CN114505543B - Involute surface enveloping ring surface worm tooth surface gear hobbing cutter confirmation method based on medium gear - Google Patents
Involute surface enveloping ring surface worm tooth surface gear hobbing cutter confirmation method based on medium gear Download PDFInfo
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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
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Abstract
The invention discloses a method for confirming a hobbing cutter for the tooth surface of an involute enveloping worm based on a medium gear, which comprises the following steps: determining a transmission pair formed by conjugate engagement of the involute enveloping worm and the external engagement involute cylindrical gear, and further determining the external engagement involute cylindrical gear; generating an involute intermediate gear with zero thickness based on the tooth surface of the external meshing involute cylindrical gear, wherein the involute intermediate gear is provided with internal teeth and external teeth, and the tooth surface parameters of the internal teeth and the external teeth are identical with the tooth surface parameters of the external meshing involute cylindrical gear; and determining an involute surface enveloping drum worm in conjugate internal engagement with the involute medium gear, and forming a cutter based on the involute surface enveloping drum worm. The method can be used for tooth surface forming processing of the enveloping worm of the constant-tooth-thickness involute gear and the enveloping worm of the variable-tooth-thickness involute gear; the method can be used for tooth surface forming processing of the enveloping worm of the equal-tooth-thickness planar gear and the enveloping worm of the variable-tooth-thickness planar gear.
Description
Technical Field
The invention relates to the technical field of gear machining, in particular to a method for confirming a gear hobbing cutter for an involute surface enveloping ring surface worm gear tooth surface based on a medium gear.
Background
The involute enveloping worm drive is used as a multi-tooth line/point contact drive mechanism, and has the advantages of large drive ratio, stable drive, small noise impact, adjustable backlash and the like.
The involute enveloping worm drive mainly has two types at present, namely constant tooth thickness involute gear enveloping worm drive and variable tooth thickness involute gear enveloping worm drive, and the drive mechanism is widely applied to the national emerging industry and strategic deployment fields of aerospace, strategic equipment, intelligent manufacturing, wind power heat energy and the like, and has important research significance and application value in the precise heavy-load drive field.
At present, the involute toroidal worm is mainly formed by turning, but cannot be formed by precise grinding, so that the machining precision and efficiency are greatly reduced. The gear hobbing is widely applied to worm gear tooth surface processing in worm drive as a high-efficiency gear processing method, and the technology is relatively mature, and the processing precision and efficiency are high. Considering the complex worm tooth surface of the involute toroidal worm drive, a hobbing method and a hob design for the involute toroidal worm tooth surface are not related at present.
Therefore, in order to solve the above problems, a method for confirming a gear hobbing cutter for an involute enveloping worm tooth surface based on a medium gear is needed to realize high-precision gear hobbing molding processing of the involute enveloping worm.
Disclosure of Invention
In view of the above, the invention provides a method for confirming a hobbing cutter for the tooth surface of an involute enveloping worm based on a medium gear, which realizes high-precision hobbing molding processing of the involute enveloping worm.
The invention relates to a method for confirming a hobbing cutter for an involute surface enveloping ring surface worm tooth surface based on a medium gear, which comprises the following steps:
s1: determining an involute surface enveloping endless screw of a target workpiece;
s2: determining a transmission pair formed by conjugate engagement of the involute enveloping worm and the external engagement involute cylindrical gear, and further determining the external engagement involute cylindrical gear;
s3: generating an involute intermediate gear with zero thickness based on the tooth surface of the external meshing involute cylindrical gear in the step S2, wherein the involute intermediate 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 the tooth surface parameters of the external meshing involute cylindrical gear;
s4: determining an involute surface enveloping drum worm in conjugate internal engagement with the involute medium-based gear;
s5: and forming the gear hobbing cutter based on the rear angle of the tooth crest machined by the involute wrap drum worm and the chip flute.
Further, the method for determining the external engagement involute cylindrical gear in the step S2 includes the steps of:
s2-1: establishing a transmission auxiliary space standard sigma u (O u -x u ,y u ,z u )、σ v (O v -x v ,y v ,z v )、σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein O is u 、O v 、O 1 And O 2 Respectively is emptySpacer sigma u 、σ v 、σ 1 Sum sigma 2 Origin of coordinates, (x) u ,y u ,z u )、(x v ,y v ,z v )、(x 1 ,y 1 ,z 1 ) And (x) 2 ,y 2 ,z 2 ) Respectively the space standard sigma u 、σ v 、σ 1 Sum sigma 2 Is defined by the coordinate axes of three directions;
space fixing standard sigma u (O u -x u ,y u ,z u ) The initial position of the external meshing involute cylindrical gear is that the bottom vector is (i) u ,j u ,k u ) Wherein i is u 、j u And k u Respectively coordinate axes x u 、y u And z u Is defined by the three vectors of (a);
space fixing standard sigma v (O v -x v ,y v ,z v ) The initial position of the involute enveloping toroidal worm is that the bottom vector is (i) v ,j v ,k v ) Wherein i is v 、j v And k v Respectively coordinate axes x v 、y v And z v Is defined by the three vectors of (a);
space motion standard sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Fixedly connected with the external meshing involute cylindrical gear and wound around z 1 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 1 ,j 1 ,k 1 ) Wherein i is 1 、j 1 And k 1 Respectively coordinate axes x 1 、y 1 And z 1 Is defined by the three vectors of (a);
space motion standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) Fixedly connected with the involute toroidal worm and wound around z 2 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 2 ,j 2 ,k 2 ) Wherein i is 2 、j 2 And k 2 Respectively coordinate axes x 2 、y 2 And z 2 Is defined by the three vectors of (a);
s2-2: based on the space differential geometry and the gear meshing principle, calculating various parameters of the external meshing involute cylindrical gear;
in space moving frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) The left tooth surface equation of the external meshing involute cylindrical gear is obtainedThe following are provided:
obtaining the right tooth surface equation of the external engagement involute cylindrical gearThe following are provided:
wherein: τ and θ are tooth surface parameters of the external meshing involute cylindrical gear; delta is the base circle half angle of the external meshing involute cylindrical gear; alpha t The end face pressure angle of the external meshing involute cylindrical gear is alpha, and the end face pressure angles of the tooth surfaces on the left side and the right side are alpha respectively if the external meshing involute cylindrical gear is a variable tooth thickness involute cylindrical gear tL And alpha tR ；The base circle radius of the external meshing involute cylindrical gear is +.A base circle radius of tooth surfaces on the left side and the right side is +.>And->Beta is the helix angle of the external meshing involute cylindrical gear, and if the external meshing involute cylindrical gear is a variable tooth thickness involute cylindrical gear, the helix angles of the tooth surfaces on the left side and the right side are respectively beta L And beta R ；
Determining tooth top of external engagement involute cylindrical gearHigh tooth root->Tooth tip clearance->The method comprises the following steps:
wherein: r is (r) II The indexing circle radius of the external meshing involute cylindrical gear; m is m n Is the normal modulus;is the normal tooth top coefficient; x is x n Normal deflection coefficient; />Normal roof clearance coefficient; z is Z 1 The tooth number of the external meshing involute cylindrical gear is the tooth number;
external engagement involute cylindrical gear tooth top circle radiusAnd root circle radius +>The method comprises the following steps:
according to the gear meshing principle, the left tooth surface of the involute toroidal worm can be obtained in the space movement standard frame sigma through coordinate transformation and bottom vector conversion 2 (O 2 -x 2 ,y 2 ,z 2 ) The tooth surface equation of (a) is:
involute toroidal worm right tooth surface spatial movement standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The tooth surface equation of (a) is:
wherein:and->The meshing functions of the left tooth surface and the right tooth surface of the involute enveloping worm are respectively; />And->The relative speeds of the left tooth surface and the right tooth surface of the involute enveloping worm are respectively; />And->Normal vectors of left and right tooth surfaces of the external meshing involute cylindrical gear respectively; m is M 21 For space movement frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) A bottom vector transformation matrix between, and:
wherein:i 21 ＝1/i 12 ＝Z 1 /Z 2 ，Z 1 for the number of teeth, Z, of the external engagement involute cylindrical gear 2 The number of heads of the toroidal worm is the involute envelope;
addendum arc radius of involute enveloping wormAnd root radius>The method comprises the following steps:
further, in step S3, the internal tooth flank equation and the external tooth flank equation of the involute intermediate gear are the same as those of the external tooth involute cylindrical gear, that is:
the thickness of the external meshing involute cylindrical gear is zero so that the internal tooth top circle radius of the corresponding internal meshing involute cylindrical gearAnd root circle radius +>The method comprises the following steps:
defining that the addendum circle of the involute intermediate gear is coincident with the root circular arc of the involute enveloping worm, and the root circle of the involute intermediate gear is smaller than or equal to the root circle of the involute cylindrical gear, then the addendum circle radius of the involute intermediate gear is coincidentAnd root circle radius +>The method comprises the following steps:
further, in step S4, the method for determining the involute wrap drum worm includes the steps of:
s4-1, establishing a space fixing calibration frame sigma u (O u -x u ,y u ,z u )、σ w (O w -x w ,y w ,z w )、σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) Wherein O is u 、O w 、O 1 And O 3 Respectively the space standard sigma u 、σ w 、σ 1 Sum sigma 3 Origin of coordinates, (x) u ,y u ,z u )、(x w ,y w ,z w )、(x 1 ,y 1 ,z 1 ) And (x) 3 ,y 3 ,z 3 ) Respectively the space standard sigma u 、σ w 、σ 1 Sum sigma 3 Is defined by the coordinate axes of three directions;
σ u (O u -x u ,y u ,z u ) Is the initial position of the inner meshing involute cylindrical gear, which is the initial position of the outer meshing involute cylindrical gearPosition coincidence, its bottom vector is (i) u ,j u ,k u ) Wherein i is u 、j u And k u Respectively coordinate axes x u 、y u And z u Is defined by the three vectors of (a);
σ w (O w -x w ,y w ,z w ) The initial position of the drum worm is enveloped by the involute surface, and the bottom vector is (i) w ,j w ,k w ) Wherein i is w 、j w And k w Respectively coordinate axes x w 、y w And z w Is defined by the three vectors of (a);
σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Fixedly connected with the inner-meshed involute cylindrical gear and winds around z 1 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 1 ,j 1 ,k 1 ) Wherein i is 1 、j 1 And k 1 Respectively coordinate axes x 1 、y 1 And z 1 Is defined by the three vectors of (a);
σ 3 (O 3 -x 3 ,y 3 ,z 3 ) Fixedly connected with the involute winding drum worm and wound around z 3 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 3 ,j 3 ,k 3 ) Wherein i is 3 、j 3 And k 3 Respectively coordinate axes x 3 、y 3 And z 3 Is defined by the three vectors of (a);
s4-2, according to the gear meshing principle, the left tooth surface of the involute wrap drum worm can be obtained in the space movement standard frame sigma through coordinate transformation and bottom vector conversion 3 (O 3 -x 3 ,y 3 ,z 3 ) The tooth surface equation of (a) is:
involute wrap drum worm right tooth surface in space movement standard frame sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) The tooth surface equation of (a) is:
wherein:and->The meshing functions of the left tooth surface and the right tooth surface of the involute winding drum worm are respectively; />And->The relative speeds of the left tooth surface and the right tooth surface of the involute winding drum worm are respectively; />And->Normal vectors of left and right tooth surfaces of the inner meshing involute cylindrical gear respectively; m is M 31 For space movement frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) A bottom vector transformation matrix between, and:
wherein:i 31 ＝1/i 13 ＝Z 1 /Z 3 ，Z 3 the number of heads of the drum worm is the involute envelope, and Z 3 ＝Z 2 。
Further, in step S5, the addendum circle radius and the dedendum arc radius of the drum hob are defined based on the involute winding drum worm:
defining that the top arc of the drum-shaped hob coincides with the top arc of the involute medium gear and the top arc of the drum-shaped hob coincides with the root circle of the involute medium gear, then the radius of the top arc of the drum-shaped hobAnd root radius>The method comprises the following steps:
the invention has the beneficial effects that:
the method provided by the invention can be used for tooth surface forming processing of the enveloping worm of the constant-tooth-thickness involute gear and the enveloping worm of the variable-tooth-thickness involute gear; the method can also be used for tooth surface forming processing of the enveloping worm of the equal-tooth-thickness planar gear and the enveloping worm of the variable-tooth-thickness planar gear; the involute toroidal worm can be formed by precise grinding, and the machining precision and efficiency can be effectively improved.
Drawings
The invention is further described below with reference to the drawings and examples.
FIG. 1 is a schematic diagram of an involute toroidal worm structure;
FIG. 2 is a schematic diagram of an involute toroidal worm drive configuration;
FIG. 3 is a schematic illustration of an involute enveloping drum worm drive configuration;
FIG. 4 is a schematic illustration of an involute intermediate gear configuration;
FIG. 5 is a schematic illustration of an involute idler gear enveloping worm drive transmission;
FIG. 6 is a schematic illustration of an involute intermediate gear enveloping worm gear drive configuration;
FIG. 7 is a schematic diagram of the meshing relationship between an involute intermediate gear and a toroidal worm, a drum worm;
FIG. 8 is a schematic diagram of the meshing relationship of a toroidal worm and a crowned worm;
FIG. 9 is a schematic diagram of an involute intermediate gear enveloping drum hob;
FIG. 10 is a schematic diagram of a drum hob hobbing generating involute toroidal worm structure;
FIG. 11 is a schematic diagram of a generic situation generalized to an intersection angle of axes not equal to zero;
Detailed Description
As shown in the figure: the involute surface enveloping worm tooth surface gear hobbing cutter confirmation method based on the medium gear comprises the following steps:
s1: determining an involute surface enveloping endless screw of a target workpiece; as shown in fig. 1, the main parameters of the involute enveloping worm 1 comprise a central axis 11 of the enveloping worm, a root arc 12 of the enveloping worm, a tip arc 13 of the enveloping worm and a right tooth surface 14 of the enveloping worm, wherein the left tooth surface of the enveloping worm is not marked to be opposite to the right tooth surface;
s2: determining a transmission pair formed by conjugate engagement of the involute enveloping worm and the external engagement involute cylindrical gear, and further determining the external engagement involute cylindrical gear;
after the target piece is determined, basic parameters of the involute toroidal worm transmission pair of the target workpiece are determined, wherein the basic parameters comprise a center distance a and a transmission ratio i 21 Number of worm heads Z 2 Normal modulus m n Normal pressure angle alpha n And the helix angle beta, if the tooth thickness is changed into the involute cylindrical gear, the helix angles of the tooth surfaces at the left side and the right side are respectively as follows: beta L And beta R And thereby calculate the face pressure angle alpha t If the tooth thickness is changed, the pressure angles of the end faces of the tooth surfaces at the left side and the right side are respectively as follows: alpha tL And alpha tR Radius of base circleIf the tooth thickness is changed, the base circle radius of the tooth surface at the left side and the right side is respectively as follows: />And->And determining parameters such as the tooth top height, the tooth root height, the tooth top clearance, the tooth root excessive fillet radius and the like of the involute toroidal worm transmission pair.
S3: generating an involute intermediate gear with zero thickness based on the tooth surface of the external-meshing involute cylindrical gear in the step S2, the involute intermediate gear having internal teeth and external teeth with the same tooth surface parameters as those of the external-meshing involute cylindrical gear;
generating a sub-body of the involute intermediate gear by taking the external meshing involute cylindrical gear as a parent body, wherein the involute intermediate gear is a sheet body with zero thickness, the tooth surface parameters of the sub-body involute intermediate gear are completely the same as those of the parent body external meshing involute cylindrical gear, the tooth top and tooth root heights of the sub-body involute intermediate gear are determined by the tooth top arc radius and the tooth root arc radius of an involute enveloping worm of a target processing body, the tooth top circle of the specific involute intermediate gear is coincident with the tooth root arc of the involute enveloping worm, and the tooth root circle of the involute intermediate gear is smaller than or equal to the tooth root circle of the involute cylindrical gear;
s4: determining an involute surface enveloping drum worm in conjugate internal engagement with the involute medium-based gear;
the involute intermediate gear moves according to complete dual generation, and an involute envelope drum worm is enveloped through internal engagement;
s5: and forming the gear hobbing cutter based on the rear angle of the tooth crest machined by the involute wrap drum worm and the chip flute.
Enveloping the drum-shaped hob based on the involute surface enveloping drum-shaped worm and establishing a hob tooth surface mathematical formula, and simultaneously forming a chip flute and a tooth top relief angle on the drum-shaped hob;
based on a drum hob mathematical model, establishing an accurate involute intermediate gear enveloping drum hob three-dimensional model through three-dimensional modeling software;
based on the established three-dimensional model of the drum hob, the drum hob formed by enveloping the involute medium gear is precisely formed and processed through equipment such as a five-axis machining center;
based on the processed drum hob entity, the hob is arranged on a gear hobbing machine tool, and the hob and a blank body of the involute enveloping worm are subjected to relative rotation according to a certain transmission ratio to generate and process tooth surfaces on two sides of the involute enveloping worm.
In this embodiment, the method for determining the external meshing involute cylindrical gear in step S2 includes the following steps:
as shown in fig. 2, the involute toroidal worm 1 is in conjugate engagement with the external engagement involute cylindrical gear 2:
s2-1: establishing a transmission auxiliary space standard sigma u (O u -x u ,y u ,z u )、σ v (O v -x v ,y v ,z v )、σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein O is u 、O v 、O 1 And O 2 Respectively the space standard sigma u 、σ v 、σ 1 Sum sigma 2 Origin of coordinates, (x) u ,y u ,z u )、(x v ,y v ,z v )、(x 1 ,y 1 ,z 1 ) And (x) 2 ,y 2 ,z 2 ) Respectively the space standard sigma u 、σ v 、σ 1 Sum sigma 2 Is defined by the coordinate axes of three directions;
space fixing standard sigma u (O u -x u ,y u ,z u ) The initial position of the external meshing involute cylindrical gear is that the bottom vector is (i) u ,j u ,k u ) Wherein i is u 、j u And k u Respectively coordinate axes x u 、y u And z u Is defined by the three vectors of (a);
space fixing standard sigma v (O v -x v ,y v ,z v ) The initial position of the involute enveloping toroidal worm is that the bottom vector is (i) v ,j v ,k v ) Wherein i is v 、j v And k v Respectively coordinate axes x v 、y v And z v Is defined by the three vectors of (a);
space motion standard sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Fixedly connected with the external meshing involute cylindrical gear and wound around z 1 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 1 ,j 1 ,k 1 ) Wherein i is 1 、j 1 And k 1 Respectively coordinate axes x 1 、y 1 And z 1 Is defined by the three vectors of (a);
space motion standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) Fixedly connected with the involute toroidal worm and wound around z 2 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 2 ,j 2 ,k 2 ) Wherein i is 2 、j 2 And k 2 Respectively coordinate axes x 2 、y 2 And z 2 Is defined by the three vectors of (a);
in the example, the center distance between the external meshing involute cylindrical gear and the involute toroidal worm is a, and the axial intersection angle is 0;
s2-2: based on the space differential geometry and the gear meshing principle, calculating various parameters of the external meshing involute cylindrical gear;
in space moving frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) The left tooth surface equation of the external meshing involute cylindrical gear is obtainedThe following are provided:
obtaining the right tooth surface equation of the external engagement involute cylindrical gearThe following are provided:
wherein: τ and θ are tooth surface parameters of the external meshing involute cylindrical gear; delta is the base circle half angle of the external meshing involute cylindrical gear; alpha t The end face pressure angle of the external meshing involute cylindrical gear is alpha, and the end face pressure angles of the tooth surfaces on the left side and the right side are alpha respectively if the external meshing involute cylindrical gear is a variable tooth thickness involute cylindrical gear tL And alpha tR ；The base circle radius of the external meshing involute cylindrical gear is +.A base circle radius of tooth surfaces on the left side and the right side is +.>And->Beta is the helix angle of the external meshing involute cylindrical gear, and if the external meshing involute cylindrical gear is a variable tooth thickness involute cylindrical gear, the helix angles of the tooth surfaces on the left side and the right side are respectively beta L And beta R ；
Determining tooth top of external engagement involute cylindrical gearHigh tooth root->Tooth tip clearance->The method comprises the following steps:
wherein: r is (r) II The indexing circle radius of the external meshing involute cylindrical gear; m is m n Is the normal modulus;is the normal tooth top coefficient; x is x n Normal deflection coefficient; />Normal roof clearance coefficient; z is Z 1 The tooth number of the external meshing involute cylindrical gear is the tooth number;
external engagement involute cylindrical gear tooth top circle radiusAnd root circle radius +>The method comprises the following steps:
according to the gear meshing principle, the left tooth surface of the involute toroidal worm can be obtained in the space movement standard frame sigma through coordinate transformation and bottom vector conversion 2 (O 2 -x 2 ,y 2 ,z 2 ) The tooth surface equation of (a) is:
the right tooth surface of the involute toroidal worm is arranged onSpace motion standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The tooth surface equation of (a) is:
wherein:and->The meshing functions of the left tooth surface and the right tooth surface of the involute enveloping worm are respectively; />And->The relative speeds of the left tooth surface and the right tooth surface of the involute enveloping worm are respectively; />And->Normal vectors of left and right tooth surfaces of the external meshing involute cylindrical gear respectively; m is M 21 For space movement frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) A bottom vector transformation matrix between, and:
wherein:i 21 ＝1/i 12 ＝Z 1 /Z 2 ，Z 1 is an external engaged involuteTooth number of cylindrical gear, Z 2 The number of heads of the toroidal worm is the involute envelope;
addendum arc radius of involute enveloping wormAnd root radius>The method comprises the following steps:
in this embodiment, as shown in fig. 4, an involute intermediate gear with zero thickness 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 intermediate gear 5 in fig. 3; the main parameters of the involute intermediate gear 5 include an intermediate gear central shaft 51, intermediate gear right tooth surfaces 52, intermediate gear left tooth surfaces 53, intermediate gear root circles 54 and intermediate gear tooth tip circles 55.
In step S3, the internal tooth flank equation and the external tooth flank equation of the involute intermediate gear are the same as those of the external tooth involute cylindrical gear, namely:
the thickness of the external meshing involute cylindrical gear is zero so that the internal tooth top circle radius of the corresponding internal meshing involute cylindrical gearAnd root circle radius +>The method comprises the following steps:
defining that the addendum circle of the involute intermediate gear is coincident with the root circular arc of the involute enveloping worm, and the root circle of the involute intermediate gear is smaller than or equal to the root circle of the involute cylindrical gear, then the addendum circle radius of the involute intermediate gear is coincidentAnd root circle radius +>The method comprises the following steps:
further, in step S4, the method of determining the involute wrap drum worm 4 includes the steps of:
for ease of discussion, fig. 3 is taken as an example, as shown in fig. 3,
s4-1, establishing a space fixing calibration frame sigma u (O u -x u ,y u ,z u )、σ w (O w -x w ,y w ,z w )、σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) Wherein O is u 、O w 、O 1 And O 3 Respectively the space standard sigma u 、σ w 、σ 1 Sum sigma 3 Origin of coordinates, (x) u ,y u ,z u )、(x w ,y w ,z w )、(x 1 ,y 1 ,z 1 ) And (x) 3 ,y 3 ,z 3 ) Respectively the space standard sigma u 、σ w 、σ 1 Sum sigma 3 Is defined by the coordinate axes of three directions;
σ u (O u -x u ,y u ,z u ) Is the initial position of the inner meshing involute cylindrical gear, which is matched with the outer meshing involute cylindrical gearThe initial positions of the wheels coincide with each other and the bottom vector thereof is (i) u ,j u ,k u ) Wherein i is u 、j u And k u Respectively coordinate axes x u 、y u And z u Is defined by the three vectors of (a);
σ w (O w -x w ,y w ,z w ) The initial position of the drum worm is enveloped by the involute surface, and the bottom vector is (i) w ,j w ,k w ) Wherein i is w 、j w And k w Respectively coordinate axes x w 、y w And z w Is defined by the three vectors of (a);
σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Fixedly connected with the inner-meshed involute cylindrical gear and winds around z 1 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 1 ,j 1 ,k 1 ) Wherein i is 1 、j 1 And k 1 Respectively coordinate axes x 1 、y 1 And z 1 Is defined by the three vectors of (a);
σ 3 (O 3 -x 3 ,y 3 ,z 3 ) Fixedly connected with the involute winding drum worm and wound around z 3 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 3 ,j 3 ,k 3 ) Wherein i is 3 、j 3 And k 3 Respectively coordinate axes x 3 、y 3 And z 3 Is defined by the three vectors of (a);
in this example, the center-to-center distance between the ring gear and the progressive-to-face enveloping crown worm is b and the angle of intersection is 0.
S4-2, according to the gear meshing principle, the left tooth surface of the involute wrap drum worm can be obtained in the space movement standard frame sigma through coordinate transformation and bottom vector conversion 3 (O 3 -x 3 ,y 3 ,z 3 ) The tooth surface equation of (a) is:
involute wrap drum worm right tooth surface in space movement standard frame sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) The tooth surface equation of (a) is:
wherein:and->The meshing functions of the left tooth surface and the right tooth surface of the involute winding drum worm are respectively; />And->The relative speeds of the left tooth surface and the right tooth surface of the involute winding drum worm are respectively; />And->Normal vectors of left and right tooth surfaces of the inner meshing involute cylindrical gear respectively; m is M 31 For space movement frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) A bottom vector transformation matrix between, and:
wherein:i 31 ＝1/i 13 ＝Z 1 /Z 3 ，Z 3 the number of heads of the drum worm is the involute envelope, and Z 3 ＝Z 2 。
In this embodiment, in step S5, the radius of the addendum circle and the radius of the dedendum arc of the drum hob are defined based on the involute winding drum worm:
considering that the involute intermediate gear 5 is a sub-body stripped from the external meshing involute cylindrical gear 2 serving as a parent body, the meshing relationship between the involute intermediate gear and the involute toroidal worm 1 is completely identical to the meshing relationship between the external meshing involute cylindrical gear 2 and the involute toroidal worm 1, and the spatial meshing relationship and the standard frame are shown in fig. 5; in the same principle, the involute intermediate gear is actually a sub-body stripped from the inner meshing involute cylindrical tooth 3 serving as a parent body, and the meshing relationship between the involute wrap drum worm 4 is completely identical to that between the inner meshing involute cylindrical gear 3 and the involute wrap drum worm 4, and the spatial meshing relationship and the standard frame are shown in fig. 6.
The meshing relationship between the involute intermediate gear and the involute enveloping worm and between the involute intermediate gear and the involute enveloping worm can be obtained as shown in fig. 7 by fusing the involute intermediate gear enveloping worm transmission in fig. 5 and the involute intermediate gear enveloping drum worm transmission in fig. 6 under the same space frame; the involute intermediate gear and the involute enveloping worm and the involute intermediate gear and the involute enveloping worm are in complete conjugate meshing relationship, and in view of the fact that the involute intermediate gear is a face gear with zero thickness, according to the lambda theorem, after the involute intermediate gear without thickness is removed, the involute enveloping worm and the involute enveloping worm are in equivalent conjugate meshing relationship, as shown in fig. 8, the wheelbase between the involute enveloping worm and the involute enveloping worm is c.
Defining that the addendum arc of drum-shaped hob is coincided with the addendum circle of involute intermediate gearThe arc of the tooth top of the drum hob is overlapped with the root circle of the involute intermediate gear, so that the arc radius of the tooth top of the drum hobAnd root radius>The method comprises the following steps:
in this embodiment, as shown in fig. 9, the drum hob 6 obtained by performing chip flute grooving and tooth top relief treatment on the above-defined involute enveloping worm is effectively defined by main parameters of the drum hob, namely, a left tooth surface 61, a right tooth surface, a tooth top relief angle 62, a tooth top arc 63, a tooth root arc 64, a chip flute 65, a central hole 66, a central shaft 67 and a key slot 68 of the drum hob.
According to the relation, in the process of generating the hobbing process of the toroidal worm blank by the drum hob to envelope the involute surface enveloping toroidal worm, the central axis of the drum hob is parallel to the central axis of the toroidal worm blank and has the same axial intersection angle with the involute intermediate gear, the axial intersection angle in the embodiment of the invention is 0, and the rotational angular speeds of the drum hob and the toroidal worm blank are equal and opposite in direction, as shown in fig. 10. At the same time, the toroidal worm blank has only feed cutting movement rotating around its own central axis, except for the drum hob around its own central axis z 1 In addition to the main cutting movement of rotation, there is also a movement along x 1 Feed cutting motion with linear motion of the shaft.
The drum hob obtained by taking the drum worm as a prototype is subjected to conjugate motion with a processed blank body through enveloping motion, so that continuous indexing generating processing of the involute enveloping 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 toroidal worm can be known as follows: the modulus, the pressure angle and the helix angle of the medium gear are the same; the number of heads is equal to that of the drum hob; the axis intersection angle between the drum hob and the intermediate gear is equal to the axis intersection angle between the drum hob and the intermediate gear. In the processing process, the installation parameters and the conjugation relation between the drum hob and the processed toroidal worm blank are required to be satisfied:
wherein: a is the center distance between the external meshing involute cylindrical gear and the involute toroidal worm; b is the center distance between the inner meshing involute cylindrical gear and the involute surface enveloping drum worm (drum hob); c is the center distance between the involute enveloping worm and the involute enveloping drum worm (drum hob); c 0 An initial wheelbase between the drum hob and the toroidal worm blank; toroidal worm blank corner(rotational speed omega) 2 ) Direction and drum hob angle->(rotation speed omega) 3 ) The directions are opposite; v 0 The speed of the cutting motion V for drum hob feed.
In this embodiment, the axial intersection angle λ between the involute toroidal worm and the external meshing involute cylindrical gear is zero, and the axial intersection angle λ between the corresponding intermediate gear and the drum hob is zero, so that the present invention is not limited to the transmission with zero axial intersection angle, but can be applied to the transmission with non-zero axial intersection angle, i.e. λ not equal to 0, where the general situation of non-zero axial intersection angle is shown in fig. 11.
Finally, it is noted that the above embodiments are only for illustrating the technical solution of the present invention and not for limiting the same, and 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 and equivalents may be made thereto without departing from the spirit and scope of the technical solution of the present invention, which is intended to be covered by the scope of the claims of the present invention.
Claims (5)
1. A method for confirming a hobbing cutter for the tooth surface of an involute enveloping 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 endless screw of a target workpiece;
s2: determining a transmission pair formed by conjugate engagement of the involute enveloping worm and the external engagement involute cylindrical gear, and further determining the external engagement involute cylindrical gear;
s3: generating an involute intermediate gear with zero thickness based on the tooth surface of the external-meshing involute cylindrical gear in the step S2, the involute intermediate gear having internal teeth and external teeth with the same tooth surface parameters as those of the external-meshing involute cylindrical gear;
s4: determining an involute surface enveloping drum worm in conjugate internal engagement with the involute medium-based gear;
s5: and forming the gear hobbing cutter based on the rear angle of the tooth crest machined by the involute wrap drum worm and the chip flute.
2. The method for confirming the hobbing cutter for the involute surface enveloping worm tooth surface based on the medium gear according to claim 1, wherein the method comprises the following steps:
the method for determining the external meshing involute cylindrical gear in the step S2 comprises the following steps:
s2-1: establishing a transmission auxiliary space standard sigma u (O u -x u ,y u ,z u )、σ v (O v -x v ,y v ,z v )、σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The method comprises the steps of carrying out a first treatment on the surface of the Wherein O is u 、O v 、O 1 And O 2 Respectively the space standard sigma u 、σ v 、σ 1 Sum sigma 2 Origin of coordinates, (x) u ,y u ,z u )、(x v ,y v ,z v )、(x 1 ,y 1 ,z 1 ) And (x) 2 ,y 2 ,z 2 ) Respectively the space standard sigma u 、σ v 、σ 1 Sum sigma 2 Is defined by the coordinate axes of three directions;
space fixing standard sigma u (O u -x u ,y u ,z u ) The initial position of the external meshing involute cylindrical gear is that the bottom vector is (i) u ,j u ,k u ) Wherein i is u 、j u And k u Respectively coordinate axes x u 、y u And z u Is defined by the three vectors of (a);
space fixing standard sigma v (O v -x v ,y v ,z v ) The initial position of the involute enveloping toroidal worm is that the bottom vector is (i) v ,j v ,k v ) Wherein i is v 、j v And k v Respectively coordinate axes x v 、y v And z v Is defined by the three vectors of (a);
space motion standard sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Fixedly connected with the external meshing involute cylindrical gear and wound around z 1 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 1 ,j 1 ,k 1 ) Wherein i is 1 、j 1 And k 1 Respectively coordinate axes x 1 、y 1 And z 1 Is defined by the three vectors of (a);
space motion standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) Fixedly connected with the involute toroidal worm and wound around z 2 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 2 ,j 2 ,k 2 ) Wherein i is 2 、j 2 And k 2 Respectively coordinate axes x 2 、y 2 And z 2 Is defined by the three vectors of (a);
s2-2: based on the space differential geometry and the gear meshing principle, calculating various parameters of the external meshing involute cylindrical gear;
in space moving frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) The left tooth surface equation of the external meshing involute cylindrical gear is obtainedThe following are provided:
obtaining the right tooth surface equation of the external engagement involute cylindrical gearThe following are provided:
wherein: τ and θ are tooth surface parameters of the external meshing involute cylindrical gear; delta is the base circle half angle of the external meshing involute cylindrical gear; alpha t The end face pressure angle of the external meshing involute cylindrical gear is alpha, and the end face pressure angles of the tooth surfaces on the left side and the right side are alpha respectively if the external meshing involute cylindrical gear is a variable tooth thickness involute cylindrical gear tL And alpha tR ；The base circle radius of the external meshing involute cylindrical gear is +.A base circle radius of tooth surfaces on the left side and the right side is +.>And->Beta is the helix angle of the external meshing involute cylindrical gear, if the helix angle is changed into teethThe helix angles of the tooth surfaces on the left side and the right side of the thick involute cylindrical gear are respectively beta L And beta R ；
Determining tooth top of external engagement involute cylindrical gearHigh tooth root->Tooth tip clearance->The method comprises the following steps:
wherein: r is (r) II The indexing circle radius of the external meshing involute cylindrical gear; m is m n Is the normal modulus;is the normal tooth top coefficient; x is x n Normal deflection coefficient; />Normal roof clearance coefficient; z is Z 1 The tooth number of the external meshing involute cylindrical gear is the tooth number;
external engagement involute cylindrical gear tooth top circle radiusAnd root circle radius +>The method comprises the following steps:
according to the gear meshing principle, the left side of the involute toroidal worm can be obtained through coordinate transformation and bottom vector transformationTooth surface in space movement standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The tooth surface equation of (a) is:
involute toroidal worm right tooth surface spatial movement standard sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) The tooth surface equation of (a) is:
wherein:and->The meshing functions of the left tooth surface and the right tooth surface of the involute enveloping worm are respectively; />And->The relative speeds of the left tooth surface and the right tooth surface of the involute enveloping worm are respectively; />And->Normal vectors of left and right tooth surfaces of the external meshing involute cylindrical gear respectively; m is M 21 For space movement frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 2 (O 2 -x 2 ,y 2 ,z 2 ) A bottom vector transformation matrix between, and:
wherein:i 21 ＝1/i 12 ＝Z 1 /Z 2 ，Z 1 for the number of teeth, Z, of the external engagement involute cylindrical gear 2 The number of heads of the toroidal worm is the involute envelope;
addendum arc radius of involute enveloping wormAnd root radius>The method comprises the following steps:
3. the method for confirming the hobbing cutter for the involute surface enveloping worm tooth surface based on the medium gear according to claim 2, characterized in that:
in step S3, the internal tooth flank equation and the external tooth flank equation of the involute intermediate gear are the same as those of the external tooth involute cylindrical gear, namely:
the thickness of the external meshing involute cylindrical gear is zero so that the internal tooth top circle radius of the corresponding internal meshing involute cylindrical gearAnd root circle radius +>The method comprises the following steps:
defining that the addendum circle of the involute intermediate gear is coincident with the root circular arc of the involute enveloping worm, and the root circle of the involute intermediate gear is smaller than or equal to the root circle of the involute cylindrical gear, then the addendum circle radius of the involute intermediate gear is coincidentAnd root circle radius +>The method comprises the following steps:
。
4. the method for confirming the hobbing cutter for the involute surface enveloping worm tooth surface based on the medium gear according to claim 3, wherein the method comprises the following steps:
in step S4, the method for determining the involute wrap drum worm includes the following steps:
s4-1, establishing a space fixing calibration frame sigma u (O u -x u ,y u ,z u )、σ w (O w -x w ,y w ,z w )、σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) Wherein O is u 、O w 、O 1 And O 3 Respectively the space standard sigma u 、σ w 、σ 1 Sum sigma 3 Origin of coordinates, (x) u ,y u ,z u )、(x w ,y w ,z w )、(x 1 ,y 1 ,z 1 ) And (x) 3 ,y 3 ,z 3 ) Respectively the space standard sigma u 、σ w 、σ 1 Sum sigma 3 Is defined by the coordinate axes of three directions;
σ u (O u -x u ,y u ,z u ) The initial position of the inner meshing involute cylindrical gear is coincident with the initial position of the outer meshing involute cylindrical gear, and the bottom vector is (i) u ,j u ,k u ) Wherein i is u 、j u And k u Respectively coordinate axes x u 、y u And z u Is defined by the three vectors of (a);
σ w (O w -x w ,y w ,z w ) The initial position of the drum worm is enveloped by the involute surface, and the bottom vector is (i) w ,j w ,k w ) Wherein i is w 、j w And k w Respectively coordinate axes x w 、y w And z w Is defined by the three vectors of (a);
σ 1 (O 1 -x 1 ,y 1 ,z 1 ) Fixedly connected with the inner-meshed involute cylindrical gear and winds around z 1 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 1 ,j 1 ,k 1 ) Wherein i is 1 、j 1 And k 1 Respectively coordinate axes x 1 、y 1 And z 1 Is defined by the three vectors of (a);
σ 3 (O 3 -x 3 ,y 3 ,z 3 ) Fixedly connected with the involute winding drum worm and wound around z 3 The shaft rotates and a certain instantaneous rotational displacement isIts bottom vector is (i) 3 ,j 3 ,k 3 ) Wherein i is 3 、j 3 And k 3 Respectively coordinate axes x 3 、y 3 And z 3 Is defined by the three vectors of (a);
s4-2, according to the gear meshing principle, the left tooth surface of the involute wrap drum worm can be obtained in the space movement standard frame sigma through coordinate transformation and bottom vector conversion 3 (O 3 -x 3 ,y 3 ,z 3 ) The tooth surface equation of (a) is:
involute wrap drum worm right tooth surface in space movement standard frame sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) The tooth surface equation of (a) is:
wherein:and->The meshing functions of the left tooth surface and the right tooth surface of the involute winding drum worm are respectively; />And->The relative speeds of the left tooth surface and the right tooth surface of the involute winding drum worm are respectively; />And->Normal vectors of left and right tooth surfaces of the inner meshing involute cylindrical gear respectively; m is M 31 For space movement frame sigma 1 (O 1 -x 1 ,y 1 ,z 1 ) Sum sigma 3 (O 3 -x 3 ,y 3 ,z 3 ) A bottom vector transformation matrix between, and:
wherein:i 31 ＝1/i 13 ＝Z 1 /Z 3 ，Z 3 the number of heads of the drum worm is the involute envelope, and Z 3 ＝Z 2 。
5. The method for confirming the hobbing cutter for the involute surface enveloping worm tooth surface based on the medium gear according to claim 4, wherein the method comprises the following steps:
in the step S5, the tooth top radius and the tooth root arc radius of the drum hob are defined based on the involute winding drum worm:
defining that the top arc of the drum-shaped hob coincides with the top arc of the involute medium gear and the top arc of the drum-shaped hob coincides with the root circle of the involute medium gear, then the radius of the top arc of the drum-shaped hobAnd root radius>The method comprises the following steps:
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