CN108875226A - A kind of ball screw assembly, random Wear Modeling method under dynamic fluctuation feeding speed conditions - Google Patents
A kind of ball screw assembly, random Wear Modeling method under dynamic fluctuation feeding speed conditions Download PDFInfo
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Abstract
The invention discloses a kind of dynamic fluctuations to feed the random Wear Modeling method of ball screw assembly, under speed conditions, which includes：According to ball screw assembly, during military service, in face of different processing operating conditions, the feeding revolving speed of ball screw assembly, has dynamic fluctuation characteristic, it is fed under speed conditions in dynamic fluctuation, establish the relative rotation speed mathematical model between ball and lead screw ball and ball and nut rolling way, according to Archard wear model, it proposes and establishes the random wear model of improved ball screw assembly, consider the dynamic fluctuation characteristic of ball screw assembly, feeding revolving speed, it establishes ball screw assembly, and feeds the random wear model under speed conditions in dynamic fluctuation, according to the random wear model of foundation, it inquires into ball screw assembly, and feeds the wear law under speed conditions in dynamic fluctuation, disclose ball screw assembly, positioning accuracy fade characteristics.
Description
Technical field
The present invention provides a kind of dynamic fluctuations to feed the random Wear Modeling method of ball screw assembly, under speed conditions, belongs to
In Precision of NC Machine Tool retentivity design field.
Background technique
Ball screw assembly, feed system is a kind of screw transmission system, by lead screw, nut, ball, pre- tabletting, reverser,
The structural elements such as dust excluder composition, it is constituted between lead screw and nut screw slot using ball as intermediate transmission body
It is recycled in closed circuit, is that the further of Ai Kemu Luo Gang extends and develop to complete the transmitting of rotation-linear motion, this
Bearing is exactly become scroll actions from sliding action by the significance of item development.Ball screw assembly, feed system is as a kind of essence
Close transmission system, has that transmission efficiency height, high sensitivity, stable drive, can to eliminate axial gap, raising axial rigidity etc. excellent
Point is widely used in fields such as modernization industry, aerospace industry, traffic engineering industry.Especially set in accurate digital control
Standby upper, ball screw assembly, feed system has become its important component part, and the expansion of application field and demand are not
Disconnected increase also constantly proposes new demand to the development and performance of ball screw assembly, feed system, wherein is used for accurate digital control equipment
On be one of urgent need to solve the problem to the technical requirements of its precision dynamic retentivity.
The country starts late to the research of the ball screw assembly, feed system American-European countries that compares, and is in response to domestic numerical control machine
What the equipment such as bed had gradually developed.Due to starting late to this area research, theoretical system is waited in perfect, manufacturing technology level
Relatively inadequate tip, compared with external of the same trade product, domestic ball screw assembly, feed system product quality, in terms of
There is also very big gaps, carry out domestic ball-screw byproduct for a long time in terms of technical parameters and performance and European and American developed countries
All there is a big difference, industry integrally fall behind, industry size very little, industrial chain numerous imbalances.Medium and high classes ball screw assembly, feeding system
Product market share unite less than 30%, compared with import ball screw assembly, feed system product, domestic ball screw assembly, is fed
System product on precision dynamic retentivity there is biggish gap, cannot reach high-precision that high end equipment design uses,
High reliability standards lead to the domestic middle and high end ball wire in the tips such as modernization industry, aerospace industry industry import foreign countries
Thick stick pair feed system product, the serious fast development for restricting the fields such as modern domestic chemical industry, aerospace industry, wherein accurate rolling
The research of ballscrew pair feed system is less, is that China's Rolling Components is hindered rapidly to send out with high-grade accurate digital control equipment industry
One of the main reason for exhibition.
Ball screw assembly, feed system is as precision transmission system, and during ball screw assembly, is on active service, abrasion is its essence
The main reason for spending retentivity loss.Since ball screw assembly, faces different processing request and processing operating condition, wherein ball wire
The feeding revolving speed of thick stick pair is one of the main operating condition factor for causing it to wear, and feeding revolving speed has the dynamic wave with processing request
Dynamic characteristic makes the relative rotation speed between ball and raceway that dynamic fluctuation characteristic be presented.Lead to ball screw assembly, feed system therewith
The fluctuation of error and the decline of positioning accuracy are fed, makes ball screw assembly, feed system precision stability that dynamic fluctuation decline be presented
Characteristic.
So establishing ball screw assembly, in the random wear model of dynamic fluctuation feeding speed conditions and then disclosing abrasion rule
Rule just seems particularly significant, how to establish ball screw assembly, and feeds the random wear model under speed conditions in dynamic fluctuation and divide
Analysing its wearing character is one of the critical issue in invention.
Summary of the invention
It is an object of the invention to provide a kind of dynamic fluctuations to feed the random Wear Modeling of ball screw assembly, under speed conditions
Method, the random Wear Modeling method include：According to ball screw assembly, during military service, in face of different processing operating conditions, rolling
The feeding revolving speed of ballscrew pair has dynamic fluctuation characteristic, in the case where dynamic fluctuation feeds speed conditions, establishes ball and lead screw rolls
Relative rotation speed mathematical model between pearl and ball and nut rolling way proposes according to Archard wear model and establishes improvement
The random wear model of ball screw assembly, consider ball screw assembly, feeding revolving speed dynamic fluctuation characteristic, establish ball screw assembly,
The random wear model under speed conditions is fed in dynamic fluctuation, according to the random wear model of foundation, inquires into ball screw assembly,
The wear law under speed conditions is fed in dynamic fluctuation, discloses ball screw assembly, positioning accuracy fade characteristics.
The technical solution adopted by the present invention is that the ball screw assembly, that a kind of dynamic fluctuation is fed under speed conditions is worn at random
Modeling method, this method include：
S1, the dynamic fluctuation characteristic for feeding revolving speed during military service according to ball screw assembly, establish ball screw assembly, and exist
Dynamic fluctuation feeds ball and raceway feeding revolving speed model under speed conditions；
S2, ball and raceway feeding revolving speed model under speed conditions are fed in dynamic fluctuation according to ball screw assembly, built
Sliding distance mathematical model between vertical ball and lead screw raceway and ball and nut rolling way, i.e. ball screw assembly, are worn at random
Model；
S3, it is based on Archard wear model, the random wear model of ball screw assembly, is improved, established respectively dynamic
Ball and the random wear model of lead screw raceway and the random mill of ball and nut rolling way under state fluctuation feeding speed conditions
Damage model；
S4, basis feed the random wear model and ball of ball and lead screw raceway under speed conditions in dynamic fluctuation
With the random wear model of nut rolling way, calculates and analyze abrasion spy of the ball screw assembly, under dynamic fluctuation feeding speed conditions
Property.
The realization process of the random wear model modeling method is as follows,
Step 1
According to the structure of ball screw assembly, space coordinates are established.
O-XYZ is the coordinate system of an entirety.O point is sat at the geometric center of lead screw and nut baseplane initial position
The position of parameter is constant, and Z axis and helical axis are parallel to each other, and X-coordinate axle and Y-coordinate axle are in the plane where nut bottom,
X, Y and Z axis are mutually perpendicular to.R is the pitch radius of ball screw assembly, as the bottom surface of multi-track spiral line where the ball centre of sphere half
Diameter.α is the helical angle of ball screw assembly,.β is angular displacement of the centre of sphere relative to lead screw of ball on multi-track spiral line.
Ob- TNB is free coordinate system, ObIt is the ball centre of sphere on multi-track spiral line, T axis is in multi-track spiral line
On the mobile tangential direction of the ball centre of sphere, perpendicular to the outer surface of lead screw, N axis crosses Z axis and perpendicular to Z for direction where N axis
The angle of axis, B axle and helical axis directions Z axis is pitch angle alpha, and T, N and B axle are mutually perpendicular to.
Local coordinate system is separately fixed at the contact point between ball and lead screw raceway, ball and nut rolling way, S-
xsyszsIndicate the local coordinate system between ball and lead screw raceway at contact point, N-xnynznIt indicates between ball and nut rolling way
The local coordinate system of contact point, S indicate the coordinate origin at ball and lead screw raceway contact point, and as ball connects with lead screw raceway
The geometric center in region is touched, the direction that the ball centre of sphere is directed toward with the geometric center point in ball and lead screw raceway contact region is zs
Direction, similarly, N indicates the coordinate origin at ball and nut rolling way contact point, as ball and nut rolling way contact area
Geometric center, the direction that the ball centre of sphere is directed toward with the geometric center point of ball and nut rolling way contact area is znSide
To.
The coordinate origin O of global coordinate system is converted to origin ObThe transformation matrix of coordinates at place is denoted as T1：
Global coordinate system exists with T axis and N axis respectively around the angular displacement of Z axis rotation β+pi/2, reference axis X-axis and Y-axis
Projection in XOY plane is conllinear, and rotational coordinates matrix is denoted as Rot1：
Global coordinate system rotates the angular displacement of 2 π-α, the throwing of reference axis X-axis, Z axis and B axle in plane XOZ around Y-axis
Shadow is conllinear, and rotational coordinates matrix is denoted as Rot2：
Global coordinate system O-XYZ and free coordinate system ObTransformational relation between-TNB is：
In formula, L is the lead of lead screw, is calculated by L=2 π Rtan α.
Free coordinate system Ob- TNB and local coordinate system S-xsyszsTransformational relation be：
Free coordinate system Ob- TNB and local coordinate system N-xnynznTransformational relation be：
ω is set as in the rotation speed of t moment, lead screw, it is assumed that within the Γ period, lead screw is under rotational speed omega rotation speed
Angular displacement be set as W.It is O-X'Y'Z' that lead screw, which rotates the reference frame after W angular displacement,.According to homogeneous coordinate transformation principle,
Coordinate transformation relation from reference frame O-X'Y'Z' to global coordinate system O-XYZ is：
The geometric center point S of contact area, the position in coordinate system O-XYZ are：
The geometric center point S of ball and lead screw raceway contact region, the position in reference frame O-X'Y'Z' are：
Butt contact S carries out solution first derivative in the position of reference frame, obtains ball and lead screw raceway contact area
The speed ν of the center in domains：
Secondly, contact point S is in ObSpeed ν in-TNBs-obFor：
Similarly, the geometric center point N of ball and nut rolling way contact area, the position in global coordinate system O-XYZ are：
The geometric center point N of ball and nut rolling way contact area, the position in reference frame O-X'Y'Z' are：
The speed ν of the center in ball and lead screw raceway contact regionnFor：
Contact point N is in ObSpeed ν in-TNBn-obFor：
Step 2
According to the relationship between ball and lead screw raceway relative velocity, obtain：
νbs-ob=νs-ob+νbs-ss (16)
In formula, νbs-ssMovement velocity for contact point S on ball relative to the contact point S on lead screw raceway, νbs-ssMeter
Operator expression formula is：
In formula, νb-sIt is movement velocity of the centre of sphere of ball relative to lead screw in the hand of spiral, ωb、ωnWith ωtIt is T respectively
The component of change in coordinate axis direction, N change in coordinate axis direction and B change in coordinate axis direction upper leading screw angular velocity of rotation.
νb-sIt is solved and is obtained by formula (18)：
The component ω of the angular speed of lead screw in three directions in ball screw assembly,b、ωnWith ωtRespectively by formula (19),
(20), (21) obtain：
In formula, ωbrIt is the angular velocity of rotation of the ball centre of sphere,It is ωbrWith the angle between two reference axis of B, T composition plane
Size.It is ωbrCorner dimension is formed by between the projection and B reference axis in the plane of B-T reference axis composition.Ball
Angular velocity of rotation ωbrIt is obtained by formula (22)：
In formula,It is projection O' of the centre of sphere of ball in the plane that X-Y coordinate axis formsbTo O coordinate origin away from
From.
Ball is denoted as R with the sliding rolling ratio at lead screw raceway contact regional center point Sbs：
According to the relative speed relationship between ball and nut rolling way, obtain：
νbn-sn=νn-ob+νbn-nn (24)
In formula, νbn-snBe on ball at the N of contact point relative to the movement velocity at the contact point N on lead screw raceway, νbn-sn
Calculation expression be：
Ball is denoted as R with the sliding rolling ratio at nut rolling way contact area central point Nbn：
Contact point S is relative to the movement velocity ν at the contact point S on lead screw raceway on ballbs-ssWith ball along lead screw
The distance l moved in a sliding manner on racewayssiBetween relationship be：
lssi=νbs-sstssi (26)
In formula, tssiMove to the position of i+1 ball from the position of i-th of ball relative to lead screw raceway for ball
The sliding time of Shi Suoyong：
In formula, tiWhen moving to the position of i+1 ball from the position of i-th of ball relative to lead screw raceway for ball
Time used.
Relative to contact point S, the sliding distance on lead screw raceway is l to ballssiWhen, sliding of the lead screw relative to contact point S
Distance is：
In formula, ωs-bIt is angular speed of the lead screw relative to ball, is calculated by (29) formula：
In formula, rsFor the distance between lead screw raceway contact central point and the ball centre of sphere, rnFor nut rolling way contact point and rolling
The distance between pearl centre of sphere.
Contact point N is relative to the movement velocity ν at the contact point N on nut rolling way on ballbn-nnWith ball along nut
The distance l moved in a sliding manner on racewaynsiBetween relationship be：
lnsi=νbn-nntnsi (30)
In formula, tnsiMove to the position of i+1 ball from the position of i-th of ball relative to nut rolling way for ball
The sliding time of Shi Suoyong：
Relative to contact point N, the sliding distance on nut rolling way is l to ballnsiWhen, sliding of the lead screw relative to contact point S
Distance is：
In formula, ωn-bIt is angular speed of the nut relative to ball, is calculated by formula (33)：
Step 3
Ball screw assembly, can be worn during military service, directly affect its positioning accuracy.According to the research of Archard
Achievement, abrasion loss V and contact load F and opposite sliding distance L are directly proportional, and softer in two kinds of different materials contacting with each other
The hardness H of material is inversely proportional：
Due to the limitation of Archard basic model, it cannot be directly used to explore ball screw assembly, in Random Load operating condition
Under wear problem.The wear law under speed conditions is fed in dynamic fluctuation for analysis ball screw assembly, predicts its feeding essence
Degree, and in the range of allowing precision, predict its life expectancy, this method provides improved under dynamic fluctuation feeding speed conditions
The random wear model of ball screw assembly,.
It is proposed that the random wear model in the case where dynamic fluctuation feeds speed conditions between ball and lead screw raceway is：
It is proposed that the random wear model in the case where dynamic fluctuation feeds speed conditions between ball and nut rolling way is：
Convolution (35) and (36) obtain ball screw assembly, and feed the random abrasion mathematical modulo under speed conditions in dynamic fluctuation
Type is：
The abrasion loss model between ball and lead screw raceway is established, according to Step 1: speed and sliding distance are analyzed in two
And formula (35) obtains the calculating formula of the abrasion loss between i-th of ball and lead screw raceway：
In formula, dimensionless coefficient of waste Kb-sWith the hardness H of lead screw roller surfaceb-sIt is constant parameter relevant to material,
FsiIt is the contact load of ball Yu lead screw raceway.
The abrasion loss model between ball and nut rolling way is established, according to Step 1: speed and sliding distance are analyzed in two
And formula (36) obtains the calculating formula of the abrasion loss between i-th of ball and nut rolling way：
In formula, dimensionless coefficient of waste Kb-nWith the hardness H in nut rolling way faceb-nIt is constant parameter relevant to material,
FniIt is the contact load of ball and nut rolling way.
Step 4
Abrasion total amount between ball and lead screw raceway is calculated, the abrasion loss between ball and lead screw raceway is rolling
The integral of pearl and the instantaneous abrasion loss of lead screw raceway.
It is calculated separately between the 1st, 2 ... i ... M balls and lead screw raceway according to formula (38) in tss1、tss2…tssi…tssM
The abrasion loss at moment：
To between the 1st, 2 ... i ... M balls and lead screw raceway in tss1、tss2…tssi…tssMThe abrasion loss at moment carries out
Integral, obtains each ball in tss1To tss2In period, tss2To tss3In period, tssiTo tssi+1In period and tssMExtremely
tss1Abrasion loss in period：
For i-th of ball in entire cycle period, the abrasion between ball and lead screw raceway carries out accumulative summation, is denoted as
In entire cycle period, the wear model of ball and lead screw raceway is denoted as all balls
Similarly, the abrasion total amount between ball and nut rolling way is calculated, the abrasion between ball and nut rolling way
Amount is the integral of ball and the instantaneous abrasion loss of nut rolling way.
The 1st, 2 ... i ... M balls in raceway are calculated separately and between nut rolling way in t according to formula (39)ns1、tns2…
tnsi…tnsMThe abrasion loss at moment：
To between the 1st, 2 ... i ... M balls and nut rolling way in tns1、tns2…tnsi…tnsMThe abrasion loss at moment carries out
Integral, obtains each ball in tns1To tns2In period, tns2To tns3In period, tnsiTo tnsi+1In period and tnsMExtremely
tns1Abrasion loss in period：
For i-th of ball in entire cycle period, the abrasion between ball and nut rolling way carries out accumulative summation, is denoted as
In entire cycle period, the wear model of ball and lead screw raceway is denoted as all balls
Convolution (46), (47) feed the random wear model of ball screw assembly, under speed conditions according to dynamic fluctuation and obtain
Total abrasion loss：
In formula,For the cycle-index between ball and lead screw raceway,Circulation between ball and nut rolling way
Number can be acquired according to the revolving speed of lead screw and effective transmission total kilometres of lead screw.
Detailed description of the invention
Fig. 1 is the random Wear Modeling method flow diagram of ball screw assembly, under dynamic fluctuation feeding revolving speed.
Fig. 2 is ball screw assembly, global coordinate system and free coordinate system.
Fig. 3 is ball screw assembly, local coordinate system
Fig. 4 is ball moving schematic diagram.
Fig. 5 is the contact condition cross-sectional view of ball and raceway.
Specific embodiment
The present invention is made into being once described in detail below in conjunction with drawings and examples.
The technical solution adopted by the present invention is that the ball screw assembly, that a kind of dynamic fluctuation is fed under speed conditions is worn at random
Modeling method, this method include：
S1, the dynamic fluctuation characteristic for feeding revolving speed during military service according to ball screw assembly, establish ball screw assembly, and exist
Dynamic fluctuation feeds ball and raceway feeding revolving speed model under speed conditions；
S2, ball and raceway feeding revolving speed model under speed conditions are fed in dynamic fluctuation according to ball screw assembly, built
Sliding distance mathematical model between vertical ball and lead screw raceway and ball and nut rolling way, i.e. ball screw assembly, are worn at random
Model；
S3, it is based on Archard wear model, the random wear model of ball screw assembly, is improved, established respectively dynamic
Ball and the random wear model of lead screw raceway and the random mill of ball and nut rolling way under state fluctuation feeding speed conditions
Damage model；
S4, basis feed the random wear model and ball of ball and lead screw raceway under speed conditions in dynamic fluctuation
With the random wear model of nut rolling way, calculates and analyze abrasion spy of the ball screw assembly, under dynamic fluctuation feeding speed conditions
Property.
The realization process of the random wear model modeling method is as follows, and Fig. 1 show the specific implementation step of this method.
Step 1 feeds revolving speed according to dynamic fluctuation and establishes ball and raceway dynamic fluctuation feeding revolving speed model
According to the structure of ball screw assembly, space coordinates are established, as shown in Figure 2 and Figure 3.
In Fig. 2, O-XYZ is the coordinate system of an entirety.O point is the geometric center of lead screw Yu nut baseplane initial position
Place, the position of reference axis is constant, and Z axis and helical axis are parallel to each other, and X-coordinate axle and Y-coordinate axle be in putting down where nut bottom
In face, X, Y and Z axis are mutually perpendicular to.R is the pitch radius of ball screw assembly, as the bottom of multi-track spiral line where the ball centre of sphere
Radius surface.α is the helical angle of ball screw assembly,.β is angular displacement of the centre of sphere relative to lead screw of ball on multi-track spiral line.
Ob- TNB is free coordinate system, ObIt is the ball centre of sphere on multi-track spiral line, T axis is in multi-track spiral line
On the mobile tangential direction of the ball centre of sphere, perpendicular to the outer surface of lead screw, N axis crosses Z axis and perpendicular to Z for direction where N axis
The angle of axis, B axle and helical axis directions Z axis is pitch angle alpha, and T, N and B axle are mutually perpendicular to.
In Fig. 3, local coordinate system is separately fixed at the contact point between ball and lead screw raceway, ball and nut rolling way
Place, S-xsyszsIndicate the local coordinate system between ball and lead screw raceway at contact point, N-xnynznIndicate ball and nut rolling
The local coordinate system of contact point between road, S indicate the coordinate origin at ball and lead screw raceway contact point, as ball and lead screw
The geometric center in raceway contact region is directed toward the direction of the ball centre of sphere with the geometric center point in ball and lead screw raceway contact region
As zsDirection, similarly, N indicates the coordinate origin at ball and nut rolling way contact point, and as ball connects with nut rolling way
The geometric center in region is touched, the direction that the ball centre of sphere is directed toward with the geometric center point of ball and nut rolling way contact area is zn
Direction.
The coordinate origin O of global coordinate system is converted to origin ObThe transformation matrix of coordinates at place is denoted as T1：
Global coordinate system exists with T axis and N axis respectively around the angular displacement of Z axis rotation β+pi/2, reference axis X-axis and Y-axis
Projection in XOY plane is conllinear, and rotational coordinates matrix is denoted as Rot1：
Global coordinate system rotates the angular displacement of 2 π-α, the throwing of reference axis X-axis, Z axis and B axle in plane XOZ around Y-axis
Shadow is conllinear, and rotational coordinates matrix is denoted as Rot2：
Global coordinate system O-XYZ and free coordinate system ObTransformational relation between-TNB is：
In formula, L is the lead of lead screw, is calculated by L=2 π Rtan α.
Free coordinate system Ob- TNB and local coordinate system S-xsyszsTransformational relation be：
Free coordinate system Ob- TNB and local coordinate system N-xnynznTransformational relation be：
ω is set as in the rotation speed of t moment, lead screw, it is assumed that within the Γ period, lead screw is under rotational speed omega rotation speed
Angular displacement be set as W.It is O-X'Y'Z' that lead screw, which rotates the reference frame after W angular displacement,.According to homogeneous coordinate transformation principle,
Coordinate transformation relation from reference frame O-X'Y'Z' to global coordinate system O-XYZ is：
The geometric center point S of contact area, the position in coordinate system O-XYZ are：
The geometric center point S of ball and lead screw raceway contact region, the position in reference frame O-X'Y'Z' are：
Butt contact S carries out solution first derivative in the position of reference frame, obtains ball and lead screw raceway contact area
The speed ν of the center in domains：
Secondly, contact point S is in ObSpeed ν in-TNBs-obFor：
Similarly, the geometric center point N of ball and nut rolling way contact area, the position in global coordinate system O-XYZ are：
The geometric center point N of ball and nut rolling way contact area, the position in reference frame O-X'Y'Z' are：
The speed ν of the center in ball and lead screw raceway contact regionnFor：
Contact point N is in ObSpeed ν in-TNBn-obFor：
Step 2 is based on the sliding distance mathematical model between feeding revolving speed model foundation ball and raceway
According to the relationship between ball and lead screw raceway relative velocity, obtain：
νbs-ob=νs-ob+νbs-ss (16)
In formula, νbs-ssMovement velocity for contact point S on ball relative to the contact point S on lead screw raceway, νbs-ssMeter
Operator expression formula is：
In formula, νb-sIt is movement velocity of the centre of sphere of ball relative to lead screw in the hand of spiral, ωb、ωnWith ωtIt is T respectively
The component of change in coordinate axis direction, N change in coordinate axis direction and B change in coordinate axis direction upper leading screw angular velocity of rotation.
νb-sIt solves to obtain by (18) formula：
The component ω of the angular speed of lead screw in three directions in ball screw assembly,b、ωnWith ωtRespectively by formula (19),
(20), (21) obtain：
In formula, ωbrIt is the angular velocity of rotation of the ball centre of sphere,It is ωbrWith the angle between two reference axis of B, T composition plane
Size.It is ωbrCorner dimension is formed by between the projection and B reference axis in the plane of B-T reference axis composition.Ball
Angular velocity of rotation ωbrIt is obtained by formula (22)：
In formula,It is projection O' of the centre of sphere of ball in the plane that X-Y coordinate axis formsbTo O coordinate origin away from
From.
Ball is denoted as R with the sliding rolling ratio at lead screw raceway contact regional center point Sbs：
According to the relative speed relationship between ball and nut rolling way, obtain：
νbn-sn=νn-ob+νbn-nn (24)
In formula, νbn-snBe on ball at the N of contact point relative to the movement velocity at the contact point N on lead screw raceway, νbn-sn
Calculation expression be：
Ball is denoted as R with the sliding rolling ratio at nut rolling way contact area central point Nbn：
Fig. 4 show spiral motion track schematic diagram of the ball in raceway.During ball-screw auxiliary driving, rolling
The contact condition cross-sectional view of pearl and raceway is as shown in Figure 5.
Contact point S is relative to the movement velocity ν at the contact point S on lead screw raceway on ballbs-ssWith ball along lead screw
The distance l moved in a sliding manner on racewayssiBetween relationship be：
lssi=νbs-sstssi (26)
In formula, tssiMove to the position of i+1 ball from the position of i-th of ball relative to lead screw raceway for ball
The sliding time of Shi Suoyong：
In formula, tiWhen moving to the position of i+1 ball from the position of i-th of ball relative to lead screw raceway for ball
Time used.
Relative to contact point S, the sliding distance on lead screw raceway is l to ballssiWhen, sliding of the lead screw relative to contact point S
Distance is：
In formula, ωs-bIt is angular speed of the lead screw relative to ball, is calculated by (29) formula：
In formula, rsFor the distance between lead screw raceway contact central point and the ball centre of sphere, rnFor nut rolling way contact point and rolling
The distance between pearl centre of sphere.
Contact point N is relative to the movement velocity ν at the contact point N on nut rolling way on ballbn-nnWith ball along nut
The distance l moved in a sliding manner on racewaynsiBetween relationship be：
lnsi=νbn-nntnsi (30)
In formula, tnsiMove to the position of i+1 ball from the position of i-th of ball relative to nut rolling way for ball
The sliding time of Shi Suoyong：
Relative to contact point N, the sliding distance on nut rolling way is l to ballnsiWhen, sliding of the lead screw relative to contact point S
Distance is：
In formula, ωn-bIt is angular speed of the nut relative to ball, is calculated by formula (33)：
Step 3 be based on Archard wear model proposes and establish dynamic fluctuation feed revolving speed under ball screw assembly, grind at random
Damage model
Ball screw assembly, can be worn during military service, directly affect its positioning accuracy.According to the research of Archard
Achievement, abrasion loss V and contact load F and opposite sliding distance L are directly proportional, and softer in two kinds of different materials contacting with each other
The hardness H of material is inversely proportional：
Due to the limitation of Archard basic model, it cannot be directly used to explore ball screw assembly, in Random Load operating condition
Under wear problem.The wear law under speed conditions is fed in dynamic fluctuation for analysis ball screw assembly, predicts its feeding essence
Degree, and in the range of allowing precision, predict its life expectancy, this method provides improved under dynamic fluctuation feeding speed conditions
The random wear model of ball screw assembly,.
It is proposed that the random wear model in the case where dynamic fluctuation feeds speed conditions between ball and lead screw raceway is：
It is proposed that the random wear model in the case where dynamic fluctuation feeds speed conditions between ball and nut rolling way is：
Convolution (35) and (36) obtain ball screw assembly, and feed the random abrasion mathematical modulo under speed conditions in dynamic fluctuation
Type is：
The abrasion loss model between ball and lead screw raceway is established, according to Step 1: speed and sliding distance are analyzed in two
And formula (35) obtains the calculating formula of the abrasion loss between i-th of ball and lead screw raceway：
In formula, dimensionless coefficient of waste Kb-sWith the hardness H of lead screw roller surfaceb-sIt is constant parameter relevant to material,
FsiIt is the contact load of ball Yu lead screw raceway.
The abrasion loss model between ball and nut rolling way is established, according to Step 1: speed and sliding distance are analyzed in two
And formula (36) obtains the calculating formula of the abrasion loss between i-th of ball and nut rolling way：
In formula, dimensionless coefficient of waste Kb-nWith the hardness H in nut rolling way faceb-nIt is constant parameter relevant to material,
FniIt is the contact load of ball and nut rolling way.
Step 4 analyzes ball screw assembly, according to random wear model and feeds the wearing character under revolving speed in dynamic fluctuation
Abrasion total amount between ball and lead screw raceway is calculated, the abrasion loss between ball and lead screw raceway is rolling
The integral of pearl and the instantaneous abrasion loss of lead screw raceway.
It is calculated separately between the 1st, 2 ... i ... M balls and lead screw raceway according to formula (38) in tss1、tss2…tssi…tssM
The abrasion loss at moment：
To between the 1st, 2 ... i ... M balls and lead screw raceway in tss1、tss2…tssi…tssMThe abrasion loss at moment carries out
Integral, obtains each ball in tss1To tss2In period, tss2To tss3In period, tssiTo tssi+1In period and tssMExtremely
tss1Abrasion loss in period：
For i-th of ball in entire cycle period, the abrasion between ball and lead screw raceway carries out accumulative summation, is denoted as
In entire cycle period, the wear model of ball and lead screw raceway is denoted as all balls
Similarly, the abrasion total amount between ball and nut rolling way is calculated, the abrasion between ball and nut rolling way
Amount is the integral of ball and the instantaneous abrasion loss of nut rolling way.
The 1st, 2 ... i ... M balls in raceway are calculated separately and between nut rolling way in t according to formula (39)ns1、tns2…
tnsi…tnsMThe abrasion loss at moment：
To between the 1st, 2 ... i ... M balls and nut rolling way in tns1、tns2…tnsi…tnsMThe abrasion loss at moment carries out
Integral, obtains each ball in tns1To tns2In period, tns2To tns3In period, tnsiTo tnsi+1In period and tnsMExtremely
tns1Abrasion loss in period：
For i-th of ball in entire cycle period, the abrasion between ball and nut rolling way carries out accumulative summation, is denoted as
In entire cycle period, the wear model of ball and lead screw raceway is denoted as all balls
Convolution (46), (47) feed the random wear model of ball screw assembly, under speed conditions according to dynamic fluctuation and obtain
Total abrasion loss：
In formula,For the cycle-index between ball and lead screw raceway,Circulation between ball and nut rolling way
Number can be acquired according to the revolving speed of lead screw and effective transmission total kilometres of lead screw.
Claims (2)
1. a kind of random Wear Modeling method of ball screw assembly, under dynamic fluctuation feeding speed conditions, it is characterised in that：The party
Method includes,
S1, the dynamic fluctuation characteristic for feeding revolving speed during military service according to ball screw assembly, establish ball screw assembly, in dynamic
Ball and raceway under fluctuation feeding speed conditions feed revolving speed model；
S2, ball and raceway feeding revolving speed model under speed conditions are fed in dynamic fluctuation according to ball screw assembly, establishes rolling
Sliding distance mathematical model between pearl and lead screw raceway and ball and nut rolling way, i.e. ball screw assembly, wear mould at random
Type；
S3, it is based on Archard wear model, the random wear model of ball screw assembly, is improved, is established respectively in dynamic wave
Ball and the random wear model of lead screw raceway and the random abrasion mould of ball and nut rolling way under dynamic feeding speed conditions
Type；
S4, random wear model and ball and spiral shell according to the ball and lead screw raceway fed in dynamic fluctuation under speed conditions
The random wear model of female raceway calculates and analyzes wearing character of the ball screw assembly, under dynamic fluctuation feeding speed conditions.
2. the random Wear Modeling side of ball screw assembly, under a kind of dynamic fluctuation feeding speed conditions according to claim 1
Method, it is characterised in that：The realization process of the random wear model modeling method is as follows,
Step 1
According to the structure of ball screw assembly, space coordinates are established；
O-XYZ is the coordinate system of an entirety；O point is reference axis at the geometric center of lead screw and nut baseplane initial position
Position it is constant, Z axis and helical axis are parallel to each other, and X-coordinate axle and Y-coordinate axle are in the plane where nut bottom, X, Y with
Z axis is mutually perpendicular to；R is the pitch radius of ball screw assembly, as the bottom surface radius of multi-track spiral line where the ball centre of sphere；α is
The helical angle of ball screw assembly,；β is angular displacement of the centre of sphere relative to lead screw of ball on multi-track spiral line；
Ob- TNB is free coordinate system, ObIt is the ball centre of sphere on multi-track spiral line, T axis is on multi-track spiral line
The mobile tangential direction of the ball centre of sphere, perpendicular to the outer surface of lead screw, N axis crosses Z axis and perpendicular to Z axis, B for direction where N axis
The angle of axis and helical axis directions Z axis is pitch angle alpha, and T, N and B axle are mutually perpendicular to；
Local coordinate system is separately fixed at the contact point between ball and lead screw raceway, ball and nut rolling way, S-xsyszsTable
Show the local coordinate system between ball and lead screw raceway at contact point, N-xnynznIndicate contact point between ball and nut rolling way
Local coordinate system, S indicates the coordinate origin at ball and lead screw raceway contact point, as ball and lead screw raceway contact region
Geometric center, the direction that the ball centre of sphere is directed toward with the geometric center point in ball and lead screw raceway contact region is zsSide
To similarly, the coordinate origin at N expression ball and nut rolling way contact point, as ball are several with nut rolling way contact area
What center, the direction that the ball centre of sphere is directed toward with the geometric center point of ball and nut rolling way contact area is znDirection；
The coordinate origin O of global coordinate system is converted to origin ObThe transformation matrix of coordinates at place is denoted as T1：
For global coordinate system around the angular displacement of Z axis rotation β+pi/2, reference axis X-axis and Y-axis are flat in XOY with T axis and N axis respectively
Projection in face is conllinear, and rotational coordinates matrix is denoted as Rot1：
Global coordinate system rotates the angular displacement of 2 π-α around Y-axis, and the projection of reference axis X-axis, Z axis and B axle in plane XOZ is total
Line, rotational coordinates matrix are denoted as Rot2：
Global coordinate system O-XYZ and free coordinate system ObTransformational relation between-TNB is：
In formula, L is the lead of lead screw, is calculated by L=2 π Rtan α；
Free coordinate system Ob- TNB and local coordinate system S-xsyszsTransformational relation be：
Free coordinate system Ob- TNB and local coordinate system N-xnynznTransformational relation be：
ω is set as in the rotation speed of t moment, lead screw, it is assumed that within the Γ period, angle of the lead screw under rotational speed omega rotation speed
Displacement is set as W；It is O-X'Y'Z' that lead screw, which rotates the reference frame after W angular displacement,；According to homogeneous coordinate transformation principle, from ginseng
The coordinate transformation relation for examining coordinate system O-X'Y'Z' to global coordinate system O-XYZ is：
The geometric center point S of contact area, the position in coordinate system O-XYZ are：
The geometric center point S of ball and lead screw raceway contact region, the position in reference frame O-X'Y'Z' are：
Butt contact S carries out solution first derivative in the position of reference frame, obtains ball and lead screw raceway contact region
The speed ν of centers：
Secondly, contact point S is in ObSpeed ν in-TNBs-obFor：
Similarly, the geometric center point N of ball and nut rolling way contact area, the position in global coordinate system O-XYZ are：
The geometric center point N of ball and nut rolling way contact area, the position in reference frame O-X'Y'Z' are：
The speed ν of the center in ball and lead screw raceway contact regionnFor：
Contact point N is in ObSpeed ν in-TNBn-obFor：
Step 2
According to the relationship between ball and lead screw raceway relative velocity, obtain：
νbs-ob=νs-ob+νbs-ss (16)
In formula, νbs-ssMovement velocity for contact point S on ball relative to the contact point S on lead screw raceway, νbs-ssComputational chart
It is up to formula：
In formula, νb-sIt is movement velocity of the centre of sphere of ball relative to lead screw in the hand of spiral, ωb、ωnWith ωtIt is T coordinate respectively
The component of axis direction, N change in coordinate axis direction and B change in coordinate axis direction upper leading screw angular velocity of rotation；
νb-sIt is solved and is obtained by formula (18)：
The component ω of the angular speed of lead screw in three directions in ball screw assembly,b、ωnWith ωtRespectively by formula (19), (20),
(21) it obtains：
In formula, ωbrIt is the angular velocity of rotation of the ball centre of sphere,It is ωbrAngle between two reference axis of B, T composition plane is big
It is small；It is ωbrCorner dimension is formed by between the projection and B reference axis in the plane of B-T reference axis composition；The rotation of ball
Tarnsition velocity ωbrIt is obtained by formula (22)：
In formula,It is projection O of the centre of sphere of ball in the plane that X-Y coordinate axis formsb' arrive O coordinate origin distance；
Ball is denoted as R with the sliding rolling ratio at lead screw raceway contact regional center point Sbs：
According to the relative speed relationship between ball and nut rolling way, obtain：
νbn-sn=νn-ob+νbn-nn (24)
In formula, νbn-snBe on ball at the N of contact point relative to the movement velocity at the contact point N on lead screw raceway, νbn-snMeter
Operator expression formula is：
Ball is denoted as R with the sliding rolling ratio at nut rolling way contact area central point Nbn：
Contact point S is relative to the movement velocity ν at the contact point S on lead screw raceway on ballbs-ssWith ball along lead screw raceway
On the distance l that moves in a sliding mannerssiBetween relationship be：
lssi=νbs-sstssi (26)
In formula, tssiMove to the position when institute of i+1 ball from the position of i-th of ball relative to lead screw raceway for ball
Sliding time：
In formula, tiIt is moved to relative to lead screw raceway from the position of i-th of ball for ball used when the position of i+1 ball
Time；
Relative to contact point S, the sliding distance on lead screw raceway is l to ballssiWhen, sliding distance of the lead screw relative to contact point S
For：
In formula, ωs-bIt is angular speed of the lead screw relative to ball, is calculated by (29) formula：
In formula, rsFor the distance between lead screw raceway contact central point and the ball centre of sphere, rnFor nut rolling way contact point and ball ball
The distance between heart；
Contact point N is relative to the movement velocity ν at the contact point N on nut rolling way on ballbn-nnWith ball along nut rolling way
On the distance l that moves in a sliding mannernsiBetween relationship be：
lnsi=νbn-nntnsi (30)
In formula, tnsiMove to the position when institute of i+1 ball from the position of i-th of ball relative to nut rolling way for ball
Sliding time：
Relative to contact point N, the sliding distance on nut rolling way is l to ballnsiWhen, sliding distance of the lead screw relative to contact point S
For：
In formula, ωn-bIt is angular speed of the nut relative to ball, is calculated by formula (33)：
Step 3
Ball screw assembly, can be worn during military service, directly affect its positioning accuracy；According to the research of Archard at
Fruit, abrasion loss V and contact load F and opposite sliding distance L are directly proportional, in two kinds of different materials contacting with each other compared with softwood
The hardness H of material is inversely proportional：
Due to the limitation of Archard basic model, it cannot be directly used to explore ball screw assembly, under Random Load operating condition
Wear problem；The wear law under speed conditions is fed in dynamic fluctuation for analysis ball screw assembly, predicts its feed accuracy, and
In the range of allowing precision, its life expectancy is predicted, this method provides improved ball under dynamic fluctuation feeding speed conditions
The random wear model of lead screw pair；
It is proposed that the random wear model in the case where dynamic fluctuation feeds speed conditions between ball and lead screw raceway is：
It is proposed that the random wear model in the case where dynamic fluctuation feeds speed conditions between ball and nut rolling way is：
Convolution (35) obtains random abrasion mathematical model of the ball screw assembly, under dynamic fluctuation feeding speed conditions with (36)：
Establish the abrasion loss model between ball and lead screw raceway, according to Step 1: in two speed and sliding distance analyze and
Formula (35) obtains the calculating formula of the abrasion loss between i-th of ball and lead screw raceway：
In formula, dimensionless coefficient of waste Kb-sWith the hardness H of lead screw roller surfaceb-sIt is constant parameter relevant to material, FsiIt is
The contact load of ball and lead screw raceway；
Establish the abrasion loss model between ball and nut rolling way, according to Step 1: in two speed and sliding distance analyze and
Formula (36) obtains the calculating formula of the abrasion loss between i-th of ball and nut rolling way：
In formula, dimensionless coefficient of waste Kb-nWith the hardness H in nut rolling way faceb-nIt is constant parameter relevant to material, FniIt is
The contact load of ball and nut rolling way；
Step 4
Abrasion total amount between ball and lead screw raceway is calculated, the abrasion loss between ball and lead screw raceway be ball with
The integral of the instantaneous abrasion loss of lead screw raceway；
It is calculated separately between the 1st, 2 ... i ... M balls and lead screw raceway according to formula (38) in tss1、tss2…tssi…tssMMoment
Abrasion loss：
To between the 1st, 2 ... i ... M balls and lead screw raceway in tss1、tss2…tssi…tssMThe abrasion loss at moment is integrated,
Each ball is obtained in tss1To tss2In period, tss2To tss3In period, tssiTo tssi+1In period and tssMTo tss1When
Between abrasion loss in section：
For i-th of ball in entire cycle period, the abrasion between ball and lead screw raceway carries out accumulative summation, is denoted as
In entire cycle period, the wear model of ball and lead screw raceway is denoted as all balls
Similarly, the abrasion total amount between ball and nut rolling way is calculated, the abrasion loss between ball and nut rolling way is
The integral of ball and the instantaneous abrasion loss of nut rolling way；
The 1st, 2 ... i ... M balls in raceway are calculated separately and between nut rolling way in t according to formula (39)ns1、tns2…tnsi…
tnsMThe abrasion loss at moment：
To between the 1st, 2 ... i ... M balls and nut rolling way in tns1、tns2…tnsi…tnsMThe abrasion loss at moment is integrated,
Each ball is obtained in tns1To tns2In period, tns2To tns3In period, tnsiTo tnsi+1In period and tnsMTo tns1When
Between abrasion loss in section：
For i-th of ball in entire cycle period, the abrasion between ball and nut rolling way carries out accumulative summation, is denoted as
In entire cycle period, the wear model of ball and lead screw raceway is denoted as all balls
Convolution (46), (47) obtain total according to the random wear model of ball screw assembly, under dynamic fluctuation feeding speed conditions
Abrasion loss：
In formula,For the cycle-index between ball and lead screw raceway,Circulation time between ball and nut rolling way
Number, can acquire according to the revolving speed of lead screw and effective transmission total kilometres of lead screw.
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| CN117131709A (en) * | 2023-10-26 | 2023-11-28 | 天津德科智控股份有限公司 | Method for analyzing shaking amount of ball entering return device |
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