CN114505543A - Involute surface enveloping ring surface worm tooth surface hobbing cutter confirming method based on medium gear - Google Patents

Involute surface enveloping ring surface worm tooth surface hobbing cutter confirming method based on medium gear Download PDF

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CN114505543A
CN114505543A CN202210182539.8A CN202210182539A CN114505543A CN 114505543 A CN114505543 A CN 114505543A CN 202210182539 A CN202210182539 A CN 202210182539A CN 114505543 A CN114505543 A CN 114505543A
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involute
worm
gear
tooth
cylindrical gear
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CN114505543B (en
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陈永洪
李忠涛
罗文军
陈兵奎
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Chongqing University
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23F—MAKING GEARS OR TOOTHED RACKS
    • B23F13/00—Making worms by methods essentially requiring the use of machines of the gear-cutting type
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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    • Y02E10/00—Energy generation through renewable energy sources
    • Y02E10/70—Wind energy
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Abstract

The invention discloses a method for confirming a hobbing cutter for the tooth surface of an involute surface enveloping ring surface worm based on a medium gear, which comprises the following steps: determining a transmission pair formed by conjugate meshing of the involute surface enveloping ring worm and an external meshing involute cylindrical gear, and further determining the external meshing involute cylindrical gear; generating an involute medium gear with zero thickness based on the tooth surface of the external-meshing involute cylindrical gear, wherein the involute medium gear is provided with internal teeth and external teeth, and the tooth surface parameters of the internal teeth and the external teeth are the same as those of the external-meshing involute cylindrical gear; and determining an involute surface enveloping drum worm which is in conjugate inner meshing with the involute medium-based gear, and forming a cutter based on the involute surface enveloping drum worm. The invention can be used for the tooth surface forming processing of the equal-tooth-thickness involute surface gear enveloping ring surface worm and the variable-tooth-thickness involute surface gear enveloping ring surface worm; the method can also be used for the tooth surface forming processing of the equi-tooth-thickness plane gear enveloping ring surface worm and the variable-tooth-thickness plane gear enveloping ring surface worm.

Description

Involute surface enveloping ring surface worm tooth surface hobbing tool confirming method based on medium gear
Technical Field
The invention relates to the technical field of gear machining, in particular to a method for confirming a hobbing cutter for the tooth surface of an involute surface enveloping ring surface worm based on a medium gear.
Background
The involute surface enveloping ring surface worm transmission is used as a multi-tooth line/point contact transmission mechanism and has the advantages of large transmission ratio, stable transmission, small noise impact, adjustable side clearance and the like.
The involute surface enveloping worm transmission mainly comprises two types, namely equal-tooth-thickness involute surface gear enveloping worm transmission and variable-tooth-thickness involute surface gear enveloping worm transmission, is widely applied to the fields of national emerging industries and strategic deployment, such as aerospace, strategic equipment, intelligent manufacturing, wind power heat energy and the like, and has important research significance and application value in the field of precise heavy-load driving transmission.
At present, the involute surface enveloping ring surface worm is mainly formed by turning, but not formed by precise grinding, so that the machining precision and efficiency are greatly reduced. The hobbing is widely applied to the processing of the tooth surface of a worm wheel in worm transmission as an efficient gear processing method, and has the advantages of relatively mature technology, high processing precision and high efficiency. In consideration of the complexity of the involute surface enveloping ring surface worm transmission worm tooth surface, no hobbing method and hobbing cutter design for the involute surface enveloping ring surface worm tooth surface is provided at present.
Therefore, in order to solve the above problems, a method for confirming a tool for determining a gear hobbing process of a tooth surface of an involute-enveloping worm based on a medium gear is needed, so as to realize high-precision hobbing forming of the involute-enveloping worm.
Disclosure of Invention
In view of this, the invention provides a method for confirming a tool for processing a gear surface of an involute surface enveloping worm based on a medium gear, so as to realize high-precision gear hobbing forming processing on the involute surface enveloping worm.
The invention discloses a method for confirming a hobbing cutter for the tooth surface of an involute surface enveloping ring surface worm based on a medium gear, which comprises the following steps:
s1: determining an involute surface enveloping ring surface worm of a target workpiece;
s2: determining a transmission pair formed by conjugate meshing of the involute surface enveloping ring worm and an external meshing involute cylindrical gear, and further determining the external meshing involute cylindrical gear;
s3: generating an involute medium gear with the thickness of zero based on the tooth surface of the external meshing involute cylindrical gear in the step S2, wherein the involute medium gear is provided with internal teeth and external teeth, and the tooth surface parameters of the internal teeth and the external teeth are the same as those of the external meshing involute cylindrical gear;
s4: determining an involute surface enveloping drum worm which is in conjugate inner engagement with the involute medium-based gear;
s5: and forming a hobbing processing cutter based on the tooth crest back angle and the chip groove of the involute surface enveloping drum worm.
Further, the method for determining the external-engagement involute cylindrical gear in step S2 includes the following steps:
s2-1: establishing transmission pair space frame sigmau(Ou-xu,yu,zu)、σv(Ov-xv,yv,zv)、σ1(O1-x1,y1,z1) And σ2(O2-x2,y2,z2) (ii) a Wherein O isu、Ov、O1And O2Respectively a space frame sigmau、σv、σ1And σ2(x) of (c)u,yu,zu)、(xv,yv,zv)、(x1,y1,z1) And (x)2,y2,z2) Respectively a space frame sigmau、σv、σ1And σ2Coordinate axes of three directions;
space fixed frame sigmau(Ou-xu,yu,zu) Is the initial position of the external-meshing involute cylindrical gear, and the base vector of the external-meshing involute cylindrical gear is (i)u,ju,ku) Wherein iu、juAnd kuAre respectively coordinate axes xu、yuAnd zuThree vectors of (a);
space fixed frame sigmav(Ov-xv,yv,zv) The initial position of the involute enveloping ring surface worm is shown as the base vector of (i)v,jv,kv) Wherein iv、jvAnd kvAre respectively coordinate axes xv、yvAnd zvThree vectors of (a);
space motion frame sigma1(O1-x1,y1,z1) Fixedly connected with an external-meshing involute cylindrical gear and wound around the gear1The shaft rotating with a certain instantaneous rotational displacement of
Figure BDA0003522067780000021
The basal vector is (i)1,j1,k1) Wherein i1、j1And k1Are respectively coordinate axes x1、y1And z1Three vectors of (a);
space motion frame sigma2(O2-x2,y2,z2) Fixedly connected with involute surface enveloping ring surface worm and wound around z2The shaft rotating with a certain instantaneous rotational displacement of
Figure BDA0003522067780000022
The basal vector is (i)2,j2,k2) Wherein i2、j2And k2Are respectively coordinate axes x2、y2And z2Three vectors of (a);
s2-2: calculating various parameters of the external engagement involute cylindrical gear based on space differential geometry and a gear engagement principle;
in space motion frame sigma1(O1-x1,y1,z1) In the method, an equation of the left flank of the external-meshing involute cylindrical gear is obtained
Figure BDA0003522067780000031
The following were used:
Figure BDA0003522067780000032
obtaining the right flank equation of the external-engagement involute cylindrical gear
Figure BDA0003522067780000033
The following were used:
Figure BDA0003522067780000034
in the formula: tau and theta are tooth surface parameters of the external-meshing involute cylindrical gear; delta is a base circle half angle of the external-meshing involute cylindrical gear; alpha is alphatThe pressure angle of the end face of the externally-engaged involute cylindrical gear is alpha respectively when the gear is a variable tooth thickness involute cylindrical geartLAnd alphatR；
Figure BDA0003522067780000035
The base radius of the external-meshing involute cylindrical gear is that of the tooth surfaces on the left side and the right side of the involute cylindrical gear respectively
Figure BDA0003522067780000036
And
Figure BDA0003522067780000037
beta is the helical angle of the external-meshing involute cylindrical gear, if the external-meshing involute cylindrical gear is a variable-tooth-thickness involute cylindrical gear, the helical angles of the tooth surfaces at the left side and the right side are respectively betaLAnd betaR；
Determining the tooth top height of an external-meshing involute cylindrical gear
Figure BDA0003522067780000038
Height of tooth root
Figure BDA0003522067780000039
Tip clearance
Figure BDA00035220677800000310
Comprises the following steps:
Figure BDA00035220677800000311
in the formula: r isIIThe radius of a reference circle of the external-meshing involute cylindrical gear; m isnIs the normal modulus;
Figure BDA00035220677800000312
normal tooth crest height coefficient; x is the number ofnNormal deflection coefficient;
Figure BDA00035220677800000313
normal headspace factor; z1The number of teeth of the externally engaged involute cylindrical gear is the number of teeth;
addendum circle radius of external-meshing involute cylindrical gear
Figure BDA00035220677800000314
Radius of tooth root
Figure BDA00035220677800000315
Comprises the following steps:
Figure BDA0003522067780000041
according to the gear meshing principle, the left tooth surface of the involute surface enveloping ring surface worm can be obtained through coordinate transformation and bottom vector conversion in a space motion frame sigma2(O2-x2,y2,z2) The tooth surface equation in (1) is:
Figure BDA0003522067780000042
scale frame sigma for spatial motion of right tooth surface of involute surface enveloping ring surface worm2(O2-x2,y2,z2) The tooth surface equation in (1) is:
Figure BDA0003522067780000043
in the formula:
Figure BDA0003522067780000044
and
Figure BDA0003522067780000045
are respectively involuteMeshing functions of left and right tooth surfaces of the surface enveloping worm;
Figure BDA0003522067780000046
and
Figure BDA0003522067780000047
the relative speeds of the left tooth surface and the right tooth surface of the involute surface enveloping ring surface worm are respectively;
Figure BDA0003522067780000048
and
Figure BDA0003522067780000049
respectively are normal vectors of left and right tooth surfaces of the external-meshing involute cylindrical gear; m21For spatial movement of frames sigma1(O1-x1,y1,z1) And σ2(O2-x2,y2,z2) And a base vector transformation matrix between, and:
Figure BDA00035220677800000410
wherein:
Figure BDA00035220677800000411
i21＝1/i12＝Z1/Z2，Z1is the number of teeth of an externally-engaged involute cylindrical gear, Z2The number of heads of the involute surface enveloping ring surface worm is;
addendum arc radius of involute surface enveloping ring surface worm
Figure BDA00035220677800000412
And root arc radius
Figure BDA00035220677800000413
Comprises the following steps:
Figure BDA00035220677800000414
further, in step S3, the internal tooth flank equation and the external tooth flank equation of the involute idler gear are the same as the tooth flank equation of the external-meshing involute cylindrical gear, that is:
Figure BDA00035220677800000415
the thickness of the external engagement involute cylindrical gear is zero, so that the radius of an internal tooth tip circle of the corresponding internal engagement involute cylindrical gear
Figure BDA0003522067780000051
Radius of tooth root
Figure BDA0003522067780000052
Comprises the following steps:
Figure BDA0003522067780000053
the radius of the top circle of the involute medium gear is defined to be coincident with the tooth root circular arc of the involute surface enveloping ring surface worm, the tooth root circle of the involute medium gear is less than or equal to the tooth root circle of the involute cylindrical gear, and then the radius of the top circle of the involute medium gear is defined to be coincident with the tooth root circular arc of the involute surface enveloping ring surface worm
Figure BDA0003522067780000054
Radius of tooth root
Figure BDA0003522067780000055
Comprises the following steps:
Figure BDA0003522067780000056
further, in step S4, the method for determining the involute-surface-enveloping worm comprises the following steps:
s4-1, establishing space fixed frame sigmau(Ou-xu,yu,zu)、σw(Ow-xw,yw,zw)、σ1(O1-x1,y1,z1) And σ3(O3-x3,y3,z3) In which O isu、Ow、O1And O3Respectively a space frame sigmau、σw、σ1And σ3(x) of (c)u,yu,zu)、(xw,yw,zw)、(x1,y1,z1) And (x)3,y3,z3) Respectively a space frame sigmau、σw、σ1And σ3Coordinate axes of three directions;
σu(Ou-xu,yu,zu) The initial position of the internally-meshed involute cylindrical gear is coincident with the initial position of the externally-meshed involute cylindrical gear, and the base vector of the internally-meshed involute cylindrical gear is (i)u,ju,ku) Wherein iu、juAnd kuAre respectively coordinate axis xu、yuAnd zuThree vectors of (a);
σw(Ow-xw,yw,zw) The initial position of the involute enveloping drum worm is that the base vector is (i)w,jw,kw) Wherein iw、jwAnd kwAre respectively coordinate axes xw、ywAnd zwThree vectors of (a);
σ1(O1-x1,y1,z1) Fixedly connected with an internal meshing involute cylindrical gear and wound around the cylindrical gear1The shaft rotates by a certain instantaneous rotational displacement of
Figure BDA0003522067780000059
The basal vector is (i)1,j1,k1) Wherein i1、j1And k1Are respectively coordinate axes x1、y1And z1Three vectors of (a);
σ3(O3-x3,y3,z3) Enveloping drum worm with involute surfaceSecured and wound z3The shaft rotates by a certain instantaneous rotational displacement of
Figure BDA0003522067780000057
The basal vector is (i)3,j3,k3) Wherein i3、j3And k3Are respectively coordinate axes x3、y3And z3Three vectors of (a);
s4-2, according to the gear meshing principle, the left flank of the involute surface envelope drum worm can be obtained through coordinate transformation and base vector transformation, and the space motion scale frame sigma is obtained3(O3-x3,y3,z3) The tooth surface equation in (1) is:
Figure BDA0003522067780000058
scale frame sigma for spatial motion of right tooth surface of involute surface envelope drum-shaped worm3(O3-x3,y3,z3) The tooth surface equation in (1) is:
Figure BDA0003522067780000061
in the formula:
Figure BDA0003522067780000062
and
Figure BDA0003522067780000063
respectively is the meshing function of the left and right tooth surfaces of the involute surface enveloping drum-shaped worm;
Figure BDA0003522067780000064
and
Figure BDA0003522067780000065
the relative speeds of the left and right tooth surfaces of the involute surface enveloping drum-shaped worm are respectively;
Figure BDA0003522067780000066
and
Figure BDA0003522067780000067
respectively are normal vectors of left and right tooth surfaces of the inner meshing involute cylindrical gear; m31For spatial movement of frames sigma1(O1-x1,y1,z1) And σ3(O3-x3,y3,z3) And a base vector transformation matrix between, and:
Figure BDA0003522067780000068
wherein:
Figure BDA0003522067780000069
i31＝1/i13＝Z1/Z3，Z3enveloping the number of heads of the worm of drum type for involute surface, and Z3＝Z2。
Further, in step S5, the addendum circle radius and the dedendum circle radius of the drum hob are defined based on the involute surface enveloping drum worm:
the radius of the addendum arc of the drum hob is defined to be coincident with the addendum circle of the involute medium gear, and the addendum arc of the drum hob is defined to be coincident with the root circle of the involute medium gear, so that the addendum arc radius of the drum hob is defined
Figure BDA00035220677800000610
And root arc radius
Figure BDA00035220677800000611
Comprises the following steps:
Figure BDA00035220677800000612
the invention has the beneficial effects that:
the method can be used for the tooth surface forming processing of the equal-tooth-thickness involute surface gear enveloping ring surface worm and the variable-tooth-thickness involute surface gear enveloping ring surface worm; the method can also be used for the tooth surface forming processing of the enveloping worm of the equal-tooth-thickness plane gear and the enveloping worm of the variable-tooth-thickness plane gear; the involute surface enveloping ring surface worm can be formed by precision grinding, and the processing precision and efficiency can be effectively improved.
Drawings
The invention is further described below with reference to the figures and examples.
FIG. 1 is a schematic view of an involute enveloping toroidal worm;
FIG. 2 is a schematic view of an involute enveloping toroidal worm drive;
FIG. 3 is a schematic view of the involute surface enveloping drum worm transmission structure;
FIG. 4 is a schematic view of an involute intermediate gear structure;
FIG. 5 is a schematic view of the involute medium gear enveloping toroidal worm drive structure;
FIG. 6 is a schematic view of an involute medium gear enveloping drum worm transmission structure;
FIG. 7 is a schematic structural view of the meshing relationship between involute medium gears and enveloping worm and drum worm;
FIG. 8 is a schematic view of the meshing relationship between the enveloping worm and the drum worm;
FIG. 9 is a schematic structural view of an involute medium gear enveloping drum hob;
FIG. 10 is a schematic structural view of a drum hob gear generating involute surface enveloping ring surface worm;
FIG. 11 is a schematic diagram of a general case generalized to an axis angle unequal to zero;
Detailed Description
As shown in the figure: the method for confirming the involute surface enveloping ring surface worm tooth surface hobbing machining cutter based on the medium gear comprises the following steps:
s1: determining an involute surface enveloping ring surface worm of a target workpiece; as shown in fig. 1, the main parameters of the involute surface enveloping toroidal worm 1 include a central axis 11 of the toroidal worm, a tooth root arc 12 of the toroidal worm, a tooth top arc 13 of the toroidal worm and a right tooth surface 14 of the toroidal worm, wherein the left tooth surface of the toroidal worm is not marked to be opposite to the right tooth surface;
s2: determining a transmission pair formed by conjugate meshing of the involute surface enveloping ring worm and an external meshing involute cylindrical gear, and further determining the external meshing involute cylindrical gear;
after the target part is determined, determining basic parameters of the involute surface enveloping ring surface worm transmission pair of the target workpiece, including center distance a and transmission ratio i21Number of worm heads Z2Normal modulus mnNormal pressure angle alphanAnd the helical angle beta, if the gear is a variable tooth thickness involute cylindrical gear, the helical angles of the left and right side tooth surfaces are respectively as follows: beta is aLAnd betaRAnd estimating the end face pressure angle alpha therefromtIf the gear is a variable tooth thickness involute cylindrical gear, the pressure angles of the end faces of the left side tooth surface and the right side tooth surface are respectively as follows: alpha is alphatLAnd alphatRRadius of base circle
Figure BDA0003522067780000071
If the gear is a variable tooth thickness involute cylindrical gear, the base radii of the tooth surfaces on the left side and the right side are respectively as follows:
Figure BDA0003522067780000072
and
Figure BDA0003522067780000073
and determining parameters such as the tooth top height, the tooth root height, the tooth top clearance, the tooth root transition fillet radius and the like of the involute surface enveloping ring surface worm transmission pair.
S3: generating an involute medium gear with the thickness of zero based on the tooth surface of the external meshing involute cylindrical gear in the step S2, wherein the involute medium gear is provided with internal teeth and external teeth, and the tooth surface parameters of the internal teeth and the external teeth are the same as those of the external meshing involute cylindrical gear;
the external-meshing involute cylindrical gear is used as a parent body to generate a sub body of the involute medium gear, the involute medium gear is a sheet body with zero thickness, tooth surface parameters of the involute medium gear are completely the same as those of the parent body external-meshing involute cylindrical gear, simultaneously, the tooth top and tooth root height of the sub body involute medium gear are determined by the tooth top arc radius and tooth root arc radius of the involute enveloping toroidal worm of a target processing body, the tooth top circle of the specific involute medium gear coincides with the tooth root arc of the involute enveloping toroidal worm, and the tooth root circle of the involute medium gear is less than or equal to the tooth root circle of the involute cylindrical gear;
s4: determining an involute surface enveloping drum worm which is in conjugate inner engagement with the involute medium-based gear;
an involute medium gear moves according to a complete dual generating motion, and an involute surface enveloping drum-shaped worm is enveloped through inner meshing;
s5: and forming a hobbing processing cutter based on the tooth crest back angle and the chip groove of the involute surface enveloping drum worm.
Enveloping the drum-shaped worm to envelop the drum-shaped hob based on the involute surface and establishing a hob tooth surface mathematical formula, wherein the drum-shaped hob is provided with a chip groove and an addendum relief angle;
establishing an accurate involute medium gear enveloping drum hob three-dimensional model through three-dimensional modeling software based on a drum hob mathematical model;
based on the established three-dimensional model of the drum hob, the accurate molding processing of the drum hob formed by enveloping the involute medium gear is realized through equipment such as a five-axis processing center;
based on the processed drum-shaped hob entity, the hob is installed on a gear hobbing machine tool, and the two side tooth surfaces of the involute surface enveloping ring worm are processed through the relative rotation motion of the drum-shaped hob and a blank body of the involute surface enveloping ring worm according to a certain transmission ratio.
In this embodiment, the method for determining the external-engagement involute cylindrical gear in step S2 includes the following steps:
specifically, as shown in fig. 2, the involute-surface enveloping worm 1 is in conjugate engagement with an external-engagement involute cylindrical gear 2:
s2-1: establishing transmission pair space frame sigmau(Ou-xu,yu,zu)、σv(Ov-xv,yv,zv)、σ1(O1-x1,y1,z1) And σ2(O2-x2,y2,z2) (ii) a Wherein O isu、Ov、O1And O2Respectively a space frame sigmau、σv、σ1And σ2(x) of (c)u,yu,zu)、(xv,yv,zv)、(x1,y1,z1) And (x)2,y2,z2) Respectively a space frame sigmau、σv、σ1And σ2Coordinate axes of three directions;
space fixed frame sigmau(Ou-xu,yu,zu) Is the initial position of the external-meshing involute cylindrical gear, and the base vector of the external-meshing involute cylindrical gear is (i)u,ju,ku) Wherein iu、juAnd kuAre respectively coordinate axes xu、yuAnd zuThree vectors of (a);
space fixed frame sigmav(Ov-xv,yv,zv) The initial position of the involute enveloping ring surface worm is shown as the base vector of (i)v,jv,kv) Wherein iv、jvAnd kvAre respectively coordinate axes xv、yvAnd zvThree vectors of (a);
space motion frame sigma1(O1-x1,y1,z1) Fixedly connected with an external-meshing involute cylindrical gear and wound around the gear1The shaft rotating with a certain instantaneous rotational displacement of
Figure BDA0003522067780000091
The basal vector is (i)1,j1,k1) Wherein i1、j1And k1Are respectively coordinate axes x1、y1And z1Three vectors of (a);
space motion frame sigma2(O2-x2,y2,z2) Fixedly connected with involute enveloping ring surface worm and wound around z2The shaft rotating with a certain instantaneous rotational displacement of
Figure BDA0003522067780000092
The base vector is (i)2,j2,k2) Wherein i2、j2And k2Are respectively coordinate axes x2、y2And z2Three vectors of (a);
in the example, the center distance between the external-meshing involute cylindrical gear and the involute surface enveloping ring surface worm is a, and the shaft intersection angle is 0;
s2-2: calculating various parameters of the external engagement involute cylindrical gear based on space differential geometry and a gear engagement principle;
in space motion frame sigma1(O1-x1,y1,z1) In the method, an equation of the left flank of the external-meshing involute cylindrical gear is obtained
Figure BDA0003522067780000093
The following were used:
Figure BDA0003522067780000094
obtaining the right flank equation of the external-engagement involute cylindrical gear
Figure BDA0003522067780000095
The following were used:
Figure BDA0003522067780000101
in the formula: tau and theta are tooth surface parameters of the external-meshing involute cylindrical gear; delta is a base circle half angle of the external-meshing involute cylindrical gear; alpha is alphatThe pressure angle of the end face of the externally-engaged involute cylindrical gear is alpha respectively when the gear is a variable tooth thickness involute cylindrical geartLAnd alphatR；
Figure BDA0003522067780000102
The base radius of the external-meshing involute cylindrical gear is the involute circle with variable tooth thicknessThe base radii of the left and right side tooth surfaces of the cylindrical gear are respectively
Figure BDA0003522067780000103
And
Figure BDA0003522067780000104
beta is the helical angle of the external-meshing involute cylindrical gear, if the external-meshing involute cylindrical gear is a variable-tooth-thickness involute cylindrical gear, the helical angles of the tooth surfaces at the left side and the right side are respectively betaLAnd betaR；
Determining the tooth crest height of an external-meshing involute cylindrical gear
Figure BDA0003522067780000105
Height of tooth root
Figure BDA0003522067780000106
Tip clearance
Figure BDA0003522067780000107
Comprises the following steps:
Figure BDA0003522067780000108
in the formula: r isIIThe radius of a reference circle of the external-meshing involute cylindrical gear; m isnIs the normal modulus;
Figure BDA0003522067780000109
normal tooth crest height coefficient; x is the number ofnNormal deflection coefficient;
Figure BDA00035220677800001010
normal headspace factor; z is a linear or branched member1The number of teeth of the externally engaged involute cylindrical gear is the number of teeth;
addendum circle radius of external-meshing involute cylindrical gear
Figure BDA00035220677800001011
Radius of tooth root
Figure BDA00035220677800001012
Comprises the following steps:
Figure BDA00035220677800001013
according to the gear meshing principle, the left tooth surface of the involute surface enveloping ring surface worm can be obtained through coordinate transformation and bottom vector conversion in a space motion frame sigma2(O2-x2,y2,z2) The tooth surface equation in (1) is:
Figure BDA00035220677800001014
scale frame sigma for spatial motion of right tooth surface of involute surface enveloping ring surface worm2(O2-x2,y2,z2) The tooth surface equation in (1) is:
Figure BDA0003522067780000111
in the formula:
Figure BDA0003522067780000112
and
Figure BDA0003522067780000113
respectively is the meshing function of the left and right tooth surfaces of the involute surface enveloping ring surface worm;
Figure BDA0003522067780000114
and
Figure BDA0003522067780000115
the relative speeds of the left tooth surface and the right tooth surface of the involute surface enveloping ring surface worm are respectively;
Figure BDA0003522067780000116
and
Figure BDA0003522067780000117
are respectively asThe normal vector of the left tooth surface and the right tooth surface of the external-meshing involute cylindrical gear; m21For spatial movement of frames sigma1(O1-x1,y1,z1) And σ2(O2-x2,y2,z2) And a base vector transformation matrix between, and:
Figure BDA0003522067780000118
wherein:
Figure BDA0003522067780000119
i21＝1/i12＝Z1/Z2，Z1is the number of teeth of an externally-engaged involute cylindrical gear, Z2The number of heads of the involute surface enveloping ring surface worm is;
addendum arc radius of involute surface enveloping ring surface worm
Figure BDA00035220677800001110
And root arc radius
Figure BDA00035220677800001111
Comprises the following steps:
Figure BDA00035220677800001112
in the present embodiment, as shown in fig. 4, an involute idler gear with a thickness of zero is formed based on an external-meshing involute cylindrical gear 2 as a parent, and for convenience of discussion, an internal-meshing involute cylindrical gear 3 with a certain thickness is formed based on internal teeth of an involute idler gear 5 in fig. 3; the main parameters of the involute medium gear 5 include a medium gear central axis 51, a medium gear right tooth surface 52, a medium gear left tooth surface 53, a medium gear tooth root circle 54 and a medium gear tooth top circle 55.
In step S3, the internal tooth flank equation and the external tooth flank equation of the involute idler gear are the same as the tooth flank equation of the external-meshing involute cylindrical gear, that is:
Figure BDA00035220677800001113
the thickness of the external engagement involute cylindrical gear is zero, so that the radius of an internal tooth tip circle of the corresponding internal engagement involute cylindrical gear
Figure BDA00035220677800001114
And root radius of tooth
Figure BDA00035220677800001115
Comprises the following steps:
Figure BDA00035220677800001116
the radius of the top circle of the involute medium gear is defined to be coincident with the tooth root circular arc of the involute surface enveloping ring surface worm, the tooth root circle of the involute medium gear is less than or equal to the tooth root circle of the involute cylindrical gear, and then the radius of the top circle of the involute medium gear is defined to be coincident with the tooth root circular arc of the involute surface enveloping ring surface worm
Figure BDA0003522067780000121
Radius of tooth root
Figure BDA0003522067780000122
Comprises the following steps:
Figure BDA0003522067780000123
further, in step S4, the method for determining the involute-surface-enveloping worm 4 includes the following steps:
for ease of discussion, reference is made to fig. 3 as an example, as shown in fig. 3,
s4-1, establishing space fixing frame sigmau(Ou-xu,yu,zu)、σw(Ow-xw,yw,zw)、σ1(O1-x1,y1,z1) And σ3(O3-x3,y3,z3) In which O isu、Ow、O1And O3Respectively a space frame sigmau、σw、σ1And σ3(x) of (c)u,yu,zu)、(xw,yw,zw)、(x1,y1,z1) And (x)3,y3,z3) Respectively a space frame sigmau、σw、σ1And σ3Coordinate axes of three directions;
σu(Ou-xu,yu,zu) The initial position of the internally-meshed involute cylindrical gear is coincident with the initial position of the externally-meshed involute cylindrical gear, and the base vector of the internally-meshed involute cylindrical gear is (i)u,ju,ku) Wherein iu、juAnd kuAre respectively coordinate axes xu、yuAnd zuThree vectors of (a);
σw(Ow-xw,yw,zw) The initial position of the involute enveloping drum worm is that the base vector is (i)w,jw,kw) Wherein iw、jwAnd kwAre respectively coordinate axes xw、ywAnd zwThree vectors of (a);
σ1(O1-x1,y1,z1) Fixedly connected with an internal meshing involute cylindrical gear and wound around the cylindrical gear1The shaft rotating with a certain instantaneous rotational displacement of
Figure BDA0003522067780000124
The basal vector is (i)1,j1,k1) Wherein i1、j1And k1Are respectively coordinate axes x1、y1And z1Three vectors of (a);
σ3(O3-x3,y3,z3) Fixedly connected with involute surface envelope drum worm and wound around z3The shaft rotating with a certain instantaneous rotational displacement of
Figure BDA0003522067780000125
The basal vector is (i)3,j3,k3) Wherein i3、j3And k3Are respectively coordinate axes x3、y3And z3Three vectors of (a);
in the example, the center distance between the internally meshed involute cylindrical gear and the gradually surface-opened involute drum-shaped worm is b, and the shaft intersection angle is 0.
S4-2, according to the gear meshing principle, the left flank of the involute surface envelope drum worm can be obtained through coordinate transformation and base vector transformation, and the space motion scale frame sigma is obtained3(O3-x3,y3,z3) The tooth surface equation in (1) is:
Figure BDA0003522067780000126
scale frame sigma for spatial motion of right tooth surface of involute surface envelope drum-shaped worm3(O3-x3,y3,z3) The tooth surface equation in (1) is:
Figure BDA0003522067780000131
in the formula:
Figure BDA0003522067780000132
and
Figure BDA0003522067780000133
respectively is the meshing function of the left and right tooth surfaces of the involute surface enveloping drum-shaped worm;
Figure BDA0003522067780000134
and
Figure BDA0003522067780000135
the relative speeds of the left and right tooth surfaces of the involute surface enveloping drum-shaped worm are respectively;
Figure BDA0003522067780000136
and
Figure BDA0003522067780000137
respectively are normal vectors of left and right tooth surfaces of the inner meshing involute cylindrical gear; m is a group of31For spatial movement of frames sigma1(O1-x1,y1,z1) And σ3(O3-x3,y3,z3) And a base vector transformation matrix between, and:
Figure BDA0003522067780000138
wherein:
Figure BDA0003522067780000139
i31＝1/i13＝Z1/Z3，Z3enveloping the number of heads of the worm of drum type for involute surface, and Z3＝Z2。
In this embodiment, in step S5, the radius of the addendum circle and the radius of the dedendum circle of the drum hob are defined based on the involute-surface enveloping drum worm:
considering that the involute medium gear 5 is a sub-body which is stripped by taking the external-meshing involute cylindrical gear 2 as a parent body, the meshing relationship between the involute medium gear and the involute surface enveloping worm 1 is completely equal to the meshing relationship between the external-meshing involute cylindrical gear 2 and the involute surface enveloping worm 1, and the space meshing relationship and the standard frame are arranged as shown in fig. 5; in the same principle, the involute intermediate gear is also a sub-body which is stripped by taking the internal meshing involute cylindrical gear 3 as a parent body, the meshing relationship between the involute intermediate gear and the involute surface enveloping drum-shaped worm 4 is completely equal to that between the internal meshing involute cylindrical gear 3 and the involute surface enveloping drum-shaped worm 4, and the space meshing relationship and the standard frame setting are shown in fig. 6.
The involute medium gear enveloping ring surface worm transmission in the figure 5 and the involute medium gear enveloping drum shape worm transmission in the figure 6 are fused under the same space frame, and the meshing relation among the involute medium gear, the involute medium ring surface worm and the involute medium drum shape worm as shown in the figure 7 can be obtained; the involute medium gear is a surface gear with zero thickness, and according to the cammus' theorem, after the involute medium gear without thickness is removed, the involute medium gear and the involute medium drum-shaped worm have an equivalent conjugate meshing relationship, as shown in fig. 8, the axial distance between the involute medium drum-shaped worm and the involute medium drum-shaped worm is c.
The radius of the addendum arc of the drum hob is defined to be coincident with the addendum circle of the involute medium gear, and the addendum arc of the drum hob is defined to be coincident with the root circle of the involute medium gear, so that the addendum arc radius of the drum hob is defined
Figure BDA0003522067780000141
And root arc radius
Figure BDA0003522067780000142
Comprises the following steps:
Figure BDA0003522067780000143
in the present embodiment, as shown in fig. 9, the drum hob 6 obtained by performing the chip groove notching and the top and back angle processing on the involute surface enveloping drum worm defined above can be effectively defined by the main parameters of the drum hob, namely, the left tooth surface 61, the right tooth surface, the top and back angles 62, the top and back arcs 63, the bottom and the grooves 64, the chip groove 65, the center hole 66, the center axis 67 and the key groove 68.
According to the relation, in the process that the drum hob performs generating hobbing on the ring surface worm blank to process the involute surface enveloping ring surface worm in an enveloping and enveloping manner, the central axis of the drum hob is parallel to the central axis of the ring surface worm blank and has the same shaft intersection angle with the involute medium gearIn the embodiment of the invention, the intersecting angle of the axes is 0, and the rotating angular speeds of the drum-shaped hob and the enveloping worm blank are equal and opposite, as shown in fig. 10. At the same time, the torroidal worm blank only has a feed cutting motion about its centerline axis, the drum hob being other than about its centerline axis z1In addition to the main cutting movement of rotation, there is also a main cutting movement in x1A feed cutting motion in which the shaft moves linearly.
The drum hob obtained by taking the drum worm as a prototype performs conjugate motion with a blank to be processed through enveloping motion, and then continuous indexing generating processing of the involute surface enveloping ring surface worm tooth surface is completed. According to the conjugate relation among the drum hob, the medium gear and the processed toroidal worm, the correct processing conditions of the involute surface enveloping toroidal worm can be known as follows: the module, the pressure angle and the helical angle of the medium gear are the same; the number of heads is equal to that of the heads of the drum-shaped hobs; the shaft angle between the intermediate gear and the drum hob is equal to the shaft angle between the drum hob and the intermediate gear. In the processing process, the installation parameters and the conjugate relation between the drum-shaped hob and the processed ring surface worm blank body need to satisfy:
Figure BDA0003522067780000151
in the formula: a is the center distance between the external-meshing involute cylindrical gear and the involute surface enveloping ring surface worm; b is the center distance between the internal meshing involute cylindrical gear and the involute surface enveloping drum worm (drum hob); c is the center distance between the involute surface enveloping ring surface worm and the involute surface enveloping drum-shaped worm (drum-shaped hob); c. C0The initial axle distance between the drum hob and the ring surface worm blank; ring surface worm blank corner
Figure BDA0003522067780000152
(rotational speed ω2) Direction and drum hob corner
Figure BDA0003522067780000153
(rotational speed is ω3) The directions are opposite; v. of0The drum hob is fed with the speed of the cutting movement V.
In this embodiment, the shaft angle λ between the involute surface enveloping worm and the external involute cylindrical gear is zero, and the shaft angle λ between the corresponding idler gear and the drum hob is zero, which is not limited to the transmission with a zero shaft angle in practical application, but also can be applied to the transmission with a non-zero shaft angle, where the shaft angle between the corresponding idler gear and the drum hob is non-zero, that is, λ ≠ 0, and the general case of the non-zero shaft angle is shown in fig. 11.
Finally, the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all of them should be covered in the claims of the present invention.

Claims (5)

1. A method for confirming a hobbing cutter for the tooth surface of an involute enveloping ring surface worm based on a medium gear is characterized by comprising the following steps: the method comprises the following steps:
s1: determining an involute surface enveloping ring surface worm of a target workpiece;
s2: determining a transmission pair formed by conjugate meshing of the involute surface enveloping ring worm and an external meshing involute cylindrical gear, and further determining the external meshing involute cylindrical gear;
s3: generating an involute medium gear with the thickness of zero based on the tooth surface of the external meshing involute cylindrical gear in the step S2, wherein the involute medium gear is provided with internal teeth and external teeth, and the tooth surface parameters of the internal teeth and the external teeth are the same as those of the external meshing involute cylindrical gear;
s4: determining an involute surface enveloping drum worm which is in conjugate inner engagement with the involute medium-based gear;
s5: and forming a hobbing processing cutter based on the tooth crest back angle and the chip groove of the involute surface enveloping drum worm.
2. The medium gear-based involute enveloping toroidal worm tooth surface hobbing tool validation method of claim 1, wherein:
the method for determining the external-meshing involute cylindrical gear in the step S2 comprises the following steps:
s2-1: establishing transmission pair space frame sigmau(Ou-xu,yu,zu)、σv(Ov-xv,yv,zv)、σ1(O1-x1,y1,z1) And σ2(O2-x2,y2,z2) (ii) a Wherein O isu、Ov、O1And O2Respectively a space frame sigmau、σv、σ1And σ2(x) of (c)u,yu,zu)、(xv,yv,zv)、(x1,y1,z1) And (x)2,y2,z2) Respectively a space frame sigmau、σv、σ1And σ2Coordinate axes of three directions;
space fixed frame sigmau(Ou-xu,yu,zu) Is the initial position of the external-meshing involute cylindrical gear, and the base vector of the external-meshing involute cylindrical gear is (i)u,ju,ku) Wherein i isu、juAnd kuAre respectively coordinate axes xu、yuAnd zuThree vectors of (a);
space fixed frame sigmav(Ov-xv,yv,zv) The initial position of the involute enveloping ring surface worm is shown as the base vector of (i)v,jv,kv) Wherein iv、jvAnd kvAre respectively coordinate axes xv、yvAnd zvThree vectors of (a);
space motion frame sigma1(O1-x1,y1,z1) Fixedly connected with an external-meshing involute cylindrical gear and wound around the gear1The shaft rotates by a certain instantaneous rotational displacement of
Figure FDA0003522067770000011
The basal vector is (i)1,j1,k1) Wherein i1、j1And k1Are respectively coordinate axes x1、y1And z1Three vectors of (a);
space motion frame sigma2(O2-x2,y2,z2) Fixedly connected with involute enveloping ring surface worm and wound around z2The shaft rotating with a certain instantaneous rotational displacement of
Figure FDA0003522067770000021
The basal vector is (i)2,j2,k2) Wherein i2、j2And k2Are respectively coordinate axis x2、y2And z2Three vectors of (a);
s2-2: calculating various parameters of the external engagement involute cylindrical gear based on space differential geometry and a gear engagement principle;
in space motion frame sigma1(O1-x1,y1,z1) In the method, an equation of the left flank of the external-meshing involute cylindrical gear is obtained
Figure FDA0003522067770000022
The following were used:
Figure FDA0003522067770000023
obtaining the right flank equation of the external-engagement involute cylindrical gear
Figure FDA0003522067770000024
The following were used:
Figure FDA0003522067770000025
in the formula: tau and theta are tooth surface parameters of the external-meshing involute cylindrical gear; delta is base circle half angle of external engagement involute cylindrical gear；αtThe pressure angle of the end face of the externally-engaged involute cylindrical gear is alpha respectively when the gear is a variable tooth thickness involute cylindrical geartLAnd alphatR；
Figure FDA0003522067770000026
The base radius of the external-meshing involute cylindrical gear is that of the tooth surfaces on the left side and the right side of the involute cylindrical gear respectively
Figure FDA0003522067770000027
And
Figure FDA0003522067770000028
beta is the helical angle of the external-meshing involute cylindrical gear, if the external-meshing involute cylindrical gear is a variable-tooth-thickness involute cylindrical gear, the helical angles of the tooth surfaces at the left side and the right side are respectively betaLAnd betaR；
Determining the tooth crest height of an external-meshing involute cylindrical gear
Figure FDA0003522067770000029
Height of tooth root
Figure FDA00035220677700000210
Tip clearance
Figure FDA00035220677700000211
Comprises the following steps:
Figure FDA0003522067770000031
in the formula: r is a radical of hydrogenIIThe radius of a reference circle of the external-meshing involute cylindrical gear; m isnIs the normal modulus;
Figure FDA0003522067770000032
normal addendum coefficient; x is the number ofnNormal deflection coefficient;
Figure FDA0003522067770000033
normal headspace factor; z1The number of teeth of the externally engaged involute cylindrical gear is the number of teeth;
addendum circle radius of external-meshing involute cylindrical gear
Figure FDA0003522067770000034
Radius of tooth root
Figure FDA0003522067770000035
Comprises the following steps:
Figure FDA0003522067770000036
according to the gear meshing principle, the left tooth surface of the involute surface enveloping ring surface worm can be obtained through coordinate transformation and bottom vector conversion in a space motion frame sigma2(O2-x2,y2,z2) The tooth surface equation in (1) is:
Figure FDA0003522067770000037
scale frame sigma for spatial motion of right tooth surface of involute surface enveloping ring surface worm2(O2-x2,y2,z2) The tooth surface equation in (1) is:
Figure FDA0003522067770000038
in the formula:
Figure FDA0003522067770000039
and
Figure FDA00035220677700000310
meshing box for enveloping left and right tooth surfaces of toroid worm with involute surfaceCounting;
Figure FDA00035220677700000311
and
Figure FDA00035220677700000312
the relative speeds of the left tooth surface and the right tooth surface of the involute surface enveloping ring surface worm are respectively;
Figure FDA00035220677700000313
and
Figure FDA00035220677700000314
respectively are normal vectors of left and right tooth surfaces of the external-meshing involute cylindrical gear; m is a group of21For spatial movement of frames sigma1(O1-x1,y1,z1) And σ2(O2-x2,y2,z2) A base vector transformation matrix therebetween, and:
Figure FDA00035220677700000315
wherein:
Figure FDA00035220677700000316
i21＝1/i12＝Z1/Z2，Z1is the number of teeth, Z, of an externally-meshed involute cylindrical gear2The number of heads of the involute surface enveloping ring surface worm is;
addendum arc radius of involute surface enveloping ring surface worm
Figure FDA0003522067770000041
And root arc radius
Figure FDA0003522067770000042
Comprises the following steps:
Figure FDA0003522067770000043
3. the medium gear-based involute enveloping toroidal worm tooth surface hobbing tool validation method of claim 2, wherein:
in step S3, the internal tooth flank equation and the external tooth flank equation of the involute idler gear are the same as the tooth flank equation of the external-meshing involute cylindrical gear, that is:
Figure FDA0003522067770000044
the thickness of the external engagement involute cylindrical gear is zero, so that the radius of an internal tooth tip circle of the corresponding internal engagement involute cylindrical gear
Figure FDA0003522067770000045
Radius of tooth root
Figure FDA0003522067770000046
Comprises the following steps:
Figure FDA0003522067770000047
the radius of the top circle of the involute medium gear is defined to be coincident with the tooth root circular arc of the involute surface enveloping ring surface worm, the tooth root circle of the involute medium gear is less than or equal to the tooth root circle of the involute cylindrical gear, and then the radius of the top circle of the involute medium gear is defined to be coincident with the tooth root circular arc of the involute surface enveloping ring surface worm
Figure FDA0003522067770000048
Radius of tooth root
Figure FDA0003522067770000049
Comprises the following steps:
Figure FDA00035220677700000410
4. the medium gear-based involute enveloping toroidal worm tooth surface hobbing tool validation method of claim 3, wherein:
in step S4, the method for determining the involute surface enveloping drum worm includes the following steps:
s4-1, establishing space fixing frame sigmau(Ou-xu,yu,zu)、σw(Ow-xw,yw,zw)、σ1(O1-x1,y1,z1) And σ3(O3-x3,y3,z3) In which O isu、Ow、O1And O3Respectively a space frame sigmau、σw、σ1And σ3(x) of (c)u,yu,zu)、(xw,yw,zw)、(x1,y1,z1) And (x)3,y3,z3) Respectively a space frame sigmau、σw、σ1And σ3Coordinate axes of three directions;
σu(Ou-xu,yu,zu) Is the initial position of the internally-meshed involute cylindrical gear, is coincident with the initial position of the externally-meshed involute cylindrical gear, and has a base vector of (i)u,ju,ku) Wherein iu、juAnd kuAre respectively coordinate axes xu、yuAnd zuThree vectors of (a);
σw(Ow-xw,yw,zw) The initial position of the involute enveloping drum worm is that the base vector is (i)w,jw,kw) Wherein iw、jwAnd kwAre respectively coordinate axes xw、ywAnd zwThree vectors of (a);
σ1(O1-x1,y1,z1) Fixedly connected with an internal meshing involute cylindrical gear and wound around the cylindrical gear1The shaft rotating with a certain instantaneous rotational displacement of
Figure FDA0003522067770000051
The basal vector is (i)1,j1,k1) Wherein i1、j1And k1Are respectively coordinate axes x1、y1And z1Three vectors of (a);
σ3(O3-x3,y3,z3) Fixedly connected with involute surface envelope drum worm and wound around z3The shaft rotates by a certain instantaneous rotational displacement of
Figure FDA0003522067770000052
The basal vector is (i)3,j3,k3) Wherein i3、j3And k3Are respectively coordinate axes x3、y3And z3Three vectors of (a);
s4-2, according to the gear meshing principle, the left flank of the involute enveloping drum worm can be obtained through coordinate transformation and base vector transformation, and the scale frame sigma of the spatial motion of the left flank of the drum worm can be obtained3(O3-x3,y3,z3) The tooth surface equation in (1) is:
Figure FDA0003522067770000053
scale frame sigma for spatial motion of right tooth surface of involute surface envelope drum-shaped worm3(O3-x3,y3,z3) The tooth surface equation in (1) is:
Figure FDA0003522067770000054
in the formula:
Figure FDA0003522067770000055
and
Figure FDA0003522067770000056
respectively is the meshing function of the left and right tooth surfaces of the involute surface enveloping drum-shaped worm;
Figure FDA0003522067770000057
and
Figure FDA0003522067770000058
the relative speeds of the left and right tooth surfaces of the involute surface enveloping drum-shaped worm are respectively;
Figure FDA0003522067770000059
and
Figure FDA00035220677700000510
respectively are normal vectors of left and right tooth surfaces of the inner meshing involute cylindrical gear; m31For spatial movement of frames sigma1(O1-x1,y1,z1) And σ3(O3-x3,y3,z3) And a base vector transformation matrix between, and:
Figure FDA00035220677700000511
wherein:
Figure FDA00035220677700000512
i31＝1/i13＝Z1/Z3，Z3enveloping the number of heads of the worm of drum type for involute surface, and Z3＝Z2。
5. The medium gear-based involute enveloping toroidal worm tooth surface hobbing tool validation method of claim 4, wherein:
in step S5, the radius of the addendum circle and the radius of the dedendum circle of the drum hob are defined based on the involute surface enveloping drum worm:
the radius of the addendum arc of the drum hob is defined to be coincident with the addendum circle of the involute medium gear, and the addendum arc of the drum hob is defined to be coincident with the root circle of the involute medium gear, so that the addendum arc radius of the drum hob is defined
Figure FDA0003522067770000061
And root arc radius
Figure FDA0003522067770000062
Comprises the following steps:
Figure FDA0003522067770000063
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