CN106292531B - Algorithm for calculating profile boundary of ZN1 worm disc-shaped forming cutter - Google Patents

Abstract

The present invention discloses an algorithm for calculating the contour boundary of a ZN1 worm disc-shaped forming tool, comprising the following steps: (1) performing iterative calculation according to basic parameters of the worm, solving a group of transcendental equations to obtain a distance A from the worm axis to a virtual point of the tool, a radius _r′b of a virtual base circle of the worm, and a lead angle _γ′b of a virtual base cylinder of the worm; (2) establishing an expression for a helical surface of the worm, and an expression for a contact line with the tool (x, y, z); (3) establishing an expression for a contact line corresponding to the tool ( _xG , _yG , _zG ) and an expression for a cutter shape ( _RG , _ZG ); (4) establishing an expression for an extreme value of the _RG condition based on a Lagrange multiplier method; (5) iteratively solving a group of transcendental equations connected by (2), (3), and (4) to obtain a maximum radius _rmax and a minimum radius _rmin of the ZN1 worm, and obtain a maximum value _RGmax and a minimum value _RGmax of the cutter shape coordinates. The beneficial effect of the present invention is that the contour boundary of the disc-shaped forming tool is accurately calculated, so that the ZN1 worm with a large lead angle can be accurately processed.

Description

An Algorithm for Calculating the Profile Boundary of ZN1 Worm Disc Forming Tool

technical field

The invention relates to the technical field of ZN1 worm processing, in particular to an algorithm for calculating and processing the contour boundary of a ZN1 worm disc-shaped forming tool.

Background technique

In the deceleration or indexing mechanism of mechanical transmission, ZN1 worm gear device is often used, and ZN1 worm is the normal straight profile worm. For the finishing of the ZN1 worm tooth profile, the disc-shaped forming tool is generally used for processing, and the disc-shaped forming tool includes a forming grinding wheel and a forming milling cutter. The "forming" of the disc-shaped forming tool is to derive the tool profile corresponding to the ZN1 worm tooth profile according to the meshing principle.

Most of the common ZN1 worms are small lead angle worms, which are processed by a disc-shaped forming tool. The tooth shape of the ZN1 worm is determined according to the profile of the disc-shaped forming tool. When the ZN1 worm with a small angle is used, the tool does not interfere with the tooth tip of the ZN1 worm, and the processed tooth shape is completed, which can meet the processing of the ZN1 worm. The top interferes, so that a part of the tooth top of the ZN1 worm is cut off, and the worm processed in this way is no longer a ZN1 worm, resulting in a mismatch between the worm and the turbine mesh. When the ZN1 worm is 30°, nearly one-third of the tooth profile of the final processed worm cannot form the ZN1 shape, which cannot meet the technical requirements of production and processing.

SUMMARY OF THE INVENTION

The purpose of the present invention is to overcome the shortcomings of the prior art, and to provide an algorithm that can accurately calculate and process the contour boundary of a ZN1 worm disc-shaped forming tool.

The object of the present invention is achieved through the following technical solutions: a kind of algorithm for calculating and processing the ZN1 worm disc-shaped forming tool profile boundary, comprising the following steps:

An algorithm for calculating the contour boundary of a ZN1 worm disc-shaped forming tool, which has the following steps:

S1. According to the known basic parameters of ZN1 worm: normal modulus m _n , head number _{z 1} , normal pressure angle α _n , lead angle γ, normal tooth thickness s _n , tip circle diameter da , root circle The diameter d _f and the maximum radius of the tool R _Ga , the thickness of the turning tool s _0n and the normal pressure angle α _0n of the turning tool are selected as unknown parameters, and the expression of the distance A from the worm axis to the virtual point on the turning tool is determined, and the virtual base circle radius r The expression of _b ′, the expression of virtual base cylinder lead angle γ _b ′;

S2. According to the geometric conditions of the ZN1 worm and the disc-shaped forming tool for processing the ZN1 worm, determine the constraint formula of the turning tool thickness s _0n and the turning tool normal pressure angle α _0n ;

S3. Iterative calculation of turning tool normal pressure angle α _0n and turning tool thickness s _0n , select normal pressure angle α _n as the initial value of turning tool normal pressure angle α _0n , and select normal tooth thickness s _n as turning tool The initial value of the thickness s _0n , the selected initial value of the normal pressure angle α _0n of the turning tool and the initial value of the turning tool thickness s _0n are substituted into the expression of step S1 and the constraint formula of step S2 for iteration, so as to solve Values of the distance A from the worm axis to the virtual point on the turning tool, the virtual base circle radius r _b ′, the virtual base cylinder lead angle γ _b ′, the normal pressure angle of the turning tool α _0n and the thickness of the turning tool s _0n ;

S4. From the values in steps S1 to S3, select the center point of the bottom of the ZN1 worm as the coordinate origin to establish the original coordinate system, and then rotate the original coordinate system by θ ₍₁₎ and move z ₍₁₎ to establish a new coordinate system, thereby The expression of the coordinates (x, y, z) of the moving point on the ZN1 worm helical surface in the new coordinate system can be obtained;

S5. According to the contact conditions of the contact line between the disk-shaped forming tool and the ZN1 worm and the expression of the moving point coordinates on the worm helical surface, determine the contact line expression for the contact between the disk-shaped forming tool and the ZN1 worm;

S6. According to the geometric relationship, the coordinates (x, y, z) on the contact line where the ZN1 worm and the disc-shaped forming tool are in contact are transformed from the ZN1 worm coordinate system to the disc-shaped forming tool coordinate system, then the point coordinates of the contact line are (x _G , y _G , z _G ), the expression of the truncated coordinates of the disc forming tool;

S7. Based on the Lagrange multiplier method, an expression for the minimum value condition of R _G is established;

S8. According to the expressions in steps S4 to S7, a set of transcendental equations is simultaneously formed, and the parameters of the position of the moving point on the ZN1 worm generatrix are selected And the angle θ≈0 that the moving point with parameter u on the ZN1 worm generatrix rotates around the Z axis on the end truncation is taken as the initial value, through the iterative method, the maximum radius r _max and the minimum radius r of the ZN1 worm tooth profile are obtained. _min , find the maximum value R _Gmax and the minimum value R _Gmax of the tool truncation coordinates;

S9. Taking the truncation expression of the disc-shaped forming tool as the constraint condition, and taking R _Gmin ≤R _G ≤R _Gmax as the range, make discrete coordinate points about (R _G , Z _G ), and then connect each point with a spline curve That is, a disk-shaped tool profile with a boundary.

The iterative calculation of the normal pressure angle α _0n of the turning tool and the thickness s _0n of the turning tool is to first select the normal pressure angle α _n as the initial value of the normal pressure angle α _0n of the turning tool, and select the normal tooth thickness s _n . As the initial value of the thickness s _0n of the turning tool, substitute the selected initial value of the normal pressure angle α _0n of the turning tool and the initial value of the thickness s _0n of the turning tool into the expression of step S1 to solve respectively: worm axis to turning tool The distance A of the virtual point, the virtual base circle radius r _b ′ and the virtual base cylinder lead angle γ _b ′; then the distance A from the worm axis to the virtual point on the turning tool, the virtual base circle radius r _b ′ and the virtual base cylinder The value of the lead angle γ _b ′ is substituted into the constraint formula in step S2. If the constraint formula of the thickness of the turning tool s _0n or the constraint formula of the normal pressure angle of the turning tool α _0n has one of the constraint formulas, the equal signs on the left and right sides are vertical. If the straight error accuracy is not within the range of 1E-15, then reselect the value of the normal pressure angle of the turning tool α _0n and the value of the thickness of the turning tool s _0n , the normal pressure angle of the turning tool α _0n value is selected near the value of α _n , and the turning tool The value of the tool thickness s _0n is selected near the value of s _n , and the above process in S3 is repeated until the constraint formula of the turning tool thickness s _0n or the equal sign of the normal pressure angle α _0n of the turning tool is the error of the numerical value on the left side If all are within the range of 1E-15, the iteration stops, so as to determine the distance A from the worm axis to the virtual point on the turning tool, the virtual base circle radius r _b ′, the virtual base cylinder lead angle γ _b ′, the normal pressure angle of the turning tool Values of α _0n and turning tool thickness s _0n .

The invention has the following advantages: by accurately calculating the equation curve of a certain point on the profile of the disc cutter, the profile of the disc cutter can be accurately obtained.

Description of drawings

Figure 1 is a schematic diagram of the calculation of the present invention.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings, but the protection scope of the present invention is not limited to the following.

A method for calculating the contour boundary of a ZN1 worm disc-shaped forming tool, comprising the following steps:

S1. According to the known basic parameters of ZN1 worm: normal modulus m _n , head number _{z 1} , normal pressure angle α _n , lead angle γ, normal tooth thickness s _n , tip circle diameter da , root circle The diameter d _f and the maximum tool radius R _Ga , the thickness of the turning tool s _0n and the normal pressure angle α _0n of the turning tool are selected as unknown parameters, and the formula (1) of the distance A from the worm axis to the virtual point on the turning tool is determined. Formula (2) for circle radius r _b ′, formula (3) for virtual base cylinder lead angle γ _b ′,

Among them, r _m : worm indexing circle radius,

S2. According to the geometric conditions of ZN1 worm and disc-shaped forming tool for machining ZN1 worm, determine the constraint formula of turning tool thickness s _0n and turning tool normal pressure angle α _0n :

Among them, p _zu : worm unit lead, And p _x is the axial pitch of the worm,

S3. Iterative calculation of turning tool normal pressure angle α _0n and turning tool thickness s _0n , select normal pressure angle α _n as the initial value of turning tool normal pressure angle α _0n , and select normal tooth thickness s _n as turning tool The initial value of the thickness s _0n , substitute the selected initial value of the normal pressure angle α _0n of the turning tool and the initial value of the turning tool thickness s _0n into formulas (1), (2) and (3) in step S1, respectively Solve: the distance A from the worm axis to the virtual point on the turning tool, the virtual base circle radius r _b ′ and the virtual base cylinder lead angle γ _b ′; then the distance A from the worm axis to the virtual point on the turning tool, the virtual base circle The values of the radius r _b ' and the virtual base cylinder lead angle γ _b ' are substituted into the formulas (4) and (5) in step S2, if there is an equal sign in the formula (4) or (5) If the vertical error accuracy on both sides is not within the range of 1E-15, then reselect the value of the normal pressure angle α _0n of the turning tool and the value of the thickness s _0n of the turning tool, and the value of the normal pressure angle α _0n of the turning tool is near the value of α _n Select, the value of turning tool thickness s _0n is selected near the value of s _n , and the above process in S3 is repeated until the errors of the left and right sides of the equations (4) and (5) are within the range of 1E-15. Then the iteration stops, and iteratively oscillates from the initial value point to the precise point, so as to determine the distance A from the worm axis to the virtual point on the turning tool, the virtual base circle radius r _b ′, the virtual base cylinder lead angle γ _b ′, the turning The value of the normal pressure angle of the tool α _0n and the thickness of the turning tool s _0n ;

S4. From the values in steps S1 to S3, select the center point of the bottom of the ZN1 worm as the coordinate origin to establish the original coordinate system, and then rotate the original coordinate system by θ ₍₁₎ and move z ₍₁₎ to establish a new coordinate system, Thus, the expression of the coordinates (x, y, z) of the moving point on the ZN1 worm helical surface in the new coordinate system can be obtained,

Among them, u: the parameter indicating the position of the moving point on the ZN1 worm busbar,

θ: represents the angle that the moving point with parameter u on the ZN1 worm busbar rotates around the Z axis on the end section,

θ ₍₁₎ : represents the angle from the original coordinate to the new coordinate,

z ₍₁₎ : represents the displacement of the original coordinate to the new coordinate movement,

and,

r: represents the radius of any point on the contact line;

S5. According to the contact conditions of the contact line between the disk-shaped forming tool and the ZN1 worm and the expression of the moving point coordinates on the worm helical surface, determine the contact line expression for the contact between the disk-shaped forming tool and the ZN1 worm:

in, is the contact line condition, the relationship between u and θ is given by

F(u, θ)=0 framed, given u in the range of tooth surface, the corresponding (x, y, z) can be obtained, and it is the only correspondence,

a ₀ : Indicates the distance from the centerline of the disc cutter to the centerline of the worm,

n _z : the normal vector representing the x direction,

n _y : the normal vector representing the y direction,

n _z : represents the normal vector in the z direction;

S6. According to the geometric relationship, the coordinates (x, y, z) on the contact line where the ZN1 worm and the disc-shaped forming tool are in contact are transformed from the ZN1 worm coordinate system to the disc-shaped forming tool coordinate system, then the point coordinates of the contact line are (x _G , y _G , z _G ), the expression for the truncated coordinates (R _G , Z _G ) of the disc forming tool,

S7. According to the equation system in step S5, the partial derivatives of x, y, z to u are obtained, and the partial derivatives of x, y, z to θ are obtained:

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, the partial derivatives of F(u, θ)=0 to u and θ are obtained:

Using the equation system in step S6, the partial derivatives of x _G , y _G , and R _G to u are obtained, and the partial derivatives of x _G , y _G , and R _G to θ are obtained:

S8. Based on the Lagrange multiplier method, establish the expression about the minimum value condition of R _G :

R _G minimum condition

Then determine the constraints:

Constraint 1: When F(u, θ)=0 and When , the maximum value r _max of the worm radius and the minimum value R _Gmin of the truncation coordinate of the tool are solved. If ra > r _max , the upper part of the worm r _max is not _a ZN1 tooth profile;

Constraint 2: When Constraint 1 has no solution, use Constraint 2. Constraint 2 is F(u, θ)=0 and R _G =R _Gf , solve r _max and R _Gmin to determine the ZN1 worm radius The maximum value r _max and the minimum value of the tool truncation coordinate R _Gmin ;

Constraint 3: When F(u, θ)=0 and R _G = R _Ga , R _Ga is the large radius of the tool, and solve the minimum value r _min of the ZN1 worm radius and the maximum value of the tool truncation coordinate R _Gmax ;

S9. Select θ≈0 is used as the initial value to be substituted into the formula in steps S4 to S8 for iteration, and then the minimum value of the tool truncation coordinate R _Gmin and the minimum value of the ZN1 worm radius r _min and the maximum ZN1 worm radius are solved according to the constraint condition 1 to the constraint condition 3. value r _max , and then solve the maximum value of the tool truncation coordinate R _Gmax and the minimum value of the tool truncation coordinate R _Gmin ; S10, with is a constraint condition, in which: R _Gmin ≤ R _G ≤ R _Gmax , make discrete coordinate points about (R _G , Z _G ), and then connect each point with a spline curve is a disk-shaped tool profile with a boundary.

The above are only preferred embodiments of the present invention, and it should be understood that the present invention is not limited to the form disclosed herein, should not be construed as an exclusion of other embodiments, but may be used in various other combinations, modifications and environments, and Modifications can be made within the scope of the concepts described herein, from the above teachings or from skill or knowledge in the relevant field. However, modifications and changes made by those skilled in the art do not depart from the spirit and scope of the present invention, and should all fall within the protection scope of the appended claims of the present invention.

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 m_nNumber of heads z₁Normal pressure angle α_nLead angle gamma, normal tooth thickness s_nDiameter d of top circle_aRoot diameter d_fAnd the maximum radius R of the tool_GaSelecting the thickness s of the turning tool_0nTurning tool normal pressure angle α_0nDetermining 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 is_m: 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 worm_0nAnd tool normal pressure angle α_0nThe constraint formula (2);

wherein p is_zu: the unit lead of the worm is that,and p is_xThe axial pitch of the worm is the axial pitch of the worm,

s3, turning tool normal pressure angle α_0nTurning tool thickness s_0nIs calculated by iteration ofThe normal pressure angle α is selected_nAs the normal pressure angle α of the turning tool_0nSelecting the normal tooth thickness s as the initial value of_nAs the thickness s of the turning tool_0nThe initial value of (A) is to select the normal pressure angle α of the turning tool_0nInitial value of (d), turning tool thickness s_0nThe 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 α_0nAnd the thickness s of the turning tool_0nA 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,

n_z: a normal vector representing the x-direction,

n_y: a normal vector representing the y-direction,

n_z: 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) coordinate_G,y_G,z_G) An expression of the truncated coordinates of the disc-shaped forming tool; section coordinates (R) of disc-shaped forming tool_G,Z_G) 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 derived_G，y_G，R_GPartial derivatives of u give x_G，y_G，R_GPartial derivatives for θ:

s7, establishing R based on Lagrange multiplier method_GExpression of minimum value condition, R_G: one of the section coordinates, R, of the disk-shaped tool_GValue range: r_Gmin≤R_G≤R_GaWherein R is_GaIs the maximum radius of the disc cutter;

R_Gminimum 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 solved_maxAnd minimum value of tool section coordinate R_GminIf r is_a＞r_maxThen worm r_maxIs 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 is_G＝R_GfTo solve for r_maxAnd R_GminTo determine the maximum value r of ZN1 worm radius_maxAnd minimum value of tool section coordinate R_Gmin；

Constraint condition three: when F (u, theta) is 0 and R_G＝R_GaWhen R is_GaSolving the minimum value r of ZN1 worm radius for large radius of the cutter_minAnd maximum value R of cutting tool section coordinate_Gmax；

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 III_GminAnd ZN1 minimum value of worm radius r_minAnd ZN1 maximum radius of worm r_maxFurther solve the maximum value R of the cutting tool section coordinate_GmaxAnd minimum value of tool section coordinate R_Gmin；

S9, using the discal forming cutter section expression as constraint condition, using R_Gmin≤R_G≤R_GmaxFor the range, make a reference to (R)_G,Z_G) 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 is_0nTurning tool thickness s_0nThe iterative calculation of (2) is to select the normal pressure angle α first_nAs the normal pressure angle α of the turning tool_0nSelecting the normal tooth thickness s as the initial value of_nAs the thickness s of the turning tool_0nThe initial value of (A) is to select the normal pressure angle α of the turning tool_0nInitial value of (d), turning tool thickness s_0nSubstituting 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 S_0nConstrained or turning tool normal pressure angle α_0nIf 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 reselected_0nValue of (d) and turning tool thickness s_0nValue of (d), turning tool normal pressure angle α_0nValue of α_nNear value selection, turning tool thickness s_0nValue of s_nSelecting the values nearby, and repeating the process in the step S3 until the turning tool thickness S_0nAnd the tool normal pressure angle α_0nAnd (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 α_0nAnd the thickness s of the turning tool_0nThe value of (c).