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CN113944728A - Unequal-pressure-angle end face double-arc gear mechanism driven by parallel shafts - Google Patents

Unequal-pressure-angle end face double-arc gear mechanism driven by parallel shafts Download PDF

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CN113944728A
CN113944728A CN202111068628.1A CN202111068628A CN113944728A CN 113944728 A CN113944728 A CN 113944728A CN 202111068628 A CN202111068628 A CN 202111068628A CN 113944728 A CN113944728 A CN 113944728A
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gear
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tooth
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CN113944728B (en
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文龙
陈祯
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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
    • F16H1/06—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members with parallel axes
    • 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
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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
    • F16H55/17—Toothed wheels
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    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H57/00—General details of gearing
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Abstract

The invention discloses an unequal pressure angle end face double-arc gear mechanism driven by parallel shafts and a design method, belonging to the field of gear transmission, wherein the double-arc gear mechanism comprises a small gear and a large gear which are parallel in axis, the end face tooth profiles of the small gear and the large gear are respectively composed of a convex arc tooth profile, a straight line tooth profile, a concave arc tooth profile and a tooth root transition curve, and the specific structure of the tooth profiles is determined by a meshing line parameter equation and parameters such as contact ratio, tooth number, transmission ratio and the like; when the driving device is correctly installed, the convex-concave arcs of the small wheel and the large wheel simultaneously realize double-point meshing contact on the end faces of the wheel teeth, the two meshing points have unequal pressure angles, and the small wheel and the large wheel rotate under the driving of the driving device to realize transmission between two shafts. The design method disclosed by the invention can be used for designing the unequal pressure angle double-arc gear mechanism for parallel shaft transmission, has the advantages of simple design, easiness in lubrication, high tooth root bending strength, large transmission ratio and contact ratio, strong bearing capacity and the like, and can be widely applied to the design of transmission systems of engineering machinery such as double-wheel milling machines and the like.

Description

Unequal-pressure-angle end face double-arc gear mechanism driven by parallel shafts
Technical Field
The invention relates to the field of gear transmission, in particular to a parallel shaft transmission double-circular-arc gear mechanism with unequal pressure angle end faces.
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. Research on core basic parts such as high-performance gears and the like is a key factor for promoting industrial transformation and upgrading and improving national industrial core competitiveness. Particularly, with the economic acceleration, the engineering machinery develops towards high power, and higher requirements are put forward on the design of core transmission parts, such as transmission gear transmission of a gearbox.
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. Parallel shaft gear drives are the most commonly used form of gear drive, with involute gears being the most widely used. However, involute gears have problems of transmission failures such as friction wear, gluing and plastic deformation caused by relative sliding of tooth surfaces until now, so that the transmission efficiency, the service life and the reliability of gear products, particularly high-speed heavy-duty gears, are seriously affected, and the performance improvement of high-precision mechanical equipment is restricted. Especially under heavy load, the relative sliding of the involute gear tooth tops is very serious, and transmission failure is easily caused.
In order to solve the problems of the involute gear transmission, domestic and foreign scholars gradually developed single-arc gears and double-arc gears including end face double-arc gears and normal double-arc gears aiming at a parallel shaft transmission mode, such as chinese patent literature with application number 202110318591.7, which discloses a double-arc small tooth difference reduction transmission device and a double-arc tooth forming method, application number 201810893876.1, which discloses a double-arc gear, application number 201620553083.1, which discloses a double-arc gear, and the like. The double-arc gear with parallel shaft transmission has larger tooth surface contact strength and tooth root bending strength and better lubricating property compared with an involute gear. However, the tooth profiles of the small wheel and the large wheel of the double-circular-arc gear are cut by a generating method based on the same hob, and in order to ensure that the large gear and the small gear are correctly meshed, the pressure angles of two meshing points of the tooth profiles of the hob are set to be equal. Therefore, the existing double-circular-arc gear mechanism has the limitation that the pressure angles of two meshing points of the tooth profile are limited to be equal, so that the structure of the double-circular-arc gear mechanism is not an optimal bearing design structure, and during heavy-load transmission of engineering machinery such as a double-wheel mill and the like in underground continuous wall construction, the root of a gear tooth can be broken to cause construction accidents. In addition, if the safety coefficient is increased simply for increasing the bending strength of the root part of the double-arc gear, the gear module is increased, so that the transmission structure of the equipment is overlarge in size, and the design and performance improvement of the whole machine are influenced.
Disclosure of Invention
The invention aims to provide an unequal pressure angle end face double-arc gear mechanism for parallel shaft transmission and a design method thereof aiming at the problems of the prior art of the double-arc gear mechanism in the field of mechanical transmission at present.
In order to achieve the purpose, the invention adopts the technical scheme that: the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft comprises a pair of gear 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 axes of the small wheel and the large wheel are parallel, the end face tooth profiles of the small wheel and the large wheel are in an axial symmetry form, and the left and right side end face tooth profiles from the tooth top to the tooth bottom respectively consist of a convex arc tooth profile, a straight line tooth profile, a concave arc tooth profile and a tooth root transition curve; the small wheel and the large wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, and the two meshing points have unequal pressure angles; the small wheel is driven by the driver to rotate, stable meshing transmission between parallel shafts is realized through continuous meshing between two pairs of convex circular arc tooth profiles and concave circular arc tooth profiles, two meshing points with different pressure angles respectively form two contact lines with the same pitch on tooth surfaces of the small wheel and the large wheel, and the two contact lines are cylindrical spiral lines; the gear tooth flanks of the small wheel and the large wheel are spiral tooth flanks obtained by cylindrical spiral motion of end face tooth profiles along respective contact lines, the screw pitch of the spiral tooth flanks is equal to that of the contact lines, and the spiral directions of the gear teeth of the small wheel and the gear teeth of the large wheel are opposite.
Further, the small wheel and the big wheel are in double-point contact concave-convex meshing transmission, and two meshing points of the small wheel and the big wheel are respectively a meshing point M of a convex circular arc tooth profile of the teeth of the small wheel and a concave circular arc tooth profile of the teeth of the big wheelR1And the meshing point M of the concave circular arc tooth profile of the adjacent gear teeth of the small gear and the convex circular arc tooth profile of the adjacent gear teeth of the large gearR2(ii) a The normal lines of the tooth profile meshing points of the two pairs of concave-convex circular arc end surfaces intersect at the same point, and the point is a tangent point of a pitch circle of the pair of double-circular-arc gears, namely a node; the horizontal distances from the two meshing points to the node are both PM; when the pair of parallel shafts drive the double-arc gear mechanism with unequal pressure angle end faces, two meshing points MR1And MR2Has the same axial movement speed and forms two spatial meshing lines K respectivelyR1-KR1And KR2-KR2And each form two contact lines of the tooth surfaces of the small wheel and the large wheel.
Further, the tooth surface contact line of the small wheel and the large wheel is determined by the following method:
at op--xp,yp,zp、ok--xk,yk,zkAnd og--xg,yg,zgIn three spatial coordinate systems, where op、okAnd ogRespectively the origin, x, of three spatial coordinate systemsp、xkAnd xgX-axis, y of three spatial coordinate systems respectivelyp、ykAnd ygX-axis, z, of three spatial coordinate systems, respectivelyp、zkAnd zgZ-axes, z, of three spatial coordinate systems, respectivelypThe axis of rotation of the shaft and the small wheel coinciding, zgThe shaft is superposed with the rotation axis of the big wheel,zkshaft-to-pass meshing point MR1Engagement line K ofR1- KR1Coincide with and zkAxis and zp、zgAxes parallel to each other, xpAnd xgThe axes being coincident, xkAnd xgAxis parallel, opAnd ogA is a; coordinate system o1--x1,y1,z1Fixedly connected with the small wheel and having a coordinate system o2--x2,y2,z2Fixedly connected with the large wheel and a small wheel coordinate system o1--x1,y1,z1And a large wheel coordinate system o2--x2,y2,z2At the starting position respectively with the coordinate system op--xp,yp,zpAnd og--xg, yg,zgCoincident, the small wheels being at a uniform angular velocity ω1Around zpThe shaft rotates clockwise and the bull wheel rotates at a uniform angular velocity ω2Around zgThe axis rotates anticlockwise, after a period of time from the starting position, the coordinate system o1--x1,y1,z1And o2--x2,y2,z2Respectively rotate when the meshing point is MR1And MR2Said small wheel winding zpThe shaft rotates through
Figure BDA0003259328080000035
Angle, said large wheel winding zgThe shaft rotates through
Figure BDA0003259328080000036
An angle;
when the small wheel and the large wheel are in meshing transmission, a meshing point M is setR1From the origin o of coordinateskBeginning along the line of engagement KR1-KR1Exercise, MR1The parametric equation for point motion is:
Figure BDA0003259328080000031
at the same time, the mesh point MR2Along the line of engagement at the same speed of movementKR2-KR2Exercise, MR2The parametric equation for point motion is:
Figure BDA0003259328080000032
in the formulae (1) and (2), t is the meshing point MR1T is more than or equal to 0 and less than or equal to delta t, and delta t is the maximum value of the motion parameter variable; c. C1The undetermined coefficient of the movement of the meshing point is represented by millimeter, and PM is the horizontal distance from the meshing point to the node; in order to ensure that the fixed gear ratio is engaged, the rotation angles of the small wheel and the large wheel and the movement of the engagement point have to be in a linear relationship, and the rotation angles of the small wheel and the large wheel and the movement of the engagement point have the following relationship:
Figure BDA0003259328080000033
in the formula (3), k is a linear proportionality coefficient of the movement of the meshing point, and the unit of k is radian; i.e. i12The transmission ratio between the small wheel and the large wheel is set;
when the point of engagement MR1Along the line of engagement KR1-KR1While in motion, point MR1Simultaneously, contact lines C are respectively formed on the convex circular arc tooth surface of the small wheel and the concave circular arc tooth surface of the large wheelR1pAnd CR1g(ii) a When the point of engagement MR2Along the line of engagement KR2-KR2While in motion, point MR2Simultaneously, contact lines C are respectively formed on the concave arc tooth surface of the small wheel and the convex arc tooth surface of the large wheelR2pAnd CR2g(ii) a Obtaining a coordinate system o according to the coordinate transformationp--xp, yp,zp、ok--xk,yk,zkAnd og--xg,yg,zg、o1--x1,y1,z1And o2--x2,y2,z2The homogeneous coordinate transformation matrix in between is:
Figure BDA0003259328080000034
wherein,
Figure BDA0003259328080000041
Figure BDA0003259328080000042
in the formulae (5) and (6), R1Is the pitch cylinder radius of the small wheel, R2Is the pitch cylinder radius of the bull wheel, and PM is the meshing point MR1And MR2Distance to node P, αt1Is a meshing point MR1End face pressure angle of alphat2Is a meshing point MR2The end face pressure angle of (1);
obtaining the contact line C of the convex arc tooth surface of the small wheel by the formulas (1) and (4)R1pThe parameter equation of (1) is as follows:
Figure BDA0003259328080000043
obtaining the contact line C of the concave circular arc tooth surface of the bull gear by the formulas (1) and (4)R1gThe parameter equation of (1) is as follows:
Figure BDA0003259328080000044
calculating a small wheel concave arc tooth surface contact line C by the formulas (2) and (4)R2pThe parameter equation of (1) is as follows:
Figure BDA0003259328080000045
obtaining a contact line C of the convex arc tooth surface of the bull wheel according to the formulas (2) and (4)R2gThe parameter equation of (1) is as follows:
Figure BDA0003259328080000046
further, the face tooth profiles of the small wheel and the large wheel are determined by:
respectively at the circle center o of the convex circular arc tooth profile ar1 of the small wheela1And the circle center o of the big wheel concave circular arc tooth profile br2b2Establishing a local coordinate system Sa1(oa1-xa1ya1za1) And Sb2(ob2-xb2yb2zb2) The obtained parameter equations of the small wheel convex circular arc tooth profile ar1 and the large wheel concave circular arc tooth profile br2 are respectively as follows:
Figure BDA0003259328080000047
Figure BDA0003259328080000051
in the formulae (11) and (12) (. rho)a1Is the arc radius, ξ of the small wheel end face convex arc tooth profile ar1a1Is the angular parameter, ξ, of ar1a1aAnd xia1bAre respectively xia1Minimum and maximum values of; rhob2Is the arc radius, xi, of the concave arc tooth profile br2 of the end surface of the bull wheelb2Angular parameter of br2, ξb2aAnd xib2bAre respectively xib2Minimum and maximum values of, wherein ξa1bThe value of the small wheel is obtained by solving the intersection point of the addendum circle of the small wheel and the convex circular arc tooth profile ar1 of the small wheel;
ξa1a＝ξa1b-π/5.5； (13)
ξb2a＝ξb2b-π/6.5； (14)
respectively at the circle center o of the small wheel concave arc tooth profile Br1b1And the circle center o of the big wheel convex circular arc tooth profile Ar2a2Establishing a local coordinate system Sb1(ob1-xb1yb1zb1) And Sa2(oa2-xa2ya2za2) The parameter equations of the small wheel convex arc tooth profile Br1 and the large wheel concave arc tooth profile Ar2 are respectively as follows:
Figure BDA0003259328080000052
Figure BDA0003259328080000053
in the formulae (15) and (16), ρb1Is the arc radius, xi, of the small wheel end surface concave arc tooth profile Br1b1Angle parameter, ξ, of Br1b1aAnd xib1bAre respectively xib1Minimum and maximum values of; rhoa2Is the arc radius, xi, of the convex arc tooth profile Ar2 of the end surface of the bull wheela2Is the angular parameter, ξ, of Ar2a2aAnd xia2bAre respectively xib2Minimum and maximum values of, wherein ξa2bThe value of the curve is obtained by solving the intersection point of the top circle of the bull gear and the convex circular arc tooth profile Ar2 of the bull gear;
ξa2a＝ξa2b-π/5.5 (17)
ξb1a＝ξb1b-π/6.5； (18)
obtaining the right convex circular arc tooth profile ar1 of the end surface of the small wheel gear tooth at S through coordinate transformationpThe parametric equation for the coordinate system is:
Figure BDA0003259328080000054
obtaining the right concave circular arc tooth profile br1 of the end surface of the small gear tooth at S through coordinate transformationpThe parametric equation for the coordinate system is:
Figure BDA0003259328080000061
obtaining the right side concave circular arc tooth profile br2 of the end surface of the bull wheel tooth at S by coordinate transformationgThe parametric equation for the coordinate system is:
Figure BDA0003259328080000062
obtaining the right convex circular arc tooth profile ar2 of the gear end surface of the bull gear at S through coordinate transformationgThe parametric equation for the coordinate system is:
Figure BDA0003259328080000063
the right transition curve hr1 from point P0PAnd P1PAnd its tangent vector T0PAnd T1PDetermine, point P0PFrom Rh1Determined so that the value xi of the tooth profile br1b1bCan be solved to obtain1PRadius R of small gear rootf1Angle delta of sum1RJointly determining, the parameter equation for solving the right side transition curve hr1 of the small gear tooth end surface is as follows:
Figure BDA0003259328080000064
Figure BDA0003259328080000065
in formulae (23) and (24), xp(P0P)，yp(P0P)，zp(P0P) Are respectively a point P0PThree coordinate axis component of (2), xp(P1P)， yp(P1P)，zp(P1P) Are respectively a point P1PThree coordinate axis component of (2), xp(T0P)，yp(T0P)，zp(T0P) Are respectively a point P0PUnit tangent vector T of0PThree coordinate axis component of (2), xp(T1P)，yp(T1P)，zp(T1P) Are respectively a point P1PUnit tangent vector T of1PThree coordinate axis component of (1), mtIs the end face modulus, b1，b2，b3，b4To calculate the parameters, THIs in the shape of tooth root transition curveShape control parameter, T is more than or equal to 0.2H≤1.5，tHFor calculating the parameter, t is more than or equal to 0H≤1；
The transition curve hr2 from the right side of the bull gear tooth end face0GAnd P1GAnd its tangent vector T0GAnd T1GDetermine, point P0GFrom Rh2Determined so that the value xi of the tooth profile br2b2bCan be solved to obtain1GBy the radius R of the root circle of the big gearf2Angle delta of sum2RJointly determining, the parameter equation for obtaining the right transition curve hr2 of the bull gear tooth end face is as follows:
Figure BDA0003259328080000071
in the formula (25), xg(P0G)，yg(P0G)，zg(P0G) Are respectively a point P0GThree coordinate axis component of (2), xg(P1G)，yg(P1G)， zg(P1G) Are respectively a point P1GThree coordinate axis component of (2), xg(T0G)，yg(T0G)，zg(T0G) Are respectively a point P0GUnit tangent vector T of0GThree coordinate axis component of (2), xg(T1G)，yg(T1G)，zg(T1G) Are respectively a point P1GUnit tangent vector T of1GThree coordinate axis components of (a);
when determining the number of teeth Z of the small gear1A transmission ratio i12Normal modulus mnCoincidence degree epsilon, linear proportionality coefficient k and pressure angle alpha of two meshing points of small wheelt1And alphat2Coefficient of diameter phidRoot transition curve shape control parameter THUndetermined coefficient c of motion of meshing point1And the motion rule, the contact line and the meshing line, the end face gear tooth profiles and the correct installation distance of the small wheel and the large wheel are correspondingly determined, and the gear tooth surface structures of the small wheel and the large wheel can be determined, so that the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is obtained.
Further, the method can be used for preparing a novel materialThe small wheel and the large wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, the two meshing points have unequal end surface pressure angles, and the end surface pressure angle of the small wheel concave circular arc tooth profile meshing point is smaller than that of the small wheel convex circular arc tooth profile meshing point so as to enhance the bending strength of a tooth root, namely alphat2＜αt1。
Furthermore, the contact ratio of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is twice of that of single-point contact meshing, the contact ratio of the single-point contact meshing needs to be more than 1, and the contact ratio calculation formula of the unequal pressure angle end face double-arc gear mechanism meshed with the double points is as follows
Figure BDA0003259328080000072
The maximum value of the motion parameter variable of the meshing point of the parallel shaft driven double-arc gear mechanism with unequal pressure angle end surfaces is obtained as
Figure BDA0003259328080000073
The design needs to be carried out according to the value epsilon of the contact ratio, the linear proportionality coefficient k and the number Z of the small gear teeth1Comprehensively determining the meshing point MR1Is measured by the motion parameter variable of (1).
Furthermore, the input shaft and the output shaft which are connected by the small wheel and the big wheel have interchangeability, namely, the small wheel is connected with the input shaft and the big wheel is connected with the output shaft, or the big wheel is connected with the input shaft and the small wheel is connected with the output shaft, and the small wheel and the big wheel are respectively corresponding to the speed reduction transmission mode or the speed increase transmission mode of a double-arc gear mechanism with unequal pressure angle end surfaces in parallel shaft transmission; the constant-speed transmission application with the transmission ratio of 1 of the mechanism is realized only when the number of teeth of the small gear and the large gear is equal.
Further, the rotation direction of the input shaft connected with the driver is clockwise or counterclockwise, so that the forward and reverse rotation transmission of the small wheel or the large wheel is realized.
The unequal pressure angle end face double-circular-arc gear mechanism driven by the parallel shaft is a gear mechanism fundamentally innovated on the theory of the traditional double-circular-arc gear transmission mechanism, and the design method of the unequal pressure angle end face double-circular-arc gear mechanism is also different from the design method of the traditional gear mechanism based on a curved surface meshing equation, and is an active design method based on a meshing line parameter equation. The meshing mode of the parallel shaft driven double-arc gear mechanism with unequal pressure angle end faces is a point meshing mode based on a meshing line parameter equation with equal sliding rate, the relative sliding speeds of all meshing points on two meshing lines are equal, and the uniform friction and abrasion of the tooth surfaces can be ensured. The invention relates to a parallel shaft transmission unequal pressure angle end face double-arc gear mechanism, which is characterized in that the bending strength of the gear root is effectively improved through the end face unequal pressure angle design and the tooth root transition curve design of the end face double-arc.
Compared with the prior art, the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts has the following beneficial effects:
1. the design of the unequal pressure angle end surface double-arc gear mechanism driven by the parallel shaft is based on an active design method of a meshing line parameter equation, an end surface double-point concave-convex meshing tooth surface is constructed, and the relative sliding speeds of all meshing points on two meshing lines are equal, so that the tooth surface abrasion loss is the same, and the lubrication is easy.
2. The two pressure angles of the end face tooth profile meshing points of the end face double-arc gear mechanism with unequal pressure angles in parallel shaft transmission are designed to be unequal, so that the bending strength of the tooth root can be increased, the service life of the gear is prolonged to the maximum extent, the structural size is reduced, and the transmission of heavy-load power is facilitated.
3. The contact ratio design of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is free, the structural shape of the tooth profile can be determined through the pre-design of the contact ratio, the uniform distribution of the load is realized, and the dynamic performance is improved.
4. The unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft has no undercut, the minimum tooth number is 1, compared with the existing parallel shaft involute gear and other mechanisms, the single-stage large transmission ratio high overlap ratio transmission can be realized, and simultaneously, as the tooth number can be designed to be smaller, and larger tooth thickness and module can be designed when the diameter of the gear is the same, the double-arc gear mechanism has higher bending strength 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.
5. According to the unequal-pressure-angle end face double-arc gear mechanism driven by the parallel shafts, the small wheel and the large wheel can have similar tooth root bending strength by adjusting the optimized design of the control parameters of the tooth root transition curve shape, the equal-strength design of the transmission mechanism is realized, and the service life of equipment is further prolonged.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
fig. 1 is a schematic structural diagram of a parallel shaft transmission double-circular-arc gear mechanism with unequal pressure angle end faces.
Fig. 2 is a schematic diagram of a space meshing coordinate system of the parallel shaft transmission unequal pressure angle end face double-circular-arc gear mechanism.
Fig. 3 shows the tooth profile composition structure and coordinate system of the end faces of the large wheel and the small wheel in fig. 1 and 2.
Fig. 4 is a partially detailed and enlarged view of fig. 3.
Fig. 5 is a three-dimensional view of the small wheel of fig. 1 of the present invention.
Fig. 6 is a three-dimensional view of the large wheel of fig. 1 in accordance with the present invention.
FIG. 7 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-driver, 2-small wheel, 3-input shaft, 4-output shaft, 5-big wheel, 6-small wheel section cylinder and 7-small wheel contact line CR2p8-large wheel contact line CR2g9-line of engagement KR2-KR210-large wheel contact line CR1g11-line of engagement KR1-KR112-small wheel contact line CR1p13-big wheel pitch cylinder, 14-small wheel tooth right side tooth root transition curve, 15-small wheel tooth right side concave circular arc tooth profile, 16-small wheel tooth right side straight line tooth profile, 17-small wheelThe gear tooth right side convex circular arc tooth profile is 18-gear tooth right side tooth root transition curve of the bull gear, 19-gear tooth right side concave circular arc tooth profile of the bull gear, 20-gear tooth right side straight line tooth profile of the bull gear, and 21-gear tooth right side convex circular arc tooth profile of the bull gear.
Detailed Description
Various exemplary embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be noted that: the relative arrangement of the components and steps, the numerical expressions and numerical values set forth in these embodiments do not limit the scope of the present invention unless specifically stated otherwise.
Meanwhile, it should be understood that the sizes of the respective portions shown in the drawings are not drawn in an actual proportional relationship for the convenience of description.
The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the invention, its application, or uses.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to specific embodiments and the accompanying drawings.
Techniques, methods, and apparatus known to those of ordinary skill in the relevant art may not be discussed in detail but are intended to be part of the specification where appropriate.
In all examples shown and discussed herein, any particular value should be construed as merely illustrative, and not limiting. Thus, other examples of the exemplary embodiments may have different values.
It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, further discussion thereof is not required in subsequent figures.
Example 1: the invention provides an unequal pressure angle end face double-arc gear mechanism for parallel shaft transmission, which is applied to transmission with the transmission ratio of 2 between parallel shafts, and the contact ratio of the parallel shafts is designed to be epsilon 4. The structure of the device is shown in figure 1, and comprises a small wheel 2 and a large wheel 5, wherein the small wheel 2 and the large wheel 5 form a pair of transmission pairs, the small wheel 2 is connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1, and the large wheel 5 is connected with an output shaft 4, namely the large wheel 5 is connected with a driven load through the output shaft 4; the axes of the small wheel 2 and the large wheel 5 are parallel to each other. FIG. 2 is a schematic diagram of a space meshing coordinate system of the unequal pressure angle end face double-circular-arc gear mechanism driven by parallel shafts.
Referring to fig. 1, 2, 3, 4 and 5, the pitch cylinder 6 of the small wheel has a radius R1The radius of the addendum circle of the small wheel is Ra1Root circle radius of Rf1The outer surface of the cylinder of the tooth root of the small wheel is uniformly provided with spiral wheel teeth, and the tooth profile of the end face of each wheel tooth is in an axisymmetric form, namely, the left tooth profile and the right tooth profile of the end face are axisymmetric. Taking the right side tooth profile of the end face of the small wheel as an example, the right side tooth profile of the small wheel tooth is composed of a right side convex circular arc tooth profile 17 of the small wheel tooth, a right side straight line tooth profile 16 of the small wheel tooth, a right side concave circular arc tooth profile 15 of the small wheel tooth and a right side tooth root transition curve 14 of the small wheel tooth sequentially from the tooth top to the tooth root.
Referring to fig. 1, 2, 3, 4 and 6, the pitch cylinder 13 of the bull wheel has a radius R2The radius of the addendum circle of the small wheel is Ra2Root circle radius of Rf2The outer surface of the cylinder of the gear tooth root is uniformly provided with spiral gear teeth, and the tooth profile of the end surface of the gear tooth is in an axisymmetric form, namely, the left tooth profile and the right tooth profile of the end surface are axisymmetric. Taking the right side tooth profile of the end face of the bull wheel as an example, the right side tooth profile of the bull wheel tooth sequentially consists of a right convex circular arc tooth profile 21 of the bull wheel tooth, a right linear tooth profile 20 of the bull wheel tooth, a right concave circular arc tooth profile 19 of the bull wheel tooth and a right tooth root transition curve 18 of the bull wheel tooth from the tooth top to the tooth root.
The teeth of the small wheel and the big wheel are both helical teeth, and the tooth surface of the small wheel is the tooth profile of the end surface of the small wheel along the contact line C of the small wheel R1p12 is obtained by a helical movement, the pitch of which is related to the contact line C R1p12 are the same; the gear surface of the bull gear is the gear profile of the end surface of the bull gear along the contact line C of the bull gear R1g10 are spirally moved, the pitch of the spiral movement is in contact with a bull wheel contact line C R1g10 are the same; at any end face, the end face tooth profiles of the small wheel and the large wheel are at two points MR1And MR2Simultaneous engagement, the spatial movement of the two points of engagement each forming a line of engagement KR1-K R111 and meshing line KR2-KR29；
The small wheel 2 is connected with an input shaft 3 and driven by the driver 1 to rotate, so that convex circular arc tooth profiles and concave circular arc tooth profiles on the small wheel and the large wheel are respectively positioned at a meshing point MR1And MR2And simultaneously, the transmission of motion and power between the parallel shafts is realized, and the driver 4 is a motor in the embodiment.
The tooth surface contact line of the small wheel and the large wheel and the end surface tooth profile structure of the small wheel and the large wheel are determined by the following method: at op--xp,yp,zp、ok--xk,yk,zkAnd og--xg,yg,zgIn three spatial coordinate systems, zpThe axis of rotation of the shaft and of the small wheel coinciding, zgThe axis of rotation of the shaft and the bull wheel coinciding, zkShaft-to-pass meshing point MR1Engagement line K ofR1-KR1Coincide with and zkAxis and zp、zgAxes parallel to each other, xpAnd xgThe axes being coincident, xkAnd xgAxis parallel, opogA is a; coordinate system o1-- x1,y1,z1Fixedly connected to the small wheel, coordinate system o2--x2,y2,z2Fixedly connected with the big wheel, and a coordinate system o of the small wheel and the big wheel1--x1,y1,z1And o2-- x2,y2,z2At the starting position respectively with the coordinate system op--xp,yp,zpAnd og--xg,yg,zgCoincident, small wheels at uniform angular velocity ω1Around zpThe shaft rotates clockwise and the bull wheel rotates at a uniform angular velocity omega2Around zgThe axis rotates anticlockwise, after a period of time from the starting position, the coordinate system o1-- x1,y1,z1And o2--x2,y2,z2Respectively rotate when the meshing point is MR1And MR2Small winding zpThe shaft rotates through
Figure BDA0003259328080000111
Corner, large wheel winding zgThe shaft rotates through
Figure BDA0003259328080000112
An angle;
when the small wheel and the big wheel are in mesh transmission, a mesh point M is setR1From the origin o of coordinateskBeginning along the line of engagement KR1-KR1Exercise, MR1The parametric equation for point motion is:
Figure BDA0003259328080000113
at the same time, the mesh point MR2Along the line of engagement K at the same speed of movementR2-KR2Exercise, MR2The parametric equation for point motion is:
Figure BDA0003259328080000114
t in the formulas (1) and (2) is a meshing point MR1T is more than or equal to 0 and less than or equal to delta t; c. C1The undetermined coefficient of the meshing point movement is expressed in millimeters (mm); in order to ensure a fixed ratio engagement, the rotational angles of the small and large wheels must be linear with the movement of the engagement point, as follows:
Figure BDA0003259328080000115
in the formula (3), k is a linear proportionality coefficient of the movement of the meshing point, and the unit of k is radian (rad); i.e. i12The transmission ratio between the small wheel and the large wheel is set;
when the point of engagement MR1Along the line of engagement KR1-KR1While in motion, point MR1Simultaneously, contact lines C are respectively formed on the convex circular arc tooth surface of the small wheel and the concave circular arc tooth surface of the large wheelR1pAnd CR1g(ii) a When the point of engagement MR2Along the line of engagement KR2-KR2While in motion, point MR2Simultaneously, the concave arc tooth surface of the small wheel and the convex of the large wheelThe arc tooth surfaces form contact lines C respectivelyR2pAnd CR2g(ii) a From the coordinate transformation, a coordinate system o can be obtainedp--xp,yp,zp、ok--xk,yk,zkAnd og—xg,yg,zg、o1—x1,y1,z1And o2—x2,y2,z2The homogeneous coordinate transformation matrix in between is:
Figure BDA0003259328080000116
wherein,
Figure BDA0003259328080000117
Figure BDA0003259328080000121
in the formulae (5) and (6), R1Is the pitch cylinder radius of the small wheel, R2Is the pitch cylinder radius of the bull wheel, and PM is the meshing point MR1And MR2Distance to node P, αt1Is a meshing point MR1End face pressure angle of alphat2Is a meshing point MR2The end face pressure angle of (1);
obtaining the contact line C of the convex arc tooth surface of the small wheel by the formulas (1) and (4)R1pThe parameter equation of (1) is as follows:
Figure BDA0003259328080000122
obtaining the contact line C of the concave circular arc tooth surface of the bull gear by the formulas (1) and (4)R1gThe parameter equation of (1) is as follows:
Figure BDA0003259328080000123
calculating a small wheel concave arc tooth surface contact line C by the formulas (2) and (4)R2pThe parameter equation of (1) is as follows:
Figure BDA0003259328080000124
obtaining a contact line C of the convex arc tooth surface of the bull wheel according to the formulas (2) and (4)R2gThe parameter equation of (1) is as follows:
Figure BDA0003259328080000125
the end tooth profiles of the small wheel and the large wheel are determined by the following method:
respectively at the circle center o of the convex circular arc tooth profile ar1 of the small wheela1And the circle center o of the big wheel concave circular arc tooth profile br2b2Establishing a local coordinate system Sa1(oa1-xa1ya1za1) And Sb2(ob2-xb2yb2zb2) The obtained parameter equations of the small wheel convex circular arc tooth profile ar1 and the large wheel concave circular arc tooth profile br2 are respectively as follows:
Figure BDA0003259328080000126
Figure BDA0003259328080000127
in the formulae (11) and (12) (. rho)a1Is the arc radius, ξ of the small wheel end face convex arc tooth profile ar1a1Is the angular parameter, ξ, of ar1a1aAnd xia1bAre respectively xia1Minimum and maximum values of; rhob2Is the arc radius, xi, of the concave arc tooth profile br2 of the end surface of the bull wheelb2Angular parameter of br2, ξb2aAnd xib2bAre respectively xib2Minimum and maximum values of, wherein ξa1bThe value of the small wheel is obtained by solving the intersection point of the addendum circle of the small wheel and the convex circular arc tooth profile ar1 of the small wheel;
ξa1a＝ξa1b-π/5.5； (13)
ξb2a＝ξb2b-π/6.5； (14)
respectively at the circle center o of the small wheel concave arc tooth profile Br1b1And the circle center o of the big wheel convex circular arc tooth profile Ar2a2Establishing a local coordinate system Sb1(ob1-xb1yb1zb1) And Sa2(oa2-xa2ya2za2) The parameter equations of the small wheel convex arc tooth profile Br1 and the large wheel concave arc tooth profile Ar2 are respectively as follows:
Figure BDA0003259328080000131
Figure BDA0003259328080000132
in the formulae (15) and (16), ρb1Is the arc radius, xi, of the small wheel end surface concave arc tooth profile Br1b1Angle parameter, ξ, of Br1b1aAnd xib1bAre respectively xib1Minimum and maximum values of; rhoa2Is the arc radius, xi, of the convex arc tooth profile Ar2 of the end surface of the bull wheela2Is the angular parameter, ξ, of Ar2a2aAnd xia2bAre respectively xib2Minimum and maximum values of, wherein ξa2bThe value of the curve is obtained by solving the intersection point of the top circle of the bull gear and the convex circular arc tooth profile Ar2 of the bull gear;
ξa2a＝ξa2b-π/5.5； (17)
ξb1a＝ξb1b-π/6.5； (18)
obtaining the right convex circular arc tooth profile ar1 of the end surface of the small wheel gear tooth at S through coordinate transformationpThe parametric equation for the coordinate system is:
Figure BDA0003259328080000133
obtaining the right concave circular arc tooth profile br1 of the end surface of the small gear tooth at S through coordinate transformationpThe parametric equation for the coordinate system is:
Figure BDA0003259328080000134
obtaining the right side concave circular arc tooth profile br2 of the end surface of the bull wheel tooth at S by coordinate transformationgThe parametric equation for the coordinate system is:
Figure BDA0003259328080000141
obtaining the right convex circular arc tooth profile ar2 of the gear end surface of the bull gear at S through coordinate transformationgThe parametric equation for the coordinate system is:
Figure BDA0003259328080000142
the right transition curve hr1 from point P0PAnd P1PAnd its tangent vector T0PAnd T1PDetermine, point P0PFrom Rh1Determined so that the value xi of the tooth profile br1b1bCan be solved to obtain1PRadius R of small gear rootf1Angle delta of sum1RJointly determining, the parameter equation for solving the right side transition curve hr1 of the small gear tooth end surface is as follows:
Figure BDA0003259328080000143
Figure BDA0003259328080000144
in formulae (23) and (24), xp(P0P)，yp(P0P)，zp(P0P) Are respectively a point P0PThree coordinate axis component of (2), xp(P1P)， yp(P1P)，zp(P1P) Are respectively a point P1PThree coordinate axis component of (2), xp(T0P)，yp(T0P)，zp(T0P) Are respectively a point P0PUnit tangent vector T of0PThree coordinate axis component of (2), xp(T1P)，yp(T1P)，zp(T1P) Are respectively a point P1PUnit tangent vector T of1PThree coordinate axis component of (1), mtIs the end face modulus, b1，b2，b3，b4To calculate the parameters, THThe control parameter is the shape of the tooth root transition curve, T is more than or equal to 0.2H≤1.5，tHFor calculating the parameter, t is more than or equal to 0H≤1；
The transition curve hr2 from the right side of the bull gear tooth end face0GAnd P1GAnd its tangent vector T0GAnd T1GDetermine, point P0GFrom Rh2Determined so that the value xi of the tooth profile br2b2bCan be solved to obtain1GBy the radius R of the root circle of the big gearf2Angle delta of sum2RJointly determining, the parameter equation for obtaining the right transition curve hr2 of the bull gear tooth end face is as follows:
Figure BDA0003259328080000145
in the formula (25), xg(P0G)，yg(P0G)，zg(P0G) Are respectively a point P0GThree coordinate axis component of (2), xg(P1G)，yg(P1G)，zg(P1G) Are respectively a point P1GThree coordinate axis component of (2), xg(T0G)，yg(T0G)，zg(T0G) Are respectively a point P0GUnit tangent vector T of0GThree coordinate axis component of (2), xg(T1G)，yg(T1G)，zg(T1G) Are respectively a point P1GUnit tangent vector T of1GThree coordinate axis components of (a);
in this embodiment:
t-motion parameter variable of the meshing point M, and t belongs to [0, delta t ];
k-is a linear proportionality coefficient;
Z1-number of small gear teeth;
Z2-a large gear tooth number;
mn-a normal modulus;
R1is the pitch cylinder radius of the small wheel, R1＝mtZ1/2； (26)
R2Is the pitch cylinder radius of the bull wheel, R2＝i12R1； (27)
i12Is the transmission ratio of the small wheel to the large wheel,
Figure BDA0003259328080000151
a-the relative positions of the axes of the small wheel and the large wheel: a ═ R1+R2； (29)
b-width of teeth of small wheel and large wheel: b ═ c1Δt＝2ΦdR1； (30)
Φd-a diameter factor;
a beta-pitch helix angle of the pitch circle,
Figure BDA0003259328080000152
mtend face modulus, mt＝mn/cosβ； (32)
Ra1Radius of addendum circle of the small wheel, Ra1＝R1+mt； (33)
Rf1Radius of root circle of small wheel, Rf1＝R1-1.25mt； (34)
Rh1-starting point P of transition curve at root of small wheel0PRadius to the centre of rotation of the small wheel, Rh1＝R1-mt； (35)
Ra2Radius of the top circle of the big gear teeth, Ra2＝R2+mt； (36)
Rf2Radius of root circle of big gear tooth, Rf2＝R2-1.25mt； (37)
Rh2-big wheel root transition curve starting point P0GRadius to centre of rotation of bull wheel, Rh2＝R2-mt； (38)
PM-horizontal distance of mesh point to node,
Figure BDA0003259328080000153
ρa1the radius of the convex arc profile of the end face of the small wheel, pa1＝PM； (40)
ρa2Radius of the convex arc profile of the bull wheel end face, pa2＝ρa1； (41)
ρb2Circular arc radius of the concave circular arc profile of the bull wheel end face, pb2＝ρa1+Δρ； (42)
Δ ρ -difference in radius of concave and convex circular arcs, Δ ρ is 0.5mt； (43)
ρb1The circular arc radius of the concave circular arc profile of the end face of the small wheel, pb1＝ρb2； (44)
δ1A central angle is formed by tooth root circular arcs of two adjacent gear teeth of the small wheel,
Figure BDA0003259328080000161
δ2a central angle is formed by tooth root circular arcs of two adjacent gear teeth of the bull wheel,
Figure BDA0003259328080000162
δ1Rthe root part transition curve hr1 on the right side of the small wheel gear toothPoint P1PCorresponding radius and xpThe acute angle of the shaft clamp is formed by the two parts,
Figure BDA0003259328080000163
δ2Rpoint P on the root transition curve hr2 on the right side of the bull gear tooth1GCorresponding radius and xpThe acute angle of the shaft clamp is formed by the two parts,
Figure BDA0003259328080000164
wherein: axes of the coordinate system, a, b, mn，mt，ρa1，Δρ，R1And R2The units of equal length, radius or distance are millimeters (mm); k is the sum of the k,
Figure BDA0003259328080000165
β，ξa1，ξb1，ξa2，ξb2the unit of angle is radians (rad); pressure angle alphat1，αt2In degrees (°);
when determining the number of teeth Z of the small gear1A transmission ratio i12Normal modulus mnCoincidence degree epsilon, linear proportionality coefficient k and pressure angle alpha of two meshing points of small wheelt1And alphat2Coefficient of diameter phidRoot transition curve shape control parameter THWhen the gear is in use, the undetermined coefficient c1 of the motion of the meshing point, the motion rule, the contact line and the meshing line, the tooth profiles of the end face gear teeth of the small wheel and the large wheel and the correct installation distance of the end face gear teeth are correspondingly determined, and the tooth surface structures of the gear teeth of the small wheel and the large wheel can be determined, so that the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is obtained.
The contact ratio of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is twice of that of single-point contact meshing, and the contact ratio of the single-point contact meshing is required to be more than 1. The contact ratio calculation formula of the double-arc gear mechanism with the end surfaces of unequal pressure angles meshed with two points is
Figure BDA0003259328080000166
The maximum value of the motion parameter variable of the meshing point of the parallel shaft driven double-arc gear mechanism with unequal pressure angle end surfaces is obtained as
Figure BDA0003259328080000167
The design needs to be carried out according to the value epsilon of the contact ratio, the linear proportionality coefficient k and the number Z of the small gear teeth1Comprehensively determining the meshing point MR1Is measured by the motion parameter variable of (1).
The small wheel and the big wheel are in end surface double-point contact concave-convex meshing transmission, the two meshing points have unequal end surface pressure angles, and the end surface pressure angle of the small wheel concave circular arc tooth profile meshing point is smaller than that of the small wheel convex circular arc tooth profile meshing point so as to enhance the bending strength of a tooth root, namely alphat2＜αt1。
When the above formula is used, the values of the relevant parameters are: z1＝18，i12＝2，m n3 millimeters (mm), 4, pi radians (rad), alphat1＝25°，αt2＝15°，Φd＝1，TH0.5, obtained by substituting the formulae (26) to (48)
Figure BDA0003259328080000171
85.7930 mm (a), 57.1953 mm (b), c1257.3789 millimeters (mm), PM 4.9912 millimeters (mm);
then, the numerical values are substituted into the formula (7) -formula (25) to obtain a contact line parameter equation and an end face tooth profile parameter equation of the small wheel and the large wheel in the example, and then the gear tooth structures of the small wheel and the large wheel are obtained according to spiral motion, and the assembly can be carried out according to the correct center distance.
When the driver 1 drives the input shaft 3 and the small wheel 2 to rotate, because two pairs of adjacent gear teeth are in a meshing state when the small wheel 2 and the large wheel 5 are correctly installed, the preset contact ratio epsilon of the pair of double-arc gears is 4, at least two pairs of gear teeth participate in meshing transmission at the same time at each moment, and the upper and lower arc tooth profiles of each pair of gear teeth are respectively provided with a meshing point, so that continuous and stable meshing transmission of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts in the rotating motion is realized. The rotation direction of an input shaft connected with the motor is clockwise, and the speed reduction transmission mode of the double-arc gear mechanism with the end faces at unequal pressure angles corresponding to the parallel shaft transmission is used for realizing speed reduction and torque increase transmission of anticlockwise rotation of the large wheel.
Example 2: the unequal pressure angle end face double-arc gear mechanism for parallel shaft transmission is applied to acceleration transmission of parallel shafts. As shown in fig. 6, a large wheel 5 is connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1, and a small wheel 2 is connected with an output shaft 4, namely, the small wheel 2 is connected with a driven load through the output shaft 4; the axes of the small wheel 2 and the large wheel 5 are parallel. In the present embodiment, the number of teeth of the large gear 5 is 36, the number of teeth of the small gear 2 is 18, and the design contact ratio ∈ is 4. When the input shaft 3 drives the large wheel 5 to rotate, because two pairs of adjacent gear teeth are in a meshing state when the large wheel 5 and the small wheel 2 are installed, the preset contact ratio epsilon of the pair of double-arc gears is 4, at least two pairs of gear teeth participate in meshing transmission at the same time at each moment, and the upper and lower two arc tooth profiles of each pair of gear teeth respectively consist of a meshing point, so that continuous and stable meshing transmission of the unequal-pressure-angle end-face double-arc gear mechanism driven by the parallel shafts in the rotating motion is realized. At this time, the speed increasing ratio of the large wheel to the small wheel is 2, namely the transmission ratio of the small wheel to the large wheel is 2.
The relevant parameters take the values as follows: z1＝18，i12＝2，m n2 mm,. epsilon.4,. alpha.t1＝28°，αt2＝14°，Φd＝1，THObtained by substituting formula (26) to formula (48) for 0.6
Figure BDA0003259328080000172
57.1953 mm (a), 38.1302 mm (b), c1171.5860, PM 3.3275 millimeters (mm);
then, the numerical values are substituted into the formula (7) -formula (25) to obtain a contact line parameter equation and an end face tooth profile parameter equation of the small wheel and the large wheel in the example, and then the distribution is moved according to the spiral, so that the tooth structures of the small wheel and the large wheel are obtained, and the assembly can be carried out according to the correct center distance.
The rotation direction of an input shaft connected with the driver is anticlockwise, and the driving mode of the double-circular-arc gear mechanism is corresponding to the speed-increasing driving mode of the parallel shaft driven unequal-pressure-angle end face double-circular-arc gear mechanism, so that the clockwise rotation of the small wheel is driven.
The design of the unequal pressure angle end surface double-arc gear mechanism driven by the parallel shaft is based on an active design method of a meshing line parameter equation, an end surface double-point concave-convex meshing tooth surface is constructed, and the relative sliding speeds of all meshing points on two meshing lines are equal, so that the tooth surface abrasion loss is the same, and the tooth surface is easy to lubricate; the pressure angles of the end surface meshing points of the end surface double-arc gear mechanism with unequal pressure angles in parallel shaft transmission are designed to be unequal, so that the bending strength of the tooth root can be increased, the service life of the gear is prolonged to the maximum extent, the structure size is reduced, and the transmission of heavy-load power is facilitated; the contact ratio design of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is free, the shape of the tooth profile structure can be determined through the pre-design of the pressure angle and the contact ratio, the uniform distribution of the load is realized, and the dynamic characteristic is improved; the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft has no undercut, the minimum tooth number is 1, compared with the existing mechanisms such as a parallel shaft involute gear, the single-stage large transmission ratio high overlap ratio transmission can be realized, and simultaneously, because the tooth number is small, larger tooth thickness can be designed when the diameter of the gear is the same, so that the parallel shaft involute gear mechanism has higher strength 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; the unequal-pressure-angle end face double-arc gear mechanism driven by the parallel shafts can also enable the small wheel and the large wheel to have similar tooth root bending strength by adjusting the optimized design of root transition curve parameter values, realize the equal-strength design of the transmission mechanism and further prolong the service life of equipment.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or system. Without further limitation, an element defined by the phrase "comprising an … …" does not exclude the presence of other like elements in a process, method, article, or system that comprises the element.
The above-mentioned serial numbers of the embodiments of the present invention are merely for description and do not represent the merits of the embodiments. In the unit claims enumerating several means, several of these means may be embodied by one and the same item of hardware. The use of the words first, second, third and the like do not denote any order, but rather the words first, second and the like may be interpreted as indicating any order.
The above description is only a preferred embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes, which are made by using the contents of the present specification and the accompanying drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (8)

1. The utility model provides a driven not equal pressure angle terminal surface double circular arc gear mechanism of parallel shaft, includes a pair of gear pair that steamboat and bull wheel are constituteed, the steamboat links firmly with the driver through the input shaft, the output shaft is connected to the bull wheel, the steamboat with the axis of bull wheel is parallel, its characterized in that: the end face tooth profiles of the small wheel and the large wheel have axial symmetry forms, and the left and right end face tooth profiles respectively consist of a convex arc tooth profile, a straight line tooth profile, a concave arc tooth profile and a tooth root transition curve from the tooth top to the tooth root; the small wheel and the large wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, and the two meshing points have unequal pressure angles; the small wheel is driven by the driver to rotate, stable meshing transmission between parallel shafts is realized through continuous meshing between two pairs of convex circular arc tooth profiles and concave circular arc tooth profiles, two meshing points with different pressure angles respectively form two contact lines with the same pitch on tooth surfaces of the small wheel and the large wheel, and the two contact lines are cylindrical spiral lines; the gear tooth flanks of the small wheel and the large wheel are spiral tooth flanks obtained by cylindrical spiral motion of end face tooth profiles along respective contact lines, the screw pitch of the spiral tooth flanks is equal to that of the contact lines, and the spiral directions of the gear teeth of the small wheel and the gear teeth of the large wheel are opposite.
2. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the small wheel and the big wheel are in double-point contact concave-convex meshing transmission, and two meshing points of the small wheel and the big wheel are respectively a meshing point M of a convex circular arc tooth profile of the teeth of the small wheel and a concave circular arc tooth profile of the teeth of the big wheelR1And the meshing point M of the concave circular arc tooth profile of the adjacent gear teeth of the small gear and the convex circular arc tooth profile of the adjacent gear teeth of the large gearR2(ii) a The normal lines of the tooth profile meshing points of the two pairs of concave-convex circular arc end surfaces intersect at the same point, and the point is a tangent point of a pitch circle of the pair of double-circular-arc gears, namely a node; the horizontal distances from the two meshing points to the node are both PM; when the pair of parallel shafts drive the double-arc gear mechanism with unequal pressure angle end faces, two meshing points MR1And MR2Has the same axial movement speed and forms two spatial meshing lines K respectivelyR1-KR1And KR2-KR2And each form two contact lines of the tooth faces of the small wheel and the large wheel.
3. The unequal pressure angle end face double-circular-arc gear mechanism of parallel shaft transmission according to claim 2, characterized in that: the tooth surface contact line of the small wheel and the large wheel is determined by the following method:
at op--xp,yp,zp、ok--xk,yk,zkAnd og--xg,yg,zgIn three spatial coordinate systems, where op、okAnd ogRespectively the origin, x, of three spatial coordinate systemsp、xkAnd xgX-axis, y of three spatial coordinate systems respectivelyp、ykAnd ygX-axis, z, of three spatial coordinate systems, respectivelyp、zkAnd zgZ-axes, z, of three spatial coordinate systems, respectivelypThe axis of rotation of the shaft and the small wheel coinciding, zgThe axis of rotation of the shaft and the bull wheel coinciding, zkShaft-to-pass meshing point MR1Engagement line K ofR1-KR1Coincide with and zkAxis and zp、zgAxes parallel to each other, xpAnd xgThe axes being coincident, xkAnd xgAxis parallel, opAnd ogA is a; coordinate system o1--x1,y1,z1Fixedly connected with the small wheel and having a coordinate system o2--x2,y2,z2Fixedly connected with the large wheel and a small wheel coordinate system o1--x1,y1,z1And a large wheel coordinate system o2--x2,y2,z2At the starting position respectively with the coordinate system op--xp,yp,zpAnd og--xg,yg,zgCoincident, the small wheels being at a uniform angular velocity ω1Around zpThe shaft rotates clockwise and the bull wheel rotates at a uniform angular velocity ω2Around zgThe axis rotates anticlockwise, after a period of time from the starting position, the coordinate system o1--x1,y1,z1And o2--x2,y2,z2Respectively rotate when the meshing point is MR1And MR2Said small wheel winding zpThe shaft rotates through
Figure FDA0003259328070000021
Angle, said large wheel winding zgThe shaft rotates through
Figure FDA0003259328070000022
An angle;
when the small wheel and the large wheel are in meshing transmission, a meshing point M is setR1From the origin o of coordinateskBeginning along the line of engagement KR1-KR1Exercise, MR1The parametric equation for point motion is:
Figure FDA0003259328070000023
at the same time, the mesh point MR2Along the line of engagement K at the same speed of movementR2-KR2Exercise, MR2The parametric equation for point motion is:
Figure FDA0003259328070000024
wherein t is the meshing point MR1T is more than or equal to 0 and less than or equal to delta t, and delta t is the maximum value of the motion parameter variable; c. C1The undetermined coefficient of the movement of the meshing point is represented by millimeter, and PM is the horizontal distance from the meshing point to the node; in order to ensure that the fixed gear ratio is engaged, the rotation angles of the small wheel and the large wheel and the movement of the engagement point have to be in a linear relationship, and the rotation angles of the small wheel and the large wheel and the movement of the engagement point have the following relationship:
Figure FDA0003259328070000025
in the formula, k is a linear proportionality coefficient of the movement of the meshing point, and the unit of k is radian; i.e. i12The transmission ratio between the small wheel and the large wheel is set;
when the point of engagement MR1Along the line of engagement KR1-KR1While in motion, point MR1Simultaneously, contact lines C are respectively formed on the convex circular arc tooth surface of the small wheel and the concave circular arc tooth surface of the large wheelR1pAnd CR1g(ii) a When the point of engagement MR2Along the line of engagement KR2-KR2While in motion, point MR2Simultaneously, contact lines C are respectively formed on the concave arc tooth surface of the small wheel and the convex arc tooth surface of the large wheelR2pAnd CR2g(ii) a Obtaining a coordinate system o according to the coordinate transformationp--xp,yp,zp、ok--xk,yk,zkAnd og--xg,yg,zg、o1--x1,y1,z1And o2--x2,y2,z2The homogeneous coordinate transformation matrix in between is:
Figure FDA0003259328070000026
wherein,
Figure FDA0003259328070000027
Figure FDA0003259328070000028
in the formula, R1Is the pitch cylinder radius of the small wheel, R2Is the pitch cylinder radius of the bull wheel, and PM is the meshing point MR1And MR2Distance to node P, αt1Is a meshing point MR1End face pressure angle of alphat2Is a meshing point MR2The end face pressure angle of (1);
from said MR1The contact line C of the convex arc tooth surface of the small wheel is obtained by the parameter equation of point motion and the homogeneous coordinate transformation matrixR1pThe parameter equation of (1) is as follows:
Figure FDA0003259328070000031
from said MR1Obtaining the contact line C of the concave circular arc tooth surface of the large wheel by the parameter equation of point motion and the homogeneous coordinate transformation matrixR1gThe parameter equation of (1) is as follows:
Figure FDA0003259328070000032
from said MR2Calculating the small wheel concave arc tooth surface contact line C by using the parameter equation of point motion and the homogeneous coordinate transformation matrixR2pThe parameter equation of (1) is as follows:
Figure FDA0003259328070000033
from said MR2Calculating the contact line C of convex arc tooth surface of large wheel by using the parameter equation of point motion and the homogeneous coordinate transformation matrixR2gThe parameter equation of (1) is as follows:
Figure FDA0003259328070000034
4. the unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the end face tooth profiles of the small wheel and the large wheel are determined by the following method:
respectively at the circle center o of the convex circular arc tooth profile ar1 of the small wheela1And the circle center o of the big wheel concave circular arc tooth profile br2b2Establishing a local coordinate system Sa1(oa1-xa1ya1za1) And Sb2(ob2-xb2yb2zb2) The obtained parameter equations of the small wheel convex circular arc tooth profile ar1 and the large wheel concave circular arc tooth profile br2 are respectively as follows:
Figure FDA0003259328070000035
Figure FDA0003259328070000041
in the formula, ρa1Is the arc radius, ξ of the small wheel end face convex arc tooth profile ar1a1Is the angular parameter, ξ, of ar1a1aAnd xia1bAre respectively xia1Minimum and maximum values of; rhob2Is the arc radius, xi, of the concave arc tooth profile br2 of the end surface of the bull wheelb2Angular parameter of br2, ξb2aAnd xib2bAre respectively xib2Minimum and maximum values of, wherein ξa1bThe value of the small wheel is obtained by solving the intersection point of the addendum circle of the small wheel and the convex circular arc tooth profile ar1 of the small wheel;
ξa1a＝ξa1b-π/5.5；
ξb2a＝ξb2b-π/6.5；
respectively at the circle center o of the small wheel concave arc tooth profile Br1b1And the circle center o of the big wheel convex circular arc tooth profile Ar2a2Establishing a local coordinate system Sb1(ob1-xb1yb1zb1) And Sa2(oa2-xa2ya2za2) The parameter equations of the small wheel convex arc tooth profile Br1 and the large wheel concave arc tooth profile Ar2 are respectively as follows:
Figure FDA0003259328070000042
Figure FDA0003259328070000043
in the formula, ρb1Is the arc radius, xi, of the small wheel end surface concave arc tooth profile Br1b1Angle parameter, ξ, of Br1b1aAnd xib1bAre respectively xib1Minimum and maximum values of; rhoa2Is the arc radius, xi, of the convex arc tooth profile Ar2 of the end surface of the bull wheela2Is the angular parameter, ξ, of Ar2a2aAnd xia2bAre respectively xib2Minimum and maximum values of, wherein ξa2bThe value of the curve is obtained by solving the intersection point of the top circle of the bull gear and the convex circular arc tooth profile Ar2 of the bull gear;
ξa2a＝ξa2b-π/5.5
ξb1a＝ξb1b-π/6.5；
coordinate transformation is used for obtaining the right convex circular arc tooth profile of the end surface of the small wheel gearar1 at SpThe parametric equation for the coordinate system is:
Figure FDA0003259328070000044
obtaining the right concave circular arc tooth profile br1 of the end surface of the small gear tooth at S through coordinate transformationpThe parametric equation for the coordinate system is:
Figure FDA0003259328070000051
obtaining the right side concave circular arc tooth profile br2 of the end surface of the bull wheel tooth at S by coordinate transformationgThe parametric equation for the coordinate system is:
Figure FDA0003259328070000052
obtaining the right convex circular arc tooth profile ar2 of the gear end surface of the bull gear at S through coordinate transformationgThe parametric equation for the coordinate system is:
Figure FDA0003259328070000053
the right transition curve hr1 from point P0PAnd P1PAnd its tangent vector T0PAnd T1PDetermine, point P0PFrom Rh1Determined so that the value xi of the tooth profile br1b1bCan be solved to obtain1PRadius R of small gear rootf1Angle delta of sum1RJointly determining, the parameter equation for solving the right side transition curve hr1 of the small gear tooth end surface is as follows:
Figure FDA0003259328070000054
Figure FDA0003259328070000055
in the formula, xp(P0P)，yp(P0P)，zp(P0P) Are respectively a point P0PThree coordinate axis component of (2), xp(P1P)，yp(P1P)，zp(P1P) Are respectively a point P1PThree coordinate axis component of (2), xp(T0P)，yp(T0P)，zp(T0P) Are respectively a point P0PUnit tangent vector T of0PThree coordinate axis component of (2), xp(T1P)，yp(T1P)，zp(T1P) Are respectively a point P1PUnit tangent vector T of1PThree coordinate axis component of (1), mtIs the end face modulus, b1，b2，b3，b4To calculate the parameters, THThe control parameter is the shape of the tooth root transition curve, T is more than or equal to 0.2H≤1.5，tHFor calculating the parameter, t is more than or equal to 0H≤1；
The transition curve hr2 from the right side of the bull gear tooth end face0GAnd P1GAnd its tangent vector T0GAnd T1GDetermine, point P0GFrom Rh2Determined so that the value xi of the tooth profile br2b2bCan be solved to obtain1GBy the radius R of the root circle of the big gearf2Angle delta of sum2RJointly determining, the parameter equation for obtaining the right transition curve hr2 of the bull gear tooth end face is as follows:
Figure FDA0003259328070000061
in the formula, xg(P0G)，yg(P0G)，zg(P0G) Are respectively a point P0GThree coordinate axis component of (2), xg(P1G)，yg(P1G)，zg(P1G) Are respectively a point P1GThree coordinate axis component of (2), xg(T0G)，yg(T0G)，zg(T0G) Are respectively a point P0GUnit tangent vector T of0GThree coordinate axis component of (2), xg(T1G)，yg(T1G)，zg(T1G) Are respectively a point P1GUnit tangent vector T of1GThree coordinate axis components of (a);
when determining the number of teeth Z of the small gear1A transmission ratio i12Normal modulus mnCoincidence degree epsilon, linear proportionality coefficient k and pressure angle alpha of two meshing points of small wheelt1And alphat2Coefficient of diameter phidRoot transition curve shape control parameter THUndetermined coefficient c of motion of meshing point1And the motion rule, the contact line and the meshing line, the end face gear tooth profiles and the correct installation distance of the small wheel and the large wheel are correspondingly determined, and the gear tooth surface structures of the small wheel and the large wheel can be determined, so that the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is obtained.
5. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the small wheel and the big wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, the two meshing points have unequal end surface pressure angles, and the end surface pressure angle of the small wheel concave circular arc tooth profile meshing point is smaller than that of the small wheel convex circular arc tooth profile meshing point so as to enhance the bending strength of a tooth root, namely alphat2＜αt1。
6. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the contact ratio of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is twice of that of single-point contact meshing, the contact ratio of the single-point contact meshing needs to be more than 1, and the contact ratio calculation formula of the unequal pressure angle end face double-arc gear mechanism meshed with two points is
Figure FDA0003259328070000062
The maximum value of the motion parameter variable of the meshing point of the parallel shaft driven double-arc gear mechanism with unequal pressure angle end surfaces is obtained as
Figure FDA0003259328070000063
The design needs to be carried out according to the value epsilon of the contact ratio, the linear proportionality coefficient k and the number Z of the small gear teeth1Comprehensively determining the meshing point MR1Is measured by the motion parameter variable of (1).
7. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the input shaft and the output shaft which are connected by 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 the speed reduction transmission mode or the speed increase transmission mode of the double-arc gear mechanism with unequal pressure angle end surfaces in parallel shaft transmission; the constant-speed transmission application with the transmission ratio of 1 of the mechanism is realized only when the number of teeth of the small gear and the large gear is equal.
8. The unequal pressure angle end face double-circular-arc gear mechanism of parallel shaft transmission according to claim 1 or 7, characterized in that: the rotation direction of the input shaft connected with the driver is clockwise or anticlockwise, so that forward and reverse rotation transmission of the small wheel or the large wheel is realized.
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