CN116592114A - Parabolic tooth trace gear mechanism with end face arc and involute combined tooth profile - Google Patents

Parabolic tooth trace gear mechanism with end face arc and involute combined tooth profile Download PDF

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Publication number
CN116592114A
CN116592114A CN202310372303.5A CN202310372303A CN116592114A CN 116592114 A CN116592114 A CN 116592114A CN 202310372303 A CN202310372303 A CN 202310372303A CN 116592114 A CN116592114 A CN 116592114A
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tooth
wheel
small
face
point
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陈祯
肖小平
陈扬枝
杨辉
李颖淇
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Guangdong Ocean University
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Guangdong Ocean University
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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
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • F16H55/0806—Involute profile

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  • Engineering & Computer Science (AREA)
  • General Engineering & Computer Science (AREA)
  • Mechanical Engineering (AREA)
  • Gears, Cams (AREA)

Abstract

The invention provides a parabolic tooth trace gear mechanism with an end face arc and involute combined tooth profile, which comprises a small wheel and a large wheel, wherein an end face tooth profile curve consists of an end face working tooth profile curve and a tooth root transition curve, and the end face tooth profile curves of the small wheel and the large wheel are symmetrical on the left side and the right side; the end face working tooth profiles of the small wheel and the large wheel are combined tooth profiles of an end face arc and an involute; the tooth surfaces of the small wheel and the large wheel are provided with parabolic tooth line structures; at least one pair of gear tooth meshing points of the small wheel and the large wheel are positioned at the node so as to realize pure rolling meshing contact, and meshing lines formed by the meshing points during rotation of the small wheel and the large wheel respectively form two contact lines on the surfaces of the small wheel and the large gear tooth. The invention has the beneficial effects that: the contact line of node engagement is constructed based on the active design of the movement rule of the engagement points, and the contact line is axisymmetric parabola after the node cylindrical surface is unfolded, so that the theoretical value of the relative sliding speed of all the engagement points on the contact line is zero, and the relative sliding and friction abrasion between tooth surfaces are effectively reduced.

Description

Parabolic tooth trace gear mechanism with end face arc and involute combined tooth profile
Technical Field
The invention relates to the technical field of transmission gears, in particular to a parabolic tooth trace gear mechanism with an end face arc and involute combined tooth profile.
Background
Gears are widely used in industrial equipment such as robot joint reducers, automotive gearboxes, wind power gearboxes, machine tool headstocks, etc., transmitting motion and power, and are considered as "hearts" in machines. At present, the conventional involute spur gear, helical gear, circular arc gear and other parallel shaft cylindrical gear transmission is difficult to overcome the problems of transmission failure such as friction wear, gluing, plastic deformation and the like, thermal deformation, vibration, noise and the like caused by relative sliding of tooth surfaces. Meanwhile, the gear lubrication system increases the weight and cost of the whole machine, and in extreme environments such as high temperature, low temperature, high pressure, vacuum, strong radiation and the like, the lubricant may fail, and the discharge of the lubricant also causes irreversible pollution to the environment. With the high-speed development of the intelligent manufacturing industry, conventional gear products cannot meet the precise transmission requirements of high-end equipment such as automatic automobile transmissions, robot reducers, wind power gearboxes, high-speed rail traffic and the like, and high-performance gear products are greatly imported. The design and manufacturing technology of high performance gears has become a key factor for restricting the development of the manufacturing field of high-end equipment, and how to avoid the relative sliding of tooth surfaces to improve the transmission performance of gears is one of the key problems to be solved in the art.
In order to solve the problems of the parallel shaft gear transmission, researchers at home and abroad sequentially invent a single circular arc gear, a double circular arc gear and a circular arc toothed line cylindrical gear, for example, chinese patent literature with application number 202110318591.7 discloses a double circular arc small tooth difference speed reduction transmission device and a double circular arc tooth forming method, and Chinese patent literature with application number 202123012746.9 discloses a double curved circular arc toothed line cylindrical gear pair structure and the like. However, the tooth profiles of the small wheel and the large wheel of the double-arc gear are cut by a generating method based on the same hob, and in order to ensure that the large and small gears are meshed correctly, the pressure angles of two meshing points of the tooth profile of the hob are set to be equal. Therefore, the limitation of the existing double-arc gear mechanism is that the pressure angles of two meshing points defining the tooth profile are equal, so that the structure of the existing double-arc gear mechanism is not an optimal bearing design structure, and when the mechanical equipment is in heavy-duty transmission, the gear teeth can be broken, so that accidents are generated; the design of the tooth surface of the hyperbolic circular arc toothed line cylindrical gear pair is limited by parameters of a processing cutter head, tooth tops at two ends of the tooth surface can be sharpened, and an effective contact area of the tooth surface is only concentrated in a limited area in the center of the tooth width, so that the risk of breakage of the tooth surface exists when the tooth surface is applied to large-load transmission, and meanwhile, the tooth surface relatively slides more, so that friction and abrasion are serious.
Disclosure of Invention
In view of the above, in order to solve the problems that in the gear mechanism in the prior art, the effective contact area of the tooth surface is only concentrated in the limited area of the center of the tooth width, the risk of breakage of the tooth surface exists, the relative sliding of the tooth surface is large, and the friction and abrasion are serious, the embodiment of the invention provides a parabolic tooth trace gear mechanism with an end surface arc and involute combined tooth profile.
The embodiment of the invention provides a parabolic tooth trace gear mechanism with an end face arc and involute combined tooth profile, which comprises a pair of gear pairs consisting of a small wheel and a large wheel which are parallel in axis, wherein the small wheel and the large wheel are in pure rolling meshing transmission, the end face tooth profile curves of the small wheel and the large wheel consist of an end face working tooth profile curve and a tooth root transition curve, and the end face tooth profile curves of the small wheel and the large wheel are both symmetrical in left and right sides; the end face working tooth profiles of the small wheel and the large wheel are combined tooth profiles of an end face arc and an involute; the tooth surfaces of the small wheel and the large wheel are provided with parabolic tooth line structures; the meshing point of at least one pair of gear teeth of the small wheel and the large wheel is positioned at a node so as to realize pure rolling meshing contact, and meshing lines formed by the meshing point during rotation of the small wheel and the large wheel form two contact lines on the small wheel and the large gear teeth respectively.
Further, tooth surfaces of the small wheel and the large wheel are provided with parabolic tooth trace structures, tooth profile curves of the end surfaces of the small wheel and the large wheel are formed by movement of tooth profile curves of the end surfaces of the small wheel and the large wheel along with contact points along tooth surface contact lines, and the contact lines are axisymmetric parabolas after being unfolded along the pitch cylindrical surfaces of the small wheel and the large wheel.
Further, the end faces of the small wheel and the large wheel workThe tooth profile is a combined tooth profile of an end face arc and an involute, and is in a symmetrical form of a left side shaft and a right side shaft, and the tooth profile of the right side of the end face can be obtained by axisymmetrical tooth profile of the left side of the end face; the left working tooth profile curve is formed by a circular arc plane curve and an involute plane curve at an interdental control point P bi Smooth connection is formed, and the tooth space control point G of the right side tooth profile is formed when the small wheel and the large wheel are installed bi And node P i Overlap, control point G bi From the left working profile curve tooth space control point P bi Axisymmetric to obtain; the curve shape of the end face working tooth profile is controlled by the tooth top control point P ai Interdental control point P bi And a tooth bottom control point P ci Determining; specifically, the combined types of the working tooth profile curves of the small wheel and the large wheel from the top to the bottom are CI, wherein C, I respectively represents an arc and an involute, the arc is a working tooth profile upper curve, and the involute is a working tooth profile lower curve; the tooth root transition curve is the tooth bottom control point P ci With root control point P di The determined Hermite curve, and the root transition curve and the working tooth profile lower curve are at the root control point P ci Smooth connection.
Further, the tooth top control points P of the left working tooth profiles of the small wheel and the large wheel ai From the tip radius R ai And offset angle χ ai Determining χ ai Tooth top datum point J for small and large wheels ai An angle of clockwise rotation about the center of the circle; tooth bottom control point P ci From the radius R of the tooth bottom circle ci Determining; wherein, the datum point J of the tops of the small wheel and the large wheel is ai Is an involute with the same base radius and end face pressure angle as the small wheel and the large wheel and the same radius R ai Intersection of addendum circle.
Further, the pinion and the large cog surface contact line is determined by the following method: at o p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g In three spatial coordinate systems, z p The axis of rotation of the shaft and the small wheel being coincident, z g The axis of rotation of the axle and the large wheel being coincident, z k Shaft and passing engagement point M a and Mb Is engaged with (a)Lines K-K coincide and z k Axis and z p 、z g Axes are parallel to each other, x p And x g Axis of coincidence, x k And x g Axes are parallel, o p o g The distance of (a) is a; coordinate system o 1 -x 1 ,y 1 ,z 1 Fixedly connected with the small wheel, the coordinate system o 2 -x 2 ,y 2 ,z 2 Fixedly connected with the large wheel, and a small wheel and large wheel coordinate system o 1 -x 1 ,y 1 ,z 1 and o2 -x 2 ,y 2 ,z 2 At the initial position and respectively with the coordinate system o p -x p ,y p ,z p O g -x g ,y g ,z g Overlap at the point of engagement M a and Mb Overlap and mark as M, the small wheel at uniform angular velocity omega 1 Around z p The shaft rotates clockwise and the large wheel rotates at a uniform angular velocity omega 2 Around z g The axis rotates anticlockwise, after a period of time from the start position, the coordinate system o 1 -x 1 ,y 1 ,z 1 O 2 -x 2 ,y 2 ,z 2 Respectively rotate, the small wheel winds around z p The shaft rotates throughAngle, large wheel around z g Shaft rotation->A corner;
when the small wheel and the large wheel are meshed for transmission, the meshing point M is set a and Mb From the origin o of coordinates respectively k Starting up and down movement along the meshing line K-K, the parametric equation describing the movement of the meshing point is:
in the formula (1), t is the meshing point M a and Mb T is more than or equal to 0 and less than or equal to deltat; b is the tooth width; "+" corresponds to the meshing point M a "-" corresponds to the meshing point M b ；
To ensure constant ratio engagement, the rotation angle of the small and large wheels must be in a linear relationship with the movement of the engagement point, as follows:
in the formula (2)Is a linear scaling factor of the meshing point motion in radians (rad); i.e 12 Is the transmission ratio between the small wheel and the large wheel;
when engaged point M a and Mb When moving along the meshing line K-K, they form contact lines C on the tooth surfaces of the pinion and the tooth surfaces of the large gear p and Cg The method comprises the steps of carrying out a first treatment on the surface of the Obtaining a coordinate system o according to the coordinate transformation p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g 、o 1 -x 1 ,y 1 ,z 1 and o2 -x 2 ,y 2 ,z 2 The homogeneous coordinate transformation matrix between the two is:
wherein ,
in the formulas (4) and (5), R 1 Is the pitch cylinder radius of the small wheel, R 2 Is the pitch cylinder radius of the large wheel, alpha i Is the end face pressure angle of the engagement point;
the small value is obtained from the formulas (1) and (4)Contact line C of tooth surface of wheel p The parametric equation for (2) is:
obtaining the contact line C of the tooth surface of the large wheel by the steps (1) and (5) g The parametric equation for (2) is:
further, the left side face tooth profiles of the small wheel and the large wheel are determined by the following method:
at the inter-tooth control point P of the large wheel and the small wheel respectively bi Establishing a local coordinate system S pbi (o pbi -x pbi y pbi z pb i) I=1, 2, where i=1 represents a small wheel, i=2 represents a large wheel, and the parametric equations for deriving the upper circular arc profile for the working profile curve combination are:
in formula (8), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; zeta type toy ai Angle parameter of arc curve, ζ aimin and ξaimax Respectively is xi ai Minimum and maximum values of the values, ρ ai Is the radius of the circular arc of the small wheel and the large wheel, when determining the offset angle χ ai Radius R of addendum circle ai ，ρ ai 、ξ aimin and ξaimax All can be solved, so that the upper arc tooth profile curve is determined;
respectively in a space coordinate system o p -x p ，y p ，z p and og -x g ，y g ，z g Is the origin o of coordinates of p and og Establishing a local coordinate system S Invi (o Invi -x Invi Y Invi z Invi ) Obtaining a curve combination for working tooth profileThe parametric equations for the lower involute profile of (2) are:
in the formula ,rbi Base radius of small and large wheels, R ci Root control point P for small and large wheels ci Radius to gear axis, i.e. tooth bottom circle radius, u i U is a parameter of involute parameter equation of small wheel and large wheel ci and upi U is respectively i Minimum and maximum values of the values are taken; when determining R i 、R ci When u pi 、u ci All can be solved, so that the lower involute profile is determined;
from the coordinate transformation, a coordinate system S can be obtained pbi (o pbi -x pbi Y pbi z pbi) and SInvi (o Invi -x Invi Y Invi z Invi ) The homogeneous coordinate transformation matrix between the two is:
wherein ,γi For node P i Is the radial vector and coordinate axis Y Invi An acute angle between the forward direction;
coordinate system S Inv1 (o Inv1 -x Inv1 Y Inv1 z Inv1) and op -x p ，y p ，z p The homogeneous coordinate transformation matrix between the two is:
coordinate system S Inv2 (o Inv2 -x Inv2 Y Inv2 z Inv2) and og -x g ，y g ，z g The homogeneous coordinate transformation matrix between the two is:
wherein ,λi The central angles corresponding to the tooth thicknesses of the indexing circles of the small wheels and the large wheels are obtained;
the transition curves of the tooth roots at the left sides of the gear tooth end surfaces of the small and large gears, namely Hermite curves, are formed by a point P ci and Pdi And its tangent vector T ci and Tdi Determining P di From root radius R di Sum angle delta i Co-determination, delta i For point P di Is defined by the radial vector and coordinate axis x k The sharp angle clamped by the tooth root control point P is obtained ci With tooth bottom control point P di The parametric equation for the determined left root transition curve, namely the Hermite curve, is:
in the formulae (14) and (15), x p (P ci )，y p (P ci )，z p (P ci ) Respectively are points P ci Three coordinate axis components, x p (P di )，y p (P di )，z p (P di ) Respectively are points P di Three coordinate axis components, x p (T ci )，y p (T ci )，z p (T ci ) Respectively are points P ci Unit tangent vector T of (2) ci Three coordinate axis components, x p (T di )，y p (T di )，z p (T di ) Respectively are points P di Unit tangent vector T of (2) di Three coordinate axis components, m t For modulus of end face，b 1 ，b 2 ，b 3 ，b 4 To calculate the parameters, T H Is the shape control parameter of the tooth root transition curve, T is more than or equal to 0.2 H ≤1.5，t H To calculate the parameters, 0 is less than or equal to t H ≤1；
In all of the above formulas:
t-engagement point M a and Mb And t.epsilon.0, Δt]；
Maximum value of the motion parameter variable of the delta t-meshing point;
one is the linear proportionality coefficient of the meshing point movement;
m t -end face modulus;
Z 1 -small number of teeth;
Z 2 -number of large gear teeth;
b-the tooth widths of the small and large wheels;
α t -an end face pressure angle;
J ai small wheel and large gear tooth top datum point
χ a1 -the angle by which the tip reference point of the small wheel rotates clockwise around the centre of the circle;
χ a2 -the angle by which the tooth top datum point of the large wheel rotates clockwise around the centre of the circle;
ρ a1 -small end face profile upper circular arc profile radius;
ρ a2 -the radius of the upper circular arc profile of the face profile of the large wheel;
k c -the start point P of the transition curve of the small and large wheel root ci Radius variation coefficient of (a);
R 1 -the pitch cylinder radius of the small wheel, R 1 ＝m t Z 1 /2； (16)
R 2 -the pitch cylinder radius of the large wheel, R 2 ＝i 12 R 1 ； (17)
i 12 For the ratio of small to large wheels,
a-relative positions of axis mounting of small and large wheels: a=r 1 +R 2 ； (19)
r b1 -small wheel base radius, r b1 ＝R 1 cosα t ； (20)
r b2 -large wheel base radius, r b2 ＝R 2 cosα t ； (21)
R a1 Pinion tooth tip radius, R a1 ＝R 1 +m t ； (22)
R c1 Pinion tooth base circle radius, i.e. root transition curve start point P c1 To the radius of the centre of rotation of the small wheel,
R c1 ＝R 1 -k c m t ； (23)
R d1 pinion root circle radius, R d1 ＝R 1 -1.25m t ； (24)
R a2 Large tooth tip radius, R a2 ＝R 2 +m t ； (25)
R c2 The radius of the base circle of the large gear tooth, i.e. the starting point P of the root transition curve c2 To the radius of the centre of rotation of the large wheel,
R c2 ＝R 2 -k c m t ； (26)
R d2 -large tooth root radius, R d2 ＝R 2 -1.25m t ； (27)
γ 1 -small wheel end face node P 1 Is defined by the radial vector and coordinate axis y Inv1 The acute angle that is clamped in the forward direction,
γ 2 -bull wheel end node P 2 Is defined by the radial vector and coordinate axis y Inv2 Acute angle clamped in forward direction，
λ 1 Central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
λ 2 central angle corresponding to the tooth thickness of the indexing circle of the large wheel,
δ 1 -small wheel left end face tooth profile point P d1 Is defined by the radial vector and coordinate axis x k The acute angle of the clamp is that of the clamp,
δ 2 -tooth profile point P of left end face of large wheel d2 Is defined by the radial vector and coordinate axis x k The acute angle of the clamp is that of the clamp,
the coincidence ratio of the parabolic tooth trace gear mechanism with the end face arc and involute combined tooth profile is required to be more than 2, and the coincidence ratio calculation formula is that
Linear scaling factor based on the value epsilon of overlapAnd the number of pinion teeth Z 1 The maximum value of the motion parameter variable of the meshing point of the parabolic tooth trace gear mechanism for solving the combination tooth profile of the end face arc and the involute is +.> (35)
When determining the number Z of pinion teeth 1 Ratio i 12 Modulus of end face m t Coincidence epsilon and linear scale factor End face pressure angle alpha t Tooth width b, tooth root transition curve shape control parameter T H The tooth top datum point of the small wheel rotates clockwise around the circle center by an angle χ a1 The tooth top datum point of the large wheel rotates clockwise around the circle center by an angle χ a2 Root transition curve starting point P of small wheel and large wheel ci Radius variation coefficient k of (2) c And when the gear mechanism is in use, the maximum value delta t of the motion parameter variable of the meshing point, the contact line and the meshing line, the end face combined tooth profile of the small wheel and the large wheel and the correct installation distance of the end face combined tooth profile and the correct installation distance of the small wheel and the large wheel are correspondingly determined, and the parabolic tooth profile structure of the gear teeth of the small wheel and the large wheel can be determined, so that the parabolic tooth profile gear mechanism of the end face circular arc and the involute combined tooth profile is obtained.
Further, the small wheel is used for being connected with an input shaft, and the large wheel is used for being connected with an output shaft.
Further, the small wheel is interchangeable with the input shaft and the output shaft connected with the large wheel.
Further, one of the small wheel and the large wheel is connected with an input shaft, the input shaft is connected with a driver, and the driver can drive the small wheel or the large wheel to rotate positively and negatively.
The technical scheme provided by the embodiment of the invention has the beneficial effects that:
1. according to the parabolic tooth line gear mechanism with the combined tooth profile of the end face arc and the involute, the contact line with the node meshing is constructed based on the active design of the movement rule of the meshing point, and the contact line is an axisymmetric parabola after the node cylindrical surface is unfolded, so that the theoretical value of the relative sliding speed of all the meshing points on the contact line is zero, and therefore the relative sliding and friction abrasion between tooth surfaces are effectively reduced, meanwhile, the parabolic tooth line gear mechanism with the combined tooth profile of the end face arc and the involute has no tooth top sharpening phenomenon, the contact area extends over the width of the tooth, and the larger tooth width can be designed and utilized to drive larger load and better movement stability; in addition, the parabolic tooth trace gear mechanism with the combined tooth profile of the end surface arc and the involute has extremely small relative difference between the maximum contact stress of the tooth surface and the maximum bending stress of the tooth root during forward and reverse transmission.
2. The parabolic toothed gear mechanism with the end surface arc and involute combined tooth profile is in pure rolling engagement theoretically, is small in friction and wear, has no axial force, has good self-centering property, is convenient to install, and has small sensitivity to installation errors; compared with the traditional involute herringbone gear transmission mechanism, the involute herringbone gear transmission mechanism has the advantages of being free of design of a tool retracting groove, capable of being formed at one time, simple in processing technology and convenient to assemble.
3. The end face tooth profile of the parabolic tooth trace gear mechanism with the end face arc and involute combined tooth profile is not a single arc isoplane curve, but is a multi-curve combined type, so that effective regulation and control of a contact ellipse and a contact area are realized, edge contact is avoided, the relative curvature radius is increased, the tooth surface contact strength and the tooth root bending strength are improved, and the bearing capacity is improved.
4. The parabolic toothed gear mechanism with the end surface arc and involute combined tooth profile has the advantages that the cylindrical surface of the contact line is axisymmetric parabola instead of an inclined straight line after being developed, so that no axial force is generated during transmission, the installation condition of a shafting is simpler, and the structure is simple.
5. The parabolic toothed gear mechanism with the end surface arc and involute combined tooth profile has no undercut, the minimum tooth number is 1, compared with the existing parallel shaft involute gear and other mechanisms and arc toothed cylindrical gear transmission mechanisms, the single-stage high-transmission ratio high-coincidence transmission can be realized, and meanwhile, as the tooth number can be designed smaller, the same gear pitch diameter can be designed with larger tooth thickness and modulus, thereby having higher bending strength and larger bearing capacity, and being suitable for popularization and application in the fields of micro/micro machinery, conventional mechanical transmission and high-speed heavy-load transmission.
6. According to the parabolic tooth trace gear mechanism with the end face circular arc and involute combined tooth profile, the small wheel and the large wheel have similar tooth root bending strength through adjusting the optimal design of the tooth root transition curve shape control parameters, the equal strength design of the transmission mechanism is realized, and the service life of equipment is further prolonged.
Drawings
Fig. 1 is a schematic structural view of a parabolic tooth trace gear mechanism with a combined tooth profile of an end face arc and an involute of the present invention.
Fig. 2 is a schematic diagram of a spatial meshing coordinate system of a parabolic tooth trace gear mechanism with a combined tooth profile of an end face arc and an involute of the present invention.
Fig. 3 shows the tooth profile composition structure of the end faces of the large wheel and the small wheel in fig. 1 and 2 and the coordinate system thereof.
FIG. 4 is a schematic diagram of the local coordinate system relationship of the combined tooth profile of the present invention.
Fig. 5 is a schematic view of the tooth top datum points and the corner of the combined tooth profile according to the present invention.
Fig. 6 is a three-dimensional view of the small wheel of fig. 1 in accordance with the present invention.
Fig. 7 is a three-dimensional view of the large wheel of fig. 1 in accordance with the present invention.
FIG. 8 is a schematic diagram of the present invention when the large wheel is connected to the input shaft to drive the small wheel to speed up.
In the above figures: 1-a driver, 2-a coupler, 3-an input shaft, 4-a small wheel, 5-an output shaft, 6-a large wheel, 7-a meshing line K-K, 8-a small wheel pitch cylinder, 9-a small wheel contact line Cp, 10-a large wheel contact line Cg, 11-a large wheel pitch cylinder, 12-a large wheel end face tooth profile left tooth root transition curve, 13-a large wheel end face working tooth profile left lower involute curve, 14-a large wheel end face working tooth profile left upper arc curve, 15-a small wheel end face tooth profile left tooth root transition curve, 16-a small wheel end face working tooth profile left lower involute curve and 17-a large wheel end face working tooth profile left upper arc curve.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be further described with reference to the accompanying drawings. The following presents a preferred one of a number of possible embodiments of the invention in order to provide a basic understanding of the invention, but is not intended to identify key or critical elements of the invention or to delineate the scope of the invention.
In all examples shown and discussed herein, any specific values should be construed as merely illustrative, and not a limitation. Thus, other examples of the exemplary embodiments may have different values.
Techniques, methods, and apparatus known to one of ordinary skill in the relevant art may not be discussed in detail, but should be considered part of the specification where appropriate.
It should be noted that: like reference numerals and letters denote like items in the following figures, and thus once an item is defined in one figure, no further discussion thereof is necessary in subsequent figures. Meanwhile, it should be understood that the sizes of the respective parts shown in the drawings are not drawn in actual scale for convenience of description.
In the description of the present invention, it should be noted that, in the present invention, circuits, electronic components, and modules are all related to the prior art, and those skilled in the art may implement the present invention completely, and it is needless to say that the protection of the present invention does not relate to improvement of internal structures and methods.
It is further noted that unless specifically stated and limited otherwise, the terms "mounted," "connected," and "connected" are to be construed broadly, and may be, for example, fixedly connected, detachably connected, or integrally connected; can be mechanically or electrically connected; can be directly connected or indirectly connected through an intermediate medium, and can be communication between two elements. The specific meaning of the above terms in the present invention will be understood in specific cases by those of ordinary skill in the art.
Example 1:
referring to fig. 1, an embodiment of the present invention provides a parabolic rack gear mechanism with a combined tooth profile of an end face arc and an involute, which is applied to a reduction transmission with a transmission ratio of 3 between parallel shafts, and is designed to have a coincidence ratio of epsilon=2.4. The structure of the device is shown in figure 1, and the device comprises a small wheel 4 and a large wheel 6, wherein the small wheel 4 and the large wheel 6 form a pair of gear pairs, the small wheel 4 is connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1 through a coupler 2, the large wheel 6 is connected with an output shaft 5, namely, the large wheel 6 is connected with a driven load through the output shaft 5; the axes of the small wheel 4 and the large wheel 6 are parallel to each other. Fig. 2 is a schematic diagram of a spatial meshing coordinate system of a parabolic rack-and-pinion gear mechanism with a combined profile of an end face arc and an involute of the present invention.
Referring to figures 1, 2, 3, 4, 5, 6, the small wheel has a pitch cylinder 8 with a radius R 1 The top circle radius of the small gear tooth is R a1 Radius of root circle R d1 The outer surface of the small gear tooth root cylinder is uniformly distributed with gear teeth with a parabolic tooth trace structure, the structure is formed by the movement of a small gear end face tooth profile curve along with a contact point along with a tooth surface contact line, and the contact line is an axisymmetric parabola after being unfolded along with a small gear section cylindrical surface. The end face tooth profile of the pinion gear tooth is in an axisymmetric form, namely, the left side tooth profile and the right side tooth profile of the end face are axisymmetric. Taking the left end face tooth profile of the small wheel as an example, the small wheel tooth from the top to the bottom sequentially comprises an upper arc curve 17 of the left end face working tooth profile of the small wheel tooth, a lower involute 16 of the left end face working tooth profile and a transition curve of the left end face tooth root, namely Hermite curve 15.
Referring to figures 1, 2, 3, 4, 5, 7, the radius of the pitch cylinder 11 of the large wheel is R 2 The radius of the top circle of the big gear tooth is R a2 Radius of root circle R d2 The outer surface of the large gear tooth root cylinder is uniformly distributed with gear teeth with a parabolic tooth trace structure, the structure is formed by the movement of a large gear end face tooth profile curve along with a contact point along with a tooth surface contact line, and the contact line is an axisymmetric parabola after being unfolded along with a large gear section cylindrical surface. The end face tooth profile of the large gear tooth is in an axisymmetric form, namely, the left side tooth profile and the right side tooth profile of the end face are axisymmetric. Taking the left end face tooth profile of a large wheel as an example, the tooth profile is sequentially formed from the top to the bottom of a small wheel gear tooth left end face working tooth profile upper arc curve 14, a left end face working tooth profile lower involute 13 and a left end face tooth root transition curve, namely Hermite curve 12.
End face working teeth of the small wheel and the large wheelThe profile is a combined tooth profile of an end face arc and an involute, and is in a symmetrical form of a left side shaft and a right side shaft, and the tooth profile of the right side of the end face can be obtained by axisymmetrical of the tooth profile of the left side of the end face; the left working tooth profile curve is formed by a circular arc plane curve and an involute plane curve at an interdental control point P bi Smooth connection is formed, and the tooth space control point G of the right side tooth profile is formed when the small wheel and the large wheel are installed bi And node P i Overlap, control point G bi From the left working profile curve tooth space control point P bi Axisymmetric to obtain; the curve shape of the end face working tooth profile is controlled by the tooth top control point P ai Interdental control point P bi And a tooth bottom control point P ci Determining; specifically, the combined types of the small and large wheel working profile curves from the top to the bottom are CI, wherein 'C, I' respectively represent an arc (C i r) which is the working profile upper curve and an involute (Inv) which is the working profile lower curve; the tooth root transition curve is the tooth bottom control point P ci With root control point P di The determined Herm i te curve (Her), and the root transition curve and the working profile lower curve are at the root control point P ci Smooth connection.
The tooth top control point P of the left working tooth profile of the small wheel and the large wheel ai From the tip radius R ai And offset angle χ ai Determining χ ai Tooth top datum point J for small and large wheels ai An angle of clockwise rotation about the center of the circle; tooth bottom control point P ci From the radius R of the tooth bottom circle ci Determining; wherein, the datum point J of the tops of the small wheel and the large wheel is ai Is an involute with the same base radius and end face pressure angle as the small wheel and the large wheel and the same radius R ai Intersection of addendum circle.
The small wheel 4 is connected with the input shaft 3, the input shaft 3 is fixedly connected with the driving motor 1 through the coupler 2, and the input shaft is driven by the driving motor 1 to rotate, so that at least one pair of meshing points of the small wheel and the large wheel are positioned at the node to realize pure rolling meshing contact, and the transmission of motion and power between parallel shafts is realized, wherein the driver 1 is a motor in the embodiment.
The small and large wheel tooth surface contact lines 9, 10 are determined by the following method: at o p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g In three spatial coordinate systems, z p The axis of rotation of the shaft and the small wheel being coincident, z g The axis of rotation of the axle and the large wheel being coincident, z k Shaft and passing engagement point M a and Mb Is coincident with the line of engagement K-K7, and z k Axis and z p 、z g Axes are parallel to each other, x p And x g Axis of coincidence, x k And x g Axes are parallel, o p o g The distance of (a) is a; coordinate system o 1 -x 1 ,y 1 ,z 1 Fixedly connected with the small wheel, the coordinate system o 2 -x 2 ,y 2 ,z 2 Fixedly connected with the large wheel, and a small wheel and large wheel coordinate system o 1 -x 1 ,y 1 ,z 1 and o2 -x 2 ,y 2 ,z 2 At the initial position and respectively with the coordinate system o p -x p ,y p ,z p O g -x g ,y g ,z g Overlap at the point of engagement M a and Mb Overlap and mark as M, the small wheel at uniform angular velocity omega 1 Around z p The shaft rotates clockwise and the large wheel rotates at a uniform angular velocity omega 2 Around z g The axis rotates anticlockwise, after a period of time from the start position, the coordinate system o 1 -x 1 ,y 1 ,z 1 O 2- x 2 ,y 2 ,z 2 Respectively rotate, the small wheel winds around z p The shaft rotates throughAngle, large wheel around z g Shaft rotation->A corner;
when the small wheel and the large wheel are meshed for transmission, the meshing point M is set a and Mb From the origin o of coordinates respectively k Starting up and down movement along the meshing line K-K, the parametric equation describing the movement of the meshing point is:
in the formula (1), t is the meshing point M a and Mb T is more than or equal to 0 and less than or equal to deltat; b is the tooth width in millimeters (mm); "+" corresponds to the meshing point M a "-" corresponds to the meshing point M b ；
To ensure constant ratio engagement, the rotation angle of the small and large wheels must be in a linear relationship with the movement of the engagement point, as follows:
in the formula (2)Is a linear scaling factor of the meshing point motion in radians (rad); i.e 12 Is the transmission ratio between the small wheel and the large wheel;
when engaged point M a and Mb When moving along the meshing line K-K, they form contact lines C on the tooth surfaces of the pinion and the tooth surfaces of the large gear p and Cg The method comprises the steps of carrying out a first treatment on the surface of the Obtaining a coordinate system o according to the coordinate transformation p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g 、o 1 -x 1 ,y 1 ,z 1 and o2 -x 2 ,y 2 ,z 2 The homogeneous coordinate transformation matrix between the two is:
wherein ,
in the formulas (4) and (5), R 1 Is the pitch cylinder radius of the small wheel, R 2 Is the pitch cylinder radius of the large wheel, alpha t Is the end face pressure angle of the engagement point;
obtaining the contact line C of the pinion tooth surface from the components (1) and (4) p The parametric equation for (2) is:
obtaining the contact line C of the tooth surface of the large wheel by the steps (1) and (5) g The parametric equation for (2) is:
the left side end face tooth profile of the small wheel and the large wheel is determined by the following method:
at the inter-tooth control point P of the large wheel and the small wheel respectively bi Establishing a local coordinate system S pbi (o pbi -x pbi y pbi z pbi ) I=1, 2, where i=1 represents a small wheel, i=2 represents a large wheel, and the parametric equations for deriving the upper circular arc profile for the working profile curve combination are:
in formula (8), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; zeta type toy ai Angle parameter of arc curve, ζ aimin and ξaimax Respectively is xi ai Minimum and maximum values of the values, ρ ai Is the radius of the circular arc of the small wheel and the large wheel, when determining the offset angle χ ai Radius R of addendum circle ai ，ρ ai 、ξ aimin and ξaimax All can be solved, so that the upper arc tooth profile curve is determined;
respectively are provided withIn the space coordinate system o p -x p ，y p ，z p and og -x g ，y g ，z g Is the origin o of coordinates of p and og Establishing a local coordinate system S Invi (o Invi -x Invi Y Invi z Invi ) The parametric equations for the lower involute profile for the working profile curve combination are respectively:
in the formula ,rbi Base radius of small and large wheels, R ci Root control point P for small and large wheels ci Radius to gear axis, i.e. tooth bottom circle radius, u i U is a parameter of involute parameter equation of small wheel and large wheel ci and upi U is respectively i Minimum and maximum values of the values are taken; when determining R i 、R ci When u pi 、u ci All can be solved, so that the lower involute profile is determined;
from the coordinate transformation, a coordinate system S can be obtained pbi (o pbi -x pbi Y pbi z pbi) and SInvi (o Invi -x Invi Y Invi z Invi ) The homogeneous coordinate transformation matrix between the two is:
wherein ,γi For node P i Is the radial vector and coordinate axis Y Invi An acute angle between the forward direction;
coordinate system S Inv1 (o Inv1 -x Inv1 Y Inv1 z Inv1) and op -x p ，y p ，z p The homogeneous coordinate transformation matrix between the two is:
coordinate system S Inv2 (o Inv2 -x Inv2 Y Inv2 z Inv2) and og -x g ，y g ，z g The homogeneous coordinate transformation matrix between the two is:
wherein ,λi The central angles corresponding to the tooth thicknesses of the indexing circles of the small wheels and the large wheels are obtained;
the left tooth root transition curve of the gear tooth end surfaces of the small and large gears, namely Herm i te curve (Her), is formed by a point P ci and Pdi And its tangent vector T ci and Tdi Determining P di From root radius R dt Sum angle delta i Co-determination, delta i For point P di Is defined by the radial vector and coordinate axis x k The sharp angle clamped by the tooth root control point P is obtained ci With tooth bottom control point P di The parametric equation for the determined left root transition curve, namely the hermi te curve, is:
in the formulae (14) and (15), x p (P ci )，y p (P ci )，z p (P ci ) Respectively are points P ci Three coordinate axis components, x p (P di )，y p (P di )，z p (P di ) Respectively are points P di Three coordinate axis components, x p (T ci )，y p (T ci )，z p (T ci ) Respectively are points P ci Unit tangent vector T of (2) ci Three coordinate axis components, x p (T di )，y p (T di )，z p (T di ) Respectively are points P di Unit tangent vector T of (2) di Three coordinate axis components, m t For end face modulus, b 1 ，b 2 ，b 3 ，b 4 To calculate the parameters, T Beating machine Is the shape control parameter of the tooth root transition curve, T is more than or equal to 0.2 Beating machine ≤1.5，t H To calculate the parameters, 0 is less than or equal to t H ≤1；
In all of the above formulas:
t-engagement point M a and Mb And t.epsilon.0, Δt]；
Maximum value of the motion parameter variable of the delta t-meshing point;
-a linear scaling factor for the meshing point movement;
m t -end face modulus;
Z 1 -small number of teeth;
Z 2 -number of large gear teeth;
b-the tooth widths of the small and large wheels;
α t -an end face pressure angle;
J ai small wheel and large gear tooth top datum point
χ a1 -the angle by which the tip reference point of the small wheel rotates clockwise around the centre of the circle;
χ a2 -the angle by which the tooth top datum point of the large wheel rotates clockwise around the centre of the circle;
ρ a1 -small end face profile upper circular arc profile radius;
ρ a2 -the radius of the upper circular arc profile of the face profile of the large wheel;
k c -the start point P of the transition curve of the small and large wheel root ci Radius variation coefficient of (a);
R 1 -the pitch cylinder radius of the small wheel, R 1 ＝m t Z 1 /2； (16)
R 2 -the pitch cylinder radius of the large wheel, R 2 ＝i 12 R 1 ； (17)
i 12 For the ratio of small to large wheels,
a-relative positions of axis mounting of small and large wheels: a=r 1 +R 2 ； (19)
r b1 -small wheel base radius, r b1 ＝R 1 cosα t ； (20)
r b2 -large wheel base radius, r b2 ＝R 2 cosα t ； (21)
R a1 Pinion tooth tip radius, R a1 ＝R 1 +m t ； (22)
R c1 Pinion tooth base circle radius, i.e. root transition curve start point P c1 To the radius of the centre of rotation of the small wheel,
R c1 ＝R 1 -k c m t ； (23)
R d1 pinion root circle radius, R d1 ＝R 1 -1.25m t ；(24)
R a2 Large tooth tip radius, R a2 ＝R 2 +m t ； (25)
R c2 The radius of the base circle of the large gear tooth, i.e. the starting point P of the root transition curve c2 To the radius of the centre of rotation of the large wheel,
R c2 ＝R 2 -k c m t ； (26)
R d2 -large tooth root radius, R d2 ＝R 2 -1.25m t ； (27)
γ 1 -small wheel end face node P 1 Is defined by the radial vector and coordinate axis y Inv1 Clamped in the forward directionAn acute angle is formed between the two surfaces,
γ 2 -bull wheel end node P 2 Is defined by the radial vector and coordinate axis y Inv2 The acute angle that is clamped in the forward direction,
λ 1 central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
λ 2 central angle corresponding to the tooth thickness of the indexing circle of the large wheel,
δ 1 -small wheel left end face tooth profile point P d1 Is defined by the radial vector and coordinate axis x k The acute angle of the clamp is that of the clamp,
δ 2 -tooth profile point P of left end face of large wheel d2 Is defined by the radial vector and coordinate axis x k The acute angle of the clamp is that of the clamp,the coincidence ratio of the parabolic tooth trace gear mechanism with the end surface arc and involute combined tooth profile is required to be more than 2, and the coincidence ratio calculation formula is +.>According to the value epsilon of the overlap ratio, the linear proportionality coefficient +.>And the number of pinion teeth Z 1 The maximum value of the motion parameter variable of the meshing point of the parabolic tooth trace gear mechanism for solving the combination tooth profile of the end face arc and the involute is +.> (35)
When determining the number Z of pinion teeth 1 Ratio i 12 Modulus of end face m t Coincidence epsilon and linear scale factorEnd face pressure angle alpha t Tooth width b, tooth root transition curve shape control parameter T H The tooth top datum point of the small wheel rotates clockwise around the circle center by an angle χ a1 The tooth top datum point of the large wheel rotates clockwise around the circle center by an angle χ a2 Root transition curve starting point P of small wheel and large wheel ci Radius variation coefficient k of (2) c When the gear mechanism is used, the maximum value delta t of the motion parameter variable of the meshing point, the contact line and the meshing line, the end face combined tooth profile of the small wheel and the large wheel and the correct installation distance of the end face combined tooth profile and the correct installation distance of the small wheel and the large wheel are correspondingly determined, and the parabolic tooth profile structure of the gear teeth of the small wheel and the large wheel can be determined, so that the parabolic tooth profile gear mechanism of the end face circular arc and the involute combined tooth profile is obtained;
in the above formula: axes of the coordinate systems, a, b, m t ，ρ a1 ，ρ a2 ，x pimi n，x pimax ，R 1 and R2 Equal length, radius or distance units are millimeters (mm);ξ ai ，ξ aimin ，ξ aimax ，δ 1 ，δ 2 ，u ci ，u i ，u pi ，χ a1 and χa2 The unit of the equal angle is radian (rad); pressure angle alpha t In degrees (°).
In the above formula, the relevant parameters are respectively as follows: z is Z 1 ＝24，i 12 ＝3，m t =4 millimeters (mm), epsilon=2.4,b=80 millimeters (mm), α t ＝20°，T H ＝0.5，χ a1 ＝0.08rad，χ a2 =0.04 rad, substituting formulae (16) - (35) to obtain Δt=0.1, a=192 millimeters (mm);
then substituting the above values into the formulas (1) - (15) can obtain the contact line parameter equation and the end face tooth profile parameter equation of the small wheel and the large wheel in the example, and then according to the spiral motion, the tooth surface structure of the small wheel and the large wheel is obtained, and the assembly can be carried out according to the correct center distance.
When the driving motor 1 drives the input shaft 3 and the small wheel 2 to rotate, the coincidence ratio epsilon=2.4 of the parabolic tooth trace pure rolling external meshing gear of the preset pair of end face circular arcs and the involute combined tooth profile is ensured to be in a meshing state when the small wheel 2 and the large wheel 5 are correctly installed, so that at least two pairs of gear teeth participate in meshing transmission at the same time at each instant, and continuous and stable meshing transmission of the parabolic tooth trace gear mechanism of the end face circular arcs and the involute combined tooth profile in 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 parabolic tooth trace pure rolling external gear corresponding to the end face circular arc and involute combined tooth profile is used for realizing the speed reduction and torque increase transmission of the anticlockwise rotation of the large wheel.
Example 2:
the parabolic toothed gear mechanism with the end surface arc and involute combined tooth profile is applied to speed-increasing transmission of parallel shafts. As shown in fig. 8, a large wheel 6 is adopted to be connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1 through a coupler 2, a small wheel 4 is connected with an output shaft 5, namely, the small wheel 4 is connected with a driven load through the output shaft 5; the axes of the small wheel 4 and the large wheel 6 are parallel. In this embodiment, the number of teeth of the large wheel 5 is 63, the number of teeth of the small wheel 2 is 21, and the overlap ratio epsilon=2.4 is designed. When the input shaft 3 drives the large wheel 6 to rotate, two pairs of adjacent gear teeth are in an engaged state when the large wheel 6 and the small wheel 4 are installed, and the coincidence ratio epsilon=2.4 of the parabolic tooth trace of the combined tooth profile of the end face circular arcs and the involute is preset, so that at least two pairs of gear teeth participate in engagement transmission at the same time at each instant, and continuous and stable engagement transmission of the parabolic tooth trace gear mechanism of the combined tooth profile of the end face circular arcs and the involute in rotation motion is realized. At this time, the step-up ratio of the large wheel set to the small wheel is 3, that is, the angular velocity ratio of the small wheel set to the large wheel is 3.
The relevant parameters are respectively as follows: z is Z 1 ＝21，i 12 ＝3，m t =3 millimeters (mm), epsilon=2.4,radian (rad), b=80 millimeters (mm), α t ＝25°，T H ＝0.6，χ a1 ＝0.06rad，χ a2 Substituting 0.03rad into equations (16) - (36) to obtain Δt=0.1, a=126 millimeters (mm);
then substituting the values into the formulas (1) - (15) 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 respectively carrying out spiral movement to obtain the tooth structures of the small wheel and the large wheel, and assembling according to the correct center distance.
The rotation direction of the input shaft connected with the driver of the embodiment is anticlockwise, and the speed-increasing transmission mode of the parabolic tooth trace gear mechanism corresponding to the end face circular arc and involute combined tooth profile is used for realizing the clockwise rotation transmission of the small wheel.
The design of the parabolic tooth trace gear mechanism of the end face arc and involute combined tooth profile is based on an active design method of an meshing line parameter equation, and an end face working tooth profile is formed by combining an arc curve and an involute, so that theoretical pure rolling meshing transmission is realized, active regulation and control of a contact area and a contact ellipse are realized, friction and friction of a tooth surface are reduced, the comprehensive curvature radius is improved, and the tooth surface contact strength and tooth root bending strength are increased; the parabolic toothed gear mechanism with the end surface arc and involute combined tooth profile 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-coincidence-ratio transmission can be realized, and simultaneously, as the tooth number is small and the same gear pitch circle diameter, the larger tooth thickness can be designed, so that the parabolic toothed 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-duty transmission; the parabolic tooth trace gear mechanism with the end surface arc and involute combined tooth profile can enable the small wheel and the large wheel to have similar tooth root bending strength through the optimized design of the root transition curve parameter value, so that the equal strength design of the transmission mechanism is realized, and the service life of equipment is further prolonged; the parabolic tooth trace pure rolling external gear mechanism with combined tooth profile of the surface arc and the involute has extremely small difference between the maximum tooth surface contact stress and the maximum tooth root bending stress of forward and backward transmission, and has similar strength of forward and backward rotation bidirectional transmission. In actual use, one of the small wheel 4 and the large wheel 6 is connected with an input shaft, the input shaft is connected with the driver 1, and the driver 1 can drive the small wheel 4 or the large wheel 6 to perform forward rotation transmission or reverse rotation transmission.
In this document, terms such as front, rear, upper, lower, etc. are defined with respect to the positions of the components in the drawings and with respect to each other, for clarity and convenience in expressing the technical solution. It should be understood that they are relative concepts and can be varied in many ways depending on the use and placement of the words, and that the use of the words should not limit the scope of the application as claimed.
The embodiments described above and features of the embodiments herein may be combined with each other without conflict.
The foregoing description of the preferred embodiments of the application is not intended to limit the application to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the application are intended to be included within the scope of the application.

Claims (9)

1. Parabolic tooth trace gear mechanism of terminal surface circular arc and involute combination tooth profile, its characterized in that: the gear pair comprises a pair of small wheels and large wheels, wherein the axes of the small wheels and the large wheels are parallel, and the small wheels and the large wheels are in pure rolling meshing transmission, and the gear pair is characterized in that: the end face tooth profile curves of the small wheel and the large wheel consist of an end face working tooth profile curve and a tooth root transition curve, and the end face tooth profile curves of the small wheel and the large wheel are symmetrical on the left side and the right side; the end face working tooth profiles of the small wheel and the large wheel are combined tooth profiles of an end face arc and an involute; the tooth surfaces of the small wheel and the large wheel are provided with parabolic tooth line structures; the meshing point of at least one pair of gear teeth of the small wheel and the large wheel is positioned at a node so as to realize pure rolling meshing contact, and meshing lines formed by the meshing point during rotation of the small wheel and the large wheel form two contact lines on the small wheel and the large gear teeth respectively.
2. The parabolic tooth trace gear mechanism of the end face arc and involute combination tooth profile according to claim 1, characterized in that: the tooth surface structures of the small wheel and the large wheel are formed by the movement of the tooth profile curves of the end surfaces of the small wheel and the large wheel along with the contact point along with the tooth surface contact line, and the contact line is in axisymmetric parabolas after being unfolded along the pitch cylindrical surfaces of the small wheel and the large wheel.
3. The parabolic tooth trace gear mechanism of the end face arc and involute combination tooth profile according to claim 1, characterized in that: the left working tooth profile curves of the small wheel and the large wheel are formed by arc and involute plane curves at an interdental control point P bi Smooth connection is formed, and the tooth space control point G of the right side tooth profile is formed when the small wheel and the large wheel are installed bi And node P i Overlap, control point G bi From the left working profile curve tooth space control point P bi Axisymmetric to obtain; the curve shape of the end face working tooth profile is controlled by the tooth top control point P ai Interdental control point P bi And a tooth bottom control point P ci Determining; specifically, the combined types of the working tooth profile curves of the small wheel and the large wheel from the top to the bottom are CI, wherein C, I respectively represents an arc and an involute, the arc is a working tooth profile upper curve, and the involute is a working tooth profile lower curve; the tooth root transition curve is the tooth bottom control point P ci With root control point P di The determined Hermite curve, and the root transition curve and the working tooth profile lower curve are at the root control point P ci Smooth connection.
4. A parabolic tooth trace gear mechanism according to claim 3, wherein the end face arc and involute combination tooth profile is characterized by: the tooth top control point P of the left working tooth profile of the small wheel and the large wheel ai From the tip radius R ai And offset angle χ ai Determining χ ai Tooth top datum point J for small and large wheels ai An angle of clockwise rotation about the center of the circle; tooth bottom control point P ci From the radius R of the tooth bottom circle ci Determining; wherein, the datum point J of the tops of the small wheel and the large wheel is ai Is an involute with the same base radius and end face pressure angle as the small wheel and the large wheel and the same radius R ai Intersection of addendum circle.
5. The parabolic tooth trace gear mechanism of the end face arc and involute combination tooth profile according to claim 1, characterized in that: the pinion and the large cog surface contact line is determined by the following method: at o p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g In three spatial coordinate systems, z p The axis of rotation of the shaft and the small wheel being coincident, z g The axis of rotation of the axle and the large wheel being coincident, z k Shaft and passing engagement point M a and Mb Is coincident with the line of engagement K-K, and z k Axis and z p 、z g Axes are parallel to each other, x p And x g Axis of coincidence, x k And x g Axes are parallel, o p o g The distance of (a) is a; coordinate system o 1 -x 1 ,y 1 ,z 1 Fixedly connected with the small wheel, the coordinate system o 2 -x 2 ,y 2 ,z 2 Fixedly connected with the large wheel, and a small wheel and large wheel coordinate system o 1 -x 1 ,y 1 ,z 1 and o2 -x 2 ,y 2 ,z 2 At the initial position and respectively with the coordinate system o p -x p ,y p ,z p O g -x g ,y g ,z g Overlap at the point of engagement M a and Mb Overlap and mark as M, the small wheel at uniform angular velocity omega 1 Around z p The shaft rotates clockwise and the large wheel rotates at a uniform angular velocity omega 2 Around z g The axis rotates anticlockwise, after a period of time from the start position, the coordinate system o 1 -x 1 ,y 1 ,z 1 O 2 -x 2 ,y 2 ,z 2 Respectively rotate, the small wheel winds around z p The axis rotates through phi 1 Angle, large wheel around z g The axis rotates through phi 2 A corner;
when the small wheel and the large wheel are meshed for transmission, the meshing point M is set a and Mb From the origin o of coordinates respectively k Starting up and down movement along the meshing line K-K, the parametric equation describing the movement of the meshing point is:
in the formula (1), t is the meshing point M a and Mb T is more than or equal to 0 and less than or equal to deltat; b is the tooth width; "+" corresponds to the meshing point M a "-" corresponds to the meshing point M b ；
To ensure constant ratio engagement, the rotation angle of the small and large wheels must be in a linear relationship with the movement of the engagement point, as follows:
in the formula (2)Is a linear scaling factor of the meshing point movement; i.e 12 Is the transmission ratio between the small wheel and the large wheel;
when engaged point M a and Mb When moving along the meshing line K-K, they form contact lines C on the tooth surfaces of the pinion and the tooth surfaces of the large gear p and Cg The method comprises the steps of carrying out a first treatment on the surface of the Obtaining a coordinate system o according to the coordinate transformation p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g 、o 1 -x 1 ,y 1 ,z 1 and o2 -x 2 ,y 2 ,z 2 The homogeneous coordinate transformation matrix between the two is:
wherein ,
in the formulas (4) and (5), R 1 Is the pitch cylinder radius of the small wheel, R 2 Is the pitch cylinder radius of the large wheel, alpha t Is the end face pressure angle of the engagement point;
obtaining the contact line C of the pinion tooth surface from the components (1) and (4) p The parametric equation for (2) is:
obtaining the contact line C of the tooth surface of the large wheel by the steps (1) and (5) g The parametric equation for (2) is:
6. the parabolic tooth trace gear mechanism of the end face arc and involute combination tooth profile according to claim 1, characterized in that: the left side end face tooth profile of the small wheel and the large wheel is determined by the following method:
at the inter-tooth control point P of the large wheel and the small wheel respectively bi Establishing a local coordinate system S pbi (o pbi -x pbi y pbi z pbi ) I=1, 2, where i=1 represents a small wheel, i=2 represents a large wheel, and the parametric equations for deriving the upper circular arc profile for the working profile curve combination are:
in formula (8), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; zeta type toy ai Angle parameter of arc curve, ζ aimin and ξaimax Respectively is xi ai Minimum and maximum values of the values, ρ ai Is the radius of the circular arc of the small wheel and the large wheel, when determining the offset angle χ ai Radius R of addendum circle ai ，ρ ai 、ξ aimin and ξaimax All can be solved, so that the upper arc tooth profile curve is determined;
Respectively in a space coordinate system o p -x p ,y p ,z p and og -x g ,y g ,z g Is the origin o of coordinates of p and og Establishing a local coordinate system S Invi (o Ihvi -x Invi u Invi z Ihvi ) The parametric equations for the lower involute profile for the working profile curve combination are respectively:
in the formula ,rbi Base radius of small and large wheels, R ci Root control point P for small and large wheels ci Radius to gear axis, i.e. tooth bottom circle radius, u i U is a parameter of involute parameter equation of small wheel and large wheel ci and upi U is respectively i Minimum and maximum values of the values are taken; when determining R i 、R ci When u pi 、u ci All can be solved, so that the lower involute profile is determined;
from the coordinate transformation, a coordinate system S can be obtained pbi (o pbi -x pbi y pbi z pbi) and SInvi (o Invi -x Invi y Invi z Invi ) The homogeneous coordinate transformation matrix between the two is:
wherein ,γi For node P i Is defined by the radial vector and coordinate axis y Invi An acute angle between the forward direction;
coordinate system S Inv1 (o Inv1 -x Inv1 y Inv1 z Inv1) and op -x p ,y p ,z p The homogeneous coordinate transformation matrix between the two is:
coordinate system S Inv2 (o Inv2 -x Inv2 y Inv2 z Inv2) and og -x g ,y g ,z g The homogeneous coordinate transformation matrix between the two is:
wherein ,λi The central angles corresponding to the tooth thicknesses of the indexing circles of the small wheels and the large wheels are obtained;
the transition curves of the tooth roots at the left sides of the gear tooth end surfaces of the small and large gears, namely Hermite curves, are formed by a point P ci and Pdi And its tangent vector T ci and Tdi Determining P di From root radius R di Sum angle delta i Co-determination, delta i For point P di Is defined by the radial vector and coordinate axis x k The sharp angle clamped by the tooth root control point P is obtained ci With tooth bottom control point P di The parametric equation for the determined left root transition curve, namely the Hermite curve, is:
in the formulae (14) and (15), x p (P ci )，y p (P ci )，z p (P ci ) Respectively are points P ci Three coordinate axis components, x p (P di )，y p (P di )，z p (P di ) Respectively are points P di Three coordinate axis components, x p (T ci )，y p (T ci )，z p (T ci ) Respectively are points P ci Unit tangent vector T of (2) ci Three coordinate axis components, x p (T di )，y p (T di )，z p (T di ) Respectively are points P di Unit tangent vector T of (2) di Three coordinate axis components, m t For end face modulus, b 1 ，b 2 ，b 3 ，b 4 To calculate the parameters, T H Is the shape control parameter of the tooth root transition curve, T is more than or equal to 0.2 H ≤1.5，t H To calculate the parameters, 0 is less than or equal to t H ≤1；
In all of the above formulas:
t-engagement point M a and Mb And t.epsilon.0, Δt]；
Δt, maximum value of motion parameter variable of meshing point;
-a linear scaling factor for the meshing point movement;
m t -end face modulus;
Z 1 -small number of teeth;
Z 2 -number of teeth of large wheel;
b-the tooth widths of the small and large wheels;
α t -face pressure angle;
J ai small wheel and large gear tooth top datum point
χ a1 -the angle by which the tip reference point of the small wheel rotates clockwise around the centre of the circle;
χ a2 -the angle by which the tooth top reference point of the large wheel rotates clockwise around the centre of the circle;
ρ a1 -small-wheel end face tooth profile upper circular arc tooth profile radius;
ρ a2 -the radius of the upper circular arc profile of the face profile of the large wheel;
k c -small and large wheel root transition curve starting point P ci Radius variation coefficient of (a);
R 1 -the pitch cylinder radius of the small wheel, R 1 ＝m t Z 1 /2； (16)
R 2 -the pitch cylinder radius of the large wheel, R 2 ＝i 12 R 1 ； (17)
i 12 For the ratio of small to large wheels,
a-relative positions of axis installation of small wheel and large wheel: a=r 1 +R 2 ； (19)
r b1 -small wheel base radius, r b1 ＝R 1 cosα t ； (20)
r b2 -large wheel base radius, r b2 ＝R 2 cosα t ； (21)
R a1 Pinion tooth top circle radius，R a1 ＝R 1 +m t ； (22)
R c1 Pinion tooth base circle radius, root transition curve start point P c1 To the radius of the centre of rotation of the small wheel,
R c1 ＝R 1 -k c m t ； (23)
R d1 radius of root circle of pinion, R d1 ＝R 1 -1.25m t ； (24)
R a2 Large tooth tip radius, R a2 ＝R 2 +m t ； (25)
R c2 -root transition curve start point P for large gear tooth base circle radius c2 To the radius of the centre of rotation of the large wheel,
R c2 ＝R 2 -k c m t ； (26)
R d2 large tooth root radius, R d2 ＝R 2 -1.25m t ； (27)
γ 1 Small wheel end face node P 1 Is defined by the radial vector and coordinate axis y Inv1 The acute angle that is clamped in the forward direction,
γ 2 large wheel end face node P 2 Is defined by the radial vector and coordinate axis y Inv2 The acute angle that is clamped in the forward direction,
λ 1 central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
λ 2 central angle corresponding to tooth thickness of indexing circle of large wheel，
δ 1 Left end face tooth profile point P of small wheel d1 Is defined by the radial vector and coordinate axis x k The acute angle of the clamp is that of the clamp,
δ 2 -left face profile point P of large wheel d2 Is defined by the radial vector and coordinate axis x k The acute angle of the clamp is that of the clamp,
the coincidence ratio of the parabolic tooth trace gear mechanism with the end face arc and involute combined tooth profile is required to be more than 2, and the coincidence ratio calculation formula is that
Linear scaling factor based on the value epsilon of overlapAnd the number of pinion teeth Z 1 The maximum value of the motion parameter variable of the meshing point of the parabolic tooth trace gear mechanism for solving the combination tooth profile of the end face arc and the involute is +.>
When determining the number Z of pinion teeth 1 Ratio i 12 Modulus of end face m t Coincidence epsilon and linear scale factorEnd face pressure angle alpha t Tooth width b, tooth root transition curve shape control parameter T H The tooth top datum point of the small wheel rotates clockwise around the circle center by an angle χ a1 The tooth top datum point of the large wheel rotates clockwise around the circle center by an angle χ a2 Root transition curve starting point P of small wheel and large wheel ci Radius variation coefficient k of (2) c And when the gear mechanism is in use, the maximum value delta t of the motion parameter variable of the meshing point, the contact line and the meshing line, the end face combined tooth profile of the small wheel and the large wheel and the correct installation distance of the end face combined tooth profile and the correct installation distance of the small wheel and the large wheel are correspondingly determined, and the parabolic tooth profile structure of the gear teeth of the small wheel and the large wheel can be determined, so that the parabolic tooth profile gear mechanism of the end face circular arc and the involute combined tooth profile is obtained.
7. The parabolic tooth trace gear mechanism of the end face arc and involute combination tooth profile according to claim 1, characterized in that: the small wheel is used for being connected with an input shaft, and the large wheel is used for being connected with an output shaft.
8. The parabolic tooth trace gear mechanism of claim 7, wherein the end face arc and involute combination profile is characterized by: the small wheel is interchangeable with the input shaft and the output shaft connected with the large wheel.
9. The parabolic tooth trace gear mechanism of claim 7, wherein the end face arc and involute combination profile is characterized by: one of the small wheel and the large wheel is connected with an input shaft, the input shaft is connected with a driver, and the driver can drive the small wheel or the large wheel to rotate positively and negatively.
CN202310372303.5A 2023-04-07 2023-04-07 Parabolic tooth trace gear mechanism with end face arc and involute combined tooth profile Pending CN116592114A (en)

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