CN113944728B - Unequal-pressure-angle end face double-arc gear mechanism driven by parallel shafts - Google Patents
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H1/00—Toothed gearings for conveying rotary motion
- F16H1/02—Toothed gearings for conveying rotary motion without gears having orbital motion
- F16H1/04—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members
- F16H1/06—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members with parallel axes
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
- F16H55/02—Toothed members; Worms
- F16H55/08—Profiling
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
- F16H55/02—Toothed members; Worms
- F16H55/17—Toothed wheels
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H57/00—General details of gearing
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
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- G06F30/17—Mechanical parametric or variational design
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
- F16H—GEARING
- F16H57/00—General details of gearing
- F16H2057/0087—Computer aided design [CAD] specially adapted for gearing features ; Analysis of gear systems
Abstract
The invention discloses an unequal pressure angle end face double-arc gear mechanism driven by parallel shafts and a design method, and belongs to the field of gear transmission, wherein the double-arc gear mechanism comprises a small gear and a large gear, the axes of the small gear and the large gear are parallel, the end face tooth profiles of the small gear and the large gear are respectively composed of a convex arc tooth profile, a straight line tooth profile, a concave arc tooth profile and a tooth root transition curve, and the specific structure of the tooth profiles is determined by a meshing line parameter equation and parameters such as contact ratio, tooth number and transmission ratio; when the driving device is correctly installed, the convex-concave arcs of the small wheel and the large wheel simultaneously realize double-point meshing contact on the end faces of the wheel teeth, the two meshing points have unequal pressure angles, and the small wheel and the large wheel rotate under the driving of the driving device to realize transmission between two shafts. The design method disclosed by the invention can be used for designing the unequal pressure angle double-arc gear mechanism for parallel shaft transmission, has the advantages of simple design, easiness in lubrication, high tooth root bending strength, large transmission ratio and contact ratio, strong bearing capacity and the like, and can be widely applied to the design of transmission systems of engineering machinery such as double-wheel milling machines and the like.
Description
Technical Field
The invention relates to the field of gear transmission, in particular to a parallel shaft transmission double-circular-arc gear mechanism with unequal pressure angle end faces.
Background
The gear is used as a basic component of a mechanical core, is widely applied to the field of equipment manufacturing industries such as machine tools, automobiles, robots, wind power, coal mines, aerospace and the like and national economy main battlefield, and the quality of the performance directly determines the quality, performance and reliability of major equipment and high-end industrial products. Research on core basic parts such as high-performance gears and the like is a key factor for promoting industrial transformation and upgrading and improving the core competitiveness of national industry. Particularly, with the economic acceleration, the engineering machinery develops towards high power, and higher requirements are put forward on the design of core transmission parts, such as transmission gear transmission of a gearbox.
At present, the main problem faced by the gear industry in China is that the design and manufacturing capability of high-performance gear products with high efficiency, large bearing capacity, light weight and high reliability is obviously insufficient. Parallel shaft gear drives are the most commonly used form of gear drive, with involute gears being the most widely used. However, involute gears have problems of transmission failures such as friction wear, gluing and plastic deformation caused by relative sliding of tooth surfaces until now, so that the transmission efficiency, the service life and the reliability of gear products, particularly high-speed heavy-duty gears, are seriously affected, and the performance improvement of high-precision mechanical equipment is restricted. Especially under heavy load, the relative sliding of the involute gear tooth tops is very serious, and transmission failure is easily caused.
In order to solve the problems of the involute gear transmission, for a parallel shaft transmission mode, domestic and foreign scholars gradually research and develop a single arc gear and a double arc gear, including a face double arc gear and a normal double arc gear, for example, chinese patent document with application number 202110318591.7 discloses a double arc small tooth difference speed reduction transmission device and a double arc tooth forming method, application number 201810893876.1 discloses a double arc gear, application number 201620553083.1 discloses a double arc gear, and the like. The double-arc gear with parallel shaft transmission has larger tooth surface contact strength and tooth root bending strength and better lubricating property compared with an involute gear. However, the tooth profiles of the small gear and the large gear of the double circular-arc gear are cut by a generating method based on the same hob, and in order to ensure that the large gear and the small gear are correctly meshed, the pressure angles of two meshing points of the hob tooth profiles are set to be equal. Therefore, the existing double-circular-arc gear mechanism has the limitation that the pressure angles of two meshing points of the tooth profile are limited to be equal, so that the structure of the double-circular-arc gear mechanism is not an optimal bearing design structure, and during heavy-load transmission of engineering machinery such as a double-wheel mill and the like in underground continuous wall construction, the root of a gear tooth can be broken to cause construction accidents. In addition, if the safety coefficient is increased simply for increasing the bending strength of the root part of the double-arc gear, the gear module is increased, so that the transmission structure of the equipment is overlarge in size, and the design and performance improvement of the whole machine are influenced.
Disclosure of Invention
The invention aims to provide an unequal pressure angle end face double-arc gear mechanism for parallel shaft transmission and a design method thereof aiming at the problems of the prior art of the double-arc gear mechanism in the field of mechanical transmission at present.
In order to achieve the purpose, the invention adopts the technical scheme that: the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft comprises a pair of gear pairs consisting of a small wheel and a large wheel, wherein the small wheel is fixedly connected with a driver through an input shaft, the large wheel is connected with an output shaft, the axes of the small wheel and the large wheel are parallel, the end face tooth profiles of the small wheel and the large wheel are in an axial symmetry form, and the left and right side end face tooth profiles from the tooth top to the tooth bottom respectively consist of a convex arc tooth profile, a straight line tooth profile, a concave arc tooth profile and a tooth root transition curve; the small wheel and the large wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, and the two meshing points have unequal pressure angles; the small wheel is driven by the driver to rotate, stable meshing transmission between parallel shafts is realized through continuous meshing between two pairs of convex circular arc tooth profiles and concave circular arc tooth profiles, two meshing points with different pressure angles respectively form two contact lines with the same pitch on tooth surfaces of the small wheel and the large wheel, and the two contact lines are cylindrical spiral lines; the gear tooth flanks of the small wheel and the large wheel are spiral tooth flanks obtained by cylindrical spiral motion of end face tooth profiles along respective contact lines, the screw pitch of the spiral tooth flanks is equal to that of the contact lines, and the spiral directions of the gear teeth of the small wheel and the gear teeth of the large wheel are opposite.
Further, the small wheel and the big wheel are in double-point contact concave-convex meshing transmission, and two meshing points of the small wheel and the big wheel are respectively a meshing point M of a convex circular arc tooth profile of the teeth of the small wheel and a concave circular arc tooth profile of the teeth of the big wheel R1 And the meshing point M of the concave circular arc tooth profile of the adjacent gear teeth of the small gear and the convex circular arc tooth profile of the adjacent gear teeth of the large gear R2 (ii) a The normal lines of the tooth profile meshing points of the two pairs of concave-convex circular arc end surfaces intersect at the same point, and the point is a tangent point of a pitch circle of the pair of double-circular-arc gears, namely a node; the horizontal distances from the two meshing points to the node are both PM; when the pair of parallel shafts drives the double-circular-arc gear mechanism with unequal pressure angle end faces, two meshing points M R1 And M R2 Have the sameAxial movement speed, respectively forming two space meshing lines K R1 -K R1 And K R2 -K R2 And each form two contact lines of the tooth surfaces of the small wheel and the large wheel.
Further, the tooth surface contact line of the small wheel and the large wheel is determined by the following method:
at o p --x p ,y p ,z p 、o k --x k ,y k ,z k And o g --x g ,y g ,z g In three spatial coordinate systems, where o p 、o k And o g Are respectively the origin, x, of three spatial coordinate systems p 、x k And x g X-axis, y of three spatial coordinate systems respectively p 、y k And y g X-axis, z, of three spatial coordinate systems, respectively p 、z k And z g Z-axes, z, of three spatial coordinate systems, respectively p The axis of rotation of the axle and the small wheel coinciding, z g The axis of rotation of the shaft and the bull wheel coinciding, z k Shaft-to-pass meshing point M R1 Engagement line K of R1 - K R1 Are superposed and z k Axis and z p 、z g Axes parallel to each other, x p And x g The axes being coincident, x k And x g Axis parallel, o p And o g A is a; coordinate system o 1 --x 1 ,y 1 ,z 1 Fixedly connected with the small wheel and having a coordinate system o 2 --x 2 ,y 2 ,z 2 Fixedly connected with the large wheel and a small wheel coordinate system o 1 --x 1 ,y 1 ,z 1 And a large wheel coordinate system o 2 --x 2 ,y 2 ,z 2 At the starting position respectively with the coordinate system o p --x p ,y p ,z p And o g --x g , y g ,z g Coincident, the small wheels being at a uniform angular velocity ω 1 Around z p The shaft rotates clockwise and the bull wheel rotates at a uniform angular velocity ω 2 Around z g The axis rotates anticlockwise, after a period of time from the starting position, the coordinate system o 1 --x 1 ,y 1 ,z 1 And o 2 --x 2 ,y 2 ,z 2 Respectively rotate when the meshing point is M R1 And M R2 Said small wheel winding z p The shaft rotates through
Angle, said bull wheel being wound z g The shaft rotates through
An angle;
when the small wheel and the large wheel are in meshing transmission, a meshing point M is set R1 From the origin of coordinates o k Beginning along the line of engagement K R1 -K R1 Exercise, M R1 The parametric equation for point motion is:
at the same time, the meshing point M R2 Along the engagement line K with the same speed of movement R2 -K R2 Exercise, M R2 The parametric equation for point motion is:
in the formulae (1) and (2), t is the meshing point M R1 T is more than or equal to 0 and less than or equal to delta t, and delta t is the maximum value of the motion parameter variable; c. C 1 The undetermined coefficient of the movement of the meshing point is represented by millimeter, and PM is the horizontal distance from the meshing point to the node; in order to ensure that the fixed gear ratio is engaged, the rotation angles of the small wheel and the large wheel and the movement of the engagement point have to be in a linear relationship, and the rotation angles of the small wheel and the large wheel and the movement of the engagement point have the following relationship:
in the formula (3), k is a linear proportionality coefficient of the meshing point motion, and the unit of k is radian; i.e. i 12 The transmission ratio between the small wheel and the large wheel is set;
when the point of engagement M R1 Along the line of engagement K R1 -K R1 While in motion, point M R1 Simultaneously, contact lines C are respectively formed on the convex circular arc tooth surface of the small wheel and the concave circular arc tooth surface of the large wheel R1p And C R1g (ii) a When the point of engagement M R2 Along the line of engagement K R2 -K R2 While in motion, point M R2 Simultaneously, contact lines C are respectively formed on the concave arc tooth surface of the small wheel and the convex arc tooth surface of the large wheel R2p And C R2g (ii) a 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 And 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 in between is:
wherein the content of the first and second substances,
in the formulae (5) and (6), R 1 Is the pitch cylinder radius of the small wheel, R 2 Is the pitch cylinder radius of the bull wheel, and PM is the meshing point M R1 And M R2 Distance to node P, α t1 Is a meshing point M R1 End face pressure angle of alpha t2 Is the meshing point M R2 The end face pressure angle of (1);
obtaining the contact line C of the convex arc tooth surface of the small wheel by the formulas (1) and (4) R1p The parametric equation of (a) is:
calculating the contact line C of the concave circular arc tooth surface of the bull wheel by the formulas (1) and (4) R1g The parametric equation of (a) is:
calculating a small wheel concave arc tooth surface contact line C by the formulas (2) and (4) R2p The parameter equation of (1) is as follows:
calculating a contact line C of the convex arc tooth surface of the bull wheel according to the formulas (2) and (4) R2g The parameter equation of (1) is as follows:
further, the end face tooth profiles of the small wheel and the large wheel are determined by:
respectively at the circle center o of the convex circular arc tooth profile ar1 of the small wheel a1 And the circle center o of the concave circular arc tooth profile br2 of the bull wheel b2 Establishing a local coordinate system S a1 (o a1 -x a1 y a1 z a1 ) And S b2 (o b2 -x b2 y b2 z b2 ) The parameter equations of the small wheel convex circular arc tooth profile ar1 and the large wheel concave circular arc tooth profile br2 are respectively as follows:
in the formulae (11) and (12) (. Rho) a1 Is the arc radius, ξ of the convex arc tooth profile ar1 of the end face of the small wheel a1 Angle parameter, ξ, of ar1 a1a And xi a1b Are respectively xi a1 Minimum and maximum values of (d); rho b2 Is the arc radius, xi, of the concave arc tooth profile br2 of the end surface of the bull wheel b2 Angle parameter, ξ, for br2 b2a And xi b2b Are respectively xi b2 Minimum and maximum values of, wherein ξ a1b The value of the small wheel is obtained by solving the intersection point of the addendum circle of the small wheel and the convex circular arc tooth profile ar1 of the small wheel;
ξ a1a ＝ξ a1b -π/5.5； (13)
ξ b2a ＝ξ b2b -π/6.5； (14)
respectively at the circle center o of the small wheel concave arc tooth profile Br1 b1 And the circle center o of the convex circular arc tooth profile Ar2 of the bull wheel a2 Establishing a local coordinate system S b1 (o b1 -x b1 y b1 z b1 ) And S a2 (o a2 -x a2 y a2 z a2 ) Then, the parameter equations of the small wheel convex arc tooth profile Br1 and the large wheel concave arc tooth profile Ar2 are respectively:
in the formulae (15) and (16), ρ b1 Is the arc radius, xi, of a small wheel end surface concave arc tooth profile Br1 b1 Angle parameter, ξ, of Br1 b1a And xi b1b Are respectively xi b1 Minimum and maximum values of; rho a2 Is the arc radius, xi, of the convex arc tooth profile Ar2 of the end surface of the bull wheel a2 Angle parameter, ξ, of Ar2 a2a And xi a2b Are respectively xi b2 Minimum and maximum values of, wherein ξ a2b The value of the numerical value is obtained by solving the intersection point of the gear tooth top circle and the gear convex circular arc tooth profile Ar2 of the big gear;
ξ a2a ＝ξ a2b -π/5.5 (17)
ξ b1a ＝ξ b1b -π/6.5； (18)
obtaining the right convex circular arc tooth profile ar1 of the end surface of the small wheel gear tooth at S through coordinate transformation p The parametric equation for the coordinate system is:
obtaining the right concave circular arc tooth profile br1 of the end surface of the small gear tooth at S through coordinate transformation p The parametric equation for the coordinate system is:
obtaining the right concave circular arc tooth profile br2 of the end surface of the bull wheel tooth at S through coordinate transformation g The parametric equation for the coordinate system is:
obtaining the right convex circular arc tooth profile ar2 of the gear end surface of the bull wheel at S through coordinate transformation g The parametric equation for the coordinate system is:
the right transition curve hr1 of the small gear tooth end surface is from point P 0P And P 1P And its tangent vector T 0P And T 1P Determine, point P 0P From R h1 Determined so that the value xi of the tooth profile br1 b1b Can be solved to obtain P 1P Radius R of small wheel tooth root f1 Angle delta of sum 1R Jointly determining, and solving a parameter equation of a transition curve hr1 on the right side of the end face of the small gear tooth as follows:
in formulae (23) and (24), x p (P 0P )，y p (P 0P )，z p (P 0P ) Are respectively a point P 0P Three coordinate axis component of (2), x p (P 1P )， y p (P 1P )，z p (P 1P ) Are respectively a point P 1P Three coordinate axis component of (2), x p (T 0P )，y p (T 0P )，z p (T 0P ) Are respectively a point P 0P Unit tangent vector T of 0P Three coordinate axis component of (2), x p (T 1P )，y p (T 1P )，z p (T 1P ) Are respectively a point P 1P Unit tangent vector T of 1P Three coordinate axis component of (1), m t Is the end face modulus, b 1 ，b 2 ，b 3 ，b 4 For calculating the parameters, T H The control parameter is the shape of the tooth root transition curve, T is more than or equal to 0.2 H ≤1.5，t H For calculating the parameter, t is 0-t H ≤1；
The right transition curve hr2 of the end face of the bull wheel gear is defined by point P 0G And P 1G And its tangent vector T 0G And T 1G Determine, point P 0G From R h2 Determining, thus, the value xi of the tooth profile br2 b2b Can be solved to obtain, P 1G The radius R of the root circle of the big gear f2 And angle delta 2R Jointly determining, solving a parameter equation of a right side transition curve hr2 of the end face of the bull wheel gear is as follows:
in the formula (25), x g (P 0G )，y g (P 0G )，z g (P 0G ) Are respectively a point P 0G Three coordinate axis component of (2), x g (P 1G )，y g (P 1G )， z g (P 1G ) Are respectively a point P 1G Three coordinate axis component of (c), x g (T 0G )，y g (T 0G )，z g (T 0G ) Are respectively a point P 0G Unit tangent vector T of 0G Three coordinate axis component of (c), x g (T 1G )，y g (T 1G )，z g (T 1G ) Are respectively a point P 1G Unit tangent vector T of 1G Three coordinate axis components of (a);
when determining the number of teeth Z of the small gear 1 A transmission ratio i 12 Normal modulus m n Coincidence degree epsilon, linear proportionality coefficient k and pressure angle alpha of two meshing points of small wheel t1 And alpha t2 Coefficient of diameter phi d Root transition curve shape control parameter T H Undetermined coefficient c of motion of meshing point 1 And the motion rule, the contact line and the meshing line, the end face gear tooth profiles and the correct installation distance of the small wheel and the large wheel are correspondingly determined, and the gear tooth surface structures of the small wheel and the large wheel can be determined, so that the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is obtained.
Furthermore, the small wheel and the large wheel are meshed in an end face double-point contact concave-convex meshing transmission mode, the two meshing points have unequal end face pressure angles, and the end face pressure angle of the small wheel concave circular arc tooth profile meshing point is smaller than that of the small wheel convex circular arc tooth profile meshing point so as to enhance the bending strength of a tooth root, namely alpha t2 ＜α t1 。
Furthermore, the contact ratio of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is twice of that of single-point contact meshing, the contact ratio of the single-point contact meshing needs to be more than 1, and the contact ratio calculation formula of the unequal pressure angle end face double-arc gear mechanism meshed with the double points is as follows
The maximum value of the motion parameter variable of the meshing point of the parallel shaft transmission unequal pressure angle end surface double-circular-arc gear mechanism is obtained as
The design needs to be carried out according to the value epsilon of the contact ratio, the linear proportionality coefficient k and the number Z of the small gear teeth 1 Comprehensive determination of the engagement point M R1 Is measured by the motion parameter variable of (1).
Furthermore, the input shaft and the output shaft which are connected by the small wheel and the big wheel have interchangeability, namely, the small wheel is connected with the input shaft and the big wheel is connected with the output shaft, or the big wheel is connected with the input shaft and the small wheel is connected with the output shaft, and the small wheel and the big wheel are respectively corresponding to the speed reduction transmission mode or the speed increase transmission mode of a double-arc gear mechanism with unequal pressure angle end surfaces in parallel shaft transmission; the constant-speed transmission application with the transmission ratio of 1 of the mechanism is realized only when the number of teeth of the small gear and the large gear is equal.
Further, the rotation direction of the input shaft connected with the driver is clockwise or counterclockwise, so that the forward and reverse rotation transmission of the small wheel or the large wheel is realized.
The invention relates to a parallel-shaft-driven unequal-pressure-angle end-face double-circular-arc gear mechanism, which is a gear mechanism fundamentally innovated on the basis of the theory of the traditional double-circular-arc gear mechanism, and the design method is different from the design method of the traditional gear mechanism based on a curved-surface meshing equation, but is an active design method based on a meshing line parameter equation. The meshing mode of the parallel shaft transmission unequal pressure angle end face double-circular-arc gear mechanism is a point meshing mode based on an equal-slip-rate meshing line parameter equation, the relative sliding speeds of all meshing points on two meshing lines are equal, and the uniform friction and abrasion of the tooth surface can be ensured. The invention relates to a parallel shaft transmission unequal pressure angle end face double-arc gear mechanism, which is characterized in that the bending strength of the gear root is effectively improved through the end face unequal pressure angle design and the tooth root transition curve design of the end face double-arc.
Compared with the prior art, the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts has the following beneficial effects:
1. the design of the unequal pressure angle end surface double-arc gear mechanism driven by the parallel shaft is based on an active design method of a meshing line parameter equation, an end surface double-point concave-convex meshing gear surface is constructed, and the relative sliding speeds of all meshing points on two meshing lines are equal, so that the gear surface abrasion loss is the same, and the gear surface is easy to lubricate.
2. The two pressure angles of the end face tooth profile meshing points of the end face double-arc gear mechanism with unequal pressure angles in parallel shaft transmission are designed to be unequal, so that the bending strength of the tooth root can be increased, the service life of the gear is prolonged to the maximum extent, the structural size is reduced, and the transmission of heavy-load power is facilitated.
3. The contact ratio design of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft is free, the structural shape of the tooth profile can be determined through the pre-design of the contact ratio, the uniform distribution of the load is realized, and the dynamic performance is improved.
4. The unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft has no undercut, the minimum tooth number is 1, compared with the existing parallel shaft involute gear and other mechanisms, the single-stage large transmission ratio high contact ratio transmission can be realized, and meanwhile, because the tooth number can be designed to be smaller, and larger tooth thickness and module can be designed when the diameter of the gear is the same, the double-arc gear mechanism has higher bending strength and larger bearing capacity, and is suitable for popularization and application in the fields of micro/micro machinery, conventional mechanical transmission and high-speed heavy-load transmission.
5. According to the unequal-pressure-angle end face double-arc gear mechanism driven by the parallel shafts, the small wheel and the large wheel can have similar tooth root bending strength by adjusting the optimized design of the control parameters of the tooth root transition curve shape, the equal-strength design of the transmission mechanism is realized, and the service life of equipment is further prolonged.
Drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
fig. 1 is a schematic structural diagram of a parallel shaft transmission double-circular-arc gear mechanism with unequal pressure angle end faces.
Fig. 2 is a schematic diagram of a space meshing coordinate system of the parallel shaft transmission unequal pressure angle end face double-circular arc gear mechanism.
Fig. 3 shows the tooth profile composition structure and coordinate system of the end faces of the large wheel and the small wheel in fig. 1 and 2.
Fig. 4 is a partially detailed and enlarged view of fig. 3.
Fig. 5 is a three-dimensional view of the small wheel of fig. 1 of the present invention.
Fig. 6 is a three-dimensional view of the large wheel of fig. 1 in accordance with the present invention.
FIG. 7 is a schematic structural view of the present invention when a large wheel is connected to an input shaft to drive a small wheel to increase speed.
In the above figures: 1-driver, 2-small wheel, 3-input shaft, 4-output shaft, 5-big wheel, 6-small wheel section cylinder and 7-small wheel contact line C R2p 8-large wheel contact line C R2g 9-line of engagement K R2 -K R2 10-large wheel contact line C R1g 11-line of engagement K R1 -K R1 12-small wheel contact line C R1p 13-large wheel pitch cylinder, 14-small wheel tooth right side tooth root transition curve, 15-small wheel tooth right side concave circular arc tooth profile, 16-small wheel tooth right side straight line tooth profile, 17-small wheel tooth right side convex circular arc tooth profile, 18-large wheel tooth right side tooth root transition curve, 19-large wheel tooth right side concave circular arc tooth profile, 20-large wheel tooth right side straight line tooth profile, and 21-large wheel tooth right side convex circular arc tooth profile.
Detailed Description
Various exemplary embodiments of the present invention will be described in detail below with reference to the accompanying drawings. It should be noted that: the relative arrangement of the components and steps, the numerical expressions and numerical values set forth in these embodiments do not limit the scope of the present invention unless specifically stated otherwise.
Meanwhile, it should be understood that the sizes of the respective portions shown in the drawings are not drawn in an actual proportional relationship for the convenience of description.
The following description of at least one exemplary embodiment is merely illustrative in nature and is in no way intended to limit the invention, its application, or uses.
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to specific embodiments and the accompanying drawings.
Techniques, methods, and apparatus known to those of ordinary skill in the relevant art may not be discussed in detail but are intended to be part of the specification where appropriate.
In all examples shown and discussed herein, any particular value should be construed as exemplary only and not as limiting. Thus, other examples of the exemplary embodiments may have different values.
It should be noted that: like reference numbers and letters refer to like items in the following figures, and thus, once an item is defined in one figure, it need not be discussed further in subsequent figures.
Example 1: the invention provides an unequal pressure angle end face double-circular-arc gear mechanism for parallel shaft transmission, which is applied to transmission with the transmission ratio of 2 between parallel shafts, and the coincidence ratio of the transmission and the transmission is designed to be epsilon =4. The structure of the device is shown in figure 1, and comprises a small wheel 2 and a large wheel 5, wherein the small wheel 2 and the large wheel 5 form a pair of transmission pairs, the small wheel 2 is connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1, and the large wheel 5 is connected with an output shaft 4, namely the large wheel 5 is connected with a driven load through the output shaft 4; the axes of the small wheel 2 and the large wheel 5 are parallel to each other. FIG. 2 is a schematic diagram of a space meshing coordinate system of the unequal pressure angle end face double-circular-arc gear mechanism driven by parallel shafts.
Referring to fig. 1, 2, 3, 4 and 5, the pitch cylinder 6 of the small wheel has a radius R 1 The addendum circle radius of the small wheel is R a1 Root circle radius of R f1 The outer surface of the cylinder of the tooth root of the small wheel is uniformly provided with spiral wheel teeth, and the tooth profile of the end face of each wheel tooth is in an axisymmetric form, namely, the left tooth profile and the right tooth profile of the end face are axisymmetric. Taking the right tooth profile of the end surface of the pinion as an example, the right tooth profile of the pinion tooth is composed of a right convex circular arc tooth profile 17, a right straight line tooth profile 16, a right concave circular arc tooth profile 15 and a right tooth root transition curve 14 sequentially from the tooth top to the tooth root.
Referring to fig. 1, 2, 3, 4 and 6, the radius of the pitch cylinder 13 of the bull wheel is R 2 The radius of the addendum circle of the small wheel is R a2 Root circle radius of R f2 Large, largeThe outer surface of the cylinder of the gear tooth root is uniformly provided with spiral gear teeth, and the tooth profile of the end surface of the gear tooth is in an axisymmetric form, namely, the left tooth profile and the right tooth profile of the end surface are axisymmetric. Taking the right side tooth profile of the end face of the bull wheel as an example, the right side tooth profile of the bull wheel tooth sequentially consists of a right convex circular arc tooth profile 21 of the bull wheel tooth, a right linear tooth profile 20 of the bull wheel tooth, a right concave circular arc tooth profile 19 of the bull wheel tooth and a right tooth root transition curve 18 of the bull wheel tooth from the tooth top to the tooth root.
The teeth of the small wheel and the big wheel are both helical teeth, and the tooth surface of the small wheel is the tooth profile of the end surface of the small wheel along the contact line C of the small wheel R1p 12. Is obtained by performing spiral motion, the pitch of the spiral motion and a contact line C R1p 12 are the same; the gear surface of the bull wheel is the gear profile of the end surface of the bull wheel along the contact line C of the bull wheel R1g 10 are spirally moved, the pitch of the spiral movement is in contact with a bull wheel contact line C R1g 10 are the same; at any end face, the tooth profiles of the end faces of the small wheel and the large wheel are at two points M R1 And M R2 Simultaneous engagement, the spatial movement of the two points of engagement each forming a line of engagement K R1 -K R1 11 and the meshing line K R2 -K R2 9；
The small wheel 2 is connected with an input shaft 3 and rotates under the driving of the driver 1, so that the convex circular arc tooth profile and the concave circular arc tooth profile on the small wheel and the large wheel are respectively positioned at the meshing point M R1 And M R2 And simultaneously, the transmission of motion and power between the parallel shafts is realized, and the driver 4 is a motor in the embodiment.
The tooth surface contact line of the small wheel and the large wheel and the end face tooth profile structure of the small wheel and the large wheel are determined by the following method: at o p --x p ,y p ,z p 、o k --x k ,y k ,z k And o g --x g ,y g ,z g In three spatial coordinate systems, z p The axis of rotation of the shaft and of the small wheel coinciding, z g The axis of rotation of the shaft and the bull wheel coinciding, z k Shaft-to-pass meshing point M R1 Engagement line K of R1 -K R1 Coincide with and z k Axis and z p 、z g Axes parallel to each other, x p And x g Axis coincidence, x k And x g Axis parallel, o p o g A is a; coordinate system o 1 -- x 1 ,y 1 ,z 1 Fixedly connected to the small wheel, coordinate system o 2 --x 2 ,y 2 ,z 2 Fixedly connected with the bull wheel, and a coordinate system o of the bull wheel and the small wheel 1 --x 1 ,y 1 ,z 1 And o 2 -- x 2 ,y 2 ,z 2 At the starting position respectively with the coordinate system o p --x p ,y p ,z p And o g --x g ,y g ,z g Coincident, small wheels at uniform angular velocity ω 1 Around z p The shaft rotates clockwise and the bull wheel rotates at a uniform angular velocity omega 2 Around z g The axis rotates anticlockwise, after a period of time from the starting position, the coordinate system o 1 -- x 1 ,y 1 ,z 1 And o 2 --x 2 ,y 2 ,z 2 Respectively rotate, at the meshing point of M R1 And M R2 Small winding z p The shaft rotates through
Corner, large wheel winding z g The shaft rotates through
An angle;
when the small wheel and the large wheel are in meshing transmission, the meshing point M is set R1 From the origin of coordinates o k Beginning along the line of engagement K R1 -K R1 Exercise, M R1 The parametric equation for point motion is:
at the same time, the mesh point M R2 Along the line of engagement K at the same speed of movement R2 -K R2 Exercise, M R2 The parametric equation for point motion is:
t in formulae (1) and (2)Is a meshing point M R1 T is more than or equal to 0 and less than or equal to delta t; c. C 1 The undetermined coefficient of the meshing point movement is expressed in millimeters (mm); in order to ensure constant ratio engagement, the rotation angles of the small and large wheels and the movement of the engagement point must be in a linear relationship, which is as follows:
in the formula (3), k is a linear proportionality coefficient of the movement of the meshing point, and the unit of k is radian (rad); i.e. i 12 The transmission ratio between the small wheel and the large wheel is set;
when point of engagement M R1 Along the meshing line K R1 -K R1 While in motion, point M R1 Simultaneously, contact lines C are respectively formed on the convex circular arc tooth surface of the small wheel and the concave circular arc tooth surface of the large wheel R1p And C R1g (ii) a When the point of engagement M R2 Along the meshing line K R2 -K R2 While in motion, point M R2 Simultaneously, contact lines C are respectively formed on the concave circular arc tooth surface of the small wheel and the convex circular arc tooth surface of the large wheel R2p And C R2g (ii) a From the coordinate transformation, a coordinate system o can be obtained p --x p ,y p ,z p 、o k --x k ,y k ,z k And 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 in between is:
wherein the content of the first and second substances,
in the formulae (5) and (6), R 1 Is the pitch cylinder radius of the small wheel, R 2 Is the pitch cylinder radius of the bull wheel, and PM is the meshing point M R1 And M R2 Distance to node P, α t1 Is the meshing point M R1 End face pressure angle of alpha t2 Is a meshing point M R2 The end face pressure angle of (1);
obtaining the contact line C of the convex arc tooth surface of the small wheel by the formulas (1) and (4) R1p The parametric equation of (a) is:
obtaining the contact line C of the concave circular arc tooth surface of the bull gear by the formulas (1) and (4) R1g The parametric equation of (a) is:
calculating a small wheel concave arc tooth surface contact line C by the formulas (2) and (4) R2p The parametric equation of (a) is:
obtaining a contact line C of the convex arc tooth surface of the bull wheel according to the formulas (2) and (4) R2g The parameter equation of (1) is as follows:
the end face tooth profiles of the small wheel and the large wheel are determined by the following method:
respectively at the circle center o of the convex circular arc tooth profile ar1 of the small wheel a1 And the circle center o of the concave circular arc tooth profile br2 of the bull wheel b2 Establishing a local coordinate system S a1 (o a1 -x a1 y a1 z a1 ) And S b2 (o b2 -x b2 y b2 z b2 ) To obtain a convex circular arc tooth profile ar1 of the small wheelAnd the parameter equations of the large wheel concave circular arc tooth profile br2 are respectively as follows:
in the formulae (11) and (12) (. Rho) a1 Is the arc radius, ξ of the convex arc tooth profile ar1 of the end face of the small wheel a1 Is the angular parameter, ξ, of ar1 a1a And xi a1b Are respectively xi a1 Minimum and maximum values of (d); ρ is a unit of a gradient b2 Is the arc radius, xi, of the concave arc tooth profile br2 of the end surface of the bull wheel b2 Angle parameter, ξ, for br2 b2a And xi b2b Are respectively xi b2 Minimum and maximum values of, wherein ξ a1b The value of the sum is obtained by solving the intersection point of the addendum circle of the small wheel and the convex circular arc tooth profile ar1 of the small wheel;
ξ a1a ＝ξ a1b -π/5.5； (13)
ξ b2a ＝ξ b2b -π/6.5； (14)
respectively at the circle center o of the small wheel concave circular arc tooth profile Br1 b1 And the circle center o of the convex circular arc tooth profile Ar2 of the bull wheel a2 Establishing a local coordinate system S b1 (o b1 -x b1 y b1 z b1 ) And S a2 (o a2 -x a2 y a2 z a2 ) Then, the parameter equations of the small wheel convex arc tooth profile Br1 and the large wheel concave arc tooth profile Ar2 are respectively:
in the formulae (15) and (16), ρ b1 Is a concave arc tooth profile of the end surface of a small wheelRadius of arc, xi, of Br1 b1 Angle parameter, ξ, of Br1 b1a And xi b1b Are respectively xi b1 Minimum and maximum values of (d); ρ is a unit of a gradient a2 Is the arc radius, xi, of the convex arc tooth profile Ar2 of the end surface of the bull wheel a2 Angle parameter, ξ, of Ar2 a2a And xi a2b Are respectively xi b2 Minimum and maximum values of, wherein ξ a2b The value of (A) is obtained by solving the intersection point of the top circle of the gear tooth of the big wheel and the convex circular arc tooth profile Ar2 of the big wheel;
ξ a2a ＝ξ a2b -π/5.5； (17)
ξ b1a ＝ξ b1b -π/6.5； (18)
obtaining the right convex circular arc tooth profile ar1 of the end surface of the small gear tooth at S through coordinate transformation p The parametric equation for the coordinate system is:
obtaining the right concave circular arc tooth profile br1 of the end surface of the small gear tooth at S through coordinate transformation p The parametric equation for the coordinate system is:
obtaining the right concave circular arc tooth profile br2 of the gear end surface of the bull wheel at S through coordinate transformation g The parametric equation for the coordinate system is:
obtaining the right convex circular arc tooth profile ar2 of the gear end surface of the bull wheel at S through coordinate transformation g The parametric equation for the coordinate system is:
small gear tooth end face right side transition curve hr1 from point P 0P And P 1P And its tangent vector T 0P And T 1P Determine point P 0P From R h1 Determined so that the value xi of the tooth profile br1 b1b Can be solved to obtain P 1P Radius R of small gear root f1 Angle delta of sum 1R Jointly determining, solving a parameter equation of a transition curve hr1 on the right side of the end face of the small gear tooth as follows:
in formulae (23) and (24), x p (P 0P )，y p (P 0P )，z p (P 0P ) Are respectively a point P 0P Three coordinate axis component of (2), x p (P 1P )， y p (P 1P )，z p (P 1P ) Are respectively a point P 1P Three coordinate axis component of (2), x p (T 0P )，y p (T 0P )，z p (T 0P ) Are respectively a point P 0P Unit tangent vector T of 0P Three coordinate axis component of (2), x p (T 1P )，y p (T 1P )，z p (T 1P ) Are respectively a point P 1P Unit tangent vector T of 1P Three coordinate axis component of (1), m t Is the end face modulus, b 1 ，b 2 ，b 3 ，b 4 To calculate the parameters, T H The control parameter is the shape of the tooth root transition curve, T is more than or equal to 0.2 H ≤1.5，t H For calculating the parameter, t is 0-t H ≤1；
The right transition curve hr2 of the end face of the bull wheel gear is defined by point P 0G And P 1G And its tangent vector T 0G And T 1G Determine, point P 0G From R h2 Determining, thus, the value xi of the tooth profile br2 b2b Can be solved to obtain P 1G By the radius R of the root circle of the big gear f2 Angle delta of sum 2R Jointly determine to obtain the large wheelThe parameter equation of the transition curve hr2 on the right side of the tooth end surface is as follows:
in the formula (25), x g (P 0G )，y g (P 0G )，z g (P 0G ) Are respectively a point P 0G Three coordinate axis component of (c), x g (P 1G )，y g (P 1G )，z g (P 1G ) Are respectively a point P 1G Three coordinate axis component of (c), x g (T 0G )，y g (T 0G )，z g (T 0G ) Are respectively a point P 0G Unit tangent vector T of 0G Three coordinate axis component of (2), x g (T 1G )，y g (T 1G )，z g (T 1G ) Are respectively a point P 1G Unit tangent vector T of 1G Three coordinate axis components of (a);
in this embodiment:
t-the motion parameter variable of the meshing point M, and t belongs to [0, delta t ];
k-is a linear proportionality coefficient;
Z 1 -number of small gear teeth;
Z 2 -a large gear tooth number;
m n -a normal modulus;
R 1 is the pitch cylinder radius of the small wheel, R 1 ＝m t Z 1 /2； (26)
R 2 Is the pitch cylinder radius of the bull wheel, R 2 ＝i 12 R 1 ； (27)
i 12 Is the transmission ratio of the small wheel to the large wheel,
a-the relative positions of the axes of the small wheel and the large wheel: a = R 1 +R 2 ； (29)
b-width of teeth of small wheel and large wheel: b = c 1 Δt＝2Φ d R 1 ； (30)
Φ d -a diameter factor;
a beta-pitch helix angle of the pitch circle,
m t end face modulus, m t ＝m n /cosβ； (32)
R a1 Radius of addendum circle of the small wheel, R a1 ＝R 1 +m t ； (33)
R f1 Radius of root circle of small wheel, R f1 ＝R 1 -1.25m t ； (34)
R h1 -starting point P of transition curve at root of small wheel 0P Radius to the centre of rotation of the small wheel, R h1 ＝R 1 -m t ； (35)
R a2 Radius of crown of large gear tooth R a2 ＝R 2 +m t ； (36)
R f2 Radius of root circle of big gear tooth, R f2 ＝R 2 -1.25m t ； (37)
R h2 -big wheel root transition curve starting point P 0G Radius to centre of rotation of bull wheel, R h2 ＝R 2 -m t ； (38)
PM-horizontal distance of mesh point to node,
ρ a1 the radius of the convex arc profile of the end face of the small wheel, p a1 ＝PM； (40)
ρ a2 Radius of the convex circular arc profile of the bull wheel end face, ρ a2 ＝ρ a1 ； (41)
ρ b2 Circular arc radius of the concave circular arc profile of the bull wheel end face, p b2 ＝ρ a1 +Δρ； (42)
Δ ρ -difference in radius of concave-convex arc, Δ ρ =0.5m t ； (43)
ρ b1 The circular arc radius of the concave circular arc profile of the end face of the small wheel, p b1 ＝ρ b2 ； (44)
δ 1 A central angle formed by the tooth root circular arcs of two adjacent small wheels,
δ 2 a central angle formed by the tooth root circular arcs of two adjacent gear teeth of the bull wheel,
δ 1R -point P on the root transition curve hr1 on the right side of the small wheel tooth 1P Corresponding radius and x p The acute angle of the shaft clamp is an acute angle,
δ 2R point P on the root transition curve hr2 on the right side of the bull gear tooth 1G Corresponding radius and x p The acute angle of the shaft clamp is an acute angle,
wherein: axes of the coordinate system, a, b, m n ，m t ，ρ a1 ，Δρ，R 1 And R 2 Equal length and radiusOr the distance units are millimeters (mm); k is the sum of the k,
β，ξ a1 ，ξ b1 ，ξ a2 ，ξ b2 the unit of angle is radians (rad); pressure angle alpha t1 ，α t2 In degrees (°);
when determining the number of teeth Z of the small gear 1 A transmission ratio i 12 Normal modulus m n Coincidence degree epsilon, linear proportionality coefficient k and pressure angle alpha of two meshing points of small wheel t1 And alpha t2 Coefficient of diameter phi d Root transition curve shape control parameter T H When the gear is used, the undetermined coefficient c1 and the motion rule of the meshing point motion, the contact line and the meshing line, the tooth profiles of the end face gear teeth of the small wheel and the large wheel and the correct installation distance of the end face gear teeth are correspondingly determined, and the tooth surface structures of the gear teeth of the small wheel and the large wheel can also be determined, so that the parallel shaft driven unequal-pressure angle end face double-arc gear mechanism is obtained.
The contact ratio of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is twice that of single-point contact meshing, and the contact ratio of the single-point contact meshing is required to be more than 1. The contact ratio calculation formula of the double-point meshed unequal pressure angle end face double-arc gear mechanism is
The maximum value of the motion parameter variable of the meshing point of the parallel shaft transmission unequal pressure angle end surface double-circular-arc gear mechanism is obtained as
The design needs to be carried out according to the numerical value epsilon of the contact ratio, the linear proportionality coefficient k and the number Z of the small gear teeth 1 Comprehensive determination of the engagement point M R1 The maximum value Δ t of the motion parameter variable.
The engaging mode of the small wheel and the big wheel is a concave-convex engaging transmission with end surfaces in double-point contactThe two meshing points have unequal end face pressure angles, and the end face pressure angle of the small wheel concave circular arc tooth profile meshing point is smaller than that of the small wheel convex circular arc tooth profile meshing point so as to enhance the bending strength of the tooth root, namely alpha t2 ＜α t1 。
When the above formula is used, the values of the relevant parameters are: z 1 ＝18，i 12 ＝2，m n =3 millimeters (mm), epsilon =4,k = pi radians (rad), alpha t1 ＝25°，α t2 ＝15°，Φ d ＝1，T H =0.5, and the equations (26) to (48) are substituted to obtain
a =85.7930 millimeters (mm), b =57.1953 millimeters (mm), c 1 =257.3789 millimeters (mm), PM =4.9912 millimeters (mm);
then, the numerical values are substituted into the formula (7) -formula (25) to obtain a contact line parameter equation and an end face tooth profile parameter equation of the small wheel and the large wheel in the example, and then the gear tooth structures of the small wheel and the large wheel are obtained according to spiral motion, and the assembly can be carried out according to the correct center distance.
When the driver 1 drives the input shaft 3 and the small wheel 2 to rotate, because two pairs of adjacent gear teeth are in a meshing state when the small wheel 2 and the large wheel 5 are correctly installed, the preset contact ratio epsilon =4 of the pair of double-arc gears ensures that at least two pairs of gear teeth participate in meshing transmission at the same time at each moment, and the upper and lower arc tooth profiles of each pair of gear teeth respectively have a meshing point, thereby realizing continuous and stable meshing transmission of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts in the rotating motion. The rotation direction of an input shaft connected with the motor is clockwise, and the speed reduction transmission mode of the double-arc gear mechanism with the end faces at unequal pressure angles corresponding to the parallel shaft transmission is used for realizing speed reduction and torque increase transmission of anticlockwise rotation of the large wheel.
Example 2: the unequal pressure angle end face double-circular-arc gear mechanism for parallel shaft transmission is applied to the speed-increasing transmission of parallel shafts. As shown in fig. 6, a large wheel 5 is connected with an input shaft 3, the input shaft 3 is fixedly connected with a driving motor 1, and a small wheel 2 is connected with an output shaft 4, namely, the small wheel 2 is connected with a driven load through the output shaft 4; the axes of the small wheel 2 and the large wheel 5 are parallel. In the present embodiment, the number of teeth of the large gear 5 is 36, the number of teeth of the small gear 2 is 18, and the design contact ratio ∈ =4. When the input shaft 3 drives the large wheel 5 to rotate, because two pairs of adjacent gear teeth are in a meshing state when the large wheel 5 and the small wheel 2 are installed, and the preset contact ratio epsilon =4 of the pair of double-arc gears, at least two pairs of gear teeth participate in meshing transmission at the same time at each moment, and the upper and lower arc tooth profiles of each pair of gear teeth respectively comprise a meshing point, so that continuous and stable meshing transmission of the unequal-pressure angle end surface double-arc gear mechanism driven by the parallel shafts in the rotating motion is realized. At this time, the speed increasing ratio of the large wheel to the small wheel is 2, namely the transmission ratio of the small wheel to the large wheel is 2.
The relevant parameters take the values as follows: z is a linear or branched member 1 ＝18，i 12 ＝2，m n =2 millimeters (mm), e =4, α t1 ＝28°，α t2 ＝14°，Φ d ＝1，T H =0.6, and is obtained by substituting the formula (26) to the formula (48)
a =57.1953 millimeters (mm), b =38.1302 millimeters (mm), c 1 =171.5860, pm =3.3275 millimeters (mm);
then, the numerical values are substituted into the formula (7) -formula (25) to obtain a contact line parameter equation and an end face tooth profile parameter equation of the small wheel and the large wheel in the example, and then the distribution is moved according to the spiral, so that the tooth structures of the small wheel and the large wheel are obtained, and the assembly can be carried out according to the correct center distance.
The rotation direction of an input shaft connected with the driver is anticlockwise, and the driving mode of the double-circular-arc gear mechanism is corresponding to the unequal pressure angle end face transmission of the parallel shaft, so that the clockwise rotation transmission of the small wheel is realized.
The design of the unequal pressure angle end face double-circular-arc gear mechanism driven by the parallel shafts is based on an active design method of a meshing line parameter equation, an end face double-point concave-convex meshing gear surface is constructed, and the relative sliding speeds of all meshing points on two meshing lines are equal, so that the gear surface abrasion loss is the same, and the gear surface is easy to lubricate; the pressure angles of the end surface meshing points of the end surface double-arc gear mechanism with unequal pressure angles in parallel shaft transmission are designed to be unequal, so that the bending strength of the tooth root can be increased, the service life of the gear is prolonged to the maximum extent, the structure size is reduced, and the transmission of heavy-load power is facilitated; the contact ratio design of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is free, the tooth profile structure shape can be determined through the pre-design of the pressure angle and the contact ratio, the uniform distribution of the load is realized, and the dynamic characteristics are improved; the unequal pressure angle end face double-arc gear mechanism driven by the parallel shaft has no undercut, the minimum tooth number is 1, compared with the existing mechanisms such as a parallel shaft involute gear, the single-stage large transmission ratio high overlap ratio transmission can be realized, and simultaneously, because the tooth number is small, larger tooth thickness can be designed when the diameter of the gear is the same, so that the parallel shaft involute gear mechanism has higher strength and larger bearing capacity, and is suitable for popularization and application in the fields of micro/micro machinery, conventional mechanical transmission and high-speed heavy-load transmission; the unequal-pressure-angle end face double-arc gear mechanism driven by the parallel shafts can also enable the small wheel and the large wheel to have similar tooth root bending strength by adjusting the optimized design of root transition curve parameter values, realize the equal-strength design of the transmission mechanism and further prolong the service life of equipment.
It should be noted that, in this document, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or system. Without further limitation, an element defined by the phrases "comprising a," "8230," "8230," or "comprising" does not exclude the presence of other like elements in a process, method, article, or system comprising the element.
The above-mentioned serial numbers of the embodiments of the present invention are merely for description and do not represent the merits of the embodiments. In the unit claims enumerating several means, several of these means may be embodied by one and the same item of hardware. The use of the words first, second, third and the like do not denote any order, but rather the words first, second and the like may be interpreted as indicating any order.
The above description is only a preferred embodiment of the present invention, and not intended to limit the scope of the present invention, and all modifications of equivalent structures and equivalent processes, which are made by using the contents of the present specification and the accompanying drawings, or directly or indirectly applied to other related technical fields, are included in the scope of the present invention.
Claims (8)
1. The utility model provides a driven not isopressure angle terminal surface double circular arc gear mechanism of parallel shaft, includes a pair of gear pair that steamboat and bull wheel are constituteed, the steamboat links firmly with the driver through the input shaft, the output shaft is connected to the bull wheel, the steamboat with the axis of bull wheel is parallel, its characterized in that: the end face tooth profiles of the small wheel and the large wheel have axial symmetry forms, and the left end face tooth profile and the right end face tooth profile from the tooth top to the tooth bottom respectively consist of a convex arc tooth profile, a straight line tooth profile, a concave arc tooth profile and a tooth bottom transition curve; the small wheel and the large wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, and the two meshing points have unequal pressure angles; the small wheel is driven by the driver to rotate, stable meshing transmission between parallel shafts is realized through continuous meshing between two pairs of convex circular arc tooth profiles and concave circular arc tooth profiles, two meshing points with different pressure angles respectively form two contact lines with the same thread pitch on the tooth surfaces of the small wheel and the large wheel, and the two contact lines are cylindrical spiral lines; the gear tooth flanks of the small wheel and the large wheel are spiral tooth flanks obtained by cylindrical spiral motion of end face tooth profiles along respective contact lines, the screw pitch of the spiral tooth flanks is equal to that of the contact lines, and the spiral directions of the gear teeth of the small wheel and the gear teeth of the large wheel are opposite.
2. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the small wheel and the big wheel are in double-point contact concave-convex meshing transmission, and two meshing points of the small wheel and the big wheel are respectively a meshing point M of a convex circular arc tooth profile of the teeth of the small wheel and a concave circular arc tooth profile of the teeth of the big wheel R1 And the meshing point M of the concave circular arc tooth profile of the adjacent gear teeth of the small gear and the convex circular arc tooth profile of the adjacent gear teeth of the large gear R2 (ii) a The normal lines of the tooth profile meshing points of the two pairs of concave-convex circular arc end surfaces intersect at the same point, and the point is a tangent point of a pitch circle of the pair of double-circular-arc gears, namely a node; the horizontal distances from the two meshing points to the node are both PM; when the pair of parallel shafts drive the double-arc gear mechanism with unequal pressure angle end faces, two meshing points M R1 And M R2 Has the same axial movement speed and respectively forms two space meshing lines K R1 -K R1 And K R2 -K R2 And each form two contact lines of the tooth faces of the small wheel and the large wheel.
3. The unequal pressure angle end face double-circular-arc gear mechanism of parallel shaft transmission according to claim 2, characterized in that: the tooth surface contact line of the small wheel and the large wheel is determined by the following method:
at o p --x p ,y p ,z p 、o k --x k ,y k ,z k And o g --x g ,y g ,z g In three spatial coordinate systems, where o p 、o k And o g Respectively the origin, x, of three spatial coordinate systems p 、x k And x g X-axis, y of three spatial coordinate systems, respectively p 、y k And y g X-axis, z of three spatial coordinate systems, respectively p 、z k And z g Z-axes, z, of three spatial coordinate systems, respectively p The axis of rotation of the shaft and the small wheel coinciding, z g The axis of rotation of the axle and the bull wheel coincide, z k Shaft-to-pass meshing point M R1 Engagement line K of R1 -K R1 Are superposed and z k Axis and z p 、z g Axes parallel to each other, x p And x g The axes being coincident, x k And x g Axis parallel, o p And o g A is a; coordinate system o 1 --x 1 ,y 1 ,z 1 Fixedly connected with the small wheel and having a coordinate system o 2 --x 2 ,y 2 ,z 2 And the placeThe large wheel is fixedly connected with the small wheel coordinate system o 1 --x 1 ,y 1 ,z 1 And a large wheel coordinate system o 2 --x 2 ,y 2 ,z 2 At the initial position and the coordinate system o p --x p ,y p ,z p And o g --x g ,y g ,z g Coincident, the small wheels being at a uniform angular velocity ω 1 Around z p The shaft rotates clockwise and the bull wheel rotates at a uniform angular velocity ω 2 Around z g The axis rotates counterclockwise, after a period of time from the starting position, the coordinate system o 1 --x 1 ,y 1 ,z 1 And o 2 --x 2 ,y 2 ,z 2 Respectively rotate, at the meshing point of M R1 And M R2 Said small wheel winding z p The shaft rotates through
Angle, said bull wheel being wound z g The shaft rotates through
An angle;
when the small wheel and the large wheel are in meshing transmission, a meshing point M is set R1 From the origin of coordinates o k Beginning along the line of engagement K R1 -K R1 Exercise, M R1 The parametric equation for point motion is:
at the same time, the mesh point M R2 Along the line of engagement K at the same speed of movement R2 -K R2 Exercise, M R2 The parametric equation for point motion is:
wherein t is the meshing point M R1 T is more than or equal to 0 and less than or equal to delta t, delta t is the motion parameter variationThe maximum value of the quantity; c. C 1 The undetermined coefficient of the movement of the meshing point is represented by millimeter, and PM is the horizontal distance from the meshing point to the node; in order to ensure a fixed gear ratio engagement, the rotation angles of the small and large wheels and the movement of the engagement point must be in a linear relationship, and the rotation angles of the small and large wheels and the engagement point are in the following relation:
in the formula, k is a linear proportionality coefficient of the meshing point motion, and the unit of k is radian; i all right angle 12 The transmission ratio between the small wheel and the large wheel is set;
when point of engagement M R1 Along the line of engagement K R1 -K R1 At the time of exercise, point M R1 Simultaneously, contact lines C are respectively formed on the convex circular arc tooth surface of the small wheel and the concave circular arc tooth surface of the large wheel R1p And C R1g (ii) a When the point of engagement M R2 Along the line of engagement K R2 -K R2 At the time of exercise, point M R2 Simultaneously, contact lines C are respectively formed on the concave circular arc tooth surface of the small wheel and the convex circular arc tooth surface of the large wheel R2p And C R2g (ii) a 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 And 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 in between is:
wherein, the first and the second end of the pipe are connected with each other,
in the formula, R 1 Is the pitch cylinder radius of the small wheel, R 2 Is the pitch cylinder radius of the bull wheel, and PM is the meshing point M R1 And M R2 Distance to node P, α t1 Is the meshing point M R1 End face pressure angle of alpha t2 Is a meshing point M R2 The end face pressure angle of (a);
by said M R1 The contact line C of the convex arc tooth surface of the small wheel is obtained by the parameter equation of point motion and the homogeneous coordinate transformation matrix R1p The parametric equation of (a) is:
by said M R1 Obtaining the contact line C of the concave circular arc tooth surface of the large wheel by the parameter equation of point motion and the homogeneous coordinate transformation matrix R1g The parametric equation of (a) is:
by said M R2 Calculating the small wheel concave arc tooth surface contact line C by using the parameter equation of point motion and the homogeneous coordinate transformation matrix R2p The parameter equation of (1) is as follows:
from said M R2 The parameter equation of point motion and the homogeneous coordinate transformation matrix are used for solving the contact line C of the convex arc tooth surface of the bull wheel R2g The parameter equation of (1) is as follows:
4. the unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the end face tooth profiles of the small wheel and the large wheel are determined by the following method:
respectively at the circle center o of the convex circular arc tooth profile ar1 of the small wheel a1 And the circle center o of the concave circular arc tooth profile br2 of the bull wheel b2 Establishing a local coordinate system S a1 (o a1 -x a1 y a1 z a1 ) And S b2 (o b2 -x b2 y b2 z b2 ) The obtained parameter equations of the small wheel convex circular arc tooth profile ar1 and the large wheel concave circular arc tooth profile br2 are respectively as follows:
in the formula, ρ a1 Is the arc radius, ξ of the convex arc tooth profile ar1 of the end face of the small wheel a1 Is the angular parameter, ξ, of ar1 a1a And xi a1b Are respectively xi a1 Minimum and maximum values of; rho b2 Is the arc radius, xi, of the concave arc tooth profile br2 of the end surface of the bull wheel b2 Is the angular parameter, ξ, of br2 b2a And xi b2b Are respectively xi b2 Minimum and maximum values of, wherein ξ a1b The value of the sum is obtained by solving the intersection point of the addendum circle of the small wheel and the convex circular arc tooth profile ar1 of the small wheel;
ξ a1a ＝ξ a1b -π/5.5；
ξ b2a ＝ξ b2b -π/6.5；
respectively at the circle center o of the small wheel concave arc tooth profile Br1 b1 And the circle center o of the convex circular arc tooth profile Ar2 of the bull wheel a2 Establishing a local coordinate system S b1 (o b1 -x b1 y b1 z b1 ) And S a2 (o a2 -x a2 y a2 z a2 ) Then small wheel convexThe parameter equations of the circular arc tooth profile Br1 and the large wheel concave circular arc tooth profile Ar2 are respectively as follows:
in the formula, ρ b1 Is the arc radius, xi, of a small wheel end surface concave arc tooth profile Br1 b1 Angle parameter, ξ, of Br1 b1a And xi b1b Are respectively xi b1 Minimum and maximum values of; rho a2 Is the arc radius, xi, of the convex arc tooth profile Ar2 of the end surface of the bull wheel a2 Angle parameter, ξ, of Ar2 a2a And xi a2b Are respectively xi b2 Minimum and maximum values of, wherein ξ a2b The value of (A) is obtained by solving the intersection point of the top circle of the gear tooth of the big wheel and the convex circular arc tooth profile Ar2 of the big wheel;
ξ a2a ＝ξ a2b -π/5.5
ξ b1a ＝ξ b1b -π/6.5；
obtaining the right convex circular arc tooth profile ar1 of the end surface of the small gear tooth at S through coordinate transformation p The parametric equation for the coordinate system is:
obtaining the right concave circular arc tooth profile br1 of the end surface of the small gear tooth at S through coordinate transformation p The parametric equation for the coordinate system is:
obtaining the right concave circular arc tooth profile br2 of the end surface of the bull wheel tooth at S through coordinate transformation g The parametric equation for the coordinate system is:
obtaining the right convex circular arc tooth profile ar2 of the gear end surface of the bull wheel at S through coordinate transformation g The parametric equation for the coordinate system is:
the right transition curve hr1 of the small gear tooth end surface is from point P 0P And P 1P And its tangent vector T 0P And T 1P Determine point P 0P From R h1 Determined so that the value xi of the tooth profile br1 b1b Can be solved to obtain 1P Radius R of small wheel tooth root f1 Angle delta of sum 1R Jointly determining, solving a parameter equation of a transition curve hr1 on the right side of the end face of the small gear tooth as follows:
in the formula, x p (P 0P )，y p (P 0P )，z p (P 0P ) Are respectively a point P 0P Three coordinate axis component of (2), x p (P 1P )，y p (P 1P )，z p (P 1P ) Are respectively a point P 1P Three coordinate axis component of (c), x p (T 0P )，y p (T 0P )，z p (T 0P ) Are respectively a point P 0P Unit tangent vector T of 0P Three coordinate axis component of (2), x p (T 1P )，y p (T 1P )，z p (T 1P ) Are respectively a point P 1P Unit tangent vector T of 1P Three coordinate axis component of (1), m t Is an endFace modulus, b 1 ，b 2 ，b 3 ，b 4 To calculate the parameters, T H The control parameter is the shape of the tooth root transition curve, T is more than or equal to 0.2 H ≤1.5，t H For calculating the parameter, t is more than or equal to 0 H ≤1；
The right transition curve hr2 of the end face of the bull wheel gear is defined by point P 0G And P 1G And its tangent vector T 0G And T 1G Determine point P 0G From R h2 Determining, thus, the value xi of the tooth profile br2 b2b Can be solved to obtain 1G By the radius R of the root circle of the big gear f2 Angle delta of sum 2R Jointly determining, the parameter equation for solving the right side transition curve hr2 of the end face of the bull wheel gear is as follows:
in the formula, x g (P 0G )，y g (P 0G )，z g (P 0G ) Are respectively a point P 0G Three coordinate axis component of (2), x g (P 1G )，y g (P 1G )，z g (P 1G ) Are respectively a point P 1G Three coordinate axis component of (2), x g (T 0G )，y g (T 0G )，z g (T 0G ) Are respectively a point P 0G Unit tangent vector T of 0G Three coordinate axis component of (2), x g (T 1G )，y g (T 1G )，z g (T 1G ) Are respectively a point P 1G Unit tangent vector T of 1G Three coordinate axis components of (a);
when determining the number of teeth of the small gear Z 1 A transmission ratio i 12 Normal modulus m n Coincidence degree epsilon, linear proportionality coefficient k and pressure angle alpha of two small wheel meshing points t1 And alpha t2 Coefficient of diameter phi d Root transition curve shape control parameter T H Undetermined coefficient c of motion of meshing point 1 The motion rule, the contact line, the meshing line, the tooth profiles of the end face teeth of the small wheel and the large wheel and the correct installation distance are correspondingly determined, and the tooth surface structures of the teeth of the small wheel and the large wheel are also determinedThe double-arc gear mechanism with the end surfaces of unequal pressure angles and the transmission of the parallel shafts can be determined.
5. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the small wheel and the big wheel are meshed in a concave-convex meshing transmission mode with end surfaces in double-point contact, the two meshing points have unequal end surface pressure angles, and the end surface pressure angle of the small wheel concave circular arc tooth profile meshing point is smaller than that of the small wheel convex circular arc tooth profile meshing point so as to enhance the bending strength of a tooth root, namely alpha t2 ＜α t1 。
6. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the contact ratio of the unequal pressure angle end face double-arc gear mechanism driven by the parallel shafts is twice of that of single-point contact meshing, the contact ratio of the single-point contact meshing needs to be more than 1, and the contact ratio calculation formula of the unequal pressure angle end face double-arc gear mechanism meshed with two points is
The maximum value of the motion parameter variable of the meshing point of the parallel shaft transmission unequal pressure angle end surface double-circular-arc gear mechanism is obtained as
The design needs to be carried out according to the value epsilon of the contact ratio, the linear proportionality coefficient k and the number Z of the small gear teeth 1 Comprehensively determining the meshing point M R1 Is measured by the motion parameter variable of (1).
7. The unequal pressure angle end face double-circular arc gear mechanism of parallel shaft transmission according to claim 1, characterized in that: the input shaft and the output shaft which are connected by the small wheel and the big wheel have interchangeability, namely, the small wheel is connected with the input shaft, the big wheel is connected with the output shaft, or the big wheel is connected with the input shaft, the small wheel is connected with the output shaft, and the speed reduction transmission mode or the speed increase transmission mode respectively corresponds to the speed reduction transmission mode or the speed increase transmission mode of the double-arc gear mechanism with unequal pressure angle end surfaces in parallel shaft transmission; the constant-speed transmission application with the transmission ratio of 1 of the mechanism is realized only when the number of teeth of the small gear and the large gear is equal.
8. The unequal pressure angle end face double-circular-arc gear mechanism of parallel shaft transmission according to claim 1 or 7, characterized in that: the rotation direction of the input shaft connected with the driver is clockwise or anticlockwise, so that forward and reverse rotation transmission of the small wheel or the large wheel is realized.
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| GB315063A (en) * | 1928-05-29 | 1929-07-11 | Nikola Trbojevich | Improvements in method of generating helical gears |
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| CN102689054A (en) * | 2012-05-25 | 2012-09-26 | 上海第二工业大学 | Processing method of tooth thickness-variable gear with intersecting axles and hob of tooth thickness-variable gear |
| CN105179600B (en) * | 2015-08-31 | 2017-07-28 | 重庆百花园齿轮传动技术研究所 | Double big negative addendum modification Involutes Gears Transmission devices |
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