CN110516345A - Planetary roller screw error source coupling modeling method - Google Patents
Planetary roller screw error source coupling modeling method Download PDFInfo
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Abstract
The invention discloses a kind of planetary roller screw error source coupling modeling method, the technical issues of the practicability is poor for solving existing method.Technical solution is the transmission error for calculating threaded portion first, driving error when acquiring the meshing point coordinate for considering error and corresponding axial gap according to continuous contact condition, then calculating lead screw continuous rotation, under each position of engagement.The meshing point coordinate and the parameters such as corresponding engagement radius and engagement drift angle of subsequent time, the driving error inscribed when then calculating corresponding are calculated again；Next establishes the finite element model of gear, divides the field of conjugate action and solves the flank of tooth softness factor matrix for considering to be bent the deformation of shear-deformable and nonlinear contact, iterative solution obtains Gear Meshing Stiffness, comprehensive meshing error and Static transmissions error；Finally establish planetary roller screw error source coupling model.The drawbacks of bringing this method solve the single analysis planetary roller screw screw thread pair error source of background technique, practicability is good.
Description
Technical field
The present invention relates to a kind of planetary roller screw error source coupling modeling methods.
Background technique
Document " Modelling of transmission accuracy of a planetary roller screw
Mechanism considering errors and elastic deformations " comes from " Mechanism and
A text establishes planetary roller screw screw thread pair error to planetary roller screw driving error in Mechine Theory " magazine
Affecting laws, this method considers the error of threaded portion, and the error of threaded portion circumferencial direction is closed by movement transmitting
System is transformed on axis, to establish planetary roller screw error model, but does not consider gear pair error, does not also consider
The coupled relation of screw thread pair and gear pair.Planetary roller screw (Planetary Roller Screw Mechanism, PRSM)
It is a kind of mechanical transmission mechanism that can convert rotational motion into linear motion.Planetary roller screw utilizes screw thread pair-gear
Secondary planet engaged transmission, carrying out lead screw rotation becomes the power transmitting of nut straight line, therefore it is special that more secondary and multi point engagements are presented
Sign has many advantages, such as that bearing capacity is big, rigidity is good, driving error is high and compact-sized.In aircraft and the full electrification hair of weaponry
In exhibition process, especially there is significant advantage in limited installation space, high-precision, high thrust and the occasions such as height is motor-driven.PRSM has
Multiple threads on single roller contacts with lead screw or nut simultaneously and spiral camber is engaged to engage with gear and is coupled
Feature, the unique engagement feature of PRSM increase the difficulty of its performance evaluation.Establish more perfect PRSM error coupler model
Structural parameters, error and applying working condition are probed into the affecting laws of the movement relation between each part in the mechanism, is into one
The basis of the performances such as the accurate analysis PRSM dynamics of step, abrasion and service life, for improving PRSM transmission performance, developing high-performance
In fields such as aerospace, ship, petrochemical industry, intelligence manufactures, application has important theory significance and engineering to PRSM and promotion PRSM
Application value.It is not ground in terms of now for the correlation between gear and thread error and to driving error influence factor
Study carefully.
Summary of the invention
In order to overcome the shortcomings of existing method, the practicability is poor, and the present invention provides a kind of coupling of planetary roller screw error source and builds
Mould method.This method calculates the transmission error of threaded portion first, and the meshing point for considering error is acquired according to continuous contact condition
Coordinate and corresponding axial gap, then biography when going out lead screw continuous rotation with analytic method COMPREHENSIVE CALCULATING, under each position of engagement
Dynamic error.After calculating the driving error under current time, further according to transmission principle and coordinate transform theoretical calculation subsequent time
Meshing point coordinate and the parameters such as corresponding engagement radius and engagement drift angle, the driving error inscribed when then calculating corresponding,
And so on driving error in entire operation process can be obtained；Next establishes the finite element model of gear, divides the field of conjugate action and asks
Solution considers the flank of tooth softness factor matrix for being bent the deformation of shear-deformable and nonlinear contact, and iterative solution obtains gear engagement
Rigidity, comprehensive meshing error and Static transmissions error；Finally by the error component of had an impact planetary roller screw transmission accuracy
It is transformed into the axis direction of nut, establishes planetary roller screw error source coupling model.It is single that this method solve background techniques
Analysis planetary roller screw screw thread pair error source the drawbacks of bringing, by the movement transitive relation of roller and nut by the mistake of the two
Poor coupled relation links together, and establishes planetary roller screw pair gear-thread error coupled relation, and practicability is good.
A kind of the technical solution adopted by the present invention to solve the technical problems: planetary roller screw error source Coupling method side
Method, its main feature is that the following steps are included:
Step 1: the rotation of lead screw 1 pushes spiral shell by the planetary motion of roller 4 according to the motion principle of planetary roller screw
Mother 2 moves along a straight line, and screw thread pair 5 engages simultaneously with gear pair 7.When lead screw 1 rotates, roller 4 had not only revolved but also rotation, in order to eliminate
The roller tooth 6 of the heeling moment that 1 helix angle of lead screw generates roller 4,4 both ends of roller is engaged with ring gear 3, makes 4 axis of roller
Line is parallel to the rolling of 1 axis of lead screw.When the driving error for defining planetary roller screw pair is that rotating motion part one-directionally rotates, straight line
The difference of the actual displacement of movement parts and theoretical displacement.Definition is as follows:
In formula, TE is driving error；It is actual displacement；θ is input corner；I is theoretical transmission ratio.
Step 2: the axis according to the definition of cumulative pitch error, in defined length, between any two same surface of screw thread
Maximum algebra difference to actual size relative to nominal dimension establishes planetary roller screw axis direction cumulative pitch error table
Up to formula Δ yl:
Δyl=Δ P (2)
Half of thread angle error delta yβThe axial displacement error of caused nut is calculated by following formula:
In formula, h is thread height；β is half of thread angle, and unit is radian.
If the theoretical amount of movement of nut is L, the practical amount of movement of nut is x, then assembling skewed error is
In formula,Angle excursion for nut axis relative to lead screw axis.
Lead screw combined deformation includes Compression and Expansion deformation and torsional deflection two parts, is calculated by following formula:
In formula, G is the coefficient of rigidity；X is lead screw by torsion partial-length；d1It is lead screw dangerouse cross-section diameter；LSIt is lead screw
Lead；F is axial service load；η is transmission efficiency.
Between lead screw and roller and roller and nut thread contact generate axial deformation belong to random error and
Normal Distribution, error mean square root are respectively σsAnd σN.Overall error are as follows:
δc=σs+σN (6)
Horizontal positioned lead screw can generate flexure, bending deformation quantity y due to self weightmaxAxial driving error δ can be generatedB。
Deflection equation in the case where two supports are as follows:
In formula, q is the gravity in unit length；L is lead screw active length；E is axial modulus of elasticity；I is lead screw inertia
Square.
The axial error that then bending deformation generates indicates are as follows:
δB≈y tan α (8)
In formula, α is the helix angle of lead screw.
XYZ is fixed coordinate system, and wherein Z axis is overlapped with the centre of gyration of lead screw, coordinate system XOY with the rotation of lead screw and
Rotation.Coordinate system XSYSZSFor threads of lead screw portion coordinate system, OSFor the center in threads of lead screw portion.Coordinate system XNYNZNFor nut coordinate
System, ONZNAxis is nut and roller axial direction mobile axis, ONPoint is nut center.XRYRZRFor roller coordinate system, ORZRCertainly for roller
Shaft axis.eSFor lead screw run-out error vector, i.e. threaded portion center ties up to XOY plane bias internal amount relative to theoretical coordinate
It is (Es, β s).ePFor nut installation error vector, i.e. nut center ties up to X relative to theoretical coordinateNONYNPlane bias internal
(En, β n).Define ωSAngular speed, ω are turned round for lead screwPFor retainer angular speed, that is, roller revolution angular speed, ωRCFor roller
Relative to the opposite angular speed of retainer, direction takes positive value, negative value is taken when opposite when identical as Z axis forward direction.
When there are installation error, lead screw coordinate system XSYSZSTo the translation matrix T of revolution coordinate system XYZS。
Coordinate system XSYSZSSpin matrix be
Lead screw coordinate system XSYSZSTo the transformation matrix of fixed coordinate system XYZ0ASeIt is indicated with following formula
0ASe=RS·TS (11)
Roller coordinate system also revolves around own axes while revolving around nut center, therefore roller coordinate system XRYRZRTo
The transformation matrix of fixed coordinate system is expressed as0ARe=TR·RR (12)
Wherein, TRFor translational coordination system, RRFor rotating coordinate system.
Mesh equation between lead screw spiral camber and the spiral camber of roller is
rrscAnd rscPosition vector, n respectively under respective coordinate systemscAnd nrscThe normal direction respectively under respective coordinate system
Vector, solve system of equation obtain on lead screw that there are axial gap δ when error for flank under flank and rollerSRe。
Due to being fixed on nut circumference direction, only translate in the axial direction, nut coordinate system XNYNZNTo fixed coordinate system
The translation matrix of XYZ is0TN, obtain transformation matrices of the nut coordinate system to fixed coordinate system0ANe:
rrNcAnd rNcPosition vector, n respectively under respective coordinate systemNcAnd nrNcThe normal direction respectively under respective coordinate system
Vector, solve system of equation obtain on roller that there are axial gap δ when error for flank under flank and nutNRe。
Step 3: gear transmission error establishes gear finite element model using hexahedral element, using finite element minor structure
Method extracts flank of tooth finite element grid connection stiffness matrix, obtains flank of tooth finite element grid nodal flexibility matrix.On this basis, root
The coordinate relationship put on sawteeth face finite element grid node and practical contact line, interpolation obtain the flexibility matrix of contact point of gear surface.
The flexibility matrix of acquisition includes that flank of tooth local deformation and gear integrally bending-are shear-deformable, by being parsed with Finite line contacts
The inaccurate flank of tooth local deformation that the localized contact deformation replacement FInite Element that formula is calculated obtains, obtains the modified flank of tooth
Practical contact point flexibility matrix, to construct face equation and solve gear pair transmission error.
Roller gear gear teeth corresponding flanks deviation includes total profile deviation, circular pitch deviation and spiral deviation.
Tooth profile total deviation (Fα) definition: in flank profil evaluation range LaIt is interior, contain two design flank profil marks of practical tooth trace
Distance between line.Total profile deviation when due to LOADED TOOTH CONTACT ANALYSIS is measured in normal section path of contact direction, conversion formula are as follows:
Fαn=Fαcosβb (16)
Wherein, FαnFor the total profile deviation in normal plane path of contact direction；βbFor Base spiral angle.
Individual circular pitch error is defined as on transverse plane, close to a circle concentric with Gear axis in the middle part of tooth height
On, the algebraic step of practical tooth pitch and theoretical tooth pitch.Since the circular pitch deviation in LOADED TOOTH CONTACT ANALYSIS is in normal section path of contact
Direction metering, so needing initial error value being transformed into normal direction, conversion formula are as follows:
fpbn=fptcosαtcosβb (17)
Wherein, fpbnFor the circular pitch deviation in normal plane path of contact direction, i.e. Normal base pitch deviation；αtFor end face reference circle pressure
Angle.
Spiral deviation is defined as between the practical helix that end face basic circle tangential direction measures and design helix
Difference.Helix total deviation (Fβ): in helix evaluation range (Lβ) in, contain two design spirals of practical helix trace
Distance between stitching line.Spiral deviation when due to LOADED TOOTH CONTACT ANALYSIS be measured in normal section path of contact direction, so
It needs initial error value being transformed into normal direction, conversion formula are as follows:
Fβn=Fβcosβb (18)
Wherein, FβnFor the spiral angular displacement in normal plane path of contact direction.
The finite element model for establishing gear obtains the softness factor square of flank of tooth node using the subsctructure method of ANSYS software
Battle array.Using multivariate interpolation method, bending-shearing softness factor matrix of each working pitch point is obtained.During iterative solution, ask
The load distribution for obtaining each working pitch point, brings the analytic sensitivity of Finite line contacts into, obtains the juxtaposition metamorphose flexibility of each point
Coefficient matrix, by itself and bending-shearing softness factor matrix assembly, obtain comprehensively considering linear bending-it is shear-deformable with it is non-thread
The flank of tooth softness factor matrix of property juxtaposition metamorphose.
It brings softness factor matrix into Coordinate deformation equation, obtains:
-[λ](k){F}+xs+ { d }={ ε } (19)
[λ](k)=[λb]+[λc] (20)
In formula, [λ b] and [λ c] is respectively to be bent-shear-deformable and juxtaposition metamorphose softness factor matrix；{ F } is each contact
Point load vector；{ d } is each contact point residual gap vector；{ ε } is each contact point primary clearance vector, and xs is close for rigid body
Amount, i.e. the macroscopic view displacement in path of contact direction.Using solution by iterative method { F } and xs, that is, gear teeth load distribution and gear tooth deformation.
Step 4: gear pair screw thread pair couples transmission error: gear can generate angular errors, gear angular errors θgIt is as follows
Formula:
The end gear of screw thread pair and roller on roller due to be roller a part, have same rotational speed.Tooth
Rotation on wheel circumferencial direction can be equivalent to the translation in the axial direction, therefore use xgIndicate the equivalent axial translation of gear,
Conversion formula is as follows:
In formula, LSIt is guide screw lead.
Driving error calculating formula in the case where every error synthesis acts on is obtained by the definition of driving error
TE=δNRe+δSRe+Δyl+Δyβ+Δyp+δS+δC+δB+xg (23)
Formula (23) is obtained based on static conditions, referred to as static driving error.
The beneficial effects of the present invention are: this method calculates the transmission error of threaded portion first, according to continuous contact condition
The meshing point coordinate for considering error and corresponding axial gap are acquired, then goes out lead screw continuous rotation with analytic method COMPREHENSIVE CALCULATING
When, the driving error under each position of engagement.After calculating the driving error under current time, further according to transmission principle and coordinate
Transformation theory calculates the meshing point coordinate and the parameters such as corresponding engagement radius and engagement drift angle of subsequent time, then calculates phase
At once the driving error inscribed, and so on driving error in entire operation process can be obtained；Next establishes the limited of gear
Meta-model divides the field of conjugate action and solves the flank of tooth softness factor matrix for considering to be bent the deformation of shear-deformable and nonlinear contact, repeatedly
In generation, solves, and obtains Gear Meshing Stiffness, comprehensive meshing error and Static transmissions error；Finally by had an impact planetary roller screw
The error component of transmission accuracy is transformed into the axis direction of nut, establishes planetary roller screw error source coupling model.This method
It solves the drawbacks of single analysis planetary roller screw screw thread pair error source of background technique is brought, passes through the movement of roller and nut
The error coupler relationship of the two together, is established planetary roller screw pair gear-thread error coupling and closed by transitive relation
System, practicability are good.
It elaborates with reference to the accompanying drawings and detailed description to the present invention.
Detailed description of the invention
Fig. 1 is the flow chart of planetary roller screw error source coupling modeling method of the present invention.
Fig. 2 is the planetary roller screw motion principle schematic diagram that the method for the present invention is related to.
Fig. 3 is the positional diagram for considering error.
In figure, 1- lead screw, 2- nut, 3- ring gear, 4- roller, 5- screw thread pair, 6- roller tooth, 7- gear pair.
Specific embodiment
Referring to Fig.1-3.Specific step is as follows for planetary roller screw error source coupling modeling method of the present invention:
Step 1: not only having included gear according to planetary roller screw but also having included the design feature of screw thread to define planetary roller silk
The transmission error of thick stick.
Step 2: all kinds of errors of screw thread pair are comprehensively considered according to the design feature of planetary roller screw screw thread pair, including
Foozle, rigging error, kinematic error, eccentric error.The meshing point coordinate for considering error is acquired using continuous contact condition
And corresponding axial gap, then when going out lead screw continuous rotation with analytic method COMPREHENSIVE CALCULATING, the transmission under each position of engagement is missed
Difference.After calculating the driving error under current time, further according to nibbling for transmission principle and coordinate transform theoretical calculation subsequent time
Chalaza coordinate and the parameters such as corresponding engagement radius and engagement drift angle, the driving error inscribed when then calculating corresponding, with this
Analogize the driving error that can be obtained in entire operation process.
Step 3: comprehensively considering all kinds of errors of gear pair according to the design feature of planetary roller screw pair gear pair, wrap
Include total profile deviation, circular pitch deviation, spiral deviation.Next establishes the finite element model of gear, and it is curved to divide field of conjugate action solution consideration
The flank of tooth softness factor matrix of bent shear-deformable and nonlinear contact deformation, iterative solution obtain the comprehensive meshing error of gear
And Static transmissions error.
Step 4: being built together with according to the error coupler relationship of roller and the movement transitive relation of nut by the two
Vertical planetary roller screw pair gear-thread error coupled relation, finally by the mistake of had an impact planetary roller screw transmission accuracy
Difference component is transformed into the axis direction of nut, establishes planetary roller screw error source coupling model.
The step 1 specifically includes the following steps:
According to the motion principle of planetary roller screw, then the rotation of lead screw 1 pushes nut 2 by the planetary motion of roller 4
Linear motion, and there are screw thread pairs 5 to engage simultaneously with gear pair 7.When lead screw 1 rotates, roller 4 had not only revolved but also rotation, together
When in order to eliminate the heeling moment that 1 helix angle of lead screw generates roller 4, therefore be machined with roller tooth 6 at 4 both ends of roller, with
Ring gear 3 engages, and is normally rolled with ensuring that 4 axis of roller is parallel to 1 axis of lead screw.The transmission for defining planetary roller screw pair misses
When difference is rotating motion part one-directional rotation, the difference of the actual displacement for the part that moves along a straight line and theoretical displacement.Definition is as follows:
In formula, TE is driving error；It is actual displacement；θ is input corner；I is theoretical transmission ratio.
The step 2 specifically includes the following steps:
According to the definition of cumulative pitch error: axial practical between any two same surface of screw thread in defined length
Maximum algebra difference of the size relative to nominal dimension establishes planetary roller screw axis direction cumulative pitch error expression formula Δ
yl:
Δyl=Δ P (2)
Half of thread angle error
Half of thread angle error delta yβRefer to the algebraic step of thread type half angle actual value and nominal value, reflects the direction of tooth side
Error.The axial displacement error of nut caused by half of thread angle error can be calculated by following formula:
In formula, h is thread height；β is half of thread angle, and unit is radian.
Assemble skewed error
If the theoretical amount of movement of nut is L, the practical amount of movement of nut is x, then skewed error is
In formula,Angle excursion for nut axis relative to lead screw axis.
Kinematic error
The error of planetary roller screw pair at runtime mainly has the combined deformation of lead screw shaft part, juxtaposition metamorphose, bending deformation
And temperature deformation.
Lead screw combined deformation includes Compression and Expansion deformation and torsional deflection two parts, can be calculated by following formula:
In formula, G is the coefficient of rigidity；X is lead screw by torsion partial-length；d1It is lead screw dangerouse cross-section diameter；LSIt is lead screw
Lead；F is axial service load；η is transmission efficiency.
Juxtaposition metamorphose
Between lead screw and roller and roller and nut thread contact generate axial deformation belong to random error and
Normal Distribution, error mean square root are respectively σsAnd σN.Overall error are as follows:
δc=σs+σN (6)
Bending deformation
Horizontal positioned lead screw can generate flexure, bending deformation quantity y due to self weightmaxAxial driving error δ can be generatedB。
Deflection equation in the case where two supports are as follows:
In formula, q is the gravity in unit length；L is lead screw active length；E is axial modulus of elasticity；I is lead screw inertia
Square.
The axial error that then bending deformation generates can indicate
δB≈y tan α (8)
In formula, α is the helix angle of lead screw.
Eccentric error
XYZ is fixed coordinate system, and wherein Z axis is overlapped with the centre of gyration of lead screw, coordinate system XOY with the rotation of lead screw and
Rotation.Coordinate system XSYSZSFor threads of lead screw portion coordinate system, OSFor the center in threads of lead screw portion.Coordinate system XNYNZNFor nut coordinate
System, ONZNAxis is nut and roller axial direction mobile axis, ONPoint is nut center.XRYRZRFor roller coordinate system, ORZRCertainly for roller
Shaft axis.eSFor lead screw run-out error vector, i.e. threaded portion center ties up to XOY plane bias internal amount relative to theoretical coordinate
It is (Es, β s).ePFor nut installation error vector, i.e. nut center ties up to X relative to theoretical coordinateNONYNPlane bias internal
(En, β n).Define ωSAngular speed, ω are turned round for lead screwPFor retainer angular speed, that is, roller revolution angular speed, ωRCFor roller
Relative to the opposite angular speed of retainer, direction takes positive value, negative value is taken when opposite when identical as Z axis forward direction.
Driving error modeling
When there are installation error, lead screw coordinate system XSYSZSTo the translation matrix T of revolution coordinate system XYZS。
Coordinate system XSYSZSSpin matrix be
Lead screw coordinate system XSYSZSTo the transformation matrix of fixed coordinate system XYZ0ASeIt can be represented by the formula
0ASe=RS·TS (11)
Roller coordinate system also revolves around own axes while revolving around nut center, therefore roller coordinate system XRYRZRTo
The transformation matrix of fixed coordinate system is represented by
0ARe=TR·RR (12)
Wherein, TRFor translational coordination system, RRFor rotating coordinate system.
Mesh equation between lead screw spiral camber and the spiral camber of roller is
rrscAnd rscPosition vector, n respectively under respective coordinate systemscAnd nrscThe normal direction respectively under respective coordinate system
Vector, solve system of equation can obtain on lead screw that there are axial gap δ when error for flank under flank and rollerSRe。
Due to being fixed on nut circumference direction, only translate in the axial direction, nut coordinate system XNYNZNTo fixed coordinate system
The translation matrix of XYZ is0TN, obtain transformation matrices of the nut coordinate system to fixed coordinate system0ANe,
rrNcAnd rNcPosition vector, n respectively under respective coordinate systemNcAnd nrNcThe normal direction respectively under respective coordinate system
Vector, solve system of equation can obtain on roller that there are axial gap δ when error for flank under flank and nutNRe。
The driving error of planetary roller screw pair is solved by the method for working out MATLAB program.When not considering the error,
The axial gap and nut of lead screw and roller-contact side and the axial gap of roller side do not change with the rotation of lead screw, but
It is when there is eccentric class error, roller will change in the gap of lead screw and nut two sides.When nut is not subject to axial direction
Load is born at light load, and the variation of axial gap is the actual displacement of nut relative to the inclined of theory displacement in motion process
Difference, i.e., static driving error.This method is acquired according to continuous contact condition first considers the meshing point coordinate of error and corresponding
Axial gap, driving error when then COMPREHENSIVE CALCULATING goes out lead screw continuous rotation, under each position of engagement.Calculate current time
Under driving error after, further according to transmission principle and the meshing point coordinate of coordinate transform theoretical calculation subsequent time and corresponding nibble
Close the parameters such as radius and engagement drift angle, the driving error inscribed when finally calculating corresponding.And so on entire operating can be obtained
Driving error in the process.
The step 3 specifically includes the following steps:
Gear transmission error establishes gear finite element model using hexahedral element, extracts tooth using finite element subsctructure method
Face finite element grid connection stiffness matrix, obtains flank of tooth finite element grid nodal flexibility matrix.On this basis, had according to the flank of tooth
The coordinate relationship put in first grid node and practical contact line is limited, interpolation obtains the flexibility matrix of contact point of gear surface.This method obtains
The flexibility matrix taken includes that flank of tooth local deformation and gear integrally bending-are shear-deformable, public by being parsed with Finite line contacts
The inaccurate flank of tooth local deformation that the localized contact deformation replacement FInite Element that formula is calculated obtains, it is real to obtain the modified flank of tooth
Border contact point flexibility matrix, to construct face equation and solve gear pair transmission error.
The classification of surface deviation
Roller gear gear teeth corresponding flanks deviation mainly includes total profile deviation, circular pitch deviation, spiral deviation, in the present invention
Gear is 5 class precisions.
Total profile deviation
Tooth profile total deviation (Fα) definition: in flank profil evaluation range LaIt is interior, contain two design flank profil marks of practical tooth trace
Distance between line.Total profile deviation when due to LOADED TOOTH CONTACT ANALYSIS is measured in normal section path of contact direction, conversion formula are as follows:
Fαn=Fαcosβb (16)
Wherein, FαnFor the total profile deviation in normal plane path of contact direction；βbFor Base spiral angle.
Circular pitch deviation
Individual circular pitch error is defined as on transverse plane, close to a circle concentric with Gear axis in the middle part of tooth height
On, the algebraic step of practical tooth pitch and theoretical tooth pitch.Since the circular pitch deviation in LOADED TOOTH CONTACT ANALYSIS is in normal section path of contact
Direction metering, so needing initial error value being transformed into normal direction, conversion formula are as follows:
fpbn=fptcosαtcosβb (17)
Wherein, fpbnFor the circular pitch deviation in normal plane path of contact direction, i.e. Normal base pitch deviation；αtFor end face reference circle pressure
Angle.
Spiral deviation
Spiral deviation is defined as between the practical helix that end face basic circle tangential direction measures and design helix
Difference.Helix total deviation (Fβ): in helix evaluation range (Lβ) in, contain two design spirals of practical helix trace
Distance between stitching line.Spiral deviation when due to LOADED TOOTH CONTACT ANALYSIS be measured in normal section path of contact direction, so
It needs initial error value being transformed into normal direction, conversion formula are as follows:
Fβn=Fβcosβb (18)
Wherein, FβnFor the spiral angular displacement in normal plane path of contact direction.
LOADED TOOTH CONTACT ANALYSIS model
The purpose of gear LOADED TOOTH CONTACT ANALYSIS is to solve mesh stiffness, comprehensive meshing error and Static transmissions error.
Its rough analysis process is as follows: establish the finite element model of gear, divide the field of conjugate action solve consider bending it is shear-deformable and
The flank of tooth softness factor matrix of nonlinear contact deformation, iterative solution obtain gear Static transmissions error.Each link summary is such as
Under.
Establish gear finite element model
Accurate gear finite element model is the basis for calculating mesh stiffness.Consider for computational accuracy, tooth contact area
Grid should densification, if establishing complete gear wheel model, the list and number of nodes that will lead to finite element model are sharply increased, calculate
Cost is excessively high.Therefore three tooth simplified model of gear is used, regular hexahedron unit is selected to carry out grid dividing to three tooth models, and
Densification is carried out to the grid of tooth contact side, to improve computational efficiency, while guaranteeing higher computational accuracy.It is compiled based on Matlab
Journey and the modeling of APDL language parameterization, establish gear physical model, grid division and are loaded under ANSYS environment.
Flank of tooth softness factor calculates
The finite element model for establishing gear obtains the softness factor square of flank of tooth node using the subsctructure method of ANSYS software
Battle array.Using multivariate interpolation method, bending-shearing softness factor matrix of available each working pitch point.In iterative solution process
In, the load distribution of each working pitch point is acquired, the analytic sensitivity of Finite line contacts is brought into, obtains the juxtaposition metamorphose of each point
Softness factor matrix, by itself and bending-shearing softness factor matrix assembly, can must comprehensively consider linear bending-it is shear-deformable with
The flank of tooth softness factor matrix of nonlinear contact deformation.
Solve comprehensive meshing error
It brings softness factor matrix into Coordinate deformation equation, can obtain:
-[λ](k){F}+xs+ { d }={ ε } (19)
[λ](k)=[λb]+[λc] (20)
In formula, [λ b] and [λ c] is respectively to be bent-shear-deformable and juxtaposition metamorphose softness factor matrix；{ F } is each contact
Point load vector；{ d } is each contact point residual gap vector；{ ε } is each contact point primary clearance vector, and xs is close for rigid body
Amount, i.e. the macroscopic view displacement in path of contact direction.Using solution by iterative method { F } and xs, that is, gear teeth load distribution and gear tooth deformation.
The step 4 specifically includes the following steps:
Gear pair screw thread pair couples transmission error: gear can generate angular errors, gear angular errors θgSuch as following formula:
The end gear of screw thread pair and roller on roller due to be roller a part, have same rotational speed.Tooth
Rotation on wheel circumferencial direction can be equivalent to the translation in the axial direction, therefore use xgIndicate the equivalent axial translation of gear,
Conversion formula is as follows:
LSIt is guide screw lead
The driving error calculating formula under every error synthesis effect can be obtained by the definition of driving error
TE=δNRe+δSRe+Δyl+Δyβ+Δyp+δS+δC+δB+xg (23)
It can be seen that driving error is the general performance of the factors such as manufacturing and fixing error and its caused gap by above formula.Formula
It (23) is obtained based on static conditions, referred to as static driving error.
The present invention establishes the transmission error of gear and screw thread respectively, by the movement transitive relation of roller and nut by the two
Error coupler relationship together, establish planetary roller screw pair gear-thread error coupling model.Reduce research people
Member's operation difficulty, saves manpower, and practicability is good.
Claims (1)
1. a kind of planetary roller screw error source coupling modeling method, it is characterised in that the following steps are included:
Step 1: lead screw (1) rotation drives the planetary motion of roller (4) to push spiral shell according to the motion principle of planetary roller screw
Female (2) linear motion, screw thread pair (5) engage simultaneously with gear pair (7)；When lead screw (1) rotation, roller (4) had not only revolved but also oneself
Turn, in order to eliminate the heeling moment that lead screw (1) helix angle generates roller (4), the roller tooth (6) at roller (4) both ends with it is interior
Gear ring (3) engagement makes roller (4) axis be parallel to the rolling of lead screw (1) axis；Define planetary roller screw pair driving error be
When rotating motion part one-directionally rotates, the difference of the actual displacement for the part that moves along a straight line and theoretical displacement；Definition is as follows:
In formula, TE is driving error；It is actual displacement；θ is input corner；I is theoretical transmission ratio；
Step 2: according to the definition of cumulative pitch error, it is axial real between any two same surface of screw thread in defined length
Maximum algebra difference of the border size relative to nominal dimension, establishes planetary roller screw axis direction cumulative pitch error expression formula
Δyl:
Δyl=Δ P (2)
Half of thread angle error delta yβThe axial displacement error of caused nut is calculated by following formula:
In formula, h is thread height；β is half of thread angle, and unit is radian；
If the theoretical amount of movement of nut is L, the practical amount of movement of nut is x, then assembling skewed error is
In formula,Angle excursion for nut axis relative to lead screw axis；
Lead screw combined deformation includes Compression and Expansion deformation and torsional deflection two parts, is calculated by following formula:
In formula, G is the coefficient of rigidity；X is lead screw by torsion partial-length；d1It is lead screw dangerouse cross-section diameter；LSIt is that lead screw is led
Journey；F is axial service load；η is transmission efficiency；
The axial deformation that the contact of thread generates between lead screw and roller and roller and nut belongs to random error and obedience
Normal distribution, error mean square root are respectively σsAnd σN；Overall error are as follows:
δc=σs+σN (6)
Horizontal positioned lead screw can generate flexure, bending deformation quantity y due to self weightmaxAxial driving error δ can be generatedB；
Deflection equation in the case where two supports are as follows:
In formula, q is the gravity in unit length；L is lead screw active length；E is axial modulus of elasticity；I is lead screw the moment of inertia；
The axial error that then bending deformation generates indicates are as follows:
δB≈y tanα (8)
In formula, α is the helix angle of lead screw；
XYZ is fixed coordinate system, and wherein Z axis is overlapped with the centre of gyration of lead screw, and coordinate system XOY is revolved with the rotation of lead screw
Turn；Coordinate system XSYSZSFor threads of lead screw portion coordinate system, OSFor the center in threads of lead screw portion；Coordinate system XNYNZNFor nut coordinate
System, ONZNAxis is nut and roller axial direction mobile axis, ONPoint is nut center；XRYRZRFor roller coordinate system, ORZRCertainly for roller
Shaft axis；eSFor lead screw run-out error vector, i.e. threaded portion center ties up to XOY plane bias internal amount relative to theoretical coordinate
It is (Es, β s)；ePFor nut installation error vector, i.e. nut center ties up to X relative to theoretical coordinateNONYNPlane bias internal
(En, β n)；Define ωSAngular speed, ω are turned round for lead screwPFor retainer angular speed, that is, roller revolution angular speed, ωRCFor roller
Relative to the opposite angular speed of retainer, direction takes positive value, negative value is taken when opposite when identical as Z axis forward direction；
When there are installation error, lead screw coordinate system XSYSZSTo the translation matrix T of revolution coordinate system XYZS；
Coordinate system XSYSZSSpin matrix be
Lead screw coordinate system XSYSZSTo the transformation matrix of fixed coordinate system XYZ0ASeIt is indicated with following formula
0ASe=RS·TS (11)
Roller coordinate system also revolves around own axes while revolving around nut center, therefore roller coordinate system XRYRZRTo fixation
The transformation matrix of coordinate system is expressed as
0ARe=TR·RR (12)
Wherein, TRFor translational coordination system, RRFor rotating coordinate system；
Mesh equation between lead screw spiral camber and the spiral camber of roller is
rrscAnd rscPosition vector, n respectively under respective coordinate systemscAnd nrscThe normal vector respectively under respective coordinate system,
Solve system of equation obtains on lead screw that there are axial gap δ when error for flank under flank and rollerSRe；
Due to being fixed on nut circumference direction, only translate in the axial direction, nut coordinate system XNYNZNTo fixed coordinate system XYZ
Translation matrix be0TN, obtain transformation matrices of the nut coordinate system to fixed coordinate system0ANe:
rrNcAnd rNcPosition vector, n respectively under respective coordinate systemNcAnd nrNcThe normal vector respectively under respective coordinate system,
Solve system of equation obtains on roller that there are axial gap δ when error for flank under flank and nutNRe；
Step 3: gear transmission error establishes gear finite element model using hexahedral element, mentioned using finite element subsctructure method
Flank of tooth finite element grid connection stiffness matrix is taken, flank of tooth finite element grid nodal flexibility matrix is obtained；On this basis, according to tooth
The coordinate relationship put on face finite element grid node and practical contact line, interpolation obtain the flexibility matrix of contact point of gear surface；It obtains
Flexibility matrix include that flank of tooth local deformation and gear integrally bending-are shear-deformable, by with Finite line contacts analytic formula
The inaccurate flank of tooth local deformation that the localized contact deformation replacement FInite Element being calculated obtains, it is practical to obtain the modified flank of tooth
Contact point flexibility matrix, to construct face equation and solve gear pair transmission error；
Roller gear gear teeth corresponding flanks deviation includes total profile deviation, circular pitch deviation and spiral deviation；
Tooth profile total deviation (Fα) definition: in flank profil evaluation range LaIt is interior, between two design flank profil traces for containing practical tooth trace
Distance；Total profile deviation when due to LOADED TOOTH CONTACT ANALYSIS is measured in normal section path of contact direction, conversion formula are as follows:
Fαn=Fαcosβb (16)
Wherein, FαnFor the total profile deviation in normal plane path of contact direction；βbFor Base spiral angle；
Individual circular pitch error is defined as on transverse plane, real on a circle concentric with Gear axis close in the middle part of tooth height
The algebraic step of border tooth pitch and theoretical tooth pitch；Since the circular pitch deviation in LOADED TOOTH CONTACT ANALYSIS is in normal section path of contact telegoniometer
Amount, so needing initial error value being transformed into normal direction, conversion formula are as follows:
fpbn=fptcosαtcosβb (17)
Wherein, fpbnFor the circular pitch deviation in normal plane path of contact direction, i.e. Normal base pitch deviation；αtFor end face pressure angle of graduated circle；
Spiral deviation is defined as the difference between the practical helix that end face basic circle tangential direction measures and design helix；
Helix total deviation (Fβ): in helix evaluation range (Lβ) in, contain two design helix traces of practical helix trace
Between distance；Spiral deviation when due to LOADED TOOTH CONTACT ANALYSIS be measured in normal section path of contact direction, so need by
Initial error value is transformed into normal direction, conversion formula are as follows:
Fβn=Fβcosβb (18)
Wherein, FβnFor the spiral angular displacement in normal plane path of contact direction；
The finite element model for establishing gear obtains the softness factor matrix of flank of tooth node using the subsctructure method of ANSYS software；Benefit
With multivariate interpolation method, bending-shearing softness factor matrix of each working pitch point is obtained；During iterative solution, acquire each
The load of working pitch point is distributed, and is brought the analytic sensitivity of Finite line contacts into, is obtained the juxtaposition metamorphose softness factor of each point
Itself and bending-shearing softness factor matrix assembly are obtained comprehensively considering that linear bending-is shear-deformable to be connect with non-linear by matrix
The flank of tooth softness factor matrix of thixotroping shape；
It brings softness factor matrix into Coordinate deformation equation, obtains:
-[λ](k){F}+xs+ { d }={ ε } (19)
[λ](k)=[λb]+[λc] (20)
In formula, [λ b] and [λ c] is respectively to be bent-shear-deformable and juxtaposition metamorphose softness factor matrix；{ F } is each contact point load
Lotus vector；{ d } is each contact point residual gap vector；{ ε } is each contact point primary clearance vector, and xs is rigid body close to amount, i.e.,
The macroscopic view displacement in path of contact direction；Using solution by iterative method { F } and xs, that is, gear teeth load distribution and gear tooth deformation；
Step 4: gear pair screw thread pair couples transmission error: gear can generate angular errors, gear angular errors θgSuch as following formula:
The end gear of screw thread pair and roller on roller due to be roller a part, have same rotational speed；Gear circle
Rotation in circumferential direction can be equivalent to the translation in the axial direction, therefore use xgIt indicates the equivalent axial translation of gear, converts
Formula is as follows:
In formula, LSIt is guide screw lead；
Driving error calculating formula in the case where every error synthesis acts on is obtained by the definition of driving error
TE=δNRe+δSRe+Δyl+Δyβ+Δyp+δS+δC+δB+xg (23)
Formula (23) is obtained based on static conditions, referred to as static driving error.
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| CN114266114A (en) * | 2021-12-14 | 2022-04-01 | 西北工业大学 | Standard planetary roller screw multi-pair dynamic coupling modeling method based on Ansys Workbench |
| CN114429004A (en) * | 2021-12-24 | 2022-05-03 | 北京精密机电控制设备研究所 | Optimization design method of curved surface tooth profile planetary roller screw |
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