CN116480755A - Arc tooth trace gear mechanism with end surface double arc combined tooth profile - Google Patents
Arc tooth trace gear mechanism with end surface double arc combined tooth profile Download PDFInfo
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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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- 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
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Abstract
The invention discloses an arc tooth trace gear mechanism with end face double arc combined tooth profiles, which belongs to the field of gear transmission, wherein the arc tooth trace pure rolling external meshing gear mechanism comprises external meshing small wheels and large wheels with parallel axes, tooth surfaces of the small wheels and the large wheels are provided with arc tooth trace structures, each tooth profile curve of the end face combined tooth profile is formed by moving along with a contact point, the contact lines are arc after being unfolded along a cylindrical surface, and the contact lines are jointly determined by a meshing line parameter equation, basic design parameters such as superposition ratio, tooth number, transmission ratio and the like; when the gear is correctly installed, at least one pair of gear teeth of the small wheel and the large wheel are in pure rolling engagement contact at the node, and the small wheel and the large wheel are driven by the driver to rotate, so that transmission between two shafts is realized. The invention has the beneficial effects that: the transmission system has the advantages of low friction and wear, high transmission efficiency, large single-stage transmission ratio, strong bearing capacity and the like, and can be widely applied to the transmission system design of mechanical equipment.
Description
Technical Field
The invention relates to the technical field of transmission gears, in particular to an arc tooth trace gear mechanism with double arc end surfaces and combined tooth profiles.
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 tooth breakage exists, the relative sliding of the tooth surface is large, and the friction and abrasion are serious, the embodiment of the invention provides an arc tooth trace gear mechanism with an end surface double arc combined tooth profile.
The embodiment of the invention provides an arc tooth trace gear mechanism with double arc combined tooth profiles on end surfaces, which comprises a pair of gear pairs consisting of small wheels and large wheels with parallel axes, wherein the small wheels and the large wheels are in pure rolling meshing transmission, the end surface tooth profile curves of the small wheels and the large wheels consist of end surface working tooth profile curves and tooth root transition curves, and the end surface tooth profile curves of the small wheels and the large wheels are symmetrical on the left side and the right side; the tooth surfaces of the small wheel and the large wheel are provided with arc tooth trace structures; the small wheel and the large wheel are positioned at a node to realize pure rolling meshing contact, and meshing lines formed by meshing points during rotation of the small wheel and the large wheel form two contact lines on the small wheel and the large wheel respectively.
Further, 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 an axisymmetric circular arc after being unfolded along the joint cylindrical surfaces of the small wheel and the large wheel.
Further I, the working tooth profile curves of the left sides of the end surfaces of the small wheel and the large wheel are formed by two arc plane curves at a control point P between teeth bi Is formed by smooth connection, 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 CC, wherein C represents an arc and is an upper arc curve of the working tooth profile and a lower arc curve of the working tooth profile; 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 And offset angle χ ci Determining χ ci Tooth base quasi point J for small wheel and large wheel ci An angle of clockwise rotation about the center of the circle; 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; small wheel and large gear tooth base quasi point J ci 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 ci Intersection of the root circles.
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 M b Is coincident with the line of engagement K-K, and zk axis and z p 、z g The 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 o 2 -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 M b 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 M b 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 M b T is more than or equal to 0 and less than or equal to deltat;linear scaling factor for engagement point movement；R 1 The pitch cylinder radius of the small wheel; r is (r) c The radius of the circular arc tooth trace after the expansion of the pitch cylinder surface of the small wheel is equal to the linear proportionality coefficient of the movement of the meshing point +.>Maximum value delta t of motion parameter variable and end surface modulus m t Parameters of small wheel Z 1 Determining the tooth width b; "+" 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:
i in formula (2) 12 Is the transmission ratio between the small wheel and the large wheel;
when engaged point M a And M b 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 C g 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 o 2 -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:
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 pbi ) I=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel, the parametric equation for the upper arc curve for the working profile curve combination is:
the parametric equation for obtaining the lower arc curve for the working tooth profile curve combination is:
in formulas (8) and (9), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; zeta type toy ai Is the angle parameter of the upper arc curve, ζ aimax Is xi ai Value takingMaximum value ρ ai Is the radius of the arc tooth profile at the upper part of the end surfaces of the small wheel and the large wheel; zeta type toy ci Is the angle parameter of the lower arc curve, ζ cimax Is xi ci Maximum value of the values ρ ci Is the radius of the arc tooth profile of the lower part of the end face of the small wheel and the large wheel; when determining the offset angle χ ai Radius R of addendum circle ai Offset angle χ ci Radius R of tooth bottom circle ci ，ρ ai 、ρ ci 、ξ aimax 、ξ cimax Can be solved, thereby determining a double-arc curve of the combined tooth profile;
from the coordinate transformation, a coordinate system S can be obtained pbi (o pbi -x pbi y pbi z pbi ) And S is Invi (o Invi -x Invi y Invi z Invi ) The homogeneous coordinate transformation matrix between the two is:
wherein, gamma i 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 o p -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 o g -x g ，y g ，z g The homogeneous coordinate transformation matrix between the two is:
wherein lambda is i Is a small wheel andcentral angle corresponding to the tooth thickness of the indexing circle of the large wheel;
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 P di And its tangent vector T ci And T di 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 (13) and (14), 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 M b 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
J ci Small wheel and large gear tooth base alignment 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;
χ c1 -the angle by which the tooth base quasi-point of the pinion rotates clockwise around the centre of the circle;
χ c2 -the angle by which the tooth base quasi-point of the wheel rotates clockwise around the centre of the circle;
ρ ai -the radius of the upper circular arc profile of the small and large wheel end profile;
ρ ci -the radius of the circular arc profile of the lower part of the end face profile of the small wheel and 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； (15)
R 2 -the pitch cylinder radius of the large wheel, R 2 ＝i 12 R 1 ； (16)
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 ； (18)
r b1 -small wheel base radius, r b1 ＝R 1 cosα t ； (19)
r b2 -large wheel base radius, r b2 ＝R 2 cosα t ； (20)
r c Radius of the circular arc tooth trace after expansion of the pitch cylinder of the pinion,
R a1 pinion tooth tip radius, R a1 ＝R 1 +m t ； (22)
R a2 Large tooth tip radius, R a2 ＝R 2 +m t ； (23)
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 ； (24)
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 ； (25)
R d1 pinion root circle radius, R d1 ＝R 1 -1.25m 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 The acute angle that is clamped in the forward direction,
into (I) 1 Central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
into (I) 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 arc tooth trace gear mechanism with the end surface double-arc 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 circular-arc tooth-line gear mechanism of the end-face double-circular-arc combined tooth profile is calculated as +. >
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 Form control parameter T of tooth width b and tooth root transition curve 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 The quasi point of the tooth base of the small wheel rotates clockwise around the center of the circle by an angle χ c1 The quasi point of the tooth base of the large wheel rotates clockwise around the center of the circle by an angle χ c2 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 small wheel and the large wheel are correspondingly determined, and the tooth surface circular arc tooth profile structure of the small wheel and the large wheel can be determined, so that the circular arc tooth profile gear mechanism with the end face double circular arc 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. the arc tooth line gear mechanism of the end surface double-arc combined tooth profile is based on the active design of the motion rule of meshing points, the contact line of node meshing is constructed, the contact line is an axisymmetric arc after the node cylindrical surface is unfolded, the theoretical value of the relative sliding speed of all meshing points on the contact line is zero during transmission, so that the relative sliding and friction abrasion between tooth surfaces are effectively reduced, meanwhile, the arc tooth line gear mechanism of the end surface double-arc combined tooth profile 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 transmit larger load and has better motion stability; in addition, the arc tooth trace gear mechanism with the end surface double-arc combined tooth profile 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 circular arc toothed gear mechanism with the end surface double circular arc combined tooth profile is in pure rolling engagement theoretically, has small 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 circular arc toothed gear mechanism with the end surface double circular arc combined tooth profile has the advantages of no need of a design of a tool withdrawal groove, one-step forming, simple processing technology and convenience in assembly.
3. The end face tooth profile of the end face double-arc combined tooth profile circular arc tooth trace gear mechanism is not a single circular arc equal plane 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 contact line of the circular arc toothed gear mechanism with the end surface double circular arc combined tooth profile is an axisymmetric circular arc instead of an inclined straight line after the cylindrical surface is developed, so that no axial force is generated during transmission, the installation condition of a shafting is simpler, and the structure is simple.
5. Compared with the existing parallel-axis involute gear and other mechanisms and arc toothed cylindrical gear transmission mechanisms, the arc toothed gear mechanism with the end surface double-arc combined tooth profile has the advantages that the minimum number of teeth is 1, single-stage large-transmission-ratio overlap ratio transmission can be realized, and meanwhile, as the number of teeth can be designed smaller, the tooth thickness and the modulus can be designed larger when the pitch diameter of the gear is the same, the bending strength is higher, the bearing capacity is higher, and the mechanism is 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 arc tooth trace gear mechanism with the end surface double-arc 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 an arc toothed gear mechanism with end surface double arc combined tooth profile.
Fig. 2 is a schematic diagram of a space meshing coordinate system of an arc tooth trace gear mechanism with end surface double arc combined tooth profile.
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 tooth profile left lower arc curve, 14-a large wheel end face 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 tooth profile left lower arc curve and 17-a large wheel end face 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 an arc-shaped toothed gear mechanism with a double arc-shaped combined tooth profile on the end surface, which is applied to a reduction transmission with a transmission ratio of 3 between parallel shafts, and is designed to have a contact 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 space meshing coordinate system of an arc tooth trace gear mechanism with end surface double arc combined tooth profile.
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 root cylinder of the small gear is uniformly distributed with gear teeth with an arc tooth trace structure, the structure is formed by the movement of the tooth profile curve of the end face of the small gear along with the contact point along with the contact line of the tooth surface, and the contact line is an axisymmetric arc after being unfolded along with the cylindrical surface of the small gear section. 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 arc curve 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 gear teeth with circular arc tooth trace structures are uniformly distributed on the outer surface of the large gear tooth root cylinder, 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 circular arc 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 upper arc curve 14 and the left end of the working tooth profile of the left end face of the small wheel tooth are sequentially arranged from the top to the bottom The lower arc curve 13 of the face working profile and the left flank root transition curve, the Hermite curve 12.
The end face working tooth profiles of the small wheel and the large wheel are end face double-arc combined tooth profiles and are in a symmetrical form of left and right shafts, and the end face right tooth profile can be obtained by the end face left tooth profile in an axisymmetrical mode; the left working tooth profile curve is formed by two arc plane curves at the 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 CC, wherein 'C' represents an arc (Cir) and is an upper arc curve of the working tooth profile and a lower arc curve of the working tooth profile; the tooth root transition curve is the tooth bottom control point P ci With root control point P di Determined Hermite curve (Her), and root transition curve and working profile lower curve at 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 And offset angle χ ci Determining χ ci Tooth base quasi point J for small wheel and large wheel ci An angle of clockwise rotation about the center of the circle; 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; small wheel and large gear tooth base quasi point J ci 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 ci Intersection of the root circles.
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.
Wherein the pinion and the bull gear interface 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 M b 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 Axis parallel, op o g The distance of (a) is a; coordinate system o 1 -x 1 ，y 1 ，z 1 Fixedly connected with the small wheel and the coordinate system o2 -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 o 2 -x 2 ，y 2 ，z 2 At the initial position and respectively with the coordinate system o p -x p ，y g ，z p O g -x g ，y g ，z g Overlap at the point of engagement M a And M b 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 M b And respectively starting to move up and down along the meshing line K-K from the origin ok of coordinates, wherein a parameter equation describing the movement of the meshing point is as follows:
in the formula (1), t is the meshing point M a And M b T is more than or equal to 0 and less than or equal to deltat;is a linear scaling factor of the meshing point motion in radians (rad); r is R 1 The pitch cylinder radius of the small wheel is in millimeters (mm); r is (r) c The radius of the circular arc tooth trace after the expansion of the pitch cylinder surface of the small wheel is equal to the linear proportionality coefficient of the movement of the meshing point +. >Maximum value delta t of motion parameter variable and end surface modulus m t Parameters of small wheel Z 1 The tooth width b is determined, and the unit is millimeter (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:
i in formula (2) 12 Is the transmission ratio between the small wheel and the large wheel;
when engaged point M a And M b 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 C g The method comprises the steps of carrying out a first treatment on the surface of the According to the change of coordinatesChanging to obtain a coordinate system o 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 o 2 -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 and i=2 represents a large wheel, the parametric equation for the upper arc curve for the working profile curve combination is:
The parametric equation for obtaining the lower arc curve for the working tooth profile curve combination is:
in formulas (8) and (9), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; zeta type toy ai Is the angle parameter of the upper arc curve, ζ aimax Is xi ai Maximum value of the values ρ ai Is the radius of the arc tooth profile at the upper part of the end surfaces of the small wheel and the large wheel; zeta type toy ci Is the angle parameter of the lower arc curve, ζ cimax Is xi ci Maximum value of the values ρ ci Is the radius of the arc tooth profile of the lower part of the end face of the small wheel and the large wheel; when determining the offset angle χ ai Radius R of addendum circle ai Offset angle χ ci Radius R of tooth bottom circle ci ，ρ ai 、ρ ci 、ξ aimax 、ξ cimax Can be solved, thereby determining a double-arc curve of the combined tooth profile;
from the coordinate transformation, a coordinate system S can be obtained pbi (o pbi -x pbi y pbi z pbi ) And S is Invi (o Invi -x Invi y Invi z Invi ) The homogeneous coordinate transformation matrix between the two is:
wherein, gamma i 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 o p -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 o g -x g ，y g ，z g The homogeneous coordinate transformation matrix between the two is:
wherein lambda is i 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 Hermite curve (Her), is formed by a point P ci And P di And its tangent vector T ci And T di 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 M b 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
J ci Small wheel and large gear tooth base alignment 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;
χ c1 -the angle by which the tooth base quasi-point of the pinion rotates clockwise around the centre of the circle;
χ c2 -the angle by which the tooth base quasi-point of the wheel rotates clockwise around the centre of the circle;
ρ ai -the radius of the upper circular arc profile of the small and large wheel end profile;
ρ ci -the radius of the circular arc profile of the lower part of the end face profile of the small wheel and 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； (15)
R 2 -the pitch cylinder radius of the large wheel, R 2 ＝i 12 R 1 ； (16)
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 ； (18)
r b1 -small wheel base radius, r b1 ＝R 1 cosα t ； (19)
r b2 -large wheel base radius, r b2 ＝R 2 cosα t ； (20)
r c Radius of the circular arc tooth trace after expansion of the pitch cylinder of the pinion,
R a1 pinion tooth tip radius, R a1 ＝R 1 +m t ； (22)
R a2 Large tooth tip radius, R a2 ＝R 2 +m t ； (23)
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 ； (24)
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 ； (25)
R d1 pinion root circle radius, R d1 ＝R 1 -1.25m 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 The acute angle that is clamped in the forward direction,
into (I) 1 Central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
into (I) 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 Pd 1 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 arc tooth trace gear mechanism with the end surface double-arc 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 circular-arc tooth-line gear mechanism of the end-face double-circular-arc combined tooth profile is calculated as +.>
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 Form control parameter T of tooth width b and tooth root transition curve 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 The quasi point of the tooth base of the small wheel rotates clockwise around the center of the circle by an angle χ c1 The quasi point of the tooth base of the large wheel rotates clockwise around the center of the circle by an angle χ c2 Root transition curve starting point P of small wheel and large wheel ci Radius variation coefficient k of (2) c 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 small wheel and the large wheel are also corresponding Determining the circular arc tooth trace structures of tooth surfaces of the small wheel and the large wheel, so as to obtain a circular arc tooth trace gear mechanism with double circular arc combined tooth profiles of the end surfaces;
in the above formula: axes of the coordinate systems, a, b, m t ，ρ ai ，ρ ci ，R 1 And R is 2 Equal length, radius or distance units are millimeters (mm);ξ aimax ，ξ cimax ，δ 1 ，δ 2 ，χ a1 ，χ c1 ，χ a2 and χ (x) c2 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 (15) - (35) to obtain Δt=0.1, a=192 millimeters (mm);
then substituting the above values into the formulas (1) - (14) 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 preset contact ratio epsilon=2.4 of the circular arc tooth trace pure rolling external meshing gear with the double circular arc combined tooth profile of the end faces when the small wheel 2 and the large wheel 5 are correctly installed, wherein two pairs of adjacent gear teeth are in a meshing state, so that at least two pairs of gear teeth participate in meshing transmission simultaneously at each instant, and continuous and stable meshing transmission of the circular arc line pure rolling external meshing gear mechanism with the double circular arc combined tooth profile of the end faces in rotating motion is realized. The rotation direction of an input shaft connected with the motor is clockwise, and the circular arc tooth trace pure rolling external gear corresponding to the double circular arc combined tooth profile of the end face is used for realizing the speed reduction and torque increase transmission of the anticlockwise rotation of the large wheel.
Example 2:
the circular arc toothed gear mechanism with the end surface double circular arc 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, as two pairs of adjacent gear teeth are in a meshed state when the large wheel 6 and the small wheel 4 are installed, the preset coincidence ratio epsilon=2.4 of the circular arc tooth trace pure rolling external meshing gear with the double circular arc combined tooth profile of the end faces ensures that at least two pairs of gear teeth participate in meshing transmission at the same time at each instant, thereby realizing continuous and stable meshing transmission of the circular arc tooth trace gear mechanism with the double circular arc combined tooth profile of the end faces in rotating motion. 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,b=80 millimeters (mm), α t ＝25°，T H ＝0.6，χ a1 ＝0.06rad，χ a2 Substituting 0.03rad into equations (15) - (35) to obtain Δt=0.1, a=126 millimeters (mm);
Then substituting the values into the formulas (1) - (14) 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 circular arc toothed line gear mechanism corresponding to the end surface double circular arc combined tooth profile is used for realizing the clockwise rotation transmission of the small wheel.
The design of the arc tooth trace gear mechanism with the end surface double-arc combined tooth profile is based on an active design method of an meshing line parameter equation, and adopts double-arc combination to form the end surface working tooth profile, 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 contact strength of the tooth surface and the bending strength of a tooth root are increased; the arc toothed gear mechanism with the end surface double-arc 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 overlap ratio transmission can be realized, and simultaneously, as the tooth number is small and the same gear pitch diameter, the tooth thickness can be designed to be larger, thereby having higher 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-duty transmission; the arc tooth trace gear mechanism with the end surface double-arc 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 arc tooth trace gear mechanism with the end surface double-arc combined tooth profile has extremely small difference between the maximum tooth surface contact stress and the maximum tooth root bending stress of forward and reverse rotation transmission, and has similar strength of forward and reverse 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 upon the application and placement, and that the use of such orientation terms should not be taken to limit the scope of protection of the present application.
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 invention is not intended to limit the invention to the precise form disclosed, and any such modifications, equivalents, and alternatives falling within the spirit and scope of the invention are intended to be included within the scope of the invention.
Claims (9)
1. The circular arc toothed gear mechanism with the end face double circular arc combined tooth profile 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, and the circular arc toothed gear mechanism 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 tooth surfaces of the small wheel and the large wheel are provided with arc tooth trace structures; the small wheel and the large wheel are positioned at a node to realize pure rolling meshing contact, and meshing lines formed by meshing points during rotation of the small wheel and the large wheel form two contact lines on the small wheel and the large wheel respectively.
2. The circular arc tooth trace gear mechanism of the end face double circular arc combined tooth profile according to claim 1, wherein: 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 an axisymmetric arc after being unfolded along the joint cylindrical surfaces of the small wheel and the large wheel.
3. The circular arc tooth trace gear mechanism of the end face double circular arc combined tooth profile according to claim 1, wherein: the working tooth profile curves of the left sides of the end surfaces of the small wheel and the large wheel are formed by two arc plane curves at an interdental control point P bi Is formed by smooth connection, 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 CC, wherein C represents an arc and is an upper arc curve of the working tooth profile and a lower arc curve of the working tooth profile; 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. The circular arc rack and pinion mechanism of claim 3 wherein: 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 And offset angle χ ci Determining χ ci Tooth base quasi point J for small wheel and large wheel ci An angle of clockwise rotation about the center of the circle; 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; small wheel and large gear tooth base quasi point J ci 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 ci Intersection of the root circles.
5. The circular arc tooth trace gear mechanism of the end face double circular arc combined tooth profile according to claim 1, wherein: 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 Three spatial coordinate systemsIn 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 M b 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 o 2 -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 M b 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 M b 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 M b T is more than or equal to 0 and less than or equal to deltat;is a linear scaling factor of the meshing point movement; r is R 1 The pitch cylinder radius of the small wheel; r is (r) c The radius of the circular arc tooth trace after the expansion of the pitch cylinder surface of the small wheel is equal to the linear proportionality coefficient of the movement of the meshing point +. >Maximum value delta t of motion parameter variable and end surface modulus m t Parameters of small wheel Z 1 Determining the tooth width b; "+" 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:
i in formula (2) 12 Is the transmission ratio between the small wheel and the large wheel;
when engaged point M a And M b 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 C g 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 o 2 -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 circular arc tooth trace gear mechanism of the end face double circular arc combined tooth profile according to claim 1, wherein: 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 and i=2 represents a large wheel, the parametric equation for the upper arc curve for the working profile curve combination is:
the parametric equation for obtaining the lower arc curve for the working tooth profile curve combination is:
in formulas (8) and (9), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; zeta type toy ai Is the angle parameter of the upper arc curve, ζ aimax Is xi ai Maximum value of the values ρ ai Is the radius of the arc tooth profile at the upper part of the end surfaces of the small wheel and the large wheel; zeta type toy ci Is the angle parameter of the lower arc curve, ζ cimax Is xi ci Maximum value of the values ρ ci Is the radius of the arc tooth profile of the lower part of the end face of the small wheel and the large wheel; when determining the offset angle χ ai Radius R of addendum circle ai Offset angle χ ci Radius R of tooth bottom circle ci ，ρ ai 、ρ ci 、ξ aimax 、ξ cimax Can be solved, thereby determining a double-arc curve of the combined tooth profile;
from the coordinate transformation, a coordinate system S can be obtained pbi (o pbi -x pbi y pbi z pbi ) And S is Invi (o Invi -x Invi y Invi z Invi ) The homogeneous coordinate transformation matrix between the two is:
wherein, gamma i 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 o p -x p ，y g ，z p The homogeneous coordinate transformation matrix between the two is:
coordinate system S Inv2 (o Inv2 -x Inv2 y Inv2 z Inv2 ) And o g -x g ，y g ，z g The homogeneous coordinate transformation matrix between the two is:
wherein lambda is i 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 P di And its tangent vector T ci And T di 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 (13) and (14), 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 of (2)T 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 M b 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
J ci Small wheel and large gear tooth base alignment 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;
χ c1 -the angle by which the tooth base quasi-point of the pinion rotates clockwise around the centre of the circle;
χ c2 -the angle by which the tooth base quasi-point of the wheel rotates clockwise around the centre of the circle;
ρ ai -the radius of the upper circular arc profile of the small and large wheel end profile;
ρ ci small and large wheel end facesArc tooth profile radius at the lower part of the tooth profile;
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； (15)
R 2 -the pitch cylinder radius of the large wheel, R 2 ＝i 12 R 1 ； (16)
i 12-is the transmission ratio of the small wheel to the large wheel,
a-relative positions of axis mounting of small and large wheels: a=r 1 +R 2 ； (18)
r b1 -small wheel base radius, r b1 ＝R 1 cosα t ； (19)
r b2 -large wheel base radius, r b2 ＝R 2 cosα t ； (20)
r c Radius of the circular arc tooth trace after expansion of the pitch cylinder of the pinion,
R a1 pinion tooth tip radius, R a1 ＝R 1 +m t ； (22)
R a2 Large tooth tip radius, R a2 ＝R 2 +m t ； (23)
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 ； (24)
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 ； (25)
R d1 pinion root circle radius, R d1 ＝R 1 -1.25m t ； (26)
R d2 A 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 The acute angle that is clamped in the forward direction,
into (I) 1 Central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
into (I) 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 a radial vector and sittingAxis x k The acute angle of the clamp is that of the clamp,
the coincidence ratio of the arc tooth trace gear mechanism with the end surface double-arc 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 small gear teeth Z1, and obtaining the maximum value of the motion parameter variable of the meshing point of the circular-arc tooth line gear mechanism of the end surface double-circular-arc combined tooth profile as +.>
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 Form control parameter T of tooth width b and tooth root transition curve 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 The quasi point of the tooth base of the small wheel rotates clockwise around the center of the circle by an angle χ c1 The quasi point of the tooth base of the large wheel rotates clockwise around the center of the circle by an angle χ c2 Root transition curve starting point P of small wheel and large wheel ci Radius variation coefficient k of (2) c In the process, 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 small wheel and the large wheel are correspondingly determined, and the tooth surface circular arc tooth line structure of the small wheel and the large wheel can be determined fromAnd the circular arc tooth trace gear mechanism with the end surface double circular arc combined tooth profile is obtained.
7. The circular arc tooth trace gear mechanism of the end face double circular arc combined tooth profile according to claim 1, wherein: 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 circular arc tooth trace gear mechanism of an end face double circular arc combined tooth profile according to claim 7, wherein: the small wheel is interchangeable with the input shaft and the output shaft connected with the large wheel.
9. The circular arc tooth trace gear mechanism of the end face double circular arc combined tooth profile according to claim 1, wherein: 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.
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