CN108875274A - The method of Tooth Contact Analysis containing error of spiral bevel gear - Google Patents
The method of Tooth Contact Analysis containing error of spiral bevel gear Download PDFInfo
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Abstract
The invention discloses a kind of Tooth Contact Analysis containing error of spiral bevel gear methods, belong to gear transmission technology field, including：(1) bull wheel flank of tooth initial value when initial contact state is solved, set the constraint condition containing rigging error, steamboat flank of tooth initial value is solved, the accurate initial value that bull wheel flank of tooth initial value, steamboat flank of tooth initial value are solved as Tooth Contact Analysis solves the initial contact point of the entire flank of tooth；(2) coordinate transformation matrix is obtained, eTCA equation group is obtained, establishes the scoring item of the eTCA equation group, completes the evaluation to spiral bevel gear face performance.Spiral bevel gear Tooth Contact Analysis method described herein, propose accurate eTCA initial value solution, and consider manufacture rigging error as basic constraint condition in solution procedure, match accurate eTCA equation group establishment process, given accurate non-linear derivation algorithm, the robustness for completing entire contact point of gear surface especially initial contact point accurately solve.
Description
Technical field
The invention belongs to gear transmission technology fields, and in particular to a kind of Tooth Contact Analysis containing error of spiral bevel gear
Method.
Background technique
Spiral bevel gear is wide because of the advantages that its overlap coefficient is big, bearing capacity is strong, transmission ratio is high, stable drive, small noise
It is general to be applied to the mechanical transmission fields such as automobile, aviation, mine.Spiral bevel gear is divided into two kinds, and one is spiral bevel gears, big
Wheel axis and the intersection of small wheel axis；Another kind is hypoid spiral bevel gear, and big wheel axis and small wheel axis have certain inclined
Set away from.The spiral bevel gear world-class company of production at present is mainly the Gleason company in the U.S. and the sharp Kanggong department of Austria of Switzerland.
Tooth Contact Analysis method (tooth contact analysis, TCA) is as simulation gear pair face mistake
Journey and one of the key technology for determining face performance, the stability of numerical solution is particularly important, it and contact point of gear surface
The determination of especially initial point contact point is especially relevant.In traditional Tooth Contact Analysis solution procedure, initial-value problem is often
It is defined as valuation problem, i.e., is all artificial estimation setting, just there is very big contingency and uncertainty；It is built in addition, solving at present
The numerical algorithm of vertical non-linear Tooth Contact Analysis equation, has ignored manufacture rigging error, and in actual gear transmission process
In, manufacture rigging error is the key factor having to take into account that, it can directly affect the contact performance such as circular tooth contact of the flank of tooth
Domain, flank of tooth driving error etc..
Summary of the invention
The purpose of the present invention is to provide a kind of Tooth Contact Analysis containing error of spiral bevel gear method, propose accurate
Initial value solution, regard manufacture rigging error as constraint condition, asked with completing the accurate of spiral bevel gear contact point of gear surface
Solution completes the evaluation to face performance by eTCA equation group.
The method of Tooth Contact Analysis containing error of this spiral bevel gear provided by the invention, includes the following steps：
(1) bull wheel flank of tooth initial value when initial contact state is solved, the constraint condition containing rigging error is set, solves steamboat
Flank of tooth initial value, the accurate initial value that bull wheel flank of tooth initial value, steamboat flank of tooth initial value are solved as Tooth Contact Analysis, solves entire tooth
The initial contact point in face；
(2) coordinate transformation matrix is obtained, eTCA equation group is obtained, establishes the scoring item of the eTCA equation group：A) flank of tooth
Contact patch；B) driving error；C) tooth surface error completes the evaluation to spiral bevel gear face performance.
In a specific embodiment, in the step (1), the specific steps are：
1.1) bull wheel flank of tooth initial value (φ is solved2,θ2)；
As a reference point with bull wheel flank of tooth midpoint M, the distance of flank of tooth midpoint M to big wheel axis is r2, flank of tooth midpoint M to axis
The subpoint of line intersects vertex O with the flank of tooth2Distance be L2, according to the parametrization discrete method of the flank of tooth, the fundamental coordinate system of foundation
{ i, j, k }, is represented by：
I=[1 0 0], j=[0 1 0], k=[0 0 1] (1)
And piIt is the cone angle δ about flank of tooth midpointMFunction, have：
It is f from flank of tooth midpoint M to the distance of large end face[t]B, b are the face width of tooth, ha2And hf2It can solve according to the following formula：
ha2=hae2-f[t]btanθa2 (3)
hf2=hfe2-f[t]btanθa2 (4)
Since flank of tooth point is set as flank of tooth midpoint M, then the distance of flank of tooth midpoint M to pitch cone is：
Flank of tooth midpoint M is to the distance that root is bored：
The distance of flank of tooth midpoint M to big wheel axis is：
r2=(Re-fb)sinδ2-△h1cosδ2=(Re-f[t] b)sinδ2-1/2(hf2-ha2)cosδ2 (7)
Coefficient f[t]It is decision condition, when taking 0.25,0.5,0.75, corresponding is gear big end, middle-end and small end
Flank of tooth reference point；
The subpoint of flank of tooth midpoint M to axis intersects vertex O with the flank of tooth2Distance be：
Using bull wheel flank of tooth midpoint M coordinate as known conditions, to solve bull wheel flank of tooth initial value (φ2,θ2), then exist with ShiShimonoseki
System：
Obviously, r2And L2It can be expressed as about φ2And θ2Parameterized function, tooth surface parametersization modeling in r2It is to have
It is nonlinear, corresponding (φ is solved by non-linear iterative2, θ2)；
1.2) steamboat flank of tooth initial value (φ is solved1,θ1)；
In steamboat flank of tooth initial value solution procedure, constraint condition when considering manufacture assembly factor as face is being pacified
In the basic definition for filling error, E is the misalignment of axe of big pinion gear, and P is steamboat along axial offset distance, and G is bull wheel along axis
The offset distance of line, α are the crossed axis angle of two axial lines；
When size gear is rationally assembled together, if bull wheel is around axis p2Rotation, steamboat is around axis p1Rotation, then they
Rigging error can be expressed as：
And two axis intersection point OpAnd OgDistance be：
The moment is initially contacted in gear, driving error is almost nil, and the transmission ratio of bull wheel and steamboat is constant and is equal to
z2/z1, the actual rotary position of output wheel and the difference of theoretical value are defined as due to driving error
△φG, export the rotation angle difference φ of bull wheelGIt is represented by the φ of input steamboat anglePNear parabolic letter
Number, thus, the assembly of the flank of tooth directly affects tooth surface error, then at the initial contact moment, steamboat flank of tooth point meets following item
Part：
Similarly, the equation is solved using non-linear iterative, obtains steamboat flank of tooth initial value (φ1,θ1)；
Then using bull wheel flank of tooth initial value, steamboat flank of tooth initial value as the accurate initial value x of Tooth Contact Analysis equation solution(0)
=(φ1 (0),φ2 (0),θ1 (0),θ2 (0))。
In a specific embodiment, it in the step (1), using convex model adaptive trust region algorithm, solves big
Take turns flank of tooth initial value, steamboat flank of tooth initial value and flank of tooth initial contact point P0*(φ1,θ1,φ2,θ2), solve the process of these three parameters
It is similar.
In a specific embodiment, steamboat flank of tooth initial value is solved using convex model adaptive trust region algorithm, specifically
Step is：
In order to solve equation (14), one can be converted it into without constrained nonlinear systems problem；
Using convex model, the subproblem of trusted zones can be expressed as：
In formula, fk=f (xk) and gk=g (xk) indicate gradient, bk∈RNIndicate horizontal vector, andWork as bk=0
OrFor convex model at a second-order model, such convex model is the generalized form of second-order model；
In order to select the parameter b of convex modelkAnd Bk, use：
If ρk>=0, have：
Otherwise, βk=1.During replacing trusted zones, for given sufficiently large positive number b*>0, have
It will be apparent that BkIt needs to be satisfied with：
In addition, it has proved that：Work as ρk>0, then dT k-1yk-1>0；Work as Bk-1>When 0, Bk>0；In current search, one is given
A trusted zones：
In formula, Rτ(rk) (τ ∈ (0,1)) be the relaxation R- function increasedd or decreased about Trust Region Radius, have：
Beta, gamma in formula1, γ2It is corresponding constant with Ω, the specific definition about R- function can be with bibliography, in addition, rk
Representative function f and fkThe ratio of gains be：
In a specific embodiment, the basic program of convex model adaptive trust region algorithm is as follows：
Input x0∈RN,△0>0,b0∈RN×1,0<β<1,0<γ1<1-β,γ2>0,M>1+γ2,ε>0,0<μ<1；
Initialization:K=0, △0=1, x=x0,b0=0, B0=I (cell matrix)；
Step 1. evaluates fk=f (xk) andif||gk| |≤ε, then are terminated, x*=xk；
K >=1 Step 2.If calculates ρkBy solving equation (17)；ifρk>=0, solve βkAccording to equation (18)；
otherwise,βk=1. solve b using equation (19)-(21) respectivelyk,yk-1, and Bk.
Step 3. sets b=bk, g=gk, and B=BkApproximate solution d* is obtained, by solving equation (17)
Step 4. assumes dk=d* calculates r according to equation (25)k.
Step 5.if rk<μ enables xk+1=xk；Otherwise enables xk+1=xk+dk.
6. corrected parameter b of StepkAnd Bk, acquire bk+1And Bk+1；△ is calculated by equation (22) and (23)k+1.
Step 7. enables k=k+1, returns to initialization
In a specific embodiment, in the step (2), the specific steps are：
2.1) coordinate transformation matrix is obtained, eTCA equation group is obtained；
It is modeled by the flank of tooth and carries out flank of tooth discretization, size flank of tooth point Pi(φi,θi) (i=1, steamboat；I=2, bull wheel) it can
In respective coordinate system Oi(Xi,Yι,Zι) (i=1,2) solve and, resultant method arrow is expressed as ri(i=1,2) and ni
(i=1,2), latter two right flank of tooth, which needs respectively to rotate by a certain angle, reaches flank of tooth conjugate point contact condition, completes connecing for the flank of tooth
Touching transmission；In this initial contact point P* (φ1,θ1,φ1,θ2) position, meet flank engagement principle, has：
Wherein, which has spin matrix Mi-fIt indicates, eTCA contact conditions mainly include：I) latter two tooth is rotated
Millet cake possesses common point arrow, i.e. (rf)1=(rf)2；Ii the method arrow for) rotating latter two flank of tooth point is conllinear, i.e. N1=χ0N2, χ0For
Constant, it can thus be concluded that spin matrix Mi-f(i=1,2)；
Firstly, considering that big pinion gear is the flank of tooth model established under respective coordinate system, then they must be rotated certain
Angle reaches unified flank engagement coordinate system.Then from gear original coordinate system Oi(Xi,Yi,Zi) arrive engagement coordinate system Of(Xf,Yf,Zf)
Transition matrix Mf-i∈R4(i=1,2) it is represented by：
Its submatrix MX-2For：
Rotating angle is：
And：
Reach the same engagement coordinate system Of(Xf,Yf,Zf) after, due to manufacturing the presence of rigging error, the size flank of tooth may
There is no point contacts, then need to further rotate transformation, constrain face motion process, and meet TCA contact conditions,
Between use following rotation mode：The steamboat flank of tooth is rotated, and rigging error is added in the bull wheel flank of tooth.Steamboat flank of tooth process is rotated herein
In, spin matrix is represented by：
(MG-X)1=(MG-A)1·(MA-B)1·(MB-C)1·MC-X (32)
It include M during realization method swears conllinear rotation transformationG-A,MA-B,MB-CAnd MC-XEqual kinematic chains, have：
In formula, angle is rotated
In realizing the conversion process that point arrow is overlapped, spin matrix (Mt-f)1With matrix MG-XProcess it is consistent, include
Sub- kinematic chain is Mt-g,Mg-a,Ma-b,Mb-cAnd Mc-f, homogeneous transformation process is：
In addition to rotating angleOutside, it is also contemplated that swing offset (△ l)1:=((△ lX)1, (△ lY)1, (△ lZ)1)T；
After entire coordinate transform, the flank of tooth point of steamboat is represented by：
Nf:N1=MX-1(MG-X)1·n1 (38)
rf:(rf)1=Mf-1(Mt-f)1·r1 (39)
It is worth noting that generally error amount is added in one of flank of tooth i.e. in the considerations of manufacturing rigging error
Can, consider that the transformation matrix of coordinates of rigging error is：
It is used for the submatrix M that calculating method swears conllinear processm-a∈R3
The bull wheel flank of tooth is added in manufacture rigging error can be obtained after all rotation transformations：
rf:(rf)2=MM-AMf-2·r2 (42)
Nf:N2=Mm-aMX-2·ni (43)
Conllinear eTCA condition is found in conclusion meeting, and can obtain following two scalar equation：
(feTCA)1:(N1)1+(N2)1=0 (44)
(feTCA)2:(N1)2+(N2)2=0 (45)
Herein, that reflection is Ni=[(Ni)1,(Ni)2,(Ni)3] two of them component functional relation；
Similarly, when meeting the eTCA condition that point arrow is overlapped, following scalar equation sets up：
(feTCA)3:[(rf)1]1-[(rf)2]1=0 (46)
(feTCA)4:[(rf)1]2-[(rf)2]2=0 (47)
(feTCA)5:[(rf)1]3-[(rf)2]3=0 (48)
At this point, about (rf)i:={ [(rf)i]1,(rf)i]2,(rf)i]3}T(i=1,2) three components will consider simultaneously；
It is assumed that the angular speed for inputting steamboat in gear drive is ω1, the angular speed of bull wheel is ω2, after rotation transformation
The angular speed of bull wheel becomes ωf2, there is following relationship to set up：
The opposite angular speed of bull wheel and steamboat is：
ω1-2=ω1-ωf2 (50)
Consider that the influence of manufacture rigging error degree has then after coordinate is converted：
So far, the linear velocity v of bull wheel2With the linear velocity v of steamboat1For：
The then relative linear velocity v of bull wheel and steamboat1-2For：
v1-2=v1-v2 (54)
State is initially contacted herein, meets the theory of engagement, can be obtained：
(feTCA)6:(N2)1×(v1-2)1+(N2)2×(v1-2)2+(N2)3×(v1-2)3=0 (55)
In summary, six scalar equations contain 6 known variables△lX, △ lYWith △ lZ, can be quick
The geometry motion transformation matrix during entire flank of tooth initial contact is accurately solved, while establishing accurate eTCA equation group feTCA
=[(feTCA)1,(feTCA)2,(feTCA)3,(feTCA)4,(feTCA)5,(feTCA)6]；
2.2) scoring item of eTCA equation group is established：A) teeth contact；B) driving error；C) tooth surface error is completed
Evaluation to spiral bevel gear face performance.
Advantageous effects of the invention：
(1) spiral bevel gear Tooth Contact Analysis method described herein proposes accurate eTCA initial value solution, and
Consider that manufacture rigging error as basic constraint condition, matches accurate eTCA equation group establishment process, gives in solution procedure
Fixed accurate non-linear derivation algorithm, the robustness for completing entire contact point of gear surface especially initial contact point accurately solve.
(2) spiral bevel gear Tooth Contact Analysis method of the present invention, using convex model adaptive trust region algorithm, tool
There is the characteristics of stability, Strong Convergence and adaptivity, the ill-conditioning problem that can effectively solve Jacobian matrix simultaneously only need to be compared with a moment
In generation, is achieved with desired quantity result；Meanwhile it being different from traditional Two-order approximation surface model, which is handling strong non-dyadic
There is better advantage in terms of the problems such as behavior, significant Curvature varying.
Detailed description of the invention
Fig. 1 be using the adaptive Trust Region of convex model (CSTAR) solve size wheel flank of tooth initial value calculate three times repeatedly
For quality evaluation.
Fig. 2 is to replace situation for the Trust Region Radius iterated to calculate three times.
Fig. 3 is to solve initial contact point P0* basic procedure.
Fig. 4 is that the result for the initial point that CSTAR, NR and PH method obtain compares.
Fig. 5 is the evaluation result of eTCA equation group：(a.1) contact point of gear surface；(a.2) flank of tooth mark；(b) driving error；
(c) tooth surface error.
Fig. 6 is the influence for manufacturing rigging error to eTCA evaluation result.
Specific embodiment
The technical scheme in the embodiments of the invention will be clearly and completely described below, it is clear that described implementation
Example is only a part of the embodiment of the present invention, rather than whole embodiments, based on the embodiments of the present invention, the common skill in this field
Art personnel every other embodiment obtained without making creative work belongs to the model that the present invention protects
It encloses, present invention will be further explained below with reference to the attached drawings and specific examples.
The embodiment of the present invention provides a kind of method of Tooth Contact Analysis containing error of spiral bevel gear, includes the following steps：
Step 1: bull wheel flank of tooth initial value when solving initial contact state, sets the constraint condition containing rigging error, solve
Steamboat flank of tooth initial value, the accurate initial value that bull wheel flank of tooth initial value, steamboat flank of tooth initial value are solved as Tooth Contact Analysis solve whole
The initial contact point P of a flank of tooth0*(φ1,θ1,φ2,θ2)；
Table 1 is the tooth surface design basic parameter of face milling spiral bevel gear, and table 2 is the machined parameters of SGM adjustment card, i.e.,
Bull wheel uses generating, and steamboat uses modified-roll method, and during gear drive, bull wheel concave surface and steamboat convex surface are connect
Touching, the parameter of the two flank of tooth will be used for the calculating of eTCA；
1 Spiral Bevel Gears of table design basic parameter
2 spiral bevel gear SGM of table adjusts card parameter
1.1) bull wheel flank of tooth initial value (φ is solved2,θ2)
As a reference point with bull wheel flank of tooth midpoint M, the distance of flank of tooth midpoint M to big wheel axis is r2, flank of tooth midpoint M to axis
The subpoint of line intersects vertex O with the flank of tooth2Distance be L2, according to the parametrization discrete method of the flank of tooth, the fundamental coordinate system of foundation
{ i, j, k }, is represented by：
I=[1 0 0], j=[0 1 0], k=[0 0 1] (1)
And piIt is the cone angle δ about flank of tooth midpointMFunction, have：
It is f from flank of tooth midpoint M to the distance of large end face[t]B, b are the face width of tooth, ha2And hf2It can solve according to the following formula：
ha2=hae2-f[t]btanθa2 (3)
hf2=hfe2-f[t]btanθa2 (4)
Since flank of tooth point is set as flank of tooth midpoint M, then the distance of flank of tooth midpoint M to pitch cone is：
Flank of tooth midpoint M is to the distance that root is bored：
The distance of flank of tooth midpoint M to big wheel axis is：
r2=(Re-fb)sinδ2-△h1cosδ2=(Re-f[t] b)sinδ2-1/2(hf2-ha2)cosδ2 (7)
Coefficient f[t]It is decision condition, when taking 0.25,0.5,0.75, corresponding is gear big end, middle-end and small end
Flank of tooth reference point；
The subpoint of flank of tooth midpoint M to axis intersects vertex O with the flank of tooth2Distance be：
Using bull wheel flank of tooth midpoint M coordinate as known conditions, to solve bull wheel flank of tooth initial value (φ2,θ2), then exist with ShiShimonoseki
System：
Obviously, r2And L2It can be expressed as about φ2And θ2Parameterized function, tooth surface parametersization modeling in r2It is to have
It is nonlinear, corresponding (φ is solved by corresponding alternative manner2,θ2)。
1.2) steamboat flank of tooth initial value (φ is solved1,θ1)
In steamboat flank of tooth initial value solution procedure, constraint condition when considering manufacture assembly factor as face is being pacified
In the basic definition for filling error, E is the misalignment of axe of big pinion gear, and P is steamboat along axial offset distance, and G is bull wheel along axis
The offset distance of line, α are the crossed axis angle of two axial lines；
When size gear is rationally assembled together, if bull wheel is around axis p2Rotation, steamboat is around axis p1Rotation, then they
Rigging error can be expressed as：
And two axis intersection point OpAnd OgDistance be：
The moment is initially contacted in gear, driving error is almost nil, and the transmission ratio of bull wheel and steamboat is constant and is equal to
z2/z1, the actual rotary position of output wheel and the difference △ φ of theoretical value are defined as due to driving errorG, export the rotation angle of bull wheel
Spend difference φGIt is represented by the φ of input steamboat anglePNear parabolic function.Thus, the assembly of the flank of tooth directly affects tooth
Surface error, then at the initial contact moment, steamboat flank of tooth point meets the following conditions：
Similarly, the equation is solved using non-linear iterative, obtains steamboat flank of tooth initial value (φ1,θ1)；
Table 3 is at the beginning of solving obtained bull wheel flank of tooth initial value, the steamboat flank of tooth using the adaptive Trust Region of convex model (CSTAR)
Value, they are used as the accurate initial value x of Tooth Contact Analysis(0)=(φ1 (0),φ2 (0),θ1 (0),θ2 (0))；
The accurate initial value of 3 Tooth Contact Analysis of table
Fig. 1 be using the adaptive Trust Region of convex model (CSTAR) solve size wheel flank of tooth initial value calculate three times repeatedly
For quality evaluation, during solving bull wheel flank of tooth initial value, steamboat flank of tooth initial value, CSTAR method calculation amount is small, and convergence is fast, closes
Preferable iteration quality can be shown in the optimization of objective function f (x), iteration step length and single order.
Solving initial contact point P0* during iterative solution, due to the complexity of calculating, bigger calculating is needed
Amount, but still can guarantee faster convergence rate.Fig. 2 is the Trust Region Radius displacement situation iterated to calculate three times, is illustrated
Trust Region Radius in current calculating, used basic parameter is γ in its initialization1=γ2=0.15, μ=0.25, β
=0.1 and ε=10-14。
Fig. 3 is to solve initial contact point P0* basic procedure will be of the invention for the correctness of further verification algorithm
(PH) algorithm is mixed with Newton-Raphson (NR) and Powell to compare, and with the BFGS (Broyden- in document
Fletcher-Goldfarb-Shanno) algorithm is compared, BFGS algorithm solve initial contact point be (0.993766,
0.1796032,0.86776021, -0.09447587) it is used as reference value, the initial point that Fig. 4 CSTAR, NR and PH method obtain
As a result compare, being compared by numerical value can obtain：CSTAR algorithm of the present invention has more advantage, the initial point P acquired0*(0.98991007,
0.0177254,0.96769247, -0.0940923) and its φ2, θ2, φ1And θ1Computational accuracy be 99.612% respectively,
98.692%, 99.993% and 99.594%.
Step 2: obtaining coordinate transformation matrix, eTCA equation group is obtained, the scoring item of eTCA equation group is established：I) flank of tooth
Contact patch；Ii) driving error；Iii) tooth surface error completes the evaluation to spiral bevel gear face performance；
It is modeled by the flank of tooth and carries out flank of tooth discretization, size flank of tooth point Pi(φi,θi) (i=1, steamboat；I=2, bull wheel) it can
In respective coordinate system Oi(Xi,Yι,Zι) (i=1,2) solve and, resultant method arrow is expressed as ri(i=1,2) and ni
(i=1,2), latter two right flank of tooth, which needs respectively to rotate by a certain angle, reaches flank of tooth conjugate point contact condition, completes connecing for the flank of tooth
Touching transmission；In this initial contact point P* (φ1,θ1,φ1,θ2) position, meet flank engagement principle, has：
Wherein, which has spin matrix Mi-fIt indicates, eTCA contact conditions mainly include：I) latter two tooth is rotated
Millet cake possesses common point arrow, i.e. (rf)1=(rf)2；Ii the method arrow for) rotating latter two flank of tooth point is conllinear, i.e. N1=χ0N2, χ0For
Constant, it can thus be concluded that spin matrix Mi-f(i=1,2)；
Firstly, considering that big pinion gear is the flank of tooth model established under respective coordinate system, then they must be rotated certain
Angle reaches unified flank engagement coordinate system.Then from gear original coordinate system Oi(Xi,Yi,Zi) arrive engagement coordinate system Of(Xf,Yf,Zf)
Transition matrix Mf-i∈R4(i=1,2) it is represented by：
Its submatrix MX-2For：
Rotating angle is：
And：
Reach the same engagement coordinate system Of(Xf,Yf,Zf) after, due to manufacturing the presence of rigging error, the size flank of tooth may
There is no point contacts, then need to further rotate transformation, constrain face motion process, and meet TCA contact conditions,
Between use following rotation mode：The steamboat flank of tooth is rotated, and rigging error is added in the bull wheel flank of tooth.Steamboat flank of tooth process is rotated herein
In, spin matrix is represented by：
(MG-X)1=(MG-A)1·(MA-B)1·(MB-C)1·MC-X (32)
It include M during realization method swears conllinear rotation transformationG-A,MA-B,MB-CAnd MC-XEqual kinematic chains, have：
In formula, angle is rotated
In realizing the conversion process that point arrow is overlapped, spin matrix (Mt-f)1With matrix MG-XProcess it is consistent, include
Sub- kinematic chain is Mt-g,Mg-a,Ma-b,Mb-cAnd Mc-f, homogeneous transformation process is：
In addition to rotating angleOutside, it is also contemplated that swing offset (△ l)1:=((△ lX)1,(△lY)1,(△lZ)1)T；
After entire coordinate transform, the flank of tooth point of steamboat is represented by：
Nf:N1=MX-1(MG-X)1·n1 (38)
rf:(rf)1=Mf-1(Mt-f)1·r1 (39)
It is worth noting that generally error amount is added in one of flank of tooth i.e. in the considerations of manufacturing rigging error
Can, consider that the transformation matrix of coordinates of rigging error is：
It is used for the submatrix M that calculating method swears conllinear processm-a∈R3
The bull wheel flank of tooth is added in manufacture rigging error can be obtained after all rotation transformations：
rf:(rf)2=MM-AMf-2·r2 (42)
Nf:N2=Mm-aMX-2·ni (43)
Conllinear eTCA condition is found in conclusion meeting, and can obtain following two scalar equation：
(feTCA)1:(N1)1+(N2)1=0 (44)
(feTCA)2:(N1)2+(N2)2=0 (45)
Herein, that reflection is Ni=[(Ni)1,(Ni)2,(Ni)3] two of them component functional relation；
Similarly, when meeting the eTCA condition that point arrow is overlapped, following scalar equation sets up：
(feTCA)3:[(rf)1]1-[(rf)2]1=0 (46)
(feTCA)4:[(rf)1]2-[(rf)2]2=0 (47)
(feTCA)5:[(rf)1]3-[(rf)2]3=0 (48)
At this point, about (rf)i:={ [(rf)i]1,(rf)i]2,(rf)i]3}T(i=1,2) three components will consider simultaneously；
It is assumed that the angular speed for inputting steamboat in gear drive is ω1, the angular speed of bull wheel is ω2, after rotation transformation
The angular speed of bull wheel becomes ωf2, there is following relationship to set up：
The opposite angular speed of bull wheel and steamboat is：
ω1-2=ω1-ωf2 (50)
Consider that the influence of manufacture rigging error degree has then after coordinate is converted：
So far, the linear velocity v of bull wheel2With the linear velocity v of steamboat1For：
The then relative linear velocity v of bull wheel and steamboat1-2For：
v1-2=v1-v2 (54)
State is initially contacted herein, meets the theory of engagement, can be obtained：
(feTCA)6:(N2)1×(v1-2)1+(N2)2×(v1-2)2+(N2)3×(v1-2)3=0 (55)
In summary, six scalar equations contain 6 known variables△lX,△lYWith △ lZ, can be quick
The geometry motion transformation matrix during entire flank of tooth initial contact is accurately solved, while establishing accurate eTCA equation group feTCA
=[(feTCA)1,(feTCA)2,(feTCA)3,(feTCA)4,(feTCA)5,(feTCA)6]。
Flank of tooth initial contact point and the solution of each face performance evaluation item are closely related, determine that the accurate flank of tooth is initial
The purpose of point as shown in figure 5, on steamboat convex surface, determining contact point of gear surface and is initially connect to carry out eTCA evaluation of result
Contact P0* generally within flank of tooth intermediate region, teeth contact is good, is mainly distributed on flank of tooth middle section, and without going out
Now excessive EDGE CONTACT is able to reflect out certain flank of tooth bearing capacity；Driving error curve is symmetric, most substantially
Value is less than 13arc sec, this is able to achieve the gear drive compared with low noise and vibration；Tooth surface error is reasonably distributed, lesser error
Flank of tooth intermediate region is concentrated on, biggish to be mainly distributed on gear small end, average absolute value is 15.523 μm, and maximum value is
23.476 μm, minimum value is -24.138 μm.
In the analytical Calculation of contact point of gear surface proposed by the present invention, manufacture rigging error is demonstrated as flank of tooth initial contact
One major influence factors of point and face performance, emphasis of the present invention provide manufacture rigging error to the shadow of eTCA result
It rings, Fig. 6 is the influence for manufacturing rigging error to eTCA evaluation result, has chosen two scoring items i.e. tooth surface error and transmission misses
Difference, wherein G=0 is as constraint condition and α is generally 0, sets 5 kinds of different operating conditions about rigging error item P and E, obtains
Data result show：Manufacture rigging error is affected to tooth surface error and driving error, especially in operating condition 2 and operating condition 3
In value interval, change most obvious.Therefore, this method can be in practical manufacturing process, consideration High-performance gear is driven and gear
The adjustment and optimization of the manufacture rigging error of product manufacturing provide thinking.
The above is only a preferred embodiment of the present invention, protection scope of the present invention is not limited merely to above-mentioned implementation
Example.To those of ordinary skill in the art, obtained improvement and change in the case where not departing from the technology of the present invention concept thereof
It changes and also should be regarded as protection scope of the present invention.
Claims (6)
1. a kind of method of Tooth Contact Analysis containing error of spiral bevel gear, which is characterized in that include the following steps：
(1) bull wheel flank of tooth initial value (φ when initial contact state is solved2,θ2), the constraint condition containing rigging error is set, is solved
Steamboat flank of tooth initial value (φ1,θ1), the accurate initial value that bull wheel flank of tooth initial value, steamboat flank of tooth initial value are solved as Tooth Contact Analysis
x(0), solve the initial contact point P of the entire flank of tooth0*；
(2) coordinate transformation matrix is obtained, eTCA equation group is obtained, establishes the scoring item of the eTCA equation group：A) face
Mark；B) driving error；C) tooth surface error completes the evaluation to spiral bevel gear face performance.
2. the method for Tooth Contact Analysis containing error of spiral bevel gear according to claim 1, which is characterized in that the step
(1) in, the specific steps are：
1.1) bull wheel flank of tooth initial value (φ is solved2,θ2)；
As a reference point with bull wheel flank of tooth midpoint M, the distance of flank of tooth midpoint M to big wheel axis is r2, flank of tooth midpoint M to axis
Subpoint intersects vertex O with the flank of tooth2Distance be L2, according to the parametrization discrete method of the flank of tooth, the fundamental coordinate system of foundation i,
J, k }, it is represented by：
I=[1 0 0], j=[0 1 0], k=[0 0 1] (1)
And piIt is the cone angle δ about flank of tooth midpointMFunction, have：
It is f from flank of tooth midpoint M to the distance of large end face[t]B, b are the face width of tooth, ha2And hf2It can solve according to the following formula：
ha2=hae2-f[t]btanθa2 (3)
hf2=hfe2-f[t]btanθa2 (4)
Since flank of tooth point is set as flank of tooth midpoint M, then the distance of flank of tooth midpoint M to pitch cone is：
Flank of tooth midpoint M is to the distance that root is bored：
The distance of flank of tooth midpoint M to big wheel axis is：
r2=(Re-fb)sinδ2-△h1cosδ2=(Re-f[t] b)sinδ2-1/2(hf2-ha2)cosδ2 (7)
Coefficient f[t]It is decision condition, when taking 0.25,0.5,0.75, corresponding is the tooth of gear big end, middle-end and small end
Face reference point；
The subpoint of flank of tooth midpoint M to axis intersects vertex O with the flank of tooth2Distance be：
Using bull wheel flank of tooth midpoint M coordinate as known conditions, to solve bull wheel flank of tooth initial value (φ2,θ2), then there is following relationship：
Obviously, r2And L2It can be expressed as about φ2And θ2Parameterized function, tooth surface parametersization modeling in r2Be have it is non-thread
Property, corresponding (φ is solved by non-linear iterative2,θ2)；
1.2) steamboat flank of tooth initial value (φ is solved1,θ1)；
In steamboat flank of tooth initial value solution procedure, constraint condition when considering manufacture assembly factor as face is missed in installation
In the basic definition of difference, E is the misalignment of axe of big pinion gear, and P is steamboat along axial offset distance, and G is bull wheel along axis
Offset distance, α are the crossed axis angle of two axial lines；
When size gear is rationally assembled together, if bull wheel is around axis p2Rotation, steamboat is around axis p1It rotates, then their dress
It can be expressed as with error：
And two axis intersection point OpAnd OgDistance be：
The moment is initially contacted in gear, driving error is almost nil, and the transmission ratio of bull wheel and steamboat is constant and is equal to z2/z1,
The actual rotary position of output wheel and the difference Δ φ of theoretical value are defined as due to driving errorG, export the rotation differential seat angle of bull wheel
Value φGIt is represented by the φ of input steamboat anglePNear parabolic function.Thus, the assembly of the flank of tooth directly affects flank of tooth mistake
Difference, then at the initial contact moment, steamboat flank of tooth point meets the following conditions：
Similarly, the equation is solved using non-linear iterative, obtains steamboat flank of tooth initial value (φ1,θ1)；
Then using bull wheel flank of tooth initial value, steamboat flank of tooth initial value as the accurate initial value x of Tooth Contact Analysis equation solution(0)=
(φ1 (0),φ2 (0),θ1 (0),θ2 (0))。
3. the method for Tooth Contact Analysis containing error of spiral bevel gear according to claim 1, which is characterized in that utilize punch-pin
Type adaptive trust region algorithm solves bull wheel flank of tooth initial value (φ2,θ2), steamboat flank of tooth initial value (φ1,θ1) and flank of tooth initial contact
Point P0*(φ1,θ1,φ2,θ2)。
4. the method for Tooth Contact Analysis containing error of spiral bevel gear according to claim 1, which is characterized in that utilize punch-pin
Type adaptive trust region algorithm solves steamboat flank of tooth initial value, the specific steps are：
In order to solve equation (14), one can be converted it into without constrained nonlinear systems problem；
Using convex model, the subproblem of trusted zones can be expressed as：
In formula, fk=f (xk) and gk=g (xk) indicate gradient, bk∈RNIndicate horizontal vector, andWork as bk=0 orFor convex model at a second-order model, such convex model is the generalized form of second-order model；
In order to select the parameter b of convex modelkAnd Bk, use：
If ρk>=0, have：
Otherwise, βk=1.During replacing trusted zones, for given sufficiently large positive number b*>0, have
It will be apparent that BkIt needs to be satisfied with：
In addition, it has proved that：Work as ρk>0, then dT k-1yk-1>0；Work as Bk-1>When 0, Bk>0；In current search, a letter is given
Rely domain：
In formula, Rτ(rk) (τ ∈ (0,1)) be the relaxation R- function increasedd or decreased about Trust Region Radius, have：
Beta, gamma in formula1, γ2It is corresponding constant with Ω, the specific definition about R- function can be with bibliography, in addition, rkIt indicates
Function f and fkThe ratio of gains be：
5. according to the method for Tooth Contact Analysis containing error of the spiral bevel gear of claim 3 or 4, which is characterized in that punch-pin
The basic program of type adaptive trust region algorithm is as follows：
Input x0∈RN,Δ0>0,b0∈RN×1,0<β<1,0<γ1<1-β,γ2>0,M>1+γ2,ε>0,0<μ<1；
Initialization:K=0, Δ0=1, x=x0,b0=0, B0=I (cell matrix)；
Step 1. evaluates fk=f (xk) andif||gk| |≤ε, then are terminated, x*=xk；
K >=1 Step 2.If calculates ρkBy solving equation (17)；ifρk>=0, solve βkAccording to equation (18)；
otherwise,βk=1. solve b using equation (19)-(21) respectivelyk,yk-1, and Bk.
Step 3. sets b=bk, g=gk, and B=BkApproximate solution d* is obtained, by solving equation (17)
Step 4. assumes dk=d* calculates r according to equation (25)k.
Step 5.if rk<μ enables xk+1=xk；Otherwise enables xk+1=xk+dk.
6. corrected parameter b of StepkAnd Bk, acquire bk+1And Bk+1；Δ is calculated by equation (22) and (23)k+1.
Step 7. enables k=k+1, returns to initialization.
6. the method for Tooth Contact Analysis containing error of spiral bevel gear according to claim 1, which is characterized in that the step
(2) in, the specific steps are：
2.1) coordinate transformation matrix is obtained, eTCA equation group is obtained；
It is modeled by the flank of tooth and carries out flank of tooth discretization, size flank of tooth point Pi(φi,θi) (i=1, steamboat；I=2, bull wheel) it can be each
From coordinate system Oi(Xi,Yι,Zι) (i=1,2) solve and, resultant method arrow is expressed as ri(i=1,2) and ni(i=
1,2), so latter two flank of tooth needs respective rotate by a certain angle to reach flank of tooth conjugate point contact condition, and the contact for completing the flank of tooth passes
It is dynamic；In this initial contact point P* (φ1,θ1,φ1,θ2) position, meet flank engagement principle, has：
Wherein, which has spin matrix Mi-fIt indicates, eTCA contact conditions mainly include：I) latter two flank of tooth point is rotated
Possess common point arrow, i.e. (rf)1=(rf)2；Ii the method arrow for) rotating latter two flank of tooth point is conllinear, i.e. N1=χ0N2, χ0For constant,
It can thus be concluded that spin matrix Mi-f(i=1,2)；
Firstly, considering that big pinion gear is the flank of tooth model established under respective coordinate system, then they must rotate a certain angle
Reach unified flank engagement coordinate system.Then from gear original coordinate system Oi(Xi,Yi,Zi) arrive engagement coordinate system Of(Xf,Yf,Zf) turn
Change matrix Mf-i∈R4(i=1,2) it is represented by：
Its submatrix MX-2For：
Rotating angle is：
And：
Reach the same engagement coordinate system Of(Xf,Yf,Zf) after, due to manufacturing the presence of rigging error, the size flank of tooth may not have
Point contact occurs, then needs to further rotate transformation, constrains face motion process, and meet TCA contact conditions, adopts therebetween
With following rotation mode：The steamboat flank of tooth is rotated, and rigging error is added in the bull wheel flank of tooth.During rotating the steamboat flank of tooth herein,
Spin matrix is represented by：
(MG-X)1=(MG-A)1·(MA-B)1·(MB-C)1·MC-X (32)
It include M during realization method swears conllinear rotation transformationG-A,MA-B,MB-CAnd MC-XEqual kinematic chains, have：
In formula, angle is rotated
In realizing the conversion process that point arrow is overlapped, spin matrix (Mt-f)1With matrix MG-XProcess it is consistent, include son fortune
Dynamic chain is Mt-g,Mg-a,Ma-b,Mb-cAnd Mc-f, homogeneous transformation process is：
Nf:N1=MX-1(MG-X)1·n1 (38)
rf:(rf)1=Mf-1(Mt-f)1·r1 (39)
It is worth noting that generally error amount is added in one of flank of tooth in the considerations of manufacturing rigging error,
Consider rigging error transformation matrix of coordinates be：
It is used for the submatrix M that calculating method swears conllinear processm-a∈R3
The bull wheel flank of tooth is added in manufacture rigging error can be obtained after all rotation transformations：
rf:(rf)2=MM-AMf-2·r2 (42)
Nf:N2=Mm-aMX-2·ni (43)
Conllinear eTCA condition is found in conclusion meeting, and can obtain following two scalar equation：
(feTCA)1:(N1)1+(N2)1=0 (44)
(feTCA)2:(N1)2+(N2)2=0 (45)
Herein, that reflection is Ni=[(Ni)1,(Ni)2,(Ni)3] two of them component functional relation；
Similarly, when meeting the eTCA condition that point arrow is overlapped, following scalar equation sets up：
(feTCA)3:[(rf)1]1-[(rf)2]1=0 (46)
(feTCA)4:[(rf)1]2-[(rf)2]2=0 (47)
(feTCA)5:[(rf)1]3-[(rf)2]3=0 (48)
At this point, about (rf)i:={ [(rf)i]1,(rf)i]2,(rf)i]3}T(i=1,2) three components will consider simultaneously；
It is assumed that the angular speed for inputting steamboat in gear drive is ω1, the angular speed of bull wheel is ω2, pass through bull wheel after rotation transformation
Angular speed become ωf2, there is following relationship to set up：
The opposite angular speed of bull wheel and steamboat is：
ω1-2=ω1-ωf2 (50)
Consider that the influence of manufacture rigging error degree has then after coordinate is converted：
So far, the linear velocity v of bull wheel2With the linear velocity v of steamboat1For：
The then relative linear velocity v of bull wheel and steamboat1-2For：
v1-2=v1-v2 (54)
State is initially contacted herein, meets the theory of engagement, can be obtained：
(feTCA)6:(N2)1×(v1-2)1+(N2)2×(v1-2)2+(N2)3×(v1-2)3=0 (55)
In summary, six scalar equations contain 6 known variablesΔlX, Δ lYWith Δ lZ, can be quickly accurate
The geometry motion transformation matrix during entire flank of tooth initial contact is solved, while establishing accurate eTCA equation group feTCA=
[(feTCA)1,(feTCA)2,(feTCA)3,(feTCA)4,(feTCA)5,(feTCA)6]；
2.2) scoring item of eTCA equation group is established：A) teeth contact；B) driving error；C) tooth surface error is completed to spiral shell
Revolve the evaluation of tooth surfaces of bevel gears contact performance.
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