CN106641105B - Method for establishing reverse gear meshing model - Google Patents

Method for establishing reverse gear meshing model Download PDF

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CN106641105B
CN106641105B CN201710054301.6A CN201710054301A CN106641105B CN 106641105 B CN106641105 B CN 106641105B CN 201710054301 A CN201710054301 A CN 201710054301A CN 106641105 B CN106641105 B CN 106641105B
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gear
meshing
point
tooth
error
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CN106641105A (en
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石照耀
李珂
于渤
林家春
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Beijing University of Technology
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Beijing University of Technology
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H1/00—Toothed gearings for conveying rotary motion
    • F16H1/02—Toothed gearings for conveying rotary motion without gears having orbital motion
    • F16H1/04—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members
    • F16H1/06—Toothed gearings for conveying rotary motion without gears having orbital motion involving only two intermeshing members with parallel axes
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
    • F16H55/02—Toothed members; Worms
    • F16H55/08—Profiling
    • F16H55/0806—Involute profile
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems

Abstract

A reverse gear meshing model belongs to the technical field of mechanical transmission. Due to the influence of gear errors, there is a significant difference between the actual gear pair transmission and the ideal gear pair transmission with smooth transition. By researching the transmission of the gear pair with the error, the invention provides the concept of reverse-order meshing of the gears and analyzes the concept. Aiming at the transmission of the gear pair with the base pitch error, a transmission error model in the negative sequence meshing process is deduced, and the blank of the research on the meshing characteristics of the error gear pair including the line external meshing section in the whole meshing process is filled.

Description

Method for establishing reverse gear meshing model
Technical Field
The invention relates to a reverse gear meshing model, and belongs to the technical field of mechanical transmission.
Background
The theoretical gear transmission is an ideal transmission with constant transmission ratio, stable gear tooth alternate transition, contact point covering all working tooth surfaces and certain backlash. However, due to the presence of gear errors, ideal gears are not present in practice. At present, kinematic analysis of gear transmission is mostly established on the basis of error-free ideal gear engagement, and in actual gear transmission, due to the influences of manufacturing errors, installation errors, elastic deformation after bearing and the like, the ideal engagement motion process of the gear is inevitably changed, and the transmission characteristics of a gear pair are influenced.
For the problem of the transmission characteristic of the gear pair, R.G.Munro et al researches the out-of-line meshing transmission error generated based on the change of the center distance, provides approximate formulas for calculating the out-of-line meshing transmission error, and the formulas can be used for analyzing experimental data obtained in a Harris graph form. However, this study was conducted for a gear without an error, and the meshing characteristics of the gear with an error were not analyzed. The Sun Yulin respectively carries out detailed analysis on the meshing errors of the top edges under the two basic joint errors that the driving gear basic joint is larger than the driven gear basic joint and the driving gear basic joint is smaller than the driven gear basic joint, and discusses the problem of eliminating the edge trimming amount of the meshing of the top edges. However, the reverse sequence phenomenon occurring in the process of off-line engagement is not indicated, and the transmission error in the whole engagement process is not deduced.
Disclosure of Invention
In order to deeply research the actual meshing characteristic of the complete meshing process of the gear pair with the error, the invention provides a reverse gear meshing model for explaining a reverse gear meshing phenomenon in the actual error gear meshing process, and a transmission error model of the reverse gear meshing process is deduced for the gear pair transmission with the pitch error.
The reverse gear meshing means that the sequence of the meshing process is opposite to that of the normal meshing process. This sequence refers to the direction in which the contact point moves on the tooth flank, and if only single-sided contact is considered, during normal meshing, the contact point moves from the tooth root to the tooth tip on the driving gear and from the tooth tip to the tooth root on the driven gear. The reverse meshing process is outside the normal meshing process and occurs when the contact point moves from the tooth top to the tooth bottom on the drive gear and from the tooth bottom to the tooth top on the driven gear. These two reverse sequences are possible when there is a gear pitch deviation.
When a pair of gears are normally meshed, the contact ratio should be greater than 1, namely when the front pair of gear teeth is not disengaged, the rear pair of gear teeth is engaged. This meshing process will continue smoothly when the gear base sections are equal. In the case of a pitch error, if the gear has a pitch error, the ideal meshing process is destroyed, and the instantaneous transmission ratio changes, resulting in shock and vibration. Taking a pair of single-sided tooth surfaces as an example, as shown in fig. 1.
In fig. 1, a gear 1 is a driving wheel, and a gear 2 is a driven wheel. Points E and F in the figure are the intersection points of addendum circles of the driving wheel and the driven wheel, and A1The point is the tooth crest and the meshing line of the gear 1A point of intersection of2Point is the intersection of gear 2 and the meshing line and point P is the node where the two gears mesh. According to the rotation direction of the two gears, the initial point of the tooth surface contact is always located at the point E and the point A2Between points, and the last point where tooth surface contact occurs is constantAt A1Between point and point F. When the actual meshing point moves to point M, the angle through which the gear 1 rotates is
Figure GDA0002266948780000021
The angle through which the gear 2 rotates is
Figure GDA0002266948780000022
When the gears 1 and 2 rotate in opposite directions as shown in the figure, the angle
Figure GDA0002266948780000023
And
Figure GDA0002266948780000024
and is noted as a negative value.
It should be noted that the negative sequence meshing phenomenon in the gear transmission is harmful, but can be utilized, and the negative sequence meshing phenomenon is utilized to carry out finishing machining and tooth profile modification in the machining process, so that the aims of improving the machining quality and improving the machining efficiency are fulfilled.
(1) Reverse order analysis of engagement process
When there is a positive base pitch deviation, the gear 1 and the gear 2 cannot directly enter an involute meshing section, and this occurs as shown in fig. 1(a)Segment meshing process at A2The point begins to enter the involute meshing section, in which the contact point gradually moves from the tooth root to the tooth tip on the tooth flank of the driving gear 1, which is the normal meshing sequence.
Figure GDA0002266948780000026
The segments belong to the meshing off-line contact process, which differs from the normal meshing sequence. In that
Figure GDA0002266948780000027
In the section, the contact point first occurs at point E, and then the contact point moves from point E to point a on the tooth flank of the gear 12As can be seen from fig. 1(a), for the gear 1,radius of point E is larger than A2The radius of the point, which is the movement from the tooth top to the tooth bottom, is the reverse of the normal meshing sequence as shown in fig. 2, which is the meshing process of the driven gear top blade, and is the reverse of the meshing process.
In fig. 1(a), the gear 1 is a driving gear and is a standard gear, and the gear 2 is a driven gear, and has positive base pitch deviation Δ fpb. The meshing process then causes a meshing line increment of
Wherein r isb20In order to be the theoretical base radius of the circle,
Figure GDA0002266948780000029
is the actual addendum circle pressure angle, ra2Is the radius of the addendum circle of the driven wheel,
Figure GDA00022669487800000210
the gear is rotated through an angle. Due to the fact that
Figure GDA00022669487800000211
Is small, therefore
Figure GDA00022669487800000212
Figure GDA00022669487800000213
In order to be the actual base circle radius,
Figure GDA00022669487800000214
rbis the base circle radius. Let λ be the driven gear rotation angle error
Figure GDA00022669487800000215
Has a period of
Figure GDA00022669487800000216
According to
Figure GDA00022669487800000217
And can obtain
Figure GDA00022669487800000218
Both sides of the equal sign of formula (2) are divided by rb20Then the error of the rotation angle is
Figure GDA00022669487800000219
Order to
Figure GDA00022669487800000220
Then
Figure GDA00022669487800000221
Because the error is the driven gear, when the angle error of the driven gear is researched, the angle of the gear 2 is from 0 toIs clockwise, therefore
Figure GDA0002266948780000032
To take
In the involute meshing part, the sum of Δ fpbCausing a transmission error of
Figure GDA0002266948780000035
For variations in radius of base circle caused by positive base pitch deviation (negative pressure angular deviation)It is known thatTherefore, make
Figure GDA0002266948780000037
Then
Figure GDA0002266948780000038
The corresponding error curve is shown in fig. 4 (a). It can be seen that the transmission error curve generated when the gear top edges are engaged is a parabola.
Let the angle of engagement of the top edge be theta because
Figure GDA00022669487800000310
For the
Figure GDA00022669487800000311
Has a period of
Figure GDA00022669487800000312
Therefore, it is
-u(-θ)-Q(-θ)2＝-u(λ-θ)
Therefore, it is not only easy to use
Figure GDA00022669487800000313
By substituting formula (u) for formula (8) and using formula (4)
Figure GDA00022669487800000314
To facilitate the study of the problem, we applied the error function
Figure GDA00022669487800000315
The coordinate transformation is performed as shown in fig. 4 (b).
Figure GDA00022669487800000316
Substituting formula (10) into formula (7) to obtain a transmission error function in a period lambda
Figure GDA00022669487800000317
Expression (2)
Figure GDA00022669487800000318
(2) Reverse order analysis of engaging process
When there is a negative base pitch deviation, the next pair of teeth cannot be engaged after the involute meshing segment of the pair of tooth flanks of the gear 1 and the gear 2 is finished, and this will appear in fig. 1(b)
Figure GDA00022669487800000319
The length of the appearing part is related to the size of the error of the base knot in the segment meshing process. In the involute mesh segment, the contact point gradually moves from the tooth tip to the tooth root on the tooth flank of the driven gear 2 in the normal mesh sequence.
Figure GDA00022669487800000320
The segments belong to the meshing off-line contact process, which differs from the normal meshing sequence. In that
Figure GDA00022669487800000321
In the section, the contact point appears first at A1Point, then contact point on the tooth surface of the gear 2 from a1The point moves to point F, and as can be seen from fig. 1(b), for gear 2, a1The radius of the point is smaller than that of the point F, the process is the movement from the tooth root to the tooth top direction, as shown in figure 3, the process is opposite to the normal meshing sequence, the process is the meshing process of the top blade of the driving gear, and is the reverse phenomenon in the meshing process.
In fig. 1(b), the gear 1 is a driving gear and is a standard gear, and the gear 2 is a driven gear, and has negative base pitch deviation Δ fpb。
The gearing-out process is similar to the gearing-in process, except that the gearing-out phase becomes such that the tooth tips of the gear 1 scrape on the tooth flanks of the gear 2, and for the change in the base radius caused by negative base pitch deviation (positive pressure angle error),
Figure GDA0002266948780000041
thus can also obtain
Wherein
Figure GDA0002266948780000043
Is the theoretical angle of rotation of the gear wheel 2,
Figure GDA0002266948780000044
Figure GDA0002266948780000045
the corresponding error curve is shown in fig. 5 (a). It can be seen from this thatpbThe resulting drive error curve for a < 0 induced top edge engagement is also parabolic.
Let the angle of engagement of the top edge be theta because
Figure GDA0002266948780000046
For the
Figure GDA0002266948780000047
Has a period of
Figure GDA0002266948780000048
Therefore, it is
u*θ-Q*θ2＝u*(θ-λ)
Therefore, it is not only easy to use
Figure GDA0002266948780000049
Will u*Is substituted by a compound of the formula (13),and obtained by the formula (4)
Figure GDA00022669487800000410
To pair
Figure GDA00022669487800000411
Coordinate transformation is performed as shown in FIG. 5(b)
Figure GDA00022669487800000412
By substituting formula (15) for formula (13), the compound
Figure GDA00022669487800000413
Drawings
Fig. 1 shows a gear meshing process with a base pitch offset.
FIG. 2 shows the engagement process in reverse order.
FIG. 3 shows the engagement process in reverse order.
FIG. 4 is a transmission error curve for a positive pitch offset gear.
FIG. 5 is a transmission error curve for a negative base pitch offset gear.
Fig. 6 is a graph of the results of the meshing parameter analysis of fig. 1.
Fig. 7 is a graph of the results of the meshing parameter analysis of fig. 2.
Detailed Description
The invention is further illustrated below with reference to specific examples:
by number of teeth Z155 standard straight spur gear as driving gear, number of teeth Z225, base node error Δ fpbThe spur gear with the diameter of 0.03mm is used as a driven wheel. The modulus m of the driving wheel and the driven wheel is equal to 2, and the reference circle pressure angle alpha is equal to 20 degrees. The processes of engagement and disengagement are analyzed in reverse order.
(1) Reverse order analysis of engagement process
When the deviation of the positive base pitch exists, the gear 1 and the gear 2 can not directly enter the involute meshing section, and then the involute tooth profile can be obtainedAppear in FIG. 1(a)
Figure GDA0002266948780000051
Segment meshing process at A2The point begins to enter the involute meshing section, in which the contact point gradually moves from the tooth root to the tooth tip on the tooth flank of the driving gear 1, which is the normal meshing sequence.
Figure GDA0002266948780000052
The segments belong to the meshing off-line contact process, which differs from the normal meshing sequence. In that
Figure GDA0002266948780000053
In the section, the contact point first occurs at point E, and then the contact point moves from point E to point a on the tooth flank of the gear 12As can be seen from FIG. 1(a), the radius of point E is greater than A for gear 12The radius of the point, which is the movement from the tooth top to the tooth bottom, is the reverse of the normal meshing sequence as shown in fig. 2, which is the meshing process of the driven gear top blade, and is the reverse of the meshing process.
In fig. 1(a), the gear 1 is a driving gear and is a standard gear, and the gear 2 is a driven gear, and has positive base pitch deviation Δ fpb. The meshing process then causes a meshing line increment of
Figure GDA0002266948780000054
Wherein r isb20In order to be the theoretical base radius of the circle,
Figure GDA0002266948780000055
the actual addendum circle pressure angle. Due to the fact that
Figure GDA0002266948780000056
Is small, therefore
Figure GDA0002266948780000057
Figure GDA0002266948780000058
In order to be the actual base circle radius,
Figure GDA0002266948780000059
according to
Figure GDA00022669487800000510
And can obtain
Figure GDA00022669487800000511
Both sides of the equal sign of formula (18) are divided by rb20Then the error of the rotation angle is
Figure GDA00022669487800000512
Order to
Figure GDA00022669487800000513
Then
Figure GDA00022669487800000514
Because the error is the driven gear, when the angle error of the driven gear is researched, the angle of the gear 2 is from 0 to
Figure GDA00022669487800000515
Is clockwise, thereforeTo take
Figure GDA00022669487800000518
In the involute meshing part, the sum of Δ fpbCausing a transmission error of
Figure GDA00022669487800000519
For the change of the base radius caused by the positive base pitch deviation (negative pressure angular deviation), it is known
Figure GDA0002266948780000061
Therefore, make
Figure GDA0002266948780000062
Then
Figure GDA0002266948780000063
Figure GDA0002266948780000064
The corresponding error curve is shown in fig. 4 (a). It can be seen that the transmission error curve generated when the gear top edges are engaged is a parabola.
Let the angle of engagement of the top edge be theta because
Figure GDA0002266948780000065
For the
Figure GDA0002266948780000066
Has a period of
Figure GDA0002266948780000067
Therefore, it is
-u(-θ)-Q(-θ)2＝-u(λ-θ)
Therefore, it is not only easy to use
Figure GDA0002266948780000068
By substituting formula (u) for formula (24) and using formula (20)
Figure GDA0002266948780000069
To facilitate the study of the problem, we applied the error functionThe coordinate transformation is performed as shown in fig. 4 (b).
Figure GDA00022669487800000611
Substituting formula (26) into formula (23) to obtain a transmission error function in a period lambda
Figure GDA00022669487800000612
Expression (2)
Figure GDA00022669487800000613
(2) Reverse order analysis of engaging process
When there is a negative base pitch deviation, the next pair of teeth cannot be engaged after the involute meshing segment of the pair of tooth flanks of the gear 1 and the gear 2 is finished, and this will appear in fig. 1(b)
Figure GDA00022669487800000614
The length of the appearing part is related to the size of the error of the base knot in the segment meshing process. In the involute mesh segment, the contact point gradually moves from the tooth tip to the tooth root on the tooth flank of the driven gear 2 in the normal mesh sequence.
Figure GDA00022669487800000615
The segments belong to the meshing off-line contact process, which differs from the normal meshing sequence. In that
Figure GDA00022669487800000616
In the section, the contact point appears first at A1Point, then contact point on the tooth surface of the gear 2 from a1The point moves to point F, and as can be seen from fig. 1(b), for gear 2, a1The radius of the point is smaller than that of the point F, the process is that the tooth root moves towards the tooth top,as shown in fig. 3, this process is a driving gear top edge engagement process, which is a reverse phenomenon in the engaging-out process, contrary to the normal engagement sequence.
In fig. 1(b), the gear 1 is a driving gear and is a standard gear, and the gear 2 is a driven gear, and has negative base pitch deviation Δ fpb。
The gearing-out process is similar to the gearing-in process, except that the gearing-out phase becomes such that the tooth tips of the gear 1 scrape on the tooth flanks of the gear 2, and for the change in the base radius caused by negative base pitch deviation (positive pressure angle error),
Figure GDA00022669487800000617
thus can also obtain
Figure GDA00022669487800000618
Wherein
Figure GDA0002266948780000071
Is the theoretical angle of rotation of the gear wheel 2,
Figure GDA0002266948780000072
Figure GDA0002266948780000073
the corresponding error curve is shown in fig. 5 (a). It can be seen from this thatpbThe resulting drive error curve for a < 0 induced top edge engagement is also parabolic.
Let the angle of engagement of the top edge be theta becauseFor the
Figure GDA0002266948780000075
Has a period of
Figure GDA0002266948780000076
Therefore, it is
u*θ-Q*θ2＝u*(θ-λ)
Therefore, it is not only easy to use
Figure GDA0002266948780000077
Will u*By substituting the formula (29) and using the formula (20)
Figure GDA0002266948780000078
To pair
Figure GDA0002266948780000079
Coordinate transformation is performed as shown in FIG. 5(b)
Figure GDA00022669487800000710
By substituting formula (31) for formula (29), a compound of formula
Figure GDA00022669487800000711
。

Claims (1)

1. A method for establishing a gear negative sequence meshing model is used for explaining a negative sequence meshing phenomenon in the actual error gear meshing process, and deducing a transmission error model in the negative sequence meshing process aiming at gear pair transmission with a pitch error; the reverse gear meshing means that the sequence of the meshing process is opposite to that of the normal meshing process; this sequence refers to the direction of movement of the contact point on the tooth flank, if only the case of single-sided contact is considered, during normal meshing, on the driving gear the contact point moves from the tooth root to the tooth tip, and on the driven gear from the tooth tip to the tooth root; the reverse meshing process is the phenomenon that the contact point moves from the tooth top to the tooth bottom on the driving gear and moves from the tooth bottom to the tooth top on the driven gear outside the normal meshing process; when the gear has base pitch deviation, the two reverse sequence processes can occur;
in the pair of unilateral tooth surfaces, the gear 1 is a driving gear, and the gear 2 is a driven gear; points E and F are the intersection points of the addendum circles of the driving and driven gears, A1The point is the tooth crest and the meshing line of the gear 1
Figure FDA0002266948770000011
A point of intersection of2The point is that the gear 2 and the meshing line
Figure FDA0002266948770000012
Point P is the node where the two gears mesh; the first point where the tooth surface contact occurs must be located at points E and A according to the rotation direction of the two gears2Between points, and the last point where tooth surface contact occurs must be located at A1Between point and point F; when the actual meshing point moves to point M, the angle through which the gear 1 rotates is
Figure FDA0002266948770000013
The angle through which the gear 2 rotates is
Figure FDA0002266948770000014
When the gear 1 and the gear 2 rotate in opposite directions, the angleAnd
Figure FDA0002266948770000016
recording as a negative value;
the method comprises the following steps:
(1) reverse order analysis of engagement process
When the deviation of the positive base pitch exists, the gear 1 and the gear 2 can not directly enter an involute meshing section, and the phenomenon that the gear 1 and the gear 2 can not directly enter the involute meshing section occurs
Figure FDA0002266948770000017
Segment meshing process at A2The point begins to enter the involute mesh segment, in which the contact point gradually moves from the tooth root to the tooth tip on the tooth flank of the gear 1, and is normalThe meshing sequence;
Figure FDA0002266948770000018
the segments belonging to the line of engagement
Figure FDA0002266948770000019
An external contact process, which is different from the normal engagement sequence; in that
Figure FDA00022669487700000110
In the section, the contact point first occurs at point E, and then the contact point moves from point E to point a on the tooth flank of the gear 12Point, E point having a radius larger than a for the gear 12The radius of the point is the movement from the tooth top to the tooth bottom, the process is opposite to the normal meshing sequence, the process is the meshing process of the driven gear top blade, and is the reverse phenomenon in the meshing process; the gear 1 is a driving gear and is a standard gear, the gear 2 is a driven gear and has positive base pitch deviation delta fpb(ii) a Then the engagement line caused by the engagement process
Figure FDA00022669487700000111
Is increased by
Figure FDA00022669487700000112
Wherein r isb20In order to be the theoretical base radius of the circle,
Figure FDA00022669487700000113
is the actual addendum circle pressure angle, ra2Is the radius of the top circle of the driven gear,rotating the gear by an angle; due to the rotation angle of the gear
Figure FDA00022669487700000115
Line of engagement
Figure FDA00022669487700000116
Angle corresponding to the increment of (a)
Figure FDA00022669487700000117
As small as negligible and therefore
Figure FDA00022669487700000119
In order to be the actual base circle radius,
Figure FDA00022669487700000120
rbis the base circle radius; let λ be the driven gear rotation angle error
Figure FDA00022669487700000121
Has a period of
Figure FDA00022669487700000122
Z2Is the number of the passive gear teeth; according to
And can obtain
Both sides of the equal sign of formula (2) are divided by rb20Then the error of the rotation angle is
Figure FDA00022669487700000125
Order to
Figure FDA00022669487700000126
Then
Figure FDA0002266948770000021
Because the error is the driven gear, when the angle error of the driven gear is researched, the angle of the gear 2 is from 0 to
Figure FDA0002266948770000022
Is clockwise, therefore
Figure FDA0002266948770000023
To take
Figure FDA0002266948770000024
Figure FDA0002266948770000025
In the involute meshing part, the sum of Δ fpbCausing a transmission error of
Figure FDA0002266948770000026
Figure FDA0002266948770000027
Is the theoretical angle of rotation, r, of the gear wheel 2b1And rb2The theoretical base circle radiuses of the driving gear and the driven gear are respectively; the change in the radius of the base circle due to the positive pitch error is known
Figure FDA0002266948770000028
Therefore, makeThen
Figure FDA00022669487700000210
The transmission error curve generated when the gear top edge is meshed is a parabola;
let the angle of engagement of the top edge be theta because
Figure FDA00022669487700000211
For theHas a period ofTherefore, it is
-u(-θ)-Q(-θ)2＝-u(λ-θ)
Therefore, it is not only easy to use
Figure FDA00022669487700000214
By substituting formula (u) for formula (8) and using formula (4)
For error function
Figure FDA00022669487700000216
Carrying out coordinate transformation;
Figure FDA00022669487700000217
substituting formula (10) into formula (7) to obtain a transmission error function in a period lambda
Figure FDA00022669487700000218
Expression (2)
Figure FDA00022669487700000219
(2) Reverse order analysis of engaging process
When negative base pitch deviation exists, after the gear 1 and the gear 2 finish the involute meshing section of the pair of tooth surfaces, the next pair of gear teeth can not enter into meshing, and the phenomenon that the next pair of gear teeth can not enter into meshing occurs
Figure FDA00022669487700000220
Segment meshing process, the length of the appearing part is related to the size of the error of the base joint; in the involute meshing section, the contact points gradually move from the tooth top to the tooth bottom on the tooth surface of the gear 2, and the normal meshing sequence is adopted;
Figure FDA00022669487700000221
the segments belonging to the line of engagement
Figure FDA00022669487700000222
An external contact process, which is different from a normal engagement sequence; in that
Figure FDA00022669487700000223
In the section, the contact point appears first at A1Point, then contact point on the tooth surface of the gear 2 from a1The point moves to point F, A for gear 21The radius of the point is smaller than that of the point F, the process is the movement from the tooth root to the tooth top direction, the sequence is opposite to the normal meshing sequence, the process is the meshing process of the top blade of the driving gear, and the process is the reverse sequence phenomenon in the meshing process;
the gear 1 is a driving gear and is a standard gear, the gear 2 is a driven gear and has negative base pitch deviation delta fpb；
The gearing-out process is similar to the gearing-in process, except that the gearing-out phase becomes such that the tooth tips of the gear 1 scrape on the tooth flanks of the gear 2, and for a change in the base radius caused by a negative base pitch deviation,
Figure FDA00022669487700000224
thus obtaining
Figure FDA0002266948770000031
Wherein
Figure FDA0002266948770000032
Is the theoretical angle of rotation of the gear wheel 2,
Figure FDA0002266948770000033
wherein r isa1Is the radius of the addendum circle of the driving gear, alphaa1The theoretical pressure angle of the addendum circle of the driving gear is shown;
the corresponding error curve is obtained; it can be seen from this thatpbThe transmission error curve generated by the meshing of the top edges caused by < 0 is also a parabola;
let the angle of engagement of the top edge be theta, sincepbInduced transmission error
Figure FDA0002266948770000035
For the
Figure FDA0002266948770000036
Has a period of
Figure FDA0002266948770000037
Therefore, it is
u*θ-Q*θ2＝u*(θ-λ)
Therefore, it is not only easy to use
Figure FDA0002266948770000038
Will u*Substituted for formula (13) and obtained by using formula (4)
Figure FDA0002266948770000039
To pair
Figure FDA00022669487700000310
The coordinate transformation is carried out to carry out the coordinate transformation,
by substituting formula (15) for formula (13), the compound
Figure FDA00022669487700000312
。
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CN101915667A (en) * 2010-07-23 2010-12-15 北京工业大学 Integrated error measuring technology and method thereof of gear pair
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