CN113536472B - Method for calculating starting point of involute of gear based on tooth profile normal method - Google Patents
Method for calculating starting point of involute of gear based on tooth profile normal method Download PDFInfo
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Abstract
A method for calculating a gear involute starting point based on a tooth profile normal method comprises the following steps: 1) Establishing a coordinate system; 2) R for calculating conversion of arbitrary point M on tooth top arc of given hob to gear tooth root curve i And a mathematical expression of the x-coordinate in the coordinate system; 3) Deriving P at any point M' on the involute from gear parameters inv(i) R at this point is obtained inv(i) And a mathematical expression of the point's x-coordinate in a coordinate system; 4) Calculating the actual data d of the initial circle of the involute of the gear by an iteration method tif The method comprises the steps of carrying out a first treatment on the surface of the 5) Data d to be derived tif Starting circle d of involute of gear marked on hob cutter design drawing tif In contrast, if D tif ≤d tif The hob meets the design requirement, if d tif ＜D tif The hob does not meet the design requirements.
Description
Technical Field
The invention relates to the field of gear machining, in particular to a method for calculating a gear involute starting point based on a tooth profile normal method.
Background
The tooth profile normal method is a common method recorded in a book of gear meshing principle of Wu Xutang, and can be used for obtaining meshing lines formed by all contact points of racks of hob cutters and gears in the moving process, and checking whether the gears machined by the hob cutters are gears wanted by users or not through the calculated meshing lines, namely, whether the machined gears meet design requirements or not.
The outer edge of the gear tooth is composed of a plurality of sections of discontinuous curved surfaces, each section of discontinuous curved surface is processed by a corresponding curved surface on the hob, the gear tooth shape can be obtained by adopting a tooth profile normal method, and each section of discontinuous curved surface is formed by a corresponding curved surface on the hob, but in order to ensure the meshing starting point of the gear, the starting point of the involute of the gear needs to be ensured, and the intersection point of the curve of the involute section on the gear and the curve of the tooth root section is an important factor for checking whether the hob meets the design of the gear.
Only if the involute starting point of the gear meets the design requirement, the gear pair meshing starting point (SAP) can be ensured, the contact ratio of the gear pair is ensured, the risk of gear meshing and squeal is reduced, and the NVH performance is improved. Currently, a technician generally uses an envelope method to determine the position of the involute starting point, but the calculation method is complex and cumbersome, not easy to understand and inaccurate in result. Therefore, how to accurately and rapidly verify whether the arc section curve of the cutter tip in the hob cutter graph designed by the cutter manufacturer meets the design requirement of a design engineer about the starting point of the involute of the gear is a problem to be solved.
Disclosure of Invention
Aiming at the corresponding defects of the prior art, the invention provides a method for calculating the starting point of a gear involute based on a tooth profile normal method, which is characterized in that the shapes of a main blade section of a rack, a curve of a cutter tip fillet blade and a curve of a plurality of sections of the gear are respectively described by using equations through the meshing relationship between the gear and a hob rack, the position of a starting circle of the gear involute is calculated and obtained through planar coordinate transformation and mathematical iteration solution, and a theoretical basis is provided for ensuring the meshing starting circle of the involute gear.
The invention is realized by adopting the following scheme: the method for calculating the starting point of the involute of the gear based on the tooth profile normal method is characterized by comprising the following steps of:
1) According to the gear plane meshing principle, a hob tooth profile is set as a first tooth profile, a workpiece gear tooth profile is set as a second tooth profile, and the following coordinate system is established:
(1) taking a meshing node P of the first tooth profile and the second tooth profile as an origin to make a coordinate system (P-x, y) with space fixed;
(2) by hob O 1 (P no /2，h ao ) Establishing a coordinate system for the origin(O 1 -x 1 ，y 1 ) X is the abscissa 1 On the hob joint line, h ao Is the tooth top height, P no For the hob normal pitch, in the starting position, the coordinate system (O 1 -x 1 ，y 1 ) Coincident with the coordinate system (P-x, y);
(3) with gear centre O 2 (S t /2，r 2 ) For the origin to make a coordinate system (O) 2 -x 2 ，y 2 )，S t Is the tooth thickness of the end face of the gear, r 2 For the gear pitch circle radius, the coordinate system (O 2 -x 2 ，y 2 ) Rotates with the rotation of the gear, y in the initial position 2 Axis at O 2 P, O 2 P is consistent with the direction of the y axis, x 2 The axis is parallel to x;
2) R converted to a gear tooth root curve at any point M on the gear tooth tip arc of the hob is calculated according to the following steps i And a coordinate system (O 2 -x 2 ，y 2 ) Lower x coordinates
Setting any point M on the first tooth profile in a coordinate system (P-x, y), (O) 1 -x 1 ，y 1 )、(O 2 -x 2 ，y 2 ) The coordinate values of (x, y), (x) 1 ，y 1 )、(x 2 ，y 2 ) And when the M point becomes the contact point, the gear rotates from the initial positionAngle, the hob is moved by a distance +.>Next, the transformation relation between M points in three coordinate systems is found.
(1) The relation equation of the meshing line and the gear hob tooth top arc is:
wherein (x, y) is the coordinates of point M in the meshing line coordinate system (P-x, y), (x) 1 ，y 1 ) Is a coordinate system (O) 1 -x 1 ，y 1 ) Coordinates of any point, r 2 Is the radius of the indexing circle of the gear, t 1 Is an auxiliary variable which is used to control the operation of the motor,is the rotation angle of the gear;
(2) the equation for the meshing line and the gear root curve is as follows:
in (x) 2 ，y 2 ) For point M in the coordinate system (O 2 -x 2 ，y 2 ) (x, y) is the coordinate of any point in the meshing line coordinate system (P-x, y), r 2 Is the radius of the reference circle of the gear,is the rotation angle of the gear, t 2 Is an auxiliary variable;
(3) according to the meshing relationship, a relationship equation of the gear tooth root curve and the hob tooth top arc is finally obtained as follows:
in (x) 2 ，y 2 ) For M points in the coordinate system (O 2 -x 2 ，y 2 ) Coordinates of (x) 1 ，y 1 ) Is a coordinate system (O) 1 -x 1 ，y 1 ) The coordinates of any one of the points in the (c),r is the rotation angle of the gear 2 Is the radius of the indexing circle of the gear, t 1 、t 2 Is an auxiliary variable;
the radius R of the circle at any point M on the gear is calculated according to the following formula M ：
Wherein x is 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The abscissa of (b), y 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The ordinate of (a);
(4) the tooth profile coordinate equation of the tooth top arc end face of the hob is shown by a hob cutter graph:
the normal tooth form equation of the hob tooth top arc is as follows:
and θ is more than or equal to 0 and less than or equal to 90 degrees alpha pn +γ 0
In (x) 1 ，y 1 ) For M points in the coordinate system (O 1 -x 1 ，y 1 ) Coordinates of (x) 0 ，y 0 ) In a coordinate system (O) for the hob top edge and the hob top arc 1 -x 1 ，y 1 ) The intersection point beta is the helix angle of the gear, alpha pn Is the pressure angle of the main cutting edge of the hob tooth shape, and theta is any point and y on the circular arc of the hob top circle 1 Included angle of axes gamma 0 Is the included angle between the hob tooth-shaped main blade and the transition blade, ρ 0 The radius of the arc is the arc of the tooth top arc of the hob;
the end face tooth profile equation of the hob tooth top arc is:
and θ is more than or equal to 0 and less than or equal to 90 degrees alpha pn +γ 0
In (x) 1 ，y 1 ) For M points in the coordinate system (O 1 -x 1 ，y 1 ) Coordinates of (x) 0 ，y 0 ) Is a hob topThe arc of the blade and the hob top are arranged in a coordinate system (O 1 -x 1 ，y 1 ) The intersection point beta is the helix angle of the gear, alpha pn Is the pressure angle of the main cutting edge of the hob tooth shape, and theta is any point and y on the circular arc of the hob top circle 1 Included angle of axes gamma 0 Is the included angle between the hob tooth-shaped main blade and the transition blade, ρ 0 The radius of the arc is the arc of the tooth top arc of the hob;
(5) according to the relation equation of the gear tooth root curve and the hob tooth top arc, the arc radius variable theta of the hob tooth top arc is in the definition domain [0, 90 ° -alpha ] pn +γ 0 ]Inner equal n parts (n.gtoreq.1), starting from the nth part and at θ 0 ＝0，Δθ＝(90°-α pn +γ 0 ) The independent variables of the series of (n-1) equal differences are mapped sequentially to the coordinate system (O 2 -x 2 ，y 2 ) Wherein θ is 0 ＝0，θ t ＝90°-α pn +γ 0 ；
When i=1000, θ i ＝90°-α pn +γ 0 From the relation equation R of the tooth root curve of the gear and the tooth top arc of the hob M(i) Obtaining the initial circle radius R of the gear i | i＝1000 and
Multiple repetition of calculations when i=999, 998, 997, … …, 2, 1, the initial radius of the gear R i and
3) R at any point M' is obtained according to the following method inv(i) And at any point M' in the coordinate system (O 2 -x 2 ，y 2 ) Lower x coordinateAnd calculating P of any point M' on involute from gear parameters inv(i) Value:
(1) the gear tooth root curve involute equation is:
in (x) 2 ，y 2 ) Is a coordinate system (O) 2 -x 2 ，y 2 ) The coordinates, ζ, of any point M M(i) Is the spread angle of the point M' on the gear, ψ b Is the half angle of the tooth thickness of the base circle of the gear, r b Is the radius of the base circle of the gear;
(2) radius R of circle establishing any point on involute of gear inv(i) With the pressure angle alpha at the arbitrary point t(i) A unitary one-time equation for a variable:
according to involute characteristics, the length of the line along the base circle is equal to the length of the arc rolled on the base circle, i.e. r b ξ M(i) ＝r b tanα t(i) ，α t(i) Is the pressure angle of any point M' on the involute of the gear, and is ζ M(i) ＝tanα t(i)
(3) Radius R of arbitrary point M' circle on involute of gear inv(i) The calculation formula of (2) is as follows:
wherein x is 2 For any point M' in the coordinate system (O 2 -x 2 ，y 2 ) The abscissa of (b), y 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The ordinate of (a);
(4) the following formula is obtained according to the definition of the pressure angle of any circle of the gear:
(5) involute equation of tooth root curve of gear and xi M(i) ＝tanα t(i) Substituting radius R of arbitrary point M' circle on involute of gear inv(i) The calculation formula is R inv (α t(i) ) Such asThe following is shown:
,
and is also provided with
Wherein r is a Is the radius of the top circle of the gear teeth, (x, y) is the coordinate of any point in the meshing line coordinate system (P-x, y), and ψ is the coordinate of any point in the meshing line coordinate system (P-x, y) b Is the half angle (known quantity) of the tooth thickness of the base circle of the gear, r b Is the radius of the base circle of the gear (known quantity), alpha tM‘ Is any point (x) 2 ，y 2 ) Is a pressure angle of (2);
4) D is calculated according to the following iterative method tif The following is shown:
(1) given value R, known as radius R of any point on the involute of the gear inv (α t(i) ) Iteration alpha using dichotomy t(k) Return alpha t(k) So that the function f (alpha t(k) )＝R-R inv (α t(k) ) Zero approximation exists with accuracy ζ=10 -5 Wherein the error of the approximation and the true value does not exceed ζ;
and then the obtained alpha t(k) The abscissa taken into the involute equation of the tooth-root curve of the gear results in point P inv(k) Point P inv(k) On the R circle, point P inv(k) Is expressed as x-coordinate of
(2) The ith point on the arc curve of the hob is given to obtain a fixed value R i and And the circle radius R of the point on the involute of the gear is obtained from the step (1) in the step 4) inv(i) Assuming i=1000 and i=999, there is +.>By step (1) in step 4) can be obtained +.>And->
(3) Construction functionSearching by newton's hill-down method to make e=10 -12 Is defined in the definition field of (a):
order theComparison of e and e 0 There are three cases:
work when e is greater than 0, e 0 > 0 and e 0 When < e, let e=e 0 Recalculate i=998 to obtainRe-iterating the comparisons e and e 0 Up to the nth point, there is e.e 0 < 0, then e N+1 ·e N Definition field of < 0, θ is [ θ ] j ，θ k ]Where j=n+1, k=n;
when e=0, the position of the element,i.e. it is sought, d tif ＝2·R i | i＝1000 The method comprises the steps of carrying out a first treatment on the surface of the When e is less than 0, the hob parameter is designed with errors and needs to be redesigned;
the third step e is monotonous and continuous, and can be guided in the definition domain [ theta ] j ，θ k ]In this case, zero must be present so that the accuracy ζ=10 -12 ；
(4) Given accuracy ζ=10 -12 The zero point approximation of the function e is calculated by a dichotomy:
determining the interval [ theta ] from step (3) in step 4) j ，θ k ]，e j ·e k < 0, howeverBased on interval [ theta ] j ，θ k ]Is defined by the midpoint theta of (2) mid Calculating to obtain e mid ；
If e mid =0, then θ mid The zero point of the function;
if e j ·e mid Let k=mid < 0;
if e mid ·e k Let j=mid < 0;
finally, it is judged whether the accuracy ζ=10 is reached -12 I.e. |theta j -θ k If the value of < xi is not equal to the zero point approximation value theta j Or theta k The calculation is repeated until the accuracy ζ=10 is satisfied -12 And calculate d tif ＝2·R i ；
5) The data d calculated in the step 4) is processed tif The diameter D of the initial circle of the involute of the gear marked on the hob cutter design drawing tif In contrast, if D tif ≤d tif If d tif ＜D tif The hob meets the design requirements.
The invention has the following beneficial effects:
1. the invention establishes the relation between a plurality of curves of the hob and the tooth profile of the gear by utilizing the gear plane meshing principle through the hob tool diagram, and draws the shape of the gear profile processed by the hob, thereby being applicable to gear products in various transmission mechanisms.
2. According to the invention, the tooth root curve processed by the arc end section of the cutter point of the hob cutter is compared with the designed gear involute at one point, the diameter of the gear involute initial circle processed by the cutter is calculated, and compared with the diameter of the designed gear involute initial circle, the diameter of the gear involute initial circle is smaller than that of the gear involute initial circle, so that the design requirement is met, and otherwise, the design requirement is not met.
3. The application product range is wide, and the numerical value of the diameter of the initial circle of the involute of the gear machined by the hob cutter can be accurately calculated: according to the method, the hob graph can be adjusted according to the checking result, or design parameters can be properly adjusted according to the development condition of hob cutters, so that the hob design difficulty is reduced, the NVH performance of the transmission is improved, and the gear delivered by a supplier is verified after the hob cutters are checked, so that the application is very accurate and convenient.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a coordinate system of rack and pinion engagement;
FIG. 3 shows a hob cutter and a point arc segment (arc 23);
FIG. 4 is a schematic view of coordinates of any point on the tooth surface of a gear;
fig. 5 is a schematic view of the starting circle of the involute gear.
Detailed Description
As shown in fig. 1 to 5, a method for calculating a starting point of an involute of a gear based on a tooth profile normal method is characterized by comprising the following steps:
1) According to the gear plane meshing principle, a hob tooth profile is set as a first tooth profile, a workpiece gear tooth profile is set as a second tooth profile, and the following coordinate system is established:
(1) taking a meshing node P of the first tooth profile and the second tooth profile as an origin to make a coordinate system (P-x, y) with space fixed;
(2) by hob O 1 (P no /2，h ao ) Establishing a coordinate system (O) 1 -x 1 ，y 1 ) X is the abscissa 1 On the hob joint line, h ao Is the tooth top height, P no For the hob normal pitch, in the starting position, the coordinate system (O 1 -x 1 ，y 1 ) Coincident with the coordinate system (P-x, y);
(3) with gear centre O 2 (S t /2，r 2 ) For the origin to make a coordinate system (O) 2 -x 2 ，y 2 )，S t Is the tooth thickness of the end face of the gear, r 2 For the gear pitch circle radius, the coordinate system (O 2 -x 2 ，y 2 ) Rotates with the rotation of the gear, y in the initial position 2 Axis at O 2 P, O 2 P is consistent with the direction of the y axis, x 2 The axis is parallel to x;
2) R converted to a gear tooth root curve at any point M on the gear tooth tip arc of the hob is calculated according to the following steps i And a coordinate system (O 2 -x 2 ，y 2 ) Lower x coordinates
Setting any point M on the first tooth profile in a coordinate system (P-x, y), (O) 1 -x 1 ，y 1 )、(O 2 -x 2 ，y 2 ) The coordinate values of (x, y), (x) 1 ，y 1 )、(x 2 ，y 2 ) And when the M point becomes the contact point, the gear rotates from the initial positionAngle, the hob is moved by a distance +.>Next, the transformation relation between M points in three coordinate systems is found.
(1) The relation equation of the meshing line and the gear hob tooth top arc is:
wherein (x, y) is the coordinates of point M in the meshing line coordinate system (P-x, y), (x) 1 ，y 1 ) Is a coordinate system (O) 1 -x 1 ，y 1 ) Coordinates of any point, r 2 Is the radius of the indexing circle of the gear, t 1 Is an auxiliary variable which is used to control the operation of the motor,is the rotation angle of the gear;
(2) the equation for the meshing line and the gear root curve is as follows:
in (x) 2 ，y 2 ) For point M in the coordinate system (O 2 -x 2 ，y 2 ) (x, y) is the coordinate of any point in the meshing line coordinate system (P-x, y), r 2 Is the radius of the reference circle of the gear,is the rotation angle of the gear, t 2 Is an auxiliary variable;
(3) according to the meshing relationship, a relationship equation of the gear tooth root curve and the hob tooth top arc is finally obtained as follows:
in (x) 2 ，y 2 ) For M points in the coordinate system (O 2 -x 2 ，y 2 ) Coordinates of (x) 1 ，y 1 ) Is a coordinate system (O) 1 -x 1 ，y 1 ) The coordinates of any one of the points in the (c),r is the rotation angle of the gear 2 Is the radius of the indexing circle of the gear, t 1 、t 2 Is an auxiliary variable;
the radius R of the circle at any point M on the gear is calculated according to the following formula M ：
Wherein x is 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The abscissa of (b), y 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The ordinate of (a);
(4) the tooth profile coordinate equation of the tooth top arc end face of the hob is shown by a hob cutter graph:
the normal tooth form equation of the hob tooth top arc is as follows:
and θ is more than or equal to 0 and less than or equal to 90 degrees alpha pn +γ 0
In (x) 1 ，y 1 ) For M points in the coordinate system (O 1 -x 1 ，y 1 ) Coordinates of (x) 0 ，y 0 ) In a coordinate system (O) for the hob top edge and the hob top arc 1 -x 1 ，y 1 ) The intersection point beta is the helix angle of the gear, alpha pn Is the pressure angle of the main cutting edge of the hob tooth shape, and theta is any point and y on the circular arc of the hob top circle 1 Included angle of axes gamma 0 Is the included angle between the hob tooth-shaped main blade and the transition blade, ρ 0 The radius of the arc is the arc of the tooth top arc of the hob;
the end face tooth profile equation of the hob tooth top arc is:
and θ is more than or equal to 0 and less than or equal to 90 degrees alpha pn +γ 0
In (x) 1 ，y 1 ) For M points in the coordinate system (O 1 -x 1 ，y 1 ) Coordinates of (x) 0 ，y 0 ) In a coordinate system (O) for the hob top edge and the hob top arc 1 -x 1 ，y 1 ) The intersection point beta is the helix angle of the gear, alpha pn Is the pressure angle of the main cutting edge of the hob tooth shape, and theta is any point and y on the circular arc of the hob top circle 1 Included angle of axes gamma 0 Is the included angle between the hob tooth-shaped main blade and the transition blade, ρ 0 The radius of the arc is the arc of the tooth top arc of the hob;
(5) according to the relation equation of the gear tooth root curve and the hob tooth top arc, the arc radius variable theta of the hob tooth top arc is in the definition domain [0, 90 ° -alpha ] pn +γ 0 ]Inner equal n parts (n.gtoreq.1), starting from the nth part and at θ 0 ＝0，Δθ＝(90°-α pn +γ 0 ) The independent variables of the series of (n-1) equal differences are mapped sequentially to the coordinate system (O 2 -x 2 ，y 2 ) Wherein θ is 0 ＝0，θ t ＝90°-α pn +γ 0 ；
When i=1000, θ i ＝90°-α pn +γ 0 From the relation equation R of the tooth root curve of the gear and the tooth top arc of the hob M(i) Obtaining the initial circle radius R of the gear i | i＝1000 and
Multiple repetition of calculations when i=999, 998, 997, … …, 2, 1, the initial radius of the gear R i and
3) R at any point M' is obtained according to the following method inv(i) And at any point M' in the coordinate system (O 2 -x 2 ，y 2 ) Lower x coordinateAnd calculating P of any point M' on involute from gear parameters inv(i) Value:
(1) the gear tooth root curve involute equation is:
in (x) 2 ，y 2 ) Is a coordinate system (O) 2 -x 2 ，y 2 ) The coordinates, ζ, of any point M M(i) Is the spread angle of the point M' on the gear, ψ b Is the half angle of the tooth thickness of the base circle of the gear, r b Is the radius of the base circle of the gear;
(2) radius R of circle establishing any point on involute of gear inv(i) With the pressure angle alpha at the arbitrary point t(i) A unitary one-time equation for a variable:
according to involute characteristics, the length of the line along the base circle is equal to the length of the arc rolled on the base circle, i.e. r b ξ M(i) ＝r b tanα t(i) ，α t(i) Is any point M on the involute of the gear' pressure angle, then ζ M(i) ＝tanα t(i)
(3) Radius R of arbitrary point M' circle on involute of gear inv(i) The calculation formula of (2) is as follows:
wherein x is 2 For any point M' in the coordinate system (O 2 -x 2 ，y 2 ) The abscissa of (b), y 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The ordinate of (a);
(4) the following formula is obtained according to the definition of the pressure angle of any circle of the gear:
(5) involute equation of tooth root curve of gear and xi M(i) ＝tanα t(i) Substituting radius R of arbitrary point M' circle on involute of gear inv(i) The calculation formula is R inv (α t(i) ) The following is shown:
,
and is also provided with
Wherein r is a Is the radius of the top circle of the gear teeth, (x, y) is the coordinate of any point in the meshing line coordinate system (P-x, y), and ψ is the coordinate of any point in the meshing line coordinate system (P-x, y) b Is the half angle (known quantity) of the tooth thickness of the base circle of the gear, r b Is the radius of the base circle of the gear (known quantity), alpha tM‘ Is any point (x) 2 ，y 2 ) Is a pressure angle of (2);
4) D is calculated according to the following iterative method tif The following is shown:
(1) given value R, any point on the involute of a known gearRadius R of (2) inv (α t(i) ) Iteration alpha using dichotomy t(k) Return alpha t(k) So that the function f (alpha t(k) )＝R-R inv (α t(k) ) Zero approximation exists with accuracy ζ=10 -5 Wherein the error of the approximation and the true value does not exceed ζ;
and then the obtained alpha t(k) The abscissa taken into the involute equation of the tooth-root curve of the gear results in point P inv ( k ) Point P inv ( k ) On the R circle, point P inv ( k ) Is expressed as x-coordinate of
(2) The ith point on the arc curve of the hob is given to obtain a fixed value R i and And the circle radius R of the point on the involute of the gear is obtained from the step (1) in the step 4) inv(i) Assuming i=1000 and i=999, there is +.>By step (1) in step 4) can be obtained +.>And->
(3) Construction functionSearching by newton's hill-down method to make e=10 -12 Is defined in the definition field of (a):
order theComparison of e and e 0 There are three cases:
when e is greater than 0,e 0 > 0 and e 0 When < e, let e=e 0 Recalculate i=998 to obtainRe-iterating the comparisons e and e 0 Up to the nth point, there is e.e 0 < 0, then e N+1 ·e N Definition field of < 0, θ is [ θ ] j ，θ k ]Where j=n+1, k=n;
when e=0, the position of the element,i.e. it is sought, d tif ＝2·R i | i＝1000 The method comprises the steps of carrying out a first treatment on the surface of the When e is less than 0, the hob parameter is designed with errors and needs to be redesigned;
the third step e is monotonous and continuous, and can be guided in the definition domain [ theta ] j ，θ k ]In this case, zero must be present so that the accuracy ζ=10 -12 ；
(4) Given accuracy ζ=10 -12 The zero point approximation of the function e is calculated by a dichotomy:
determining the interval [ theta ] from step (3) in step 4) j ，θ k ]，e j ·e k < 0, then according to interval [ theta ] j ，θ k ]Is defined by the midpoint theta of (2) mid Calculating to obtain e mid ；
If e mid =0, then θ mid The zero point of the function;
if e j ·e mid Let k=mid < 0;
if e mid ·e k Let j=mid < 0;
finally, it is judged whether the accuracy ζ=10 is reached -12 I.e. |theta j -θ k If the value of < xi is not equal to the zero point approximation value theta j Or theta k The calculation is repeated until the accuracy ζ=10 is satisfied -12 And calculate d tif ＝2·R i ；
5) The data d calculated in the step 4) is processed tif The diameter of the initial circle of the involute of the gear marked on the design drawing of the hob cutterD tif In contrast, if D tif ≤d tif If d tif ＜D tif The hob meets the design requirements.
In this embodiment, taking a certain intermediate shaft three-gear of a DCT as an example, basic parameters of a hob cutter are shown in the following table:
| hob rack parameter | Rack parameter |
| Full tooth height of hob | 6.589 |
| Tooth top of hob | 2.888 |
| Main blade pressure angle | 14° |
| Pitch circle tooth thickness of hob | 2.68 |
| Radius of tool fillet | 0.7 |
| Pressure angle of transition blade | 9° |
| Included angle of main blade and transition blade | 5° |
| Axial normal protrusion quantity | 0.069281 |
| Chamfering blade pressure angle | 45 |
| Chamfer edge starting point height | 5.84 |
| Helix angle | 33.1° |
| The two sides are provided with a shaving allowance | 0.12 |
The hob cutter parameters and the structure parameters in the table are shown in fig. 3;
the basic parameters of a certain DCT intermediate shaft three-gear corresponding to hob cutter are shown in the following table:
| gear parameters | Design value |
| Tooth number | 50 |
| Modulus of | 2.05 |
| Pressure angle | 18.00 |
| Helix angle | 33.10 |
| Diameter of base circle | 114.08 |
| Measurement mode | Span rod distance |
| Post-finishing stick span | 127.395 |
| Diameter of measuring rod | 4 |
| Nominal value of root circle diameter | 113.240 |
| Nominal value of tip circle diameter | 125.760 |
| Involute initial circle diameter after finishing | 116.490 |
| Nominal value of chamfer starting circle diameter | 125.360 |
| Tooth top chamfer height | 0.2 |
| Tooth tip chamfer width | 0.2 |
D obtained from hob cutter drawing of certain DCT three-gear tif Is 116.15 in design D tif = 116.49 belowBy comparison, the tool diagram meets the design requirements.
The above description is only of the preferred embodiments of the present invention, and is not intended to limit the invention, and those skilled in the art will appreciate that the modifications made to the invention fall within the scope of the invention without departing from the spirit of the invention.
Claims (1)
1. The method for calculating the starting point of the involute of the gear based on the tooth profile normal method is characterized by comprising the following steps of:
1) According to the gear plane meshing principle, a hob tooth profile is set as a first tooth profile, a workpiece gear tooth profile is set as a second tooth profile, and the following coordinate system is established:
(1) taking a meshing node P of the first tooth profile and the second tooth profile as an origin to make a coordinate system (P-x, y) with space fixed;
(2) by hob O 1 (P no /2，h ao ) Establishing a coordinate system (O) 1 -x 1 ，y 1 ) X is the abscissa 1 On the hob joint line, h ao Is the tooth top height, P no For the hob normal pitch, in the starting position, the coordinate system (O 1 -x 1 ，y 1 ) Coincident with the coordinate system (P-x, y);
(3) with gear centre O 2 (S t /2，r 2 ) For the origin to make a coordinate system (O) 2 -x 2 ，y 2 )，S t Is the tooth thickness of the end face of the gear, r 2 For the gear pitch circle radius, the coordinate system (O 2 -x 2 ，y 2 ) Rotates with the rotation of the gear, y in the initial position 2 Axis at O 2 P, O 2 P is consistent with the direction of the y axis, x 2 The axis is parallel to x;
2) R converted to a gear tooth root curve at any point M on the gear tooth tip arc of the hob is calculated according to the following steps i And a coordinate system (O 2 -x 2 ，y 2 ) Lower x coordinates
Set the firstAny point M on a tooth profile is in a coordinate system (P-x, y), (O) 1 -x 1 ，y 1 )、(O 2 -x 2 ，y 2 ) The coordinate values of (x, y), (x) 1 ，y 1 )、(x 2 ，y 2 ) And when the M point becomes the contact point, the gear rotates from the initial positionAngle, the hob is moved by a distance +.>Next, the transformation relation between M points in three coordinate systems is found.
(1) The relation equation of the meshing line and the gear hob tooth top arc is:
wherein (x, y) is the coordinates of point M in the meshing line coordinate system (P-x, y), (x) 1 ，y 1 ) Is a coordinate system (O) 1 -x 1 ，y 1 ) Coordinates of any point, r 2 Is the radius of the indexing circle of the gear, t 1 Is an auxiliary variable which is used to control the operation of the motor,is the rotation angle of the gear;
(2) the equation for the meshing line and the gear root curve is as follows:
in (x) 2 ，y 2 ) For point M in the coordinate system (O 2 -x 2 ，y 2 ) (x, y) is the coordinate of any point in the meshing line coordinate system (P-x, y), r 2 Is the radius of the reference circle of the gear,is the rotation angle of the gear, t 2 Is an auxiliary variable;
(3) according to the meshing relationship, a relationship equation of the gear tooth root curve and the hob tooth top arc is finally obtained as follows:
in (x) 2 ，y 2 ) For M points in the coordinate system (O 2 -x 2 ，y 2 ) Coordinates of (x) 1 ，y 1 ) Is a coordinate system (O) 1 -x 1 ，y 1 ) The coordinates of any one of the points in the (c),r is the rotation angle of the gear 2 Is the radius of the indexing circle of the gear, t 1 、t 2 Is an auxiliary variable;
the radius R of the circle at any point M on the gear is calculated according to the following formula M ：
Wherein x is 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The abscissa of (b), y 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The ordinate of (a);
(4) the tooth profile coordinate equation of the tooth top arc end face of the hob is shown by a hob cutter graph:
the normal tooth form equation of the hob tooth top arc is as follows:
in the method, in the process of the invention,(x 1 ，y 1 ) For M points in the coordinate system (O 1 -x 1 ，y 1 ) Coordinates of (x) 0 ，y 0 ) In a coordinate system (O) for the hob top edge and the hob top arc 1 -x 1 ，y 1 ) The intersection point beta is the helix angle of the gear, alpha pn Is the pressure angle of the main cutting edge of the hob tooth shape, and theta is any point and y on the circular arc of the hob top circle 1 Included angle of axes gamma 0 Is the included angle between the hob tooth-shaped main blade and the transition blade, ρ 0 The radius of the arc is the arc of the tooth top arc of the hob;
the end face tooth profile equation of the hob tooth top arc is:
in (x) 1 ，y 1 ) For M points in the coordinate system (O 1 -x 1 ，y 1 ) Coordinates of (x) 0 ，y 0 ) In a coordinate system (O) for the hob top edge and the hob top arc 1 -x 1 ，y 1 ) The intersection point beta is the helix angle of the gear, alpha pn Is the pressure angle of the main cutting edge of the hob tooth shape, and theta is any point and y on the circular arc of the hob top circle 1 Included angle of axes gamma 0 Is the included angle between the hob tooth-shaped main blade and the transition blade, ρ 0 The radius of the arc is the arc of the tooth top arc of the hob;
(5) according to the relation equation of the gear tooth root curve and the hob tooth top arc, the arc radius variable theta of the hob tooth top arc is in the definition domain [0, 90 ° -alpha ] pn +γ 0 ]Inner equal n parts (n.gtoreq.1), starting from the nth part and at θ 0 ＝0，Δθ＝(90°-α pn +γ 0 ) The independent variables of the series of (n-1) equal differences are mapped sequentially to the coordinate system (O 2 -x 2 ，y 2 ) Wherein θ is 0 ＝0，θ t ＝90°-α pn +γ 0 ；
When i=1000, θ i ＝90°-α pn +γ 0 From the relation equation R of the tooth root curve of the gear and the tooth top arc of the hob M(i) To obtain teethWheel initial circle radius R i | i＝1000 and
Multiple repetition of calculations when i=999, 998, 997, … …, 2, 1, the initial radius of the gear R i and
3) R at any point M' is obtained according to the following method inv(i) And at any point M' in the coordinate system (O 2 -x 2 ，y 2 ) Lower x coordinate x Pinv(i) And calculating P of any point M' on involute from gear parameters inv(i) Value:
(1) the gear tooth root curve involute equation is:
in (x) 2 ，y 2 ) Is a coordinate system (O) 2 -x 2 ，y 2 ) The coordinates, ζ, of any point M M(i) Is the spread angle of the point M' on the gear, ψ b Is the half angle of the tooth thickness of the base circle of the gear, r b Is the radius of the base circle of the gear;
(2) radius R of circle establishing any point on involute of gear inv(i) With the pressure angle alpha at the arbitrary point t(i) A unitary one-time equation for a variable:
according to involute characteristics, the length of the line along the base circle is equal to the length of the arc rolled on the base circle, i.e. r b ξ M(i) ＝r b tanα t(i) ，α t(i) Is the pressure angle of any point M' on the involute of the gear, and is ζ M(i) ＝tanα t(i)
(3) Radius R of arbitrary point M' circle on involute of gear inv(i) The calculation formula of (2) is as follows:
wherein x is 2 For any point M' in the coordinate system (O 2 -x 2 ，y 2 ) The abscissa of (b), y 2 For M points in the coordinate system (O 2 -x 2 ，y 2 ) The ordinate of (a);
(4) the following formula is obtained according to the definition of the pressure angle of any circle of the gear:
(5) involute equation of tooth root curve of gear and xi M(i) ＝tanα t(i) Substituting radius R of arbitrary point M' circle on involute of gear inv(i) The calculation formula is R inv (α t(i) ) The following is shown:
,
wherein r is a Is the radius of the top circle of the gear teeth, (x, y) is the coordinate of any point in the meshing line coordinate system (P-x, y), and ψ is the coordinate of any point in the meshing line coordinate system (P-x, y) b Is the half angle (known quantity) of the tooth thickness of the base circle of the gear, r b Is the radius of the base circle of the gear (known quantity), alpha tM‘ Is any point (x) 2 ，y 2 ) Is a pressure angle of (2);
4) D is calculated according to the following iterative method tif The following is shown:
(1) given value R, known as radius R of any point on the involute of the gear inv (α t(i) ) Iteration alpha using dichotomy t(k) Return alpha t(k) So that the function f (alpha t(k) )＝R-R inv (α t(k) ) Zero approximation exists with accuracy ζ=10 -5 Wherein the error of the approximation and the true value does not exceed ζ;
and then the obtained alpha t(k) The abscissa taken into the involute equation of the tooth-root curve of the gear results in point P inv(k) Point P inv(k) On the R circle, point P inv(k) Is expressed as x-coordinate of
(2) The ith point on the arc curve of the hob is given to obtain a fixed value R i and And the circle radius R of the point on the involute of the gear is obtained from the step (1) in the step 4) inv(i) Assuming i=1000 and i=999, there is +.>By step (1) in step 4) can be obtained +.>And->
(3) Construction functionSearching by newton's hill-down method to make e=10 -12 Is defined in the definition field of (a):
order theComparison of e and e 0 There are three cases:
work when e is greater than 0, e 0 > 0 and e 0 When < e, let e=e 0 Recalculate i=998 to obtainRe-iterating the comparisons e and e 0 Up to the nth point, there is e.e 0 < 0, then e N+1 ·e N Definition field of < 0, θ is [ θ ] j ，θ k ]Where j=n+1, k=n;
when e=0, the position of the element,i.e. it is sought, d tif ＝2·R i | i＝1000 The method comprises the steps of carrying out a first treatment on the surface of the When e is less than 0, the hob parameter is designed with errors and needs to be redesigned;
the third step e is monotonous and continuous, and can be guided in the definition domain [ theta ] j ，θ k ]In this case, zero must be present so that the accuracy ζ=10 -12 ；
(4) Given accuracy ζ=10 -12 The zero point approximation of the function e is calculated by a dichotomy:
determining the interval [ theta ] from step (3) in step 4) j ，θ k ]，e j ·e k < 0, then according to interval [ theta ] j ，θ k ]Is defined by the midpoint theta of (2) mid Calculating to obtain e mid ；
If e mid =0, then θ mid The zero point of the function;
if e j ·e mid Let k=mid < 0;
if e mid ·e k Let j=mid < 0;
finally, it is judged whether the accuracy ζ=10 is reached -12 I.e. |theta j -θ k If the value of < xi is not equal to the zero point approximation value theta j Or theta k The calculation is repeated until the accuracy ζ=10 is satisfied -12 And calculate d tif ＝2·R i ；
5) The data d calculated in the step 4) is processed tif The diameter D of the initial circle of the involute of the gear marked on the hob cutter design drawing tif In contrast, if D tif ≤d tif If d tif ＜D tif The hob meets the design requirements.
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| GB160447A (en) * | 1920-03-13 | 1921-12-08 | Vickers Electrical Co Ltd | Improvements in hobs for cutting gear teeth |
| CN102699449A (en) * | 2012-06-21 | 2012-10-03 | 浙江工商职业技术学院 | Design method of hobbing cutter with special circular tooth shape |
| WO2016197909A1 (en) * | 2015-06-08 | 2016-12-15 | 中车戚墅堰机车车辆工艺研究所有限公司 | Non-fully-symmetric involute gear and machining method therefor |
| WO2016197905A1 (en) * | 2015-06-08 | 2016-12-15 | 中车戚墅堰机车车辆工艺研究所有限公司 | Gear-cutting hob and designing method therefor, and non-fully-symmetric involute gear and machining method therefor |
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|---|---|---|---|---|
| GB160447A (en) * | 1920-03-13 | 1921-12-08 | Vickers Electrical Co Ltd | Improvements in hobs for cutting gear teeth |
| CN102699449A (en) * | 2012-06-21 | 2012-10-03 | 浙江工商职业技术学院 | Design method of hobbing cutter with special circular tooth shape |
| WO2016197909A1 (en) * | 2015-06-08 | 2016-12-15 | 中车戚墅堰机车车辆工艺研究所有限公司 | Non-fully-symmetric involute gear and machining method therefor |
| WO2016197905A1 (en) * | 2015-06-08 | 2016-12-15 | 中车戚墅堰机车车辆工艺研究所有限公司 | Gear-cutting hob and designing method therefor, and non-fully-symmetric involute gear and machining method therefor |
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