CN104915468B - The force deformation error calculation method of roller line slideway auxiliary under a kind of vertical load - Google Patents

The force deformation error calculation method of roller line slideway auxiliary under a kind of vertical load Download PDF

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CN104915468B
CN104915468B CN201510159553.6A CN201510159553A CN104915468B CN 104915468 B CN104915468 B CN 104915468B CN 201510159553 A CN201510159553 A CN 201510159553A CN 104915468 B CN104915468 B CN 104915468B
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mrow
msub
mfrac
msup
line slideway
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蔡力钢
李丹丹
刘志峰
冯秋男
赵宏伟
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Beijing University of Technology
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Abstract

The force deformation error calculation method of roller line slideway auxiliary belongs to the accuracy Design field of line slideway auxiliary under a kind of vertical load, to obtain deformation of the line slideway auxiliary under load effect, computational efficiency is improved, the present invention is considered under external load function, the force deformation error of roller line slideway auxiliary.The present invention carries out parametric modeling using Three-dimensional CAD Software, analyzes the stressing conditions of roller line slideway auxiliary, establishes in the secondary mechanics analysis model of external load function lower guideway.Line slideway auxiliary contact model is studied again, the relation between elastic deformation and load during roller-trench contact single with improved Palmgren formula analysis.Load according to suffered by the size of external applied load solves each rolling element, and the elastic deformation amount of each rolling element is obtained, then derive the displacement of sliding block, the force deformation error of as whole line slideway auxiliary.Finally modeled with Matlab program parametersization, its conclusion calculates for the kinematic accuracy of rolling linear guide and life-span and provides theoretical foundation.

Description

The force deformation error calculation method of roller line slideway auxiliary under a kind of vertical load
Technical field
The present invention is a kind of force deformation error approach for calculating roller line slideway auxiliary under vertical load effect, belongs to straight The accuracy Design field of line guideway.
Background technology
Roller line slideway auxiliary is a kind of rolling bearing of the relative linear reciprocating motion of new work, between guide rail and sliding block Roller roll to avoid directly contacting for guide rail and sliding block, sliding friction instead of with rolling friction, have positioning precision it is high, The advantages that static and dynamic friction coefficient is small, maintainable good, it has also become the industry such as numerically controlled processing equipment, Aero-Space and industrial robot In a key feature.The bearing capacity and ratio of rigidity spherical guide of roller line slideway auxiliary are high, but to guide pass Parallelism error requires more sensitive, easy laterally offset and slip, is mainly used on the larger lathe of bearing capacity, calculates rolling The force deformation error of post line slideway auxiliary is significant for improving machine tool accuracy.
So far, line slideway auxiliary is in terms of load metamorphism analysis, and existing research is mainly by analyzing line slideway The motion in the trench of secondary rolling element and stressing conditions, its vibration and frictional behavior in the process of running is calculated or Modeling.HUNG is based on Hertzian contact theory, by establishing the lower rolling guide FEM model of different loads effect, have studied load Influence to guide rail rigidity；The clock Yankee of Institutes Of Technology Of Nanjing is built in Palmgren contact stiffness formula and dynamics of rigid bodies Vertical line slideway rigidity model；Sun Jianli of the Central China University of Science and Technology et al. to the kinematic accuracy of rolling linear guide, frictional behavior, The theoretical calculation of various contact conditions of groove physical dimension with the relation of guide rail performance and on linear rolling guide etc. All studied, the Rigidity Calculation and guide rail life-span for carrying out the rolling linear guide of various structures calculate, and are studies in China rollings Dynamic line slideway comparison system.
Research either carries out force analysis for line slideway auxiliary external applied load or uses for reference the analysis side of rolling bearing above Method, using Elasticity, carry out contact force for guideway interior contact situation and finite element deformation calculates, seldom analyze outside Under load effect, the deformation of whole line slideway auxiliary calculates.Make the invention provides a kind of roller line slideway in vertical load Distortion inaccuracy computational methods under, based on mechanical equilibrium condition and Elastic Contact Theory, consider under external load function, The force deformation error of roller line slideway auxiliary.And programmed with Matlab, parameterize each variable, there is provided a kind of quick computing power The method of distortion inaccuracy.
The content of the invention
Roller line slideway auxiliary is mainly acted on by two kinds of load, and one kind is to be used as a stress using whole line slideway auxiliary External load suffered by body；Another kind is in the presence of external applied load, and each element of guide rail interior is suffered when interacting Internal load.The present invention is provided under a kind of consideration external load function, calculates roller line slideway auxiliary under vertical load effect Force deformation error approach.
The technical problem to be solved in the present invention is the force deformation error to obtain roller line slideway auxiliary under vertical load The calculating of amount, ensure further to improve operation efficiency on the basis of precision.The load type roller lines such as four directions with classics are led Rail pair is basic research object, by parametric modeling and program composition, calculates the force deformation error under vertical load effect.
Fig. 1 show the specific implementation step of this method.
Step 1：Threedimensional model is established, calculates external force suffered by roller line slideway auxiliary.
Modeling is digitized to the part on roller line slideway auxiliary with Three-dimensional CAD Software, and sets each zero The material properties of part, using qualitative attribute button, calculate the gravity and position of centre of gravity suffered by guideway.
Step 2：The force analysis of each sliding block of guideway.
In the analysis to roller line slideway auxiliary external load, the guide rail elastic deformation amount caused by external load For size with guide rail than relatively small, the research of external magnitude of load influences very little, using workbench and sliding block as rigid body, rolling Kinetoplast is studied as elastomer.
Using the load type roller line slideway auxiliary such as four directions as research object, it is straight to analyze common Four-slider workbench roller The stressing conditions of each sliding block in line guideway.External applied load analysis is mainly according to the equilibrium condition and compatibility of deformation bar of power system Part, space coordinates O is established by origin of the center of line slideway auxiliary1XYZ, such as Fig. 2, list equilibrium equation, obtain and act on Power on each sliding block.
Step 3：The contact analysis of single roller-groove.
When roller line slideway auxiliary is by external load function, roller and groove are in contact deformation.Managed according to Elastic Contact By the contact problems that can be equivalent to the contact between roller and groove between elastic cylinder and two rigid planes.
Hertz linear contact lay theoretical researches are that two cylinders with smooth surface contact with each other along plain line. Know two elastic cylinders a diameter of R, length le, then under load Q effects, it is 2a, length l that contact, which produces width,eRectangle connect Touch area.
As shown in figure 3, the stress distribution in two cylinders contact regions is rounded, and along any point x of cylinder radial direction Hertz contact stress be：
o1, o2Point between Elastic approach be：
For there is the situation that limit for length's roller moves in the trench, the relation between its elastic deformation and load can use the present invention Palmgren empirical equations obtain, and are shown below.
Wherein：
δi--- the elastic deformation amount of the i-th row roller；Qi--- the contact surface normal load of the i-th row roller；
E1、E2--- the modulus of elasticity of two contacts；The equivalent elastic modulus of E ' --- contact；
μ1、μ2--- the Poisson's ratio of two contacts；The equivalent Poisson's ratio of μ --- contact；
a1、a2--- the width of two contact areas；le--- the contact length of two rollers.
The formula shows that Elastic approach is unrelated with the diameter of roller and raceway, and this does not obviously square with the fact.In order that Palmgren formula are applied to the situation of roller diameter change, introduce the fit term of δ~D curve.
K is an item undetermined, and it is equivalent to roller diameter, in the case where load is certain, in order that the knot of numerical computations Fruit is identical, then roller diameter D and length leIt must is fulfilled for certain relation：
Substitution formula (6), is obtained：
Step 4：Consider the influence of interior external applied load, calculate the force deformation error of roller line slideway auxiliary.
According to step 2, the radial load F perpendicular to each sliding block upper surface can be calculated.If 4 row roller in single sliding block Numbering 1,2,3,4, as shown in figure 4, establishing coordinate system O in symmetrical centre point2XYZ, the sliding block that the i-th each roller arranged is subject to The normal load Q of contact surfacei, roller elastic deformation amount δi.Under pretightning force effect, the initial compression deflection δ of each row roller0, Under Radial Loads, the deformation of contact zone is δV, the angle of Contact normal direction and vertical direction is θ.
Consider the dynamic balance of roller line slideway auxiliary origin of coordinates under radial load F effects, can obtain
Wherein, neRepresent to contact ball number simultaneously in roller line slideway auxiliary.
The Q according to the characteristics of the load type roller line slideway auxiliary such as four directions1=Q2, Q3=Q4, then：
Aggregative formula (7) (8) (9), obtains the relation between sliding block radial load and displacement：
F=f (δV) (10)
F is obtained by external applied load force analysis, displacement δ of the sliding block under load effect can be obtained by formula (8)V, this Displacement is force deformation error of the line slideway auxiliary under vertical load.
Further, using Matlab softwares, to the pass between step 4 roller line slideway auxiliary external applied load and distortion inaccuracy It is that expression formula is programmed.Each variable is parameterized, the quick force deformation error calculated under vertical load effect.
Improvement is step 1, and the parameter model of parts is carried out with three-dimensional software, can be changed by changing parameter Become the model for studying roller line slideway auxiliary.
Improvement is step 2, when studying external applied load, using workbench and sliding block as rigid body, rolling element conduct Elastomer is studied.
Improvement is step 3, and relative to Hertz contact theories are used, improved Palmgren empirical equations more can be quick Calculate stress and the deformation of roller and plane contact.
Improvement is step 4, considers influence of the interior external applied load to roller line slideway auxiliary, establishes external applied load and power Relational expression between distortion inaccuracy.
Improvement is step 5, is programmed by Matlab, the parameter and stand under load feelings of parametrization setting roller line slideway auxiliary Condition, the quick force deformation error for calculating roller line slideway auxiliary under vertical load effect.
By result of calculation of the present invention compared with Finite element analysis results, relative error is shown in Fig. 6,7 within 10%.It is limited During meta analysis, first three-dimensional CAD hypostazation model is imported into finite element software and is converted into FEM model；To FEM model When carrying out mesh generation, using multi grid integration method, in grid interpolation, elder generation's interpolation goes out even-numbered nodes number, then recycles The unusual several sections of points of even-numbered nodes number interpolation；The material properties and constraints of each part are defined, apply vertical load, are carried out quiet Mechanical analysis, solve the force deformation error of roller line slideway auxiliary.
Understand, the present invention can accurately calculate force deformation error of the roller line slideway auxiliary under vertical load, mould The parametrization application of type, computational efficiency is greatly improved, and there is certain applicability.
Brief description of the drawings
Fig. 1 is patented method implementing procedure figure.
Fig. 2 is force analysis of the Four-slider line slideway auxiliary under vertical load effect.
Fig. 3 is elastic roller and plane contact schematic diagram.
Fig. 4 analyzes for roller line slideway auxiliary contact force.
Fig. 5 is Digit Control Machine Tool and roller line slideway auxiliary model.
Fig. 6 is the finite element analysis of roller line slideway auxiliary.
Fig. 7 is model calculation value figure compared with FEM calculation value.
Embodiment
Using Four-slider workbench roller line slideway auxiliary as research object, the power calculated under vertical load effect becomes the present invention Shape error, verify the practicality of the present invention.
Step 1：Threedimensional model is established, calculates external force suffered by roller line slideway auxiliary.
Using base rail sub-center as the origin of coordinates, the threedimensional model of the lathe upper part of the body is established, as shown in Fig. 5.Set zero The material of the material of part, guide rail and sliding block is GCr15, and isotropism, homogeneous media, the density of material is 7800kg/m3；It is vertical The body element such as post and crossbeam material HT200, density 7200kg/m3。
Using qualitative attribute, it is 1731.41kg to obtain lathe upper part of the body mass M；The lathe upper half that then base rail is born Body gravity G is 16967.82N.
By three-dimensional software, measuring lathe upper part of the body center of gravity (unit mm) is：
xG, yG, zG)=(6,235.17,806.84)
Step 2：The external load calculation of each sliding block of guideway.
Assuming that the power being subject on each sliding block is Fi(l=1,2,3,4), as shown in Fig. 2 the gravity G of the lathe upper part of the body, matter Heart position is (xG, yG, zG), according to the equilibrium condition of space force system, Σ Fz=0, Σ Mx=0, Σ My=0.FzRepresent guideway In the stress of Z-direction；Mx、MyRepresent torque of the guideway in X Y-directions.
List following equilibrium equation：
Wherein, L1Represent the sliding block of workbench two in the distance of X-direction, L2Represent the distance of two sliding blocks in the Y direction.Have in formula Four unknown quantitys, three equations.Because using workbench and sliding block as rigid body, rolling element is as elastomer here.Observe work Make displacement of the platform along reference axis Z-direction, before and after worktable displacement, with the contact point of sliding block 1,2,3,4 all the time in one plane, then The deflection ω of each sliding blockiRelation is：
ω3-ω1=ω4-ω2 (12)
The linear rolling guide preload for being generally used for workbench is bigger, it is believed that its rigidity is linear.So：
F3 F1=F4 F2 (13)
Known L1=630mm, L2=505mm, try to achieve four sliding block bearing capacities：F1=8048.25N, F2=8337.35N, F3=285.96N, F4=296.25N.Wherein FiRepresent the power that i-th of sliding block is subject to.
Step 3：Single roller line slideway contact analysis.
The parameter for carrying out roller line slideway auxiliary first determines that roller line slideway auxiliary GZB35 parameters as shown in table 1 are：
The roller line slideway auxiliary GZB35 of table 1 basic parameter
With improved Palmgren formula, the elastic deformation amount of the i-th row roller：
The then contact normal load between roller body and plane：
Step 4：Consider the influence of interior external applied load, calculate the force deformation error of roller line slideway auxiliary.
Consider the dynamic balance of roller line slideway auxiliary origin of coordinates under radial load F effects, can obtain
(Q1cosθ+Q2cosθ-Q3cosθ-Q4cosθ)nc=F (15)
Then the relational expression between sliding block radial load and displacement is：
It is the deformation that research object analyzes contact area with sliding block 1, it is known that the radial load F suffered by sliding block 11For 8048.25N θ, ne、le、P0As shown in table 1, δ0It can be tried to achieve by formula (7), and then calculate δV=0.94 μm, as roller line Force deformation error of the guideway under vertical load effect.
Step 5：Parameterized variables, programmed using Matlab, effective computing power distortion inaccuracy.
The present invention can be programmed with Matlab, quick to calculate by each variable parameter of studied roller line slideway auxiliary Force deformation error condition under vertical load effect, to calculate every error of ball line slideway auxiliary, improve adding for lathe Work precision provides more easily method.
The present invention considers the effect of Four-slider line slideway auxiliary external force and the elastic contact force between roller and groove, carries A kind of force deformation error calculation method of roller line slideway auxiliary under vertical load effect is gone out.It is soft with three-dimensional CAD first Part establishes threedimensional model, estimates external force suffered by roller line slideway auxiliary.Recycle mechanical balance equation and deformation compatibility condition, meter Calculate the bearing capacity of each sliding block in Four-slider workbench.Established followed by contact theory and improved Palmgren empirical equations Relation between roller, the normal load in trench contact face and juxtaposition metamorphose amount, then establish sliding block bearing capacity and deformed with sliding block Measure model.The influence of interior external applied load is considered again, calculates the force deformation error of roller line slideway auxiliary.Also Matlab can be passed through Parameterize each variable, quick computing power distortion inaccuracy, can be clearly seen that proposed by the present invention can relatively accurately count Force deformation error of the roller line slideway auxiliary under vertical load effect is calculated, has frontier nature work to the precision for improving line slideway auxiliary With.

Claims (1)

  1. A kind of 1. force deformation error calculation method of roller line slideway auxiliary under vertical load, with load types such as four directions of classics Roller line slideway auxiliary is basic research object, by parametric modeling and program composition, is calculated under vertical load effect Force deformation error；
    It is characterized in that：
    Step 1：Threedimensional model is established, calculates external force suffered by roller line slideway auxiliary
    Modeling is digitized to the part on roller line slideway auxiliary with Three-dimensional CAD Software, and sets each part Material properties, using qualitative attribute button, calculate the gravity and position of centre of gravity suffered by guideway；
    Step 2：The force analysis of each sliding block of guideway
    In the analysis to roller line slideway auxiliary external load, using workbench and sliding block as rigid body, rolling element is as bullet Property body is studied；Using the load type roller line slideway auxiliary such as four directions as research object, common Four-slider workbench rolling is analyzed The stressing conditions of each sliding block in post line slideway auxiliary；According to the equilibrium condition and deformation compatibility condition of power system, with line slideway Space coordinates O is established in secondary center for origin1XYZ, equilibrium equation is listed, obtain the power acted on each sliding block；
    Step 3：The contact analysis of single roller-groove
    The known a diameter of R of two elastic cylinders, length le, then under load Q effects, it is 2a, length l that contact, which produces width,e's Rectangular contact area；
    <mrow> <mi>a</mi> <mo>=</mo> <msqrt> <mrow> <mn>4</mn> <mi>R</mi> <mi>Q</mi> <mo>/</mo> <mrow> <mo>(</mo> <msub> <mi>&amp;pi;l</mi> <mi>e</mi> </msub> <msup> <mi>E</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    <mrow> <mfrac> <mn>1</mn> <msup> <mi>E</mi> <mo>&amp;prime;</mo> </msup> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msubsup> <mi>&amp;mu;</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msubsup> <mi>&amp;mu;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    Any point x of the contact area along cylinder radial direction Hertz contact stress is：
    <mrow> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>Q</mi> </mrow> <mrow> <msub> <mi>&amp;pi;a</mi> <mn>1</mn> </msub> <msub> <mi>l</mi> <mi>e</mi> </msub> </mrow> </mfrac> <msqrt> <mrow> <mn>1</mn> <mo>-</mo> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>/</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    o1, o2Point between Elastic approach be：
    <mrow> <mi>&amp;delta;</mi> <mo>=</mo> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mi>Q</mi> </mrow> <mrow> <msup> <mi>&amp;pi;E</mi> <mi>l</mi> </msup> <msub> <mi>l</mi> <mi>e</mi> </msub> </mrow> </mfrac> <mo>&amp;lsqb;</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>4</mn> <mi>R</mi> </mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mi>l</mi> <mi>n</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>4</mn> <mi>R</mi> </mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> <mo>&amp;rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    Relation between elastic deformation and load is shown below：
    <mrow> <msub> <mi>&amp;delta;</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>1.36</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&amp;mu;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&amp;mu;</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mn>0.9</mn> </msup> <mfrac> <msubsup> <mi>Q</mi> <mi>i</mi> <mn>0.9</mn> </msubsup> <msubsup> <mi>l</mi> <mi>e</mi> <mn>0.74</mn> </msubsup> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mi>D</mi> </mfrac> <mo>)</mo> </mrow> <mn>0.1</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Wherein：
    δi--- the elastic deformation amount of the i-th row roller；Qi--- the contact surface normal load of the i-th row roller；
    E1、E2--- the modulus of elasticity of two contacts；The equivalent elastic modulus of E ' --- contact；
    μ1、μ2--- the Poisson's ratio of two contacts；The equivalent Poisson's ratio of μ --- contact；
    a1、a2--- the width of two contact areas；le--- the contact length of two rollers；
    The formula (5) shows that Elastic approach is unrelated with the diameter of roller and raceway, and this does not obviously square with the fact；In order that Palmgren formula are applied to the situation of roller diameter change, introduce the fit term of δ~D curve；
    <mrow> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1.36</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&amp;mu;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&amp;mu;</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mn>0.9</mn> </msup> <mfrac> <msubsup> <mi>Q</mi> <mi>i</mi> <mn>0.9</mn> </msubsup> <msubsup> <mi>l</mi> <mi>e</mi> <mn>0.2</mn> </msubsup> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mi>R</mi> <mi>D</mi> </mfrac> <mo>)</mo> </mrow> <mn>0.1</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    K is an item undetermined, and it is equivalent to roller diameter, in the case where load is certain, in order that the result phase of numerical computations Together, then roller diameter D and length leIt must is fulfilled for certain relation：
    Substitution formula (6), is obtained：
    <mrow> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1.36</mn> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&amp;mu;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> <msub> <mi>E</mi> <mn>1</mn> </msub> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>1</mn> <mo>-</mo> <msup> <msub> <mi>&amp;mu;</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> <msub> <mi>E</mi> <mn>2</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mn>0.9</mn> </msup> <mfrac> <msubsup> <mi>Q</mi> <mi>i</mi> <mn>0.9</mn> </msubsup> <msubsup> <mi>l</mi> <mi>e</mi> <mn>0.24</mn> </msubsup> </mfrac> <msup> <mrow> <mo>(</mo> <mfrac> <mn>1</mn> <mi>D</mi> </mfrac> <mo>)</mo> </mrow> <mn>0.1</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    Step 4：Consider the influence of interior external applied load, calculate the force deformation error of roller line slideway auxiliary；
    If the numbering 1,2,3,4 of 4 row rollers in single sliding block, coordinate system O is established in symmetrical centre point2XYZ, each rolling of the i-th row The normal load Q for the slider contact surface that post is subject toi, roller elastic deformation amount δi；Pretightning force effect under, each row roller it is initial Compress variation δ0, under Radial Loads, the deformation of contact zone is δV, the angle of Contact normal direction and vertical direction is θ；
    Consider the dynamic balance of roller line slideway auxiliary origin of coordinates under radial load F effects, obtain
    (Q1cosθ+Q2cosθ-Q3cosθ-Q4cosθ)ne=F (8)
    Wherein, neRepresent to contact ball number simultaneously in roller line slideway auxiliary；
    The Q according to the characteristics of the load type roller line slideway auxiliary such as four directions1=Q2, Q3=Q4, then：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>&amp;delta;</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>&amp;delta;</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>&amp;delta;</mi> <mi>V</mi> </msub> <mi>cos</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&amp;delta;</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>&amp;delta;</mi> <mn>4</mn> </msub> <mo>=</mo> <msub> <mi>&amp;delta;</mi> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>&amp;delta;</mi> <mi>V</mi> </msub> <mi>cos</mi> <mi>&amp;theta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
    Aggregative formula (7) (8) (9), obtains the relation between sliding block radial load and displacement：
    F=f (δV) (10)
    F is obtained by external applied load force analysis, displacement δ of the sliding block under load effect is obtained by formula (8)V, this displacement is The force deformation error for being roller line slideway auxiliary under vertical load.
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