CN116522484A - Helical gear transmission error calculation method considering shape correction quantity - Google Patents
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Abstract
The invention discloses a helical gear transmission error calculation method considering a modification amount, which solves the helical gear transmission error by establishing a multi-tooth load balance equation on the basis of solving the time-varying meshing stiffness and the tooth profile nonlinear modification amount of a helical gear. The implementation steps are as follows: dividing the helical gear into a plurality of slicing teeth along the tooth width direction, and obtaining time-varying meshing stiffness of a single slicing tooth; dispersing meshing points of the tooth surfaces of the bevel gears into r multiplied by s along the end surfaces and the tooth directions of the bevel gears to obtain the rigidity value of each meshing point; according to the change rule of the meshing line of the bevel gears, solving the multi-tooth time-varying meshing stiffness of the bevel gears under the large overlap ratio; establishing a tooth profile modification mathematical model along the end face of the helical gear, and calculating the deformation of each meshing point of the gear; and establishing a helical gear load balance equation by utilizing the helical gear multi-tooth time-varying meshing stiffness and the deformation of each meshing point, and solving a helical gear transmission error.
Description
Technical Field
The invention relates to the technical field of mechanical dynamics, in particular to a transmission error calculation method of a bevel gear taking shape modifying quantity into consideration.
Background
The new energy automobile industry is the main stream of future development nowadays, the new energy automobile adopts a driving motor to replace an engine on the traditional automobile, so that the masking effect in the traditional fuel automobile is eliminated, the noise of a gear transmission device is highlighted, the comfort of the whole automobile is poor, the riding experience of a driver is influenced, so that the optimization of vibration noise is needed urgently, the means of optimization mainly come from shape correction, the main internal excitation source of the vibration noise comes from rigidity and transmission error, the optimization calculation of the rigidity and the transmission error is very important, the conventional calculation method regards any position on gear teeth as the same rigidity, the calculation precision is greatly reduced, and the correction amount along the tooth profile and the tooth direction is not accurate, so the calculation method of the transmission error considering the correction of any position of the tooth surface is designed, and the calculation precision of the transmission error of the shape correction helical gear is comprehensively improved.
Disclosure of Invention
Aiming at the problems existing in the prior art, the invention aims to solve the technical problems that: how to provide a transmission error calculation method of a helical gear with high precision.
In order to solve the technical problems, the invention adopts the following technical scheme:
a transmission error calculation method of a helical gear considering the shape correction quantity comprises the following steps:
step 1, dividing a single helical tooth into a plurality of slicing teeth along the tooth direction based on a helical gear model;
step 2, calculating the meshing stiffness k of the slicing teeth total (α K )；
Step 3, dispersing the end surface of the helical gear into s meshing points, and calculating the corresponding meshing rigidity K of each meshing point total (t), (t=1····s), the previous step is repeated along the tooth direction to divide the meshing tooth surface into r x s meshing points, obtaining the meshing rigidity of each point;
step 4, determining the meshing stiffness K of the bevel gear according to the change rule of the meshing line dc (i)；
Step 5, substituting the tooth profile modification parameters Ca and La to obtain the modification quantity of the corresponding position on the tooth profile;
and 6, setting a minimum deviation value of the total transferred load, solving a tooth surface bearing contact equation based on an iteration method to obtain a transfer error of the helical gear, introducing a shape correction amount, and finally obtaining the transfer error after shape correction.
Preferably, the engagement stiffness k of the slicing teeth is calculated in the step S2 total (α K ) The specific steps of (a) are as follows:
and calculating the meshing stiffness of the slicing teeth based on a potential energy method:
at this time, bending-shearing-radial compression stiffness k b i 、k s i 、k a i Stiffness of Hertz contact k h Rigidity k of wheel body f i Expressed as:
wherein i=p, g; i=p denotes a driving wheel, i=g denotes a driven wheel, k total Being a single-piece gearThe meshing stiffness, F, is the load acting on the monolithic gear,for the bending stiffness of the gear>For shear stiffness>For radial compression stiffness, k h For the Hz contact stiffness, < >>For rigidity of the gear body, E is elastic modulus of gear material, r b Radius of base circle, r f Radius of root circle, v is poisson ratio, b 0 For effective tooth width, alpha is the included angle between the corresponding tangent point at different positions of tooth profile and X axis, alpha 1 Alpha is the angle between the meshing force and the Y axis 2 Is the corresponding half angle of the base circle, alpha 4 Is the half angle corresponding to the root circle, alpha 5 ＝arccos(r b /r f )+α 4 ，α 5 An included angle corresponding to the X axis at the tooth root circle; />For the distance between the point of the root circle on the tooth symmetry line and the point of intersection of the line of action of the meshing force and the tooth symmetry line,/>For the length of the circular arc corresponding to a single gear tooth on the root circle, L * ,M * ,P * ,Q * All represent coefficients, X i * Representing coefficient L * ,M * ,P * ,Q * ，h f Is the ratio of the radius of the gear tooth root circle to the radius of the gear aperture, theta f Half of the corresponding central angle of the arc occupied by the single gear tooth;
establishment of alpha 1 And alpha is K Transforming the engagement stiffness function into a corresponding pressure at the engagement pointForce angle alpha K As a function of the variables,
wherein K is total (α K ) As to the pressure angle alpha K Is a meshing stiffness of (2);
definition alpha 1 And alpha is K The relationship of (2) is as follows:
α 1 ＝α K -(π/2·z-(inv(α K )-inv(α t )))；
wherein alpha is t The engagement angle of the end faces is represented, and z represents the number of teeth;
preferably, the specific step of calculating the engagement stiffness of each point in the step 3 is as follows:
based on a slicing method, cutting the tooth profile curved surface into r spur gears, wherein the rotation angle of each equal part is as follows:
Δλ＝(B/p)/r
the angle rotated by the start and end positions is divided into s parts, namely, the bevel gear tooth profile is divided into s parts from the tooth root to the tooth top at the end surface, the angle rotated by each part is,
Δω＝|ω B2 -ω B1 |/s
let Δω=Δλ, then
Wherein omega B2 Indicating the corresponding angle omega when the end surfaces start to mesh B1 The corresponding angle when the end face finishes meshing is represented, B represents the tooth width, and p represents the lead parameter;
the continuous rotation time is dispersed into a plurality of time points, t is the sequence number of the engagement position on the tooth profile in the rotation process, N 1 The common tangent point of the base circle of the driving wheel and the meshing line is adopted, and P' is a node; pressure angle alpha at point t K (t)The expression is:
α K (t)＝arctan(N 1 B 2 +B 2 B 1 ·t/s)/N 1 O 1
wherein t=0, 1 2. S, B (B) 2 Represents the engagement starting point of the front end face on the engagement plane, B 1 Indicating the end point of engagement of the front end face on the engagement plane, N 1 Represents the tangent point of the meshing plane and the base circle of the driving wheel, O 1 Representing the center of the circle of the driving wheel;
combining the above formula, converting the straight gear slice meshing stiffness expression of the above formula into a function related to a discrete quantity t to obtain the stiffness of each slice tooth corresponding to different positions on the divided tooth profile:
preferably, the engagement stiffness K is calculated in said step 4 dc (i) The specific steps of (a) are as follows:
the helical gear meshing line can be discretized into different meshing points, each meshing point is distributed in meshing areas with different colors, the rigidity of the meshing points is accumulated, and the single-tooth time-varying meshing rigidity in one period is expressed as follows in combination with the meshing line change rule:
wherein i represents the meshing time corresponding to the ith meshing line when single teeth are meshed;
the multi-tooth engagement stiffness formula for one cycle is as follows:
wherein: k (K) dc (i) A calculation formula of meshing stiffness at the ith moment of five teeth and four teeth in one period is represented, wherein alpha is as follows m Corner corresponding to interval from front tooth to rear tooth；
α m ＝2·π/z
α z ＝ω B2 -(ω B1 -B/p)。
Preferably, the specific step of calculating the modification amount of the corresponding position on the tooth profile in step 5 is as follows:
tooth profile modification amount ε (t) expression:
wherein Ca represents the modification amount, la represents the modification length, r b Represents the radius of the base circle, r g Represents the radius of the root circle, r d Indicating the tip circle radius.
Preferably, the specific steps of the step 6 are as follows, and the total load of five teeth is:
wherein F (x) is:
calculating an allowable deviation value of the total load and the actual total load through a given gear, and iteratively solving a transmission error of the helical gear under one meshing period; the method comprises the following specific steps:
in the above formula, for a contact point t, if delta > epsilon (t) is satisfied, the contact point is contacted, u (t) takes a positive value, otherwise u (t) takes 0;
1) Let k=1, given the initial value of the transmission error δ (1) ；
2) Determining deformation delta at each contact point i in turn (k) -epsilon (t) is of a size such that it is equal to 0 if it is smaller than 0;
3) Obtaining the total load F Z (k)；
4) Judging |F Z (k)-F|<F 0 Whether or not to establish; if not, let delta (k+1) ＝δ (k) -(F Z (k)-F)/k s ，k＝k+1, returning to step 2); if so, the iteration is stopped, output δ=δ (k) ；
Where k is the number of iterations, delta (k+1) To transfer error, delta, for step k+1 (k) To transfer errors for the kth step, F Z (k) Calculate the meshing force for the kth step, k s The average meshing stiffness of a plurality of gear teeth at a certain moment of the gear pair is obtained;
δ＝u(t)+ε(t)
where u (t) is the deformation, δ is the transmission error after the modification, and ε (t) is the modification amount of each contact point.
Compared with the prior art, the invention has at least the following advantages:
by utilizing the characteristic that the meshing stiffness at different positions on the tooth profile is different, the stiffness of different slice teeth is overlapped according to meshing lines at different moments, so that the time-varying high-precision meshing stiffness is obtained, and the transmission error obtained by solving the load balance equation on the basis has higher precision.
Drawings
FIG. 1 is a flow chart of the present invention.
FIG. 2 is a schematic diagram of a single slice gear tooth engagement stiffness calculation parameter.
Fig. 3 is a schematic view of a meshing process, fig. 3 (a) is a schematic view of an end face meshing process, and fig. 3 (b) is a schematic view of a tooth face meshing line.
Fig. 4 is a graph showing the relationship between meshing stiffness and meshing line length in one cycle of a single tooth, fig. 4 (a) shows that the first section stiffness increases with the increase of the meshing line length, fig. 4 (b) shows that the second section stiffness changes with the position of the meshing line length, and fig. 4 (c) shows that the third section stiffness decreases with the decrease of the meshing line length.
Fig. 5 is a superimposed graph of multiple tooth meshing line lengths.
Fig. 6 is a diagram comparing a multi-tooth meshing stiffness calculation model with a finite element model and a national standard calculation method.
Fig. 7 is a schematic view of helical gear single tooth slicing and tooth profile modification.
Fig. 8 is a graph comparing gear transmission error calculation model and finite element model results.
Fig. 9 is a graph comparing gear transmission errors before and after modification.
Detailed Description
The present invention will be described in further detail below.
A transmission error calculation method of a helical gear considering the shape correction quantity comprises the following steps:
and step 1, dividing a single helical tooth into a plurality of slicing teeth along the tooth direction based on a helical gear model.
Step 2, calculating the meshing stiffness k of the slicing teeth total (α K ) The method comprises the following specific steps:
and calculating the meshing stiffness of the slicing teeth based on a potential energy method:
at this time, bending-shearing-radial compression stiffness k b i 、k s i 、k a i Stiffness of Hertz contact k h Rigidity of wheel bodyExpressed as:
wherein i=p, g; i=p denotes a driving wheel, i=g denotes a driven wheel, k total For the meshing stiffness of the monolithic gear, F is the load acting on the monolithic gear,for the bending stiffness of the gear>For shear stiffness>For radial compression stiffness, k h For the Hz contact stiffness, < >>For rigidity of the gear body, E is elastic modulus of gear material, r b Radius of base circle, r f Radius of root circle, v is poisson ratio, b 0 For effective tooth width, alpha is the included angle between the corresponding tangent point at different positions of tooth profile and X axis, alpha 1 Alpha is the angle between the meshing force and the Y axis 2 Is the corresponding half angle of the base circle, alpha 4 Is the half angle corresponding to the root circle, alpha 5 ＝arccos(r b /r f )+α 4 ，α 5 An included angle corresponding to the X axis at the tooth root circle; />For the distance between the point of the root circle on the tooth symmetry line and the point of intersection of the line of action of the meshing force and the tooth symmetry line,/>For the length of the circular arc corresponding to a single gear tooth on the root circle, L * ,M * ,P * ,Q * All represent coefficients, X i * Representing coefficient L * ,M * ,P * ,Q * ，h f Is the tooth root of the gearRatio of circle radius to gear aperture radius, θ f The arc occupied by a single gear tooth corresponds to half of the central angle.
Establishment of alpha 1 And alpha is K Transforming the engagement stiffness function to a corresponding pressure angle alpha at the engagement point K As a function of the variables so that alpha can be subsequently represented by the gear parameters K 。
Wherein K is total (α K ) As to the pressure angle alpha K Is provided.
Definition alpha 1 And alpha is K The relationship of (2) is as follows:
α 1 ＝α K -(π/2·z-(inv(α K )-inv(α t )))；
wherein alpha is t The end face engagement angle is represented, and z represents the number of teeth.
Step 3, dispersing the end surface of the helical gear into s meshing points, and calculating the corresponding meshing rigidity K of each meshing point total (t), (t=1····s), the previous step is repeated along the tooth direction to divide the meshing tooth surface into r x s meshing points, the method for obtaining the meshing rigidity of each point comprises the following specific steps:
calculating single tooth time-varying meshing stiffness according to the corresponding stiffness of each tooth surface position, wherein for the helical gear, the meshing process is shown as a figure (3 b), and the meshing line on a single driving gear is from A 1 A 2 Continuously gets longer and grows to B 1 B 1 ", the meshing process of the helical gear is finished when the end face is observed, but the meshing line continues to move in the tooth profile direction, and the helical gear is formed by B 1 B 1 "shorten to C 1 C 2 . Throughout the meshing process, the rotation angle of the gear axis is omega B2 To omega B1 -B/p, (consider |omega) B2 -ω B1 I > B/p). Wherein B isIs the tooth width, p is the lead parameter
P＝P z /(2·π)
P z ＝π·d/cotβ
P in the formula z The lead, d is the pitch diameter and β is the helix angle.
ω B1 ＝π/2+α t -atan(N 1 B 1 /r b1 )-N 1 B 1 /r b1 +α B1
ω B2 ＝π/2+α t -atan(N 1 B 2 /r b1 )-N 1 B 2 /r b1 +α B2
Alpha in the formula t Is the engagement angle of the end face, theta B2 、θ B2 Respectively B 2 、B 1 Spread angle of point alpha B1 α B2 Respectively B 2 、B 1 Pressure angle, rb of (2) 1 Base radius, omega of the positioning driving wheel B1 、ω B2 Represents B 1 、B 2 Rotational position of the point.
In order to obtain the meshing stiffness corresponding to the meshing lines at different moments, the meshing stiffness values corresponding to the meshing points on the meshing lines need to be obtained first. Based on the slicing method, the tooth profile curved surface is along the edge B 2 B 2 The' direction is split into r spur gears, and the rotation angle of each equal part is as follows:
Δλ＝(B/p)/r
will start and end position B 2 To B 1 The rotated angle is divided into s parts, namely, the bevel gear tooth profile is divided into s parts from the tooth root to the tooth top at the end face, the rotated angle of each part is,
Δω＝|ω B2 -ω B1 |/s
let Δω=Δλ, then
Wherein omega B2 Indicating the corresponding angle omega when the end surfaces start to mesh B1 The angle corresponding to the end face when the engagement is completed is represented by B, the tooth width is represented by B, and the lead parameter is represented by p.
The position of the meshing point of the end surface of the helical gear wheel also changes along with the change of the rotation angle, wherein continuous rotation time is dispersed into a plurality of time points, t is the meshing position serial number on the tooth profile in the rotation process, a solid line represents the driving wheel, a dotted line represents the driven wheel and N 1 And P' is a node, which is the common tangent point of the base circle of the driving wheel and the meshing line. Pressure angle alpha at point t K The expression of (t) is:
α K (t)＝arctan(N 1 B 2 +B 2 B 1 ·t/s)/N 1 O 1
wherein t=0, 1 2. S, B (B) 2 Represents the engagement starting point of the front end face on the engagement plane, B 1 Indicating the end point of engagement of the front end face on the engagement plane, N 1 Represents the tangent point of the meshing plane and the base circle of the driving wheel, O 1 Representing the center of the circle of the driving wheel.
Combining the above formula, converting the straight gear slice meshing stiffness expression of the above formula into a function related to a discrete quantity t to obtain the stiffness of each slice tooth corresponding to different positions on the divided tooth profile:
the change in the meshing stiffness from root to tip for each spur gear is approximately considered to be the same, as the differently colored regions in FIG. 3b represent meshing stiffness values at different locations on the tooth profile surface.
Step 4, determining the meshing stiffness K of the bevel gear according to the change rule of the meshing line dc (i) The method comprises the following specific steps:
the helical gear meshing line can be discretized into different meshing points, each meshing point is distributed in meshing areas with different colors, the rigidity of the meshing points is accumulated, and the single-tooth time-varying meshing rigidity in one period is expressed as follows in combination with the meshing line change rule:
wherein i represents the meshing time corresponding to the ith meshing line when the single teeth are meshed, as shown in FIG. 3b, the r+s-1 common meshing line (A 1 A 2 -C 1 C 2 ) Each meshing line corresponds to one moment, and r+s-1 moments are divided into three sections A 1 Point to H, H Point to B 1 Point, then from B 1 "to C 2 The points are respectively corresponding to three conditions of i.ltoreq.r, r.ltoreq.s and s.ltoreq.i.ltoreq.r+s in the formula, and respectively corresponding to three conditions as shown in a graph (4), wherein the rigidity of the first graph is increased along with the increase of the length of the meshing line, the rigidity of the second graph is firstly increased and then reduced along with the change of the rotation angle when the length of the meshing line is unchanged, and the rigidity of the third graph is also reduced along with the decrease of the length of the meshing line.
According to the single-tooth time-varying meshing rigidity, the multi-tooth time-varying meshing rigidity is calculated, and the bevel gears are alternately meshed in the meshing process, wherein the bevel gear pair with the contact ratio of 4-5 is taken as an example, four teeth and five teeth are alternately meshed, and the lengths of the multi-tooth meshing lines are all overlapped on the basis of the single tooth, as shown in a figure (5). Similarly, the meshing stiffness is formed by superposition on the basis, the meshing line length of each corner is different based on a multi-tooth meshing line length forming diagram, the multi-tooth meshing stiffness diagram of each corner is formed by superposition of four teeth or five teeth stiffness, different lines represent different gear teeth, and the intersection point between the vertical dotted lines and the vertical dotted lines in the diagram represents that a plurality of contact lines positioned at different positions on different teeth exist when gears mesh at a certain moment.
The multi-tooth meshing stiffness is alpha m As a function of period, the multi-tooth engagement stiffness equation for one period is deduced as follows:
wherein: k (K) dc (i) A calculation formula of meshing stiffness at the ith moment of five teeth and four teeth in one period is represented, wherein alpha is as follows m A corner corresponding to the interval from the front tooth to the rear tooth;
α m ＝2·π/z
α z ＝ω B2 -(ω B1 -B/p)。
the periodic time-varying meshing stiffness value of the helical gear with the large overlap ratio, which is obtained based on a slicing method, is not only related to the length of the meshing line, but also considers the meshing stiffness change condition when each meshing point on the meshing line is positioned at different positions on the tooth profile. Compared with the meshing stiffness calculated by the ISO-6336 method, the meshing stiffness value of the gears in the meshing process can be reflected more accurately, the meshing stiffness value is matched with the finite element simulation result, and the method has important significance for subsequent gear shaping and vibration reduction research. The results of the comparison of the multi-tooth meshing stiffness of the main bevel gear obtained by solving the ISO-6336 method and the method are shown in a graph (6).
And 5, substituting the tooth profile modification parameters Ca and La to obtain the modification quantity of the corresponding position on the tooth profile, wherein the specific steps are as follows: taking the modification of the tooth top in the modification of the tooth profile as an example, the modification curve is a parabolic modification curve; the profile is modified within the margin of the profile, wherein the hatched portion is a modified portion, as shown in fig. 7. Tooth profile modification amount ε (t) expression:
wherein Ca represents the modification amount, la represents the modification length, r b Represents the radius of the base circle, r g Represents the radius of the root circle, r d Indicating the tip circle radius.
Step 6, setting a minimum deviation value of a transmission total load, solving a tooth surface bearing contact equation based on an iteration method to obtain a transmission error of the helical gear, introducing a shape correction amount, and finally obtaining the transmission error after shape correction, wherein the method comprises the following specific steps of:
in the loading process of the bevel gear, each meshing point on the meshing line can generate contact deformation, and the product of the meshing rigidity of each point and the loading deformation quantity of each point is the total load borne by the gear, and the calculation formula is as follows (taking five teeth as an example):
the total five-tooth load is:
wherein F (x) is
Calculating an allowable deviation value of the total load and the actual total load through a given gear, and iteratively solving a transmission error of the helical gear under one meshing period; the method comprises the following specific steps:
in the above formula, for the contact point t, if δ > ε (t) is satisfied, the point contact, u (t), takes a positive value, otherwise u (t) takes 0.
1) Let k=1, given the initial value of the transmission error δ (1) ；
2) Determining deformation delta at each contact point i in turn (k) -epsilon (t) is of a size such that it is equal to 0 if it is smaller than 0;
3) Obtaining the total load F Z (k)；
4) Judging |F Z (k)-F|<F 0 (F 0 To allow a small amount) is established. If not, let delta (k+1) ＝δ (k) -(F Z (k)-F)/k s K=k+1, returning to step 2); if so, the iteration is stopped, output δ=δ (k) ；
Where k is the number of iterations, delta (k+1) To transfer error, delta, for step k+1 (k) To transfer errors for the kth step, F Z (k) Calculate the meshing force for the kth step, k s The average meshing stiffness of a plurality of gear teeth at a certain moment of the gear pair is obtained;
the transmission error generated in the transmission process of the helical gear is related to the initial gear side clearance amount, the gear load deformation amount, the gear shaping amount, the tooth surface error and the like. The tooth surface error is assumed to be small relative to the macroscopic structure of the gear, and the influence caused by assembly problems is not considered, so that the normal directions are not changed after contact.
δ＝u(t)+ε(t)
Where u (t) is the deformation, δ is the transmission error after the modification, and ε (t) is the modification amount of each contact point. The initial gap amount is ignored here.
When a group of gear parameters with the overlap ratio between 4 and 5 are combined, a comparison graph of the transmission error of the calculation method of the invention and the transmission error of the finite element simulation can be obtained without modification (namely epsilon (t) =0), the transmission error is shown in a graph (8), the transmission error after the modification quantity Ca=6um and La=9mm is introduced, and the amplitude of the transmission error is obviously reduced as shown in a graph (9).
| Parameters (parameters) | Driving wheel | Driven wheel |
| Tooth number | 53 | 48 |
| Modulus of | 2.25 | |
| Pressure angle (°) | 14.5 | |
| Helix angle (°) | 27 | |
| Tooth width (mm) | 23.6 | 24 |
| Diameter of tip circle (mm) | 141.6 | 129.2 |
| Root diameter (mm) | 129.43 | 117 |
| Aperture (mm) | 95 | 36.8 |
| Coefficient of displacement | 0.3144 | 0.1456 |
| Material | 20CrMoH | |
| Young's modulus (GPa) | 207 |
Finally, it is noted that the above embodiments are only for illustrating the technical solution of the present invention and not for limiting the same, and although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications and equivalents may be made thereto without departing from the spirit and scope of the technical solution of the present invention, which is intended to be covered by the scope of the claims of the present invention.
Claims (6)
1. The transmission error calculation method of the helical gear taking the shape correction amount into consideration is characterized by comprising the following steps of:
step 1, dividing a single helical tooth into a plurality of slicing teeth along the tooth direction based on a helical gear model;
step 2, calculating the meshing stiffness k of the slicing teeth total (α K )；
Step 3, dispersing the end surface of the helical gear into s meshing points, and calculating the corresponding meshing rigidity K of each meshing point total (t), (t= … s), and then repeating the previous step along the tooth direction to divide the meshing tooth surface into r×s meshing points, so as to obtain the meshing rigidity of each point;
step 4, determining the meshing stiffness K of the bevel gear according to the change rule of the meshing line dc (i)；
Step 5, substituting the tooth profile modification parameters Ca and La to obtain the modification quantity of the corresponding position on the tooth profile;
and 6, setting a minimum deviation value of the total transferred load, solving a tooth surface bearing contact equation based on an iteration method to obtain a transfer error of the helical gear, introducing a shape correction amount, and finally obtaining the transfer error after shape correction.
2. The method for calculating transmission error of helical gear in consideration of shaping quantity according to claim 1, wherein said step S2 calculates engagement stiffness k of said slicing teeth total (α K ) The specific steps of (a) are as follows:
and calculating the meshing stiffness of the slicing teeth based on a potential energy method:
at this time, bending-shearing-radial compression stiffness k b i 、k s i 、k a i Stiffness of Hertz contact k h Rigidity of wheel bodyExpressed as:
wherein i=p, g; i=p denotes a driving wheel, i=g denotes a driven wheel, k total For the meshing stiffness of the monolithic gear, F is the load acting on the monolithic gear,for the bending stiffness of the gear>For shear stiffness>For radial compression stiffness, k h For the Hz contact stiffness, < >>For rigidity of the gear body, E is elastic modulus of gear material, r b Radius of base circle, r f Radius of root circle, v is poisson ratio, b 0 For effective tooth width, alpha is the included angle between the corresponding tangent point at different positions of tooth profile and X axis, alpha 1 Alpha is the angle between the meshing force and the Y axis 2 Is the corresponding half angle of the base circle, alpha 4 Is the half angle corresponding to the root circle, alpha 5 ＝arccos(r b /r f )+α 4 ，α 5 An included angle corresponding to the X axis at the tooth root circle; />For the distance between the point of the root circle on the tooth symmetry line and the point of intersection of the line of action of the meshing force and the tooth symmetry line,/>For the length of the circular arc corresponding to a single gear tooth on the root circle, L * ,M * ,P * ,Q * All represent coefficients, X i * Representing coefficient L * ,M * ,P * ,Q * ，h f Is the ratio of the radius of the gear tooth root circle to the radius of the gear aperture, theta f Half of the corresponding central angle of the arc occupied by the single gear tooth;
establishment of alpha 1 And alpha is K Transforming the engagement stiffness function to a corresponding pressure angle alpha at the engagement point K As a function of the variables,
wherein K is total (α K ) As to the pressure angle alpha K Is a meshing stiffness of (2);
definition alpha 1 And alpha is K The relationship of (2) is as follows:
α 1 ＝α K -(π/2·z-(inv(α K )-inv(α t )))；
wherein alpha is t The end face engagement angle is represented, and z represents the number of teeth.
3. The method for calculating the transmission error of the helical gear taking the correction into consideration as set forth in claim 2, wherein the specific steps of calculating the meshing stiffness of each point in the step 3 are as follows:
based on a slicing method, cutting the tooth profile curved surface into r spur gears, wherein the rotation angle of each equal part is as follows:
Δλ＝(B/p)/r
the angle rotated by the start and end positions is divided into s parts, namely, the bevel gear tooth profile is divided into s parts from the tooth root to the tooth top at the end surface, the angle rotated by each part is,
Δω＝|ω B2 -ω B1 |/s
let Δω=Δλ, then
Wherein omega B2 Indicating the corresponding angle omega when the end surfaces start to mesh B1 The corresponding angle when the end face finishes meshing is represented, B represents the tooth width, and p represents the lead parameter;
the continuous rotation time is dispersed into a plurality of time points, t is the sequence number of the engagement position on the tooth profile in the rotation process, N 1 The common tangent point of the base circle of the driving wheel and the meshing line is adopted, and P' is a node; pressure angle alpha at point t K The expression of (t) is:
α K (t)＝arctan(N 1 B 2 +B 2 B 1 ·t/s)/N 1 O 1
wherein t=0, 1 2. S, B (B) 2 Represents the engagement starting point of the front end face on the engagement plane, B 1 Indicating the end point of engagement of the front end face on the engagement plane, N 1 Represents the tangent point of the meshing plane and the base circle of the driving wheel, O 1 Representing the center of the circle of the driving wheel;
combining the above formula, converting the straight gear slice meshing stiffness expression of the above formula into a function related to a discrete quantity t to obtain the stiffness of each slice tooth corresponding to different positions on the divided tooth profile:
4. the transmission of helical gears according to claim 3, wherein said helical gears take into account the amount of shapingThe error calculation method is characterized in that the engagement stiffness K is calculated in the step 4 dc (i) The specific steps of (a) are as follows:
the helical gear meshing line can be discretized into different meshing points, each meshing point is distributed in meshing areas with different colors, the rigidity of the meshing points is accumulated, and the single-tooth time-varying meshing rigidity in one period is expressed as follows in combination with the meshing line change rule:
wherein i represents the meshing time corresponding to the ith meshing line when single teeth are meshed;
the multi-tooth engagement stiffness formula for one cycle is as follows:
wherein: k (K) dc (i) A calculation formula of meshing stiffness at the ith moment of five teeth and four teeth in one period is represented, wherein alpha is as follows m A corner corresponding to the interval from the front tooth to the rear tooth;
α m ＝2·π/z
α z ＝ω B2 -(ω B1 -B/p)。
5. the method for calculating the transmission error of the helical gear taking the correction into consideration according to claim 4, wherein the step 5 of calculating the correction of the corresponding position on the tooth profile comprises the following steps:
tooth profile modification amount ε (t) expression:
wherein Ca represents the modification amount, la represents the modification length, r b Represents the radius of the base circle, r g Represents the radius of the root circle, r d Indicating the tip circle radius.
6. The method for calculating the transmission error of the helical gear taking the correction into consideration according to claim 5, wherein the specific steps of the step 6 are as follows:
the total five-tooth load is:
wherein F (x) is:
calculating an allowable deviation value of the total load and the actual total load through a given gear, and iteratively solving a transmission error of the helical gear under one meshing period; the method comprises the following specific steps:
in the above formula, for a contact point t, if delta > epsilon (t) is satisfied, the contact point is contacted, u (t) takes a positive value, otherwise u (t) takes 0;
1) Let k=1, given the initial value of the transmission error δ (1) ；
2) Determining deformation delta at each contact point i in turn (k) -epsilon (t) is of a size such that it is equal to 0 if it is smaller than 0;
3) Obtaining the total load F Z (k)；
4) Judging |F Z (k)-F|<F 0 Whether or not to establish; if not, let delta (k+1) ＝δ (k) -(F Z (k)-F)/k s K=k+1, returning to step 2); if so, the iteration is stopped, output δ=δ (k) ；
Where k is the number of iterations, delta (k+1) To transfer error, delta, for step k+1 (k) To transfer errors for the kth step, F Z (k) Calculate the meshing force for the kth step, k s The average meshing stiffness of a plurality of gear teeth at a certain moment of the gear pair is obtained;
δ＝u(t)+ε(t)
where u (t) is the deformation, δ is the transmission error after the modification, and ε (t) is the modification amount of each contact point.
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