CN109446711A - Contact force driving error numerical computation method of the spiral bevel gear containing installation error - Google Patents
Contact force driving error numerical computation method of the spiral bevel gear containing installation error Download PDFInfo
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Abstract
A kind of contact force driving error numerical computation method the invention discloses spiral bevel gear containing installation error, flank of tooth driving error of spiral bevel gear under the conditions of loading and being not loaded with is solved on the basis of Tooth Contact Analysis, and influence of the installation error to driving error is considered in solution procedure, it can guarantee and point contact really occurs between two flank of tooth.Significant thinking is provided for the design and analysis of spiral bevel gear.Entire solution procedure calculates auxiliary software using numerical value and is achieved, and there is no contingency and uncertainty caused by human factor, calculating process can pass through software realization.
Description
Technical field
The invention belongs to spiral bevel gear design field, specially a kind of contact force of the spiral bevel gear containing installation error
Driving error numerical computation method.
Background technique
Spiral Bevel Gear Transmission is that the fields such as mechanical engineering are most common, one of most common driving form, is held due to having
The features such as loading capability is big, stable drive, rotational noise is low, transmission ratio is big, registration is high, can be with high efficiency, highly reliable transmitting
The movement and power of any between centers in space.And the driving error of spiral bevel gear commenting as Spiral Bevel Gears contact performance
One of price card standard, it is closely bound up with shock and vibration, the noise of spiral bevel gear etc., directly affect the fortune of spiral bevel gear
Turn stationarity.
Excellent driving error amplitude and driving error curve in order to obtain, it is necessary to consider installation error, profile error
Influence etc. a variety of errors to driving error.The driving error of spiral bevel gear is calculated, mainly uses face at present
Analysis and solution be not loaded in the case of driving error and using the driving error under finite element method loading environment, all deposit
In artificial contingency and uncertainty.
Summary of the invention
The purpose of the present invention is to provide a kind of spiral bevel gears that can guarantee solution procedure and determine there is no man's activity
Contact force driving error numerical computation method containing error.
This Spiral Bevel Gears load driving error numerical computation method provided by the invention, calculates separately monodentate and nibbles
Close be not loaded with, load and bi-tooth gearing be not loaded with, under stress state consider installation error driving error, to obtain list respectively
Tooth engagement load driving error and bi-tooth gearing load rotation error, comprising the following steps:
A, driving error defines
What driving error indicated is difference of the actual rotational angle of driven wheel relative to theoretical corner during engagement rotation
Value, i.e. driving error are as follows:
Wherein, φ1For the corner of steamboat；φ2For the corner of bull wheel；N1 and N2 is respectively the number of teeth of steamboat and bull wheel, steamboat
It is driven wheel for driving wheel, bull wheel；
B, monodentate engages the driving error Δ φ being not loaded under state1It calculates
B.1 Summary of machine settings parameter, basic gear blank design parameter and cutter parameters) are inputted, are modeled by tooth surface parameters,
Determine that the engagement contact condition and coordinate conversion matrix of large and small gear, bull wheel consider the coordinate conversion matrix of installation error, ask
Solve the accurate flank of tooth initial point of large and small gear；
B.2 it) is overlapped according to an arrow, method arrow is overlapped and meshing condition establishes engagement contact nonlinear equation group (TCA equation
Group)；
B.3 bull wheel theory corner) is calculated, bull wheel actual rotational angle is solved, driving error is calculated, acquires unknown parameter；
B.4) change steamboat corner, repeat step 2.2) and 2.3), such loop iteration, until find out required corner it
Under bull wheel actual rotational angle, Δ φ can be calculated1；
C, under monodentate engagement stress state, the driving error Δ due to caused by loaded deformation1 *It calculates
C.1 contact point K to the distance rk of rotation axis, the helical angle of the flank of tooth principal curvatures δ k and contact point K on the flank of tooth) are solved
βkWith face power F；
C.2) according to principal curvatures δ k and face power F calculate the elliptical major semiaxis a of face, semi-minor axis b and and
Deformation of tooth surface amount w；
C.3) the parameter obtained according to above-mentioned solution solves the caused driving error △ under load effect1 *；
Monodentate engages the driving error under loaded-up condition
D, the driving error calculating process and principle that bi-tooth gearing is not loaded under state engage under the state of being not loaded with monodentate
Calculating as；
E, under bi-tooth gearing stress state, two teeth calculate the driving error due to caused by loaded deformation
E.1) first determine that one of tooth to 0 engagement contact condition and coordinate conversion matrix, solves according to input parameter
Initial point, and establish TCA equation group；
E.2 tooth) is obtained into tooth to 1 around the angle that own axes rotate a tooth by coordinate transform to 0 again, again basis
Input parameter determines that it engages contact condition and transformation matrix of coordinates, establishes TCA equation group；
E.3) the unknown parameter variable according to involved in the solving equations equation of foundation, including bull wheel actual rotational angle and
Steamboat actual rotational angle；
E.4) based on the bull wheel actual rotational angle and steamboat actual rotational angle acquired, public affairs when calculating are not loaded with using monodentate engagement
What formula was calculated two tooth pair of bi-tooth gearing is not loaded with driving error；
E.5 flank of tooth principal curvatures δ k0, contact point helixangleβ) are solved to 0 TCA equation group according to toothk0, contact point to axis
The linear distance rk0 and driving error STE0 due to caused by loaded deformation；According to tooth to 1 TCA equation group, flank of tooth principal curvatures is solved
δ k1, contact point helixangleβk1, contact point to axial line distance rk1 and the driving error STE1 due to caused by loaded deformation；
E.6 step (E.5) calculated parameter) is utilized, establishes that driving error is equal and counterweight balance equation, is calculated separately
Face power F0 and F1 on two rodent populations out, back substitution solution can obtain two rodent populations under bi-tooth gearing stress state and draw
The driving error risen；
Monodentate engagement is not loaded with driving error Δ φ1It is specific calculating it is as follows:
Lathe coordinate system Sg, the bad coordinate system S1 of wheel rigidly fixed with wheel blank that setting is rigidly fixed with cutting lathe, with
The coordinate system St that cage chair rigidly fixes；The cutterhead flank of tooth of processing spiral bevel gear is a circular cone under lathe fixed coordinate system
Face, equation may be expressed as:
Corresponding cutterhead per unit system arrow are as follows:
Wherein, (up, θp) it is surface coordinates, α is the profile angle of cutter, rcIt is cutter radius, the vector with positive α and negative α
Function respectively indicates the flank of tooth of two cutterheads for processing steamboat concave and convex surface；
Firstly, large and small wheel establishes flank of tooth model under respective coordinate system, the transition matrix from cutterhead to wheel blank:
Wherein, φ1=mcφc1, mcFor cutting rolling ratio；γm1, Δ Em1, Δ XB1, Δ XD2, Sr1, q1It is setting for machine ginseng
Number, can be obtained by Summary of machine settings；φc1It is the corner of cage chair.
For the calculating of subsequent normal vector, transition matrix is removed into last line and last column obtains its sub- square
Battle array:
From wheel blank coordinate system to the transition matrix of engagement coordinate system are as follows:
In formula, angle is rotated
Swing offset (△ l)1=((△ lX)1,(△lY)1,(△lZ)1)；
Its corresponding submatrix are as follows:
After reaching same engagement coordinate system, due to the presence of installation error, two flank of tooth may need to be examined there is no point contact
Consider installation error and further rotates transformation；Installation error is added on the flank of tooth of bull wheel, steamboat remains unchanged, and bull wheel considers installation
The transformation matrix of coordinates of error are as follows:
Its corresponding submatrix are as follows:
Here, the installation error of eT angle between two Gear axis；EAX is to miss along the installation in Gear axis direction
Difference；EOS is gear distance between axles installation error；
After entire coordinate transform, tooth surface equation and normal vector of the steamboat in the case where engaging coordinate system be may be expressed as:
Rm1(θ1,φ1)=(Mt-f)1×M1p·Rp(θ1,φ1) (10)
Nm1(θ1,φ1)=(Lt-f)1×L1p·np(θ1,φ1) (11)
The installation error addition bull wheel flank of tooth can be obtained into the bull wheel flank of tooth in the case where engaging coordinate system after all coordinate transforms
Tooth surface equation and the flank of tooth method arrow:
Rm2(θ2,φ2)=MM-A×(Mt-f)2×M2p·Rp(θ2,φ2) (12)
Nm2(θ2,φ2)=Lm-a(Lt-f)2×L2p·np(θ2,φ2) (13)
The first and second primitive forms of the steamboat flank of tooth can be solved according to above-mentioned two formula；
First primitive form:
Second primitive form:
In formula, Rθ1And Rφ1Respectively two tangent lines of the flank of tooth；
The first and second primitive forms for acquiring flank of tooth substitution following equation can be solved into tooth principal curvature of a surface Rk1And Rk2；
Similarly, tooth surface equation R of the bull wheel in the case where engaging coordinate system can be acquiredm2With normal vector Nm2And the bull wheel flank of tooth
Principal curvatures；
Two flank of tooth will reach engaged transmission, and contact surface necessarily is in continuous contact state, it is desirable that the vector sum of two flank of tooth
Method arrow in office one is instantaneously all overlapped, and meets mesh equation；
Nm1(θp,φc1)=Nm2(θg,φc2) (19)
Rm1(up,θp,φc1)=Rm2(ug,θg,φc2) (20)
It can export from above-mentioned two vector equation containing there are six unknown number up, θp, φc1, ug, θg, φc2Five independences
Vector equation；To solve unknown parameter, it is necessary to which, plus mesh equation as supplement, mesh equation may be expressed as:
Nv=Nm1(θp,φc1)·v12=f (up,θp,φc1)=0 (21)
In formula, v12It is speed of related movement；
Here
In summary the equation group of six nonlinear equations composition, can solve above-mentioned six unknown parameter (up, θp,
φc1, ug, θg, φc2)；
To solve driving error, it need to give steamboat one initial corner, each corner carries out a Tooth Contact Analysis
(solving TCA equation group), obtains one group of parameter, and by successive ignition, contact analysis solves a series of reality that can obtain bull wheel wheels
Corner and theoretical corner, the actual rotational angle of bull wheel subtract driving error of the available gear of theoretical corner when being not loaded with
Δφ1。
Monodentate engages caused driving error Δ under stress state1 *It calculates specific as follows:
The moment of flexure according to suffered by gear can calculate power suffered on the flank of tooth:
In formula, M is moment of flexure suffered by gear；rkIt is contact point of gear surface K the distance between to gear rotary shaft, it can be by formula
It calculates；α is tool-tooth profile angle；βkIt is the helical angle of contact point of gear surface K；Since the rotary shaft of steamboat is z-axis, face can be obtained
Distance of the point K to z-axis are as follows:
The helical angle of the contact point position K can be found out by formula:
In formula, r0For cutter radius；R ' is the pitch cone radius at the K of contact point；β is nominal helical angle；R is mean cone distance；R0It is outer
Pitch cone radius；B is the facewidth；
The elliptical length of face half can be calculated according to power and tooth principal curvature of a surface suffered on the flank of tooth of above-mentioned solution
Axial length and deformation of tooth surface:
Major semiaxis:
Semi-minor axis: b=λ a (32)
In formula, λ is the root of equation (33):
δk2J1(λ)-δk1J2(λ)=0 (33)
Deformation of tooth surface amount:
Wherein, E1、E2The elasticity modulus of respectively small bull wheel；u1、u2The Poisson's ratio of respectively small bull wheel；E*It is comprehensive elasticity
Modulus, ξ are general machined parameters swivel angle；
So far, deformation of tooth surface w, the long axial length a of Contact Ellipse, flank of tooth any point helixangleβk, flank of tooth any point to rotation
The distance r of shaft axiskParameter has all been found out；In summary required parameter, can calculate biography of the flank of tooth under loading conditions
Dynamic error:
Entire solution procedure can use numerical value and calculate auxiliary software realization.
Step 5.6) establish driving error is equal and counterweight balance equation are as follows:
δφ0(F0)=δ φ1(F1) (40)
F0rk0cosαcosβ0+F1rk1cosαcosβ1=M (41).
The present invention solves the flank of tooth of spiral bevel gear under the conditions of loading and being not loaded on the basis of Tooth Contact Analysis
Driving error, and influence of the installation error to driving error is considered in solution procedure, it can guarantee and really sent out between two flank of tooth
Raw point contact.Significant thinking is provided for the design and analysis of spiral bevel gear.Entire solution procedure is using numerical value meter
It calculates auxiliary software to be achieved, there is no contingency and uncertainty caused by human factor, calculating process can pass through software reality
It is existing.
Detailed description of the invention
Fig. 1 is that spiral bevel gear manufactures assembling schematic diagram.
Fig. 2 is that monodentate engaged transmission error solves flow chart in the case of considering installation error.
Fig. 3 is that bi-tooth gearing driving error solves flow chart in the case of considering installation error.
Fig. 4 is not loaded with driving error curve graph when being one example monodentate engagement of the present invention.
Fig. 5 is that this example monodentate loads driving error curve graph when engaging.
Driving error curve graph is loaded when Fig. 6 is this example bi-tooth gearing.
Specific embodiment
The present invention is different from traditional driving error calculation method, proposes a kind of spiral bevel gear for considering installation error
Driving error numerical computation method is loaded, speed transmission under meshing state and bi-tooth gearing state of peace is missed respectively using this method
Difference carries out numerical value calculating, specific as follows:
What driving error indicated is difference of the actual rotational angle of driven wheel relative to theoretical corner during engagement rotation
Value.That is driving error are as follows:
Wherein, φ1For the corner of steamboat；φ2For the corner of bull wheel；N1 and N2 is respectively the number of teeth of steamboat and bull wheel.
It should be noted that since the modeling process of steamboat is all more complicated for bull wheel, so
For Machining Analysis process of the invention mainly based on steamboat, the calculating of bull wheel solves the method and process that can refer to steamboat.
One, it is calculated about the driving error under monodentate engagement
The cutterhead flank of tooth of processing spiral bevel gear is a circular conical surface under lathe fixed coordinate system, and equation can indicate
Are as follows:
Corresponding cutterhead per unit system arrow are as follows:
Wherein, (up, θp) it is surface coordinates, α is the oblique angle of cutter, rcIt is cutter radius, the vector letter with positive α and negative α
Number respectively indicates the flank of tooth of two cutterheads for processing steamboat concave and convex surface.
It is by corresponding coordinate transform that the cutterhead under lathe fixed coordinate system is equations turned under wheel blank coordinate system, it obtains
Tooth surface equation under wheel blank coordinate system, the present invention used in slave lathe coordinate system to the transition matrix of wheel blank coordinate system are as follows:
Wherein, φ1=mcφc1, mcRatio, φ are rolled for cuttingc1It is the corner of cage chair.γm1, Δ Em1, Δ XD2, Sr1, q1, Δ
EB1It is lathe adjusting parameter, can be obtained by Summary of machine settings.
For the calculating of subsequent flank of tooth normal vector, transition matrix need to be removed to last line and last column, obtain its correspondence
Submatrix:
Two gears will realize engagement, it is necessary to rotate a certain angle and reach same flank engagement coordinate system, by flank of tooth coordinate
Under tooth surface equation and flank of tooth unit normal direction vector median filters to engagement coordinate system under system, the tooth surface equation under engagement coordinate system is obtained
Rm1(θ1,φ1) and flank of tooth normal vector nm1(θ1,φ1).It is used herein slave wheel blank coordinate system to the conversion square of engagement coordinate system
Battle array are as follows:
In formula, angle is rotatedSwing offset (△ l)1=((△ lX)1,
(△lY)1,(△lZ)1)。
Its corresponding submatrix are as follows:
After reaching same engagement coordinate system, due to the presence of installation error, two flank of tooth may there is no point contact, need by
Installation error is also taken into account, and transformation is further rotated.
It should be noted that general need to only be added to installation error on one of flank of tooth considers when considering installation error
, method used herein is that installation error is added on the flank of tooth of bull wheel, and steamboat remains unchanged.It considers installation error
Transformation matrix of coordinates are as follows:
Its corresponding submatrix are as follows:
In formula, the installation error of eT angle between two Gear axis, i.e. α in Fig. 1；EAX is along Gear axis side
To installation error, i.e. P and G in Fig. 1；EOS is gear distance between axles installation error, i.e. E in Fig. 1.
After entire coordinate transform, tooth surface equation and flank of tooth method arrow of the steamboat flank of tooth in the case where engaging coordinate system can be indicated
Are as follows:
Rm1(θ1,φ1)=(Mt-f)1×M1p·Rp(θ1,φ1) (10)
Nm1(θ1,φ1)=(Lt-f)1×L1p·np(θ1,φ1) (11)
After the bull wheel flank of tooth is added in installation error, after all coordinate transforms, the bull wheel flank of tooth can be obtained in engagement coordinate
Tooth surface equation and flank of tooth method arrow under system:
Rm2(θ2,φ2)=MM-A×(Mt-f)2×M2p·Rp(θ2,φ2) (12)
Nm2(θ2,φ2)=Lm-a(Lt-f)2×L2p·np(θ2,φ2) (13)
The first and second primitive forms of the steamboat flank of tooth can be solved according to above-mentioned two formula.
First primitive form:
Second primitive form:
In formula, Rθ1And Rφ1Respectively two tangent lines of the flank of tooth.
The first and second primitive forms for acquiring flank of tooth substitution following equation can be solved into tooth principal curvature of a surface Rk1And Rk2；
Similarly, tooth surface equation R of the bull wheel in the case where engaging coordinate system can be acquiredm2With normal vector Nm2And the bull wheel flank of tooth
Principal curvatures；
The flank of tooth will reach engagement, and contact surface necessarily is in continuous contact state, it is desirable that the position vector and method of two flank of tooth
Line vector in office one is instantaneously all overlapped, and two flank of tooth meet theory of engagement equation.
Nm1(θp,φc1)=Nm2(θg,φc2) (19)
Rm1(up,θp,φc1)=Rm2(ug,θg,φc2) (20)
Nv=Nm1(θp,φc1)·v12=f (up,θp,φc1)=0 (21)
In formula, v12 is speed of related movement.
Wherein,
Six effective equations can be exported according to above equation group, solution can obtain six unknown parameter up, θp, φp, ug,
θg, φg.It, can be in the hope of principal curvatures δ k1 and the δ k2 at any point on the flank of tooth according to the six of above-mentioned solution parameters.
For solution driving error, it need to give steamboat one initial corner, each corner carries out a Tooth Contact Analysis,
One group of parameter is obtained, by successive ignition, contact analysis solves a series of actual rotational angle that can obtain bull wheels and theoretical corner, bull wheel
Actual rotational angle subtract driving error Δ φ of the available gear of theoretical corner when being not loaded with.Its rough solution stream
Journey is as shown in Figure 2.
For the influence for considering load factor, it is necessary to calculate flank of tooth driving error Δ * due to caused by loaded deformation.By gear
Suffered load can calculate the contact force on the flank of tooth:
Wherein, M is moment of flexure suffered by gear；rkIt is contact point of gear surface K the distance between to gear rotary shaft；α is cutter
Profile angle；βkIt is the helical angle of contact point of gear surface K.
Due to steamboat rotary shaft be z-axis, can obtain contact point of gear surface K to z-axis distance are as follows:
In formula, r0For cutter radius；R ' is the pitch cone radius at the K of contact point；β is nominal helical angle；R is mean cone distance.R0It is outer
Pitch cone radius；B is the facewidth.
In summary the face power and tooth principal curvature of a surface solved can calculate the elliptical length of face by formula
Semiaxis and semi-minor axis:
Major semiaxis:
Semi-minor axis: b=λ a (32)
In formula, λ is the root of equation (34):
δk2J1(λ)-δk1J2(λ)=0 (33)
Deformation of tooth surface amount:
Wherein, E*It is synthetical elastic modulus；E1、E2The elasticity modulus of respectively small bull wheel；u1、u2The pool of respectively small bull wheel
Loose ratio.
So far, deformation of tooth surface, face power, face ellipse major semiaxis, some helical angles of the flank of tooth, the flank of tooth a little arrive
The parameters such as the distance of gear rotation axis have all been found out, and can calculate biography of the gear under load effect according to the parameter of solution
Dynamic error:
Above-mentioned entire solution procedure can use numerical value calculating auxiliary software and be achieved.In conjunction with it is above-mentioned acquire be not added
Driving error Δ φ under the conditions of load and the driving error Δ under load condition*, spiral bevel gear is obtained under load effect
Driving error:
δ φ=△ φ+△*
Two, it is calculated about the driving error under bi-tooth gearing
Driving error under bi-tooth gearing is by the driving error two under the driving error and loading environment that are not loaded under state
Part forms, wherein the driving error calculating process and principle under the conditions of being not loaded with are as the calculating under monodentate engagement.Specifically
Process can be referring to the calculating process of the driving error under monodentate meshing state.About the calculating of the driving error under loading environment,
Unlike monodentate engagement, when bi-tooth gearing, there is the total torque of two pairs of teeth (tooth is to 0 and tooth to 1) shared, need to distinguish
Calculate the face power on the respective flank of tooth.As shown in figure 3, substantially calculation process is as follows:
A) calculation method first is engaged according to monodentate, solves tooth to the contact point helixangleβ k on 00, contact point to gear is revolved
The distance rk of shaft axis0And driving error Δ0 *Etc. parameters.
B) again to tooth to 1 carry out it is same calculate analysis, obtain tooth to contact point helixangleβ k on 11, contact point to gear
The distance rk of rotation axis1And driving error Δ1 *Etc. parameters.
C) above-mentioned parameter is utilized, establishes equation with counterweight balance relationship according to driving error is equal, when obtaining bidentate contact,
Respective face power F0 and F1, back substitution enter equation (28) and (34) on two pairs of flank of tooth, can acquire under loaded conditions, arc
The driving error of bevel gear bi-tooth gearing.
δφ0(F0)=δ φ1(F1) (40)
F0rk0cosαcosβ0+F1rk1cosαcosβ1=M (41)
D) being not loaded with and the driving error under load two states, available curved-tooth bevel gear under comprehensive bi-tooth gearing
Take turns the driving error in bi-tooth gearing.
So far, spiral bevel gear is not loaded with and misses with the transmission under load condition under monodentate engagement and bi-tooth gearing state
Difference can all determine that the design for high-precision spiral bevel gear provides important evidence.
Three, example
Below by taking a pair of of high-speed overload aviation spiral bevel gear as an example, with calculation method set forth above, count respectively
It calculates under conditions of considering installation error, driving error numerical value of the spiral bevel gear under monodentate engagement and under bi-tooth gearing is big
It is small.
Table 1 gives the tooth surface design basic parameter of face milling spiral bevel gear；Table 2 gives spiral bevel gear bull wheel
Summary of machine settings machined parameters；Table 3 gives spiral bevel gear steamboat Summary of machine settings machined parameters.
1 Spiral Bevel Gears of table design basic parameter
2 spiral bevel gear bull wheel of table adjusts card parameter
3 spiral bevel gear steamboat of table adjusts card parameter
Fig. 4 gives the driving error curve being not loaded under spiral bevel gear monodentate meshing state；Fig. 5 shows curved tooth
The driving error curve loaded under bevel gear monodentate meshing state；What Fig. 6 was provided is added under spiral bevel gear bi-tooth gearing state
Carry driving error curve.
Comparative analysis result is it is found that the driving error under stress state is bigger than the driving error under the conditions of being not loaded with；And
Driving error under monodentate meshing condition is bigger than the driving error under bi-tooth gearing state.Main reason is that: when load, tooth
The loaded deformation in face, can cause additional driving error；When bi-tooth gearing, since two pairs of teeth contact simultaneously, load is undertaken, on the flank of tooth
Contact force than monodentate engage when it is small, cause deformation of tooth surface small, caused driving error is small, so transmission when bi-tooth gearing
Error is less than driving error when monodentate engages.It follows that Spiral Bevel Gear Transmission error value proposed in this paper calculating side
Method has great significance for design of gears and analysis.
Claims (4)
1. a kind of contact force driving error numerical computation method of spiral bevel gear containing installation error calculates separately monodentate engagement
Be not loaded with, load and bi-tooth gearing be not loaded with, under stress state consider installation error Spiral Bevel Gear Transmission error, thus point
Monodentate engagement load driving error and bi-tooth gearing load rotation error are not obtained, comprising the following steps:
1, driving error defines
What driving error indicated is difference of the actual rotational angle of driven wheel relative to theoretical corner during engagement rotation, i.e.,
Driving error are as follows:
Wherein, φ1For the corner of steamboat；φ2For the corner of bull wheel；N1 and N2 is respectively the number of teeth of steamboat and bull wheel；
2, monodentate engages the driving error Δ φ being not loaded under state1It calculates
2.1) Summary of machine settings parameter, basic gear blank design parameter and cutter parameters are inputted, are modeled by tooth surface parameters, are determined
The engagement contact condition and coordinate conversion matrix of large and small gear, bull wheel consider the coordinate conversion matrix of installation error, solve
The large and small accurate flank of tooth initial point of gear；
2.2) it is overlapped according to an arrow, method arrow is overlapped and meshing condition establishes engagement contact nonlinear equation group (TCA equation group)；
2.3) bull wheel theory corner is calculated, bull wheel theory corner is solved, driving error is calculated, acquires unknown parameter；
2.4) change steamboat corner, repeat step 2.2) and 2.3), such loop iteration, until finding out in required bracket at corner
Bull wheel actual rotational angle can calculate Δ φ1；
3, under monodentate engagement stress state, the driving error Δ due to caused by loaded deformation1 *It calculates
3.1) contact point K to the distance rk of rotation axis, the helixangleβ of the flank of tooth principal curvatures δ k and contact point K on the flank of tooth are solvedkWith
Face power F；
3.2) the elliptical major semiaxis a of face, semi-minor axis b and and the flank of tooth are calculated according to principal curvatures δ k and face power F
Deflection w；
3.3) parameter obtained according to above-mentioned solution solves the caused driving error △ under load effect1 *；
4, the driving error under monodentate engagement loaded-up condition
5, the driving error calculating process and principle that bi-tooth gearing is not loaded under state engage down under the state of being not loaded with monodentate
It calculates the same；
6, the driving error under bi-tooth gearing stress state calculates
6.1) first determine that one of tooth to 0 engagement contact condition and coordinate conversion matrix, solves initial according to input parameter
Point, and establish TCA equation group；
6.2) tooth is obtained into tooth to 1, again according to input around the angle that own axes rotate a tooth by coordinate transform to 0 again
Parameter determines that it engages contact condition and transformation matrix of coordinates, establishes TCA equation group；
6.3) the unknown parameter variable according to involved in the solving equations equation of foundation, including bull wheel actual rotational angle and steamboat
Actual rotational angle；
6.4) based on the bull wheel actual rotational angle and steamboat actual rotational angle acquired, formula meter when calculating is not loaded with using monodentate engagement
Calculate obtain bi-tooth gearing transmission be not loaded with driving error；
6.5) flank of tooth principal curvatures δ k0, contact point helixangleβ are solved to 0 TCA equation group according to toothk0, contact point to axis away from
From rk0 and the driving error STE0 due to caused by loaded deformation；According to tooth to 1 TCA equation group, solve flank of tooth principal curvatures δ k1,
Contact point helixangleβk1, contact point to axial line distance rk1 and the driving error STE1 due to caused by loaded deformation；
6.5) the calculated parameter of step (i) is utilized, establishes that driving error is equal and counterweight balance equation, calculates separately out two
Face power F0 and F1 on rodent population, back substitution solution can be obtained and be passed caused by two rodent populations under bi-tooth gearing stress state
Dynamic error.
2. contact force driving error numerical computation method of the spiral bevel gear as described in claim 1 containing installation error,
It is characterized in that, monodentate engagement is not loaded with driving error Δ φ1It is specific calculating it is as follows:
The lathe coordinate system Sg, the bad coordinate system S1 of wheel rigidly fixed with wheel blank, with cage chair that setting is rigidly fixed with cutting lathe
The coordinate system St rigidly fixed；The cutterhead flank of tooth of processing spiral bevel gear is a circular conical surface under lathe fixed coordinate system,
Equation may be expressed as:
Corresponding cutterhead per unit system arrow are as follows:
Wherein, (up, θp) it is surface coordinates, α is the profile angle of cutter, rcIt is cutter radius, the phasor function with positive α and negative α
Respectively indicate the flank of tooth of two cutterheads for processing steamboat concave and convex surface；
Firstly, large and small wheel establishes flank of tooth model under respective coordinate system, the transition matrix from cutterhead to wheel blank:
Wherein, φ1=mcφc1, mcFor cutting rolling ratio；γm1, Δ Em1, Δ XB1, Δ XD2, Sr1, q1It is lathe adjusting parameter,
It can be obtained by Summary of machine settings；φc1It is the corner of cage chair；
For the calculating of subsequent normal vector, transition matrix is removed into last line and last column obtains its submatrix:
From wheel blank coordinate system to the transition matrix of engagement coordinate system are as follows:
In formula, angle is rotated
Swing offset
Its corresponding submatrix are as follows:
After reaching same engagement coordinate system, due to the presence of installation error, two flank of tooth may need to will be installed there is no point contact
Error is also taken into account, and transformation is further rotated；Installation error is added on the flank of tooth of bull wheel, steamboat remains unchanged, and bull wheel is examined
Consider the transformation matrix of coordinates of installation error are as follows:
Its corresponding submatrix are as follows:
In formula, the installation error of eT angle between two Gear axis；EAX is the installation error along Gear axis direction；eOS
For gear distance between axles installation error；
After entire coordinate transform, tooth surface equation and normal vector of the steamboat in the case where engaging coordinate system be may be expressed as:
Rm1(θ1,φ1)=(Mt-f)1×M1p·Rp(θ1,φ1) (10)
Nm1(θ1,φ1)=(Lt-f)1×L1p·np(θ1,φ1) (11)
The installation error addition bull wheel flank of tooth can be obtained into tooth of the bull wheel flank of tooth in the case where engaging coordinate system after all coordinate transforms
Face equation and flank of tooth method arrow:
Rm2(θ2,φ2)=MM-A×(Mt-f)2×M2p·Rp(θ2,φ2) (12)
Nm2(θ2,φ2)=Lm-a(Lt-f)2×L2p·np(θ2,φ2) (13)
The first and second primitive forms of the steamboat flank of tooth can be solved according to above-mentioned two formula；
First primitive form:
Second primitive form:
In formula, Rθ1And Rφ1Respectively two tangent lines of the flank of tooth；
The first and second primitive forms for acquiring flank of tooth substitution following equation can be solved into tooth principal curvature of a surface Rk1And Rk2；
Similarly, tooth surface equation R of the bull wheel in the case where engaging coordinate system can be acquiredm2With normal vector Nm2And the main song of the bull wheel flank of tooth
Rate；
Two flank of tooth will reach engaged transmission, and contact surface necessarily is in continuous contact state, it is desirable that the vector sum method of two flank of tooth is sweared
In office one is instantaneously all overlapped, and meets mesh equation；
Nm1(θp,φc1)=Nm2(θg,φc2) (19)
Rm1(up,θp,φc1)=Rm2(ug,θg,φc2) (20)
It can export from above-mentioned two vector equation containing there are six unknown number up, θp, φc1, ug, θg, φc2Five independent arrows
Measure equation；To solve unknown parameter, it is necessary to which, plus mesh equation as supplement, mesh equation may be expressed as:
Nv=Nm1(θp,φc1)·v12=f (up,θp,φc1)=0 (21)
In formula, v12It is speed of related movement；
Here
In summary the equation group of six nonlinear equations composition, can solve above-mentioned six unknown parameter (up, θp, φc1, ug,
θg, φc2)；
To solve driving error, it need to give steamboat one initial corner, each corner carries out a Tooth Contact Analysis and (solves
TCA equation group), obtain one group of parameter, by successive ignition, contact analysis solve can obtain a series of bull wheel wheels actual rotational angle and
Theoretical corner, the actual rotational angle of bull wheel subtract driving error Δ φ 1 of the available gear of theoretical corner when being not loaded with.
3. contact force driving error numerical computation method of the spiral bevel gear as claimed in claim 2 containing installation error,
It is characterized in that, monodentate engages caused driving error Δ under stress state1 *Calculate specific as follows: the moment of flexure according to suffered by gear can
Calculate power suffered on the flank of tooth:
In formula, M is moment of flexure suffered by gear；rkIt is contact point of gear surface K the distance between to gear rotary shaft, can be calculated by formula；
α is tool-tooth profile angle；βkIt is the helical angle of contact point of gear surface K；Since the rotary shaft of steamboat is z-axis, contact point of gear surface K can be obtained and arrived
The distance of z-axis are as follows:
The helical angle of the contact point position K can be found out by formula:
In formula, r0For cutter radius；R ' is the pitch cone radius at the K of contact point；β is nominal helical angle；R is mean cone distance；R0For outer cone
Away from；B is the facewidth；
It is long that the elliptical major semiaxis of face can be calculated according to power and tooth principal curvature of a surface suffered on the flank of tooth of above-mentioned solution
And deformation of tooth surface:
Major semiaxis:
Semi-minor axis: b=λ a (32)
In formula, λ is the root of equation (33):
δk2J1(λ)-δk1J2(λ)=0 (33)
Deformation of tooth surface amount:
Wherein, E1、E2The elasticity modulus of respectively small bull wheel；u1、u2The Poisson's ratio of respectively small bull wheel；E*It is comprehensive springform
Amount, ξ are general machined parameters swivel angle；
So far, deformation of tooth surface w, the long axial length a of Contact Ellipse, flank of tooth any point helixangleβk, flank of tooth any point to rotation axis
Distance rkParameter has all been found out；In summary required parameter, can calculate driving error of the flank of tooth under loading conditions:
Entire solution procedure can use numerical value and calculate auxiliary software realization.
4. contact force driving error numerical computation method of the spiral bevel gear as claimed in claim 3 containing installation error,
Be characterized in that, step 6.2) establish driving error is equal and counterweight balance equation are as follows:
δφ0(F0)=δ φ1(F1) (40)
F0rk0cosαcosβ0+F1rk1cosαcosβ1=M (41).
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Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2010096242A (en) * | 2008-10-15 | 2010-04-30 | Ricoh Co Ltd | Gear design support method, gear design support device, gear design support program and storage medium |
| CN104408220A (en) * | 2014-10-08 | 2015-03-11 | 西北工业大学 | A modified method for gear teeth loading contact analysis |
| CN104615878A (en) * | 2015-01-29 | 2015-05-13 | 西北工业大学 | Method for calculating between-tooth load distribution with edge contact |
| CN106599335A (en) * | 2016-09-22 | 2017-04-26 | 北京航空航天大学 | Tooth surface modification method capable of reducing sensitivity of installation error to gear transmission pair |
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2018
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Patent Citations (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| JP2010096242A (en) * | 2008-10-15 | 2010-04-30 | Ricoh Co Ltd | Gear design support method, gear design support device, gear design support program and storage medium |
| CN104408220A (en) * | 2014-10-08 | 2015-03-11 | 西北工业大学 | A modified method for gear teeth loading contact analysis |
| CN104615878A (en) * | 2015-01-29 | 2015-05-13 | 西北工业大学 | Method for calculating between-tooth load distribution with edge contact |
| CN106599335A (en) * | 2016-09-22 | 2017-04-26 | 北京航空航天大学 | Tooth surface modification method capable of reducing sensitivity of installation error to gear transmission pair |
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