CN105760662A - Machine tool machining precision reliability and sensitivity analyzing method based on quick Markov chain - Google Patents

Abstract

The invention discloses a machine tool machining precision reliability and sensitivity analyzing method based on a quick Markov chain.According to the method, a whole space error model of a machine tool is built through a multi-body system theory on the basis of a topological structure of the machine tool, and then the machine tool machining precision reliability is analyzed according to a quick Markov chain stimulation method; due to the fact that geometric errors of the machine tool are correlate to one another, firstly, the machine tool machining precision reliability is defined, seven failure modes of the machine tool are built, and the correlated geometric errors are converted into independent standard normal random variables; secondly, the failure probability of the machining precision is estimated through a reliability analyzing method for an independent space, and on the basis of a failure probability method for integral, machine tool machining precision reliability and sensitivity analyzing is achieved, and the main geometric errors influencing the machining precision reliability are obtained; lastly, the validity of the method is proved by taking a four-shaft machine tool as an example, and multi-shaft machine tool machining precision reliability and sensitivity analyzing can be performed through the method.

Description

A kind of machine finish Reliability Sensitivity Method based on quick Markov chain

Technical field

The invention provides a kind of machine finish Reliability Sensitivity Method based on quick Markov chain, belong to machine tool accuracy design field.

Background technology

Multi-axis NC Machine Tools is typical electromechanical integration equipment, has higher added value and is widely applied, and this is the performance needing to improve lathe.Machining accuracy is the key of the quality of engineering goods and performance, is the key factor needing to consider of any manufacturer.Affect the factor of machining accuracy, such as kinematic error, Thermal Error, the error that cutting force causes, servo error and tool wear error etc..In various source of errors, the geometric error of lathe is one of maximum error source, accounts for more than 40%.The machining accuracy how improving Digit Control Machine Tool has become the hot issue of Chinese scholars research.

The main accuracy of manufacture deriving from its functional part of the geometric error of lathe also has installation accuracy equal error.In order to improve the machining accuracy of Digit Control Machine Tool, error modeling is the key improving machine tool capability.That error modeling technology can provide a system, suitable method is to set up error model.In recent years, substantial amounts of research work is being studied gang tool error modeling always and is completed.And from different angles geometric error having been modeled, modeling method rough classification is as follows: triangle relation modeling, the error moments tactical deployment of troops, secondary relational model method, theory of mechanisms modeling, rigid body kinematics method, Multi-system modeling method etc..Wherein, Multi-system modeling has become the best approach of lathe geometrical error modeling

Along with requirement on machining accuracy is more and more higher, the reliability of machining accuracy also becomes a measurement index of lathe.But, current lathe reliability consideration is concentrated mainly on intensity and life-span aspect, seldom the machining accuracy aspect of lathe is studied.As it has been described above, through long-term operation, machining accuracy can reduce, it is impossible to meet the specification of lathe.In the reason that machining accuracy declines, along with abrasion and the deformation of contact surface and structure, the increase of geometric error is a main cause.

The geometric error of lathe is mainly derived from the site error and the error of perpendicularity that manufacture or assemble defect and each moving component of each moving component of lathe.Due to the randomness manufactured and assemble, therefore, geometric error is different in different positions.For the situation that geometric error is uncertain variables, the definition giving machining accuracy reliability is as follows: the reliability of machining accuracy be a kind of within the given time, perform the ability of machining accuracy of its regulation when regulation.In general, Space processing error can be analyzed to tri-axles of X, Y, Z, if machining accuracy is respectively smaller than regulation requirement in X, Y, Z-direction, then can be regarded as machining accuracy and is unsatisfactory for requiring in the direction.

How improving the machining accuracy of lathe, improve the machining accuracy of lathe, be manufacturer and the required important goal considered of user, the important method completing this work is as follows: one is the reliability of the mechanical precision how expressing and measuring lathe；Another kind is how to determine the crucial geometric error that machining accuracy is had the greatest impact, the machining accuracy reliability of each failure mode.During this investigation it turned out, by the method for Failure Probability Integration, sensitivity analysis is used for providing information for reliability design.

The geometric error of Digit Control Machine Tool is to intercouple, for instance, the geometric error of three axle lathes is 21, and they influence each other.Therefore, finding out the crucial geometric error affecting machine finish reliability is a very difficult job.Fortunately, susceptibility assays can be used to identify that the uncertain of geometric error changes the impact on machining accuracy reliability and identify the crucial geometric error item that various failure modes are had the greatest impact.In the present invention, in order to extract the crucial geometric error affecting machining accuracy, on theory of multi body system basis, a kind of machine finish Reliability Sensitivity Method based on quick Markov chain is suggested.

Many research work have been carried out the precision sensitivity analysis of relevant lathe or other mechanisms.At present, Local sensitivity analysis method and global sensitivity analysis method all have this well to apply in respective evolution, however in machine finish reliability sensitive analysis problem, the but solution of neither one comparison system.Patent of the present invention is based on this kind of starting point, it is proposed that a kind of machine finish Reliability Sensitivity Method based on quick Markov chain.The advantages such as the method has steadily and surely, calculate efficiently and the sample number that needs is relatively low.The method can be used to extract the crucial geometric error item that machine finish impact is bigger.By example calculation it is shown that method proposed by the invention is effective.

So, set up accurate error model and just seem particularly significant, how utilizing error model to analyze accurately the error term that machine finish reliability effect is bigger is one of key issue in the present invention.

Summary of the invention

It is an object of the invention to provide a kind of machine finish Reliability Sensitivity Method based on quick Markov chain, pass through theory of multi body system, on the basis of the topological structure of lathe, set up the spatial error model that lathe is overall, machine finish fail-safe analysis is carried out according to quick Markov chain emulation mode, owing to the geometric error of lathe is to be mutually related, first, machine finish is defined, set up seven kinds of fault modes of lathe, relevant geometric error is converted into independent standard normal random variable, then with the analysis method for reliability of separate space, the failure probability of machining accuracy is estimated.Failure probability method based on integration, achieve the machining accuracy reliability sensitivity analysis of lathe, obtain the main geometric error affecting machining accuracy reliability, the advantages such as the design for precise numerical control machine provides important theoretical foundation, and the method that the present invention proposes has steadily and surely, to calculate sample number that is efficient and that need relatively low.Finally, carry out proving the effectiveness of method proposed by the invention with four axle lathe actually examples, and this invention can carry out gang tool machining accuracy reliability sensitivity analysis.

For achieving the above object, the technical solution used in the present invention is a kind of machine finish Reliability Sensitivity Method based on quick Markov chain, for solving the technical problem in Digit Control Machine Tool machining accuracy reliability sensitivity analysis process.The method to realize process as follows.

Step one sets up the space error modeling of lathe according to theory of multi body system

Root theory of multi body system ultimate principle of the present invention, by abstract for each motion parts of lathe be the vector form of 4 × 1；By forms of motion and synthetic error modularized processing, set up the spatial synthesis error model of lathe according to the topological structure of lathe.

Step 1.1 machine tool structure figure

During this investigation it turned out, four-shaft numerically controlled lathe, its 3 d structure model is as it is shown in figure 1, its important technological parameters is listed in table 1, and its geometric error item has been listed in table 2.

Step 1.2 topological structure and geometric error

Four axle lathes have 4 moving components, and each parts move relative to each other, and cutter and workpiece are fixed on lathe.Table 3 describes the restriction of the degree of freedom between each unit, and wherein " 0 " refers to do not have degree of freedom and " 1 " to refer to there is one degree of freedom.

Based on theory of multi body system, each moving component of lathe, being conceptualized as topological structure, as in figure 2 it is shown, four axle lathes are described as in a topological structure with double; two branch, the first branch is by lathe bed, Y-axis moving component, X-axis moving component and cutter.Second branch is by lathe bed, Z axis moving component, A axle moving component and workpiece.Lathe bed is inertial reference system, uses B₀Representing that the order according to growth naturally numbers in order, table 4 is the lower body array of selected precise horizontal machining center.

One rigid body has 6 degree of freedom.6 coordinates uniquely specify rigid body position in three dimensions.Four axle lathes have 4 moving components, move relative to each other, and other 2 bodies being fixed on lathe are cutter and workpiece.Each moving component has 6 geometric errors, Δ x_h, Δ y_h, Δ z_h, Δ α_h, Δ β_hWith Δ γ_h。Δx_h, Δ y_hWith Δ z_hBelong to translational error.Δα_h, Δ β_hWith Δ γ_hBelong to rotation error, under be designated as the direction of motion, subscript letter h will take following letter respectively: X, Y, Z and A, between each kinematic axis, has kinematic error between 5 bodies, Δ γ_XY, Δ β_XZ, Δ α_YZ, Δ γ_YAWith Δ β_ZA.Utilize theory of multi body system by abstract for each moving component of lathe be one group of rigid body time, four axle lathes have 24 geometric errors relevant to position, kinematic error between 5 independent bodies, and above-mentioned error has been listed in table 2 all.

Step 1.3 generalized coordinates is arranged and eigenmatrix

In order to make machine tool accuracy modeling more convenient, it is necessary to coordinate system is carried out special agreement.Used herein of agreeing as follows: (1) establishes rectangular coordinate system, for all of inertance element and moving component.These coordinate systems are generalized coordinates systems, and inertial coodinate system is called coordinate system, and the coordinate system of other motions is called coordinate system.(2) X of each coordinate system, Y, Z axis should be parallel.

According to multi-body system ultimate principle, the eigenmatrix of selected machining center is listed in table 5.

Assuming that cutter becomes the form point coordinate in tool coordinate system to be:

P_t=[P_tx,P_ty,P_tz,1]^T(1)

Component shaping point coordinate in workpiece coordinate system can be expressed as follows:

P_w=[P_wx,P_wy,P_wz,1]^T(2)

Ideally, lathe does not have error, cutter to become form point and component shaping point to coincide together, and under ideal conditions, precision machined constraint equation can be expressed as follows:

$\[\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\Π}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\]P_t=\[\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\Π}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\]P_{wideal}---\left(3\right)$

By variation, formula (3) is written as following form:

$P_{wideal}={\[\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\Π}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\]}^{-1}\[\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\Π}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\]P_t---\left(4\right)$

Machining accuracy be last with actual building motion time, cutter become the relative position error between form point and component shaping point be correlated with.In practical situations both, precision machined constraint equation is write as following formula:

$\begin{array}{c}{P}_{wactual}={\[\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\Π}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{\mathrm{\ΔM}}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}{\mathrm{\ΔM}}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\]}^{-1}\\ \×\[\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\Π}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{\mathrm{\ΔM}}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}{\mathrm{\ΔM}}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\]P_t\end{array}---\left(5\right)$

The general space error caused by the gap between form point and ideal forming point is become to be written as by reality:

$\begin{array}{c}E=\[\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\Π}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{\mathrm{\ΔM}}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}{\mathrm{\ΔM}}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\]P_{wideal}\\ -\[\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\Π}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{\mathrm{\ΔM}}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}{\mathrm{\ΔM}}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\]P_t\end{array}---\left(6\right)$

By the synthetic error model of the precise horizontal machining center that table 4 eigenmatrix and formula (6) obtain.Equally, general lathe spatial error model is established as follows:

E=E (G, P_t,P)(7)

In formula (7), E=[E_X,E_Y,E_Z,0]^TRepresent space error vector；G=[g₁,g₂,…,g₂₁]^TRepresent the vector being made up of 29 geometric errors, be written as Δ x_X,Δy_X,Δz_X,Δα_X,Δβ_X,Δγ_X,Δx_Y,Δy_Y,Δz_Y,Δα_Y,Δβ_Y,Δγ_Y,Δx_Z,Δy_Z,Δz_Z,Δα_Z,Δβ_Z,1Δγ_Z,Δx_A,Δy_A,Δz_A,Δα_A,Δβ_A,Δγ_A,Δγ_XY,Δβ_XZ,Δα_YZ,Δγ_YA,Δβ_ZA=g₁,g₂,g₃,g₄,g₅,g₆,g₇,g₈,g₉,g₁₀,g₁₁,g₁₂,g₁₃,g₁₄,g₁₅,g₁₆,g₁₇,g₁₈,g₁₉,g₂₀,g₂₁,g₂₂,g₂₂,g₂₃,g₂₄,g₂₅,g₂₆,g₂₇,g₂₈,g₂₉；P=[x, y, z, 0]^TRepresent the position vector of the kinematic axis at lathe center.

Step 2 is based on the machining accuracy fail-safe analysis of Markov chain

2.1 machining accuracy reliability definition

Along with requirement on machining accuracy is more and more higher, the reliability of machining accuracy also becomes a measurement index of lathe.But, current lathe reliability consideration is concentrated mainly on intensity and life-span aspect, seldom the machining accuracy aspect of lathe is studied.As it has been described above, through long-term operation, machining accuracy can reduce, it is impossible to meet the specification of lathe.In the reason that machining accuracy declines, along with abrasion and the deformation of contact surface and structure, the increase of geometric error is a main cause.

The geometric error of lathe is mainly derived from the site error and the error of perpendicularity that manufacture or assemble defect and each moving component of each moving component of lathe.Due to the randomness manufactured and assemble, therefore, geometric error is different in different positions.For the situation that geometric error is uncertain variables, give the definition of machining accuracy reliability.

The reliability of machining accuracy be a kind of within the given time, perform the ability of machining accuracy of its regulation when regulation.In general, Space processing error can be analyzed to tri-axles of X, Y, Z, if machining accuracy is respectively smaller than regulation requirement in X, Y, Z-direction, then can be regarded as machining accuracy and is unsatisfactory for requiring in the direction.

2.2 fault modes and failure probability

The Synthetic Volumetric Error Model of machining center is expressed as:

E=E (G)=[E_X(G),E_Y(G),E_Z(G),0]^T(8)

Assuming that the maximum allowable space error of lathe is e=(e_X,e_Y,e_Z,0)^T, wherein e_X,e_Y,e_ZIt is illustrated respectively in the maximum allowable space error in X-, Y-, Z-direction, as follows with function representation:

$F=\[E-e\]={\[E_X\left(G\right)-e_X,E_Y\left(G\right)-e_Y,E_Z\left(G\right)-e_Z,0\]}^T=\left[\begin{array}{c}{H}_X\left(G\right)\\ H_Y\left(G\right)\\ H_Z\left(G\right)\\ 0\end{array}\right]---\left(9\right)$

The machining accuracy of Digit Control Machine Tool has seven kinds of failure modes as follows:

M₁={ H_X≥0,H_Y≤0andH_Z≤0}(10)

M₂={ H_X≤0,H_Y≥0andH_Z≤0}(11)

M₃={ H_X≤0,H_Y≤0andH_Z≥0}(12)

M₄={ H_X≥0,H_Y≥0andH_Z≤0}(13)

M₅={ H_X≥0,H_Y≤0andH_Z≥0}(14)

M₆={ H_X≤0,H_Y≥0andH_Z≥0}(15)

M₇={ H_X≥0,H_Y≥0andH_Z≥0}(16)

In formula (10) to (12), M₁、M₂And M₃The machining accuracy representing lathe respectively is unsatisfactory for maximum allowable space error in X-, Y-and Z-direction.In formula (13) to (15), M₄、M₅And M₆The machining accuracy both direction in X-, Y-and Z-direction representing lathe respectively is unsatisfactory for maximum allowable space error.In formula (13), M₇The machining accuracy representing lathe is all unsatisfactory for maximum allowable space error in X-, Y-and Z-direction.

The inefficacy territory corresponding with every kind of failure mode is as follows:

F₁={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≤0,G∈H_Z(G)≤0}(17)

F₂={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≥0,G∈H_Z(G)≤0}(18)

F₃={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≤0,G∈H_Z(G)≥0}(19)

F₄={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≥0,G∈H_Z(G)≤0}(20)

F₅={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≤0,G∈H_Z(G)≥0}(21)

F₆={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≥0,G∈H_Z(G)≥0}(22)

F₇={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≥0,G∈H_Z(G)≥0}(23)

In machining accuracy fail-safe analysis process, failure probability P is defined as the joint probability density function f (G) of geometric error integration on the F of inefficacy territory, and therefore the failure probability of every kind of failure mode is represented as follows:

$P_F^{\left(i\right)}=P\{G\∈F_i\}=\∫...{\∫}_{F_i}f\left(G\right)dG---\left(24\right)$

Wherein, i=1,2 ... 7, i is the sequence number of failure mode.

The failure probability P that machining accuracy is total_FIt is written as follows:

$P_F=P_F^{\left(1\right)}+P_F^{\left(2\right)}+P_F^{\left(3\right)}+P_F^{\left(4\right)}+P_F^{\left(5\right)}+P_F^{\left(6\right)}+P_F^{\left(7\right)}---\left(25\right)$

2.3 by the relevant conversion of normal variate to independent standard normal variable

In the actual course of processing, the geometric error of lathe is to be mutually related.The impact of machining accuracy failure probability is very important by this dependency.So in actual analysis of Machining, in order to tally with the actual situation, it is necessary to consider the relation between the geometric error of lathe.First, relevant geometric error is converted into independent standard normal random variable.Then the failure probability of machining accuracy is obtained with the analysis method for reliability of separate space.

The n item geometric error of lathe can be write a n and be tieed up basic random variables: G=(g₁,g₂,…g_n)^T, owing to every geometric error is relevant, n ties up the joint probability density function f (G) of normal variate G and is represented as follows:

$f\left(G\right)={\left(2\mathrm{\π}\right)}^{-\frac{n}{2}}{\left|C_G\right|}^{-\frac{1}{2}}\exp \[-\frac{1}{2}{(G-{\mathrm{\μ}}_G)}^T{C}_G^{-1}(G-{\mathrm{\μ}}_G)\]---\left(26\right)$

Wherein,

$C_G=\left[\begin{array}{ccccc}{\mathrm{\σ}}_{g_1}^2& {\mathrm{\ρ}}_{g_1{g}_2}{\mathrm{\σ}}_{g_1}{\mathrm{\σ}}_{g_2}& {\mathrm{\ρ}}_{g_1{g}_3}{\mathrm{\σ}}_{g_1}{\mathrm{\σ}}_{g_3}& ...& {\mathrm{\ρ}}_{g_1{g}_n}{\mathrm{\σ}}_{g_1}{\mathrm{\σ}}_{g_n}\\ {\mathrm{\ρ}}_{g_1{g}_2}{\mathrm{\σ}}_{g_1}{\mathrm{\σ}}_{g_2}& {\mathrm{\σ}}_{g_2}^2& {\mathrm{\ρ}}_{g_2{g}_3}{\mathrm{\σ}}_{g_2}{\mathrm{\σ}}_{g_3}& ...& {\mathrm{\ρ}}_{g_2{g}_n}{\mathrm{\σ}}_{g_2}{\mathrm{\σ}}_{g_n}\\ {\mathrm{\ρ}}_{g_1{g}_3}{\mathrm{\σ}}_{g_1}{\mathrm{\σ}}_{g_3}& {\mathrm{\ρ}}_{g_2{g}_3}{\mathrm{\σ}}_{g_2}{\mathrm{\σ}}_{g_3}& {\mathrm{\σ}}_{g_3}^2& & {\mathrm{\ρ}}_{g_3{g}_n}{\mathrm{\σ}}_{g_2}{\mathrm{\σ}}_{g_n}\\ .& .& .& & .\\ .& .& .& & .\\ .& .& .& & .\\ {\mathrm{\ρ}}_{g_1{g}_n}{\mathrm{\σ}}_{g_1}{\mathrm{\σ}}_{g_n}& {\mathrm{\ρ}}_{g_2{g}_n}{\mathrm{\σ}}_{g_2}{\mathrm{\σ}}_{g_n}& {\mathrm{\ρ}}_{g_3{g}_n}{\mathrm{\σ}}_{g_2}{\mathrm{\σ}}_{g_n}& ...& {\mathrm{\σ}}_{g_n}^2\end{array}\right]---\left(27\right)$

Represent the covariance matrix of aggregate error G；For C_GInverse matrix；|C_G| represent C_GDeterminant；μ_G=(μ_g1,μ_g2,…μ_gn)^TRepresent the mean vector of geometric error, μ_giAnd σ_giRepresent geometric error g_i(i=1,2,3 ..., average n) and variance,Represent g_iAnd g_jCorrelation coefficient.

Ultimate principle according to linear algebra, there will necessarily be an orthogonal matrix A and n ties up correlated random variables G=(g₁,g₂,…g_n)^TBe converted to n and tie up uncorrelated random variables y=(y₁,y₂,…y_n)^T, have:

$f_Y\left(y\right)=f_G(A^{-1}y+{\mathrm{\μ}}_G)={\left(2\mathrm{\π}\right)}^{-\frac{n}{2}}{({\mathrm{\λ}}_1{\mathrm{\λ}}_2...{\mathrm{\λ}}_n)}^{-\frac{1}{2}}\exp (-\frac{1}{2}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}\frac{y_i^2}{{\mathrm{\λ}}_i})---\left(28\right)$

With

Y=A^T(G-μ_G), y_i～N (0, λ_i)(29)

Wherein, λ₁,λ₂,…λ_nIt is covariance matrix C_GCharacteristic root.The column vector of A orthogonal matrix is equal to covariance matrix C_GOrthogonal eigenvectors.

According to above principle, relevant n is tieed up random vector G=(g₁,g₂,…g_n)^TBe converted to n and tie up incoherent random vector y=(y₁,y₂,…y_n)^T.By following formula, independent normal variate y=(y₁,y₂,…y_n)^TIt is converted into standard independent normal variate u=(u₁,u₂,…u_n)^T, it may be assumed that

$u_i=\frac{y_i-{\mathrm{\μ}}_{y_i}}{{\mathrm{\σ}}_{y_i}}=\frac{y_i}{\sqrt{{\mathrm{\λ}}_i}},(i=1,2,...n)---\left(30\right)$

Therefore the space corresponding in inefficacy territory F (G) and constraint function H (G) is converted into inefficacy territory F (u) and constraint function H (u) standard absolute version.The failure probability of every kind of failure mode is written as following form:

$P_F^{\left(i\right)}=\∫...{\∫}_{F_i\left(G\right)}{f}_G\left(G\right)dG=\∫...{\∫}_{F_i\left(y\right)}{f}_Y\left(y\right)dy=\∫...{\∫}_{F_i\left(u\right)}{f}_U\left(u\right)d\left(u\right)---\left(31\right)$

The quick Markov chain emulation mode that 2.4 failure probabilities are estimated

Based on the method for Digital Simulation, the method having many calculating processing precision reliabilitys, it is applied not only to single failure mode fail-safe analysis and is also used for the fail-safe analysis of Multiple Failure Modes.But up to now, but without the problem of the fail-safe analysis carrying out machining accuracy with Markov chain.

Inefficacy sample point is effectively simulated, for general nonlinear limit state equation due to Markov chain $H_U\left(u\right)=H_G\left(G\right)=\left\{\begin{array}{c}{H}_X\left(G\right)=0\\ H_Y\left(G\right)=0\\ H_Z\left(G\right)=0\end{array}\right.,$ Simulation of Markov chains is utilized quickly to obtain the point that in inefficacy territory, most probable lost efficacy, i.e. design point.By design point, introduce a linear limit state equation with L (u)=0 with same design point $H_U\left(u\right)=H_G\left(G\right)=\left\{\begin{array}{c}{H}_X\left(G\right)=0\\ H_Y\left(G\right)=0\\ H_Z\left(G\right)=0\end{array}\right.,$ The equation obtains easily in standard normal space.

Based on the multiplication theorem in theory of probability, set up following 2 equations:

P{F_H∩F_L}=P{F_H}P{F_L|F_H}(32)

P{F_H∩F_L}=P{F_L}P{F_H|F_L}(33)

Wherein, F_H={ u:u → G ∈ F_i, F_L={ u:L (u)≤0}, P{F_L}=P{L (u)≤0} and P{F_H}=P{F_i}。P{F_L|F_HAnd P{F_H|F_LIt is two conditional probability.

Therefore failure probability P_FIt is expressed as form:

$P_F^{\left(i\right)}=P\left(F_i\right)=P\left\{F_H\right\}=P\left\{F_L\right\}\frac{P\left\{F_H\right|F_L\}}{P\left\{F_L\right|F_H\}}---\left(34\right)$

WhereinCharacteristic ratio factor S can be defined as,

$S=\frac{P\left\{F_H\right|F_L\}}{P\left\{F_L\right|F_H\}}---\left(35\right)$

Formula (34) is reduced to following form:

$P_F^{\left(i\right)}=P\left\{F_L\right\}\·S---\left(36\right)$

The probability density function F of inefficacy sample point_HIt is represented as following form:

$q_H\left(u\right|F_H)=\frac{I_H\left(u\right)f_U\left(u\right)}{P_H}---\left(37\right)$

Wherein, I_HU () is the indicator of non-linear behaviour function H (u),

Wherein,

$I_H\left(u\right)=\left\{\begin{array}{cc}1,& H\left(u\right)<0\\ 0,& H\left(u\right)\&GreaterEqual;0\end{array}\right.---\left(38\right)$

Ultimate principle according to Markov chain, Markov chain is passed through from a state to another state to advise distribution function f^*(ε | u) control.

There is symmetric n dimension normal distribution and n dimension be uniformly distributed and all as markovian suggestion distribution, in this article, can select have symmetric n dimension normal distribution as suggestion distribution, it may be assumed that

Wherein, ε_kAnd u_kThe respectively kth component of n-dimensional vector ε and u；l_kRepresent the n centered by u and tie up hyperpolyhedron u_kThe length of side in direction, l_kDecide next sample and deviate the maximum allowable range of current sample..

Based on practical engineering experience and numerical method, select inefficacy territory F_HInterior 1 u₀As markovian original state.Markovian jth state u_j, it is at preceding state u_j-1On basis, it is proposed that substep and Metropolis-Hastings criterion are determined.First by suggestion distribution f^*(ε|u_j-1) produce alternative state ε.

Then, alternative state conditional probability density function q (ε | F_H) it is expressed as with the ratio of the conditional probability density function of Markov Chain preceding state:

R=q (ε | F_H)/q(u_j-1|F_H)(40)

Finally, jth state u is determined according to Metropolis-Hastings criterion_j:

$u_j=\left\{\begin{array}{cc}\mathrm{\&epsiv;},& min\{1,r\}>random\&lsqb;0,1\&rsqb;\\ u_{j-1},& min\{1,r\}\&le;random\&lsqb;0,1\&rsqb;\end{array}\right.----\left(41\right)$ In formula, random [0,1] represents [0,1] interval equally distributed random number.

Repeat above method, N_HIndividual Markov Chain stateIt is obtained, as inefficacy territory F_HMiddle probability density is q_H(u|F_H) condition sample point.

From inefficacy territory F_HN_HSelecting in individual sample point, employing probability density function is q_H(u|F_H) Markov Chain simulate inefficacy territory F_HN_HIndividual sample point, therefrom obtains F in standard normal space_HThe approximate maximum likelihood point in region $u^{*}=(u_1^{*},u_2^{*},...,u_n^{*}).$

In standard normal space, with F_HRegion has the linear limit state equation of same pole maximum-likelihood point to be expressed as follows:

L (u)=(0-u^*)(u-u^*)^T=0 (42)

Corresponding failure probability is:

$P\left\{F_L\right\}=\mathrm{\&Phi;}(-\sqrt{{\left(u_1^{*}\right)}^2+{\left(u_2^{*}\right)}^2+...+{\left(u_n^{*}\right)}^2})---\left(43\right)$

In formula, the distribution function that Φ () is standard normal variable.

By N_HSample point substitutes into formula (42), falls into region F_L={ u:L (u)≤0} sample size is designated as N_L|H。

Conditional probability P{F_L|F_HEstimated value obtained by following formula, it may be assumed that

$\hat{P}\left\{F_L\right|F_H\}=\frac{N_{L|H}}{N_H}---\left(44\right)$

In like manner, conditional probability P{F_H|F_LAlso there is Markov Chain simulation F_LThe sample in region obtains, F_LThe joint probability density function of zone sample point can be expressed as follows:

$q_L\left(u\right|F_L)=\frac{I_L\left(u\right)f_U\left(u\right)}{P_L}---\left(45\right)$

N in inefficacy territory is obtained by Markov Chain simulation_LSample point.These sample points brought into H (u) and calculates the value of H (u) respectively, calculating sample point and fall into inefficacy territory F_H={ quantity of u:H (u)≤0}, is designated as N_H|L。

Conditional probability P{F_L|F_HEstimated value through type (46) obtain, characteristic ratio factor S through type (47) obtains,

Then have:

$\hat{P}\left\{F_H\right|F_L\}=\frac{N_{H|L}}{N_L}---\left(46\right)$

$\hat{S}=\frac{\hat{P}\left\{F_H\right|F_L\}}{\hat{P}\left\{F_L\right|F_H\}}=\frac{N_{H|L}}{N_L}\&CenterDot;\frac{N_H}{N_{L|H}}---\left(47\right)$

Owing to lathe has Multiple Failure Modes, the failure probability of each failure mode should be calculated respectively.

Another F_H=F_i, i=1,2 ... 7, then withAnd S⁽ⁱ⁾The failure probability through type (48) of corresponding failure mode respectively obtains, it may be assumed that

$P_F^{\left(i\right)}=P\left\{F_L^{\left(i\right)}\right\}\&CenterDot;S^{\left(i\right)},i=1,2,...7---\left(48\right)$

The composite failure probability of machining accuracy is expressed as:

${\hat{P}}_F=P_F^{\left(1\right)}+P_F^{\left(2\right)}+P_F^{\left(3\right)}+P_F^{\left(4\right)}+P_F^{\left(5\right)}+P_F^{\left(6\right)}+P_F^{\left(7\right)}---\left(49\right)$

Step 3 is based on the machining accuracy reliability sensitivity analysis of Failure Probability Integration

Machining accuracy reliability sensitivity coefficient is commonly defined as the failure probability of the every kind of failure mode partial derivative to the probability distribution of the geometric error parameter of kth item, is expressed as form:

$S_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}=\&Integral;...{\&Integral;}_{F_i}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}dG---\left(50\right)$

$S_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}=\&Integral;...{\&Integral;}_{F_i}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}dG---\left(51\right)$

In formula, i=1,2 ..., 7；K=1,2 ..., n, n is the number of geometric error. μ_kRepresent the average of kth item geometric error.σ_kRepresent the standard deviation of kth item geometric error.Representing in i-th kind of failure mode, failure probability is to mean μ_kMachining accuracy reliability sensitivity coefficient.Represent in i-th kind of failure mode, the failure probability machining accuracy reliability sensitivity coefficient to standard deviation.

The definition of regularization reliability sensitivity coefficient is as follows:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}\frac{{\mathrm{\&sigma;}}_k}{P_F^{\left(i\right)}}---\left(52\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}\frac{{\mathrm{\&sigma;}}_k}{P_F^{\left(i\right)}}---\left(53\right)$

Formula (52) and formula (53) are converted to integrated form as follows:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\left(\frac{f\left(G\right)}{P_F^{\left(i\right)}}\right)dG---\left(54\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}\left(\frac{f\left(G\right)}{P_F^{\left(i\right)}}\right)dG---\left(55\right)$

Obviously, formula 54 and 53 can be expressed as at inefficacy territory F_iMathematic expectaion:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=E_{F_i}\&lsqb;\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\&rsqb;---\left(56\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=E_{F_i}\&lsqb;\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}\&rsqb;---\left(57\right)$

In formula,Represent inefficacy territory F_iMathematic expectaion.

By formula (29) and formula (30), sample pointCan be converted intoWillSubstituting in equation below, normalized reliability sensitivity coefficient can obtain, it may be assumed that

${\widehat{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{1}{N_H}\underset{0}{\overset{N_H-1}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}---\left(58\right)$

${\widehat{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{1}{N_H}\underset{0}{\overset{N_H-1}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}---\left(59\right)$

Therefore, general reliability sensitivity coefficient is as follows:

$S_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}={\widehat{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}\&CenterDot;\frac{P_F^{\left(i\right)}}{{\mathrm{\&sigma;}}_k}---\left(60\right)$

$S_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}={\widehat{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}\&CenterDot;\frac{P_F^{\left(i\right)}}{{\mathrm{\&sigma;}}_k}---\left(61\right)$

Accompanying drawing explanation

The structure chart of Fig. 1 precise horizontal machining center.

The topological diagram of Fig. 2 precise horizontal machining center.

The distribution of Fig. 3 error measure point.

Fig. 4 processing site picture.

Fig. 5 is the implementing procedure figure of this method.

Detailed description of the invention

Example: for four-axle linked numerical control machine tool (Fig. 1)

Step one sets up the space error modeling of lathe according to theory of multi body system

The present invention according to theory of multi body system ultimate principle, by abstract for each motion parts of lathe be the vector form of 4 × 1；By forms of motion and synthetic error modularized processing, set up the spatial synthesis error model of lathe according to the topological structure of lathe.

Step 1.1 machine tool structure figure

During this investigation it turned out, four-shaft numerically controlled lathe, its 3 d structure model is as it is shown in figure 1, its important technological parameters is listed in table 1, and its geometric error item has been listed in table 2.

Step 1.2 topological structure and geometric error

Four axle lathes have 4 moving components, it is possible to move relative to each other, and cutter and workpiece are fixed on lathe.Table 3 describes the restriction of the degree of freedom between each unit, and wherein " 0 " refers to do not have degree of freedom and " 1 " to refer to there is one degree of freedom.

Based on theory of multi body system, each moving component of lathe, it is possible to be conceptualized as topological structure, as shown in Figure 2, four axle lathes can be described as in a topological structure with double; two branch, and the first branch is by lathe bed, Y-axis moving component, X-axis moving component and cutter.Second branch is by lathe bed, Z axis moving component, A axle moving component and workpiece.Lathe bed is inertial reference system, uses B₀Representing that the order according to growth naturally numbers in order, table 4 is the lower body array of selected precise horizontal machining center.

One rigid body has 6 degree of freedom.6 coordinates uniquely specify rigid body position in three dimensions.Four axle lathes have 4 moving components, it is possible to move relative to each other, and other 2 bodies being fixed on lathe are cutter and workpiece.Each moving component has 6 geometric errors, Δ x_h, Δ y_h, Δ z_h, Δ α_h, Δ β_hWith Δ γ_h。Δx_h, Δ y_hWith Δ z_hBelong to translational error.Δα_h, Δ β_hWith Δ γ_hBelong to rotation error, under be designated as the direction of motion, subscript letter h will take following letter respectively: X, Y, Z and A, between each kinematic axis, has kinematic error between 5 bodies, Δ γ_XY, Δ β_XZ, Δ α_YZ, Δ γ_YAWith Δ β_ZA.Utilize theory of multi body system by abstract for each moving component of lathe be one group of rigid body time, four axle lathes have 24 geometric errors relevant to position, and kinematic error between 5 independent bodies, all these errors have been listed in table 2.

Step 1.3 generalized coordinates is arranged and eigenmatrix

In order to make machine tool accuracy modeling more convenient, it is necessary to coordinate system is carried out special agreement.Used herein of agreeing as follows: (1) establishes rectangular coordinate system, for all of inertance element and moving component.These coordinate systems are generalized coordinates systems, and inertial coodinate system is called coordinate system, and the coordinate system of other motions is called coordinate system.(2) X of each coordinate system, Y, Z axis should be parallel.

According to multi-body system ultimate principle, the eigenmatrix of selected machining center is listed in table 5.

Assuming that cutter becomes the form point coordinate in tool coordinate system to be:

P_t=[P_tx,P_ty,P_tz,1]^T(62)

Component shaping point coordinate in workpiece coordinate system can be expressed as follows:

P_w=[P_wx,P_wy,P_wz,1]^T(63)

Ideally, lathe does not have error, cutter to become form point and component shaping point to coincide together, and under ideal conditions, precision machined constraint equation can be expressed as follows:

$\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\&Pi;}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t=\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\&Pi;}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;P_{wideal}---\left(64\right)$

By variation, formula (3) can be written as following form:

$P_{wideal}={\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\&Pi;}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;}^{-1}\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\&Pi;}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t---\left(65\right)$

Machining accuracy be last with actual building motion time, cutter become the relative position error between form point and component shaping point be correlated with.In practical situations both, it is possible to write precision machined constraint equation as following formula:

$\begin{array}{c}{P}_{wactual}={\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\&Pi;}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;}^{-1}\\ \&times;\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\&Pi;}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t\end{array}---\left(66\right)$

The general space error caused by the gap between form point and ideal forming point is become to be written as by reality:

$\begin{array}{c}E=\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\mathrm{\&Pi;}}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;P_{wideal}\\ -\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\mathrm{\&Pi;}}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t\end{array}---\left(67\right)$

Synthetic error model by table 4 eigenmatrix and formula (67) available precise horizontal machining center.Equally, general lathe spatial error model can be established as follows:

E=E (G, P_t,P)(68)

In formula (68), E=[E_X,E_Y,E_Z,0]^TRepresent space error vector；G=[g₁,g₂,…,g₂₁]^TRepresent the vector being made up of 29 geometric errors, be written as Δ x_X,Δy_X,Δz_X,Δα_X,Δβ_X,Δγ_X,Δx_Y,Δy_Y,Δz_Y,Δα_Y,Δβ_Y,Δγ_Y,Δx_Z,Δy_Z,Δz_Z,Δα_Z,Δβ_Z,1Δγ_Z,Δx_A,Δy_A,Δz_A,Δα_A,Δβ_A,Δγ_A,Δγ_XY,Δβ_XZ,Δα_YZ,Δγ_YA,Δβ_ZA=g₁,g₂,g₃,g₄,g₅,g₆,g₇,g₈,g₉,g₁₀,g₁₁,g₁₂,g₁₃,g₁₄,g₁₅,g₁₆,g₁₇,g₁₈,g₁₉,g₂₀,g₂₁,g₂₂,g₂₂,g₂₃,g₂₄,g₂₅,g₂₆,g₂₇,g₂₈,g₂₉；P=[x, y, z, 0]^TRepresent the position vector of the kinematic axis at lathe center.

Step 2 is based on the machining accuracy fail-safe analysis of Markov chain

2.1 machining accuracy reliability definition

Along with requirement on machining accuracy is more and more higher, the reliability of machining accuracy also becomes a measurement index of lathe.But, current lathe reliability consideration is concentrated mainly on intensity and life-span aspect, seldom the machining accuracy aspect of lathe is studied.As it has been described above, through long-term operation, machining accuracy can reduce, it is impossible to meet the specification of lathe.In the reason that machining accuracy declines, along with abrasion and the deformation of contact surface and structure, the increase of geometric error is a main cause.

The geometric error of lathe is mainly derived from the site error and the error of perpendicularity that manufacture or assemble defect and each moving component of each moving component of lathe.Due to the randomness manufactured and assemble, therefore, geometric error is different in different positions.For the situation that geometric error is uncertain variables, give the definition of machining accuracy reliability.

The reliability of machining accuracy be a kind of within the given time, perform the ability of machining accuracy of its regulation when regulation.In general, Space processing error can be analyzed to tri-axles of X, Y, Z, if machining accuracy is respectively smaller than regulation requirement in X, Y, Z-direction, then can be regarded as machining accuracy and is unsatisfactory for requiring in the direction.

2.2 fault modes and failure probability

The Synthetic Volumetric Error Model of machining center can be expressed as:

E=E (G)=[E_X(G),E_Y(G),E_Z(G),0]^T(69)

Assuming that the maximum allowable space error of lathe is e=(e_X,e_Y,e_Z,0)^T, wherein e_X,e_Y,e_ZIt is illustrated respectively in the maximum allowable space error in X-, Y-, Z-direction, as follows with function representation:

$F=\&lsqb;E-e\&rsqb;={\&lsqb;E_X\left(G\right)-e_X,E_Y\left(G\right)-e_Y,E_Z\left(G\right)-e_Z,0\&rsqb;}^T=\left[\begin{array}{c}{H}_X\left(G\right)\\ H_Y\left(G\right)\\ H_Z\left(G\right)\\ 0\end{array}\right]---\left(70\right)$

The machining accuracy of Digit Control Machine Tool has seven kinds of failure modes as follows:

M₁={ H_X≥0,H_Y≤0andH_Z≤0}(71)

M₂={ H_X≤0,H_Y≥0andH_Z≤0}(72)

M₃={ H_X≤0,H_Y≤0andH_Z≥0}(73)

M₄={ H_X≥0,H_Y≥0andH_Z≤0}(74)

M₅={ H_X≥0,H_Y≤0andH_Z≥0}(75)

M₆={ H_X≤0,H_Y≥0andH_Z≥0}(76)

M₇={ H_X≥0,H_Y≥0andH_Z≥0}(77)

In formula (71) to (73), M₁、M₂And M₃The machining accuracy representing lathe respectively is unsatisfactory for maximum allowable space error in X-, Y-and Z-direction.In formula (74) to (76), M₄、M₅And M₆The machining accuracy both direction in X-, Y-and Z-direction representing lathe respectively is unsatisfactory for maximum allowable space error.In formula (77), M₇The machining accuracy representing lathe is all unsatisfactory for maximum allowable space error in X-, Y-and Z-direction.

The inefficacy territory corresponding with every kind of failure mode is as follows:

F₁={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≤0,G∈H_Z(G)≤0}(78)

F₂={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≥0,G∈H_Z(G)≤0}(79)

F₃={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≤0,G∈H_Z(G)≥0}(80)

F₄={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≥0,G∈H_Z(G)≤0}(81)

F₅={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≤0,G∈H_Z(G)≥0}(82)

F₆={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≥0,G∈H_Z(G)≥0}(83)

F₇={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≥0,G∈H_Z(G)≥0}(84)

In machining accuracy fail-safe analysis process, failure probability P can be defined as the joint probability density function f (G) of geometric error integration on the F of inefficacy territory, and therefore the failure probability of every kind of failure mode can be represented as follows:

$P_F^{\left(i\right)}=P\{G\&Element;F_i\}=\&Integral;...{\&Integral;}_{F_i}f\left(G\right)dG---\left(85\right)$

Wherein, i=1,2 ... 7, i is the sequence number of failure mode.

The failure probability P that machining accuracy is total_FCan be written as follows:

$P_F=P_F^{\left(1\right)}+P_F^{\left(2\right)}+P_F^{\left(3\right)}+P_F^{\left(4\right)}+P_F^{\left(5\right)}+P_F^{\left(6\right)}+P_F^{\left(7\right)}---\left(86\right)$

2.3 by the relevant conversion of normal variate to independent standard normal variable

In the actual course of processing, the geometric error of lathe is to be mutually related.The impact of machining accuracy failure probability is very important by this dependency.So in actual analysis of Machining, in order to tally with the actual situation, it is necessary to consider the relation between the geometric error of lathe.First, relevant geometric error is converted into independent standard normal random variable.Then the failure probability of machining accuracy is obtained with the analysis method for reliability of separate space.

The n item geometric error of lathe can be write a n and be tieed up basic random variables: G=(g₁,g₂,…g_n)^T, owing to every geometric error is relevant, n ties up the joint probability density function f (G) of normal variate G and can be represented as follows:

$f\left(G\right)={\left(2\mathrm{\&pi;}\right)}^{-\frac{n}{2}}{\left|C_G\right|}^{-\frac{1}{2}}\exp \&lsqb;-\frac{1}{2}{(G-{\mathrm{\&mu;}}_G)}^T{C}_G^{-1}(G-{\mathrm{\&mu;}}_G)\&rsqb;---\left(87\right)$

Wherein,

$C_G=\left[\begin{array}{ccccc}{\mathrm{\&sigma;}}_{g_1}^2& {\mathrm{\&rho;}}_{g_1{g}_2}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_2}& {\mathrm{\&rho;}}_{g_1{g}_3}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_3}& ...& {\mathrm{\&rho;}}_{g_1{g}_n}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_n}\\ {\mathrm{\&rho;}}_{g_1{g}_2}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_2}& {\mathrm{\&sigma;}}_{g_2}^2& {\mathrm{\&rho;}}_{g_2{g}_3}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_3}& ...& {\mathrm{\&rho;}}_{g_2{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}\\ {\mathrm{\&rho;}}_{g_1{g}_3}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_3}& {\mathrm{\&rho;}}_{g_2{g}_3}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_3}& {\mathrm{\&sigma;}}_{g_3}^2& & {\mathrm{\&rho;}}_{g_3{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}\\ .& .& .& & .\\ .& .& .& & .\\ .& .& .& & .\\ {\mathrm{\&rho;}}_{g_1{g}_n}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_n}& {\mathrm{\&rho;}}_{g_2{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}& {\mathrm{\&rho;}}_{g_3{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}& ...& {\mathrm{\&sigma;}}_{g_n}^2\end{array}\right]---\left(88\right)$

Represent the covariance matrix of aggregate error G；For C_GInverse matrix；|C_G| represent C_GDeterminant；Represent the mean vector of geometric error,WithRepresent geometric error g_i(i=1,2,3 ..., average n) and variance,Represent g_iAnd g_jCorrelation coefficient.

Ultimate principle according to linear algebra, there will necessarily be an orthogonal matrix A and n ties up correlated random variables G=(g₁,g₂,…g_n)^TBe converted to n and tie up uncorrelated random variables y=(y₁,y₂,…y_n)^T, have:

$f_Y\left(y\right)=f_G(A^{-1}y+{\mathrm{\&mu;}}_G)={\left(2\mathrm{\&pi;}\right)}^{-\frac{n}{2}}{({\mathrm{\&lambda;}}_1{\mathrm{\&lambda;}}_2...{\mathrm{\&lambda;}}_n)}^{-\frac{1}{2}}\exp (-\frac{1}{2}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{y_i^2}{{\mathrm{\&lambda;}}_i})---\left(89\right)$

With

Y=A^T(G-μ_G), y_i～N (0, λ_i)(90)

Wherein, λ₁,λ₂,…λ_nIt is covariance matrix C_GCharacteristic root.The column vector of A orthogonal matrix is equal to covariance matrix C_GOrthogonal eigenvectors.

According to above principle, it is possible to relevant n is tieed up random vector G=(g₁,g₂,…g_n)^TBe converted to n and tie up incoherent random vector y=(y₁,y₂,…y_n)^T.By following formula, independent normal variate y=(y₁,y₂,…y_n)^TStandard independent normal variate u=(u can be converted into₁,u₂,…u_n)^T, it may be assumed that

$u_i=\frac{y_i-{\mathrm{\&mu;}}_{y_i}}{{\mathrm{\&sigma;}}_{y_i}}=\frac{y_i}{\sqrt{{\mathrm{\&lambda;}}_i}},(i=1,2,...n)---\left(91\right)$

Therefore the space corresponding in inefficacy territory F (G) and constraint function H (G) can be converted into inefficacy territory F (u) and constraint function H (u) standard absolute version.The failure probability of every kind of failure mode can be written as following form:

$P_F^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i\left(G\right)}{f}_G\left(G\right)dG=\&Integral;...{\&Integral;}_{F_i\left(y\right)}{f}_Y\left(y\right)dy=\&Integral;...{\&Integral;}_{F_i\left(u\right)}{f}_U\left(u\right)d\left(u\right)---\left(92\right)$

The quick Markov chain emulation mode that 2.4 failure probabilities are estimated

Based on the method for Digital Simulation, the method having many calculating processing precision reliabilitys, can be not only used for single failure mode fail-safe analysis and can be used for the fail-safe analysis of Multiple Failure Modes.But up to now, but without the problem of the fail-safe analysis carrying out machining accuracy with Markov chain.

Owing to Markov chain can simulate inefficacy sample point effectively, for general nonlinear limit state equation $H_U\left(u\right)-H_G\left(G\right)=\left\{\begin{array}{c}{H}_X\left(G\right)=0\\ H_Y\left(G\right)=0\\ H_Z\left(G\right)=0\end{array}\right.,$ Simulation of Markov chains can be utilized quickly to obtain the point that in inefficacy territory, most probable lost efficacy, i.e. design point.By design point, introduce a linear limit state equation with L (u)=0 with same design point $H_U\left(u\right)=H_G\left(G\right)=\left\{\begin{array}{c}{H}_X\left(G\right)=0\\ H_Y\left(G\right)=0\\ H_Z\left(G\right)=0\end{array}\right.,$ The equation can obtain easily in standard normal space.

Based on the multiplication theorem in theory of probability, it is possible to set up following 2 equations:

P{F_H∩F_L}=P{F_H}P{F_L|F_H}(93)

P{F_H∩F_L}=P{F_L}P{F_H|F_L}(94)

Wherein, F_H={ u:u → G ∈ F_i, F_L={ u:L (u)≤0}, P{F_L}=P{L (u)≤0} and P{F_H}=P{F_i}。P{F_L|F_HAnd P{F_H|F_LIt is two conditional probability.

Therefore failure probability P_FForm can be expressed as:

$P_F^{\left(i\right)}=P\left(F_i\right)=P\left\{F_H\right\}=P\left\{F_L\right\}\frac{P\left\{F_H\right|F_L\}}{P\left\{F_L\right|F_H\}}---\left(95\right)$

WhereinCharacteristic ratio factor S can be defined as,

$S=\frac{P\left\{F_H\right|F_L\}}{P\left\{F_L\right|F_H\}}---\left(96\right)$

Formula (95) can be reduced to following form:

$P_F^{\left(i\right)}=P\left\{F_L\right\}\&CenterDot;S---\left(97\right)$

The probability density function F of inefficacy sample point_HFollowing form can be represented as:

$q_H\left(u\right|F_H)=\frac{I_H\left(u\right)f_U\left(u\right)}{P_H}---\left(98\right)$

Wherein, I_HU () is the indicator of non-linear behaviour function H (u),

Wherein,

$I_H\left(u\right)=\left\{\begin{array}{cc}1,& H\left(u\right)<0\\ 0,& H\left(u\right)\&GreaterEqual;0\end{array}\right.---\left(99\right)$

Ultimate principle according to Markov chain, Markov chain is passed through from a state to another state to advise distribution function f^*(ε | u) control.

There is symmetric n dimension normal distribution and n dimension be uniformly distributed and all as markovian suggestion distribution, in this article, can select have symmetric n dimension normal distribution as suggestion distribution, it may be assumed that

Wherein, ε_kAnd u_kThe respectively kth component of n-dimensional vector ε and u；l_kRepresent the n centered by u and tie up hyperpolyhedron u_kThe length of side in direction, l_kDecide next sample and deviate the maximum allowable range of current sample.

Based on practical engineering experience and numerical method, select inefficacy territory F_HInterior 1 u₀As markovian original state.Markovian jth state u_j, it is at preceding state u_j-1On basis, determine according to suggestion substep and Metropolis-Hastings criterion, first by suggestion distribution f^*(ε|u_j-1) produce alternative state ε.

Then, alternative state conditional probability density function q (ε | F_H) can be expressed as with the ratio of the conditional probability density function of Markov Chain preceding state:

R=q (ε | F_H)/q(u_j-1|F_H)(101)

Finally, jth state u is determined according to Metropolis-Hastings criterion_j:

$u_j=\left\{\begin{array}{cc}\mathrm{\&epsiv;},& min\{1,r\}>random\&lsqb;0,1\&rsqb;\\ u_{j-1},& min\{1,r\}\&le;random\&lsqb;0,1\&rsqb;\end{array}\right.----\left(102\right)$

In formula, random [0,1] represents [0,1] interval equally distributed random number.

Repeat above method, N_HIndividual Markov Chain stateCan be obtained, as inefficacy territory F_HMiddle probability density is q_H(u|F_H) condition sample point.

From inefficacy territory F_HN_HSelecting in individual sample point, employing probability density function is q_H(u|F_H) Markov Chain simulate inefficacy territory F_HN_HIndividual sample point, therefrom can obtain F in standard normal space_HThe approximate maximum likelihood point in region $u^{*}=(u_1^{*},u_2^{*},...,u_n^{*}).$

In standard normal space, with F_HRegion has the linear limit state equation of same pole maximum-likelihood point to be expressed as follows:

L (u)=(0-u^*)(u-u^*)^T=0 (103)

Corresponding failure probability is:

$P\left\{F_L\right\}=\mathrm{\&Phi;}(-\sqrt{{\left(u_1^{*}\right)}^2+{\left(u_2^{*}\right)}^2+...+{\left(u_n^{*}\right)}^2})---\left(104\right)$

In formula, the distribution function that Φ () is standard normal variable.

By N_HSample point substitutes into formula (103), falls into region F_L={ u:L (u)≤0} sample size is designated as N_L|H。

Conditional probability P{F_L|F_HEstimated value can be obtained by following formula, it may be assumed that

$\hat{P}\left\{F_L\right|F_H\}=\frac{N_{L|H}}{N_H}---\left(105\right)$

In like manner, conditional probability P{F_H|F_LCan also there is Markov Chain simulation F_LThe sample in region obtains, F_LThe joint probability density function of zone sample point can be expressed as follows:

$q_L\left(u\right|F_L)=\frac{I_L\left(u\right)f_U\left(u\right)}{P_L}---\left(106\right)$

N in inefficacy territory can be obtained by Markov Chain simulation_LSample point.These sample points brought into H (u) and calculates the value of H (u) respectively, calculating sample point and fall into inefficacy territory F_H={ quantity of u:H (u)≤0}, is designated as N_H|L。

Conditional probability P{F_L|F_HEstimated value can obtain by through type (46), characteristic ratio factor S can obtain by through type (108), then have:

$\hat{P}\left\{F_H\right|F_L\}=\frac{N_{H|L}}{N_L}---\left(107\right)$

$\hat{S}=\frac{\hat{P}\left\{F_H\right|F_L\}}{\hat{P}\left\{F_L\right|F_H\}}=\frac{N_{H|L}}{N_L}\&CenterDot;\frac{N_H}{N_{L|H}}---\left(108\right)$

Owing to lathe has Multiple Failure Modes, the failure probability of each failure mode should be calculated respectively.

Another F_H=F_i, i=1,2 ... 7, then withAnd S⁽ⁱ⁾The failure probability of corresponding failure mode can respectively obtain by through type (109), it may be assumed that

$P_F^{\left(i\right)}=P\left\{F_L^{\left(i\right)}\right\}\&CenterDot;S^{\left(i\right)},i=1,2,...7---\left(109\right)$

The composite failure probability of machining accuracy can be expressed as:

${\hat{P}}_F=P_F^{\left(1\right)}+P_F^{\left(2\right)}+P_F^{\left(3\right)}+P_F^{\left(4\right)}+P_F^{\left(5\right)}+P_F^{\left(6\right)}+P_F^{\left(7\right)}---\left(110\right)$

Step 3 is based on the machining accuracy reliability sensitivity analysis of Failure Probability Integration

Machining accuracy reliability sensitivity coefficient is commonly defined as the failure probability of the every kind of failure mode partial derivative to the probability distribution of the geometric error parameter of kth item, it is possible to be expressed as form:

$S_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}=\&Integral;...{\&Integral;}_{F_i}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}dG---\left(111\right)$

$S_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}=\&Integral;...{\&Integral;}_{F_i}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}dG---\left(112\right)$

In formula, i=1,2 ..., 7；K=1,2 ..., n, n is the number of geometric error. μ_kRepresent the average of kth item geometric error.σ_kRepresent the standard deviation of kth item geometric error.Representing in i-th kind of failure mode, failure probability is to mean μ_kMachining accuracy reliability sensitivity coefficient.Represent in i-th kind of failure mode, the failure probability machining accuracy reliability sensitivity coefficient to standard deviation.

The definition of regularization reliability sensitivity coefficient is as follows:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}\frac{{\mathrm{\&sigma;}}_k}{P_F^{\left(i\right)}}---\left(113\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}\frac{{\mathrm{\&sigma;}}_k}{P_F^{\left(i\right)}}---\left(114\right)$

Formula (113) and formula (114) are converted to integrated form as follows:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\left(\frac{f\left(G\right)}{P_F^{\left(i\right)}}\right)dG---\left(115\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}\left(\frac{f\left(G\right)}{P_F^{\left(i\right)}}\right)dG---\left(116\right)$

Obviously, formula (115) and formula (116) can be expressed as at inefficacy territory F_iMathematic expectaion:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=E_{F_i}\&lsqb;\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\&rsqb;---\left(117\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=E_{F_i}\&lsqb;\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}\&rsqb;---\left(118\right)$

In formula,Represent inefficacy territory F_iMathematic expectaion.

By formula (29) and formula (30), sample pointCan be converted intoWillSubstituting in equation below, normalized reliability sensitivity coefficient can obtain, it may be assumed that

${\widehat{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{1}{N_H}\underset{0}{\overset{N_H-1}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}---\left(119\right)$

${\widehat{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{1}{N_H}\underset{0}{\overset{N_H-1}{\mathrm{\&Sigma;}}}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}---\left(120\right)$

Therefore, general reliability sensitivity coefficient is as follows:

$S_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}={\widehat{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}\&CenterDot;\frac{P_F^{\left(i\right)}}{{\mathrm{\&sigma;}}_k}---\left(121\right)$

$S_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}={\widehat{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}\&CenterDot;\frac{P_F^{\left(i\right)}}{{\mathrm{\&sigma;}}_k}---\left(122\right)$

Analyze the fail-safe analysis of the machining accuracy of lathe in the proposed method of use, the probability distribution of parameter should provide.K data measurement points is selected from the working area of processing, for each geometric error relevant to position, at each measurement point, measure 100 times in a like fashion, adopt two-frequency laser interferometer to measure six geometric errors relevant to position, measure the error of perpendicularity and parallelism error with geometric error measuring instrument.

By the sample data obtained is carried out statistical analysis, obtain the probability distribution of geometric error.At spatial point (x=200mm, y=400mm, z=300mm) place, with position error Δ x_xFor example, this position error substantially belongs to normal distribution.Test result indicate that, every geometric error relevant to position belongs to normal distribution.Table 6 gives the value of the error of perpendicularity.Table 7 gives the geometric error probability distribution at spatial point (x=200mm, y=400mm, z=300mm) place, including meansigma methods (M) and variance yields (V).

By above-mentioned method, at spatial point (x=200mm, y=400mm, z=300mm), the failure probability of every kind of failure mode is as shown in table 8.The result of machining accuracy reliability sensitive analysis is as shown in table 9.

At each body diagonal of lathe work space, nine test points of evenly spaced selection (totally 33 test points) is as shown in Figure 3.In the result of the sensitive analysis of each testing site, can be obtained by previously mentioned method.Then, utilize weighted average method, calculate the sensitivity analysis of whole work space.

For whole work space, for i-th kind of failure mode, geometric error g_kMean μ_kAnd variances sigma_kMachining accuracy reliability sensitivity coefficient can be defined asWithThat is:

${\hat{S}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{1}{j}\underset{j=1}{\overset{33}{\mathrm{\&Sigma;}}}S_{{\mathrm{\&mu;}}_k}^{\left(i\right)}{}_j---\left(123\right)$

${\hat{S}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{1}{j}\underset{j=1}{\overset{33}{\mathrm{\&Sigma;}}}S_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}{}_j---\left(124\right)$

Table 10 and table 11 list the probability of the generation of various failure mode and the result of calculation of the machining accuracy reliability sensitivity of whole work space respectively.

Step 3 application and improvement

Selecting lathe as shown in Figure 1 is example, and the method is described.Machining center important technological parameters is as shown in table 1.In order to study the machining accuracy reliability of selected machining center in actual processing environment, machining center is used for processing a typical workpiece.Fig. 4 is processing site picture.

On modern advanced machining production line, in order to improve efficiency and the productive temp of automatic production line, a machine only need to be performed one or several procedure of processing.For selected machine, in production line, only need to reduction box finished surface as shown in Figure 4 being processed, the crudy of finished surface is limited primarily by the impact of machining center Z axis machining accuracy reliability.As shown in table 9, by formula (77) to (77) it can be seen that only have failure mode M₃, M₅, M₆And M₇Relation is had with machining center Z-direction machining accuracy reliability.It is further known that, failure mode M₃And M₆Failure probability value than failure mode M₅And M₇Failure probability value big, therefore failure mode M₃And M₆It it is the dominant failure mode affecting machined surface quality.

As shown in table 8, for failure mode M₃, Δ γ_zWith Δ α_ASensitivity coefficient is maximum, and for failure mode M₆, Δ z_xWith Δ y_zSensitivity coefficient is maximum, thus, it can be known that Δ γ_z, Δ α_A, Δ z_xWith Δ y_zCan be determined that and affect failure mode M₃And M₆Critical error.

Owing to the geometric error of lathe is to be caused by the geometric accuracy of infeed mean, so the mapping relations also existed between basic geometric error and the precision parameter of infeed mean.List the corresponding relation between basic geometric error and the precision parameter of assembly in table 12.

So we take following amendment measure:

(1) X direction guiding rail straightness error in vertical is improved；

(2) Z-direction guide rail straightness error in horizontal plane is improved；

(3) Z-direction guide rail is improved at parallelism error；

(4) change as higher Precision Lead-Screw for A axle.

The failure probability of each failure mode in amended whole work space is carried out analysis result as shown in table 13.By comparing, it can be deduced that the failure probability of each failure mode of modified precise horizontal machining center substantially reduces, failure mode M₃,₅,₆And M₇Failure probability transformation after be greatly reduced.From this it can be concluded that the machining accuracy Reliability Sensitivity Method that the present invention proposes is feasible and effective.

The important technological parameters of the four-shaft numerically controlled lathe of table 1

The geometric error of table 2 precise horizontal machining center

The degree of freedom of table 3 precise horizontal machining center adjacent motion parts

The lower body array of table 4 precise horizontal machining center

The eigenmatrix of table 5 precise horizontal machining center

The numerical value of table 6 error of perpendicularity

The probability distribution of table 7 spatial point (x=200mm, y=400mm, z=300mm) place geometric error

The probability of happening of the 7 kinds of failure modes in table 8 spatial point (x=200mm, y=400mm, z=300mm) place

The result of table 9 spatial point (x=200mm, y=400mm, z=300mm) place machining accuracy reliability sensitive analysis

The probability of happening of each failure mode of the whole work space of table 10

Table 11 whole work space machining accuracy reliability sensitivity analysis result

Corresponding relation between basic geometric error and the precision parameter of table 12 assembly

For the probability of happening of failure mode each in whole work space after table 13 correction

Claims (1)

1. the machine finish Reliability Sensitivity Method based on quick Markov chain, it is characterised in that:

The method to realize process as follows；

Step one sets up the space error modeling of lathe according to theory of multi body system

Root theory of multi body system ultimate principle of the present invention, by abstract for each motion parts of lathe be the vector form of 4 × 1；By forms of motion and synthetic error modularized processing, set up the spatial synthesis error model of lathe according to the topological structure of lathe；

Step 1.1 machine tool structure figure

During this investigation it turned out, four-shaft numerically controlled lathe, its important technological parameters is listed in table 1, and its geometric error item has been listed in table 2；

Step 1.2 topological structure and geometric error

Four axle lathes have 4 moving components, and each parts move relative to each other, and cutter and workpiece are fixed on lathe；Table 3 describes the restriction of the degree of freedom between each unit, and wherein " 0 " refers to do not have degree of freedom and " 1 " to refer to there is one degree of freedom；

Based on theory of multi body system, each moving component of lathe, being conceptualized as topological structure, four axle lathes are described as in a topological structure with double; two branch, and the first branch is by lathe bed, Y-axis moving component, X-axis moving component and cutter；Second branch is by lathe bed, Z axis moving component, A axle moving component and workpiece；Lathe bed is inertial reference system, uses B₀Representing that the order according to growth naturally numbers in order, table 4 is the lower body array of selected precise horizontal machining center；

One rigid body has 6 degree of freedom；6 coordinates uniquely specify rigid body position in three dimensions；Four axle lathes have 4 moving components, move relative to each other, and other 2 bodies being fixed on lathe are cutter and workpiece；Each moving component has 6 geometric errors, Δ x_h, Δ y_h, Δ z_h, Δ α_h, Δ β_hWith Δ γ_h；Δx_h, Δ y_hWith Δ z_hBelong to translational error；Δα_h, Δ β_hWith Δ γ_hBelong to rotation error, under be designated as the direction of motion, subscript letter h will take following letter respectively: X, Y, Z and A, between each kinematic axis, has kinematic error between 5 bodies, Δ γ_XY, Δ β_XZ, Δ α_YZ, Δ γ_YAWith Δ β_ZA；Utilize theory of multi body system by abstract for each moving component of lathe be one group of rigid body time, four axle lathes have 24 geometric errors relevant to position, kinematic error between 5 independent bodies, and above-mentioned error has been listed in table 2 all；

Step 1.3 generalized coordinates is arranged and eigenmatrix

In order to make machine tool accuracy modeling more convenient, it is necessary to coordinate system is carried out special agreement；Used herein of agreeing as follows: (1) establishes rectangular coordinate system, for all of inertance element and moving component；These coordinate systems are generalized coordinates systems, and inertial coodinate system is called coordinate system, and the coordinate system of other motions is called coordinate system；(2) X of each coordinate system, Y, Z axis should be parallel；

According to multi-body system ultimate principle, the eigenmatrix of selected machining center is listed in table 5；

Assuming that cutter becomes the form point coordinate in tool coordinate system to be:

P_t=[P_tx,P_ty,P_tz,1]^T(1)

Component shaping point coordinate in workpiece coordinate system can be expressed as follows:

P_w=[P_wx,P_wy,P_wz,1]^T(2)

Ideally, lathe does not have error, cutter to become form point and component shaping point to coincide together, and under ideal conditions, precision machined constraint equation can be expressed as follows:

$\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\&Pi;}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t=\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\&Pi;}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;P_{wideal}---\left(3\right)$

By variation, formula (3) is written as following form:

$P_{wideal}={\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\&Pi;}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;}^{-1}\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\&Pi;}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t---\left(4\right)$

Machining accuracy be last with actual building motion time, cutter become the relative position error between form point and component shaping point be correlated with；In practical situations both, precision machined constraint equation is write as following formula:

$\begin{array}{c}{P}_{wactual}={\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\&Pi;}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;}^{-1}\\ \&times;\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\&Pi;}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t\end{array}---\left(5\right)$

The general space error caused by the gap between form point and ideal forming point is become to be written as by reality:

$\begin{array}{c}E=\&lsqb;\underset{u=n,L^n\left(w\right)=0}{\overset{u=1}{\&Pi;}}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)p}{M}_{L^u\left(w\right)L^{u-1}\left(w\right)s}{\mathrm{\&Delta;M}}_{L^u\left(w\right)L^{u-1}\left(w\right)s}\&rsqb;P_{wideal}\\ -\&lsqb;\underset{k=n,L^n\left(t\right)=0}{\overset{k=1}{\&Pi;}}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)p}{M}_{L^k\left(t\right)L^{k-1}\left(t\right)s}{\mathrm{\&Delta;M}}_{L^k\left(t\right)L^{k-1}\left(t\right)s}\&rsqb;P_t\end{array}---\left(6\right)$

By the synthetic error model of the precise horizontal machining center that table 4 eigenmatrix and formula (6) obtain；Equally, general lathe spatial error model is established as follows:

E=E (G, P_t,P)(7)

In formula (7), E=[E_X,E_Y,E_Z,0]^TRepresent space error vector；G=[g₁,g₂,...,g₂₁]^TRepresent the vector being made up of 29 geometric errors, be written as Δ x_X,Δy_X,Δz_X,Δα_X,Δβ_X,Δγ_X,Δx_Y,Δy_Y,Δz_Y,Δα_Y,Δβ_Y,Δγ_Y,Δx_Z,Δy_Z,Δz_Z,Δα_Z,Δβ_Z,1Δγ_Z,Δx_A,Δy_A,Δz_A,Δα_A,Δβ_A,Δγ_A,Δγ_XY,Δβ_XZ,Δα_YZ,Δγ_YA,Δβ_ZA=g₁,g₂,g₃,g₄,g₅,g₆,g₇,g₈,g₉,g₁₀,g₁₁,g₁₂,g₁₃,g₁₄,g₁₅,g₁₆,g₁₇,g₁₈,g₁₉,g₂₀,g₂₁,g₂₂,g₂₂,g₂₃,g₂₄,g₂₅,g₂₆,g₂₇,g₂₈,g₂₉；P=[x, y, z, 0]^TRepresent the position vector of the kinematic axis at lathe center；

Step 2 is based on the machining accuracy fail-safe analysis of Markov chain

2.1 machining accuracy reliability definition

Along with requirement on machining accuracy is more and more higher, the reliability of machining accuracy also becomes a measurement index of lathe；But, current lathe reliability consideration is concentrated mainly on intensity and life-span aspect, seldom the machining accuracy aspect of lathe is studied；As it has been described above, through long-term operation, machining accuracy can reduce, it is impossible to meet the specification of lathe；In the reason that machining accuracy declines, along with abrasion and the deformation of contact surface and structure, the increase of geometric error is a main cause；

The geometric error of lathe is mainly derived from the site error and the error of perpendicularity that manufacture or assemble defect and each moving component of each moving component of lathe；Due to the randomness manufactured and assemble, therefore, geometric error is different in different positions；For the situation that geometric error is uncertain variables, give the definition of machining accuracy reliability；

The reliability of machining accuracy be a kind of within the given time, perform the ability of machining accuracy of its regulation when regulation；In general, Space processing error can be analyzed to tri-axles of X, Y, Z, if machining accuracy is respectively smaller than regulation requirement in X, Y, Z-direction, then can be regarded as machining accuracy and is unsatisfactory for requiring in the direction；

2.2 fault modes and failure probability

The Synthetic Volumetric Error Model of machining center is expressed as:

E=E (G)=[E_X(G),E_Y(G),E_Z(G),0]^T(8)

Assuming that the maximum allowable space error of lathe is e=(e_X,e_Y,e_Z,0)^T, wherein e_X,e_Y,e_ZIt is illustrated respectively in the maximum allowable space error in X-, Y-, Z-direction, as follows with function representation:

$F=\&lsqb;E-e\&rsqb;={\&lsqb;E_X\left(G\right)-e_X,E_Y\left(G\right)-e_Y,E_Z\left(G\right)-e_Z,0\&rsqb;}^T=\left[\begin{array}{c}{H}_X\left(G\right)\\ H_Y\left(G\right)\\ H_Z\left(G\right)\\ 0\end{array}\right]---\left(9\right)$

The machining accuracy of Digit Control Machine Tool has seven kinds of failure modes as follows:

M₁={ H_X≥0,H_Y≤0andH_Z≤0}(10)

M₂={ H_X≤0,H_Y≥0andH_Z≤0}(11)

M₃={ H_X≤0,H_Y≤0andH_Z≥0}(12)

M₄={ H_X≥0,H_Y≥0andH_Z≤0}(13)

M₅={ H_X≥0,H_Y≤0andH_Z≥0}(14)

M₆={ H_X≤0,H_Y≥0andH_Z≥0}(15)

M₇={ H_X≥0,H_Y≥0andH_Z≥0}(16)

In formula (10) to (12), M₁、M₂And M₃The machining accuracy representing lathe respectively is unsatisfactory for maximum allowable space error in X-, Y-and Z-direction；In formula (13) to (15), M₄、M₅And M₆The machining accuracy both direction in X-, Y-and Z-direction representing lathe respectively is unsatisfactory for maximum allowable space error；In formula (13), M₇The machining accuracy representing lathe is all unsatisfactory for maximum allowable space error in X-, Y-and Z-direction；

The inefficacy territory corresponding with every kind of failure mode is as follows:

F₁={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≤0,G∈H_Z(G)≤0}(17)

F₂={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≥0,G∈H_Z(G)≤0}(18)

F₃={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≤0,G∈H_Z(G)≥0}(19)

F₄={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≥0,G∈H_Z(G)≤0}(20)

F₅={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≤0,G∈H_Z(G)≥0}(21)

F₆={ G:G ∈ H_X(G)≤0,G∈H_Y(G)≥0,G∈H_Z(G)≥0}(22)

F₇={ G:G ∈ H_X(G)≥0,G∈H_Y(G)≥0,G∈H_Z(G)≥0}(23)

In machining accuracy fail-safe analysis process, failure probability P is defined as the joint probability density function f (G) of geometric error integration on the F of inefficacy territory, and therefore the failure probability of every kind of failure mode is represented as follows:

$P_F^{\left(i\right)}=P\{G\&Element;F_i\}=\&Integral;...{\&Integral;}_{F_i}f\left(G\right)dG---\left(24\right)$

Wherein, i=1,2...7, i is the sequence number of failure mode；

The failure probability P that machining accuracy is total_FIt is written as follows:

$P_F=P_F^{\left(1\right)}+P_F^{\left(2\right)}+P_F^{\left(3\right)}+P_F^{\left(4\right)}+P_F^{\left(5\right)}+P_F^{\left(6\right)}+P_F^{\left(7\right)}---\left(25\right)$

2.3 by the relevant conversion of normal variate to independent standard normal variable

In the actual course of processing, the geometric error of lathe is to be mutually related；The impact of machining accuracy failure probability is very important by this dependency；So in actual analysis of Machining, in order to tally with the actual situation, it is necessary to consider the relation between the geometric error of lathe；First, relevant geometric error is converted into independent standard normal random variable；Then the failure probability of machining accuracy is obtained with the analysis method for reliability of separate space；

The n item geometric error of lathe can be write a n and be tieed up basic random variables: G=(g₁,g₂,...g_n)^T, owing to every geometric error is relevant, n ties up the joint probability density function f (G) of normal variate G and is represented as follows:

$f\left(G\right)={\left(2\mathrm{\&pi;}\right)}^{-\frac{n}{2}}{\left|C_G\right|}^{-\frac{1}{2}}\exp \&lsqb;-\frac{1}{2}{(G-{\mathrm{\&mu;}}_G)}^T{C}_G^{-1}(G-{\mathrm{\&mu;}}_G)\&rsqb;---\left(26\right)$

Wherein,

$C_G=\left[\begin{array}{ccccc}{\mathrm{\&sigma;}}_{g_1}^2& {\mathrm{\&rho;}}_{g_1{g}_2}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_2}& {\mathrm{\&rho;}}_{g_1{g}_3}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_3}& \mathrm{...}& {\mathrm{\&rho;}}_{g_1{g}_n}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_n}\\ {\mathrm{\&rho;}}_{g_1{g}_2}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_2}& {\mathrm{\&sigma;}}_{g_2}^2& {\mathrm{\&rho;}}_{g_2{g}_3}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_3}& \mathrm{...}& {\mathrm{\&rho;}}_{g_2{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}\\ {\mathrm{\&rho;}}_{g_1{g}_3}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_3}& {\mathrm{\&rho;}}_{g_2{g}_3}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_3}& {\mathrm{\&sigma;}}_{g_3}^2& & {\mathrm{\&rho;}}_{g_3{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}\\ .& .& .& & .\\ .& .& .& & .\\ .& .& .& & .\\ {\mathrm{\&rho;}}_{g_1{g}_n}{\mathrm{\&sigma;}}_{g_1}{\mathrm{\&sigma;}}_{g_n}& {\mathrm{\&rho;}}_{g_2{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}& {\mathrm{\&rho;}}_{g_3{g}_n}{\mathrm{\&sigma;}}_{g_2}{\mathrm{\&sigma;}}_{g_n}& \mathrm{...}& {\mathrm{\&sigma;}}_{g_n}^2\end{array}\right]---\left(27\right)$

Represent the covariance matrix of aggregate error G；For C_GInverse matrix；|C_G| represent C_GDeterminant；Represent the mean vector of geometric error,WithRepresent geometric error g_i(i=1,2,3 ..., average n) and variance,Represent g_iAnd g_jCorrelation coefficient；

Ultimate principle according to linear algebra, there will necessarily be an orthogonal matrix A and n ties up correlated random variables G=(g₁,g₂,…g_n)^TBe converted to n and tie up uncorrelated random variables y=(y₁,y₂,…y_n)^T, have:

$f_Y\left(y\right)=f_G(A^{-1}y+{\mathrm{\&mu;}}_G)={\left(2\mathrm{\&pi;}\right)}^{-\frac{n}{2}}{({\mathrm{\&lambda;}}_1{\mathrm{\&lambda;}}_2...{\mathrm{\&lambda;}}_n)}^{-\frac{1}{2}}\exp (-\frac{1}{2}\underset{i=1}{\overset{n}{\&Sigma;}}\frac{y_i^2}{{\mathrm{\&lambda;}}_i})---\left(28\right)$

With

Y=A^T(G-μ_G), y_i～N (0, λ_i)(29)

Wherein, λ₁,λ₂,…λ_nIt is covariance matrix C_GCharacteristic root；The column vector of A orthogonal matrix is equal to covariance matrix C_GOrthogonal eigenvectors；

According to above principle, relevant n is tieed up random vector G=(g₁,g₂,…g_n)^TBe converted to n and tie up incoherent random vector y=(y₁,y₂,…y_n)^T；By following formula, independent normal variate y=(y₁,y₂,…y_n)^TIt is converted into standard independent normal variate u=(u₁,u₂,…u_n)^T, it may be assumed that

$u_i=\frac{y_i-{\mathrm{\&mu;}}_{y_i}}{{\mathrm{\&sigma;}}_{y_i}}=\frac{y_i}{\sqrt{{\mathrm{\&lambda;}}_i}},(i=1,2,...n)---\left(30\right)$

Therefore the space corresponding in inefficacy territory F (G) and constraint function H (G) is converted into inefficacy territory F (u) and constraint function H (u) standard absolute version；The failure probability of every kind of failure mode is written as following form:

$P_F^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i\left(G\right)}{f}_G\left(G\right)dG=\&Integral;...{\&Integral;}_{F_i\left(y\right)}{f}_Y\left(y\right)dy=\&Integral;...{\&Integral;}_{F_i\left(u\right)}{f}_U\left(u\right)d\left(u\right)---\left(31\right)$

The quick Markov chain emulation mode that 2.4 failure probabilities are estimated

Based on the method for Digital Simulation, the method having many calculating processing precision reliabilitys, it is applied not only to single failure mode fail-safe analysis and is also used for the fail-safe analysis of Multiple Failure Modes；But up to now, but without the problem of the fail-safe analysis carrying out machining accuracy with Markov chain；

Inefficacy sample point is effectively simulated, for general nonlinear limit state equation due to Markov chain $H_U\left(u\right)=H_G\left(G\right)=\left\{\begin{array}{c}{H}_X\left(G\right)=0\\ H_Y\left(G\right)=0\\ H_Z\left(G\right)=0\end{array}\right.,$ Simulation of Markov chains is utilized quickly to obtain the point that in inefficacy territory, most probable lost efficacy, i.e. design point；By design point, introduce a linear limit state equation with L (u)=0 with same design point $H_U\left(u\right)=H_G\left(G\right)=\left\{\begin{array}{c}{H}_X\left(G\right)=0\\ H_Y\left(G\right)=0\\ H_Z\left(G\right)=0\end{array}\right.,$ The equation obtains easily in standard normal space；

Based on the multiplication theorem in theory of probability, set up following 2 equations:

P{F_H∩F_L}=P{F_H}P{F_L|F_H}(32)

P{F_H∩F_L}=P{F_L}P{F_H|F_L}(33)

Wherein, F_H={ u:u → G ∈ F_i, F_L={ u:L (u)≤0}, P{F_L}=P{L (u)≤0} and P{F_H}=P{F_i}；P{F_L|F_HAnd P{F_H|F_LIt is two conditional probability；

Therefore failure probability P_FIt is expressed as form:

$P_F^{\left(i\right)}=P\left(F_i\right)=P\left\{F_H\right\}=P\left\{F_L\right\}\frac{P\left\{F_H\right|F_L\}}{P\left\{F_L\right|F_H\}}---\left(34\right)$

WhereinCharacteristic ratio factor S can be defined as,

$S=\frac{P\left\{F_H\right|F_L\}}{P\left\{F_L\right|F_H\}}---\left(35\right)$

Formula (34) is reduced to following form:

$P_F^{\left(i\right)}=P\left\{F_L\right\}\&CenterDot;S---\left(36\right)$

The probability density function F of inefficacy sample point_HIt is represented as following form:

$q_H\left(u\right|F_H)=\frac{I_H\left(u\right)f_U\left(u\right)}{P_H}---\left(37\right)$

Wherein, I_HU () is the indicator of non-linear behaviour function H (u),

Wherein,

$I_H\left(u\right)=\left\{\begin{array}{cc}1,& H\left(u\right)<0\\ 0,& H\left(u\right)\&GreaterEqual;0\end{array}\right.---\left(38\right)$

Ultimate principle according to Markov chain, Markov chain is passed through from a state to another state to advise distribution function f^*(ε | u) control；

There is symmetric n dimension normal distribution and n dimension be uniformly distributed and all as markovian suggestion distribution, in this article, can select have symmetric n dimension normal distribution as suggestion distribution, it may be assumed that

Wherein, ε_kAnd u_kThe respectively kth component of n-dimensional vector ε and u；l_kRepresent the n centered by u and tie up hyperpolyhedron u_kThe length of side in direction, l_kDecide next sample and deviate the maximum allowable range of current sample；.

Based on practical engineering experience and numerical method, select inefficacy territory F_HInterior 1 u₀As markovian original state；Markovian jth state u_j, it is at preceding state u_j-1On basis, it is proposed that substep and Metropolis-Hastings criterion are determined；First by suggestion distribution f^*(ε|u_j-1) produce alternative state ε；

Then, alternative state conditional probability density function q (ε | F_H) it is expressed as with the ratio of the conditional probability density function of Markov Chain preceding state:

R=q (ε | F_H)/q(u_j-1|F_H)(40)

Finally, jth state u is determined according to Metropolis-Hastings criterion_j:

$u_j=\left\{\begin{array}{cc}\mathrm{\&epsiv;},& min\{1,r\}\&le;random\&lsqb;0,1\&rsqb;\\ u_{j-1},& min\{1,r\}\&le;random\&lsqb;0,1\&rsqb;\end{array}\right.---\left(41\right)$ In formula, random [0,1] represents [0,1] interval equally distributed random number；

Repeat above method, N_HIndividual Markov Chain stateIt is obtained, as inefficacy territory F_HMiddle probability density is q_H(u|F_H) condition sample point；

From inefficacy territory F_HN_HSelecting in individual sample point, employing probability density function is q_H(u|F_H) Markov Chain simulate inefficacy territory F_HN_HIndividual sample point, therefrom obtains F in standard normal space_HThe approximate maximum likelihood point in region $u^{*}=(u_1^{*},u_2^{*},...,u_n^{*});$

In standard normal space, with F_HRegion has the linear limit state equation of same pole maximum-likelihood point to be expressed as follows:

L (u)=(0-u^*)(u-u^*)^T=0 (42)

Corresponding failure probability is:

$P\left\{F_L\right\}=\mathrm{\&Phi;}(-\sqrt{{\left(u_1^{*}\right)}^2+{\left(u_1^{*}\right)}^2+\mathrm{...}+{\left(u_1^{*}\right)}^2})---\left(43\right)$

In formula, the distribution function that Φ () is standard normal variable；

By N_HSample point substitutes into formula (42), falls into region F_L={ u:L (u)≤0} sample size is designated as N_LH；

Conditional probability P{F_L|F_HEstimated value obtained by following formula, it may be assumed that

$\hat{P}\left\{F_L\right|F_H\}=\frac{N_{L|H}}{N_H}---\left(44\right)$

In like manner, conditional probability P{F_H|F_LAlso there is Markov Chain simulation F_LThe sample in region obtains, F_LThe joint probability density function of zone sample point can be expressed as follows:

$q_L\left(u\right|F_L)=\frac{I_L\left(u\right)f_U\left(u\right)}{P_L}---\left(45\right)$

N in inefficacy territory is obtained by Markov Chain simulation_LSample point；These sample points brought into H (u) and calculates the value of H (u) respectively, calculating sample point and fall into inefficacy territory F_H={ quantity of u:H (u)≤0}, is designated as N_HL；

Conditional probability P{F_L|F_HEstimated value through type (46) obtain, characteristic ratio factor S through type (47) obtains, then have:

$\hat{P}\left\{F_H\right|F_L\}=\frac{N_{H|L}}{N_L}---\left(46\right)$

$\hat{S}=\frac{\hat{P}\left\{F_H\right|F_L\}}{\hat{P}\left\{F_L\right|F_H\}}=\frac{N_{H|L}}{N_L}\&CenterDot;\frac{N_H}{N_{L|H}}---\left(47\right)$

Owing to lathe has Multiple Failure Modes, the failure probability of each failure mode should be calculated respectively；

Another F_H=F_i, i=1,2 ... 7, then withAnd S⁽ⁱ⁾The failure probability through type (48) of corresponding failure mode respectively obtains, it may be assumed that

$P_F^{\left(i\right)}=P\left\{F_L^{\left(i\right)}\right\}\&CenterDot;S^{\left(i\right)},i=1,2,...7---\left(48\right)$

The composite failure probability of machining accuracy is expressed as:

${\hat{P}}_F=P_F^{\left(1\right)}+P_F^{\left(2\right)}+P_F^{\left(3\right)}+P_F^{\left(4\right)}+P_F^{\left(5\right)}+P_F^{\left(6\right)}+P_F^{\left(7\right)}---\left(49\right)$

Step 3 is based on the machining accuracy reliability sensitivity analysis of Failure Probability Integration

Machining accuracy reliability sensitivity coefficient is commonly defined as the failure probability of the every kind of failure mode partial derivative to the probability distribution of the geometric error parameter of kth item, is expressed as form:

$S_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}=\&Integral;...{\&Integral;}_{F_i}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}dG---\left(50\right)$

$S_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}=\&Integral;...{\&Integral;}_{F_i}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}dG---\left(51\right)$

In formula, i=1,2 ..., 7；K=1,2 ..., n, n is the number of geometric error. μ_kRepresent the average of kth item geometric error；σ_kRepresent the standard deviation of kth item geometric error；Representing in i-th kind of failure mode, failure probability is to mean μ_kMachining accuracy reliability sensitivity coefficient；Represent in i-th kind of failure mode, the failure probability machining accuracy reliability sensitivity coefficient to standard deviation；

The definition of regularization reliability sensitivity coefficient is as follows:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&mu;}}_k}\frac{{\mathrm{\&sigma;}}_k}{P_F^{\left(i\right)}}---\left(52\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\frac{\&part;P_F^{\left(i\right)}}{\&part;{\mathrm{\&sigma;}}_k}\frac{{\mathrm{\&sigma;}}_k}{P_F^{\left(i\right)}}---\left(53\right)$

Formula (52) and formula (53) are converted to integrated form as follows:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\left(\frac{f\left(G\right)}{P_F^{\left(i\right)}}\right)dG---\left(54\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=\&Integral;...{\&Integral;}_{F_i}\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\left(\frac{f\left(G\right)}{P_F^{\left(i\right)}}\right)dG---\left(55\right)$

Obviously, formula 54 and 53 can be expressed as at inefficacy territory F_iMathematic expectaion:

${\mathrm{SA}}_{{\mathrm{\&mu;}}_k}^{\left(i\right)}=E_{F_i}\&lsqb;\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&mu;}}_k}\&rsqb;---\left(56\right)$

${\mathrm{SA}}_{{\mathrm{\&sigma;}}_k}^{\left(i\right)}=E_{F_i}\&lsqb;\frac{{\mathrm{\&sigma;}}_k}{f\left(G\right)}\frac{\&part;f\left(G\right)}{\&part;{\mathrm{\&sigma;}}_k}\&rsqb;---\left(57\right)$

In formula,Represent inefficacy territory F_iMathematic expectaion；

By formula (29) and formula (30), sample pointCan be converted intoWillSubstituting in equation below, normalized reliability sensitivity coefficient can obtain, it may be assumed that

Therefore, general reliability sensitivity coefficient is as follows:

The important technological parameters of the four-shaft numerically controlled lathe of table 1

The geometric error of table 2 precise horizontal machining center

The degree of freedom of table 3 precise horizontal machining center adjacent motion parts

The lower body array of table 4 precise horizontal machining center

The eigenmatrix of table 5 precise horizontal machining center