CN112212758A

CN112212758A - Method for evaluating digitalized common reference radial total run-out error

Info

Publication number: CN112212758A
Application number: CN202011087310.3A
Authority: CN
Inventors: 黄美发; 王家浩; 唐哲敏
Original assignee: Guilin University of Electronic Technology
Current assignee: Guilin University of Electronic Technology
Priority date: 2020-10-13
Filing date: 2020-10-13
Publication date: 2021-01-12

Abstract

The invention belongs to the field of precision metering and computer application, and relates to a quick, stable, simple and digital method for evaluating a common reference radial total run-out error. The invention comprises the following steps: acquiring an initial measuring point set of a measured section and a reference section; pre-positioning is carried out, and a pre-positioned reference section measuring point set, a measured section measuring point set and a reference section initial circle center coordinate set are obtained; constructing a parameter matrix according to measuring points of each section of the reference section, constructing an analysis matrix by the key point set for analysis, and determining an optimization strategy; updating a coordinate set of the center of the circle of the section of the reference section according to the optimizing result; constructing a parameter matrix by a reference section circle center coordinate set, constructing an analysis matrix by a key point set for analysis, and determining an optimization strategy; updating the measuring point sets and related parameters of the reference segment and the measured segment according to the optimizing result; judging whether the reference section and the tested section meet the design size requirement or not; and calculating and judging whether the radial total run-out error of the measured section of the measured shaft is qualified or not.

Description

Method for evaluating digitalized common reference radial total run-out error

Technical Field

The invention belongs to the field of precision metering and computer application, and relates to a quick, stable, simple and digital evaluation method for common-reference radial total run-out errors, which can be used for evaluating the radial total run-out errors of shaft parts taking the axes of specified shaft sections at two sides of a shaft section to be measured as common reference axes, and provides guidance for the improvement of the processing technology of the shaft parts.

Background

The shaft part is one of common parts in machining, the size error and the form and position error (short for shape error and position error) generated in the machining process directly influence the product quality, the assembly and the service life of the assembly, the part error is calculated quickly and accurately, and the method has important significance.

The total radial run-out is one of important indexes of geometric tolerance of shaft parts, and a definition and detection method of the total radial run-out error is given in national standards GBT 1182-2008 and ISO 1101:2012 (E), but a method for calculating the total radial run-out error value from specific measurement data of the parts is not given. Moreover, the five currently popular evaluation methods are difficult to be directly used for evaluating the radial full run-out error of the shaft parts.

First, a specialized geometric assessment method. And gradually searching the radial total run-out error meeting the definition and/or judgment conditions of the national standard and the ISO standard by utilizing the geometric properties of the cylinder and according to the translation and deformation strategies of the circumscribed cylinder. The method has high speed, but the mathematical modeling is complex, and the method is not easy to popularize and use.

Second, convex hull or convex hull-like evaluation methods. And constructing a convex hull or a similar convex hull by using the properties of the convex hull, acquiring effective measurement data, reducing the scale of the data to be evaluated, and finally acquiring the radial total run-out error meeting the definition and/or judgment conditions of the national standard and the ISO standard by using an enumeration method. This type of approach has significant advantages when dealing with medium scale station data. For larger data sizes, the data size can also be reduced by constructing convex hulls, but when such methods are used for direct evaluation, the efficiency is insufficient.

And in the third category, a linear or nonlinear target optimization function is constructed, optimization solution is carried out by adopting a general optimization method, and the optimization value of the target optimization function is used as a radial total run-out error. The method is simple and easy to understand, realizes a standard solution method in a plurality of software, and is easy to popularize. However, the efficiency of this type of method is generally not high, since no geometric features for the evaluation of the full radial run-out error are added, and the situation of large data scale in the evaluation task is not considered.

The fourth category, artificial intelligence/biological intelligence algorithms. The advantage of this type of method over the third type of method is to analyze the "objective function with complex gradient or no apparent analytic expression" and to find the "global optimum". The method also realizes a standard solution in a plurality of software at present, and is easy to popularize. Although this type of method is popular at present, it is not suitable for the evaluation of full radial run-out error. This is because the gradient of the objective function for full run-out error assessment is the sum of a large number of simple analytical expressions, and the "local optimum" of the objective function is the "global optimum". Therefore, this type of method does not have significant advantages over the third type of method.

The fifth category, active set methods. The active set method is a method specially used for processing large-scale planning problems and is characterized in that the processing of 'invalid constraint' is reduced as much as possible in the optimization process. When the method is applied to the evaluation of the common reference radial total run-out error, the efficiency is equivalent to that of the first method, the algorithm maturity and the software integration are equivalent to that of the third method and the fourth method, and the method is a rapid and simple radial total run-out error evaluation method at present. However, this method is very sensitive to initial values and does not always perform the geometric assessment task stably.

In summary, the existing geometric evaluation method is applied to evaluation of the common reference radial total run-out error of the shaft parts, and cannot simultaneously give consideration to rapidness, stability and simple form.

Disclosure of Invention

The purpose of the invention is:

aiming at the problems and the defects in the prior art, the invention provides the method for evaluating the radial total run-out error of the shaft part, which is rapid, stable, simple in form and digital and takes the axes of the designated shaft sections at the two sides of the shaft section to be measured as the common reference axis, and the evaluation result can provide guidance for the improvement of the processing technology of the shaft part.

The method is suitable for evaluating the radial total run-out error of the shaft parts which have higher requirements on the processing precision of the cylindrical surfaces of the shaft parts such as a high-precision linear guide shaft and the like and use the axes of the designated shaft sections at the two sides of the shaft section to be measured as the common reference axis.

The invention takes a high-precision linear guide shaft as an example, and verifies an evaluation method of common reference radial total run-out errors of shaft parts.

The scheme adopted by the invention is as follows:

the method for evaluating the common reference radial total run-out error of the shaft comprises the following steps:

step 1: obtaining initial measuring point set of measured segmentQ _k*}: taking the cylindrical surface of the shaft section to be measured as the section to be measured, and intercepting the section on the cylinder at equal intervals along the axial directionkEach section is perpendicular to the axis of the cylinder, and a single measuring point is selected on each section circle, so that the measuring points are approximately spirally arranged along the axis direction of the cylinder, and the measuring points are arranged in a spiral manneriInitial measuring point set for a segment to be measured consisting of measuring pointsQ _k}; wherein:

k=1, 2, 3, …, K；kthe serial number of the cross section is shown,Kis the total number of the sections;

Q _k*={x _k*，y _k*，z _kand the original space rectangular coordinate of the measured segment is marked.

After step 1, step 2 is performed.

Step 2: obtaining initial measuring point set of reference sectionP _i，j*}: the cylindrical surfaces of the designated shaft sections on the two sides of the shaft section to be detected are respectively used as a reference section 1 and a reference section 2, and the cylindrical surfaces of the reference section 1 are cut at equal intervals along the axial directionMA cross section equally spaced in the axial direction on the cylinder of the reference section 2NEach cross section being uniformly selected over the circumference of each cross-sectional circlejA measuring point, each cross section being perpendicular to the axis, thei×jInitial measuring point set for forming reference section by measuring pointsP _i，j}; wherein:

i=1, 2, 3, … , 2M；iis a section number, 2MThe total number of the sections of the reference section;

j=1, 2, 3, … , N；jthe serial numbers of the measuring points on the single cross section,Nthe total number of the measuring points of a single section is;

P _i，j*={x _i，j*，y _i，j*，z _i，jand the original space rectangular coordinate of the measuring points of the reference segment.

After step 2, step 3 is performed.

And step 3: the shaft is pre-positioned as follows: selecting coordinates of measuring points of all reference segmentsx _i，j*_maxAndx _i，j*_mincalculating the average valuex _o* =（x _i，j*_max+ x _i，j*_min) Selecting coordinates of measuring points of all reference segmentsy _i，j *_maxAndy _i，j *_mincalculating the average valuey _o*=（y _i，j*_max+ y _i，j*_min) 2; obtaining a pre-positioned measured section measuring pointQ _kAnd using it to form a measuring point set of the segment to be measuredQ _k}; obtaining a pre-positioned reference segment measuring pointP _i，jAnd using it to form a reference segment measuring point setP _i，j}; obtaining the initial circle center coordinates of each section circle of the pre-positioned reference sectionO _iAnd using it to form a reference section with a preset circle center coordinate setO _i}; wherein:

Q _k={x _k，y _k，z _kand the space rectangular coordinate of the measured point of the measured section after pre-positioning, wherein:x _k= x _k*- x _o*，y _k= y _k*-y _o*，z _k= z _kand the cylinder axis of the measured segment approaches the coordinate systemzOf the shaft, both side end faces of the section to be measuredThe central plane is approximately parallel to the XOY plane of the coordinate system;

P _i，j={x _i，j，y _i，j，z _i，jand the space rectangular coordinate of the measuring point of the pre-positioned reference segment is shown, wherein:x _i，j= x _i，j*- x _o*，y _i，j= y _i，j*- y _o*，z _i，j= z _i，jand the cylindrical axis of the reference segment approximates the coordinate systemzThe central planes of the end surfaces on the two sides of the reference section are approximately parallel to the XOY plane of the coordinate system;

O _i={x _i，y _i，z _ithe preset center of each cross-section circle of the pre-positioned reference section is a spatial rectangular coordinate, wherein:x _i= y _i=0，z _i= z _i，j*。

after step 3, step 3.1 is performed.

Step 3.1: in each section, a great deal is made from a set of reference segment measuring pointsP _i，jRespectively establishing a characteristic line vector setW _i，jGreat, set of boundary elementsb _i，jGreat Chinese character and state element sett _i，j}; obtaining the center coordinates of each section circleO _i' and using it to form coordinate set of centre of circle of section of reference segmentO _i' }; wherein:

W _i，j =（[x _i，j/t _i，j，y _i，j/t _i，j]) Is a feature row vector, all feature row vectorsW _i，jIs a set of characteristic line vectorsW _i，j}；

b _i，j=bIs a real number greater than 0, all boundary elementsb _i，jIs a set of boundary elementsb _i，j}；

O _i’={x _i’，y _i’，z _i' is the space rectangular coordinate of the center of each section of the reference section, and initially,x _i’= x _i，y _i’= y _i，z _i’= z _i；

status elements of all measurement points in the reference segmentt _i，jThe set of (1) is a state element set of a reference segment measuring pointt _i，j}。

After step 3.1, step 3.2 is performed.

Step 3.2: in each cross section, the cross section is taken ont _i，jMaximum value oft _maxCorresponding serial numbere ₁Is a key serial number, and wille ₁The key serial number sets respectively added to respective sectionseIn (c) }.

Step 3.3 is performed after step 3.2 is completed.

Step 3.3: according to the key sequence number set of each sectioneRespectively establishing analysis matrixesWAnd analyzing the column vectorsbWherein:

W= […，W _m ^T，…，W _n ^T，…]^Tis aEA matrix of rows and 2 columns of,Eis a critical sequence number seteThe number of the elements in the (C),m，nis a critical sequence number seteThe elements in (1);

b= […，b _m，…，b _n，…]^Tis aEA column vector of rows.

Step 3.4 is performed after step 3.3 is completed.

Step 3.4: for analysis matrixWAnd an augmented analysis matrixW，b]Rank analysis was performed.

Computingr _W=rank(W)，r _Wb=rank([W，b]) And comparer _WAndr _Wbthere are two cases:

the first condition is as follows: if it is notr _W=r _WbIf the optimization is to be continued, step 3.5 is executed;

case two: if it is notr _W<r _WbAttempting to derive a secondary analysis matrixWAnd analyzing the column vectorsbMiddle removed key serial number seteSome element offCorresponding rows, resulting in a reduced matrixW _f-And reducing the column vectorb _f-According toW _f- U _f-= b _f-Solved to obtainU _f-=U _f-0Then calculateb _f-=W _f U _f-0(ii) a If the key sequence number seteAll the elements in the Chinese character have been tried, and none of them is obtainedb _f->bThen the optimization should be ended, jumping to step 3.7; if the critical sequence number set is triedeElements in (b) }fWhen it is obtainedb _f->bThen the matrix will be reducedW _f-And reducing the column vectorb _f-Respectively as analysis matrixWAnd analyzing the column vectorsbWill elementfMovable key serial number seteAnd jumping to step 3.5; wherein:U _f-=[w _f-,1，w _f-,2]^T，U _f-0= [w _{f- ,01}，w _{f- ,02}]^T。

step 3.5: to findSystem of linear equationsWU=bSolution of (2)U=U ₀WhereinU=[U ₁，U ₂]^T，U ₀=[U _,01，U _,02]^T。

After step 3.5, step 3.6 is performed.

Step 3.6: in each section, calculate separatelyu _i，j= W _i，j U ₀Then calculateτ _g= (t _max-t _i，j)/(b-u _i，j) (ii) a Respectively takeτ _gMinimum value of the part ofτ _minCorresponding serial numbere ₂Is a new key serial number and wille ₂Key serial number set added to each sectioneIn (1) };

according to the cross sectionτ _minAndU ₀the center coordinates of each cross-sectional circleO _iIs updated toO _i’+τ _min∙[U ₁，U ₂，0]^T；

In each section, the center coordinates of the section circle after updating are respectively used as the basisO _i' and coordinates of each measuring point on the sectionP _i，jUpdating state element sett _i，j}，t _maxIs updated tot _i，jIs measured.

And 3.6, finishing one-time optimization after the step 3.6 is finished, and performing the step 3.3.

Step 3.7: extracting the center coordinates of the section circle of each section which is finally updatedO _i' and using it to update coordinate set of center of circle of section of reference segmentO _i’}。

Step 4 is performed after step 3.7 is completed.

And 4, step 4: obtaining a final updated reference segment section circle center coordinate setO _i' }, according toO _i' } set up feature lineVector setA _iGreat, set of boundary elementsb _iGreat, state element sets _i}; extracting measured section measuring point set of the step 3Q _kAccording toQ _kEstablishing state element setT _k}; extracting the reference segment measuring point set in the step 3P _i，jAccording toP _i，jRe-establishing a set of state elementst’_i，j}; wherein:

O _i’={x _i’，y _i’，z _i' } is the space rectangular coordinate of the center of each section of the reference section;

A _i=（[-x _i’/s _i，- y _i’/s _i，y _i’ z _i’/s _i，- x _i’ z _i’/s _i]) Is a feature row vector, all feature row vectorsA _iIs a characteristic row vectorA _i}；

b _i=bIs a real number greater than 0, all boundary elementsb _iIs a set of boundary elementsb _i}；

State elements of the centers of all cross-sections in the reference sections _iThe set of the state elements is the center of the circle of the section of the reference segments _i}；

In the section to be measuredAll state elementsT _kThe set of (a) is a state element set of a measured segment measuring pointT _k}；

All state elements in the reference segmentt’_i，jThe set of (1) is a state element set of a reference segment measuring pointt’_i，j }。

After step 4, step 4.1 is performed.

Step 4.1: gets _iMaximum values _maxCorresponding serial numberl ₁Is a key serial number, and willl ₁Last page added to key serial number setlIn (c) }.

After step 4.1, step 4.2 is performed.

Step 4.2: according to the key sequence numberlEstablishment of an analysis matrixAAnd analyzing the column vectorsb', wherein:

A=[…，A _p ^T，…，A _q ^T，…]^Tis aLA matrix of rows and 4 columns,Lis a critical sequence number setlThe number of the elements in the (C),p，qis a critical sequence number setlThe elements in (1);

b’=[…，b _p，…，b _q，…]^Tis aLA column vector of rows.

After step 4.2, step 4.3 is performed.

Step 4.3: for analysis matrixAAnd the broadening matrix [ alpha ]A，b’]Rank analysis was performed.

Computingr _A= rank(A)，r _Ab’= rank([A，b’]) And comparer _AAndr _Ab’there are two cases:

the first condition is as follows: if it is notr _A= r _Ab’Should, shouldContinuing optimizing and executing the step 4.4;

case two: if it is notr _A<r _AbAttempting to derive a secondary analysis matrixAAnd analyzing the column vectorsb' middle removing key serial number setlSome element oflCorresponding rows, resulting in a reduced matrixA _l-And reducing the column vectorb’_l-According toA _l- Ψ _l-= b’_l-Solved to obtainΨ _l-=Ψ _l-0Then calculate b’_l-=A _l Ψ _l-0(ii) a If the key sequence number setlAll the elements in the Chinese character have been tried, and none of them is obtainedb’_l->bThen the optimization should be ended, jumping to step 5; if the critical sequence number set is triedlElements in (b) }lWhen it is obtainedb’_l->bThen the matrix will be reducedA _l-And reducing the column vectorb’_l-Respectively as analysis matrixAAnd analyzing the column vectorsb', will elementlMovable key serial number setlAnd jumping to step 4.4; whereinΨ _l-=[v _l-,1，v _l-,2，v _l-,3，v _l-,4]^T，Ψ _l-0=[v _{l- ,01}，v _{l- ,02}，v _{l- ,03}，v _{l- ,04}]^T。

Step 4.4: solving a system of linear equationsAΨ=b' solution ofΨ=Ψ ₀WhereinΨ=[Ψ ₁，Ψ ₂，Ψ ₃，Ψ ₄]^T，Ψ ₀=[Ψ _0,1，Ψ _0,2，Ψ _0,3，Ψ _0,4]^T。

After step 4.4, step 4.5 is performed.

Step 4.5: computingv _i=A _i Ψ ₀Then calculateτ’_g=(s _max-s _i)/(b-v _i) (ii) a Getτ’_gMinimum value of the part ofτ’_minCorresponding serial numberl ₂Is a new key serial number and willl ₂Last page added to key serial number setlIn (1) };

the coordinate set of the center of a circle of the section of the reference segmentO _i' }is updated toO _i’ +τ’_min∙V ₁Wherein:

；

according to the updatedO _i' update state element sets _i}，s _maxIs updated tos _iMaximum value of (d);

set of measured segment measuring pointsQ _kIs updated toQ _k+τ’_min∙V ₂Wherein:

；

according to the updatedQ _kUpdating state element setT _k}；

Set the reference segment measuring pointP _i，jIs updated toP _i，j+τ’_min∙V ₃Wherein:

；

according to the updatedP _i，jUpdate status elementCollection checkt’_i，j}。

And 4.5, finishing one-time optimization after the step 4.5 is finished, and performing the step 4.2.

And 5: obtaining the final reference segment measuring point state element sett’_i，jComparing all the reference section measuring pointst’_i，jMaximum value thereoft’_{i，j max}And minimum valuet’_{i，j min}And (6) judging whether the reference shaft section meets the design size requirement of the measured shaft, if so, performing the step 6, otherwise, judging that the reference section of the measured shaft does not meet the design size requirement, and directly ending.

Step 6: obtaining the final state element set of the measured segment measuring pointT _kComparing the measured points of all the measured segmentsT _kMaximum value thereofT _{k max}And minimum valueT _{k min}Namely the minimum circumscribed cylinder radius and the maximum inscribed cylinder radius of the measured shaft section, judging whether the measured shaft section meets the design size requirement of the measured shaft, if so, performing the step 7, otherwise, judging that the measured section of the measured shaft does not meet the design size requirement, and directly ending.

And 7: comparing all points of the section to be measuredT _kCalculatingT=T _{k max}-T _{k min}Namely the radial total run-out error of the measured section; and judging whether the measured shaft meets the requirement of the radial full run-out tolerance.

To get a more accurate solution, the following optimization can be done:

in step 4.5, ifτ ^’ _min ∙V _iOf single or several iterationsτ ^’ _min∙ V _iGreater than a given thresholdaThen, according to the updated coordinate set of the circle center of the section of the reference segmentO _i' } jump to step 4 to update characteristic line vector setA _i}, set of boundary elements{b _iGreat, state element sets _i}. To facilitate numerical calculation, can makebTaking a specific value greater than 0, but not limited to 1.

The invention has the beneficial effects that:

1. the geometric characteristics of the radial total run-out are fully considered, and the evaluation form is simplified, so that the method is easier to popularize than the first type of evaluation method. 2. The geometric characteristics of the radial total run-out are fully considered, a better value is obtained through mature linear operation in each iteration, and the minimum radial total run-out error can be finally obtained, so that the algorithm is stable, and the problem of initial value sensitivity of the fifth method does not exist. 3. The fact that most of the measuring points are invalid measuring points in the radial total run-out error evaluation is implied, and the invalid measuring points are not added with iteration, so that the iteration times of the method are fewer and are equivalent to the first type evaluation method and the fifth type evaluation method. 4. When calculating optimizing direction, only considering key sequence number setlAnd (4) corresponding measuring points, so that the operation amount of each iteration is small, and the method is equivalent to the fifth type evaluation method. 5. Because the iteration times are less and the operation amount of each iteration is less, the total operation speed is equivalent to the first type evaluation method and the fifth type evaluation method.

The invention provides a rapid, stable, simple and digital evaluation method for the full run-out error in the radial direction, which can be used for evaluating the full run-out error in the radial direction of shaft parts taking the axes of the designated shaft sections at the two sides of the shaft section to be measured as a common reference axis and providing guidance for the improvement of the processing technology of the shaft parts, thereby having industrial possibility.

Drawings

FIG. 1 is a flow chart of the present invention.

FIG. 2 is a flow chart of the present invention.

FIG. 3 is a tolerance layout for parts in an exemplary embodiment.

Detailed Description

The following are specific embodiments of the present invention, and the embodiments of the present invention will be further described with reference to the drawings, but the present invention is not limited to these embodiments.

The common reference full radial run-out error for one axis is evaluated and its tolerance design specification is shown in figure 3.

Step 1: obtaining initial measuring point set of measured segmentQ _kThe following:

after step 1, step 2 is performed.

Step 2: obtaining initial measuring point set of reference sectionP _i，jThe following:

after step 2, step 3 is performed.

And step 3: obtaining a predetermined measured segment measuring point setQ _kThe method comprises the following steps:

obtaining pre-positioned reference segment measuring point setP _i，jThe method comprises the following steps:

obtaining pre-positioned reference section initial circle center coordinate setO _iThe method comprises the following steps:

after step 3, step 3.1 is performed.

Step 3.1: taking the section 1 as an example, according to the reference segment measuring point set corresponding to each measuring point on the section 1P _i，jGreat atP _，j1Construction of characteristic line vector setW _，j1The method comprises the following steps:

establishing a set of boundary elementsb _，j1The method comprises the following steps:

{b _，j1}=[1，1，1，1，1，1]^T；

establishing a set of state elementst _，j1The method comprises the following steps:

obtaining the center coordinates of the section 1 of the reference sectionO ₁' the following:

O ₁’={0，0，15}。

after step 3.1, step 3.2 is performed.

Step 3.2: in section 1, taket _，j1Maximum value oft _maxCorresponding serial number 2 is a key serial number, and 2 is added to the key serial number set of the section 1eChinese, a large aperturee}={2}。

Step 3.3 is performed after step 3.2 is completed.

Step 3.3: according to the key sequence number set of section 1eEstablishment of an analysis matrixWAnd analyzing the column vectorsbWherein:

W= W ₂=[-0.49999，0.86603]^T，

Wis a matrix with 1 row and 2 columns, and a key sequence number seteElement ofThe number is 1, the element is 2;

b=1。

step 3.4 is performed after step 3.3 is completed.

Computingr _W=rank(W)=1，r _Wb=rank([W，b]) =1, and comparingr _WAndr _Wb. Because of the fact thatr _W=r _WbSo the optimization should continue, step 3.5 is performed.

Step 3.5: solving a system of linear equationsWU=bSolution of (2)U=U ₀=[-0.49999，0.86603]^T。

After step 3.5, step 3.6 is performed.

Step 3.6: in section 1, calculateu _，j1= W _，j1 U ₀The results are as follows:

then calculateτ _g= (t _max-t _i，j)/(b-u _i，j) Get itτ _gThe results for the portion of the series greater than zero are as follows:

minimum value thereofτ _minThe corresponding serial number 5 is a new key serial number, and 5 is added to the key serial number set of the section 1eAt this time, a retaining openinge}={2，5}。

According to section 1τ _minAndU ₀coordinates of the center of a circle in the cross section 1O ₁Is updated toO ₁’+τ _min∙[U ₁，U ₂，0]^TThe results are as follows:

O ₁’={-0.00086，0.00150，15}；

according to the updated center coordinates of the cross-section circleO ₁' and coordinates of each measuring point on the sectionP _，j1Updating state element sett _，j1}，t _maxIs updated tot _，j1The results are as follows:

By analogy, after the second optimization, the key sequence number sete}={2，5，4}，O ₁’={0.00211，0.00321 , 15}。

At this time, step 3.3 is performed first: according to the key sequence number set of section 1e} = {2, 5, 4} establishing analysis matrixWAnd analyzing the column vectorsbWherein:

，

b=[1，1，1]^T。

step 3.4 is performed after step 3.3 is completed.

Is calculated tor _W=rank(W)=2，r _Wb=rank([W，b])=3，r _W< r _Wb(ii) a First, an attempt is made to analyze the matrix fromWAnd analyzing the column vectorsbMiddle deleted key serial number seteRow corresponding to the first element 2 in = {2, 5, 4}, resulting in a reduced matrixW _-2：

Andb _2-column vector:

b _2-=[1，1]^T。

according toW _2- U _-2= b _2-Solved to obtainU _-2= U _-20=[-0.00002，-1.15472]^TThen calculate to obtainb _2-=W ₂ U _-20=

-1.00002 <1=b。

Similarly, respectively obtainb _5-= -1.00005 <1=b，b _4-= 0.99997<1=bNone is obtainedb _f->bSo the seek should be ended, jumping to step 3.7.

Step 3.7: extracting the center coordinates of the finally updated section circle of the section 1O ₁’={-0.00211，-0.00321，15}。

By parity of reasoning, the center coordinates of the finally updated section circles of the other sections are obtainedO _i' and using it to update coordinate set of center of circle of section of reference segmentO _i' } as follows:

step 4 is performed after step 3.7 is completed.

And 4, step 4: extracting the finally updated reference segment section circle center coordinate setO _i' }, according toO _i' establishing a characteristic line vector setA _iThe method comprises the following steps:

build formForm element sets _iThe method comprises the following steps:

establishing a set of boundary elementsb _iThe method comprises the following steps:

{b _i}=[1，1，1，1，1，1，1，1，1，1]^T。

extracting measured section measuring point set of the step 3Q _kAccording toQ _kEstablishing state element setT _kThe method comprises the following steps:

extracting the reference segment measuring point set in the step 3P _i，jAccording toP _i，jRe-establishing a set of state elementst’_i，jThe method comprises the following steps:

after step 4, step 4.1 is performed.

Step 4.1: gets _iMaximum values _maxThe corresponding serial number 8 is the key serial number, and 8 is added into the key serial number setlChinese, a large aperturel}=8。

After step 4.1, step 4.2 is performed.

A= A ₈=[0.72885，-0.68468，143.78185，153.05810]^T；

b’=1。

after step 4.2, step 4.3 is performed.

Computingr _A= rank(A)=1，r _Ab’= rank([A，b’]) =1, comparisonr _AAndr _Ab’(ii) a Because of the fact thatr _A= r _Ab’So the optimization should continue, step 4.4 is performed.

Step 4.4: solving a system of linear equationsAΨ=b' solution to getΨ=Ψ ₀=[0.00002，-0.00002，0.00326，0.00347]^T。

After step 4.4, step 4.5 is performed.

Step 4.5: computingv _i=A _i Ψ ₀The results are as follows:

then calculateτ’_g=(s _max-s _i)/(b-v _i) Get itτ’_gThe results for the portion of the series greater than zero are as follows:

minimum value thereofτ’_minThe corresponding serial number 7 is a new key serial number, and 7 is added into the key serial number setlIn (c) }. At this moment, the openingl}={8，7}。

，

obtaining updatedO _i' } as follows:

according to the updatedO _i' update state element sets _i}，s _maxIs updated tos _iThe results are as follows:

set of measured segment measuring pointsQ _kIs updated toQ _k+τ’_min∙ V ₂Wherein:

，

obtaining updatedQ _kThe method comprises the following steps:

according to the updatedQ _kUpdating state element setT _kResults are as follows:

set the reference segment measuring pointP _i，jIs updated toP _i，j+τ’_min∙ V ₃Wherein:

，

obtaining updatedP _i，jThe method comprises the following steps:

according to the updatedP _i，jUpdating state element set t’_i，jResults are as follows:

By analogy, after the 4 th optimization, the key sequence number is collectedl}={8，7，6，4，1}。

At this time, step 4.2 is performed first: establishing an analysis matrix according to the key sequence number set {8, 7, 6, 4, 1}AAnd analyzing the column vectorsb', wherein:

，

b’=[1，1，1，1，1]^T。

Computingr _A= rank(A)=4，r _Ab’= rank([A，b’])=5， r _A< r _Ab’. First, an attempt is made to analyze the matrix fromAAnd analyzing the column vectorsb' middle removing key serial number setlRow corresponding to the first element 8 in = {8, 7, 6, 4, 1}, resulting in a reduced matrixA _-8：

And reducing the column vectorb’_-8：

b’_-8=[1，1，1，1]^T。

According toA _-8 Ψ _-8= b’_-8Solved to obtainΨ _-8=Ψ _8-0=[3.15016，-2.14150，-0.03632，-0.16944]^TThen calculate to obtainb’_8-= A ₈ Ψ _8-0= -27.39477<1=b。

Similarly, respectively obtainb’_7-=-1.18356 <1=b，b’_6-=-1.03268 <1=b，b’_4-=-79.63369 <1=b，b’_1-=0.99459 <1=bNone is obtainedb’_l->bSo the seek should be ended, jumping to step 5.

And 5: obtaining the final reference segment measuring point state element set t’_i，jThe method comprises the following steps:

comparing measured points of all reference sectionst’_i，jWhich isMaximum value oft’_{i，j max}And minimum valuet’_{i，j min}Namely the minimum circumscribed cylindrical radius and the maximum inscribed cylindrical radius of the reference shaft segment,t’_{i，j min}=35.00268>35.000，t’_{i，j max}=35.02443 <35.025, the measured shaft reference section meets the design size requirement of 35.000-35.025, and step 6 is performed.

Step 6: obtaining the final state element set of the measured segment measuring pointT _kThe method comprises the following steps:

comparing measured points of all measured sectionsT _kMaximum value thereofT _{k max}And minimum valueT _{k min}Namely the minimum circumscribed cylinder radius and the maximum inscribed cylinder radius of the measured shaft section,T _{k max}=49.99059<50.000，T _{k min}=49.96790>49.961, the section to be measured of the measured shaft meets the design size requirement 49.961-50.000, and step 7 is performed.

And 7: comparing all points of the section to be measuredT _kCalculatingT=T _{k max}-T _{k min}=0.02269 radial full run-out error of the measured section of the measured shaft, 0.02269<0.030, so the measured shaft meets its full radial runout tolerance requirements.

In the above description, the present invention has been described by way of specific embodiments, but those skilled in the art will appreciate that various modifications and variations can be made within the spirit and scope of the invention as hereinafter claimed.

The results obtained by the measurement are in accordance with GBT 1182-.

Claims

1. The invention takes a high-precision linear guide shaft as an example, and verifies an evaluation method of common reference radial total run-out errors of shaft parts.

2. The method for evaluating the common reference radial total run-out error of the shaft comprises the following steps:

Q _k*={x _k*，y _k*，z _kthe coordinate is the initial space rectangular coordinate of the measured point of the measured segment;

after the step 1 is finished, performing a step 2;

j=1, 2, 3, … , N；jis a single cross sectionThe serial number of the measuring point on the measuring point,Nthe total number of the measuring points of a single section is;

P _i，j*={x _i，j*，y _i，j*，z _i，jthe original space rectangular coordinate of the measuring point of the reference segment is marked;

step 3 is carried out after step 2 is finished;

Q _k={x _k，y _k，z _kand the space rectangular coordinate of the measured point of the measured section after pre-positioning, wherein:x _k= x _k*- x _o*，y _k= y _k*-y _o*，z _k= z _kand the cylinder axis of the measured segment approaches the coordinate systemzThe center of the end faces of both sides of the shaft and the measured sectionThe plane is approximately parallel to the XOY plane of the coordinate system;

O _i={x _i，y _i，z _ithe preset center of each cross-section circle of the pre-positioned reference section is a spatial rectangular coordinate, wherein:x _i= y _i=0，z _i= z _i，j*；

step 3.1 is carried out after step 3 is finished;

status elements of all measurement points in the reference segmentt _i，jThe set of (1) is a state element set of a reference segment measuring pointt _i，j}；

Step 3.2 is carried out after step 3.1 is finished;

step 3.2: in each cross section, the cross section is taken ont _i，jMaximum value oft _maxCorresponding serial numbere ₁Is a key serial number, and wille ₁The key serial number sets respectively added to respective sectionseIn (1) };

step 3.3 is carried out after step 3.2 is finished;

b= […，b _m，…，b _n，…]^Tis aEA column vector of rows;

step 3.4 is performed after step 3.3 is finished;

step 3.4: for analysis matrixWAnd an augmented analysis matrixW，b]Performing rank analysis;

case two: if it is notr _W<r _WbAttempting to derive a secondary analysis matrixWAnd analyzing the column vectorsbMiddle removed key serial number seteSome element offCorresponding rows, resulting in a reduced matrixW _f-And reducing the column vectorb _f-According toW _f- U _f-= b _f-Solved to obtainU _f-=U _f-0Then calculateb _f-=W _f U _f-0(ii) a If the key sequence number seteAll the elements in the Chinese character have been tried, and none of them is obtainedb _f->bThen the optimization should be ended, jumping to step 3.7; if the critical sequence number set is triedeElements in (b) }fWhen it is obtainedb _f->bThen the matrix will be reducedW _f-And reducing the column vectorb _f-Respectively as analysis matrixWAnd analyzing the column vectorsbWill elementfMovable key serial number seteAnd jumping to step 3.5; wherein:U _f-=[w _f-,1，w _f-,2]^T，U _f-0= [w _{f- ,01}，w _{f- ,02}]^T；

step 3.5: solving a system of linear equationsWU=bSolution of (2)U=U ₀WhereinU=[U ₁，U ₂]^T，U ₀=[U _,01，U _,02]^T；

Step 3.6 is performed after step 3.5 is finished;

In each section, the center coordinates of the section circle after updating are respectively used as the basisO _i' and coordinates of each measuring point on the sectionP _i，jUpdating state element sett _i，j}，t _maxIs updated tot _i，jMaximum value of (d);

completing one optimization after the step 3.6 is finished, and performing the step 3.3;

step 3.7: extracting the center coordinates of the section circle of each section which is finally updatedO _i' and using it to update coordinate set of center of circle of section of reference segmentO _i’}；

Step 4 is carried out after step 3.7 is finished;

and 4, step 4: obtaining a final updated reference segment section circle center coordinate setO _i' }, according toO _i' establish characteristic line directionVolume setA _iGreat, set of boundary elementsb _iGreat, state element sets _i}; extracting measured section measuring point set of the step 3Q _kAccording toQ _kEstablishing state element setT _k}; extracting the reference segment measuring point set in the step 3P _i，jAccording toP _i，jRe-establishing a set of state elementst’_i，j}; wherein:

All state elements in the segment under testT _kThe set of (a) is a state element set of a measured segment measuring pointT _k}；

All state elements in the reference segmentt’_i，jThe set of (1) is a state element set of a reference segment measuring pointt’_i，j }；

Step 4.1 is carried out after step 4 is finished;

step 4.1: gets _iMaximum values _maxCorresponding serial numberl ₁Is a key serial number, and willl ₁Last page added to key serial number setlIn (1) };

step 4.2 is carried out after step 4.1 is finished;

b’=[…，b _p，…，b _q，…]^Tis aLA column vector of rows;

step 4.3 is carried out after step 4.2 is finished;

step 4.3: for analysis matrixAAnd the broadening matrix [ alpha ]A，b’]Performing rank analysis;

the first condition is as follows: if it is notr _A= r _Ab’If the optimization is to be continued, step 4.4 is executed;

case two: if it is notr _A<r _AbAttempting to derive a secondary analysis matrixAAnd analyzing the column vectorsb' middle removing key serial number setlSome element oflCorresponding rows, resulting in a reduced matrixA _l-And reducing the column vectorb’_l-According toA _l- Ψ _l-= b’_l-Solved to obtainΨ _l-=Ψ _l-0Then calculate b’_l-=A _l Ψ _l-0(ii) a If the key sequence number setlAll the elements in the Chinese character have been tried, and none of them is obtainedb’_l->bThen the optimization should be ended, jumping to step 5; if the critical sequence number set is triedlElements in (b) }lWhen it is obtainedb’_l->bThen the matrix will be reducedA _l-And reducing the column vectorb’_l-Respectively as analysis matrixAAnd analyzing the column vectorsb', will elementlMovable key serial number setlAnd jumping to step 4.4; whereinΨ _l-=[v _l-,1，v _l-,2，v _l-,3，v _l-,4]^T，Ψ _l-0=[v _{l- ,01}，v _{l- ,02}，v _{l- ,03}，v _{l- ,04}]^T；

Step 4.4: solving a system of linear equationsAΨ=b' solution ofΨ=Ψ ₀WhereinΨ=[Ψ ₁，Ψ ₂，Ψ ₃，Ψ ₄]^T，Ψ ₀=[Ψ _0,1，Ψ _0,2，Ψ _0,3，Ψ _0,4]^T；

Step 4.5 is carried out after step 4.4 is finished;

；

；

according to the updatedQ _kUpdating state element setT _k}；

；

according to the updatedP _i，jUpdating state element sett’_i，j}；

Completing one optimization after the step 4.5 is finished, and performing the step 4.2;

and 5: obtaining the final reference segment measuring point state element sett’_i，jComparing all the reference section measuring pointst’_i，jMaximum value thereoft’_{i，j max}And minimum valuet’_{i，j min}Judging whether the reference shaft section meets the design size requirement of the measured shaft or not, if so, performing the step 6, otherwise, judging that the reference section of the measured shaft does not meet the design size requirement, and directly ending;

step 6: obtaining the final state element set of the measured segment measuring pointT _kComparing the measured points of all the measured segmentsT _kMaximum value thereofT _{k max}And minimum valueT _{k min}Judging whether the measured shaft section meets the design size requirement of the measured shaft or not, if so, performing step 7, otherwise, judging that the measured section of the measured shaft does not meet the design size requirement, and directly ending;

3. The method of claim 2, wherein said coordinate transformation moves by an average of coordinates.

4. A method of assessing the common reference radial run out error of a shaft as claimed in claim 2, wherein in step 3.6, if soτ _min∙[U ₁，U ₂，0]^TOf single or several iterationsτ _min∙[U ₁，U ₂，0]^TGreater than a given thresholdaThen, based on the updated center coordinates of the cross-sectional circle of the reference sectionO _i' jump to step 3.1 to update feature row vector setW _i，jGreat, set of boundary elementsb _i，jGreat Chinese character and state element sett _i，j}。

5. A method of assessing the common reference radial run out error of a shaft as claimed in claim 2, wherein in step 4.5, if soτ’_min ∙V _iOf single or several iterationsτ ^’ _min∙ V _iGreater than a given thresholdbThen, according to the updated coordinate set of the circle center of the section of the reference segmentO _i' } jump to step 4 to update characteristic line vector setA _iGreat, set of boundary elementsb _iGreat, state element sets _i}。

6. The method of assessing common reference full radial runout error of a shaft of claim 2,b=1。

7. the measured result conforms to GBT 1182 + 2008 and ISO 1101:2012 (E); the method is suitable for evaluating the radial total run-out error of the shaft parts which have higher requirements on the processing precision of the cylindrical surfaces of the shaft parts such as a high-precision linear guide shaft and the like and use the axes of the designated shaft sections at the two sides of the shaft section to be measured as the common reference axis.