CN106292531B - Algorithm for calculating profile boundary of ZN1 worm disc-shaped forming cutter - Google Patents
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Abstract
The invention discloses an algorithm for calculating the boundary of a profile of a disc-shaped forming cutter for processing a ZN1 worm, which comprises the following steps: (1) iterative calculation is carried out according to basic parameters of the worm, the transcendental equation set is solved to obtain the distance A from the axis of the worm to the virtual point of the cutter, and the radius r 'of the virtual base circle of the worm'bAnd a worm virtual base cylinder lead angle γ'b(ii) a (2) Establishing a worm helicoid expression and establishing a contact line expression (x, y, z) with a cutter; (3) establishing the expression (x) of the corresponding contact line of the cutterG,yG,zG) And tool section expression (R)G,ZG) (ii) a (4) Establishing a relationship for R based on the Lagrange multiplier methodGAn expression of a conditional extremum; (5) solving the maximum radius r of the ZN1 worm by iteratively solving the simultaneous transcendental equations of (2), (3) and (4)maxAnd a minimum radius rminDetermining the maximum value R of the tool section coordinatesGmaxAnd a minimum value RGmax. The invention has the beneficial effects that: the profile boundary of the disc-shaped forming tool is accurately calculated, so that the ZN1 worm with a large lead angle can be accurately machined.
Description
Technical Field
The invention relates to the technical field of ZN1 worm machining, in particular to an algorithm for calculating the boundary of a disc-shaped forming cutter profile of a ZN1 worm.
Background
In a mechanical transmission speed reducing or indexing mechanism, a ZN1 worm gear is commonly used, and a ZN1 worm is a normal straight profile worm. The ZN1 worm tooth profile finish machining is generally carried out by adopting a disc-shaped forming tool, and the disc-shaped forming tool comprises a forming grinding wheel and a forming milling cutter. The "forming" of a disc-shaped forming tool, i.e., deriving a tool profile corresponding to the ZN1 worm tooth profile based on the meshing principle.
Most of common ZN1 worms are worms with small lead angles and are machined by a disc-shaped forming tool, the tooth form of the ZN1 worm is determined according to the profile of the disc-shaped forming tool, when the common disc-shaped forming tool machines a ZN1 worm with a small lead angle, the tool does not interfere with the tooth crest of the ZN1 worm, the machined tooth form is completed, the machining of the ZN1 worm can be met, when the ZN1 worm with a large lead angle is machined, the tool interferes with the tooth crest of the ZN1 worm, so that a part of the tooth crest of the ZN1 worm is cut off, the machined worm is no longer the ZN1 worm, the worm is not matched with the worm wheel in meshing, for example, when the ZN1 with a 30-degree lead angle is machined by the common disc-shaped forming tool, one-third tooth form of the tooth face of the finally machined worm cannot form the ZN1 shape, and the technical requirements of production and machining cannot be met.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide an algorithm capable of accurately calculating the boundary of the profile of a disc-shaped forming cutter for machining a ZN1 worm.
The purpose of the invention is realized by the following technical scheme: an algorithm for calculating the boundary of a profile for a ZN1 worm disc-like forming tool comprising the steps of:
an algorithm for calculating the boundary of a profile for a ZN1 worm disc-shaped forming tool, comprising the steps of:
s1, according to basic parameters of a known ZN1 worm: normal modulus mnNumber of heads z1Normal pressure angle αnLead angle gamma, normal tooth thickness snDiameter d of top circleaRoot diameter dfAnd the maximum radius R of the toolGaSelecting the thickness s of the turning tool0nTurning tool normal pressure angle α0nAs unknown parameters, determining the distance A expression from the worm axis to the virtual point on the turning tool, the virtual base radius rb' expression, virtual base cylinder lead angle γbThe expression of' is used;
s2, determining the turning tool thickness S according to the geometrical conditions of the ZN1 worm and the disc-shaped forming tool for machining the ZN1 worm0nAnd tool normal pressure angle α0nThe constraint formula (2);
s3, turning tool normal pressure angle α0nTurning tool thickness s0nIs calculated by selecting the normal pressure angle αnAs the normal pressure angle α of the turning tool0nSelecting the normal tooth thickness s as the initial value ofnAs the thickness s of the turning tool0nThe initial value of (A) is to select the normal pressure angle α of the turning tool0nInitial value of (d), turning tool thickness s0nIs substituted into the expression of step S1 and the constraint of step S2 to iterate, thereby solvingDistance A from worm axis to virtual point on turning tool, and virtual base radius rb', virtual base cylinder lead angle gammab' turning tool normal pressure angle α0nAnd the thickness s of the turning tool0nA value of (d);
s4, selecting the central point of the bottom of the ZN1 worm as the origin of coordinates to establish an original coordinate system according to the numerical values in the steps S1-S3, and rotating the original coordinate system by theta(1)Z is moved(1)Establishing a new coordinate system, thereby obtaining an expression of coordinates (x, y, z) of a moving point on a spiral surface of the ZN1 worm in the new coordinate system;
s5, determining a contact line expression of the disc-shaped forming tool in contact with the ZN1 worm according to the contact condition of the disc-shaped forming tool and the ZN1 worm contact line and the expression of the coordinates of the operating point on the spiral surface of the worm;
s6, according to the geometrical relation, the coordinate (x, y, z) on the contact line of the ZN1 worm and the disc-shaped forming tool is converted from the ZN1 worm coordinate system to the disc-shaped forming tool coordinate system, and then the point coordinate of the contact line is (x) coordinateG,yG,zG) An expression of the truncated coordinates of the disc-shaped forming tool;
s7, establishing R based on Lagrange multiplier methodGAn expression of a minimum condition;
s8, establishing an transcendental equation set in a simultaneous way according to the expressions in the steps S4-S7, and selecting the parameter variable of the position of the action point on the ZN1 worm generatrixAnd the angle theta of the moving point with the parameter number u on the ZN1 worm generatrix rotating around the Z axis on the end section shape is approximately equal to 0 as an initial value, and the maximum radius r of the ZN1 worm tooth shape is obtained by an iteration methodmaxAnd a minimum radius rminDetermining the maximum value R of the tool section coordinatesGmaxAnd a minimum value RGmax；
S9, using the discal forming cutter section expression as constraint condition, using RGmin≤RG≤RGmaxFor the range, make a reference to (R)G,ZG) And connecting each point by a spline curve to form the profile of the disc-shaped cutter with the boundary.
The normal pressure angle α of the turning tool0nTurning tool thickness s0nThe iterative calculation of (2) is to select the normal pressure angle α firstnAs the normal pressure angle α of the turning tool0nSelecting the normal tooth thickness s as the initial value ofnAs the thickness s of the turning tool0nThe initial value of (A) is to select the normal pressure angle α of the turning tool0nInitial value of (d), turning tool thickness s0nSubstituting the initial values into the expression of step S1 to respectively solve: distance A from worm axis to virtual point on turning tool, and virtual base radius rb' and virtual base cylindrical lead angle gammab'; then the distance A from the worm axis to the virtual point on the turning tool and the virtual base circle radius rb' and virtual base cylindrical lead angle gammab' into the constraint equation in step S2, if the lathe tool thickness S0nConstrained or turning tool normal pressure angle α0nIf the vertical error precision of the left side and the right side of the equal sign of one of the constraint formulas is not within the range of 1E-15, the normal pressure angle α of the turning tool is reselected0nValue of (d) and turning tool thickness s0nValue of (d), turning tool normal pressure angle α0nValue of αnNear value selection, turning tool thickness s0nValue of snSelecting the values nearby, and repeating the process in the step S3 until the turning tool thickness S0nConstrained or turning tool normal pressure angle α0nIf the errors of the numerical values of the left side and the right side of the constraint equal sign are within the range of 1E-15, the iteration is stopped, and the distance A from the worm axis to the virtual point on the turning tool and the virtual base circle radius r are determinedb', virtual base cylinder lead angle gammab' turning tool normal pressure angle α0nAnd the thickness s of the turning tool0nThe value of (c).
The invention has the following advantages: the profile of the disc cutter can be accurately obtained by accurately calculating the equation curve of a certain point on the profile of the disc cutter.
Drawings
FIG. 1 is a schematic diagram of the calculations of the present invention.
Detailed Description
The invention will be further described with reference to the accompanying drawings, but the scope of the invention is not limited to the following.
A method for calculating the boundary of a profile of a disc-shaped forming tool for machining a ZN1 worm, comprising the steps of:
s1, according to basic parameters of a known ZN1 worm: normal modulus mnNumber of heads z1Normal pressure angle αnLead angle gamma, normal tooth thickness snDiameter d of top circleaRoot diameter dfAnd the maximum radius R of the toolGaSelecting the thickness s of the turning tool0nTurning tool normal pressure angle α0nEquation (1) for determining the distance A from the worm axis to a virtual point on the tool, the virtual base radius r, as an unknown parameterb' s (2), virtual base cylinder lead angle γb' the formula (3),
wherein r ism: the worm is divided into a radius of a circle,
s2, determining the turning tool thickness S according to the geometrical conditions of the ZN1 worm and the disc-shaped forming tool for machining the ZN1 worm0nAnd tool normal pressure angle α0nThe constraint formula of (1):
wherein p iszu: the unit lead of the worm is that,and p isxThe axial pitch of the worm is the axial pitch of the worm,
s3, turning tool normal pressure angle α0nTurning tool thickness s0nIs calculated by selecting the normal pressure angle αnAs the normal pressure angle α of the turning tool0nSelecting the normal tooth thickness s as the initial value ofnAs the thickness s of the turning tool0nThe initial value of (A) is to select the normal pressure angle α of the turning tool0nInitial value of (d), turning tool thickness s0nSubstituting the initial value of (a) into the equations (1), (2) and (3) in step S1 to solve: distance A from worm axis to virtual point on turning tool, and virtual base radius rb' and virtual base cylindrical lead angle gammab'; then the distance A from the worm axis to the virtual point on the turning tool and the virtual base circle radius rb' and virtual base cylindrical lead angle gammabThe value of' is substituted into the equations (4) and (5) in the step S2, and if the error accuracy of the left and right vertical sides of the equal sign of one equation in the equations (4) or (5) is not in the range of 1E-15, the turning tool normal pressure angle α is reselected0nValue of (d) and turning tool thickness s0nValue of (d), turning tool normal pressure angle α0nValue of αnNear value selection, turning tool thickness s0nValue of snSelecting values nearby, repeatedly calculating the process in S3 until the errors of the values on the left side and the right side of the equal sign of the formula (4) and the formula (5) are within the range of 1E-15, stopping iteration, and oscillating from an initial value point to an accurate point in an iteration mode to determine the distance A between the worm axis and a virtual point on the turning tool and the radius r of a virtual base circleb', virtual base cylinder lead angle gammab' turning tool normal pressure angle α0nAnd the thickness s of the turning tool0nA value of (d);
s4, selecting the central point of the bottom of the ZN1 worm as the origin of coordinates to establish an original coordinate system according to the numerical values in the steps S1-S3, and rotating the original coordinate system by theta(1)Z is moved(1)Establishing a new coordinate system, thereby obtaining an expression of the coordinates (x, y, z) of the moving point on the spiral surface of the ZN1 worm in the new coordinate system,
wherein, u: a parameter number representing the position of the action point on the ZN1 worm generatrix,
θ: indicating the angle of the moving point of the parameter u on the ZN1 worm generatrix rotating around the Z axis on the end section,
θ(1): representing the angle through which the original coordinates were rotated to the new coordinates,
z(1): a displacement representing the movement of the original coordinates to the new coordinates,
and,
r: represents the radius of any point on the contact line;
s5, determining the contact line expression of the disc-shaped forming tool in contact with the ZN1 worm according to the contact condition of the disc-shaped forming tool and the ZN1 worm contact line and the expression of the coordinates of the operating point on the spiral surface of the worm:
wherein,for the contact line condition, the relationship between u and θ is defined by
F (u, θ) is framed as 0, given u in the range of the tooth surface, a corresponding (x, y, z) is obtained, and is the only correspondence,
a0: indicating the distance of the disc cutter centerline from the worm centerline,
nz: a normal vector representing the x-direction,
ny: a normal vector representing the y-direction,
nz: a normal vector representing the z direction;
s6, according to the geometrical relation, the coordinate (x, y, z) on the contact line of the ZN1 worm and the disc-shaped forming tool is converted from the ZN1 worm coordinate system to the disc-shaped forming tool coordinate system, and then the point coordinate of the contact line is (x) coordinateG,yG,zG) Section coordinates (R) of the disc-shaped forming toolG,ZG) The expression (c) of (a),
s7, obtaining the partial derivatives of x, y and z to u according to the equation set in the step S5, and obtaining the partial derivatives of x, y and z to theta:
thus determining the normal vector:
thus, the partial derivative of the normal vector to u is obtained, and the partial derivative of the normal vector to θ is obtained:
thus, we derive the partial derivative of u and θ for F (u, θ) ═ 0:
using the system of equations in step S6, x is derivedG，yG，RGPartial derivatives of u give xG，yG，RGPartial derivatives for θ:
s8, establishing R based on Lagrange multiplier methodGExpression for the minimum condition:
RGminimum value condition
And then determining the constraint conditions:
constraint one: when F (u, theta) is 0 andthen, the maximum value r of the worm radius is solvedmaxAnd minimum value of tool section coordinate RGminIf r isa＞rmaxThen worm rmaxIs not ZN1 tooth form;
constraint two: when constraint one has no solution, the condition of constraint two is used, and the condition of constraint two is that F (u, theta) is 0 and R isG＝RGfTo solve for rmaxAnd RGminTo determine the maximum value r of ZN1 worm radiusmaxAnd minimum value of tool section coordinate RGmin；
Constraint condition three: when F (u, theta) is 0 and RG＝RGaWhen R isGaFor large radius of the cutter, the minimum value of ZN1 worm radius is solvedrminAnd maximum value R of cutting tool section coordinateGmax；
S9, selectingSubstituting theta ≈ 0 as an initial value into the formula from step S4 to step S8 for iteration, and solving the minimum value R of the truncated coordinate of the tool according to the constraint conditions from one to threeGminAnd ZN1 minimum value of worm radius rminAnd ZN1 maximum radius of worm rmaxFurther solve the maximum value R of the cutting tool section coordinateGmaxAnd minimum value of tool section coordinate RGmin(ii) a S10, andis a constraint, wherein: rGmin≤RG≤RGmaxMake a reference to (R)G,ZG) And connecting each point by a spline curve to form the profile of the disc-shaped cutter with the boundary.
The foregoing is illustrative of the preferred embodiments of this invention, and it is to be understood that the invention is not limited to the precise form disclosed herein and that various other combinations, modifications, and environments may be resorted to, falling within the scope of the concept as disclosed herein, either as described above or as apparent to those skilled in the relevant art. And that modifications and variations may be effected by those skilled in the art without departing from the spirit and scope of the invention as defined by the appended claims.
Claims (2)
1. An algorithm for calculating the boundary of a profile for a ZN1 worm disc-like forming tool, comprising the steps of:
s1, according to basic parameters of a known ZN1 worm: normal modulus mnNumber of heads z1Normal pressure angle αnLead angle gamma, normal tooth thickness snDiameter d of top circleaRoot diameter dfAnd the maximum radius R of the toolGaSelecting the thickness s of the turning tool0nTurning tool normal pressure angle α0nDetermining the distance from the worm axis to a virtual point on the tool as an unknown parameterExpression from A, virtual base radius r'bExpression of (1), virtual base cylinder lead angle γ'bThe expression of (1);
wherein r ism: the worm is divided into a radius of a circle,
s2, determining the turning tool thickness S according to the geometrical conditions of the ZN1 worm and the disc-shaped forming tool for machining the ZN1 worm0nAnd tool normal pressure angle α0nThe constraint formula (2);
wherein p iszu: the unit lead of the worm is that,and p isxThe axial pitch of the worm is the axial pitch of the worm,
s3, turning tool normal pressure angle α0nTurning tool thickness s0nIs calculated by iteration ofThe normal pressure angle α is selectednAs the normal pressure angle α of the turning tool0nSelecting the normal tooth thickness s as the initial value ofnAs the thickness s of the turning tool0nThe initial value of (A) is to select the normal pressure angle α of the turning tool0nInitial value of (d), turning tool thickness s0nThe initial value of (d) is substituted into the expression of step S1 and the constraint equation of step S2, and the distance A from the worm axis to the virtual point on the turning tool and the virtual base radius r 'are solved'bAnd a virtual base cylinder lead angle γ'bTurning tool normal pressure angle α0nAnd the thickness s of the turning tool0nA value of (d);
s4, selecting the central point of the bottom of the ZN1 worm as the origin of coordinates to establish a coordinate system according to the numerical values in the steps S1-S3, and rotating the original coordinate system by theta(1)Z is moved(1)Establishing a new coordinate system, thereby obtaining an expression of coordinates (x, y, z) of a moving point on a spiral surface of the ZN1 worm in the new coordinate system; the expression for coordinates (x, y, z) is:
wherein, u: a parameter number representing the position of the action point on the ZN1 worm generatrix,
θ: indicating the angle of the moving point of the parameter u on the ZN1 worm generatrix rotating around the Z axis on the end section,
θ(1): representing the angle through which the original coordinates were rotated to the new coordinates,
z(1): a displacement representing the movement of the original coordinates to the new coordinates,
and,
r: represents the radius of any point on the contact line;
s5, determining a contact line expression of the disc-shaped forming tool in contact with the ZN1 worm according to the contact condition of the disc-shaped forming tool and the ZN1 worm contact line and the expression of the coordinates of the operating point on the spiral surface of the worm; the contact line expression of the disc-shaped forming tool in contact with the ZN1 worm is:
wherein,for the contact line condition, the relationship between u and θ is framed by F (u, θ) being 0, given u in the range of the tooth surface a corresponding (x, y, z) is obtained, and is a unique correspondence,
a0: indicating the distance of the disc cutter centerline from the worm centerline,
nz: a normal vector representing the x-direction,
ny: a normal vector representing the y-direction,
nz: a normal vector representing the z direction;
s6, according to the geometrical relation, the coordinate (x, y, z) on the contact line of the ZN1 worm and the disc-shaped forming tool is converted from the ZN1 worm coordinate system to the disc-shaped forming tool coordinate system, and then the point coordinate of the contact line is (x) coordinateG,yG,zG) An expression of the truncated coordinates of the disc-shaped forming tool; section coordinates (R) of disc-shaped forming toolG,ZG) The expression (c) of (a),
from the system of equations in step S5, we derive the partial derivatives of x, y, z versus u, we derive the partial derivatives of x, y, z versus θ:
thus determining the normal vector:
thus, the partial derivative of the normal vector to u is obtained, and the partial derivative of the normal vector to θ is obtained:
thus, we derive the partial derivative of u and θ for F (u, θ) ═ 0:
using the system of equations in step S6, x is derivedG，yG，RGPartial derivatives of u give xG，yG，RGPartial derivatives for θ:
s7, establishing R based on Lagrange multiplier methodGExpression of minimum value condition, RG: one of the section coordinates, R, of the disk-shaped toolGValue range: rGmin≤RG≤RGaWherein R isGaIs the maximum radius of the disc cutter;
RGminimum value condition
And then determining the constraint conditions:
constraint one: when F (u, theta) is 0 andthen, the maximum value r of the worm radius is solvedmaxAnd minimum value of tool section coordinate RGminIf r isa＞rmaxThen worm rmaxIs not ZN1 tooth form;
constraint two: when constraint one has no solution, the condition of constraint two is used, and the condition of constraint two is that F (u, theta) is 0 and R isG＝RGfTo solve for rmaxAnd RGminTo determine the maximum value r of ZN1 worm radiusmaxAnd minimum value of tool section coordinate RGmin；
Constraint condition three: when F (u, theta) is 0 and RG＝RGaWhen R isGaSolving the minimum value r of ZN1 worm radius for large radius of the cutterminAnd maximum value R of cutting tool section coordinateGmax；
S8, selecting the parameter number of the position of the action point on the ZN1 worm generatrixTheta ≈ 0 as an initial value is substituted into those in steps S4 to S8Iterating the formula, and solving the minimum value R of the cutting coordinate of the cutter according to the constraint condition I to the constraint condition IIIGminAnd ZN1 minimum value of worm radius rminAnd ZN1 maximum radius of worm rmaxFurther solve the maximum value R of the cutting tool section coordinateGmaxAnd minimum value of tool section coordinate RGmin；
S9, using the discal forming cutter section expression as constraint condition, using RGmin≤RG≤RGmaxFor the range, make a reference to (R)G,ZG) And connecting each point by a spline curve to form the profile of the disc-shaped cutter with the boundary.
2. The algorithm for calculating the boundary of the profile of a disc-shaped forming tool for machining ZN1 worm according to claim 1, wherein the normal turning tool pressure angle α is0nTurning tool thickness s0nThe iterative calculation of (2) is to select the normal pressure angle α firstnAs the normal pressure angle α of the turning tool0nSelecting the normal tooth thickness s as the initial value ofnAs the thickness s of the turning tool0nThe initial value of (A) is to select the normal pressure angle α of the turning tool0nInitial value of (d), turning tool thickness s0nSubstituting the initial values into the expression of step S1 to respectively solve: distance A from worm axis to virtual point on turning tool, and virtual base radius r'bAnd a virtual base cylinder lead angle γ'b(ii) a The distance A from the worm axis to the virtual point on the turning tool and the virtual base radius r'bAnd a virtual base cylinder lead angle γ'bThe value of (b) is substituted into the constraint formula in step S2, if the lathe tool thickness S0nConstrained or turning tool normal pressure angle α0nIf the error precision of the numerical values on the left side and the right side of the equal sign of one of the constraint formulas is not within the range of 1E-15, the normal pressure angle α of the turning tool is reselected0nValue of (d) and turning tool thickness s0nValue of (d), turning tool normal pressure angle α0nValue of αnNear value selection, turning tool thickness s0nValue of snSelecting the values nearby, and repeating the process in the step S3 until the turning tool thickness S0nAnd the tool normal pressure angle α0nAnd (4) if the errors of the numerical values of the left side and the rear side of the bound equal sign are within the range of 1E-15, stopping iteration, and determining the distance A from the worm axis to the virtual point on the turning tool and the virtual base circle radius r'bAnd a virtual base cylinder lead angle γ'bTurning tool normal pressure angle α0nAnd the thickness s of the turning tool0nThe value of (c).
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| CN112797119B (en) * | 2018-06-29 | 2023-02-07 | 吉林大学 | Machining method of multi-head double-lead line contact offset worm |
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| CN1962186A (en) * | 2006-11-24 | 2007-05-16 | 陕西法士特齿轮有限责任公司 | Method for trimming shaving cutter tooth profile using error compensation method |
| CN103093054A (en) * | 2013-01-29 | 2013-05-08 | 福州大学 | Modeling method of plane secondary envelope torus worm-drive worm gear hob tooth profile |
| CN103942396A (en) * | 2014-04-30 | 2014-07-23 | 武汉理工大学 | Helical-gear precise modeling method involving tooth alignment errors |
| CN104156948A (en) * | 2014-07-25 | 2014-11-19 | 中国航空综合技术研究所 | Method for assessing surface profile tolerance of tooth surface of face gear |
| CN105138748A (en) * | 2015-08-10 | 2015-12-09 | 清华大学 | Design method of face gear pair |
| CN105631131A (en) * | 2015-12-29 | 2016-06-01 | 重庆大学 | Form grinding axial modification error compensation method |
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| CN106292531A (en) | 2017-01-04 |
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