CN113742976A - High-speed gear finite element analysis tooth profile mesh discretization method based on sliding energy loss - Google Patents

High-speed gear finite element analysis tooth profile mesh discretization method based on sliding energy loss Download PDF

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CN113742976A
CN113742976A CN202111079924.1A CN202111079924A CN113742976A CN 113742976 A CN113742976 A CN 113742976A CN 202111079924 A CN202111079924 A CN 202111079924A CN 113742976 A CN113742976 A CN 113742976A
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黄一伦
陈旭
胡玉梅
富丽娟
杨英
宋洁
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Chongqing University of Technology
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Abstract

The invention relates to the technical field of gear meshing finite element simulation analysis, and particularly discloses a tooth profile mesh discretization method for high-speed gear finite element analysis based on sliding energy loss. The method solves the problem of accuracy of the tooth profile finite element mesh dispersion of the high-speed light heavy-duty straight toothed spur gear, and effectively reduces the tooth profile error caused by the finite element mesh dispersion, thereby ensuring the reliability of the subsequent finite element simulation analysis result.

Description

High-speed gear finite element analysis tooth profile mesh discretization method based on sliding energy loss
Technical Field
The invention belongs to the technical field of gear meshing finite element simulation analysis, and particularly relates to a tooth profile mesh discrete method for high-speed gear finite element analysis based on sliding energy loss.
Background
The finite element is used as a common gear meshing simulation analysis method and is widely applied to the research of the dynamic performance of gears, wherein when the tooth profile of a gear is subjected to finite element mesh dispersion, the size of the mesh has great influence on the correctness of the gear meshing simulation analysis result under high linear speed, along with the continuous development of the modern aerospace technology, the reliability of the gear by the aeromachinery is higher and higher, wherein the high linear speed spur gear is used as a key part for the power transmission of the aeroengine, the reliability directly influences the realization of the functionality and the structural safety of the aeroengine, the service process of the high linear speed gear presents strong nonlinearity and transient performance due to the characteristics of light weight, high rotary speed, large load and the like, the speed and the acceleration of all parts of a gear transmission system and the interaction force among all contact surfaces are more instantaneous changes, the method has no rules, and higher requirements are provided for the mesh discretization technology of the tooth profile in the finite element mesh model of the gear pair.
The prior mainstream gear tooth profile mesh discrete method is to carry out mesh discrete on a tooth surface according to the computing capability of a computer, the number of meshes which are discrete from a tooth root to a tooth top is usually 7-12, the method is suitable for the mesh discrete of the tooth surface of a low-speed gear, and the tooth surface mesh size determined by the method is a fixed value and cannot adapt to the changes of the tooth surface rolling speed and the tooth surface curvature radius in the processes of engaging and disengaging the tooth surface.
Disclosure of Invention
In view of the above situation, in order to overcome the defects in the prior art, the invention provides a tooth profile mesh discretization method for finite element analysis of a high-speed gear based on sliding energy loss, which effectively solves the problems proposed in the background art.
In order to solve the problem of accurate dispersion of a tooth surface finite element mesh of a high-speed straight-tooth cylindrical gear, a tooth profile mesh dispersion method for finite element analysis of the high-speed gear based on sliding energy loss is necessary, the method adopts a two-disc curved surface contact finite element mesh model to simulate a tooth surface contact finite element mesh model, and determines the optimal value of the mesh size of the tooth profile finite element mesh model by comparing and analyzing simulation analysis results and theoretical calculated values of the two-disc curved surface contact finite element mesh model under different parameter conditions, and specifically comprises the following steps:
deducing a curvature calculation formula at a meshing point of the driving wheel and the driven wheel according to a geometric relation in the meshing process of the driving wheel and the driven wheel, and respectively calculating curvature radiuses of the driving wheel and the driven wheel at the meshing point and the meshing point according to the curvature calculation formula;
deducing a calculation method of tangential speed and relative slip speed of the driving wheel and the driven wheel at the meshing point according to the speed vector relation in the meshing process of the driving wheel and the driven wheel, so as to respectively obtain the maximum tooth surface tangential speed and the relative slip speed in the meshing process of the driving wheel and the driven wheel;
step three, establishing a plurality of two-disc curved surface contact finite element mesh models according to the curvature radius, the maximum tooth surface tangential speed, the relative slip speed and different mesh sizes obtained in the step one and the step two, and performing simulation calculation on the slip energy loss through the plurality of two-disc curved surface contact finite element mesh models, and comparing the slip energy loss with theoretical calculated values respectively to obtain the optimal mesh size;
and step four, establishing a discrete tooth profile model of the straight spur gear according to the grid size determined in the step three.
Further, in the first step, according to the curvature radius relation of any meshing position of the driving wheel and the driven wheel, the curvature radius of a meshing point of the driving wheel and the driven wheel is obtained, a gear involute tooth profile is formed when the driving wheel and the driven wheel are meshed, the gear involute tooth profile is a smooth curve formed by a series of circles with different curvature radii, and the curvature radius is gradually increased from a tooth root to a tooth top.
Furthermore, on the involute, the curvature radius of any meshing point on the tooth surface is equal to the length of the generating line of the involute tooth surface, namely the distance from the driving wheel to the meshing point to N1, the distance from the driven wheel to the meshing point to N2,
the minimum and maximum meshing points of the primary sheave tooth surface curvature radius are labeled B1 and B2, respectively, for the minimum curvature radius ρ'1Equal to B1N1, maximum radius of curvature ρ'2Equal to the sum of B2N1,
the point where the radius of curvature of the driven wheel tooth surface is smallest and the point where the radius of curvature is largest are B2 and B1, respectively, corresponding to the smallest radius of curvature ρ ″1Equal to B2N2, maximum radius of curvature ρ ″)2Equal to the sum of B1N2,
when the driving wheel and the driven wheel are both installed according to the standard center, the pitch circle radius is equal to the index table radius, which can be obtained according to the perpendicular relation between O1N1 and the meshing line N1N2,
N1P＝PO1×sinα＝r1sinα (1)
Figure BDA0003263550480000031
Figure BDA0003263550480000032
in formulae (1) to (3):
rk1、rk2the radius of any engagement point of K1 and K2 on the driving wheel;
rb1-the base radius of the capstan;
r1-the driving wheel reference circle radius;
alpha-the angle of engagement;
any position near the root of the driven or primary pulley is marked as K1, then K1 ═ dk1p，dk1pThe distance on the meshing line of the node P and the meshing point K1,
Figure BDA0003263550480000033
the engagement point K1 at any position near the tooth crest of the driving wheel gear or driven wheel is at the driving wheelRadius of curvature ρ 'on the wheel'k1And radius of curvature ρ ″' on the driven wheelk1Can be represented as, respectively, a,
ρ′k1＝K1N1＝N1P+K1P＝r1sinα-dk1p (5)
ρ″k1＝K1N2＝N2P+K1P＝r2sinα+dk1p (6)
in formulae (4) to (6):
dk1p-distance of mesh point K1 from node P on the mesh line;
r1、r2-the main and driven wheel main reference circle radii;
ρ′k1、ρ″k1-radius of curvature of the meshing points on the driving and driven wheels;
any position engaging point K2 near the root of the driven wheel or the top of the driving wheel, K2 is equal to dk2p，dk2pThe distance of the meshing point K2 from the node C on the meshing line,
Figure BDA0003263550480000041
the radius of curvature ρ 'of the capstan at any position of the meshing point K2 near the capstan tip or the idler root'k2And radius of curvature ρ ″' on the driven wheelk2Can be respectively expressed as:
ρ′k2＝K2N1＝N1P+K2P＝r1sinα+dk2p (8)
ρ″k2＝K2N2＝N2P-K2P＝r2sinα-dk2p (9)
in formulae (7) to (9):
dk2p-distance of mesh point K2 from node C on the mesh line;
according to the formulas (1) to (9), arbitrary meshing points K1 and K2 on two sides of the O1O2 line are alignedThe corresponding relational expressions are integrated together, and the position d of the meshing point at any position on the tooth surface on the meshing linekpCan be expressed as:
Figure BDA0003263550480000042
in the formula (10)
dkp-distance of any meshing point from node P on the meshing line;
rk-radius of any engagement point on the driving wheel;
the curvature radiuses ρ', ρ ″ of the engagement point at any position on the driving wheel and the driven wheel can be expressed as:
ρ′＝r1siaα±dkp (11)
Figure BDA0003263550480000043
in equations (10) to (12), the lower symbols are used when the meshing point is close to the driven wheel tooth crest or the driving wheel tooth crest, and the upper symbols are used when the meshing point is close to the driven wheel tooth crest or the driving wheel tooth crest.
Further, when there is a backlash between the primary pulley and the secondary pulley, i.e., the addendum portion is fully engaged in tooth engagement and only a partial region of the dedendum is engaged in tooth engagement, the point B2 is located near the addendum of the primary pulley and the dedendum of the secondary pulley, the point B1 is located near the dedendum of the primary pulley and the addendum of the secondary pulley, and the radius of the primary pulley at B2 is r'kEqual to the radius of the addendum circle of the driving wheel,
Figure BDA0003263550480000051
and the radius of the driving wheel at the meshing point B1 is r ″)kCan be based on a right triangle delta O2N2B1And a right triangle delta O1N1B1The geometric relationship between the two components is obtained,
N1N2＝N1P+PN2＝r1sinα+r2sinα (14)
Figure BDA0003263550480000052
Figure BDA0003263550480000053
Figure BDA0003263550480000054
in the formula (10), r at B2kValue is r 'from size'kAnd r at B1kThe value is r ″)k，
And the calculated maximum and minimum curvature radius is used as the basis for determining the size of the tooth surface mesh.
Further, in the second step, according to the velocity vector triangle and the right triangle delta K of any engagement point K2 on the driving wheel2O1N1The similarity relationship of (a) to (b) can be obtained,
V1t、V1t、V1nthe absolute speed, the tangential speed and the normal speed of the meshing point K2 on the driving wheel respectively;
Figure BDA0003263550480000055
so that the tangential speed V of any meshing point on the driving wheel along the tooth surface1tCan be expressed as:
Figure BDA0003263550480000056
in formulae (18) to (19):
n1-the rotational speed of the driving wheel in r/min;
ρ' -radius of curvature of any meshing point K2 on the capstan in mm;
V1t-tangential speed of any meshing point K2 on the capstan along the tooth surface, in m/s;
according to the speed vector triangle and the right triangle delta K of any meshing point K2 on the driven wheel2O2N2The similarity relationship of (a) to (b) can be obtained,
V2k、V2t、V2nabsolute, tangential and normal speeds of the engagement point K2 on the driven wheel
Figure BDA0003263550480000061
The tangential velocity V of any meshing point on the driven wheel along the tooth surface2tCan be expressed as:
Figure BDA0003263550480000062
in formulae (20) to (21):
n2-driven wheel speed, in r/min;
ρ "— the radius of curvature of any meshing point K2 on the driven wheel, in mm;
V2t-tangential speed of any meshing point K2 on the driven wheel along the tooth surface, in m/s;
relative slip speed V at any meshing point of driving wheel and driven wheeltCan be expressed as:
Vt＝|V1t-V2t| (23)，
the relative slip speed between the driving wheel and the driven wheel at the point of tooth flank engagement and the maximum tangential speed of the tooth flank, namely the relative slip speed at the point of B2 or B1 and the tangential speed of the tooth flank are obtained according to the formulas (18) to (23),
wherein the curvature radius rho 'of the point B2 on the driving wheel is B2N1, the curvature radius rho' of the driven wheel is B2N2, the curvature radius rho 'of the point B1 on the driving wheel is B1N1, the curvature radius rho' of the driven wheel is B1N2,
and the calculated relative slip speed is used as the basis for determining the rotating speed of the disc in the simplified disc model.
Further, in the third step, the mesh size of the two-disc curved surface contact finite element mesh model is determined according to the mesh size standard.
Further, the grid size criteria are: for a tooth surface maximum tangential velocity greater than 25m/s for a gear span, the grid size ranges from R/50 to R/110, R being the radius of curvature of the disk.
Further, the grid size criteria are: the maximum tangential speed of the tooth surface is 25m/s-40m/s, and the mesh size range of the tooth surface is R/75; the maximum tangential speed of the tooth surface is 40m/s-55m/s, and the mesh size range of the tooth surface is R/85; the maximum tangential speed of the tooth surface is 55m/s-70m/s, and the mesh size range of the tooth surface is R/100; and the maximum tangential speed of the tooth surface is 70m/s-90m/s, the size range of the tooth surface grid is R/110, and R is the curvature radius of the disc.
Compared with the prior art, the invention has the beneficial effects that:
the method solves the problem of accuracy of the tooth profile finite element mesh dispersion of the high-speed light heavy-duty straight toothed spur gear, and effectively reduces the tooth profile error caused by the finite element mesh dispersion, thereby ensuring the reliability of the subsequent finite element simulation analysis result.
Drawings
The accompanying drawings, which are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 is a schematic illustration of the gear tooth profile meshing process of the present invention;
FIG. 2 is a velocity vector diagram of the engagement point K2 on the driving wheel according to the present invention;
FIG. 3 is a velocity vector diagram of the mesh point K2 of the present invention on the driven wheel;
FIG. 4 is a schematic view of a contact model of a disc surface according to the present invention;
FIG. 5 is a disk surface contact mesh model of the present invention;
FIG. 6 is a graph showing the slip energy curves of the upper and lower disks of the present invention at linear speeds of 20m/s and 0 m/s;
FIG. 7 is a graph of slip energy at 25m/s and 5m/s linear speeds for the upper and lower discs of the present invention;
FIG. 8 is a graph showing the slip energy curves of the upper and lower disks of the present invention at linear speeds of 40m/s and 20 m/s;
FIG. 9 is a graph of slip energy for 55m/s and 35m/s linear speeds for upper and lower disks in accordance with the present invention;
FIG. 10 is a graph of slip energy at linear speeds of 70m/s and 50m/s for upper and lower disks in accordance with the present invention;
FIG. 11 is a graph showing the slip energy curves of the upper and lower disks of the present invention at linear speeds of 90m/s and 70 m/s;
FIG. 12 is a finite element mesh discretization of a gear tooth profile according to the present invention;
FIG. 13 is a flow chart of the present invention for a method for accurately discretizing a finite element mesh for a high linear speed spur gear tooth profile.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments; all other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in fig. 1 to 13, the present embodiment provides a tooth profile mesh discretization method for finite element analysis of high-speed gears based on sliding energy loss, which includes the following steps:
the method comprises the following steps:
the involute tooth profile of gear is a smooth curve composed of a series of circles with different curvature radiuses, and the curvature radiuses are gradually increased from the tooth root to the tooth tip. A theoretical calculation formula of the curvature radius of any contact point of the tooth surface can be deduced according to the gear meshing process. As shown in figure 1, the tooth profile meshing process is a schematic view, P is a node point, K2 is an engagement point at any position close to the tooth root of a driven wheel or the tooth crest of a driving wheel, and K1 is a position close to the tooth crest of the driven wheel or the tooth root of the driving wheelAny position of the engagement point. N1N2 is the theoretical meshing line, B1B2 is the actual meshing line, and the driving wheel enters into the meshing from the point B1 and pushes the tooth crest of the driven wheel to rotate, and the driven wheel is disengaged from the tooth crest of the driving wheel at the point B2. According to the involute theory, the curvature radius of any meshing point on the tooth surface is equal to the length of the generating line of the involute tooth surface, namely the distance from the meshing point to N1 (the distance from the driven wheel to the N2), and the curvature radius is gradually increased from the tooth root to the tooth crest, so that the meshing point with the minimum curvature radius of the tooth surface of the driving wheel and the maximum meshing point are respectively B1 and B2, and the corresponding minimum curvature radius rho'1Equal to B1N1, maximum radius of curvature ρ'2Equal to B2N1, the points where the radius of curvature of the face contact portion of the driven wheel is smallest and largest are B2 and B1, respectively, corresponding to the smallest radius of curvature ρ ″1Equal to B2N2, maximum radius of curvature ρ ″)2Equal to B1N 2.
Assuming that the gears are mounted at a standard center-to-center distance, the pitch radius is equal to the pitch radius. Using the perpendicular relationship of O1N1 to the meshing line N1N2, one can see:
N1P＝PO1×sinα＝r1sinα (1)
Figure BDA0003263550480000091
Figure BDA0003263550480000092
in formulae (1) to (3):
rk1、rk2the radius of any engagement point of K1 and K2 on the driving wheel;
rb1-the base radius of the capstan;
r1-the driving wheel reference circle radius;
alpha-the angle of engagement;
according to the geometrical relation in the tooth profile meshing process schematic diagram, the meshing point K1 at any position close to the driven wheel tooth crest or the driving wheel tooth root (namely the position in figure 1)In O1O2The right-hand mesh point) on the mesh line (i.e., the distance d between the mesh point K1 and the node P on the mesh linek1p) Can be expressed as:
Figure BDA0003263550480000093
so any location near the capstan root or the idler tip mesh point K1 radius of curvature ρ 'on the capstan'k1And radius of curvature ρ ″' on the driven wheelk1Can be respectively expressed as:
ρ′k1＝K1N1＝N1P-K1P＝r1sinα-dk1p (5)
ρ″k1＝K1N2＝N2P+K1P＝r2sinα+dk1p (6)
in formulae (4) to (6):
dk1p-distance of mesh point K1 from node P on the mesh line;
r1、r2-the main and driven wheel main reference circle radii;
ρ′k1、ρ″k1-radius of curvature of the meshing points on the driving and driven wheels;
engaging point K2 at any position near the driven wheel tooth root or the driving wheel tooth crest (i.e. at O in FIG. 1)1O2Mesh point on left) on the mesh line (i.e., distance d between mesh point K2 and node C on the mesh linek2p) Can be expressed as:
Figure BDA0003263550480000101
so the radius of curvature ρ 'of the meshing point K2 at any position on the capstan near the capstan tip or the idler root'k2And radius of curvature ρ ″' on the driven wheelk2Can be respectively expressed as:
ρ′k2＝K2N1＝N1P+K2P＝r1sinα+dk2p (8)
ρ″k2＝K2N2＝N2P-K2P＝r2sinα-dk2p (9)
in formulae (7) to (9):
dk2p-distance of mesh point K2 from node C on the mesh line;
according to the formulas (1) - (9), the relational expressions corresponding to any meshing points K1 and K2 on two sides of the O1O2 line are integrated, and the position d of the meshing point at any position on the tooth surface on the meshing linekpCan be expressed as:
Figure BDA0003263550480000102
in the formula (10)
dkp-distance of any meshing point from node P on the meshing line;
rk-radius of any engagement point on the driving wheel;
therefore, the curvature radiuses ρ', ρ ″ of the engagement point at any position on the driving wheel and the driven wheel can be expressed as:
ρ′＝r1sinα±dkp (11)
Figure BDA0003263550480000111
in equations (10) to (12), the lower symbols are used when the meshing point is close to the driven wheel tooth crest or the driving wheel tooth crest, and the upper symbols are used when the meshing point is close to the driven wheel tooth crest or the driving wheel tooth crest.
The minimum radius of curvature ρ 'of the tooth surface of the primary sheave can be obtained by equations (10) to (12)'1And a maximum radius of curvature ρ'2And the minimum curvature radius rho ″' of the driven wheel tooth surface1And a maximum radius of curvature ρ ″)2That is, when the maximum engagement points B1 and B2 are found, B1N1, B2N1, B2N2 and B1N2 are largeIs small. Due to the backlash between the drive and driven wheel addendum and dedendum, typically the tooth flank addendum portion is substantially fully engaged in tooth engagement, while only a partial region of the dedendum is engaged in tooth engagement. According to the tooth profile meshing process schematic diagram, the B2 point is located near the tooth crest of the driving wheel and the tooth root of the driven wheel, the B1 point is located near the tooth root of the driving wheel and the tooth crest of the driven wheel, and the radius r of the driving wheel at the meshing point B2k' equal to the radius of the addendum circle of the driving wheel:
Figure BDA0003263550480000112
and the radius of the driving wheel at the meshing point B1 is r ″)kCan be based on a right triangle delta O2N2B1And a right triangle delta O1N1B1The geometrical relationship between them yields:
N1N2＝N1P+PN2＝r1sinα+r2sinα (14)
Figure BDA0003263550480000113
Figure BDA0003263550480000114
Figure BDA0003263550480000115
so in equation (10), r at B2kValue is r 'from size'kAnd r at B1kThe value is r ″)k. And taking the calculated maximum and minimum curvature radius as a determination basis of the subsequent tooth surface mesh size.
The basic parameters of each gear in an aircraft engine are substituted into the formulas (10) - (12), and the curvature radius of the tooth surface of the gear of the aircraft engine is generally between 10 and 37mm, so that the curvature radius of an upper disc and a lower disc in a subsequent disc simplified model is between 10 and 37 mm.
Step two:
the disc simplified model is mainly used for simulating the contact of the tooth surface of the driving wheel and the tooth surface of the driven wheel in the gear tooth meshing process of the gear, so that the relative slip speed of the tooth surfaces in the gear meshing process needs to be calculated according to a theoretical formula before the finite element grids of the contact interface of the two discs are dispersed, and the rotating speed of the discs in the disc simplified model is determined.
Since the tooth surface tangential velocity and the relative slip velocity at any meshing point on the meshing line are calculated in the same manner, the theoretical formula derivation of the tooth surface tangential velocity and the relative slip velocity is performed below using any meshing point K2 as an example.
As shown in fig. 2, which is a schematic view of tooth profile meshing containing a velocity vector of a K2 meshing point on a driving wheel, B1B2 is an actual meshing line, the driving wheel enters into meshing at a point B1 and pushes the tooth crest of a driven wheel to rotate, and the driven wheel is disengaged from the tooth crest of the driving wheel at a point B2. The tooth surface relative slip speed changes in the process from engagement to engagement, and the tooth surface relative slip speed changes in a decreasing and increasing mode, so that the gear teeth have the most serious relative slip during engagement and the point with the highest relative slip speed is B2 or B1.
V in FIG. 21k、V1t、V1nThe absolute speed, the tangential speed and the normal speed of the engagement point K2 on the driving wheel respectively are determined according to the speed vector triangle and the right-angle triangle delta K of any engagement point K2 (the same principle is adopted in the method for calculating the relative slip speed of the K1 point) on the driving wheel2O1N1The similarity relationship of (a) can be found in:
Figure BDA0003263550480000121
so that the tangential speed V of any meshing point on the driving wheel along the tooth surface1tCan be expressed as:
Figure BDA0003263550480000131
in formulae (18) to (19):
n1-the rotational speed of the driving wheel in r/min;
ρ' -radius of curvature of any meshing point K2 on the capstan in mm;
V1t-tangential speed of any meshing point K2 on the capstan along the tooth surface, in m/s;
FIG. 3 is a schematic diagram of tooth profile engagement including a velocity vector of the engagement point K2 on the driven wheel, V2k、V2t、V2nThe absolute speed, tangential speed and normal speed of the mesh point K2 on the driven wheel, respectively.
According to the speed vector triangle and the right triangle delta K of any meshing point K2 (the same principle of the relative slip speed calculation method of the K1 point) on the driven wheel2O2N2The similarity relationship of (a) can be found in:
Figure BDA0003263550480000132
therefore, the tangential speed V of any meshing point on the driven wheel along the tooth surface2tCan be expressed as:
Figure BDA0003263550480000133
in formulae (20) to (21):
n2-driven wheel speed, in r/min;
ρ "— the radius of curvature of any meshing point K2 on the driven wheel, in mm;
V2t-tangential speed of any meshing point K2 on the driven wheel along the tooth surface, in m/s;
relative slip speed V at any meshing point of driving wheel and driven wheeltCan be expressed as:
Vt＝|V1t-V2t| (23)
from equations (18) to (23), the relative slip velocity between the driving and driven wheels at the point where the tooth flanks mesh in, and the maximum tangential velocity of the tooth flanks, that is, the relative slip velocity and the tooth flank tangential velocity at the point B2 or B1, can be found. The curvature radius rho 'of the point B2 on the driving wheel is B2N1, the curvature radius rho' of the driven wheel is B2N2, the curvature radius rho 'of the point B1 on the driving wheel is B1N1, and the curvature radius rho' of the driven wheel is B1N 2. And taking the calculated relative slip speed as a basis for determining the rotating speed of the disc in the subsequent simplified disc model.
The size of the tooth surface relative slip speed is related to various parameters of the gear, wherein the tooth surface tangential speed of the driving wheel and the driven wheel directly influences the tooth surface relative slip speed. It can be seen from the equations (18) to (23) that for low speed gears with a linear speed of less than 25m/s, the tooth flank tangential speed at the point of engagement is between about 5m/s and 25 m/s; and the linear speed is more than 25m/s and less than 145m/s, and the tooth surface tangential speed is between 10m/s and 90 m/s. For aviation gears, the linear velocity is typically 80m/s to 145m/s, and the relative slip velocity at the point of engagement is typically around 15m/s to 40 m/s. So that the relative speed of the upper and lower disks in the subsequent disk simplified model is kept between 15m/s and 40m/s, and the tangential speed does not exceed 90 m/s.
Step three:
and (3) establishing a curved surface contact finite element model of the two disks to simulate the tooth surface contact according to the parameters of the curvature radius, the maximum tooth surface tangential speed, the relative sliding speed and the like calculated in the first step and the second step, analyzing the sliding energy loss generated by the contact friction of the two disks at different rotating speeds but with a rotating speed difference and a constant load through finite element simulation, and comparing the sliding energy loss with the theoretical calculated value to obtain the corresponding optimal mesh size at different rotating speeds. The sliding energy loss is the energy exchange caused by the collision of two surfaces sliding relatively, and is inevitable. The magnitude of the slip energy of the curved surface contact is related to the friction coefficient and the positive pressure, and the magnitude of the slip energy is equal to the product of the friction force and the slip distance. If the precision requirement is high, the slip energy error needs to be ensured within 5 percent, and for the general precision requirement, the slip energy error needs to be ensured within 10 percent. And (4) according to data such as contact force, rotating speed, slip energy and the like output by finite element simulation analysis, programming and calculating a slip energy theoretical value, and analyzing and comparing the error between the slip energy theoretical value and a simulation value to draw a conclusion.
In order to explore the relationship between the size of the tooth surface mesh and the curvature radius under different tooth surface tangential speeds, the relative sliding speed of an upper disc and a lower disc in a disc curved surface contact model is kept to be 20m/s, 6 disc curved surface contact models with different linear speeds are respectively established, specific parameters are shown in table 1, and R in table 1 represents the disc curvature radius.
TABLE 1 disc surface contact model parameter table
Figure BDA0003263550480000151
The schematic diagram of the disc curved surface contact model is shown in fig. 4, the curvature radius of the upper disc is 15mm, the curvature radius of the lower disc is 20mm, the upper disc and the lower disc respectively apply different rotating speeds, and a vertical downward load is applied to the upper disc, and the size of the load is 3000N. And (3) constraining the translational freedom degree of the lower disc X, Y, Z in the axial direction and the rotational freedom degree of the lower disc X, Y in the axial direction, releasing only the translational freedom degree of the load direction and the rotational freedom degree of the Z axis by the upper disc, finally establishing contact between the contact surfaces of the upper disc and the lower disc, and completing establishment of a disc curved surface contact grid model, as shown in fig. 5.
As shown in FIG. 6, the slip energy curves in the case of the load stabilization phase, the linear speeds of the upper and lower disks being 20m/s and 0m/s, the mesh sizes of the disk contact surfaces being R/50, R/35, R/25 and R/10 (where R and R are both the disk curvature radius and the tooth surface curvature radius for the gear), and the like are indicated, and the slip energy errors at 0.02s, 0.05s and 0.08s are indicated. As can be seen from the figure, as the grid is refined, the percentage error between the simulated value and the theoretical value of the slip performance is gradually reduced as a whole. When the grid size is R/10, the percentage error value reaches about 23 percent, and is obviously larger; the percentage error value when the grid size is about R/25 is between 5% and 6%, and is relatively reasonable; and the percentage error values of the grid sizes from R/50 to R/35 are all less than 5%. Therefore, at the rotating speed, the grid sizes of R/25-R/50 all meet the engineering precision requirement.
As shown in FIG. 7, the slip energy curves are shown in the load stabilization stage when the linear speeds of the upper and lower disks are 25m/s and 5m/s, and the grid sizes of the contact surfaces of the disks are R/50, R/35, R/25, R/10, etc. When the grid size is R/10-R/25, the slip energy error value is more than 10 percent, and the precision is lower; when the grid size is about R/25-R/35, the error value of the slippage energy is between 5% and 10%, and the precision is improved; and when the grid size is R/50-R/35, the slip energy error value is 4% -6%, and the precision is higher. Therefore, at the rotating speed, the grid sizes of R/25-R/50 all meet the engineering precision requirement.
As shown in FIG. 8, the slip energy curves are shown in the case of the load stabilization phase, where the linear speeds of the upper and lower disks are 40m/s and 20m/s, and the mesh sizes of the contact surfaces of the disks are R/35, R/50, R/60, R/75, etc. When the grid size is R/75, the percentage error value of the disk slip energy is less than 10 percent, the precision is high, and the percentage error value when the grid size is R/35-R/60 exceeds 10 percent along with the increase of time, and the precision is relatively low. Therefore, at the rotating speed, the grid size of R/75 meets the engineering precision requirement.
FIG. 9 shows the slip energy curves in the case of the load stabilization phase, the linear speeds of the upper and lower disks being 55m/s and 35m/s, and the mesh sizes of the contact surfaces of the disks being R/75 and R/85, respectively. When the grid size is R/75, the percentage error of the slip energy is larger than 10 percent, the precision is lower, and when the grid size is R/85, the percentage error of the slip energy is about 10 percent but less than 10 percent, and the precision requirement is met. Therefore, at the rotating speed, the grid size of R/85 meets the engineering precision requirement.
FIG. 10 shows the slip energy curves in the case of the load stabilization phase, the linear speeds of the upper and lower disks being 70m/s and 50m/s, and the mesh sizes of the contact surfaces of the disks being R/85 and R/100, respectively. When the grid size is R/85, the percentage error of the slip energy is larger than 10%, the precision is poor, and when the grid size is R/100, the percentage error of the slip energy is kept about 5.4%, the contact error is small, and the precision is high. Therefore, at the rotating speed, the grid size of R/100 meets the engineering precision requirement.
FIG. 11 shows the slip energy curves in the case of the load stabilization phase, the linear speeds of the upper and lower disks being 90m/s and 70m/s, the grid sizes being R/100 and R/110, respectively. When the grid size is R/100, the percentage error of the slippage energy reaches 22.3 percent, the error is large, and the precision is very poor; and when the grid size is R/110, the percentage error of the slip energy is kept at 8.8 percent, and the precision is relatively good. Therefore, at the rotating speed, the grid size of R/110 meets the engineering precision requirement.
According to the simulation result analysis, the percentage value of the slip energy error shows a gradually decreasing trend along with the refinement of the grid, and the deviation between the slip energy simulation value and the theoretical value is gradually increased along with the time, but the percentage error is basically kept constant; meanwhile, under the same grid refinement degree, the slip energy error and the absolute value of the maximum linear velocity between the two contact curved surfaces are in a direct proportion relation, and the larger the linear velocity is, the larger the percentage value of the slip energy error is. When the speed is medium or low, the error percentage value is small when the grid size of the disc contact surface is R/50-R/25, and the requirement of modeling precision is met; at high speed, due to the fact that the rotating speed is higher, the centrifugal force of the grids on the contact surface is larger, the curved surface contact of the upper disc and the lower disc is not stable, and the size of the grids on the contact surface of the discs needs to be thinned to be R/50-R/110 or even thinner, so that the slippage error can be guaranteed within a reasonable range.
Combining various factors such as errors, gear tooth surface parameters and the like, and for the medium and low linear speed gear, the tooth surface grid size recommends to use R/25-R/50. When the maximum tangential speed of the tooth surface is less than 20m/s, the optimal tooth surface grid size is R/25; while for gear pairs with tooth surfaces having a maximum tangential velocity of 20m/s to 25m/s, the tooth surface mesh size recommends the use of R/35.
For high linear speed gears, the tooth surface mesh size recommends the use of R/50-R/110. The maximum tangential speed of the tooth surface of the gear pair is 25m/s-40m/s, and the grid size of the tooth surface recommends using R/75; the maximum tangential speed of the tooth surface is 40m/s-55m/s, and the mesh size of the tooth surface is recommended to use R/85; the maximum tangential speed of the tooth surface is 55m/s-70m/s, and the mesh size of the tooth surface is recommended to use R/100; and the maximum tangential speed of the tooth surface is 70m/s-90m/s, and the tooth surface grid size recommends using R/110.
The tooth surface mesh size determination criteria are shown in table 1, where R represents the minimum radius of curvature of the tooth surface. It is noted that the minimum refinement required for the tooth surface mesh to be achieved while ensuring that the gear contact error meets the engineering accuracy requirements is given in table 1. If the refinement degree of the tooth surface mesh of the gear is smaller than the standard given in table 1 in the actual operation process, the mesh dispersion of the tooth surface of the gear is not accurate, the result obtained by the simulation of the model has no reference value, but if higher tooth surface mesh dispersion accuracy is pursued, the refinement degree of the tooth surface mesh is allowed to be larger than the standard given in table 2.
TABLE 2 tooth flank mesh size determination criteria
Figure BDA0003263550480000181
Step four:
taking a pair of straight-tooth cylindrical gear pairs in a certain aircraft engine gearbox as an example, a discrete process of a tooth profile finite element grid of a high-speed straight-tooth cylindrical gear is explained. The gear basic parameters are shown in table 3. Since the tooth profile finite element mesh discrete methods of the driving wheel and the driven wheel are the same, the following description focuses on the discrete process of the tooth profile finite element mesh of the driving wheel.
Table 3: spur gear parameters in a gearbox of an aircraft engine
Figure BDA0003263550480000182
Figure BDA0003263550480000191
According to the calculation formula of the curvature radius of the tooth surface in the step one, the curvature radius of the tooth root limit meshing point and the curvature radius of the tooth top limit meshing point can be calculated, and the curvature radius of the tooth root limit meshing point and the curvature radius of the tooth top limit meshing point of the driving wheel in the embodiment are respectively 13.3mm and 26 mm.
And calculating the maximum tooth surface tangential speed of the main tooth surface and the driven tooth surface from the engaging to the engaging according to a calculation formula of the tooth surface tangential speed in the step two. The maximum tangential speed of the tooth flanks of the gear in this embodiment is 89 m/s.
And determining the optimal mesh sizes of the tooth root and the tooth top of the gear at the maximum tooth surface tangential speed according to the tooth surface mesh determination standard of the straight toothed spur gear given in the third step. The optimum tooth surface mesh refinement degree of the gear in the present embodiment is R/110 (where R represents the tooth surface curvature radius at different meshing points), so the mesh sizes at the driving wheel root limit meshing point and the tooth crest limit meshing point are 0.13mm and 0.24mm, respectively.
In order to ensure that the subsequent tooth surface body meshes are all hexahedral meshes, the tooth profile curve is shifted to the inside of the tooth end surface by a certain distance so as to form a tooth surface mesh guideline, and the finally generated tooth profile mesh is shown in fig. 12.
It is noted that, herein, relational terms such as first and second, and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that changes, modifications, substitutions and alterations can be made in these embodiments without departing from the principles and spirit of the invention, the scope of which is defined in the appended claims and their equivalents.

Claims (8)

1. A tooth profile mesh discrete method for finite element analysis of a high-speed gear based on sliding energy loss is characterized by comprising the following steps: the method comprises the following steps of simulating a tooth surface contact finite element mesh model by adopting a two-disc curved surface contact finite element mesh model, and determining the optimal value of the mesh size of the tooth profile finite element mesh model by comparing and analyzing a sliding energy loss simulation result and a theoretical calculation result of the two-disc curved surface contact finite element mesh model under different parameter conditions, wherein the method specifically comprises the following steps:
deducing a curvature calculation formula at a meshing point of the driving wheel and the driven wheel according to a geometric relation in the meshing process of the driving wheel and the driven wheel, and respectively calculating curvature radiuses of the driving wheel and the driven wheel at the meshing point and the meshing point according to the curvature calculation formula;
deducing a calculation method of tangential speed and relative slip speed of the driving wheel and the driven wheel at the meshing point according to the speed vector relation in the meshing process of the driving wheel and the driven wheel, so as to respectively obtain the maximum tooth surface tangential speed and the relative slip speed in the meshing process of the driving wheel and the driven wheel;
step three, establishing a plurality of two-disc curved surface contact finite element mesh models according to the curvature radius, the maximum tooth surface tangential speed, the relative slip speed and different mesh sizes obtained in the step one and the step two, and performing simulation calculation on the slip energy loss through the plurality of two-disc curved surface contact finite element mesh models, and comparing the slip energy loss with theoretical calculated values respectively to obtain the optimal mesh size;
and step four, establishing a discrete tooth profile model of the straight spur gear according to the grid size determined in the step three.
2. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 1, wherein the method comprises the following steps: in the first step, the curvature radius of a meshing point of the driving wheel and the driven wheel is obtained according to the curvature radius relation of any meshing position of the driving wheel and the driven wheel, a gear involute tooth profile is formed when the driving wheel and the driven wheel are meshed, the gear involute tooth profile is a smooth curve formed by a series of circles with different curvature radii, and the curvature radius is gradually increased from a tooth root to a tooth top.
3. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 2, wherein the method comprises the following steps: on the involute, the curvature radius of any meshing point on the tooth surface is equal to the length of the generating line of the involute tooth surface, namely the distance from the driving wheel to the meshing point N1, the distance from the driven wheel to the meshing point N2,
the minimum and maximum meshing points of the primary sheave tooth surface curvature radius are labeled B1 and B2, respectively, for the minimum curvature radius ρ'1Equal to B1N1, maximum radius of curvature ρ'2Equal to the sum of B2N1,
the point where the radius of curvature of the driven wheel tooth surface is smallest and the point where the radius of curvature is largest are B2 and B1, respectively, corresponding to the smallest radius of curvature ρ ″1Equal to B2N2, maximum radius of curvature ρ ″)2Equal to the sum of B1N2,
when the driving wheel and the driven wheel are both installed according to the standard center, the pitch circle radius is equal to the index table radius, which can be obtained according to the perpendicular relation between 01N1 and the meshing line N1N2,
N1P＝PO1×sinα＝r1sinα (1)
Figure FDA0003263550470000021
Figure FDA0003263550470000022
in formulae (1) to (3):
rk1、rk2-radius of any engagement point of K1, K2 on the capstan;
rb1-a capstan base radius;
r1-a driving wheel reference circle radius;
an alpha-engagement angle;
any position near the root of the driven or primary pulley is marked as K1, then K1 ═ dk1p，dk1pThe distance on the meshing line of the node P and the meshing point K1,
Figure FDA0003263550470000023
the radius of curvature ρ 'of the drive pulley at any position of the meshing point K1 near the tooth tip of the drive or driven gear'k1And radius of curvature ρ ″' on the driven wheelk1Can be represented as, respectively, a,
ρ′k1＝K1N1＝N1P-K1P＝r1sinα-dk1p (5)
ρ″k1＝K1N2＝N2P+K1P＝r2sinα+dk1p (6)
in formulae (4) to (6):
dk1pthe distance of the meshing point K1 from the node P on the meshing line;
r1、r2-the primary and secondary wheel primary reference circle radii;
ρ′k1、ρ″k1-radius of curvature of the meshing points on the driving and driven wheels;
any position engaging point K2 near the root of the driven wheel or the top of the driving wheel, K2 is equal to dk2p，dk2pThe distance of the meshing point K2 from the node C on the meshing line,
Figure FDA0003263550470000031
the radius of curvature ρ 'of the capstan at any position of the meshing point K2 near the capstan tip or the idler root'k2And radius of curvature ρ ″' on the driven wheelk2Can be respectively expressed as:
ρ′k2＝K2N1＝N1P+K2P＝r1sinα+dk2p (8)
ρ″k2＝K2N2＝N2P-K2P＝r2sinα-dk2p (9)
in formulae (7) to (9):
dk2pthe distance of the meshing point K2 from the node C on the meshing line;
according to the formulas (1) - (9), the relational expressions corresponding to the arbitrary meshing points K1 and K2 on the two sides of the 0102 line are integrated, and the position d of the arbitrary position meshing point on the tooth surface on the meshing linekpCan be expressed as:
Figure FDA0003263550470000032
in the formula (10)
dkp-distance of any meshing point from node P on the meshing line;
rk-radius of any engagement point on the driving wheel;
the curvature radiuses ρ', ρ ″ of the engagement point at any position on the driving wheel and the driven wheel can be expressed as:
ρ′＝r1sinα±dkp (11)
Figure FDA0003263550470000041
in equations (10) to (12), the lower symbols are used when the meshing point is close to the driven wheel tooth crest or the driving wheel tooth crest, and the upper symbols are used when the meshing point is close to the driven wheel tooth crest or the driving wheel tooth crest.
4. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 3, wherein the method comprises the following steps: when there is backlash between the primary and secondary wheels, i.e., the addendum portion is fully engaged in tooth engagement and only a partial region of the dedendum is engaged in tooth engagement, point B2 is located near the primary and secondary wheel dedendum, point B1 is located near the primary and secondary wheel dedendum, and the primary wheel radius r 'at B2'kEqual to the radius of the addendum circle of the driving wheel,
Figure FDA0003263550470000042
and the radius of the driving wheel at the meshing point B1 is r ″)kCan be based on a right triangle delta O2N2B1And a right triangle delta O1N1B1The geometric relationship between the two components is obtained,
N1N2＝N1P+PN2＝r1sinα+r2sinα (14)
Figure FDA0003263550470000043
Figure FDA0003263550470000044
Figure FDA0003263550470000045
in the formula (10), r at B2kValue is r 'from size'kAnd r at B1kThe value is r ″)k，
And the calculated maximum and minimum curvature radius is used as the basis for determining the size of the tooth surface mesh.
5. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 1, wherein the method comprises the following steps: in the second step, according to the velocity vector triangle and the right triangle delta K of any meshing point K2 on the driving wheel2O1N1The similarity relationship of (a) to (b) can be obtained,
V1k、V1t、V1nthe absolute speed, the tangential speed and the normal speed of the meshing point K2 on the driving wheel respectively;
Figure FDA0003263550470000051
so that the tangential speed V of any meshing point on the driving wheel along the tooth surface1tCan be expressed as:
Figure FDA0003263550470000052
in formulae (18) to (19):
n1-the rotational speed of the driving wheel in r/min;
ρ' a radius of curvature in mm of an arbitrary engagement point K2 on the capstan;
V1tthe tangential speed of any meshing point K2 on the capstan along the tooth surface, in m/s;
according to the speed vector triangle and the right triangle delta K of any meshing point K2 on the driven wheel2O2N2The similarity relationship of (a) to (b) can be obtained,
V2k、V2t、V2nabsolute, tangential and normal speeds of the engagement point K2 on the driven wheel
Figure FDA0003263550470000053
The tangential velocity V of any meshing point on the driven wheel along the tooth surface2tCan be expressed as:
Figure FDA0003263550470000054
in formulae (20) to (21):
n2-driven wheel speed in r/min;
ρ "-radius of curvature of any meshing point K2 on the driven wheel in mm;
V2ttangential speed of any meshing point K2 on the driven wheel along the tooth surface, in m/s;
master and slaveRelative slip velocity V at any engagement point of moving wheeltCan be expressed as:
Vt＝|V1t-V2t| (23)，
the relative slip speed between the driving wheel and the driven wheel at the point of tooth flank engagement and the maximum tangential speed of the tooth flank, namely the relative slip speed at the point of B2 or B1 and the tangential speed of the tooth flank are obtained according to the formulas (18) to (23),
wherein the curvature radius rho 'of the point B2 on the driving wheel is B2N1, the curvature radius rho' of the driven wheel is B2N2, the curvature radius rho 'of the point B1 on the driving wheel is B1N1, the curvature radius rho' of the driven wheel is B1N2,
and the calculated relative slip speed is used as the basis for determining the rotating speed of the disc in the simplified disc model.
6. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 1, wherein the method comprises the following steps: and in the third step, determining the mesh size of the two-disc curved surface contact finite element mesh model according to the mesh size standard.
7. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 6, wherein the method comprises the following steps: the grid size standard is as follows: for a tooth surface maximum tangential velocity greater than 25m/s for a gear span, the grid size ranges from R/50 to R/110, R being the radius of curvature of the disk.
8. The method for discretizing tooth profile mesh for finite element analysis of high-speed gears based on sliding energy loss according to claim 7, wherein the method comprises the following steps: the grid size standard is as follows: the maximum tangential speed of the tooth surface is 25m/s-40m/s, and the mesh size range of the tooth surface is R/75; the maximum tangential speed of the tooth surface is 40m/s-55m/s, and the mesh size range of the tooth surface is R/85; the maximum tangential speed of the tooth surface is 55m/s-70m/s, and the mesh size range of the tooth surface is R/100; and the maximum tangential speed of the tooth surface is 70m/s-90m/s, the size range of the tooth surface grid is R/110, and R is the curvature radius of the disc.
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