CN101629801A - Method for confirming guide rail thermal error of numerical control grinder - Google Patents

Abstract

The invention relates to a method for confirming guide rail thermal error of a numerical control grinder, comprising the concrete steps: 1. measuring guide rail thermal error on a limit position of the numerical control grinder; 2. establishing a numerical control grinder guide rail thermal feature analysis response surface model; 3. optimizing the numerical control grinder guide rail limit element model and calculating the thermal difference. On the basis of using measurement of numerical control grinder guide rail thermal error, the response surface method is adopted to establish a similar model of designed variable and target function with the reduction of error of thermal error simulation value as the optimization aim and a margin condition of limit element analysis as the design variable. During optimization of limit element analysis parameters, the measurement data of limit position guide rail thermal deformation is used to rectify the limit element calculation margin conditions, thus obtaining the guide rail thermal deformation error value close to the actual measurement value. The invention can solve the measurement problem of numerical control grinder thermal deformation, can provides an optimization method for guide rail limit element model, thus obtaining the guide rail thermal deformation difference close to the actual measurement value through limit element calculation.

Description

Guide rail thermal error of numerical control grinder is determined method

Technical field

The present invention relates to a kind of machine tool guideway error and determine method, especially a kind of guideway of grinding machine heat mistake method for determining difference.

Background technology

In the cut of precision machine tool, thermal source is very big to the influence of machining precision, improves the machining precision of workpiece and must do quantitative examination to the thermal deformation of lathe, and do control rationally and compensation in process.Track segment is one of main thermal source of lathe, and the relative motion meeting of machine tool guideway and worktable produces a large amount of heat of friction, causes the thermal deformation of guide rail and worktable, influences the relative position relation between cutter and the workpiece, finally influences the machining precision of workpiece.Especially for the numerical control main axle grinding machine, because worktable is carrying very heavy part of the grinding head, and the stroke of worktable is very big, and the guide rail thermal deformation that is caused by friction is the one of the main reasons of steering error, and the guide rail thermal deformation is very big for the machining precision influence of part.Therefore,, need to obtain the interior thermal deformation dynamic variable quantity of the whole stroke range of guide rail, so that carry out the Real-time Error compensation for improving its machining precision.But, also do not have effective method at present to carrying out the measurement in real time of hot error in the guide rail total travel scope.

Along with constantly improving of finite element theory and reaching its maturity of numerical simulation technology, make numerical simulation technology obtain widespread use at the thermal characteristics analysis field, become the important means of thermal deformation analysis.The finite element numerical simulation technology can calculate temperature distribution state and data such as consequent thermal walking, stress and strain quantitatively.But because the uncertainty (for example thermal source and convection coefficient) of finite element analysis boundary condition can't provide precise calculation result to concrete machine tool guideway, and can only provide analysis result qualitatively.

Summary of the invention

The present invention will provide a kind of guide rail thermal error of numerical control grinder to determine method, is used to solve guide rail thermal error of numerical control grinder and measures and compensation problem.

For solving the problems of the technologies described above, the technical solution used in the present invention is: a kind of guide rail thermal error of numerical control grinder is determined method, comprises following concrete steps:

(1) measures numerically control grinder the hot error of extreme position upper rail is arranged

Along the guide rail movement Z-direction several displacement transducers are installed in the Y direction, wherein two displacement transducers should be placed on two target locations, and other displacement transducers are evenly distributed between the target location:

If sensor 1,2 ... the reading of r is respectively δ ₁(i), δ ₂(i) ... δ _r(i), i=0,1,2 ... N is for measuring sequence number, establish when measuring beginning the main shaft thermal deformation and be 0, the i measurement Y direction 1,2 ... the heat distortion amount at r sensing station place is respectively:

δ _1i＝δ ₁(i)-δ ₁(0)δ _2i＝δ ₂(i)-δ ₂(0)……δ _ri＝δ _r(i)-δ _r(0)

(2) set up numerically control grinder guide rail thermal characteristics analyzing responding surface model

Utilize finite element analysis software to set up numerically control grinder guide rail finite element model, the boundary condition of finite element model is comprised guide rail surface convection coefficient (ξ ₁, ξ ₂ξ _M) and the thermal value (Q of thermal source ₁, Q ₂Q _M), represent that the Calculation of Thermal Deformation error sum of squares of getting r sensor installation site on the guide rail is as characteristic quantity, with F (X) expression, as the computation optimization index of design variable X as design variable with X;

Thermal deformation errors simulation value Δ to r measuring point _1i, Δ _2i... Δ _RiWith actual measured value δ _1i, δ _2i... δ _RiCompare, get the computation optimization index of its error sum of squares structure design variable X

$F\left(X\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{({\mathrm{\Δ}}_{1i}-{\mathrm{\δ}}_{1i})}^2+\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{({\mathrm{\Δ}}_{2i}-{\mathrm{\δ}}_{2i})}^2+\·\·\·\·\·\·+\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{({\mathrm{\Δ}}_{\mathrm{ri}}-{\mathrm{\δ}}_{\mathrm{ri}})}^2$

Adopt above-mentioned response surface method to set up approximate model between computation optimization index F (X) and the design variable X, the surface function expression formula meets with a response:

F(X)＝f(ξ ₁，ξ ₂，…ξ _M，Q ₁，Q ₂，…Q _M)

(3) optimization and the hot error of calculating numerically control grinder guide rail finite element model

The mathematical model of guide rail finite element analysis model optimization is:

$\left\{\begin{array}{c}\min F\left(X\right)\\ s.t.X=[{\mathrm{\ξ}}_1,{\mathrm{\ξ}}_2,\·\·\·{\mathrm{\ξ}}_M,Q_1,Q_2,\·\·\·Q_M]\\ X_{\min }\≤X\≤X_{\max }\end{array}\right.$

X wherein _Min, X _MaxBe respectively the span of design variable; Adopt above-mentioned optimization mathematical model and select optimized Algorithm, guide rail finite element model boundary condition is optimized design, obtain one group of optimal design variable X=[ξ ₁, ξ ₂... ξ _M, Q ₁, Q ₂... Q _M]; According to the boundary condition after optimizing, utilize finite element analysis software to calculate the heat distortion amount of guide rail in whole stroke range.

The beneficial effect that the present invention has:

Can solve the problems of measurement of numerically control grinder guide rail thermal deformation by the present invention, and provide the optimization method of guide rail finite element model, thereby can obtain guide rail thermal deformation errors by FEM (finite element) calculation near measured value.The present invention combines numerical simulation technology and actual tests, utilize experimental test data correction finite element analysis boundary condition, thereby obtain guide rail Calculation of Thermal Deformation result accurately, make its thermal deformation real-Time Compensation that can be applied to machine tool guideway, improve the machining precision of lathe.

Description of drawings

Fig. 1 is numerically control grinder moves the Y direction thermal deformation measurement that causes along the Z axle a sensor scheme of installation; Wherein: left target measurement position 1, middle measuring position 2, right target measurement position 3, workpiece erecting bed 4, sensor mount 5, displacement transducer 6.

Embodiment

The present invention will solve the technical matters that numerically control grinder heat error compensation modeling middle guide thermal deformation errors is measured, and on the basis of limited position measurement guide rail thermal deformation, carries out a kind of method of the thermal deformation errors calculating of numerical control machine slide rail with method for numerical simulation.

For the numerical control main axle grinding machine, axial deformation (Z to) and radially (directions X) little to the influence of workpiece processing precision, so mainly consider that radially (Y direction) thermal deformation errors is seen Fig. 1 along the distribution of guide rail movement direction.

On the basis that utilizes the hot error of numerical control machine slide rail, be optimization aim with the error that reduces the hot error numerical value analogue value, be design variable with the boundary condition of finite element analysis, adopt the response surface method to set up the approximate model of design variable and objective function.In finite element analysis Parameter Optimization process, utilize the measurement data that the thermal deformation of extreme position guide rail is arranged, repair the FEM (finite element) calculation boundary condition, thereby obtain guide rail thermal deformation errors value near actual measured value.

1. response surface approximate model method general introduction

The response surface method is the cover statistical treatment technology that is used to handle modeling of multivariate problem and analysis based on test design, and its basic thought is by polynomial expression y=f (x with clear and definite expression-form of approximate structure ₁, x ₂..., x _k), approach implicit relationship complicated between characteristic quantity and design variable with explicit response surface approximate model, and on this basis original model is revised.The mathematic(al) representation of response surface method is a multiple linear regression equations.In the present invention, this method is used to the actual measurement data according to the guide rail thermal deformation, revises the boundary condition (convective heat-transfer coefficient and thermal source) of finite element model.

In the engineering the response surface approximate function of extensive employing be second-order model:

$\hat{y}={\mathrm{\&alpha;}}_0+\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_i{x}_i+\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{\mathrm{ii}}{x_i}^2+\underset{\mathrm{ij}(i<j)}{\mathrm{\&Sigma;}}{\mathrm{\&alpha;}}_{\mathrm{ij}}{x}_i{x}_j---\left(1\right)$

Wherein: n is the design variable number; x _iBe design variable; α ₀, α _i, α _Ii, α _Ij(i＜j) is polynomial undetermined coefficient.

For example: for the situation of 4 design variables, the second-order response surface function expands into:

$\hat{y}={\mathrm{\&alpha;}}_0+{\mathrm{\&alpha;}}_1{x}_1+{\mathrm{\&alpha;}}_2{x}_2+{\mathrm{\&alpha;}}_3{x}_3+{\mathrm{\&alpha;}}_4{x}_4+{\mathrm{\&alpha;}}_{11}{x}_1^2+{\mathrm{\&alpha;}}_{22}{x}_2^2+{\mathrm{\&alpha;}}_{33}{x}_3^2+{\mathrm{\&alpha;}}_{44}{x}_4^2---\left(2\right)$

$+{\mathrm{\&alpha;}}_{12}{x}_1{x}_2+{\mathrm{\&alpha;}}_{13}{x}_1{x}_3+{\mathrm{\&alpha;}}_{14}{x}_1{x}_4+{\mathrm{\&alpha;}}_{23}{x}_2{x}_3+{\mathrm{\&alpha;}}_{24}{x}_2{x}_4+{\mathrm{\&alpha;}}_{34}{x}_3{x}_4$

Make x ₅=x ₁ ², x ₆=x ₂ ², x ₇=x ₃ ², x ₈=x ₄ ², x ₉=x ₁x ₂, x ₁₀=x ₁x ₃, x ₁₁=x ₁x ₄, x ₁₂=x ₂x ₃, x ₁₃=x ₂x ₄, x ₁₄=x ₃x ₄, β ₀=α ₀, β ₁=α ₁, β ₂=α ₂, β ₃=α ₃, β ₄=α ₄, β ₅=α ₁₁, β ₆=α ₂₂, β ₇=α ₃₃, β ₈=α ₄₄, β ₉=α ₁₂, β ₁₀=α ₁₃, β ₁₁=α ₁₄, β ₁₂=α ₂₃, β ₁₃=α ₂₄, β ₁₄=α ₃₄, then following formula is convertible into polynary line style regression model:

$\hat{y}={\mathrm{\&beta;}}_0+{\mathrm{\&beta;}}_1{x}_1+{\mathrm{\&beta;}}_2{x}_2+{\mathrm{\&beta;}}_3{x}_3+{\mathrm{\&beta;}}_4{x}_4+{\mathrm{\&beta;}}_5{x}_5+{\mathrm{\&beta;}}_6{x}_6+{\mathrm{\&beta;}}_7{x}_7+{\mathrm{\&beta;}}_8{x}_8---\left(3\right)$

$+{\mathrm{\&beta;}}_9{x}_9+{\mathrm{\&beta;}}_{10}{x}_{10}+{\mathrm{\&beta;}}_{11}{x}_{11}+{\mathrm{\&beta;}}_{12}{x}_{12}+{\mathrm{\&beta;}}_{13}{x}_{13}+{\mathrm{\&beta;}}_{14}{x}_{14}$

Simple form that can be unified:

$\hat{y}={\mathrm{\&beta;}}_0+\underset{i=1}{\overset{k-1}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_i{x}_i---\left(4\right)$

Wherein k is undetermined coefficient β _iNumber.In order to determine β _i, need do the independent experiment of m 〉=k time, obtain corresponding factor beta by finding the solution _iResponse surface can be expressed as with matrix form:

$y=\mathrm{x\&beta;}+\mathrm{\&epsiv;}=\hat{y}+\mathrm{\&epsiv;}---\left(5\right)$

$y=\left[\begin{array}{c}{y}^{\left(1\right)}\\ y^{\left(2\right)}\\ .\\ .\\ .\\ y^{\left(m\right)}\end{array}\right],$ $x=\left[\begin{array}{cccc}1& {x_1}^{\left(1\right)}& ...& x_{k-1}^{\left(1\right)}\\ 1& {x_1}^{\left(2\right)}& ...& x_{k-1}^{\left(2\right)}\\ .& .& & \\ .& .& & \\ .& .& ...& \\ 1& {x_1}^{\left(m\right)}& ...& x_{k-1}^{\left(m\right)}\end{array}\right],$ $\mathrm{\&beta;}=\left[\begin{array}{c}{\mathrm{\&beta;}}_0\\ {\mathrm{\&beta;}}_1\\ .\\ .\\ .\\ {\mathrm{\&beta;}}_{k-1}\end{array}\right],$ $\mathrm{\&epsiv;}=\left[\begin{array}{c}{\mathrm{\&epsiv;}}_1\\ {\mathrm{\&epsiv;}}_2\\ .\\ .\\ .\\ {\mathrm{\&epsiv;}}_m\end{array}\right]$

Wherein, y is actual trial value,

Be response surface approximate function value, ε be Normal Distribution N (0, σ ²) error of fitting.

In order to estimate the factor beta of quadratic polynomial, can use least square method, make the error sum of squares minimum, that is:

$S_{\mathrm{\&beta;}}=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_j^2=\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}{(\underset{i=0}{\overset{k-1}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_i{x}_i^{\left(j\right)}-y^{\left(j\right)})}^2\&RightArrow;\min ---\left(6\right)$

According to the infinitesimal calculus extreme value theorem, order:

$\frac{\&PartialD;S}{{\&PartialD;\mathrm{\&beta;}}_l}=2\underset{j=1}{\overset{m}{\mathrm{\&Sigma;}}}\left(x_l^{\left(j\right)}(\underset{i=0}{\overset{k-1}{\mathrm{\&Sigma;}}}{\mathrm{\&beta;}}_i{x}_i^{\left(j\right)}-y^{\left(j\right)})\right)=0,(l=\mathrm{0,1},\&CenterDot;\&CenterDot;\&CenterDot;k-1)---\left(7\right)$

Obviously, following formula is a system of linear equations with k unknown number and k equation, and it is as follows to be write as matrix form:

(xβ-y) ^Tx＝0 (8)

Separate following formula and can get β, the surface function expression formula promptly meets with a response.

More than be the ultimate principle of response surface method and the concrete grammar that the response surface coefficient is found the solution.

2 for solving the problems of the technologies described above, and concrete grammar of the present invention is:

(1) numerically control grinder has the measurement of the hot error of extreme position upper rail

The guide rail structure synoptic diagram of numerically control grinder and the sensor mounting means of hot error measure are as shown in Figure 1.Along the guide rail movement Z-direction r displacement transducer is installed in the Y direction, wherein 2 displacement transducers should be placed on two target locations, and other displacement transducers are evenly distributed between the target location.The selection of target location should consider to process stroke, as close as possible stroke maximum point.Read the measured value of displacement transducer for the measurement requirement timing (for example per 5 minutes) of guide rail thermal deformation, and calculate the deflection error of Y direction according to measured value.Consider the motion problems of worktable, sensor can be selected non-contact displacement sensors such as capacitive transducer or current vortex sensor for use.

If sensor 1,2 ... the reading of r is respectively δ ₁(i), δ ₂(i) ... δ _r(i), i=0,1,2 ... N is for measuring sequence number.If when measuring beginning the main shaft thermal deformation be 0, the i measurement Y direction 1,2 ... the heat distortion amount at r sensing station place is respectively

δ _1i＝δ ₁(i)-δ ₁(0)δ _2i＝δ ₂(i)-δ ₂(0)……δ _ri＝δ _r(i)-δ _r(0)(9)

(2) foundation of numerically control grinder guide rail thermal characteristics analyzing responding surface model

Utilize finite element analysis software to set up numerically control grinder guide rail finite element model.

In process to research guide rail temperature field and thermal deformation errors finite element analysis, because the boundary condition (convective heat-transfer coefficient and thermal source) of finite element model is difficult to theoretical accurately definite, make and often have tangible error between result of finite element and the test findings, in order to dwindle this error, can carry out the model correction by the response surface method.The advantage of response surface method is and can obtains between design variable and the characteristic quantity enough mutual relationship accurately by less test (FEM (finite element) calculation), and can only show with simple algebraic expression.

Therefore the boundary condition of finite element model is comprised guide rail surface convection coefficient (ξ ₁, ξ ₂... ξ _M) and the thermal value (Q of thermal source ₁, Q ₂... Q _M) as design variable, represent with X.On the basis that theory is calculated, choose the factor level scope of design variable, that is: ξ in conjunction with experience _1min≤ ξ ₁≤ ξ _1max, ξ _2min≤ ξ ₂≤ ξ _2max... ξ _Mmin≤ ξ _M≤ ξ _Mmax, Q _1min≤ Q ₁≤ Q _1max, Q _2min≤ Q ₂≤ Q _2max... Q _Mmin≤ Q _M≤ Q _MmaxIn order to make numerical simulation result approach experimental result, the Calculation of Thermal Deformation error sum of squares of getting r sensor installation site on the guide rail is as characteristic quantity, with F (X) expression, as the computation optimization index of design variable X.

Thermal deformation errors simulation value Δ to r measuring point _1i, Δ _2i... Δ _RiWith actual measured value δ _1i, δ _2i... δ _RiCompare, get the computation optimization index of its error sum of squares structure design variable X

$F\left(X\right)=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&Delta;}}_{1i}-{\mathrm{\&delta;}}_{1i})}^2+\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&Delta;}}_{2i}-{\mathrm{\&delta;}}_{2i})}^2+\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;+\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&Delta;}}_{\mathrm{ri}}-{\mathrm{\&delta;}}_{\mathrm{ri}})}^2---\left(10\right)$

Adopt experimental design method to carry out data sampling (experimental design method has factorial experiment design, Central Composite design (CCD), orthogonal experiment design and D-optimal design etc.), utilize finite element model to calculate computation optimization index F (X) calculated value in one group of design variable X span, adopt above-mentioned response surface method (formula 1-8) to set up approximate model between computation optimization index F (X) and the design variable X.The surface function expression formula meets with a response:

F(X)＝f(ξ ₁，ξ ₂，…ξ _M，Q ₁，Q ₂，…Q _M)

(3) optimization of numerically control grinder guide rail finite element model and hot Error Calculation

For the numerical control machine slide rail finite element model, in order to make numerical simulation result approach actual measured results, its optimization aim should reduce as far as possible because the finite element model boundary condition is difficult to accurately definite FEM (finite element) calculation error that is caused, so the mathematical model of guide rail finite element analysis model optimization is:

$\left\{\begin{array}{c}\min F\left(X\right)\\ s.t.X=[{\mathrm{\&xi;}}_1,{\mathrm{\&xi;}}_2,\&CenterDot;\&CenterDot;\&CenterDot;{\mathrm{\&xi;}}_M,Q_1,Q_2,\&CenterDot;\&CenterDot;\&CenterDot;Q_M]\\ X_{\min }\&le;X\&le;X_{\max }\end{array}\right.---\left(11\right)$

X wherein _Min, X _MaxBe respectively the span of design variable.

Like this, the finite element model optimization problem just is converted into finding the solution formula (11).

Adopt above-mentioned optimization mathematical model and select optimized Algorithm, guide rail finite element model boundary condition is optimized design.Obtain one group of optimal design variable X=[ξ ₁, ξ ₂... ξ _M, Q ₁, Q ₂... Q _M]

According to the boundary condition after optimizing, utilize finite element analysis software to calculate the heat distortion amount of guide rail in whole stroke range.

Claims (1)

1. a guide rail thermal error of numerical control grinder is determined method, it is characterized in that: comprise following concrete steps:

(1) measures numerically control grinder the hot error of extreme position upper rail is arranged

Along the guide rail movement Z-direction several displacement transducers are installed in the Y direction, wherein two displacement transducers should be placed on two target locations, and other displacement transducers are evenly distributed between the target location;

If sensor 1,2 ... the reading of r is respectively δ ₁(i), δ ₂(i) ... δ _r(i), i=0,1,2 ... N is for measuring sequence number, establish when measuring beginning the main shaft thermal deformation and be 0, the i measurement Y direction 1,2 ... the heat distortion amount at r sensing station place is respectively:

δ _1i＝δ ₁(i)-δ ₁(0)?δ _2i＝δ ₂(i)-δ ₂……?δ _ri＝δ _r(i)-δ _r(0)

(2) set up numerically control grinder guide rail thermal characteristics analyzing responding surface model

Utilize finite element analysis software to set up numerically control grinder guide rail finite element model, the boundary condition of finite element model is comprised guide rail surface convection coefficient (ξ ₁, ξ ₂ξ _M) and the thermal value (Q of thermal source ₁, Q ₂Q _M), represent that the Calculation of Thermal Deformation error sum of squares of getting r sensor installation site on the guide rail is as characteristic quantity, with F (X) expression, as the computation optimization index of design variable X as design variable with X;

Thermal deformation errors simulation value Δ to r measuring point _1i, Δ _2i... Δ _RiWith actual measured value δ _1i, δ _2i... δ _RiCompare, get the computation optimization index of its error sum of squares structure design variable X:

$F\left(X\right)=\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&Delta;}}_{1i}-{\mathrm{\&delta;}}_{1i})}^2+\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&Delta;}}_{2i}-{\mathrm{\&delta;}}_{2i})}^2+\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;+\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{({\mathrm{\&Delta;}}_{\mathrm{ri}}-{\mathrm{\&delta;}}_{\mathrm{ri}})}^2$

Employing response surface method is set up the approximate model between computation optimization index F (X) and the design variable X, and the surface function expression formula meets with a response:

F(X)＝f(ξ ₁，ξ ₂，…ξ _M，Q ₁，Q ₂，…Q _M)

(3) optimization of numerically control grinder guide rail finite element model and hot Error Calculation

The mathematical model of guide rail finite element analysis model optimization is:

$\left\{\begin{array}{c}\min F\left(X\right)\\ s.t.X=[{\mathrm{\&xi;}}_1,{\mathrm{\&xi;}}_2,\\ X_{\min }\&le;X\&le;X_{\max }\end{array},\&CenterDot;\&CenterDot;\&CenterDot;{\mathrm{\&xi;}}_M,Q_1,Q_2,\&CenterDot;\&CenterDot;\&CenterDot;Q_M]\right.$

X wherein _Min, X _MaxBe respectively the span of design variable; Adopt above-mentioned optimization mathematical model and select optimized Algorithm, guide rail finite element model boundary condition is optimized design, obtain one group of optimal design variable X=[ξ ₁, ξ _2,ξ _M, Q ₁, Q ₂... Q _M]; According to the boundary condition after optimizing, utilize finite element analysis software to calculate the heat distortion amount of guide rail in whole stroke range.