CN116498728A - Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile - Google Patents
Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile Download PDFInfo
- Publication number
- CN116498728A CN116498728A CN202310372272.3A CN202310372272A CN116498728A CN 116498728 A CN116498728 A CN 116498728A CN 202310372272 A CN202310372272 A CN 202310372272A CN 116498728 A CN116498728 A CN 116498728A
- Authority
- CN
- China
- Prior art keywords
- tooth
- wheel
- small
- point
- 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
- 230000033001 locomotion Effects 0.000 claims abstract description 44
- 230000007704 transition Effects 0.000 claims abstract description 39
- 238000005096 rolling process Methods 0.000 claims abstract description 13
- 230000005540 biological transmission Effects 0.000 claims description 46
- 230000009466 transformation Effects 0.000 claims description 18
- 238000000034 method Methods 0.000 claims description 16
- 230000001154 acute effect Effects 0.000 claims description 15
- 239000011159 matrix material Substances 0.000 claims description 12
- 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
- 244000309464 bull Species 0.000 claims description 6
- 238000009434 installation Methods 0.000 claims description 5
- 238000004364 calculation method Methods 0.000 claims description 3
- 239000013256 coordination polymer Substances 0.000 claims description 3
- 238000013461 design Methods 0.000 abstract description 12
- 238000005299 abrasion Methods 0.000 abstract description 4
- 230000009286 beneficial effect Effects 0.000 abstract description 2
- 238000005452 bending Methods 0.000 description 7
- 238000010586 diagram Methods 0.000 description 4
- 230000009467 reduction Effects 0.000 description 4
- 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
- 230000002457 bidirectional effect Effects 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
Landscapes
- Engineering & Computer Science (AREA)
- General Engineering & Computer Science (AREA)
- Mechanical Engineering (AREA)
- Gear Transmission (AREA)
- Gears, Cams (AREA)
Abstract
The invention provides an arc tooth trace gear mechanism with combined tooth profiles of an end face arc and a parabola, which comprises a small wheel and a large wheel, wherein 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 a parabola; the tooth surfaces of the small wheel and the large wheel are provided with arc tooth trace structures; at least one pair of gear tooth meshing points of the small wheel and the large wheel are positioned at the node so as to realize pure rolling meshing contact, and meshing lines formed by the 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. The invention has the beneficial effects that: based on the active design of the motion rule of the engagement points, the contact line of node engagement 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 the engagement points on the contact line is zero, and the relative sliding and friction abrasion between tooth surfaces are effectively reduced.
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 parabolic 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 arc and parabolic combined tooth profile.
The embodiment of the invention provides an arc tooth trace gear mechanism with an end face arc and parabolic combined tooth profile, which comprises a pair of gear pairs consisting of a small wheel and a large wheel which are parallel in axis, wherein the small wheel and the large wheel are in pure rolling meshing transmission, 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 both 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 a parabola; 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 respectively formed by the movement of respective tooth profile curves along contact points along with contact lines, and the contact lines are circular arcs with z-axis symmetry after being unfolded along the joint cylindrical surfaces of the small wheel and the large wheel.
Further, the left working tooth profile curves of the small wheel and the large wheel are both arc-shaped and parabolic plane curves at an interdental control point P bi Smooth connection to formInterdental control point G of right side profile 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 that the combination types of the working tooth profile curves of the small wheel and the large wheel from the top to the bottom are CP, wherein C, P respectively represent an arc and a parabola, the arc is a working tooth profile upper curve, and the parabola 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 For the tooth top datum point J of the small wheel and the large wheel 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 For the tooth base quasi-point J of the small wheel and the large wheel ci An angle of clockwise rotation about the center of the circle; wherein the small wheel and the large wheel tooth top datum point J ai Is an involute with the same base radius, end face pressure angle and the same radius R as the small wheel and the large wheel respectively ai Intersection of addendum circle; the small wheel and the big gear tooth base quasi point J ci Is an involute with the same base radius, end face pressure angle and the same radius R as the small wheel and the large wheel respectively 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 Rotation of the shaft with the small wheelAxis is coincident with z g The axis coincides with the rotation axis of the large wheel, 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 and a coordinate system o 2 -x 2 ,y 2 ,z 2 Fixedly connected with the large wheel, and the small wheel and the 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 The overlap is denoted as M, the small wheels are at a uniform angular velocity omega 1 Around z p The shaft rotates clockwise, 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 wheels wind 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 and driven, a 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 Motion of (a)Parameter variable, t is more than or equal to 0 and less than or equal to delta t;is a linear scaling factor of the meshing point movement; r is R 1 A pitch cylinder radius for the small wheel; r is (r) c For the radius of the circular arc tooth trace after the expansion of the pitch cylinder surface of the small wheel, the linear proportionality coefficient of the movement of the meshing point is +.>Maximum value delta t of motion parameter variable and end surface modulus m t Said small wheel parameter 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 rotational 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 A transmission ratio between the small wheel and the large wheel;
when engaged point M a And M b Moving along the meshing line K-K, they form contact lines C at the pinion flanks and the bull flanks, respectively 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 For the pitch cylinder radius of the small wheel, R 2 For 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:
the contact line C of the tooth surface of the large wheel is obtained 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:
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;
the parametric equation for the lower parabolic curve for the working profile curve combination is:
in formula (9), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; a, a pi The quadratic term coefficient of parabolic curve of small wheel and large wheel, x pi Is the coordinate axis x pbi Parameter of (x) pimin And x pimax Respectively x pi Minimum and maximum values of the values are taken; when determining the radius R of the tooth bottom circle ci Offset angle χ ci When a is pi 、x pimin And x pimax All can be solved, so that a lower parabolic tooth profile curve 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 A central angle corresponding to the indexing circular tooth thickness of the small wheel and the large wheel is obtained;
the tooth root transition curve at the left side of the gear tooth end surfaces of the small gear and the large gear is Hermite curve, which 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 transition curve of the tooth rootShape control parameter, 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 -said small number of teeth;
Z 2 -said large number of teeth;
a pi -the quadratic coefficients of the parabolic curves of the small and large wheels;
x pimin -x pi the minimum value of the values is taken;
x pimax -x pi the maximum value of the values is taken;
b-the tooth widths of the small and large wheels;
α t -an end face pressure angle;
J ai -said small wheel and said large wheel tooth top datum point
J ci -said small wheel and said large gear tooth base quasi-points
χ a1 -an angle by which the tip reference point of the small wheel rotates clockwise around the center of the circle;
χ a2 -an angle by which the tooth top datum point of the large wheel rotates clockwise around the centre of a 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 large wheel rotates clockwise around the centre of the circle;
ρ a1 -the small end face profile upper circular arc profile radius;
ρ a2 -said large wheelRadius of arc tooth profile at upper part of end face tooth profile;
k c -the small wheel and the large wheel root transition curve starting point P ci Radius variation coefficient of (a);
R 1 -for the pitch cylinder radius of the small wheel, R 1 =m t Z 1 /2; (15)
R 2 -for the pitch cylinder radius of the large wheel, R 2 =i 12 R 1 ; (16)
i 12 For the transmission ratio of the small wheel to the large wheel,
a-relative positions of axis mounting of the small wheel and the large wheel: a=r 1 +R 2 ; (18)
r b1 -said small wheel base radius, r b1 =R 1 cosα t ; (19)
r b2 -said 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 -said pinion tooth tip radius, R a1 =R 1 +m t ; (22)
R c1 -the pinion tooth base radius, i.e. root transition curve start point P c1 Radius to the centre of rotation of the trolley, R c1 =R 1 -k c m t ; (23)
R d1 -said pinion tooth root circle radius, R d1 =R 1 -1.25m t ; (24)
R a2 -said large tooth tip radius, R a2 =R 2 +m t ; (25)
R c2 -the large tooth bottom radius, i.e. root transition curve start point P c2 Radius to the centre of rotation of the large wheel, R c2 =R 2 -k c m t ; (26)
R d2 -said large tooth root radius, R d2 =R 2 -1.25m t ; (27)
γ 1 One said 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 one of the large wheel end face nodes 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 -the central angle corresponding to the small wheel pitch circle tooth thickness,
λ 2 -a central angle corresponding to the tooth thickness of the indexing circle of the large wheel,
δ 1 -the 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 -the left face profile point P of the bull 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 of the end face arc and the parabolic combined tooth profile is required to be more than 2, and the coincidence ratio calculation formula is as follows:
linear scaling factor based on the value epsilon of overlapAnd the number of teeth Z of the small wheel 1 The maximum value of the motion parameter variable of the meshing point of the circular arc tooth trace gear mechanism of the end face circular arc and parabolic combined tooth profile is obtained by the following steps:
when determining the number Z of the small gear 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 An angle χ of clockwise rotation of the tooth top datum point of the large wheel around the circle center a2 The tooth base quasi point of the small wheel rotates clockwise around the circle center by an angle χ c1 The tooth base quasi point of the large wheel rotates clockwise around the circle center by an angle χ c2 The root transition curve starting point P of the small wheel and the 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 line structure of the small wheel and the large wheel can be determined, so that the circular arc tooth line gear mechanism of the end face circular arc and parabolic 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 face arc and parabolic 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, and therefore the relative sliding and friction abrasion between tooth surfaces are effectively reduced, meanwhile, the arc tooth line gear mechanism of the end face arc and parabolic combined tooth profile is free of the phenomenon of tooth top sharpening, a contact area extends over the width of the tooth, and larger tooth width can be designed and utilized to drive larger load and has better motion stability; in addition, the parabolic tooth trace pure rolling external gear mechanism with the end face circular arc and circular 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 rotation transmission.
2. The circular arc tooth trace gear mechanism with the end surface circular arc and parabolic combined tooth profile has the advantages that the cylindrical surface of the contact line is an axisymmetric circular arc instead of an inclined straight line after being developed, so that no axial force is generated during transmission, the installation condition of a shafting is simpler, and the structure is simple.
3. The circular arc toothed gear mechanism with the end surface circular arc and parabolic combined tooth profile is in pure rolling engagement in theory, 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 circular arc and parabolic 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.
4. The end face tooth profile of the arc tooth line gear mechanism with the end face arc and parabolic combined tooth profile is not a single arc equiplanar 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.
5. Compared with the existing parallel-axis involute gear and other mechanisms and arc toothed line cylindrical gear transmission mechanisms, the circular arc toothed line gear mechanism with the end face circular arc and parabolic combined tooth profile has no undercut, has the minimum tooth number of 1, can realize single-stage large-transmission-ratio overlap ratio transmission, and simultaneously has higher bending strength and larger bearing capacity due to the fact that the tooth number can be designed smaller and the tooth thickness and modulus can be designed larger when the pitch diameter of the gear is the same, and 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 face arc and parabolic 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 a combination of end arc and parabola tooth profile.
Fig. 2 is a schematic diagram of a spatial meshing coordinate system of a circular arc tooth trace gear mechanism with a combination of end circular arc and parabolic tooth profile according to the present invention.
Fig. 3 shows the tooth profile composition structure of the end faces of the large wheel and the small wheel in fig. 1 and 2 and the coordinate system thereof.
FIG. 4 is a schematic diagram of the local coordinate system relationship of the combined tooth profile of the present invention.
Fig. 5 is a schematic view of the tooth top datum points and the corner of the combined tooth profile according to the present invention.
Fig. 6 is a three-dimensional view of the small wheel of fig. 1 in accordance with the present invention.
Fig. 7 is a three-dimensional view of the large wheel of fig. 1 in accordance with the present invention.
FIG. 8 is a schematic diagram of the present invention when the large wheel is connected to the input shaft to drive the small wheel to speed up.
In the above figures: 1-a driver, 2-a coupler, 3-an input shaft, 4-a small wheel, 5-an output shaft, 6-a large wheel, 7-a meshing line K-K, 8-a small wheel pitch cylinder, 9-a small wheel contact line Cp, 10-a large wheel contact line Cg, 11-a large wheel pitch cylinder, 12-a large wheel end face tooth profile left tooth root transition curve, 13-a large wheel end face tooth profile left lower parabolic 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 parabolic 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:
the invention provides an arc tooth trace gear mechanism with an end face arc and parabolic combined tooth profile, which is applied to speed reduction transmission with a transmission ratio of 3 between parallel shafts and is designed to have the coincidence ratio of epsilon=2.4. The structure of the device is shown in figure 1, and the device comprises a small wheel 4 and a large wheel 6, wherein the small wheel 4 and the large wheel 6 form a pair of gear pairs, the small wheel 4 is connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1 through a coupler 2, the large wheel 6 is connected with an output shaft 5, namely, the large wheel 6 is connected with a driven load through the output shaft 5; the axes of the small wheel 4 and the large wheel 6 are parallel to each other. Fig. 2 is a schematic diagram of a spatial meshing coordinate system of a circular arc tooth trace gear mechanism with a combination of end surface circular arcs and parabolas.
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 left end face work of the small wheel gear teeth is carried out sequentially from the top to the bottomThe upper arc curve 17, the lower flank profile parabola 16 and the root transition curve, i.e., 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 tooth profile of the left end face working tooth profile of the small wheel tooth from the top to the bottom sequentially comprises an upper arc curve 14, a lower parabola 13 and a transition curve of the left end face and the bottom, namely a 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 a parabola and are in a symmetrical form of left and right axes, and the end face right tooth profile can be obtained by the axial symmetry of the end face left tooth profile; the left working tooth profile curve is formed by a circular arc plane curve and a parabolic 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 type of the small-wheel and large-wheel working profile curves from the top to the bottom is CP, wherein "C, P" represents an arc (Cir) and a parabola (Par), respectively, the arc being the working profile upper curve and the parabola being the working 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.
Left working tooth profile of the small wheel and the large wheelTip control point P of (2) 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 small and large wheel tooth surface contact lines 9, 10 are determined by the following method: at o p -x p ,y p ,z p 、o k -x k ,y k ,z k O g -x g ,y g ,z g In three spatial coordinate systems, z p The axis of rotation of the shaft and the small wheel being coincident, z g The axle coincides with the axis of rotation of the large wheel, zk 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 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 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 in mm(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, the parametric equation for the upper arc curve for the working profile curve combination is:
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;
the parametric equation for the lower parabolic curve for the working profile curve combination is:
in formula (9), i=1, 2, where i=1 represents a small wheel and i=2 represents a large wheel; a, a pi The quadratic term coefficient of parabolic curve of small wheel and large wheel, x pi Is the coordinate axis x pbi Parameter of (x) pimin And x pimax Respectively arex pi Minimum and maximum values of the values are taken; when determining the radius R of the tooth bottom circle ci Offset angle χ ci When a is pi 、x pimin And x pimax All can be solved, so that a lower parabolic tooth profile curve 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 A central angle corresponding to the indexing circular tooth thickness of the small wheel and the large wheel is 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;
a pi -quadratic coefficients of parabolic curves of small and large wheels;
x pimin -x pi the minimum value of the values is taken;
x pimax -x pi the maximum value of the values is taken;
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 a1 -small end face profile upper circular arc profile radius;
ρ a2 -the radius of the upper circular arc profile of the face profile of the large wheel;
k c -the start point P of the transition curve of the small and large wheel root ci Radius variation coefficient of (a);
R 1 -for the pitch cylinder radius of the small wheel, R 1 =m t Z 1 /2; (15)
R 2 -for the pitch cylinder radius of the large wheel, R 2 =i 12 R 1 ; (16)
i 12 For the transmission ratio of the small wheel to the large wheel,
a-relative positions of axis mounting of the small wheel and the large wheel: a=r 1 +R 2 ; (18)
r b1 -said small wheel base radius, r b1 =R 1 cosα t ; (19)
r b2 -said 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 -said pinion tooth tip radius, R a1 =R 1 +m t ; (22)
R c1 -the pinion tooth base radius, i.e. root transition curve start point P c1 Radius to the centre of rotation of the trolley, R c1 =R 1 -k c m t ; (23)
R d1 -said pinion tooth root circle radius, R d1 =R 1 -1.25m t ;(24)
R a2 -said large tooth tip radius, R a2 =R 2 +m t ; (25)
R c2 -the large tooth bottom radius, i.e. root transition curve start point P c2 Radius to the centre of rotation of the large wheel, R c2 =R 2 One k c m t ; (26)
R d2 -said large tooth root radius, R d2 =R 2 -1.25m t ; (27)
γ 1 -said 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,
β 2 -said 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 a central angle corresponding to the tooth thickness of the indexing circle of the small wheel,
λ 2 -a central angle corresponding to the tooth thickness of the indexing circle of the large wheel,
δ 1 -the 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 -the left face profile point P of the bull 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 of the end face arc and parabolic 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 teeth Z of the small wheel 1 The maximum value of the motion parameter variable of the meshing point of the circular arc tooth trace gear mechanism of the end face circular arc and parabolic combined tooth profile is obtained by the following steps:
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 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 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 of the end face circular arc and parabolic combined tooth profile is obtained;
in the above formula: axes of the coordinate systems, a, b, m t ,ρ a1 ,ρ a2 ,x pimin ,x pimax ,r c ,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 ,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 of the combined tooth profile of the end face circular arc and the parabola is ensured to be in a meshing state because of the fact that at least two pairs of adjacent gear teeth are simultaneously engaged in meshing transmission at each instant when the small wheel 2 and the large wheel 5 are correctly installed, so that 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 parabola in rotary motion is realized. The rotation direction of an input shaft connected with the motor is clockwise, and the speed reduction transmission mode of the circular arc tooth trace pure rolling external gear corresponding to the end face circular arc and parabolic combined tooth profile is used for realizing the speed reduction and torque increase transmission of the anticlockwise rotation of the large wheel.
Example 2:
the circular arc tooth trace gear mechanism of the end face circular arc and parabolic 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 combined tooth profile of the end face circular arc and the parabola 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 combined tooth profile of the end face circular arc and the parabola 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, mt=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 (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 tooth trace gear mechanism corresponding to the end face circular arc and parabolic 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 parabolic combined tooth profile is based on an active design method of a meshing line parameter equation, and an end face working tooth profile is formed by combining an arc curve and a parabolic, 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 parabolic 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 pitch diameter of the same gear is the same, the tooth thickness can be designed, so that the arc tooth trace gear mechanism has higher strength and larger bearing capacity, and is suitable for popularization and application in the fields of micro/micro machinery, conventional mechanical transmission and high-speed heavy-duty transmission; the arc tooth trace gear mechanism with the end surface arc and parabolic 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 pure rolling external gear mechanism with combined tooth profile of arc and parabola has extremely small difference between maximum tooth surface contact stress and maximum tooth root bending stress of forward and backward transmission, and has similar strength of forward and backward 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 with the end face circular arc and parabolic combined tooth profile 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, 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 a parabola; 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 circular arc rack and pinion mechanism of the end face circular arc and parabolic combined tooth profile of claim 1, wherein: the tooth surface structures of the small wheel and the large wheel are respectively formed by the movement of respective tooth profile curves along contact points along contact lines, and the contact lines are axisymmetric circular arcs after being unfolded along the joint cylindrical surfaces of the small wheel and the large wheel.
3. The circular arc rack and pinion mechanism of the end face circular arc and parabolic combined tooth profile of claim 1, wherein: the left working tooth profile curves of the small wheel and the large wheel are both arc-shaped and parabolic 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 that the combination types of the working tooth profile curves of the small wheel and the large wheel from the top to the bottom are CP, wherein C, P respectively represent an arc and a parabola, the arc is a working tooth profile upper curve, and the parabola 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. The circular arc tooth trace gear mechanism with combined end surface circular arc and parabolic tooth profile as in claim 3The method is characterized in that: 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 For the tooth top datum point J of the small wheel and the large wheel 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 For the tooth base quasi-point J of the small wheel and the large wheel ci An angle of clockwise rotation about the center of the circle; wherein the small wheel and the large wheel tooth top datum point J ai Is an involute with the same base radius, end face pressure angle and the same radius R as the small wheel and the large wheel respectively ai Intersection of addendum circle; the small wheel and the big gear tooth base quasi point J ci Is an involute with the same base radius, end face pressure angle and the same radius R as the small wheel and the large wheel respectively ci Intersection of the root circles.
5. The circular arc rack and pinion mechanism of the end face circular arc and parabolic combined tooth profile of 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 In three spatial coordinate systems, z p The axis coincides with the axis of rotation of the small wheel, z g The axis coincides with the rotation axis of the large wheel, 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 and a coordinate system o 2 -x 2 ,y 2 ,z 2 Fixedly connected with the large wheel, and the small wheel and the 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 The overlap is denoted as M, the small wheels are at a uniform angular velocity omega 1 Around z p The shaft rotates clockwise, 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 wheels wind 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 and driven, a 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 A pitch cylinder radius for the small wheel; r is (r) c For the radius of the circular arc tooth trace after the expansion of the pitch cylinder surface of the small wheel, the linear proportionality coefficient of the movement of the meshing point is +.>Maximum value delta t of motion parameter variable and end surface modulus m t Said small wheel parameter 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 rotational 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 A transmission ratio between the small wheel and the large wheel;
when engaged point M a And M b Moving along the meshing line K-K, they form contact lines C at the pinion flanks and the bull flanks, respectively 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 For the pitch cylinder radius of the small wheel, R 2 For 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:
the contact line C of the tooth surface of the large wheel is obtained by the steps (1) and (5) g The parametric equation for (2) is:
6. the circular arc rack and pinion mechanism of the end face circular arc and parabolic combined tooth profile of 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 the small wheel, i=2 represents the large wheel, and the parametric equations for obtaining the upper arc curve for the working profile curve combination are respectively:
in formula (8), i=1, 2, wherein i=1 represents the small wheel and i=2 represents the 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 arcs 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;
the parametric equations for the lower parabolic curve for the working profile curve combination are respectively:
in the formula (9), a pi A quadratic term coefficient, x, of the parabolic curve of the small wheel and the large wheel pi Is the coordinate axis x pbi Parameter of (x) pimin And x pimax Respectively x pbi Minimum and maximum values of the values are taken; when determining the radius R of the tooth bottom circle ci Offset angle χ ci When a is pi 、x pimin And x pimax All can be solved, so that a lower parabolic tooth profile curve 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 lnvi 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 A central angle corresponding to the indexing circular tooth thickness of the small wheel and the large wheel is obtained;
the tooth root transition curve at the left side of the gear tooth end surfaces of the small gear and the large gear is Hermite curve, which 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 -said small number of teeth;
Z 2 -said large number of teeth;
a pi -the quadratic coefficients of the parabolic curves of the small and large wheels;
x pimin -x pi the minimum value of the values is taken;
x pimax -x pi the maximum value of the values is taken;
b-the tooth widths of the small and large wheels;
α t -an end face pressure angle;
J ai -said small wheel and said large wheel tooth top datum point
J ci -said small wheel and said large gear tooth base quasi-points
χ a1 -an angle by which the tip reference point of the small wheel rotates clockwise around the center of the circle;
χ a2 -an angle by which the tooth top datum point of the large wheel rotates clockwise around the centre of a 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 large wheel rotates clockwise around the centre of the circle;
ρ a1 -the small end face profile upper circular arc profile radius;
ρ a2 -the large face profile upper circular arc profile radius;
k c -the small wheel and the large wheel root transition curve starting point P ci Radius variation coefficient of (a);
R 1 -for the pitch cylinder radius of the small wheel, R 1 =m t Z 1 /2; (15)
R 2 -for the pitch cylinder radius of the large wheel, R 2 =i 12 R 1 ; (16)
i 12 For the transmission ratio of the small wheel to the large wheel,
a-relative positions of axis mounting of the small wheel and the large wheel: a=r 1 +R 2 ; (18)
r b1 -said small wheel base radius, r b1 =R 1 cosα t ; (19)
r b2 -said 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 -said pinion tooth tip radius, R a1 =R 1 +m t ; (22)
R c1 -the pinion tooth base radius, i.e. root transition curve start point P c1 Radius to the centre of rotation of the trolley, R c1 =R 1 -k c m t ; (23)
R d1 -said pinion tooth root circle radius, R d1 =R 1 -1.25m t ; (24)
R a2 -said large tooth tip radius, R a2 =R 2 +m t ; (25)
R c2 -the large tooth bottom radius, i.e. root transition curve start point P c2 Radius to the centre of rotation of the large wheel, R c2 =R 2 -k c m t ; (26)
R d2 -said large tooth root radius, R d2 =R 2 -1.25m t ; (27)
γ 1 One said 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 one of the large wheel end face nodes 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 -the central angle corresponding to the small wheel pitch circle tooth thickness,
λ 2 -a central angle corresponding to the tooth thickness of the indexing circle of the large wheel,
δ 1 -the 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 -the left face profile point P of the bull 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 of the end face arc and parabolic 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 overlap And the number of teeth Z of the small wheel 1 The maximum value of the motion parameter variable of the meshing point of the circular arc tooth trace gear mechanism of the end face circular arc and parabolic combined tooth profile is obtainedWhen determining the number Z of the small gear teeth 1 Ratio i 12 Modulus of end face m t Overlap epsilon and linear scaling factor ∈>End 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 An angle χ of clockwise rotation of the tooth top datum point of the large wheel around the circle center a2 The tooth base quasi point of the small wheel rotates clockwise around the circle center by an angle χ c1 The tooth base quasi point of the large wheel rotates clockwise around the circle center by an angle χ c2 The root transition curve starting point P of the small wheel and the 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 line structure of the small wheel and the large wheel can be determined, so that the circular arc tooth line gear mechanism of the end face circular arc and parabolic combined tooth profile is obtained.
7. The circular arc rack and pinion mechanism of the end face circular arc and parabolic combined tooth profile of 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 rack and pinion mechanism of claim 7 wherein the end circular arc and parabolic combination profile is characterized by: the small wheel is interchangeable with the input shaft and the output shaft connected with the large wheel.
9. The circular arc rack and pinion mechanism of the end face circular arc and parabolic combined tooth profile of 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.
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN202310372272.3A CN116498728A (en) | 2023-04-07 | 2023-04-07 | Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile |
Applications Claiming Priority (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
CN202310372272.3A CN116498728A (en) | 2023-04-07 | 2023-04-07 | Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile |
Publications (1)
Publication Number | Publication Date |
---|---|
CN116498728A true CN116498728A (en) | 2023-07-28 |
Family
ID=87321002
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
CN202310372272.3A Pending CN116498728A (en) | 2023-04-07 | 2023-04-07 | Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile |
Country Status (1)
Country | Link |
---|---|
CN (1) | CN116498728A (en) |
-
2023
- 2023-04-07 CN CN202310372272.3A patent/CN116498728A/en active Pending
Similar Documents
Publication | Publication Date | Title |
---|---|---|
US8967012B2 (en) | Double involute pinion-face gear drive system | |
US8070640B2 (en) | Fluctuating gear ratio limited slip differential | |
US20060288809A1 (en) | Rack and pinion transmission | |
WO2011093744A1 (en) | Eccentrically cycloidal engagement of toothed profiles having curved teeth | |
US20060005653A1 (en) | Enveloping worm transmission | |
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 | |
Jang et al. | Geometry design and dynamic analysis of a modified cycloid reducer with epitrochoid tooth profile | |
CN108533683B (en) | Convex-convex meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission | |
CN111237397A (en) | Two-tooth differential close-packed combined tooth surface cycloid oscillating tooth transmission unit | |
CN108533681B (en) | flat-convex meshing pure rolling gear mechanism with internal meshing transmission of parallel shafts | |
CN116498728A (en) | Arc tooth trace gear mechanism with end face arc and parabolic combined tooth profile | |
US20040221672A1 (en) | Enveloping worm transmission | |
CN116480754A (en) | Parabolic tooth trace gear mechanism with end face arc and parabolic combined tooth profile | |
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 | |
CN116480752A (en) | Parabolic tooth trace gear mechanism with end surface double-arc combined tooth profile | |
CN108533679B (en) | Parallel shaft external engagement transmission convex-Ping Niege pure rolling gear mechanism | |
CN108533686B (en) | Concave-convex mesh pure rolling bevel gear mechanism for crossed shaft transmission | |
CN108533685B (en) | Convex-concave meshing pure rolling spiral bevel gear mechanism for crossed shaft transmission | |
KR101375980B1 (en) | Gear pump | |
CN108691954B (en) | flat-convex meshing pure rolling bevel gear mechanism for crossed shaft transmission | |
CN210153157U (en) | Double-arc planetary transmission device with small tooth difference and meshed with gaps | |
CN113944728B (en) | Unequal-pressure-angle end face double-arc gear mechanism driven by parallel shafts |
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 |