CN111322373B - Gear pair design method based on claw helix rack knife - Google Patents

Gear pair design method based on claw helix rack knife Download PDF

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CN111322373B
CN111322373B CN202010172485.8A CN202010172485A CN111322373B CN 111322373 B CN111322373 B CN 111322373B CN 202010172485 A CN202010172485 A CN 202010172485A CN 111322373 B CN111322373 B CN 111322373B
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rack
tooth
claw
knife
coordinate system
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CN111322373A (en
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苏进展
常乐浩
封硕
贺朝霞
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Changan University
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • 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
    • F16H57/00—General details of gearing
    • F16H2057/0087—Computer aided design [CAD] specially adapted for gearing features; Analysis of gear systems

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

Abstract

The invention discloses a method for designing a gear pair based on a claw helix rack knife, which comprises the following steps: the tooth profile of the conjugate gear pair is obtained by adopting a claw helix blade to replace a straight blade and an arc blade of a rack knife and by means of the generating principle of an involute tooth profile; establishing geometric constraint conditions according to the gear tooth basic parameters, and calculating the shape parameters of the cavel spiral line; and establishing a tooth profile coordinate system and a moving coordinate system of the rack cutter to obtain a position vector and a normal vector of the rack cutter. By controlling the motion relation between the moving speed of the rack knife along the pitch line and the rotating speed of the workpiece around the axis of the rack knife, a tooth surface equation of the large wheel and the small wheel is deduced by means of homogeneous coordinate transformation and a plane meshing principle, and a gear pair model is established. According to the invention, concave-convex contact of the gear pair is obtained through the concave-convex rack knife, so that the tooth surface contact strength is improved; the tooth root is processed through the concave tooth top of the rack knife, the thickness of the tooth root of the gear tooth is increased, the bending strength of the gear tooth is improved, and the number of teeth which are not undercut at least is reduced.

Description

Gear pair design method based on claw helix rack knife
Technical Field
The invention belongs to the technical field of gear transmission, and particularly relates to a gear pair design method based on a claw helical rack knife.
Background
The tooth profile and the tooth trace are two key factors determining the shape of the gear tooth and the meshing performance. The development of gear transmission technology has been mainly developed around these two aspects. In high-speed heavy-load occasions, in order to improve the transmission performance of the involute cylindrical gear, the gear teeth are required to be modified, and the modified involute cylindrical gear is in meshing transmission of convex teeth to convex teeth in the tooth profile and tooth trace directions, so that the induced curvature radius of a gear pair is small, the tooth surface contact stress is large, and the bearing capacity is limited to a certain extent. The gear pair with the concave-convex tooth profile in contact can increase the induced curvature radius and improve the tooth surface contact strength. Increase root fillet radius is the effective measure who improves root bending strength, and the fillet radius increase can cause the reduction of actual work tooth height to produce the tooth top tooth root and interfere the phenomenon, take place marginal contact phenomenon, cause the teeth of a cogwheel to become invalid even. The cavel spiral line is used as a moderation curve, moderates the curve change between a straight line and a circular curve line, and is characterized in that: the initial curvature is 0 from the origin of coordinates, the curvature of the function curve is increased linearly, the function curve is used for smooth connection of a straight line and any circular arc curve, the connection point is ensured to be completely consistent with the curvature of the circular arc curve, and the function curve is widely applied to the turning of road design, such as expressways, high-speed railways and the like. Therefore, if a claw helix is applied to the cutting edge of the rack bar for smoothly connecting the straight top blade and the straight root blade, the engagement performance of the gear pair can be improved.
Disclosure of Invention
In order to solve the defects of tooth surface contact strength, weak tooth root bending, limitation of the minimum undercut tooth number and the like in the transmission of an involute gear, the invention provides a gear pair design method based on a claw helical rack knife, wherein concave-convex contact of the gear pair is obtained through a concave-convex rack knife, and the tooth surface contact strength is improved; the tooth root is processed through the concave tooth top of the rack knife, the thickness of the tooth root of the gear is increased, the bending strength of the gear is improved, the number of teeth which are not undercut at least is reduced, and therefore the novel toothed gear pair is suitable for occasions of heavy load and large speed ratio transmission.
In order to achieve the purpose, the invention adopts the following technical scheme:
a gear pair design method based on a claw helix rack knife comprises the following steps:
(1) designing the normal tooth profile of the rack knife into a cleat spiral line, and determining the shape parameters of the cleat spiral line through a geometric constraint relation; establishing a tooth profile coordinate system and a rack knife moving coordinate system, and obtaining a position vector and a normal vector of a rack knife by virtue of coordinate transformation;
(2) by means of a standard involute cylindrical gear generating principle, by controlling the relation between the moving speed of a rack cutter along a pitch line and the rotating speed of a workpiece around the axis of the rack cutter, and by means of homogeneous coordinate transformation and a plane meshing principle, a tooth surface equation of a large wheel and a small wheel is deduced, and a gear pair model is established;
(3) the concave-convex degree of the claw helix is adjusted through the position coefficient, so that the concave-convex contact degree of the gear pair is controlled.
As a further improvement of the invention, the normal tooth profile is the tooth profile of the conjugate gear pair obtained by adopting a claw helix blade to replace a straight blade and an arc blade of a rack knife and by means of the generating principle of an involute tooth profile.
As a further improvement of the invention, in the step (1), the cutting edge of the rack knife is divided into an upper part and a lower part, wherein the upper part and the lower part are respectively a cleat spiral line for processing the tooth top and a cleat spiral line for processing the tooth bottom, and the upper part and the lower part are curves symmetrical about the node; the lower horn helix begins at the node and terminates at a tangent to the root edge, while the upper horn helix is a portion of a curve that is symmetrical about the node.
As a further improvement of the present invention, in the step (1), the obtaining of the position vector and the normal vector of the rack knife specifically includes:
at the node O of the cutting edgetEstablishing a tooth profile reference coordinate system, xtDirection and xcThe included angle of the direction is the pressure angle alpha of the rack knifen，ytPerpendicular xtPointing to the tool entity to establish a coordinate system StIn a coordinate system StIn, the upper and lower cavel helices are represented as:
Figure GDA0002997928370000031
in the formula, the upper and lower marks respectively represent upper and lower horn spiral lines on the left side of the rack knife, the right rack knife is obtained through symmetry, z is an integral upper limit, and t is an integral variable; parameter kxAnd kyThe shape parameters of the helix of the claw are obtained by homogeneous coordinate transformationcIs represented by:
rc(z,lc)＝Mct(lc)rt(z)
in the formula, MctIs a slave StTo ScIs expressed as:
Figure GDA0002997928370000032
in the matrix, the upper and lower tables represent the corresponding transformations on the left and right sides, respectively,/cTo be in a coordinate system ScOrigin O ofcAs a length of the starting point in the width direction of the rack bar, the corresponding unit normal vector expression is:
Figure GDA0002997928370000033
the shape parameter k of the cavel spiral line is obtained by the following nonlinear equation systemxAnd ky，
Figure GDA0002997928370000034
The first and second equations in the set of equations are used to determine the position, x, of the endpoint of the cleat spiralcAnd ycIs a rack knife position vector rcCoordinate component of, mnThe value range of the gear modulus and the position coefficient lambda is as follows:
1.25tgαnλ is not less than pi/4, when λ is 1.25tg αnWhen the cutting edge is a straight line, when lambda is pi/4, the end point of the cutting edge is on the middle point of the tooth space;
obtaining the parameter k of the cavel spiral line by solving the nonlinear equation setxAnd kyThereby obtaining a position vector r of the rack toolcAnd unit normal vector nc。
As a further improvement of the present invention, the step (2) specifically comprises:
small wheel moving coordinate system S1Reference coordinate system ShAnd rack knife coordinate system Sc(ii) a Similar to the standard involute tooth profile generating motion, a small wheel moving coordinate system S1About its own axis of rotation z1Rotate
Figure GDA0002997928370000041
While rack and knife coordinate system ScMove
Figure GDA0002997928370000042
rp1Is the pitch circle radius of the small wheel; by the meshing principle, at the contact point of the rack tooth surface and the tooth surface of the small wheel to be generated, the unit normal vector must pass through the instantaneous revolution axis I-I, and the following meshing conditions are obtained:
Figure GDA0002997928370000043
in the formula Xc、Yc、ZcFor instantaneous axis of rotation I-I in rack-and-pinion coordinate system ScCoordinate of (1), xc、yc、zcFor instantaneous contact point of two tooth surfaces at ScCoordinate of (1), ncx，ncy，nczThe component of the unit normal vector of the instantaneous contact point is used for obtaining the machining rotation angle of the small wheel
Figure GDA0002997928370000044
Is expressed as
Figure GDA0002997928370000045
In the formula, Xc＝0，
Figure GDA0002997928370000046
rp1＝mnN12 is the pitch circle radius, N1The number of teeth of the small gear;
the flank position vector and the normal vector of the small wheel are expressed as,
Figure GDA0002997928370000047
Figure GDA0002997928370000048
in the formula, M1cIs ScTo S1The transformation matrix of (2);
and obtaining the equation of the big gear tooth surface matched with the same equation.
Compared with the prior art, the invention has the following technical effects:
the tooth surface equations of the driving wheel and the driven wheel of the gear pair are consistent, a pair of completely conjugate gear pairs can be processed by using a rack cutter or a hob, and the cutter is simple in design; the generating principle of the claw spiral gear is similar to that of an involute, the gear transmission ratio is only related to the base radius of a measuring gear, the center distance and the transmission ratio of a gear pair are not affected by the error of the center distance, and therefore the claw spiral gear has center distance separability, the working tooth surface and the transition curved surface are integrally designed, the working edge, the top edge and the root edge are all represented by claw spiral lines, and tooth top fillets do not need to be poured. The tooth root thickness in the gear pair is bigger than the involute gear pair, can improve the tooth root bending strength of the teeth of a cogwheel and reduce the number of teeth that do not take place the undercut, can be applied to in the gear mechanism of big velocity ratio. The tooth profile of the gear pair is in concave-convex contact, and compared with the convex-convex contact of the involute gear pair, the tooth surface contact strength is improved, and the gear pair is more suitable for heavy-load transmission occasions.
Drawings
FIG. 1 is a rack knife normal profile of the present invention;
FIG. 2 is a gear tooth generating coordinate system of the present invention;
FIG. 3 is a straight gear pair meshing tooth profile processed by the claw helix rack knife of the invention;
FIG. 4 is a bevel gear machined by the claw helix rack cutter of the present invention;
FIG. 5 is a meshing tooth form of a small tooth number spur gear pair processed by the claw helical rack knife of the invention;
fig. 6 shows a gear pair based on a claw helical rack knife according to the invention.
Detailed Description
So that the manner in which the features and advantages of the invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings.
As shown in fig. 1, the gear pair design based on the claw helical rack knife of the invention comprises the following steps:
(1) designing the normal tooth profile of a rack cutter into a claw helix, and determining the parameters of the claw helix according to the geometric constraint relations of the tooth top height, the tooth root height, the tooth thickness, the tooth groove width and the like; and establishing a tooth profile coordinate system and a rack knife moving coordinate system, and obtaining the position vector and the normal vector of the rack knife by virtue of coordinate transformation.
(2) By means of the generating principle of a standard involute cylindrical gear, the relationship between the moving speed of a rack cutter along a pitch line and the rotating speed of a workpiece around the axis of the rack cutter is controlled, and by means of coordinate transformation and the plane meshing principle, a tooth surface equation of a large wheel and a small wheel is deduced, and a gear pair model is established.
(3) The concave-convex degree of the claw helix is adjusted through the position coefficient, so that the concave-convex contact degree of the gear pair is controlled, and the strength of the gear teeth is improved.
The specific steps are explained as follows:
(1) the cutting edge of the rack knife is divided into an upper part and a lower part, namely an upper part cleat spiral line for processing a tooth top and a lower part cleat spiral line for processing a tooth root. The upper and lower horn spiral lines are symmetrical curves about the node, the height coefficient of the upper curve is 1.0, and the height coefficient of the lower curve is 1.25. The lower horn helix starts from the node and ends tangent to the root edge, while the upper horn helix is part of a curve that is symmetrical about the node for the lower horn helix, reducing the root coefficient by 0.25. The position coefficient lambda is used for controlling the concave-convex degree and the tooth root tooth thickness of a tooth profile curve, when the value of lambda is larger, concave-convex contact is more obvious, the tooth root is thicker, and the value of lambda is pi/4-1.25 tg alphanIn which α isnIs the normal pressure angle. On the graduation line of the rack knife, the midpoint O of the tooth groove is takencAs a coordinate system ScOrigin of (a), ycDirection along the tooth thickness direction, xcAlong the tooth height direction. At the node O of the cutting edgetEstablishing a tooth profile reference coordinate system, xtDirection and xcThe included angle of the direction is the pressure angle alpha of the rack knifen，ytPerpendicular xtPointing to the tool entity to establish a coordinate systemSt. In a coordinate system StIn, the upper and lower cavel helices are represented as:
Figure GDA0002997928370000061
in the formula, the upper and lower marks respectively represent upper and lower horn helices at the left side of the rack knife, the rack knife at the right side can be obtained through symmetry, z is an integral upper limit, and t is an integral variable. Parameter kxAnd kyIs the shape parameter of the cavel helix, and the value of the shape parameter is related to the position coefficient lambda. The cutting edge can be obtained in a rack coordinate system S through homogeneous coordinate transformationcIs shown in the drawing (a) and (b),
rc(z,lc)＝Mct(lc)rt(z)
in the formula, MctIs a slave StTo ScIs expressed as
Figure GDA0002997928370000071
In the matrix, the top and bottom tables represent the corresponding transformations on the left and right sides, respectively. lcTo be in a coordinate system ScOrigin O ofcIs the length along the width direction of the rack bar as a starting point. The corresponding unit normal vector expression is
Figure GDA0002997928370000072
The shape parameter k of the cavel spiral line is obtained by the following nonlinear equation systemxAnd ky，
Figure GDA0002997928370000073
The first and second equations in the set of equations are used to determine the position, x, of the endpoint of the cleat spiralcAnd ycIs a rack knife position vector rcIs divided intoAmount, mnThe value range of the gear modulus and the position coefficient lambda is as follows:
1.25tgαnλ is not less than pi/4, when λ is 1.25tg αnWhen the cutting edge is a straight line, when lambda is pi/4, the end point of the cutting edge is on the middle point of the tooth slot. Obtaining the parameter k of the cavel spiral line by solving the nonlinear equation setxAnd kyThereby obtaining a position vector r of the rack toolcAnd unit normal vector nc。
(2) Fig. 2 is a gear generating coordinate system. Small wheel moving coordinate system S1Reference coordinate system ShAnd rack knife coordinate system Sc. Similar to the standard involute tooth profile generating motion, a small wheel moving coordinate system S1About its own axis of rotation z1Rotate
Figure GDA0002997928370000074
While rack and knife coordinate system ScMove
Figure GDA0002997928370000075
rp1Is the pitch circle radius of the small wheel. By the meshing principle, at the contact point of the rack tooth surface and the tooth surface of the small wheel to be generated, the unit normal vector must pass through the instantaneous revolution axis I-I, and the following meshing conditions are obtained:
Figure GDA0002997928370000081
in the formula Xc、Yc、ZcFor instantaneous axis of rotation I-I in rack-and-pinion coordinate system ScCoordinate of (1), xc、yc、zcFor instantaneous contact point of two tooth surfaces at ScCoordinate of (1), ncx，ncy，nczThe component of the unit normal vector of the instantaneous contact point is used for obtaining the machining rotation angle of the small wheel
Figure GDA0002997928370000082
Is expressed as
Figure GDA0002997928370000083
In the formula, Xc＝0，
Figure GDA0002997928370000084
rp1＝mnN12 is the pitch circle radius, N1And the number of teeth of the small gear.
The tooth surface position vector and normal vector of the small wheel can be expressed as,
Figure GDA0002997928370000085
Figure GDA0002997928370000086
in the formula, M1cIs ScTo S1The transformation matrix of (2).
The same equation can be used to obtain the equation of the big gear tooth surface matched with the same equation.
[ examples of design ]
The number of teeth of the small gear 21, the number of teeth of the large gear 30, the modulus 3.0mm, the pressure angle 20 degrees, the addendum coefficient 1.0, the dedendum coefficient 1.25 and the lambda equal to pi/4. Fig. 3 is a meshing tooth form of a claw helix tooth profile spur gear pair, and the gear pair is in concave-convex contact. Fig. 4 shows helical gears with a cavel helix profile, the basic parameters of which are as follows: the tooth number 21, the modulus 3.0mm, the pressure angle 20 °, the crest factor 1.0, the root factor 1.25, λ ═ pi/4, the helix angle 10 °, the tooth width 30.0 mm. Fig. 5 shows the meshing tooth profile of a small-tooth-number cavel spiral tooth profile spur gear pair, and the basic parameters are as follows: the number of teeth of the small gear is 5, the number of teeth of the large gear is 30, the modulus is 3.0mm, the pressure angle is 20 degrees, the addendum coefficient is 1.0, the dedendum coefficient is 1.25, the lambda is pi/4, and the transmission ratio is 6.8, so that the gear pair processed by the claw spiral rack knife can be applied to the occasion of large transmission ratio.
The above is a detailed description of the present invention with reference to specific preferred embodiments, and it should not be considered that the present invention is limited to the specific embodiments, but that the present invention can be easily derived or substituted by those skilled in the art without departing from the spirit of the present invention, and all of them should be considered as falling within the scope of the patent protection defined by the claims of the present invention.

Claims (5)

1. A gear pair design method based on a claw helix rack knife is characterized by comprising the following steps:
(1) designing the normal tooth profile of the rack knife into a cleat spiral line, and determining the shape parameters of the cleat spiral line through a geometric constraint relation; establishing a tooth profile coordinate system and a rack knife moving coordinate system, and obtaining a position vector and a normal vector of a rack knife by virtue of coordinate transformation;
(2) by means of a standard involute cylindrical gear generating principle, by controlling the relation between the moving speed of a rack cutter along a pitch line and the rotating speed of a workpiece around the axis of the rack cutter, and by means of homogeneous coordinate transformation and a plane meshing principle, a tooth surface equation of a large wheel and a small wheel is deduced, and a gear pair model is established;
(3) the concave-convex degree of the claw helix is adjusted through the position coefficient, so that the concave-convex contact degree of the gear pair is controlled.
2. The method for designing a pinion pair based on a claw helix rack knife according to claim 1, wherein the normal tooth profile is obtained by replacing a straight edge and an arc edge of the rack knife with a claw helix edge and by means of an involute tooth profile generating principle.
3. The method for designing a gear pair based on a claw helical rack knife according to claim 1, wherein in the step (1), the cutting edge of the rack knife is divided into an upper claw helical line for processing the top of the tooth and a lower claw helical line for processing the bottom of the tooth, and the upper claw helical line and the lower claw helical line are curves symmetrical about the node; the lower horn helix begins at the node and terminates at a tangent to the root edge, while the upper horn helix is a portion of a curve that is symmetrical about the node.
4. The method for designing a pinion based on a claw helical rack knife according to claim 1, wherein in the step (1), the obtaining of the position vector and the normal vector of the rack knife specifically comprises:
at the node O of the cutting edgetEstablishing a tooth profile reference coordinate system, xtDirection and xcThe included angle of the direction is the pressure angle alpha of the rack knifen，ytPerpendicular xtPointing to the tool entity to establish a coordinate system StIn a coordinate system StIn, the upper and lower cavel helices are represented as:
Figure FDA0002997928360000021
in the formula, the upper and lower marks respectively represent upper and lower horn spiral lines on the left side of the rack knife, the right rack knife is obtained through symmetry, z is an integral upper limit, and t is an integral variable; parameter kxAnd kyThe shape parameters of the helix of the claw are obtained by homogeneous coordinate transformationcIs represented by:
rc(z,lc)＝Mct(lc)rt(z)
in the formula, MctIs a slave StTo ScIs expressed as:
Figure FDA0002997928360000022
in the matrix, the upper and lower tables represent the corresponding transformations on the left and right sides, respectively,/cTo be in a coordinate system ScOrigin O ofcAs a length of the starting point in the width direction of the rack bar, the corresponding unit normal vector expression is:
Figure FDA0002997928360000023
the shape parameter k of the cavel spiral line is obtained by the following nonlinear equation systemxAnd ky，
Figure FDA0002997928360000024
The first and second equations in the set of equations are used to determine the position, x, of the endpoint of the cleat spiralcAnd ycIs a rack knife position vector rcCoordinate component of, mnThe value range of the gear modulus and the position coefficient lambda is as follows: 1.25tg αnλ is not less than pi/4, when λ is 1.25tg αnWhen the cutting edge is a straight line, when lambda is pi/4, the end point of the cutting edge is on the middle point of the tooth space;
obtaining the parameter k of the cavel spiral line by solving the nonlinear equation setxAnd kyThereby obtaining a position vector r of the rack toolcAnd unit normal vector nc。
5. The method for designing a pinion pair based on a claw helical rack cutter according to claim 1, wherein the step (2) specifically comprises:
small wheel moving coordinate system S1Reference coordinate system ShAnd rack knife coordinate system Sc(ii) a Similar to the standard involute tooth profile generating motion, a small wheel moving coordinate system S1Around z1Rotate
Figure FDA0002997928360000031
While rack and knife coordinate system ScMove
Figure FDA0002997928360000032
rp1Is the pitch circle radius of the small wheel; by the meshing principle, at the contact point of the rack tooth surface and the tooth surface of the small wheel to be generated, the unit normal vector must pass through the instantaneous revolution axis I-I, and the following meshing conditions are obtained:
Figure FDA0002997928360000033
in the formula Xc、Yc、ZcFor instantaneous axis of rotation I-I in rack-and-pinion coordinate system ScCoordinate of (1), xc、yc、zcFor instantaneous contact point of two tooth surfaces at ScCoordinate of (1), ncx，ncy，nczThe component of the unit normal vector of the instantaneous contact point is used for obtaining the machining rotation angle of the small wheel
Figure FDA0002997928360000034
Is expressed as
Figure FDA0002997928360000035
In the formula, Xc＝0，
Figure FDA0002997928360000036
rp1＝mnN12 is the pitch circle radius, N1The number of teeth of the small gear;
the flank position vector and the normal vector of the small wheel are expressed as,
Figure FDA0002997928360000037
Figure FDA0002997928360000038
in the formula, M1cIs ScTo S1The transformation matrix of (2);
and obtaining the equation of the big gear tooth surface matched with the same equation.
CN202010172485.8A 2020-03-12 2020-03-12 Gear pair design method based on claw helix rack knife Expired - Fee Related CN111322373B (en)

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