CN114036701A - Method for accurately calculating maximum grinding wheel diameter of shaping herringbone gear shaping mill - Google Patents
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Abstract
The invention aims to provide a method for accurately calculating the maximum grinding wheel diameter of a shaping herringbone gear shaping mill, which comprises the following steps: establishing a gear tooth profile three-section parabolic modified curved surface model, and solving an involute function to obtain position vectors and normal vectors of tooth surfaces of left and right hand gears so as to obtain a mathematical model of the tooth surfaces of the herringbone gears; deducing an engagement equation of the grinding wheel and the workpiece through the position vector and the normal vector of the workpiece and the installation parameters of the grinding wheel, and further obtaining an equation of a contact line between the grinding wheel and the workpiece; the contact line is projected to the cross section of the grinding wheel in a rotating mode around the axis of the grinding wheel to obtain the cross section of the formed grinding wheel, and meanwhile the change of the center distance of the grinding wheel is considered, so that the bidirectional shaping of the herringbone gear is achieved; and calculating the maximum diameter of the grinding wheel according to the geometric condition of the grinding wheel when the grinding wheel grinds the tooth surface termination point. The method can accurately calculate the maximum grinding wheel diameter of the shaping mill of the shaping herringbone gear, solves the problems of experience and approximation of determining the grinding wheel diameter in the past engineering, and meets the requirement of selecting the grinding wheel diameter in high-precision shaping gear processing.
Description
Technical Field
The invention relates to a gear machining method, in particular to a gear machining parameter calculation method.
Background
A driving power gear transmission device of a ship mostly adopts herringbone gears, the herringbone gears are composed of two single bevel gears with larger helical angles (generally 25-45 degrees), so that axial force generated by the bevel gears can be counteracted, and a cutter pushing groove with a certain length is reserved between the two single bevel gears for the cutter retraction during gear machining. The gear is generally a high-precision carburized and quenched hard-tooth-surface gear, and the final machining process of the gear is gear grinding. The grinding mode can be divided into a plurality of types such as a butterfly double grinding wheel type, a conical grinding wheel type, a worm grinding wheel type, a forming grinding wheel type and the like according to the grinding wheel type, wherein the forming grinding wheel type grinding tooth has high machining precision and machining efficiency and is widely applied to gear grinding tooth finish machining. There are many factors that affect the machining accuracy of the formed gear grinding, such as the accuracy of the gear grinding machine, the degree of tooth surface distortion, the grinding wheel, etc. The selection of the maximum diameter of the grinding wheel is an important parameter in the gear forming mill, and the gear grinding precision is influenced. The diameter of the grinding wheel is not easy to be selected to be too small, and the diameter of the installation inner hole of the small grinding wheel is small, so that the rigidity of a grinding wheel shaft for fixing the grinding wheel is poor, and the grinding precision is reduced. But the diameter of the grinding wheel cannot be too large, and the phenomenon that the grinding wheel interferes with the gear teeth on the other side of the tool withdrawal groove is caused by the too large diameter of the grinding wheel. The diameter of the forming grinding wheel in the prior engineering is mostly determined by experience or is approximately determined by a simple drawing method, and the machining requirement of a high-precision shaping gear is difficult to adapt.
Disclosure of Invention
The invention aims to provide a method for accurately calculating the maximum grinding wheel diameter of a shaping herringbone gear shaping mill, which is used for calculating the maximum diameter of a grinding wheel according to the geometric conditions of grinding of the grinding wheel by establishing a bidirectional shaping mathematical model of the herringbone gear shaping and grinding method.
The purpose of the invention is realized as follows:
the invention discloses a method for accurately calculating the maximum grinding wheel diameter of a shaping herringbone gear shaping mill, which is characterized by comprising the following steps of:
(1) establishing a gear tooth profile three-section parabolic modified curved surface model, and solving an involute function to obtain position vectors and normal vectors of tooth surfaces of left and right hand gears so as to obtain a mathematical model of the tooth surfaces of the herringbone gears;
(2) deducing an engagement equation of the grinding wheel and the workpiece through the position vector and the normal vector of the workpiece and the installation parameters of the grinding wheel, and further obtaining an equation of a contact line between the grinding wheel and the workpiece; the contact line is projected to the cross section of the grinding wheel in a rotating mode around the axis of the grinding wheel to obtain the cross section of the formed grinding wheel, and meanwhile the change of the center distance of the grinding wheel is considered, so that the bidirectional shaping of the herringbone gear is achieved;
(3) and calculating the maximum diameter of the grinding wheel according to the geometric condition of the grinding wheel when the grinding wheel grinds the tooth surface termination point.
The present invention may further comprise:
1. the step (1) is specifically as follows:
a. according to the differential geometric tooth surface conjugate theory, the determined position vector and normal vector of the gear tooth surface are respectively
Wherein rb1 is a basic radius, μ 1 is a half angle corresponding to the width of a groove on a base circle, θ 1 is a flare angle of a point on an involute, a spiral parameter p1 is rb1tg λ b1, and λ b1 is a lead angle on a gear base circle; the superscript represents a right side tooth surface, the subscript represents a left side tooth surface, the position vectors and normal vectors of the left and right tooth surfaces of the left-handed gear are obtained through the method, the rotation direction of the gear teeth is changed, and a mathematical model of the other side tooth surface of the herringbone gear is obtained;
b. on a gear rotation projection plane, establishing a function expression of three-section parabola modification of the tooth profile according to a geometric relation:
in the formula I1Indicating root relief length, /)2Indicates the middle drum length,/3Representing addendum relief length, a1, a2 and a3 are parabolic coefficients of section I, b1 is a parabolic coefficient of section II, and c1, c2 and c3 are parabolic coefficients of section III; determining coefficients according to the maximum modification amount of each segment of parabola and the continuous tangency condition of adjacent parabolas at a connecting point, wherein the length parameter of the tooth profile on a rotating projection plane is (x12+ y12)1/2-rp1, and rp1 is the radius of a reference circle;
c. superposing the theoretical standard tooth surface and the modified curved surface to obtain the bit error rm of the tooth profile modified tooth surface, wherein the expression is
rm(u1,θ1)＝r1(u1,θ1)+n1(u1,θ1)δ(l)
The unit normal vector of the tooth profile modified tooth surface is
Wherein, the two tangent vectors of the modified tooth surface are respectively
2. The step (2) is specifically as follows:
a. deducing an engagement equation of the grinding wheel and the workpiece as
f(u1,θ1)＝(Ec-xm+p1 cotΣ)nz+
Ec cotΣny+zmnx＝0
In the formula, xm, ym, zm, nx, ny and nz are three components of a gear bit vector rm and a unit normal vector nm after tooth profile modification respectively; Σ is a grinding wheel mounting angle, Ec is the shortest distance between the grinding wheel spindle and the workpiece spindle, Ec is rp1+ rg, rp1 is the workpiece reference circle radius, rg is the grinding wheel radius;
b. deducing the contact line equation of the grinding wheel and the gear, and simultaneously solving the formula rm(u1,θ1)＝r1(u1,θ1)+n1(u1,θ1)δ(l)、
Determining the angle of rotation
Contact line L1 on the workpiece; converting the coordinate of the contact line L1 on the workpiece into the position vector and the normal vector of the grinding wheel contact line Lc through the coordinate transformation to Sc under the grinding wheel reference coordinate system
rc(u1,θ1)＝McfMf1r1(u1,θ1)
In the formula, the matrix Mf1 is a unitary matrix
c. Deducing a section equation of the formed grinding wheel, and projecting the contact line to the section of the grinding wheel in a rotating way around the axis of the grinding wheel to obtain the section equation of the formed grinding wheel
Making the contact line Lc on the grinding wheel make spiral motion around the axis of the workpiece, adopting the motion law of symmetrical parabola, and the expression is
In the formula, alpha p is a tooth parabola modification coefficient; z1 is the axial coordinate of the workpiece, the maximum dressing at the tooth tip is δ z, and the corresponding wheel center variation is
In the formula, beta b is a base circle helix angle, alpha n is a normal pressure angle, a tooth direction parabolic motion coefficient alpha p is determined, and after tooth profile and tooth direction bidirectional modification grinding, a small wheel tooth surface equation is expressed as
In the form of matrix
3. The step (3) is specifically as follows:
a. the limiting conditions for grinding the herringbone gear are as follows: taking a right-hand gear as an example, the grinding termination point on the right side of the tooth slot satisfies the following equation
In the formula, the first equation and the second equation represent the end point of the right tooth surface ground at the time, and the third equation is the contact line equation at the time; ra1 is addendum circle radius, B0 is tool withdrawal groove size, x ' 1, y ' 1 and z ' 1 are three components of bidirectional profile grinding tooth surface position vector respectively, and the corresponding contact line equation on the outer diameter of the grinding wheel is
Wherein rf1 is root circle radius, and when the grinding wheel grinds the termination point, the maximum grinding wheel radius is determined by the intersection of the track line formed by the rotation of the lowest point of the grinding wheel contact line around the grinding wheel axis and the tooth top circle of the tool withdrawal end face of the gear on the other side, and the geometrical condition is that
In the formula, a first equation represents that after tooth direction modification is considered, a grinding wheel contact line Lc performs spiral motion around the axis of a workpiece to obtain the track of the outer diameter of the grinding wheel; the second equation is the axial position relation of the outer diameter of the grinding wheel contacting the end surface of the tooth groove on the other side of the gear tooth; the third equation is the condition that the grinding wheel interferes with the gear teeth on the other side at the tooth tops;
(2) determining the maximum diameter of the grinding wheel: the maximum radius rgr of the grinding wheel when grinding the right side of the tooth space is obtained by the quasi-Newton method, and similarly, when grinding the left side of the tooth space, the formula is replaced by rf1
And (3) obtaining the maximum radius rgl of the grinding wheel when the right tooth surface end point of the grinding tooth socket is ground by using the ra1 in the middle without changing other equations, wherein the maximum diameter of the final grinding wheel is equal to
dg＝2max(rgr,rgl)。
The invention has the advantages that: the method can accurately calculate the maximum grinding wheel diameter of the shaping mill of the shaping herringbone gear, effectively solves the problems of experience and approximation of determining the grinding wheel diameter in the past engineering, and meets the requirement of selecting the grinding wheel diameter in high-precision shaping gear processing.
Drawings
FIG. 1 is an involute end profile;
FIG. 2 is a profile modification curve on a rotational projection plane;
FIG. 3 is a formed wheel grinding principle;
fig. 4 is a profile of a tooth profile.
Detailed Description
The invention will now be described in more detail by way of example with reference to the accompanying drawings in which:
the technical scheme adopted by the invention is as follows: a method for accurately designing the diameter of a maximum grinding wheel of a shaping herringbone gear shaping mill comprises the following steps:
1. the method comprises the following steps: and establishing a gear tooth profile three-section parabolic modified curved surface model. And solving an involute function to obtain position vectors and normal vectors of the tooth surfaces of the left and right hand gears so as to obtain a mathematical model of the tooth surfaces of the herringbone gears.
1.1, according to the differential geometric tooth surface conjugate theory, the determined position vector and normal vector of the gear tooth surface are respectively
In the formula, rb1Is a basic radius; mu.s1Is a half angle corresponding to the width of the tooth groove on the base circle; theta1The spread angle is the point on the involute; screw parameter p1＝rb1tgλb1，λb1The lead angle on the gear base circle; the superscript indicates the right flank and the subscript indicates the left flank. By analogy, the position vector and the normal vector of the left tooth surface and the right tooth surface of the left-handed gear can be obtained. And changing the rotation direction of the gear teeth to obtain a mathematical model of the tooth surface on the other side of the herringbone gear.
1.2, on a gear rotation projection plane, establishing a function expression of three-section tooth profile parabolic modification according to a geometric relation:
in the formula I1Indicating root relief length, /)2Indicating the intermediate drum length,/3Indicates the tooth tip relief length, a1、a2And a3Is the parabolic coefficient of the section I; b1Is a parabolic coefficient of the section II; c. C1、c2And c3Is a parabolic coefficient of the third section. Maximum modification according to each segment of parabolaThe shape quantity and the continuous tangency condition of adjacent parabolas at the connecting point determine the coefficient, and the length parameter l ═ x on the projection plane of rotation of the tooth profile1 2+y1 2)1/2-rp1，rp1Is the reference circle radius.
1.3, superposing the theoretical standard tooth surface and the modification curved surface to obtain the position error r of the tooth profile modification tooth surfacemThe expression is
rm(u1,θ1)＝r1(u1,θ1)+n1(u1,θ1)δ(l) (4)
The unit normal vector of the tooth profile modified tooth surface is
Wherein, the two tangent vectors of the modified tooth surface are respectively
2. Step two: deducing an engagement equation of the grinding wheel and the workpiece according to the position vector and the normal vector of the workpiece and the installation parameters of the grinding wheel, and further obtaining an equation of a contact line between the grinding wheel and the workpiece; and the contact line is projected to the cross section of the grinding wheel in a rotating way around the axis of the grinding wheel, so that the cross section of the formed grinding wheel is obtained. Meanwhile, the bidirectional dressing of the herringbone gear is realized by considering the change of the center distance of the grinding wheel.
2.1, deducing an engagement equation of the grinding wheel and the workpiece as
f(u1,θ1)＝(Ec-xm+p1cotΣ)nz+
EccotΣny+zmnx＝0 (6)
In the formula, xm，ym，zmAnd nx，ny，nzRespectively is a gear after the modification of tooth profileVector rmAnd unit normal vector nmThree components of (a); Σ is the grinding wheel mounting angle, EcIs the shortest distance between the grinding spindle and the workpiece spindle, Ec＝rp1+rg，rp1Radius of reference circle, r, for the workpiecegIs the grinding wheel radius.
And 2.2, deriving an equation of a contact line between the grinding wheel and the gear. By solving equations (4) - (6) simultaneously, a certain rotation angle can be obtained
Time (hypothesis)
) Line of contact L on a workpiece1(ii) a Contact line L on workpiece1Obtaining the bit vector and the normal vector of a grinding wheel contact line Lc by converting the coordinate into Sc under a grinding wheel reference coordinate system
rc(u1,θ1)＝McfMf1r1(u1,θ1) (7)
In the formula, matrix Mf1Is a unitary matrix
And 2.3, deriving a section equation of the formed grinding wheel. The contact line is projected to the cross section of the grinding wheel in a rotating way around the axis of the grinding wheel, and the cross section equation of the formed grinding wheel can be obtained
The contact line Lc on the grinding wheel makes spiral motion around the axis of the workpiece. If the axial modification is considered, the axial modification is realized by changing the center distance between the axis of the grinding wheel and the axis of the workpiece, the motion track of the center of the grinding wheel during grinding adopts a symmetrical parabolic motion law, and the expression is
In the formula, alphapThe modification coefficient is a tooth parabola modification coefficient; z is a radical of1Is the axial coordinate of the workpiece. Maximum modification delta at tooth tipzCorresponding to a grinding wheel center variation of
In the formula, betabIs a base circle helix angle, alphanIs the normal pressure angle. From this, the tooth parabolic motion coefficient α can be determinedp. After the tooth profile and the tooth direction bidirectional modification grinding, the equation of the tooth surface of the small wheel is expressed as
In the form of matrix
3. Step three: and calculating the maximum diameter of the grinding wheel according to the geometric condition of the grinding wheel when the grinding wheel grinds the tooth surface termination point.
3.1, grinding limit conditions of herringbone gears. Taking a right-hand gear as an example, the grinding termination point on the right side of the tooth slot should satisfy the following equation
In the formula, the first equation and the second equation represent the end point of the right tooth surface ground at the time, and the third equation is the contact line equation at the time; r isa1Radius of addendum circle, B0Is the relief groove size, x'1，y′1，z′1Three components of the bidirectional profile grinding tooth surface position vector are respectively. At this time, the equation of the contact line corresponding to the outer diameter of the grinding wheel is
In the formula, rf1The root circle radius. When the grinding wheel grinds the termination point, the maximum grinding wheel radius is determined by the intersection of the track line formed by the rotation of the lowest point of the grinding wheel contact line around the grinding wheel axis and the tooth top circle of the tool withdrawal groove end surface of the gear on the other side, and the geometrical condition is that
In the formula, the first equation represents the contact line L of the grinding wheel after considering the axial modificationcPerforming spiral motion around the axis of the workpiece to obtain the track of the outer diameter of the grinding wheel; the second equation is the axial position relation of the outer diameter of the grinding wheel contacting the end surface of the tooth groove on the other side of the gear tooth; the third equation is the condition for the wheel to interfere with the other side tooth at the tooth tip.
And 3.2, determining the maximum diameter of the grinding wheel. The equations (12) to (14) comprise 7 unknowns and have 7 constraint equations in total, so that the equations have solutions, and the maximum radius r of the grinding wheel when the right tooth surface end point of the grinding tooth socket is ground can be obtained by adopting a quasi-Newton methodgr. Similarly, when grinding the left flank end point of the tooth slot, r is usedf1Instead of r in the formula (12)a1And other equations are not changed, and the maximum radius r of the grinding wheel when the right tooth surface end point of the grinding tooth socket is obtainedglAnd the maximum diameter of the final grinding wheel is
dg＝2max(rgr,rgl) (15)
The following is specifically described with reference to fig. 1 to 4:
the method for accurately designing the maximum grinding wheel diameter of the shaping herringbone gear shaping mill comprises the following steps:
the method comprises the following steps: FIG. 1 is a view taken at X1Y1Z1In three-dimensional right-angle planes, by plane Z1The tooth profile of the tooth surface at one side of the herringbone gear in the end section is obtained by 0 cutting, and the axis X1Is the axis of symmetry of the tooth socket, beta1Is the intersection point of the tooth profile curve I and the base circle, and the half angle mu corresponding to the tooth groove width on the base circle1Is formed by the axis x1And a position vector O1β1Is constructed of and mu1＝π/2z-invαtWherein z is the number of gear teeth, αtThe involute function inv α being the pressure angle of the reference circlet＝tanαt-αtThen, the position vector and normal vector of the gear tooth surface are respectively:
in the formula, rb1Is a basic radius; mu.s1Is a half angle corresponding to the width of the tooth groove on the base circle; theta1The spread angle is the point on the involute; screw parameter p1＝rb1tgλb1，λb1The lead angle on the gear base circle; the superscript indicates the right flank and the subscript indicates the left flank. By analogy, the position vector and the normal vector of the left tooth surface and the right tooth surface of the left-handed gear can be obtained. And changing the rotation direction of the gear teeth to obtain a mathematical model of the tooth surface on the other side of the herringbone gear.
Fig. 2 is a functional expression for establishing three-segment parabolic modification of a tooth profile on a gear rotation projection plane, wherein the abscissa is the length from an involute rotation radius to a reference circle, and the ordinate is the modification amount. Dividing the whole tooth profile into three sections, respectively l1，l2And l3Wherein l is1Indicating root relief length, /)2Indicating the intermediate drum length,/3The maximum modification amount corresponding to the three-section modification curve is h1，h2And h3. According to the geometric relation, a function expression of three-section parabola modification of the tooth profile is established as
In the formula I1Indicating root relief length, /)2Indicating the intermediate drum length,/3Indicates the tooth tip relief length, a1、a2And a3Is the parabolic coefficient of the section I; b1Is a parabolic coefficient of the section II; c. C1、c2And c3Is a parabolic coefficient of the third section. Determining coefficients according to the maximum modification amount of each segment of parabola and the continuous tangency condition of adjacent parabolas at a connecting point, wherein the length parameter l of the tooth profile on a rotating projection plane is (x)1 2+y1 2)1/2-rp1，rp1Is the reference circle radius.
The theoretical standard tooth surface and the modification curved surface are superposed to obtain the position error r of the tooth profile modification tooth surfacemThe expression is
rm(u1,θ1)＝r1(u1,θ1)+n1(u1,θ1)δ(l) (4)
The unit normal vector of the tooth profile modified tooth surface is
Wherein, the two tangent vectors of the modified tooth surface are respectively
Step two: and solving the section shape of the formed grinding wheel. FIG. 3 is a coordinate system of a formed wheel grinding helical gear, wherein the coordinate system Sc、SfAnd S1A grinding wheel reference coordinate system, a workpiece reference coordinate system and a workpiece moving coordinate system. r isp1Radius of reference circle, r, for the workpiecegIs the grinding wheel radius, EcIs the shortest distance between the grinding wheel shaft and the workpiece shaft, the mounting angle sigma of the grinding wheel is pi/2-beta, and beta is the helical angle of the workpiece on the reference circle。
The principle of grinding herringbone gears by using the formed grinding wheel is basically the same as that of grinding helical gears, and the difference is that the influence of the size of a tool withdrawal groove on the selection of the maximum diameter of the grinding wheel needs to be considered. The meshing condition when grinding a workpiece with a formed grinding wheel is that the normal vector of the contact line between the tooth surface Σ 1 having the tooth profile modification and the grinding wheel surface Σ g passes through the grinding wheel revolution axis. Deducing an engagement equation of the grinding wheel and the workpiece as
f(u1,θ1)＝(Ec-xm+p1cotΣ)nz+
EccotΣny+zmnx＝0 (6)
In the formula, xm，ym，zmAnd nx，ny，nzRespectively is a gear position vector r after the modification of tooth profilemAnd unit normal vector nmThe center-to-center distance Ec ═ rp1+rg。
By solving equations (4) - (6) simultaneously, a certain rotation angle can be obtained
Time (hypothesis)
) Line of contact L on a workpiece1(ii) a Contact line L on workpiece1Obtaining the bit vector and the normal vector of a grinding wheel contact line Lc by converting the coordinate into Sc under a grinding wheel reference coordinate system
rc(u1,θ1)＝McfMf1r1(u1,θ1) (7)
In the formula, matrix Mf1Is a unitary matrix of the first phase,
the contact line is rotated around the axis of the grinding wheel and projected to the cross section of the grinding wheel, so as to obtain the cross section of the formed grinding wheel,
the cross section of the grinding wheel can be used for calculating the dressing track of the diamond roller arc dressing and forming grinding wheel profile. And (3) making the contact line Lc on the grinding wheel perform spiral motion around the axis of the workpiece to obtain the workpiece tooth surface with the tooth profile modification ground by the forming grinding wheel. If the axial modification is considered, the axial modification is realized by changing the center distance between the axis of the grinding wheel and the axis of the workpiece, the figure 4 is the motion track of the center of the grinding wheel during grinding, a symmetrical parabolic motion law is adopted, and the expression is
In the formula, alphapThe modification coefficient is a tooth parabola modification coefficient; z is a radical of1Is the axial coordinate of the workpiece. Maximum modification delta at tooth tipzCorresponding to a grinding wheel center variation of
In the formula, betabIs a base circle helix angle, alphanIs the normal pressure angle. From this, the tooth parabolic motion coefficient α can be determinedp. After the tooth profile and the tooth direction bidirectional modification grinding, the equation of the tooth surface of the small wheel is expressed as
In the form of matrix
Step three: the maximum grinding wheel diameter is determined. When grinding a herringbone gear, when a grinding wheel is close to the tooth end of the tool withdrawal groove, the relation between the size of the tool withdrawal groove and the diameter of the grinding wheel needs to be considered, and the grinding of the whole tooth surface can be completed on the premise of no interference. During grinding, the contact line of the grinding wheel and the workpiece is approximately the same as the spiral line direction, and only the inclination angle and the bending degree of the contact line are different. For right-hand gears, the tooth flank top point on the right side of the tooth slot is the right-side grinding termination point, and the tooth flank root point on the left side of the tooth slot is the grinding termination point of the left-side tooth flank; for a left-hand gear, the tooth flank root point on the right side of the tooth slot is the grinding termination point, and the tooth flank tip point on the left side of the tooth slot is the grinding termination point. Taking a right-hand gear as an example, the grinding termination point on the right side of the tooth slot should satisfy the following equation
In the equation (12), the first and second equations represent the end point of the right tooth surface to be ground at this time, and the third equation is the contact line equation at this time; r isa1Radius of addendum circle, B0Is the relief groove size, x'1，y′1，z′1Three components of the bidirectional profile grinding tooth surface position vector are respectively. At this time, the equation of the contact line corresponding to the outer diameter of the grinding wheel is
In the formula, rf1The root circle radius. When the grinding wheel grinds the termination point, the maximum grinding wheel radius is determined by the intersection of the track line formed by the rotation of the lowest point of the grinding wheel contact line around the grinding wheel axis and the tooth top circle of the tool withdrawal groove end surface of the gear on the other side, and the geometrical condition is that
In the formula, the first equation represents the contact line L of the grinding wheel after considering the axial modificationcPerforming spiral motion around the axis of the workpiece to obtain the track of the outer diameter of the grinding wheel(ii) a The second equation is the axial position relation of the outer diameter of the grinding wheel contacting the end surface of the tooth groove on the other side of the gear tooth; the third equation is the condition for the wheel to interfere with the other side tooth at the tooth tip. The equations (12) to (14) comprise 7 unknowns and have 7 constraint equations in total, so that the equations have solutions, and the maximum radius r of the grinding wheel when the right tooth surface end point of the grinding tooth socket is ground can be obtained by adopting a quasi-Newton methodgr. Similarly, when grinding the left flank end point of the tooth slot, r is usedf1Instead of r in the formula (12)a1And other equations are not changed, and the maximum radius r of the grinding wheel when the right tooth surface end point of the grinding tooth socket is obtainedglAnd the maximum diameter of the final grinding wheel is
dg＝2max(rgr,rgl) (15)。
Claims (4)
1. A method for accurately calculating the diameter of a maximum grinding wheel of a shaping herringbone gear shaping mill is characterized by comprising the following steps:
(1) establishing a gear tooth profile three-section parabolic modified curved surface model, and solving an involute function to obtain position vectors and normal vectors of tooth surfaces of left and right hand gears so as to obtain a mathematical model of the tooth surfaces of the herringbone gears;
(2) deducing an engagement equation of the grinding wheel and the workpiece through the position vector and the normal vector of the workpiece and the installation parameters of the grinding wheel, and further obtaining an equation of a contact line between the grinding wheel and the workpiece; the contact line is projected to the cross section of the grinding wheel in a rotating mode around the axis of the grinding wheel to obtain the cross section of the formed grinding wheel, and meanwhile the change of the center distance of the grinding wheel is considered, so that the bidirectional shaping of the herringbone gear is achieved;
(3) and calculating the maximum diameter of the grinding wheel according to the geometric condition of the grinding wheel when the grinding wheel grinds the tooth surface termination point.
2. The method for accurately calculating the maximum grinding wheel diameter of the shaping herringbone gear shaping mill according to claim 1, wherein the method comprises the following steps of: the step (1) is specifically as follows:
a. according to the differential geometric tooth surface conjugate theory, the determined position vector and normal vector of the gear tooth surface are respectively
Wherein rb1 is a basic radius, μ 1 is a half angle corresponding to the width of a groove on a base circle, θ 1 is a flare angle of a point on an involute, a spiral parameter p1 is rb1tg λ b1, and λ b1 is a lead angle on a gear base circle; the superscript represents a right side tooth surface, the subscript represents a left side tooth surface, the position vectors and normal vectors of the left and right tooth surfaces of the left-handed gear are obtained through the method, the rotation direction of the gear teeth is changed, and a mathematical model of the other side tooth surface of the herringbone gear is obtained;
b. on a gear rotation projection plane, establishing a function expression of three-section parabola modification of the tooth profile according to a geometric relation:
in the formula I1Indicating root relief length, /)2Indicates the middle drum length,/3Representing addendum relief length, a1, a2 and a3 are parabolic coefficients of section I, b1 is a parabolic coefficient of section II, and c1, c2 and c3 are parabolic coefficients of section III; determining coefficients according to the maximum modification amount of each segment of parabola and the continuous tangency condition of adjacent parabolas at a connecting point, wherein the length parameter of the tooth profile on a rotating projection plane is (x12+ y12)1/2-rp1, and rp1 is the radius of a reference circle;
c. superposing the theoretical standard tooth surface and the modified curved surface to obtain the bit error rm of the tooth profile modified tooth surface, wherein the expression is
rm(u1,θ1)＝r1(u1,θ1)+n1(u1,θ1)δ(l)
The unit normal vector of the tooth profile modified tooth surface is
Wherein, the two tangent vectors of the modified tooth surface are respectively
3. The method for accurately calculating the maximum grinding wheel diameter of the shaping herringbone gear shaping mill according to claim 1, wherein the method comprises the following steps of: the step (2) is specifically as follows:
a. deducing an engagement equation of the grinding wheel and the workpiece as
f(u1,θ1)＝(Ec-xm+p1cotΣ)nz+
EccotΣny+zmnx＝0
In the formula, xm, ym, zm, nx, ny and nz are three components of a gear bit vector rm and a unit normal vector nm after tooth profile modification respectively; Σ is a grinding wheel mounting angle, Ec is the shortest distance between the grinding wheel spindle and the workpiece spindle, Ec is rp1+ rg, rp1 is the workpiece reference circle radius, rg is the grinding wheel radius;
b. deducing the contact line equation of the grinding wheel and the gear, and simultaneously solving the formula rm(u1,θ1)＝r1(u1,θ1)+n1(u1,θ1)δ(l)、
Determining the angle of rotation
Contact line L1 on the workpiece; converting the contact line L1 on the workpiece into the grinding wheel reference coordinate system by coordinate transformation Sc, obtaining the position vector and normal vector of the grinding wheel contact line Lc
rc(u1,θ1)＝McfMf1r1(u1,θ1)
In the formula, the matrix Mf1 is a unitary matrix
c. Deducing a section equation of the formed grinding wheel, and projecting the contact line to the section of the grinding wheel in a rotating way around the axis of the grinding wheel to obtain the section equation of the formed grinding wheel
Making the contact line Lc on the grinding wheel make spiral motion around the axis of the workpiece, adopting the motion law of symmetrical parabola, and the expression is
In the formula, alpha p is a tooth parabola modification coefficient; z1 is the axial coordinate of the workpiece, the maximum dressing at the tooth tip is δ z, and the corresponding wheel center variation is
In the formula, beta b is a base circle helix angle, alpha n is a normal pressure angle, a tooth direction parabolic motion coefficient alpha p is determined, and after tooth profile and tooth direction bidirectional modification grinding, a small wheel tooth surface equation is expressed as
In the form of matrix
4. The method for accurately calculating the maximum grinding wheel diameter of the shaping herringbone gear shaping mill according to claim 1, wherein the method comprises the following steps of: the step (3) is specifically as follows:
a. the limiting conditions for grinding the herringbone gear are as follows: taking a right-hand gear as an example, the grinding termination point on the right side of the tooth slot satisfies the following equation
In the formula, the first equation and the second equation represent the end point of the right tooth surface ground at the time, and the third equation is the contact line equation at the time; ra1 is addendum circle radius, B0 is tool withdrawal groove size, x ' 1, y ' 1 and z ' 1 are three components of bidirectional profile grinding tooth surface position vector respectively, and the corresponding contact line equation on the outer diameter of the grinding wheel is
Wherein rf1 is root circle radius, and when the grinding wheel grinds the termination point, the maximum grinding wheel radius is determined by the intersection of the track line formed by the rotation of the lowest point of the grinding wheel contact line around the grinding wheel axis and the tooth top circle of the tool withdrawal end face of the gear on the other side, and the geometrical condition is that
In the formula, a first equation represents that after tooth direction modification is considered, a grinding wheel contact line Lc performs spiral motion around the axis of a workpiece to obtain the track of the outer diameter of the grinding wheel; the second equation is the axial position relation of the outer diameter of the grinding wheel contacting the end surface of the tooth groove on the other side of the gear tooth; the third equation is the condition that the grinding wheel interferes with the gear teeth on the other side at the tooth tops;
(2) determining the maximum diameter of the grinding wheel: the maximum radius rgr of the grinding wheel when grinding the right side of the tooth space is obtained by the quasi-Newton method, and similarly, when grinding the left side of the tooth space, the formula is replaced by rf1
And (3) obtaining the maximum radius rgl of the grinding wheel when the right tooth surface end point of the grinding tooth socket is ground by using the ra1 in the middle without changing other equations, wherein the maximum diameter of the final grinding wheel is equal to
dg＝2max(rgr,rgl)。
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