CN116480753A - Arc tooth trace gear mechanism with end face arc and involute combined tooth profile - Google Patents
Arc tooth trace gear mechanism with end face arc and involute combined tooth profile Download PDFInfo
- Publication number
- CN116480753A CN116480753A CN202310371991.3A CN202310371991A CN116480753A CN 116480753 A CN116480753 A CN 116480753A CN 202310371991 A CN202310371991 A CN 202310371991A CN 116480753 A CN116480753 A CN 116480753A
- Authority
- CN
- China
- Prior art keywords
- tooth
- wheel
- small
- arc
- face
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
- 230000007246 mechanism Effects 0.000 title claims abstract description 59
- 230000005540 biological transmission Effects 0.000 claims abstract description 51
- 238000005096 rolling process Methods 0.000 claims abstract description 12
- 230000033001 locomotion Effects 0.000 claims description 41
- 230000007704 transition Effects 0.000 claims description 38
- 230000009466 transformation Effects 0.000 claims description 18
- 230000001154 acute effect Effects 0.000 claims description 15
- 238000000034 method Methods 0.000 claims description 15
- 239000011159 matrix material Substances 0.000 claims description 12
- 238000009434 installation Methods 0.000 claims description 9
- CGFFKDRVHZIQHL-UHFFFAOYSA-N 1-but-3-en-2-yl-3-(methylcarbamothioylamino)thiourea Chemical compound CNC(=S)NNC(=S)NC(C)C=C CGFFKDRVHZIQHL-UHFFFAOYSA-N 0.000 claims description 7
- 238000004364 calculation method Methods 0.000 claims description 3
- 238000013461 design Methods 0.000 abstract description 13
- 238000005452 bending Methods 0.000 description 7
- 238000010586 diagram Methods 0.000 description 4
- 230000009467 reduction Effects 0.000 description 4
- 238000005299 abrasion Methods 0.000 description 3
- 238000004519 manufacturing process Methods 0.000 description 3
- 239000003638 chemical reducing agent Substances 0.000 description 2
- 238000011161 development Methods 0.000 description 2
- 238000005516 engineering process Methods 0.000 description 2
- 239000000314 lubricant Substances 0.000 description 2
- 230000009347 mechanical transmission Effects 0.000 description 2
- 238000012545 processing Methods 0.000 description 2
- 230000002035 prolonged effect Effects 0.000 description 2
- 230000002441 reversible effect Effects 0.000 description 2
- 238000004026 adhesive bonding Methods 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 230000002457 bidirectional effect Effects 0.000 description 1
- 244000309464 bull Species 0.000 description 1
- 238000004891 communication Methods 0.000 description 1
- 230000006872 improvement Effects 0.000 description 1
- 230000002427 irreversible effect Effects 0.000 description 1
- 238000005461 lubrication Methods 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 230000005855 radiation Effects 0.000 description 1
- 230000035945 sensitivity Effects 0.000 description 1
Classifications
-
- 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
-
- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
- Y02T90/00—Enabling technologies or technologies with a potential or indirect contribution to GHG emissions mitigation
Landscapes
- Engineering & Computer Science (AREA)
- General Engineering & Computer Science (AREA)
- Mechanical Engineering (AREA)
- Gear Transmission (AREA)
Abstract
The invention provides an arc tooth trace gear mechanism with an end face arc and involute combined tooth profile, which belongs to the field of gear transmission and comprises an external meshing small wheel and a large wheel with parallel axes, wherein tooth surfaces of the small wheel and the large wheel are respectively provided with an arc tooth trace structure, each tooth surface of the small wheel and the large wheel is formed by moving an end face combined tooth profile curve along with a contact point, the contact line is an arc after being unfolded along a cylindrical surface, and the arc is jointly determined by a meshing line parameter equation and basic design parameters such as the superposition ratio, the tooth number, the 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 arc tooth trace gear mechanism with the end face arc and involute combined tooth profile 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 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 an arc tooth trace gear mechanism with an end face arc and involute combined tooth profile.
The embodiment of the invention provides an arc tooth trace gear mechanism with combined tooth profile of a face arc and an involute, 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, an end face tooth profile curve of the small wheel and an end face tooth profile curve of 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 arc tooth trace structures; the meshing points of at least one pair of gear teeth of the small wheel and the large wheel are positioned at nodes so as to realize pure rolling meshing contact, and meshing lines formed by the relative rotation meshing points of the small wheel and the large wheel respectively form two contact lines on tooth surfaces of the small wheel and the large wheel.
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 pitch cylindrical surfaces of the small wheel and the large wheel.
Further, the left working tooth profile curves of the end surfaces 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, the small wheel and the large wheelInterdental control point G of right flank profile when the wheel is mounted 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 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 engaged withThe end face pressure angle of the 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 equations for obtaining the upper arc curve for the working profile curve combination are respectively:
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 xi aimax 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 xi aimax 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 o g -x g ,y g ,z g Is the origin o of coordinates of p And o g Establishing a local coordinate system S Invi (o Invi -x Invi y Invi z Invi ) The parameter equations for obtaining the lower involute for the working profile curve combination are respectively:
wherein r is bi Base radius of small and large wheels, R ci Root control point P for small and large wheels c 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 u pi U is respectively i Minimum and maximum values of the values are taken; when determining R ci ,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 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:
coordinates ofS series 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 (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];
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 of small and large wheels;
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 1a -small-wheel end face tooth profile upper circular arc tooth profile radius;
ρ 2a -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 c Radius of the circular arc tooth trace after expansion of the pitch cylinder of the pinion,
R a1 pinion tooth top circle radius, R a1 =R 1 +m t ; (23)
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 ; (24)
R d1 radius of root circle of pinion, R d1 =R 1 -1.25m t ; (25)
R a2 Large tooth tip radius, R a2 =R 2 +m t ; (26)
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 ; (27)
R d2 large tooth root radius, R d2 =R 2 -1.25m t ; (28)
γ 1 -small wheel endPlane 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 the tooth thickness of the indexing circle of the 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 arc 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 circular arc tooth trace gear mechanism for solving the combined tooth profile of the end face circular 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 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 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 arc tooth line structure of the tooth surfaces of the small wheel and the large wheel can be determined, so that the arc tooth line gear mechanism of the end face arc and 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 arc 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 method of the meshing line parameter equation, the contact line is a symmetrical arc after the node cylindrical surface is unfolded, so that the theoretical value of the relative sliding speed of all meshing points on the contact line is zero, and therefore the relative sliding and friction abrasion between tooth surfaces are effectively reduced, meanwhile, the arc 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 has better motion stability; in addition, the arc tooth trace gear mechanism with the combined tooth profile of the end face 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 arc tooth trace gear mechanism with the end surface arc and involute 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 arc tooth trace gear mechanism with the combined tooth profile of the end face arc and the involute has the advantages of no need of design of a tool withdrawal groove, one-step forming, simple processing technology and convenience in assembly.
3. The end face tooth profile of the arc tooth line gear mechanism with the end face arc and involute combined tooth profile is not a single 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 arc tooth trace gear mechanism of the end face arc and involute combined tooth profile is an axisymmetric arc instead of an inclined straight line after the cylindrical surface of the contact line 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 shaft involute gear and other mechanisms and arc toothed line cylindrical gear transmission mechanisms, the single-stage large-transmission-ratio high-coincidence-ratio transmission can be realized, and meanwhile, as the number of teeth can be designed smaller, the same pitch circle diameter of the gear 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 arc tooth trace gear mechanism with the end face 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 an arc tooth trace gear mechanism with an end arc and involute combined tooth profile.
Fig. 2 is a schematic diagram of a space meshing coordinate system of an arc tooth trace gear mechanism with an end arc and involute 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 involute 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 involute 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 tooth trace gear mechanism with a combined tooth profile of an end 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 space meshing coordinate system of an arc tooth trace gear mechanism with an end arc and involute 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 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 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. Big sizeThe end face tooth profile of the 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.
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, and are in a symmetrical form of left and right shafts, and the end face right tooth profile can be obtained by axisymmetrical end face left tooth profile; 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 are CI from the top to the bottom, wherein 'C, I' respectively represents an arc (Cir) and an involute (Inv), 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 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 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.
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 The axis coincides with the meshing line K-K7 passing through the meshing point M 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 Overlapping, the small wheels 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 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 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:
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, resulting in a parameter of the upper arc curve for the working profile curve combinationThe numerical equations are respectively:
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 xi aimax 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 xi aimax 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 o g -x g ,y g ,z g Is the origin o of coordinates of p And o g Establishing a local coordinate system S Invi (o Invi -x Invi y Invi z Invi ) The parameter equations for obtaining the lower involute for the working profile curve combination are respectively:
wherein r is bi Base radius of small and large wheels, R ci Root control point P for small and large wheels c 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 u pi U is respectively i Minimum and maximum values of the values are taken; when determining R ci ,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 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 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 (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];
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 of small and large wheels;
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 1a -small-wheel end face tooth profile upper circular arc tooth profile radius;
ρ 2a -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 c Radius of the circular arc tooth trace after expansion of the pitch cylinder of the pinion,
R a1 pinion tooth top circle radius, R a1 =R 1 +m t ; (23)
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 ; (24)
R d1 radius of root circle of pinion, R d1 =R 1 -1.25m t ; (25)
R a2 Large tooth tip radius, R a2 =R 2 +m t ; (26)
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 ; (27)
R d2 large tooth root radius, R d2 =R 2 -1.25m t ; (28)
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 the tooth thickness of the indexing circle of the 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 arc 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 circular arc tooth trace gear mechanism for solving the combined tooth profile of the end face circular 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 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 Small and large wheelsRoot transition curve starting point P 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 small wheel and the large wheel are correspondingly determined, and the arc tooth line structure of the tooth surfaces of the small wheel and the large wheel can be determined, so that an arc tooth line gear mechanism of the end face arc and involute combined tooth profile is obtained;
in the above formula: axes of the coordinate systems, a, b, m t ,ρ a1 ,ρ a2 ,R 1 And R is 2 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 χ (x) a2 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,radian (rad), b=80 millimeters (mm), α t =20°,T H =0.5,χ a1 =0.08rad,χ a2 =0.04 rad, substituting formulae (16) - (36) 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 preset contact ratio epsilon=2.4 of the circular arc tooth trace pure rolling external meshing gear of the combined tooth profile of the end face circular arc and the involute 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 circular arc tooth trace gear mechanism of the combined tooth profile of the end face circular arc and the involute is realized in the rotating motion. The rotation direction of the input shaft connected with the motor is clockwise, and the circular arc tooth trace corresponding to the end face circular arc and involute combined tooth profile is in a speed reduction transmission mode of the pure rolling external gear, so that the speed reduction torque increase transmission of anticlockwise rotation of the large wheel is realized.
Example 2:
the circular arc tooth trace gear mechanism of the end face circular 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, 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 degree epsilon=2.4 of the circular arc tooth trace pure rolling external gear with the combined tooth profile of the end face circular arc and the involute is guaranteed, at least two pairs of gear teeth participate in meshing transmission at the same time at each instant, and therefore continuous and stable meshing transmission of the circular arc tooth trace gear mechanism with the combined tooth profile of the end face circular arc and the involute in rotating 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 circular arc 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 arc 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 arc tooth trace gear mechanism with the end face 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 diameter, the tooth thickness can be designed to be larger, so that the mechanism has higher strength and larger bearing capacity, and is suitable for popularization and application in the fields of micro/micro machinery, conventional mechanical transmission and high-speed heavy-load transmission; the arc 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 root transition curve parameter values, 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 pure rolling external gear mechanism with combined tooth profile of the face 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 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 tooth trace gear mechanism of the end face circular arc and involute combined tooth profile comprises a pair of gear pairs consisting of a small wheel and a large wheel with parallel axes, wherein the small wheel and the large wheel are in pure rolling meshing transmission, and the circular arc tooth trace 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 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 arc tooth trace structures; the meshing points of at least one pair of gear teeth of the small wheel and the large wheel are positioned at nodes so as to realize pure rolling meshing contact, and meshing lines formed by the relative rotation meshing points of the small wheel and the large wheel respectively form two contact lines on tooth surfaces of the small wheel and the large wheel.
2. The arc tooth trace gear mechanism of the end face arc and involute combined 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 an axisymmetric circular arc after being unfolded along the joint cylindrical surfaces of the small wheel and the large wheel.
3. The arc tooth trace gear mechanism of the end face arc and involute combined tooth profile according to claim 1, characterized in that: the left working tooth profile curve of the end surfaces of the small wheel and the large wheel is formed by a circular arc plane curve and an involute plane curve 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 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 circular arc tooth trace gear mechanism according to claim 3, wherein the end face circular 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 arc tooth trace gear mechanism of the end face arc and involute combined tooth profile according to claim 1, characterized in that: the contact line between the small wheel and the large wheel tooth surface is formed by the following methodAnd (3) determining: 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 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 arc tooth trace gear mechanism of the end face arc and involute combined 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 and i=2 represents a large wheel, the parametric equations for obtaining the upper arc curve for the working profile curve combination are respectively:
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 xi aimax 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 xi aimax 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 o g -x g ,y g ,z g Is the origin o of coordinates of p And o g Establishing a local coordinate system S Invi (o Invi -x Invi y Invi z Invi ) The parameter equations for obtaining the lower involute for the working profile curve combination are respectively:
wherein r is bi Base radius of small and large wheels, R ci Root control point P for small and large wheels c 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 u pi U is respectively i Minimum and maximum values of the values are taken; when determining R ci ,u pi 、u ci All can be solved, so that the lower involute profile is determined;
from the coordinate transformation, can be obtainedCoordinate system S 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 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 (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 -end face pressure angles of the small and large wheels;
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 1a -small end face profile upper circular arc profile radius;
ρ 2a -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 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 ; (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 d1 pinion root circle radius, R d1 =R 1 -1.25m t ; (25)
R a2 Large tooth tip radius, R a2 =R 2 +m t ; (26)
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 ; (27)
R d2 -large tooth root radius, R d2 =R 2 -1.25m t ; (28)
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,
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 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 Pd of left end face of large wheel 2 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 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 circular arc tooth trace gear mechanism for solving the combined tooth profile of the end face circular 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 Form control parameter T of tooth width b and tooth root transition curve H Small wheelAn angle χ of clockwise rotation of the tooth top datum point around the center of the circle 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 arc tooth line structure of the tooth surfaces of the small wheel and the large wheel can be determined, so that the arc tooth line gear mechanism of the end face arc and involute combined tooth profile is obtained.
7. The arc tooth trace gear mechanism of the end face arc and involute combined 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 arc tooth trace gear mechanism of the end face arc and involute combined tooth profile according to claim 7, characterized in that: the small wheel is interchangeable with the input shaft and the output shaft connected with the large wheel.
9. The arc tooth trace gear mechanism of the end face arc and involute combined tooth profile according to claim 1, characterized in that: 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.
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN202310371991.3A CN116480753A (en) | 2023-04-07 | 2023-04-07 | Arc tooth trace gear mechanism with end face arc and involute combined tooth profile |
Applications Claiming Priority (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN202310371991.3A CN116480753A (en) | 2023-04-07 | 2023-04-07 | Arc tooth trace gear mechanism with end face arc and involute combined tooth profile |
Publications (1)
Publication Number | Publication Date |
---|---|
CN116480753A true CN116480753A (en) | 2023-07-25 |
Family
ID=87224361
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
CN202310371991.3A Pending CN116480753A (en) | 2023-04-07 | 2023-04-07 | Arc tooth trace gear mechanism with end face arc and involute combined tooth profile |
Country Status (1)
Country | Link |
---|---|
CN (1) | CN116480753A (en) |
-
2023
- 2023-04-07 CN CN202310371991.3A patent/CN116480753A/en active Pending
Similar Documents
Publication | Publication Date | Title |
---|---|---|
US8967012B2 (en) | Double involute pinion-face gear drive system | |
EP2584224B1 (en) | Gear with free curved surfaces | |
US20060288809A1 (en) | Rack and pinion transmission | |
CN104819266B (en) | Without escape arc spiral line mixed type herringbone bear and its processing method | |
US8070640B2 (en) | Fluctuating gear ratio limited slip differential | |
US20060005653A1 (en) | Enveloping worm transmission | |
US20020134184A1 (en) | Non-involute gears with conformal contact | |
CN111120622B (en) | Modified wave cam and design method thereof, wave generator and wave reducer | |
CN116592114A (en) | Parabolic tooth trace gear mechanism with end face arc and involute combined tooth profile | |
CN110848332B (en) | Intersecting-axis non-circular-face gear transmission mechanism | |
CN113446377B (en) | Conjugate cycloid tooth profile harmonic speed reducer | |
CN202690900U (en) | Novel tooth-shaped gear eccentric drive mechanism | |
CN111237397A (en) | Two-tooth differential close-packed combined tooth surface cycloid oscillating tooth transmission unit | |
US20040221672A1 (en) | Enveloping worm transmission | |
CN108533683B (en) | Convex-convex meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission | |
CN108533681B (en) | flat-convex meshing pure rolling gear mechanism with internal meshing transmission of parallel shafts | |
CN116480753A (en) | Arc tooth trace gear mechanism with end face arc and involute combined tooth profile | |
CN116480755A (en) | Arc tooth trace gear mechanism with end surface double arc combined tooth profile | |
CN116498728A (en) | Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile | |
CN116480754A (en) | Parabolic tooth trace gear mechanism with end face arc and parabolic combined tooth profile | |
CN116480752A (en) | Parabolic tooth trace gear mechanism with end surface double-arc combined tooth profile | |
CN201496509U (en) | Male and female full rolling gear | |
CN108533679B (en) | Parallel shaft external engagement transmission convex-Ping Niege pure rolling gear mechanism | |
WO2004102036A2 (en) | Enveloping worm transmission and machining of enveloping worm transmission | |
WO2010033090A2 (en) | Helical cylindrical gear pair for uniform power transmission |
Legal Events
Date | Code | Title | Description |
---|---|---|---|
PB01 | Publication | ||
PB01 | Publication | ||
SE01 | Entry into force of request for substantive examination | ||
SE01 | Entry into force of request for substantive examination |