conventional machines which are difficult to lubricate.

Description

Convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission
Technical Field
the invention relates to a bevel gear transmission device, in particular to a convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission.
Background
The gear is used as a basic component of a mechanical core, is widely applied to the field of equipment manufacturing industries such as machine tools, automobiles, robots, wind power, coal mines, aerospace and the like and national economy main battlefield, and the quality of the performance directly determines the quality, performance and reliability of major equipment and high-end industrial products.
The main problem faced by the gear industry in China at present is that the design and manufacturing capability of high-performance gear products with high efficiency, large bearing capacity, light weight and high reliability is obviously insufficient. The traditional straight gear, helical gear and bevel gear pair widely applied in the field of industrial production and manufacturing never thoroughly solve the problems of transmission failures such as friction wear, gluing, plastic deformation and the like caused by relative sliding of tooth surfaces, seriously affect the transmission efficiency, service life and reliability of gear products, particularly high-speed heavy-duty gears, and restrict the performance improvement of high-precision mechanical equipment. A common way to reduce tooth surface wear is to use lubricants such as lubricating oils, greases, etc., but these lubricants can fail in certain extreme environments, such as high temperature, low temperature, high pressure, high radiation, etc. Moreover, the gear lubrication system provided for improving the wear of the tooth surfaces increases the overall cost and weight of the machine, and the emission of lubricating oil and grease also causes environmental pollution. The development trend of modern equipment manufacturing industry 'lightweight, modularization and intellectualization' puts higher requirements on gear transmission performance, weight, volume and green gear design and manufacture. How to realize the green design and manufacture of a high-performance gear mechanism with resource saving and environmental friendliness, reduce or avoid transmission failure caused by relative sliding of tooth surfaces, and further improve the transmission efficiency and the bearing capacity is one of the important and urgent problems in the field of gear research at present.
The design of the pure rolling meshing tooth surface has great significance for gear transmission, particularly high-speed, heavy-load and precise gear transmission, and can effectively reduce or even eliminate relative sliding between the tooth surfaces, so that the transmission failures such as tooth surface friction abrasion, gluing, plastic deformation and the like caused by the relative sliding can be effectively controlled, the friction loss between the tooth surfaces of the high-speed gear can be reduced, heat and vibration are reduced, the gear transmission service life can be prolonged, the transmission efficiency is improved, the transmission precision and stability are ensured, the tooth surface meshing performance is better, and the gear system has a great positive effect on improving the comprehensive performance of a gear pair and a gear train.
At present, the transmission of motion and power between two crossed shafts in a plane is the involute bevel gear mechanism which is most widely applied in industry. However, the meshing principle of the involute bevel gear mechanism follows the curved surface meshing theory, and relative sliding between tooth surfaces inevitably exists in the design theory, so that common failure modes of gear transmission such as tooth surface abrasion, tooth surface gluing and tooth surface plastic deformation cannot be avoided, and the service life and reliability of a gear pair are influenced.
in recent years, a novel micro transmission mechanism with original characteristics is innovated in the field of gear meshing theory at home and abroad. As in chinese patent document, application No. 201510054843.4 discloses "a helical circular-arc gear mechanism for parallel-axis external meshing transmission", and application No. 201510051923.4 discloses "a helical circular-arc gear mechanism for parallel-axis internal meshing transmission". The two transmission mechanisms are limited in that the design methods of the two transmission mechanisms are based on a space curve meshing theory, the meshing tooth surface is calculated and solved by a curve meshing equation, the meshing mode is a convex-flat meshing mode, the meshing point is located at the edge of the tooth profile of the concave tooth, excessive local stress can be generated due to edge contact during transmission, the tooth crest of the concave tooth is easy to break to cause transmission failure, and the two transmission mechanisms cannot be used for conventional power and high-speed heavy-load transmission in industrial production. In addition, the design methods of the two mechanisms cannot realize strict design of the contact ratio, so that the contact ratio value of the transmission pair is uncertain, and the uniform distribution of the load is not facilitated. Moreover, they can only realize the motion and power transmission between two parallel axes in a plane, but cannot realize the motion and power transmission between two orthogonal axes in the plane. Therefore, their range of use is greatly limited. Chinese patent document, application number 201310049845.5, discloses a bevel gear meshing pair based on conjugate curves, comprising a bevel gear I and a bevel gear II which are meshed with each other at points and have circular-arc tooth profile curves, and the bevel gear mechanism has high transmission efficiency; the tooth surface is easy to process and manufacture, the transmission error is small, and the service life is long; however, in the bevel gear, the tooth surfaces move along a conjugate curve when the bevel gear I and the bevel gear II are meshed, so that relative sliding exists between the tooth surfaces, and the tooth surfaces have failure modes such as gluing, abrasion, plastic deformation and the like.
Disclosure of Invention
The invention aims to solve the problems in the prior art in the field of mechanical transmission, provides a convex-flat meshing pure rolling spiral bevel gear mechanism for cross shaft transmission at any angle of a plane, and has the advantages of simple design, easiness in processing, no relative sliding between tooth surfaces during transmission, high transmission efficiency, predefined design of contact ratio, strong bearing capacity and the like, and can be widely applied to the fields of micro machinery and conventional machinery which are difficult to lubricate.
In order to achieve the purpose, the technical measures adopted by the invention are as follows: the utility model provides a be used for crossing driven protruding-flat meshing pure spiral bevel gear mechanism that rolls, constitute a pair of transmission pair including steamboat and bull wheel, the steamboat links firmly with the driver through the input shaft, and the output shaft is connected to the bull wheel, and the axis of steamboat and bull wheel is alternately its characterized in that: convex spiral arc teeth are uniformly distributed on the outer surface of the small wheel cone, spiral trapezoidal grooves are uniformly distributed on the outer surface of the large wheel cone, the central line of the spiral arc teeth and the reference line of the spiral trapezoidal grooves are both equal-lift-distance conical spiral lines, and the spiral arc teeth of the small wheel are matched with the spiral trapezoidal grooves of the large wheel; a transition fillet is arranged between the spiral arc tooth of the small wheel and the outer surface of the cone of the small wheel so as to reduce the stress concentration of the tooth root; the meshing mode of the spiral arc teeth and the spiral trapezoidal grooves is pure rolling meshing transmission of point contact, the small wheel rotates under the driving of the driver, stable meshing transmission between the crossed shafts is realized through the continuous meshing action between the spiral arc teeth and the spiral trapezoidal grooves, all meshing points are positioned on the tangent line of a theoretical indexing cone of the small wheel and the large wheel, the relative movement speed of all the meshing points is zero, and the contact lines of the meshing points respectively formed on the small wheel and the large wheel are equal-lift-distance conical spiral lines;
the structure of the spiral arc tooth and the shape of the central line thereof, the structure of the spiral trapezoidal groove and the shape of the reference line thereof are determined by the following method: at o- -x, y, z, ok--xk,yk,zkand op--xp,yp,zpIn three space coordinate systems, the z axis is coincident with the rotation axis of the small wheel, and z ispThe axis of rotation of the shaft and the bull wheel coinciding, zkThe axis coincides with the line of engagement of the small and large wheels, and the z-axis coincides with the z-axisp、zkThe axes intersect at a point; coordinate system o1--x1,y1,z1fixedly connected to the small wheel, coordinate system o2--x2,y2,z2fixedly connected with the big wheel, the small wheel and the big wheel are respectively connected with the coordinate system o-x, y, z and o at the initial positionsp--xp,yp,zpCoincidence, ookA distance R1，opokA distance R2，zkThe acute angle between the axis and the z-axis is delta1，zkAxis and zpthe acute angle included by the shaft is delta2With small wheels at uniform angular speedDegree omega1Rotating about the z-axis, the bull wheel at a uniform angular velocity ω2around zpThe axes are rotated, the angular velocity vector included angle of the rotation axes of the small wheel and the large wheel is theta, and after a period of time from the initial position, the coordinate system o1--x1,y1,z1and o2--x2,y2,z2Move respectively, at the meshing point M, the small wheel rotates around the axis zcorner, large wheel winding zpthe shaft rotates throughAn angle;
When the small wheel and the large wheel are in mesh transmission, the mesh point M is from the coordinate origin okstarting to move linearly at a constant speed along the meshing line k-k, and defining a parameter equation of M point motion as follows:
In the formula (1), t is a motion parameter variable of the meshing point M, and t is more than or equal to 0 and less than or equal to Δ t; c. C1The undetermined coefficient of the meshing point movement is expressed in millimeters (mm); in order to ensure pure rolling engagement of the small and large wheels, the rotation angle of the small and large wheels and the movement of the engagement point must be in a linear relationship, which is as follows:
In the formula (2), k is a linear proportionality coefficient of the movement of the meshing point, and the unit is radian (rad); i.e. i12the transmission ratio between the small wheel and the large wheel is set;
When the meshing point M moves along the meshing line k-k, the point M simultaneously forms contact lines C on the surfaces of the small wheel and the large wheel respectively1and C2(ii) a According to the coordinate transformation, the coordinate system o-x, y, z, o can be obtainedk--xk,yk,zk、op--xp,yp,zp、o1--x1,y1,z1And o2--x2,y2,z2The homogeneous coordinate transformation matrix in between is:
Wherein:
obtaining:
from the homogeneous coordinate transformation, equation (6) yields:
Calculating the contact line C on the tooth surface of the small wheel from the formula (8)1the pitch-equaling conical spiral line has the parameter equation:
The following equation (2) is taken into equation (9):
In the formula (10), T is an angle parameter variable of the conical spiral line with equal lift distance, wherein the T is kt, and the T is more than or equal to 0 and less than or equal to delta T;
From the homogeneous coordinate transformation, equation (7) yields:
Obtaining a contact line C on the tooth surface of the bull gear from the formula (11)2the pitch-equaling conical spiral line has the parameter equation:
The following equation (2) is taken into equation (12):
and the transmission ratio of the small wheel to the large wheel is as follows:
obtained by substituting formula (14) for formula (13):
the theoretical reference cone angles of the small wheel and the large wheel are respectively delta1and delta2Their relationship is:
The convex tooth surface of the helical arc tooth of the small wheel is in a shape of a section L consisting of an axial arc tooth profile containing a meshing point M1generated by right-handed helical motion, of circular-arc-tooth-shaped cross-section L1Is a generating bus of a small wheel tooth surface, and the axial pitch parameter and the contact line C of the spiral motion of the generating bus1The parameters of the axial screw pitches are consistent, and the right-handed screw motion track of the meshing point M and the contact line C are ensured1Overlapping; in a coordinate system o-x, y and z, a generatrix parameter equation of the convex tooth surface of the small wheel is as follows:
Deducing and obtaining convex tooth surface of helical circular arc tooth of small wheel in coordinate system o by right-handed helical motion1–x1,y1,z1The parameter equation is:
at the moment, the equation of the central line of the convex tooth surface of the spiral circular arc tooth of the small wheel is as follows:
the concave tooth surface of the spiral trapezoidal groove of the bull wheel is formed by an axial isosceles trapezoid section L containing a meshing point M2Generated by left-handed helical motion, isosceles trapezoid truncated L2The profile generating bus is a profile generating bus of a concave tooth surface of a bull wheel, the section height of the bull wheel in the axial direction of an isosceles trapezoid is h, the distance from an meshing line of an M point passing through the isosceles trapezoid to a big bottom of the trapezoid is h', an acute angle formed by the waist of the isosceles trapezoid and the big bottom of the trapezoid is pi/2-gamma, the distance from the meshing line of the M point passing through the isosceles trapezoid to the intersection point of the two waists of the isosceles trapezoid is w, and the profile generating bus meet:
In the formula (18), e is a gap between the large wheel and the small wheel cylinder;
Axial isosceles trapezoid section L of spiral trapezoid groove2is noted as theta2，Θ2Is the middle point of the line connecting the meshing line of the M points and the intersection point of the two waists of the isosceles trapezoid, theta2Isosceles trapezoid section L following spiral trapezoid groove2The parameter equation of the reference line formed by the spiral motion is as follows:
Therefore, the axial pitch parameter and the contact line C of the generating generatrix spiral motion of the big gear tooth surface2axial screw ofThe distance parameters are consistent, and the left-handed spiral motion track of the meshing point M and the contact line C are ensured2Overlapping;
The length of the meshing line of the small wheel and the large wheel is as follows:
The axial height of the small wheel is as follows:
△z1＝△zk sinδ2 (22)
The height of the axis of the bull wheel is:
△z2＝△zk sinδ1 (23)
The cone clearance of the big wheel and the small wheel is as follows:
e＝r1 (24)
In all the above formulae:
t is the motion parameter variable of the meshing point M, and t belongs to [0, [ delta ] t ];
T is a parameter variable of the equal-lift-distance conical spiral line, and T is formed by [0 and delta T ], and the delta T is k and delta T; (26)
k is linear proportionality coefficient of the movement of the meshing point;
R1-the theoretical indexing cone large end radius for the small wheel;
R1a-the radius of the large end of the cone being a small wheel; r1a＝R1-(ρ1 sinγ/cosδ1)； (27)
R2-the radius of the large end of the theoretical indexing cylinder of the bull wheel;
R2aThe radius of the large end of the cone, R, of the large wheel2a＝R2+[(ρ1 sinγ-e)/cosδ2)]； (28)
δ1-is the theoretical reference cone angle of the small wheel;
δ2-is the theoretical indexing cone angle of the bull wheel;
i12-is the transmission ratio of the small wheel to the large wheel;
r1transition fillet half of spiral circular arc tooth on small wheelDiameter;
ρ1The circular arc radius of the spiral circular arc tooth of the small wheel;
Gamma is the axial meshing angle of the small wheel and the big wheel;
△zk-length of meshing line of small and large wheels;
△z1-the axial height of the small wheel;
△z2-the axial height of the large wheel;
Delta T-angle parameter variable value range of the conical spiral line;
delta t is the value range of the motion parameter variable of the meshing point M;
delta T-angle parameter variable value range of the conical spiral line;
n1the number of the small gear teeth is the number of the spiral circular arc teeth of the small gear;
n2the number of teeth of the big wheel is the number of spiral trapezoidal grooves of the big wheel;
c1-meshing point motion undetermined coefficients;
Wherein: axes of each coordinate system, e, r1，ρ1，h，h'，w，R1，R2And c1the units of equal length or distance are millimeters (mm);δ1，δ2，ξ1The angular units of T, Delta T, k, gamma, theta and the like are radians (rads);
when the angular speed vector included angle theta and the transmission ratio i of the two crossed axes are determined12Radius R of big end of theoretical indexing cone of small wheel1Small gear tooth number n1Arc radius rho of helical arc tooth of small wheel1coincidence degree epsilon, axial meshing angle gamma and meshing point motion undetermined coefficient c1when the linear proportional parameter k of the movement of the meshing point and the clearance e of the small wheel and the big wheel cone are adopted, the cone structures of the small wheel and the big wheel, the central line of the spiral arc tooth of the small wheel, the tooth surface structure and the shape are determined, the reference line of the spiral trapezoidal groove of the big wheel, the tooth surface structure and the shape are determined, and the installation positions of the small wheel and the big wheel are also determinedAccordingly, the convex-flat meshing pure rolling spiral bevel gear mechanism for the crossed shaft transmission is obtained.
the small wheel and the large wheel form a transmission pair, and the design and calculation formula of the contact ratio is as follows:
Then, the result is obtained,
The design needs to be carried out according to the numerical value epsilon of the contact ratio, the linear proportionality coefficient k and the number n of the small gear teeth1And comprehensively determining the value range delta t of the motion parameter variable t of the meshing point M.
The spiral arc teeth uniformly distributed on the outer surface of the cone of the small wheel are in a shape of a section L in the form of an axial arc tooth1And make the center theta1Moving along the central line of the circular arc teeth of the small wheel to form spiral circular arc teeth; the spiral trapezoidal grooves uniformly distributed on the outer surface of the big wheel cone are in the shape of an axial isosceles trapezoid truncated L2And let it reference the point theta2A spiral trapezoidal groove formed by moving along the trapezoidal groove reference line of the large wheel.
the input shaft and the output shaft which are correspondingly connected with the small wheel and the big wheel have interchangeability, namely, the small wheel is connected with the input shaft, the big wheel is connected with the output shaft, or the big wheel is connected with the input shaft, the small wheel is connected with the output shaft, and the speed reduction transmission mode or the speed increase transmission mode respectively corresponds to a convex-flat meshing pure rolling spiral bevel gear mechanism used for cross shaft transmission; the constant-speed transmission application with the transmission ratio of 1 of the convex-flat meshing pure rolling bevel gear mechanism is realized only when the number of teeth of the small gear and the large gear is equal.
The rotation direction of an input shaft connected with the driver is clockwise or anticlockwise, so that forward and reverse rotation transmission of a small wheel or a large wheel is realized.
The convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission is a gear mechanism which is fundamentally innovated on the basis of the theory of the traditional gear transmission mechanism, and the design method of the convex-flat meshing pure rolling spiral bevel gear mechanism is also different from the design method of the traditional gear mechanism based on the curved surface meshing equation. The convex-flat meshing pure rolling spiral bevel gear mechanism for the crossed shaft transmission is a node meshing mode based on a pure rolling meshing line equation, the relative motion speed of all meshing points is zero, and a continuous stable meshing transmission method can be provided for micro, micro-mechanical and conventional mechanical devices for the crossed shaft transmission at any angle in a plane.
compared with the prior art, the convex-flat meshing pure rolling spiral bevel gear mechanism for the crossed shaft transmission has the advantages that:
1. The convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission has the greatest advantages that a meshing tooth surface without relative sliding is constructed by an active design method of a pure rolling meshing line parameter equation, the relative motion speed of all meshing points is zero, common failure modes such as tooth surface abrasion, gluing and tooth surface plastic deformation in gear transmission can be avoided, and the transmission efficiency is high.
2. The convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission has free contact ratio design, the structural shape of the gear body can be determined through the pre-design of the contact ratio, the uniform distribution of load is realized, and the dynamic characteristic is improved.
3. The convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission has the advantages that the tooth surface structure shape is simple, the small gear is a spiral concave trapezoidal groove tooth surface, the large gear is a spiral convex circular arc tooth surface, the processing and the manufacturing are easy, parameters such as a meshing angle and the like can be designed and adjusted at will, and the mechanical property of the tooth profile is optimized.
4. the convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission has no undercut, the minimum tooth number is 1, compared with the existing involute bevel gear and other mechanisms, the single-stage large transmission ratio high contact ratio transmission can be realized, the structure is compact, the installation space is greatly saved, and meanwhile, as the tooth number is small, larger tooth thickness can be designed, so that the convex-flat meshing pure rolling spiral bevel gear mechanism has higher strength and rigidity and larger bearing capacity, and is suitable for popularization and application in the fields of micro/micro machinery, conventional mechanical transmission and high-speed heavy-load transmission.
Drawings
FIG. 1 is a schematic structural diagram of a male-plain meshing pure rolling helical bevel gear mechanism for a crossed-axis transmission according to the present invention;
FIG. 2 is a schematic diagram of the spatial coordinate system of the convex-flat meshing pure rolling helical bevel gear mechanism for cross-shaft transmission of the present invention;
FIG. 3 is an axial cross-sectional view of the engaged spiral arc teeth and spiral trapezoidal grooves of FIG. 1;
FIG. 4 is a front view of the small wheel of FIG. 1 and its helical circular arc teeth;
FIG. 5 is a top view of the small wheel of FIG. 1 and its spiral-arc teeth;
FIG. 6 is an axial section L of the helical circular arc teeth of the small wheel of FIG. 11a schematic diagram of structural parameters;
FIG. 7 is a front view of the bull wheel of FIG. 1 and its helical trapezoidal groove;
FIG. 8 is a top view of the bull wheel of FIG. 1 and its helical trapezoidal groove;
FIG. 9 is an axial cross-sectional L of the spiral trapezoidal groove of the bull wheel of FIG. 12A schematic diagram of structural parameters;
FIG. 10 is a schematic structural view of the present invention when a large wheel is connected to an input shaft to drive a small wheel to increase speed.
In the above figures: 1-small wheel, 2-spiral arc tooth, 3-input shaft, 4-driver, 5-transition fillet, 6-output shaft, 7-spiral trapezoidal groove, 8-big wheel, 9-spiral arc tooth central line, 10-spiral trapezoidal groove reference line, 11-small wheel theory indexing cone, 12-big wheel theory indexing cone, 13-small wheel contact line, 14-big wheel contact line, 15-small wheel mounting hole and 16-big wheel mounting hole.
Detailed Description
the invention is further described with reference to the following drawings and specific examples, but the practice of the invention is not limited thereto.
Example 1: the invention provides a convex-flat meshing pure rolling spiral bevel gear mechanism for transmission of crossed shafts, which is applied to transmission with the transmission ratio of 1 between two crossed shafts in a plane, and the structure of the mechanism is shown in figure 1, and the mechanism comprises a small wheel 1 and a large wheel 8, wherein the small wheel 1 and the large wheel 8 form a pair of transmission pairs, the small wheel 1 is connected with an input shaft 3, the large wheel 8 is connected with an output shaft 6, namely the large wheel 8 is connected with a driven load through the output shaft 6; the axes of the small wheel 1 and the large wheel 8 are intersected, and the angular velocity vector included angle of the small wheel and the large wheel is theta, wherein theta is 2 pi/3 radian (rad) in the example. Fig. 2 is a schematic space coordinate system diagram of the convex-flat meshing pure rolling spiral bevel gear mechanism for the crossed shaft transmission.
Referring to fig. 1, 2, 3, 4, 5 and 6, the radius of the large end of the theoretical indexing cone of the small wheel is R1theoretical reference cone angle of the small wheel is delta1convex spiral arc teeth 2 are uniformly distributed on the outer surface of the cone of the small wheel 1, and the radius of the large end of the cone of the small wheel is R1athe axial meshing angle gamma, a transition fillet 5 is arranged between the spiral trapezoidal groove of the small wheel and the cone of the small wheel, and the radius of the transition fillet is r1the radius of the arc of the spiral trapezoidal tooth of the small wheel is rho1And (4) millimeter.
Referring to fig. 1, 2, 3, 7, 8 and 9, the radius of the large end of the theoretical indexing cone of the bull wheel is R2the theoretical reference cone angle of the bull wheel is delta2the outer surface of the cone of the bull wheel 8 is evenly provided with spiral trapezoidal grooves 7, and the radius of the big end of the cone of the bull wheel is R1aaxial engagement angle γ. The axial section type in spiral dovetail groove on the bull wheel is isosceles trapezoid, and its dimensional parameter includes: the height h, the isosceles trapezoid crosses the distance h' from the meshing line of M point to the big end of the trapezoid, the acute angle Pi/2-gamma between the waist of the isosceles trapezoid and the big end of the trapezoid is included, and the distance w between the meshing line of M point and the intersection point of the two waists of the isosceles trapezoid is respectively as follows:
the small wheel 1 is connected with an input shaft 3 and is driven by a driver 4 to rotate, so that the spiral arc teeth 2 of the small wheel 1 are continuously meshed with the spiral trapezoidal grooves 7 of the large wheel 8, and the motion and power transmission between crossed shafts in a plane is realized, wherein the driver 4 is a motor in the embodiment.
The central line of the spiral arc tooth 2 of the small wheel and the reference line of the spiral trapezoidal groove 7 of the large wheel are both equal-lift-distance conical spiral lines; the spiral arc teeth 2 are continuously meshed with the spiral trapezoidal grooves 7, so that continuous and stable meshing transmission between two crossed shafts in a plane is realized.
the structure of the spiral arc tooth 2 and the shape of the central line thereof, the structure of the spiral trapezoidal groove 7 and the shape of the reference line thereof are determined by the following method: see FIG. 2, at o- -x, y, z, ok--xk,yk,zkAnd op--xp,yp,zpIn three space coordinate systems, the z axis is coincident with the rotation axis of the small wheel, and z ispthe axis of rotation of the shaft and the bull wheel coinciding, zkThe axis coincides with the line of engagement of the small and large wheels, and the z-axis coincides with the z-axisp、zkThe axes intersect at a point; coordinate system o1--x1,y1,z1Fixedly connected to the small wheel, coordinate system o2--x2,y2,z2Fixedly connected with the big wheel, the small wheel and the big wheel are respectively connected with the coordinate system o-x, y, z and o at the initial positionsp--xp,yp,zpCoincidence, ookA distance R1，opokA distance R2，zkThe acute angle between the axis and the z-axis is delta1，zkAxis and zpThe acute angle included by the shaft is delta2With small wheels at uniform angular velocity omega1rotating about the z-axis, the bull wheel at a uniform angular velocity ω2Around zpThe axes are rotated, the angular velocity vector included angle of the rotation axes of the small wheel and the large wheel is theta, and after a period of time from the initial position, the coordinate system o1--x1,y1,z1And o2--x2,y2,z2Move respectively, at the meshing point M, the small wheel rotates around the axis zcorner, large wheel winding zpthe shaft rotates throughan angle;
When the small wheel is engaged withWhen the bull wheel is in meshing transmission, the meshing point M is from the origin of coordinates okStarting to move linearly at a constant speed along the meshing line k-k, and defining a parameter equation of M point motion as follows:
in the formula (1), t is a motion parameter variable of the meshing point M, and t is more than or equal to 0 and less than or equal to Δ t; c. C1the undetermined coefficient of the meshing point movement is expressed in millimeters (mm); in order to ensure pure rolling engagement of the small and large wheels, the rotation angle of the small and large wheels and the movement of the engagement point must be in a linear relationship, which is as follows:
In the formula (2), k is a linear proportionality coefficient of the movement of the meshing point, and the unit is radian (rad); i.e. i12The transmission ratio between the small wheel and the large wheel is set;
when the meshing point M moves along the meshing line k-k, the point M simultaneously forms contact lines C on the surfaces of the small wheel and the large wheel respectively1and C2(ii) a According to the coordinate transformation, the coordinate system o-x, y, z, o can be obtainedk--xk,yk,zk、op--xp,yp,zp、o1--x1,y1,z1And o2--x2,y2,z2the homogeneous coordinate transformation matrix in between is:
wherein:
Obtaining:
From the homogeneous coordinate transformation, equation (6) yields:
calculating the contact line C on the tooth surface of the small wheel from the formula (8)1The pitch-equaling conical spiral line has the parameter equation:
The following equation (2) is taken into equation (9):
In the formula (10), T is an angle parameter variable of the conical spiral line with equal lift distance, wherein the T is kt, and the T is more than or equal to 0 and less than or equal to delta T;
from the homogeneous coordinate transformation, equation (7) yields:
obtaining a contact line C on the tooth surface of the bull gear from the formula (11)2The pitch-equaling conical spiral line has the parameter equation:
The following equation (2) is taken into equation (12):
and the transmission ratio of the small wheel to the large wheel is as follows:
obtained by substituting formula (14) for formula (13):
The theoretical reference cone angles of the small wheel and the large wheel are respectively delta1And delta2their relationship is:
The convex tooth surface of the helical arc tooth of the small wheel is in a shape of a section L consisting of an axial arc tooth profile containing a meshing point M1Generated by right-handed helical motion, of circular-arc-tooth-shaped cross-section L1Is a generating bus of a small wheel tooth surface, and the axial pitch parameter and the contact line C of the spiral motion of the generating bus1The parameters of the axial screw pitches are consistent, and the right-handed screw motion track of the meshing point M and the contact line C are ensured1overlapping; in a coordinate system o-x, y and z, a generatrix parameter equation of the convex tooth surface of the small wheel is as follows:
Deducing and obtaining convex tooth surface of helical circular arc tooth of small wheel in coordinate system o by right-handed helical motion1–x1,y1,z1the parameter equation is:
At the moment, the equation of the central line of the convex tooth surface of the spiral circular arc tooth of the small wheel is as follows:
Spiral trapezoid of bull wheelthe concave flank of the groove is formed by an axial isosceles trapezoid cross-section L containing the meshing point M2Generated by left-handed helical motion, isosceles trapezoid truncated L2the profile generating bus is a profile generating bus of a concave tooth surface of a bull wheel, the section height of the bull wheel in the axial direction of an isosceles trapezoid is h, the distance from an meshing line of an M point passing through the isosceles trapezoid to a big bottom of the trapezoid is h', an acute angle formed by the waist of the isosceles trapezoid and the big bottom of the trapezoid is pi/2-gamma, the distance from the meshing line of the M point passing through the isosceles trapezoid to the intersection point of the two waists of the isosceles trapezoid is w, and the profile generating bus meet:
in the formula (18), e is a gap between the large wheel and the small wheel cylinder;
Axial isosceles trapezoid section L of spiral trapezoid groove2Is noted as theta2，Θ2is the middle point of the line connecting the meshing line of the M points and the intersection point of the two waists of the isosceles trapezoid, theta2Isosceles trapezoid section L following spiral trapezoid groove2The parameter equation of the reference line formed by the spiral motion is as follows:
Therefore, the axial pitch parameter and the contact line C of the generating generatrix spiral motion of the big gear tooth surface2the parameters of the axial thread pitches are consistent, and the left-handed spiral motion track of the meshing point M and the contact line C are ensured2Overlapping;
the length of the meshing line of the small wheel and the large wheel is as follows:
The axial height of the small wheel is as follows:
△z1＝△zk sinδ2 (22)
the height of the axis of the bull wheel is:
△z2＝△zk sinδ1 (23)
The cone clearance of the big wheel and the small wheel is as follows:
e＝r1 (24)
in all the above formulae:
t is the motion parameter variable of the meshing point M, and t belongs to [0, [ delta ] t ];
t is a parameter variable of the equal-lift-distance conical spiral line, and T is formed by [0 and delta T ], and the delta T is k and delta T; (26)
k is linear proportionality coefficient of the movement of the meshing point;
R1-the theoretical indexing cone large end radius for the small wheel;
R1a-the radius of the large end of the cone being a small wheel; r1a＝R1-(ρ1 sinγ/cosδ1)； (27)
R2-the radius of the large end of the theoretical indexing cylinder of the bull wheel;
R2aThe radius of the large end of the cone, R, of the large wheel2a＝R2+[(ρ1 sinγ-e)/cosδ2)]； (28)
δ1-is the theoretical reference cone angle of the small wheel;
δ2-is the theoretical indexing cone angle of the bull wheel;
i12-is the transmission ratio of the small wheel to the large wheel;
r1-radius of transition fillet of spiral circular arc tooth on small wheel;
ρ1The circular arc radius of the spiral circular arc tooth of the small wheel;
Gamma is the axial meshing angle of the small wheel and the big wheel;
△zk-length of meshing line of small and large wheels;
△z1-the axial height of the small wheel;
△z2-the axial height of the large wheel;
delta T-angle parameter variable value range of the conical spiral line;
delta t is the value range of the motion parameter variable of the meshing point M;
Delta T-angle parameter variable value range of the conical spiral line;
n1the number of the small gear teeth is the number of the spiral circular arc teeth of the small gear;
n2The number of teeth of the big wheel is the number of spiral trapezoidal grooves of the big wheel;
c1-meshing point motion undetermined coefficients;
wherein: axes of each coordinate system, e, r1，ρ1，h，h'，w，R1，R2and c1the units of equal length or distance are millimeters (mm);δ1，δ2，ξ1The angular units of T, Delta T, k, gamma, theta and the like are radians (rads);
The small wheel and the large wheel form a transmission pair, and the design and calculation formula of the contact ratio is as follows:
Then, the result is obtained,
the design needs to be carried out according to the numerical value epsilon of the contact ratio, the linear proportionality coefficient k and the number n of the small gear teeth1And comprehensively determining the value range delta t of the motion parameter variable t of the meshing point M.
When the angular speed vector included angle theta and the transmission ratio i of the two crossed axes are determined12Radius R of big end of theoretical indexing cone of small wheel1small gear tooth number n1Arc radius rho of helical arc tooth of small wheel1Coincidence degree epsilon, axial meshing angle gamma and meshing point motion undetermined coefficient c1The linear proportional parameter k of the motion of the meshing point and the clearance e of the small wheel and the big wheel cone are determined, the cone structures of the small wheel and the big wheel, the central line of the spiral arc tooth, the tooth surface structure and the shape of the small wheel are also determined, the reference line of the spiral trapezoidal groove, the tooth surface structure and the shape of the big wheel are also determined, and the installation positions of the small wheel and the big wheel are also correspondingly determined, so that the convex-flat meshing pure rolling spiral bevel gear for cross shaft transmission is obtainedAnd (4) a mechanism.
when in the above formula: the relevant parameters take the values as follows:ε＝2，i12＝1，c1＝30，k＝π，R125 millimeters (mm), ρ23 mm (mm) and e 0.5 mm (mm), and the value obtained by substituting the formula (16)Δ T is determined to be 1 in substitution for formula (30), and Δ T is determined to be pi in substitution for formula (26).
The equation for obtaining the central line of the spiral circular arc tooth of the small wheel in the embodiment is obtained by substituting the formula (19):
the equation of the reference line of the helical trapezoidal groove of the large wheel in the embodiment is obtained by taking the numerical values into formula (21):
the substitution formula (20) obtains the height of the section of the big wheel axial isosceles trapezoid as h ═ 3 millimeters (mm), the distance from the meshing line of the M point of the isosceles trapezoid to the big bottom of the trapezoid as h ═ 1 millimeter (mm), the acute angle between the waist of the isosceles trapezoid and the big bottom of the trapezoid is pi/3 radian (rad), and the distance from the meshing line of the M point to the intersection point of the two waists of the isosceles trapezoid is
Calculating the length of the meshing line of the small wheel and the large wheel as delta z by substituting formula (22)kThe axial height of the small wheel is calculated as delta z by substituting formula (23) as 30 millimeters (mm)1The axial height of the large wheel is calculated as Δ z by the equation 24, which is 15 millimeters (mm)215 millimeters (mm); the radius of the large end of the cone of the small wheel is R by the formula (27)1aThe radius of the large end of the cone of the large wheel obtained by the equation (28) is R (23.268 mm)2a26.154 mm(mm); the transition fillet radius r of the small wheel is obtained from the equation (25)10.5 millimeters (mm).
Setting the number of spiral arc teeth as n1When the number of spiral grooves is n, the number of spiral grooves is obtained from the formula (14) as 42And (4) determining the shapes of the pair of spiral arc bevel gear transmission pairs of the small wheel 1 and the large wheel 8 according to the reference line equation of the spiral trapezoidal groove 7, the central line equation of the spiral arc teeth 2 and the data of the cone structure parameters of the small wheel and the large wheel respectively, so as to obtain the shape of the pure rolling convex-flat meshing bevel gear mechanism and perform correct assembly.
when the driver 4 drives the input shaft 3 and the small wheel 1 to rotate, because a pair of spiral arc teeth 2 and the spiral trapezoidal groove 7 are in a meshed state when the small wheel 1 and the large wheel 8 are installed, and the contact ratio of the spiral arc bevel gear is defined to be epsilon, 2, which is larger than 1 when the small wheel 1 and the large wheel 8 are designed, when the pair of spiral arc teeth 2 and the spiral trapezoidal groove 7 rotate, namely, are disengaged but are not completely disengaged, another pair of adjacent spiral arc teeth 2 and spiral trapezoidal grooves 7 are engaged again, so that continuous and stable meshing transmission of the spiral arc bevel gear mechanism in the rotating motion is realized. The rotation direction of an input shaft connected with the driver of the embodiment is clockwise, and the constant-speed transmission of the spiral circular arc bevel gear mechanism is corresponding to the constant-speed transmission of the spiral circular arc bevel gear mechanism, so that the constant-speed transmission of the anticlockwise rotation of the large wheel is realized.
example 2: the convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission is applied to speed-up transmission between two vertical crossed shafts, and theta is 2/pi radian (rad). As shown in fig. 10, a large wheel 8 is connected with an input shaft 3, and a small wheel 1 is connected with an output shaft 6, namely, the small wheel 1 is connected with a driven load through the output shaft 6; the axes of the small wheel 1 and the large wheel 8 are perpendicular to each other, and the angular speed included angle theta is 2/pi radian (rad). In the embodiment, eight spiral trapezoidal grooves 7 are formed in the large wheel 8, four spiral circular arc teeth 2 are formed in the small wheel 1, and when the input shaft 3 drives the large wheel 8 to rotate, the designed contact ratio epsilon is 2. When the big wheel 8 and the small wheel 1 are installed, the spiral trapezoidal groove 7 on the big wheel 8 and the spiral arc tooth 2 on the small wheel are in a meshed state, and when the big wheel 8 rotates, the big wheel and the small wheel rotate to keep the meshed contact ratio of the spiral trapezoidal groove and the spiral arc tooth to be larger than 1, so that continuous and stable meshing transmission of a spiral arc bevel gear mechanism is realized. At this time, the transmission ratio of the small wheels to the large wheels is 2, that is, the speed increasing ratio of the large wheels to the small wheels is 2.
The relevant parameters take the values as follows:ε＝2，i12＝2，c1＝30，k＝π，R125 millimeters (mm), ρ23 millimeters (mm), e 0.5 millimeters (mm). Calculating δ by substituting formula (16)10.4636 radians (rad), δ21.1071 radians (rad). Δ T is determined to be 1 in substitution for formula (30), and Δ T is determined to be pi in substitution for formula (26).
The equation of the reference line of the helical trapezoidal groove of the large wheel in the embodiment is obtained by taking the numerical values into formula (21):
The equation for obtaining the central line of the spiral circular arc tooth of the small wheel in the embodiment is obtained by substituting the formula (19):
the substitution formula (20) obtains the height of the section of the big wheel axial isosceles trapezoid as h ═ 3 millimeters (mm), the distance from the meshing line of the M point of the isosceles trapezoid to the big bottom of the trapezoid as h ═ 1 millimeter (mm), the acute angle between the waist of the isosceles trapezoid and the big bottom of the trapezoid is pi/3 radian (rad), and the distance from the meshing line of the M point to the intersection point of the two waists of the isosceles trapezoid is
Calculating the length of the meshing line of the small wheel and the large wheel as delta z by substituting formula (22)kthe axial height of the small wheel is calculated as delta z by substituting formula (23) as 30 millimeters (mm)1the axial height Δ z of the large wheel obtained by the equation (24) is 26.8328 mm (mm)213.4164 millimeters (mm); the radius of the large end of the cone of the small wheel is R by the formula (27)1a＝23323 millimeters (mm), the radius of the large end of the cone of the bull wheel obtained by the equation (28) given below is R2a47.764 millimeters (mm); the transition fillet radius r of the small wheel is obtained from the equation (25)10.5 millimeters (mm).
Because the number of the spiral arc teeth is 4 and the number of the spiral trapezoidal grooves is 8, the shapes of the pair of spiral arc bevel gear transmission pairs of the small wheel 1 and the large wheel 8 can be determined according to the reference line equation of the spiral trapezoidal groove 7, the central line equation of the spiral arc teeth 7 and the data of the cone structure parameters of the large wheel and the small wheel respectively, and therefore the shape of the pure rolling convex-flat meshing bevel gear mechanism is obtained and the correct assembly is carried out.
The rotation direction of an input shaft connected with the driver of the embodiment is clockwise, and the driving device corresponds to a speed-increasing driving mode of the spiral arc bevel gear mechanism and is used for realizing the transmission of anticlockwise rotation of the small wheel.
The invention is used for the protruding-flat meshing pure rolling spiral bevel gear mechanism of the crossed shaft transmission, because there is no undercut, there is no restriction of the minimum number of teeth, can carry on the design of the large tooth thickness, have higher bending strength, contact strength and greater rigidity, the invention has also provided the design method of the bevel gear mechanism of the continuous stable meshing transmission between two crossed shafts of arbitrary angle in the level, can design the protruding-flat meshing pure rolling bevel gear mechanism parameter according to the numerical value of the degree of contact ratio, it is high to have tooth profile intensity, tooth surface have relative slip, have no undercut, the single-stage drive ratio is large, high transmission efficiency, greatly reduce the tooth surface and glue, wear and plastic deformation, etc. and invalid probability, can simplify the structure of the conventional gear mechanism and micromechanical drive unit, suitable for the application in the fields of the small, miniature machinery and conventional machinery.

Claims (8)

1. A protruding-flat meshing pure rolling spiral bevel gear mechanism for transmission of crossing shaft, constitute a pair of transmission pair including steamboat and bull wheel, the steamboat links firmly with the driver through the input shaft, the bull wheel connects the output shaft, the axis of steamboat with the axis of bull wheel is alternately its characterized in that: the outer surface of the cone of the small wheel is provided with n1The central lines of all the spiral arc teeth are equal-lift-distance circlesA transition fillet is arranged between the spiral circular arc tooth and the outer surface of the cone of the small wheel, and n is arranged on the outer surface of the cone of the large wheel2The spiral trapezoidal groove of strip evenly distributed, the central line in all spiral trapezoidal grooves is the circular cone helix of equal lift-off distance, spiral trapezoidal groove with there is transition fillet between the bull wheel cone surface, the spiral arc tooth of steamboat with the pure roll engagement transmission that the spiral trapezoidal groove meshing mode of bull wheel was the point contact, through spiral arc tooth on the bull wheel with continuous meshing effect between the spiral trapezoidal groove on the bull wheel drives the bull wheel rotates, spiral arc tooth structure with spiral trapezoidal groove structure is confirmed by following method:
s1, constructing a coordinate system, and defining the position relation between the small wheel and the large wheel and each coordinate system;
construction of o- -x, y, z, ok--xk,yk,zkAnd op--xp,yp,zpThree spatial coordinate systems, with the z-axis coinciding with the axis of rotation of the small wheel, zpthe axis of rotation of the shaft and the bull wheel coinciding, zkThe axis coincides with the line of engagement of the small wheel and the large wheel, and the z-axis coincides with zp、zkThe axes intersect at a point to construct a coordinate system o fixedly connected with the small wheel1--x1,y1,z1A coordinate system o fixedly connected to said large wheel2--x2,y2,z2The small wheel and the large wheel respectively have initial positions in a coordinate system o-x, y, z and op--xp,yp,zpcoincidence, ookA distance R1，opoka distance R2，zkThe acute angle between the axis and the z-axis is delta1，zkAxis and zpThe acute angle included by the shaft is delta2The small wheel has a uniform angular speed omega1rotating around the z-axis, the large wheel being at a uniform angular velocity ω2around zpthe shaft rotates, the angular velocity vector included angle of the rotation axes of the small wheel and the large wheel is theta, and after a period of time from the initial position, a coordinate system o1--x1,y1,z1Rotating about the z-axis, coordinate system o2--x2,y2,z2Around zpThe axis rotates, the meshing point is M, and the small wheel rotates around the z axisAngle, said large wheel winding zpThe shaft rotates throughAn angle;
s2, determining a motion parameter equation of the meshing point M:
when the small wheel and the large wheel are in meshing transmission, the meshing point M is from the coordinate origin okThe constant-speed linear motion is carried out along the meshing line k-k, and the parameter equation of M point motion is defined as follows:
wherein t is a motion parameter variable of the meshing point M, and t is more than or equal to 0 and less than or equal to delta t; c. C1determining a undetermined coefficient for the movement of the meshing point, wherein the small wheel is in pure rolling meshing with the large wheel, and the rotation angles of the small wheel and the large wheel and the movement of the meshing point are necessarily in a linear relation, wherein the relation is as follows:
where k is the linear proportionality coefficient of the movement of the engagement point, i12the transmission ratio between the small wheel and the large wheel is set;
S3 determining contact line C on small wheel tooth surface1and the contact line C on the tooth surface of the bull wheel2the parameter equation of (2):
When the meshing point M moves along the meshing line k-k, the point M simultaneously forms contact lines C on the surfaces of the small wheel and the large wheel respectively1And C2according to the coordinate transformation, the coordinate system o-x, y, z, o can be obtainedk--xk,yk,zk、op--xp,yp,zp、o1--x1,y1,z1And o2--x2,y2,z2The homogeneous coordinate transformation matrix in between is:
Respectively solving the contact line C on the tooth surface of the small wheel by homogeneous coordinate transformation1And the contact line C on the tooth surface of the bull wheel2Equation of parameters, contact line C1Is a conical spiral line with equal lift distance and the parameter equation is as follows:
contact wire C2is a conical spiral line with equal lift distance and the parameter equation is as follows:
in the formula, T is an angle parameter variable of the conical spiral line with equal lift distance, wherein T is kt, and T is more than or equal to 0 and less than or equal to delta T;
S4, determining a generating generatrix parameter equation and a convex tooth surface parameter equation of the small wheel convex tooth surface:
the convex tooth surface of the helical arc tooth of the small wheel is in a shape of a section L consisting of an axial arc tooth profile containing a meshing point M1Generated by right-handed helical motion, of circular-arc-tooth-shaped cross-section L1is a generating bus of a small wheel tooth surface, and the axial pitch parameter and the contact line C of the spiral motion of the generating bus1the parameters of the axial screw pitches are consistent, and the right-handed screw motion track of the meshing point M and the contact line C are ensured1And (3) coincidence, wherein the parameter equation of the shape generating generatrix of the convex tooth surface of the small wheel in a coordinate system o-x, y and z is as follows:
Deducing and obtaining the convex tooth surface of the helical circular arc tooth of the small wheel in a coordinate system o by right-handed helical motion1--x1,y1,z1The parameter equation is:
s5, determining a parameter equation of a generating generatrix and a parameter equation of a reference line of the large-wheel spiral trapezoidal groove:
The concave tooth surface of the spiral trapezoidal groove of the bull wheel is formed by an axial isosceles trapezoid section L containing a meshing point M2Generated by left-handed helical motion, isosceles trapezoid truncated L2The profile generating bus is a profile generating bus of a concave tooth surface of a bull wheel, the section height of the bull wheel in the axial direction of an isosceles trapezoid is h, the distance from an meshing line of an M point passing through the isosceles trapezoid to a big bottom of the trapezoid is h', an acute angle formed by the waist of the isosceles trapezoid and the big bottom of the trapezoid is pi/2-gamma, the distance from the meshing line of the M point passing through the isosceles trapezoid to the intersection point of the two waists of the isosceles trapezoid is w, and the profile generating bus meet:
Wherein e is the gap between the cones of the large wheel and the small wheel,
cutting the meshing line with an isosceles trapezoid2the middle point of the connecting line of the two waist intersections is a reference point, and the reference point moves along the reference line to enable the isosceles trapezoid to be in a section shape L2make spiral motion to form a spiral trapezoidal groove, and ensure the left-handed spiral motion track of the meshing point M and the contact line C2Are superimposed on a coordinate system o2--x2,y2,z2The parameter equation of the middle reference line is as follows:
In all the above formulae:
k is the linear proportionality coefficient of the meshing point motion,
Epsilon-is the contact ratio of the small wheel and the large wheel;
R1-the theoretical indexing cone large end radius for the small wheel;
R2-the radius of the large end of the theoretical indexing cylinder of the bull wheel;
δ1-is the theoretical reference cone angle of the small wheel;
δ2-is the theoretical indexing cone angle of the bull wheel;
Rho is the arc radius of the spiral arc teeth of the small wheel and the big wheel;
Gamma is the axial meshing angle of the small wheel and the big wheel;
i12-is the transmission ratio of the small wheel to the large wheel;
delta T-angle parameter variable value range of the conical spiral line;
delta t is the value range of the motion parameter variable of the meshing point M;
c1undetermined coefficient of motion of meshing point
ξ1Is an angle parameter.
2. The male-plain mesh pure rolling helical bevel gear mechanism for crossed-axis transmission according to claim 1 wherein the centerline of the convex flank of the helical circular arc tooth of the small wheel is in the coordinate system o1--x1,y1,z1The parameter equation in (1) is:
3. the male-plain meshing pure rolling helical bevel gear mechanism for a crossed-axis transmission according to claim 1, wherein the meshing line length of the small wheel and the large wheel is:
4. the male-plain mesh pure rolling helical bevel gear mechanism for a crossed-axis transmission according to claim 3, wherein:
the axial height of the small wheel is as follows: delta z1＝△zksinδ2，
The axial height of the bull wheel is as follows: delta z2＝△zksinδ1。
5. the male-plain mesh pure rolling helical bevel gear mechanism for a crossed-axis transmission according to claim 1, wherein: the clearance between the cones of the big wheel and the small wheel is e-r1，r1And the transition fillet radius of the spiral circular arc teeth on the small wheel is shown.
6. The male-plain mesh pure rolling helical bevel gear mechanism for a crossed-axis transmission according to claim 1, wherein: and the input shaft and the output shaft which are correspondingly connected with the small wheel and the large wheel have interchangeability.
7. the male-plain mesh pure rolling helical bevel gear mechanism for a crossed-axis transmission according to claim 1, wherein: the number of teeth n of the small gear1And the number n of teeth of said large gear2Are equal.
8. the male-plain mesh pure rolling helical bevel gear mechanism for a crossed-axis transmission according to claim 1, wherein: the rotation direction of the input shaft is clockwise or anticlockwise.
CN201810603089.9A 2018-06-12 2018-06-12 Convex-flat meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission Expired - Fee Related CN108533684B (en)

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