CN104636543A - Heavy planomiller beam gravity deformation predicting method based on finite difference method - Google Patents

Abstract

The invention provides and relates to a heavy planomiller beam gravity deformation predicting method based on the finite difference method. The heavy planomiller beam gravity deformation predicting method aims at solving the problems that by means of an existing finite element analyzing and calculating method, the beam gravity deformation curve can not be accurately calculated under the condition that actual material properties are not uniform and then the difference between the calculation result and the actual deformation value is large. The method includes the steps of firstly, obtaining the beam gravity deformation curve; secondly, simplifying the beam gravity deformation curve into a beam gravity deformation model and a beam torsional deformation model; thirdly, establishing a beam gravity deformation discrete model; fourthly, calculating the equivalent flexural rigidity; fifthly, obtaining a beam finite element gravity deformation curve; sixthly, separating the finite element gravity deformation curve; seventhly, obtaining the final beam gravity deformation curve. The method is applied to the prediction of the heavy planomiller beam gravity deformation.

Description

A kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference

Technical field

The present invention relates to heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology, particularly a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference.

Background technology

Heavy digital control machine tool is widely used in the major fields [1] such as national defence, Aero-Space, the energy, boats and ships, metallurgy as processing machine tool, and the quality of its precision has been reflected based on the heavy planer-type milling machine crossbeam gravity deformation curve computing method of method of finite difference with a manufacturing level of country.Due to structural factors such as heavy self large scale of planer-type milling machine crossbeam, large spans, under self gravitation effect, distortion to a certain degree can be caused, and the distortion inaccuracy that this gravity causes cannot be ignored.

Crossbeam is as the core component of heavy planer-type milling machine, and the depth of parallelism (G5 item precision) that rail head moves work top is its most important precision index.By compensating crossbeam loading end processing reversible deformation curve, the G5 item precision of lathe effectively can be improved.But due to the uncontrollability of casting process, inevitably there is the various defect such as burning into sand, pore in the structural member of heavy machine tool, cause crossbeam material properties, size etc. inconsistent, the Finite Element Method making current crossbeam reversible deformation calculate employing calculates accuracy only can reach 40% ~ 50%, crossbeam need be checked through many experiments, repeated disassembled and assembled repair could meet accuracy requirement, and cost is higher and very consuming time.Therefore, accurately calculate crossbeam gravity deformation curve, crossbeam dismounting number of times can be reduced, reduce costs.Fig. 2 is heavy planer-type milling machine crossbeam vertical view.

Zhang Yanting (" compensation method of double column vertical lathes crossbeam elastic deformation ") obtains the elastic deformation curve of double column vertical lathes crossbeam by approximate treatment, propose to obtain the required predeformation method adopted of rational guide rail geometric configuration, improve the precision of lathe.The method computation process is too loaded down with trivial details, and owing to adopting approximate simplified model, computational accuracy is poor.Guo Tieneng (" the large span crossbeam endurance curves of heavy duty is analyzed and experimental study ") etc. utilizes ANSYS to carry out finite element analysis to heavy planer-type milling machine, obtain the deflection of 25 equidistant working positions on crossbeam, drawing the endurance curves obtaining crossbeam, showing the pre-appraisal needing increase by 7% ~ 16% when predicting crossbeam endurance curves by experiment.To provide when processing loading end pre-estimates and lack theory support by means of only contrast finite element analysis and experimental result for the method, and wide usage is poor.King, Thomas Boyces (" copying of Longmen machine tool crossbeam ") etc. propose a kind of based on finite element analysis, combine the actual method detected simultaneously and draw crossbeam reversible deformation Processing Curve, reduce cost, improve efficiency of assembling.The method does not consider the inhomogeneity of material properties comprehensively, and only crossbeam deformation induced by gravity curve is obtained by experiment, and External Force Acting curve is obtained by finite element simulation.

In sum, theoretical calculation method process is too loaded down with trivial details, and computational accuracy is poor, but reflects crossbeam practical distortion situation by the crossbeam material properties in formula; Use finite element analysis fast and easy, computational accuracy is higher, but pretreatment process only can the material properties of definition component entirety, cannot consider the inhomogeneity of real material, not meet actual conditions, cause result of calculation and practical distortion value to differ greatly.

Summary of the invention

The object of the invention is in the inhomogenous situation of real material attribute, accurately to calculate crossbeam gravity deformation curve to solve existing finite element analysis computation method, cause the problem that result of calculation and practical distortion value differ greatly, and propose a kind of heavy twin columns based on method of finite difference and found car crossbeam gravity deformation Forecasting Methodology.

Above-mentioned goal of the invention is achieved through the following technical solutions:

Step one, by placing parallels at crossbeam and column assembling place, simulate practical set condition design heavy machine tool crossbeam deformation induced by gravity and test, obtain crossbeam deformation induced by gravity curve;

Step 2, utilize theory of mechanics of materials, according to the stressing conditions of crossbeam under Gravitative Loads, crossbeam is reduced to crossbeam deformation induced by gravity model and crossbeam torsional deflection model;

Crossbeam gravity deformation discretization model is set up in conjunction with method of finite difference after step 3, crossbeam deformation induced by gravity model discretize that step 2 is obtained:

Crossbeam gravity deformation discretization model described in heavy machine tool crossbeam deformation induced by gravity curve described in step 4, integrating step one and step 3, calculates the equivalent bendind rigidity of discrete micro-section of crossbeam; Wherein, equivalent bendind rigidity characterizes the material properties of crossbeam;

Step 5, practical set condition by finite element method for simulating heavy machine tool crossbeam, calculate crossbeam finite element gravity deformation curve after crossbeam and rail head being assembled; Wherein, crossbeam finite element gravity deformation comprises crossbeam bend distortion and torsional deflection;

Step 6, utilize crossbeam gravity deformation finite element simulation separation method to be separated by finite element gravity deformation curve to obtain rail head assembling rear cross beam flexural deformation finite element curve and torsional deflection finite element curve;

Step 7, the equivalent bendind rigidity utilizing step 4 to calculate, based on method of finite difference, the finite element curve that step 6 calculates rail head assembling rear cross beam flexural deformation and crossbeam torsional deflection is corrected, obtain final crossbeam gravity deformation curve; Namely a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference is completed.

Invention effect

Because theoretical calculation method and finite element method have good complementarity, therefore causing the inaccurate problem of Finite element analysis results for solving due to factors such as crossbeam material, manufacturing process, the present invention is based on the crossbeam gravity deformation curve computing method that method of finite difference proposes a kind of Combining material mechanics, deformation induced by gravity experiment and Finite Element Method.

The flexural deformation caused due to gravity and torsional deflection all have larger impact to the precision of planer-type milling machine, need to consider its impact on planer-type milling machine precision simultaneously.Realize as follows based on the main flow of the planer-type milling machine crossbeam gravity deformation curve computing method of method of finite difference: by the experiment of simulation practical set condition design heavy machine tool crossbeam deformation induced by gravity, obtain the crossbeam deformation induced by gravity curve considering material inhomogeneity.Utilize theory of mechanics of materials crossbeam to be reduced to free beam flexural deformation mechanical model and clamped beam torsional deflection mechanical model, this model is separated into micro-section, sets up the gravity deformation discretization model of heavy machine tool crossbeam in conjunction with method of finite difference.In conjunction with the equivalent bendind rigidity of the experiment of crossbeam deformation induced by gravity, each discrete segments of gravity deformation discretization model calculating crossbeam, characterize the real material attribute of crossbeam.By the practical set condition of finite element method for simulating crossbeam, calculate the finite element gravity deformation curve of rail head assembling rear cross beam, and utilize the finite element simulation data separation method of crossbeam bend distortion and torsional deflection to obtain the emulated data of flexural deformation and torsional deflection.The equivalent bendind rigidity that computation obtains, corrects the gravity deformation curve of Finite Element Method calculating, obtains crossbeam gravity deformation curve accurately for flexural deformation and torsional deflection based on method of finite difference.Crossbeam gravity deformation simulation curve, practical distortion curve and the gravity deformation curve computing method based on method of finite difference are corrected result contrast, as shown in figure 15.Verify the correctness of these computing method.

Curve after correcting as can be seen from Figure 16 compares the beam deformation situation of finite element simulation curve more closing to reality.As calculated, the average error rate of former result of finite element and actual beam deformation is 26.86%, and the crossbeam Z-direction distortion obtained based on the crossbeam gravity deformation computing method of method of finite difference is 8.37% with the average error rate of actual beam deformation, the error amount of main machining area is 0.0564mm to the maximum.Demonstrate based on method of finite difference and the correctness reversing the finite element result bearing calibration calculated.

Accompanying drawing explanation

Fig. 1 is the crossbeam gravity reversible deformation curve calculation flow chart that embodiment one proposes;

Fig. 2 is the heavy planer-type milling machine crossbeam vertical view that embodiment four proposes;

Fig. 3 is the heavy planer-type milling machine crossbeam vertical view coordinate system schematic diagram of the foundation that embodiment two proposes;

Fig. 4 is the crossbeam constraint condition schematic diagram that embodiment proposes; Wherein, the numbering of constrained type is added when A, B and C represent and carry out finite element analysis;

Fig. 5 is the rail loads definition schematic diagram that embodiment proposes;

Fig. 6 is that the crossbeam gravity load that embodiment four proposes bends computation model sketch; Wherein, L is the half of length between free beam fulcrum, and 2L is the length between free beam fulcrum; L1 is the length of two ends rectangular beam; L2 is the half of the length of stage casing rectangular beam, and 2L2 is the length of stage casing rectangular beam; A is the length of overhanging beam; QI is the gravity load intensity value in rectangular beam cross section, two ends; QII is the gravity load intensity value in rectangular beam cross section, stage casing; A and B is the name code of two position of the fulcrum, is easy to statement;

Fig. 7 is that the crossbeam knife rest gravity load that embodiment four proposes reverses computation model sketch; L is the half of length between clamped beam fulcrum, and 2L is the length between clamped beam fulcrum, identical with the L in free beam mechanical model; What S represented is the name code that knife rest moves to this position, is easy to statement, as rest position S; S is the distance of knife rest range coordinate system initial point O in the X-axis direction;

Fig. 8 is the crossbeam discretization model schematic diagram that embodiment five proposes;

Fig. 9 is the Z-direction distortion schematic diagram under the Action of Gravity Field of embodiment proposition;

Figure 10 is the crossbeam point of a knife point Z-direction gravity deformation simulation curve schematic diagram that embodiment proposes;

Figure 11 is the crossbeam floor cross section torsional centre FEM (finite element) calculation schematic diagram that embodiment seven proposes;

Figure 12 is that the crossbeam of embodiment seven proposition is without floor cross section torsional centre FEM (finite element) calculation schematic diagram;

Figure 13 is the crossbeam torsional centre observer sitting schematic diagram that embodiment seven proposes;

Figure 14 is that the crossbeam gravity deformation curve based on finite difference distribution that embodiment proposes calculates result schematic diagram;

Figure 15 is the crossbeam actual measurement G5 item precision curve synoptic diagram that embodiment proposes;

Figure 16 is that the crossbeam gravity deformation curve that embodiment proposes calculates methods and results checking schematic diagram.

Embodiment

Embodiment one: a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference of present embodiment, specifically prepare according to following steps:

Step one, by placing parallels at crossbeam and column assembling place, simulation practical set condition design heavy machine tool crossbeam deformation induced by gravity is tested, and obtains the crossbeam deformation induced by gravity curve of consideration material inhomogeneity;

Step 2, utilize theory of mechanics of materials, according to the stressing conditions of crossbeam under Gravitative Loads, crossbeam is reduced to crossbeam deformation induced by gravity model and crossbeam torsional deflection model;

Step 3, crossbeam is separated into one group discrete micro-section, after the crossbeam deformation induced by gravity model discretize that step 2 is obtained, is set up to crossbeam gravity deformation discretization model in conjunction with method of finite difference:

Crossbeam gravity deformation discretization model described in heavy machine tool crossbeam deformation induced by gravity curve described in step 4, integrating step one and step 3, calculates the equivalent bendind rigidity of discrete micro-section of crossbeam; Wherein, equivalent bendind rigidity characterizes the material properties of crossbeam;

Step 5, practical set condition by finite element method for simulating heavy machine tool crossbeam, calculate crossbeam finite element gravity deformation curve after crossbeam and rail head being assembled; Wherein, crossbeam finite element gravity deformation comprises crossbeam bend distortion and torsional deflection;

Step 6, utilize crossbeam gravity deformation finite element simulation separation method to be separated by finite element gravity deformation curve to obtain rail head assembling rear cross beam flexural deformation finite element curve and torsional deflection finite element curve;

Step 7, the equivalent bendind rigidity utilizing step 4 to calculate, based on method of finite difference, the finite element curve that step 6 calculates rail head assembling rear cross beam flexural deformation and crossbeam torsional deflection is corrected, obtain final crossbeam gravity deformation curve; As namely Fig. 1 completes a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference.

Present embodiment effect:

Because theoretical calculation method and finite element method have good complementarity, therefore for solving because the factors such as crossbeam material, manufacturing process cause the inaccurate problem of Finite element analysis results, present embodiment proposes a kind of Combining material mechanics based on method of finite difference, deformation induced by gravity tests the crossbeam gravity deformation curve computing method with Finite Element Method.

The flexural deformation caused due to gravity and torsional deflection all have larger impact to the precision of planer-type milling machine, need to consider its impact on planer-type milling machine precision simultaneously.Realize as follows based on the main flow of the planer-type milling machine crossbeam gravity deformation curve computing method of method of finite difference: by the experiment of simulation practical set condition design heavy machine tool crossbeam deformation induced by gravity, obtain the crossbeam deformation induced by gravity curve considering material inhomogeneity.Utilize theory of mechanics of materials crossbeam to be reduced to free beam flexural deformation mechanical model and clamped beam torsional deflection mechanical model, this model is separated into micro-section, sets up the gravity deformation discretization model of heavy machine tool crossbeam in conjunction with method of finite difference.In conjunction with the equivalent bendind rigidity of the experiment of crossbeam deformation induced by gravity, each discrete segments of gravity deformation discretization model calculating crossbeam, characterize the real material attribute of crossbeam.By the practical set condition of finite element method for simulating crossbeam, calculate the finite element gravity deformation curve of rail head assembling rear cross beam, and utilize the finite element simulation data separation method of crossbeam bend distortion and torsional deflection to obtain the emulated data of flexural deformation and torsional deflection.The equivalent bendind rigidity that computation obtains, corrects the gravity deformation curve of Finite Element Method calculating, obtains crossbeam gravity deformation curve accurately for flexural deformation and torsional deflection based on method of finite difference.Crossbeam gravity deformation simulation curve, practical distortion curve and the gravity deformation curve computing method based on method of finite difference are corrected result contrast, as shown in figure 15.Verify the correctness of these computing method.

Curve after correcting as can be seen from Figure 16 compares the beam deformation situation of finite element simulation curve more closing to reality.As calculated, the average error rate of former result of finite element and actual beam deformation is 26.86%, and the crossbeam Z-direction distortion obtained based on the crossbeam gravity deformation computing method of method of finite difference is 8.37% with the average error rate of actual beam deformation, the error amount of main machining area is 0.0564mm to the maximum.Demonstrate based on method of finite difference and the correctness reversing the finite element result bearing calibration calculated.

Embodiment two: present embodiment and embodiment one unlike: by placing parallels at crossbeam and column assembling place in step one, the experiment of simulation practical set condition design heavy machine tool crossbeam deformation induced by gravity, obtains considering that the crossbeam deformation induced by gravity curve detailed process of material inhomogeneity is:

(1), according to heavy planer-type milling machine crossbeam vertical view, using the crossbeam mid point in the surface level of crossbeam lower guideway place as coordinate origin O, set up cartesian coordinate system, X-direction is along beam guideway direction, and be to the right just, Y-axis perpendicular to X-axis, and upwards for just and Z axis positive dirction meet the right-hand rule as shown in Figure 3;

(2) the equivalent S-curve that the crossbeam Finite element analysis results, calculated using level measurement spacing (i.e. step-length) obtains in conjunction with the actual processing experiential modification of crossbeam processes line style as crossbeam;

(3), by crossbeam keep flat, according to crossbeam processing line style, utilize planer-type milling machine to process the beam guideway after keeping flat, and adopt autocollimator to measure the Z-direction linearity data on flat condition sill lower guideway surface; Keep flat processing crossbeam to eliminate the impact of gravity factor on processing;

(4), crossbeam be sidelong and place parallels at crossbeam and assembling place of column leading screw, simulate actual machining state, adopt level meter or autocollimator to measure crossbeam lower guideway in the linearity data of Z-direction, obtain the lower guideway Z-axis direction linearity data that crossbeam is sidelong;

(5), calculate the difference of being sidelong the Z-direction linearity data recorded after the Z-direction linearity that records to stabilization and crossbeam keep flat, utilize this difference to obtain crossbeam deformation induced by gravity curve.Other step and parameter identical with embodiment one.

Embodiment three: present embodiment and embodiment one or two unlike: utilize theory of mechanics of materials in step 2, according to the stressing conditions of crossbeam under Gravitative Loads crossbeam be reduced to crossbeam deformation induced by gravity model and crossbeam torsional deflection model process is:

(1) utilize the computing method of the mechanics of materials to simplify computation model according to the crossbeam profile of Fig. 2, according to working environment and the assembly constraint condition of machine tool beam, crossbeam bend crushed element is reduced to free beam;

(2) gravity puts on free beam as uniformly distributed load, represents uniformly distributed load with the gravity size in crossbeam unit length and gravity load intensity; Obtain the free beam mechanical model of Fig. 6;

(3) according to free beam mechanical model, application straight beam distortion line of deflection approximate differential equation (1) calculates the Z-axis direction flexural deformation of crossbeam, obtains theoretical deformation induced by gravity curve and the crossbeam deformation induced by gravity model of crossbeam bend part;

$z^{\′\′}\left(x\right)=\frac{M\left(x\right)}{\mathrm{EI}\left(x\right)}---\left(1\right)$

In formula, x is the coordinate figure of crossbeam along guide rail direction;

The deformation curve that z (x) is crossbeam;

The moment of flexure of M (x) suffered by crossbeam bend distortion;

E is the elastic modulus of crossbeam material;

The distribution function that I (x) is cross sectional moment of inertia;

When knife rest moves on crossbeam, gravity, except causing crossbeam and bending, also can make crossbeam twist distortion;

(4) according to working environment and the assembly constraint condition of machine tool beam, be the beam that two ends are fixed by crossbeam torsional deflection simplified partial; Obtain clamped beam mechanical model as shown in Figure 7;

(5) angle of twist per unit length α when section of beam twists under gravity is obtained according to clamped beam mechanical model by straight beam torsional deflection unit torsion angle computing formula (2):

$\stackrel{\·}{\mathrm{\θ}}\left(x\right)=\mathrm{\α}\left(x\right)=\frac{T\left(x\right)}{{\mathrm{GI}}_p\left(x\right)}---\left(2\right)$

In formula, the torsion angle of each position when θ (x) is out of shape for crossbeam twists;

I _px () is for section of beam is to the polar moment of inertia of its centre of form;

The modulus of shearing that G (x) is material;

The moment of torsion of each position when T (x) is out of shape for crossbeam twists;

α (x) is unit length torsion angle

Under utilizing formula (3) to calculate twisting action, namely the displacement of point of a knife point in coordinate system Z-direction obtain crossbeam torsional deflection model (3);

$z_t\left(x\right)=\mathrm{\θ}\left(x\right)\·y_c^t---\left(3\right)$

In formula, z _tx () is point of a knife point displacement in z-direction under torsional deflection effect;

Y _c ^tfor knife rest center of gravity is to section of beam centre of form place side-play amount in the Y direction;

So far heavy planer-type milling machine crossbeam deformation induced by gravity model (1) and crossbeam torsional deflection model (3) is established.Other step and parameter identical with embodiment one or two.

Embodiment four: one of present embodiment and embodiment one to three unlike: the process setting up crossbeam gravity deformation discretization model in conjunction with method of finite difference after the crossbeam deformation induced by gravity model discretize obtained step 2 in step 3 is:

(1) consider that problem that Finite element analysis results and practical distortion differ greatly causes due to crossbeam material inhomogeneity, can think that beam part material properties is everywhere different, therefore need crossbeam deformation induced by gravity model (1) discretize under single material, set up the crossbeam gravity deformation discretization model considering material inhomogeneity; As shown in Figure 8, crossbeam is equidistantly divided into n section and discrete micro-section of n crossbeam, and the coordinate x of discrete micro-section of i-th section of crossbeam _imeet:

x _i＝x ₀+ih,i＝0,1,...,n (4)

In formula, h is step-length, h=2L/n;

L is the half of crossbeam total length;

X ₀for the starting point coordinate of crossbeam left end;

(2) for crossbeam bend crushed element, obtain according to the difference formula of second derivative and crossbeam deflection differential equation (1) the crossbeam gravity deformation discretization model (5) considering material inhomogeneity;

$\left\{\begin{array}{cc}{z}_i{|}_{i=0}=z_0,z_i{|}_{i=n}=z_n& i=0,n\\ z_{i+1}-{2z}_i+z_{i-1}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_i}& i=1,...,n-1\end{array}\right.---\left(5\right)$

In formula,

Z ₀, z ₁..., z _i..., z _nfor the Z-direction Deformation Theory value of each discrete segments of crossbeam;

M _ifor the moment suffered by each discrete segments i of crossbeam;

(EI) _ifor the bendind rigidity of crossbeam discrete segments i.Other step and parameter identical with one of embodiment one to three.

Embodiment five: one of present embodiment and embodiment one to four are unlike crossbeam gravity deformation discretization model described in heavy machine tool crossbeam deformation induced by gravity curve described in integrating step one and step 3 in step 4, and the equivalent bendind rigidity detailed process calculating discrete micro-section of crossbeam is:

(1) according to heavy machine tool crossbeam deformation induced by gravity experiment measuring obtain Z-direction linearity data and crossbeam gravity deformation discretization model by crossbeam gravity deformation discretization model (5) arrange after formula (6);

$\left\{\begin{array}{cc}{z}_i=z_{\mathrm{ri}}& i=0,...,n\\ {\left(\mathrm{EI}\right)}_{\mathrm{vi}}=h^2\frac{M_i}{z_{i+1}-{2z}_i+z_{i-1}}& i=1,...,n-1\end{array}\right.---\left(6\right)$

In formula, z _rifor the linearity of the actual measurement Z-direction of crossbeam discrete segments i in deformation induced by gravity experiment;

(EI) _vifor the equivalent bendind rigidity of the discrete micro-section of i of crossbeam;

(2) by deformation induced by gravity being tested the moment M that the measurement spacing (i.e. step-length), Z-direction linearity data and the free beam mechanical model that obtain calculate _isubstitute into formula (6) and calculate crossbeam each discrete segments equivalent bendind rigidity (EI) _v.Other step and parameter identical with one of embodiment one to four.

Embodiment six: present embodiment and one of embodiment one to five unlike: utilize crossbeam gravity deformation finite element simulation separation method to be separated by finite element gravity deformation curve in step 6 to obtain rail head to assemble rear cross beam flexural deformation finite element curve and torsional deflection finite element curve detailed process is:

(1) flexural deformation caused due to gravity and torsional deflection all have larger impact to the precision of planer-type milling machine, need to consider its impact on planer-type milling machine precision simultaneously; Crossbeam distortion is under gravity made up of bending and torsion; Because the finite difference calibration model of bending and torsion is different, when using the method to calculate crossbeam gravity deformation curve, need the bending and torsion data in finite element simulation to be separated, could correct respectively; There is a torsional centre by the known section of beam of theory of mechanics of materials, can think in this torsional centre position and only occur bending and deformation, and torsional centre position and external load have nothing to do, only relevant with section of beam own form; Experiment section of beam input ANSYS software is carried out analyzing and obtains crossbeam floor cross section and the crossbeam torsional centre position without floor cross section, as is illustrated by figs. 11 and 12;

(2) because crossbeam torsional centre position does not exist solid model, need under the prerequisite not affecting gravity deformation simulation result, manual interpolation is by the entity of crossbeam torsional centre, therefore the shaft-like entity that interpolation material parameter is identical with crossbeam is selected, material end is in torsional centre position, arrange observation point to emulate Z-direction distortion in torsional centre position, obtain the finite element curve of rail head assembling rear cross beam flexural deformation curve and crossbeam torsional deflection; The setting of torsional centre observation point as shown in figure 13; Obtain rail head assembling rear cross beam flexural deformation curve and rail head assembling rear cross beam torsional deflection curve data as table 4.Other step and parameter identical with one of embodiment one to five.

Embodiment seven: one of present embodiment and embodiment one to six unlike: utilize the equivalent bendind rigidity that step 4 calculates in step 7, correct the finite element curve that step 6 calculates rail head assembling rear cross beam flexural deformation and crossbeam torsional deflection based on method of finite difference, obtaining final crossbeam gravity deformation curve detailed process is:

(1) rail head assembling rear cross beam flexural deformation curve is corrected

(1) for flexural deformation part, there is the relation such as formula (7) in the Z-direction deformation values of crossbeam coordinate system and bendind rigidity:

$\left\{\begin{array}{cc}{z}_i{|}_{i=0}=z_0,z_i{|}_{i=n}=z_n& i=0,n\\ z_{i+1}-{2z}_i+z_{i-1}=\frac{h^2{M}_i}{{\left(\mathrm{EI}\right)}_i}& i=1,...,n-1\end{array}\right.---\left(7\right)$

(2) because the material properties parameter of finite-element preprocessing process input is certain value, truly cannot reflect the material inhomogeneity of actual crossbeam, therefore need the crossbeam equivalent bendind rigidity utilizing step 4 to calculate to revise the bendind rigidity that finite element analysis inputs, calculate the crossbeam gravity deformation curve considering material;

(3) utilize finite difference method, the emulated data of finite element is carried out data processing according to the left side of equation (5), obtain the finite difference fraction (8) of finite element simulation;

$\left\{\begin{array}{cc}{z}_i^{\mathrm{bs}}{|}_{i=0}=z_0^{\mathrm{bs}},z_i^{\mathrm{bs}}{|}_{i=n}=z_n^{\mathrm{bs}}& i=0,n\\ z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{bs}}+z_{i+1}^{\mathrm{bs}}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}& i=1,...,n-1\end{array}\right.---\left(8\right)$

In formula, z ^bs _ifor crossbeam each section of flexural deformation Z-direction deformation values that finite element simulation obtains;

(EI) _inputfor the theoretical bendind rigidity of the theoretical bendind rigidity that inputs during FEM (finite element) calculation and the discrete micro-section of i of crossbeam;

(4) actual correction rear cross beam each section of flexural deformation Z-direction deformation values z ^br _imeet formula (9);

$\left\{\begin{array}{cc}{z}_0^{\mathrm{br}}=z_0^{\mathrm{bs}},z_n^{\mathrm{br}}=z_n^{\mathrm{bs}}& i=0,n\\ z_{i-1}^{\mathrm{br}}-{2z}_i^{\mathrm{br}}+z_{i+1}^{\mathrm{br}}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& \\ =h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}\·\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& \\ =(z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{bs}}+z_{i+1}^{\mathrm{bs}})\·\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& i=1,...,n-1\end{array}\right.---\left(9\right)$

(5) the theoretical bendind rigidity (EI) of the discrete micro-section of i of crossbeam is made _inputwith equivalent bendind rigidity (EI) _viratio be the correction factor k of this discrete segments ⁱ _i; I.e. formula (10):

$\left\{\begin{array}{cc}{k}_0^I=k_n^I=0& i=0,n\\ k_i^I=\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& i=1,...,n-1\end{array}\right.---\left(10\right)$

(6) because method of finite difference is based on the diastrophic approximate differential equation of mechanics of materials middle cross beam, near bearing, the calculating of equivalent bendind rigidity causes correction factor excessive due to the simplification of load in computation model, does not meet actual conditions; According to the constraint condition that crossbeam assembles in head tree position, 6 degree of freedom are all limited, and auxiliary strut place Z-direction is limited, then when practical distortion and simulation calculation, the amount of deflection of crossbeam both sides constraint portions is identical with deformation extent, and namely the starting condition of formula is:

Make z ^bs ₀=z ^br ₀, z ^bs _n=z ^br _n, k ⁱ ₁=k ⁱ _n-1=1;

Δz ^br _i＝z ^br _i-1-2z ^br _i+z ^br _i+1，Δz ^bs _i＝z ^bs _i-1-2z ^bs _i+z ^bs _i+1，

Then formula is;

$\left\{\begin{array}{cc}{z}_0^{\mathrm{br}}=z_0^{\mathrm{bs}},z_n^{\mathrm{br}}=z_n^{\mathrm{bs}}& i=0,n\\ {\mathrm{\Δz}}_i^{\mathrm{br}}={\mathrm{\Δz}}_i^{\mathrm{bs}}\·k_i^I& i=1,...,n-1\end{array}\right.---\left(11\right)$

Formula (11) is crossbeam gravity deformation curvature correction model;

(2) the finite element curve of rail head assembling rear cross beam torsional deflection is corrected:

(1) for crossbeam torsional deflection part, by the relational expression (12) of formula and shear modulus G and elastic modulus E;

$G=\frac{E}{2(1+v)}---\left(12\right)$

Wherein, ν is Poisson ratio;

Therefore crossbeam coordinate system Z-axis direction deformation values and bendind rigidity exist such as formula relation:

$z^{\′}\left(x\right)=z\left({\mathrm{\θ}}^{\′}\left(x\right)\right)=z\left(\mathrm{\α}\left(x\right)\right)=f\left(\frac{1}{\mathrm{EI}}\right)---\left(13\right);$

Wherein, θ (x) crossbeam twist distortion time each position torsion angle; α (x) is unit length torsion angle;

(2) utilize the thought of finite difference, the calculating data of finite element are carried out data processing according to the molecular moiety of first order difference form;

$z^{\′}\≈\frac{z_i-z_{i-1}}{h}---\left(14\right)$

(3) starting condition that torsional deflection corrects corrects the identical starting condition Z being torsional deflection and correcting with flexural deformation ^ts ₀=z ^tr ₀, z ^ts _n=z ^tr _n, need move left and right knife rest respectively in conjunction with during actual measurement G5 item precision, deformation curve, in the discontinuous situation of mid point, obtains the finite difference fraction (15) of finite element simulation;

$\left\{\begin{array}{cc}{z}_0^{\mathrm{ts}},z_n^{\mathrm{ts}}& i=0,n\\ z_i^{\mathrm{ts}}-z_{i-1}^{\mathrm{ts}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_i}\right)& i=1~\left[\frac{n}{2}\right]\\ z_{i+1}^{\mathrm{ts}}-z_i^{\mathrm{ts}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_i}\right)& i=\left[\frac{n}{2}\right]+1~n-1\end{array}\right.---\left(15\right)$

In formula, z ^ts _ifor crossbeam each section of torsional deflection Z-direction deformation values that finite element simulation obtains;

(4) according to the trend of crossbeam practical distortion, the correction factor k of reverse part ^iI _ifor:

$\left\{\begin{array}{cc}{k}_0^{\mathrm{II}}=k_n^{\mathrm{II}}=0& i=0,n\\ k_i^{\mathrm{II}}=\left|\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}\right|& i=1,...,n-1\end{array}\right.---\left(16\right)$

(5) then the result on the right of equation in formula (16) is corrected, if the Z-direction deformation values correcting each section of rear cross beam is z ^tr _i(i=1 ~ n-1); Make z ^tr _i-z ^tr _i-1=Δ z ^tr _i(i=1 ~ [n/2]) and z ^tr _i+1-z ^tr _i=Δ z ^tr _i(i=[n/2]+1 ~ n-1), then formula (15) is:

$\left\{\begin{array}{cc}{z}_0^{\mathrm{tr}}=z_0^{\mathrm{ts}},z_n^{\mathrm{tr}}=z_n^{\mathrm{ts}}& i=0,n\\ {\mathrm{\Δz}}_i^{\mathrm{tr}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}\right)=k_i^{\mathrm{II}}\·f\left(\frac{1}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}\right)& i=1,...,n-1\end{array}\right.---\left(17\right)$

Formula (17) is crossbeam torsional deflection calibration model; So far obtain based on the namely final crossbeam gravity deformation curve of crossbeam gravity deformation curve computation model (18) of method of finite difference:

$z_i^r=z_i^{\mathrm{br}}+z_i^{\mathrm{tr}}---\left(18\right)$

Z in formula ^r _ifor the crossbeam each section of Z-direction deformation values after finite element result correction.Other step and parameter identical with one of embodiment one to six.

Following examples are adopted to verify beneficial effect of the present invention:

Embodiment one:

A kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference of the present embodiment, specifically prepare according to following steps:

Step one, by placing parallels at crossbeam and column assembling place, simulation practical set condition design heavy machine tool crossbeam deformation induced by gravity is tested, and obtains the crossbeam deformation induced by gravity curve of consideration material inhomogeneity;

(1), according to heavy planer-type milling machine crossbeam vertical view, using the crossbeam mid point in the surface level of crossbeam lower guideway place as coordinate origin O, set up cartesian coordinate system, X-direction is along beam guideway direction, and be to the right just, Y-axis perpendicular to X-axis, and upwards for just and Z axis positive dirction meet the right-hand rule as shown in Figure 3;

(2) the equivalent S-curve that the crossbeam Finite element analysis results, calculated using level measurement spacing (i.e. step-length) 460mm obtains in conjunction with the actual processing experiential modification of crossbeam processes line style as crossbeam;

(3), by crossbeam keep flat, according to crossbeam processing line style, utilize planer-type milling machine to process the beam guideway after keeping flat, and adopt autocollimator to measure the Z-direction linearity data on flat condition sill lower guideway surface; Keep flat processing crossbeam to eliminate the impact of gravity factor on processing;

(4), by crossbeam be sidelong 68 hours and place parallels at crossbeam and assembling place of column leading screw, simulating actual machining state, adopting level meter or autocollimator to measure the linearity data of crossbeam lower guideway in Z-direction, as shown in table 1; Obtain the lower guideway Z-axis direction linearity data that crossbeam is sidelong;

(5), calculate the difference of being sidelong the Z-direction linearity data recorded after the Z-direction linearity that records to stabilization keeps flat with crossbeam, utilize this difference obtain crossbeam deformation induced by gravity curve as after table 1 three arrange shown in:

Table 1 deformation induced by gravity experiment crossbeam lower guideway Z-direction linearity data

Step 2, utilize theory of mechanics of materials, according to the stressing conditions of crossbeam under Gravitative Loads, crossbeam is reduced to crossbeam deformation induced by gravity model and crossbeam torsional deflection model;

(1) utilize the computing method of the mechanics of materials to simplify computation model according to the crossbeam profile of Fig. 2, according to working environment and the assembly constraint condition of machine tool beam, crossbeam bend crushed element is reduced to free beam;

(2) gravity puts on free beam as uniformly distributed load, represents uniformly distributed load with the gravity size in crossbeam unit length and gravity load intensity; Obtain the free beam mechanical model of Fig. 6;

(3) because this heavy machine tool crossbeam span is 9500mm, be highly 1350mm, span-depth radio is greater than 5, according to free beam mechanical model, application straight beam distortion line of deflection approximate differential equation (1) calculates the Z-axis direction flexural deformation of crossbeam, obtains theoretical deformation induced by gravity curve and the crossbeam deformation induced by gravity model of crossbeam bend part;

$z^{\′\′}\left(x\right)=\frac{M\left(x\right)}{\mathrm{EI}\left(x\right)}---\left(1\right)$

In formula, x is the coordinate figure of crossbeam along guide rail direction;

The deformation curve that z (x) is crossbeam;

The moment of flexure of M (x) suffered by crossbeam bend distortion;

E is the elastic modulus of crossbeam material;

The distribution function that I (x) is cross sectional moment of inertia;

When knife rest moves on crossbeam, gravity, except causing crossbeam and bending, also can make crossbeam twist distortion;

(4) according to working environment and the assembly constraint condition of machine tool beam, be the beam that two ends are fixed by crossbeam torsional deflection simplified partial; Obtain clamped beam mechanical model as shown in Figure 7;

(5) angle of twist per unit length α when section of beam twists under gravity is obtained according to clamped beam mechanical model by straight beam torsional deflection unit torsion angle computing formula (2):

$\stackrel{\·}{\mathrm{\θ}}\left(x\right)=\mathrm{\α}\left(x\right)=\frac{T\left(x\right)}{{\mathrm{GI}}_p\left(x\right)}---\left(2\right)$

In formula, the torsion angle of each position when θ (x) is out of shape for crossbeam twists;

I _px () is for section of beam is to the polar moment of inertia of its centre of form;

The modulus of shearing that G (x) is material;

The moment of torsion of each position when T (x) is out of shape for crossbeam twists;

α (x) is unit length torsion angle

Under utilizing formula (3) to calculate twisting action, namely the displacement of point of a knife point in coordinate system Z-direction obtain crossbeam torsional deflection model (3);

$z_t\left(x\right)=\mathrm{\θ}\left(x\right)\·y_c^t---\left(3\right)$

In formula, z _tx () is point of a knife point displacement in z-direction under torsional deflection effect;

Y _c ^tfor knife rest center of gravity is to section of beam centre of form place side-play amount in the Y direction;

So far heavy planer-type milling machine crossbeam deformation induced by gravity model (1) and crossbeam torsional deflection model (3) is established.

Step 3, crossbeam is separated into one group discrete micro-section, after the crossbeam deformation induced by gravity model discretize that step 2 is obtained, is set up to crossbeam gravity deformation discretization model in conjunction with method of finite difference:

(1) consider that problem that Finite element analysis results and practical distortion differ greatly causes due to crossbeam material inhomogeneity, can think that beam part material properties is everywhere different, therefore need crossbeam deformation induced by gravity model (1) discretize under single material, set up the crossbeam gravity deformation discretization model considering material inhomogeneity; As shown in Figure 8, crossbeam is equidistantly divided into n section and discrete micro-section of n crossbeam, and the coordinate x of discrete micro-section of i-th section of crossbeam _imeet:

x _i＝x ₀+ih,i＝0,1,...,n (4)

In formula, h is step-length, h=2L/n;

L is the half of crossbeam total length;

X ₀for the starting point coordinate of crossbeam left end;

(2) for crossbeam bend crushed element, obtain according to the difference formula of second derivative and crossbeam deflection differential equation (1) the crossbeam gravity deformation discretization model (5) considering material inhomogeneity;

$\left\{\begin{array}{cc}{z}_i{|}_{i=0}=z_0,z_i{|}_{i=n}=z_n& i=0,n\\ z_{i+1}-{2z}_i+z_{i-1}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_i}& i=1,...,n-1\end{array}\right.---\left(5\right)$

In formula,

Z ₀, z ₁..., z _i..., z _nfor the Z-direction Deformation Theory value of each discrete segments of crossbeam;

M _ifor the moment suffered by each discrete segments i of crossbeam;

(EI) _ifor the bendind rigidity of crossbeam discrete segments i.

Crossbeam gravity deformation discretization model described in heavy machine tool crossbeam deformation induced by gravity curve described in step 4, integrating step one and step 3, calculates the equivalent bendind rigidity of discrete micro-section of crossbeam; Wherein, equivalent bendind rigidity characterizes the material properties of crossbeam;

(1) according to heavy machine tool crossbeam deformation induced by gravity experiment measuring obtain Z-direction linearity data and crossbeam gravity deformation discretization model by crossbeam gravity deformation discretization model (5) arrange after formula (6);

$\left\{\begin{array}{cc}{z}_i=z_{\mathrm{ri}}& i=0,...,n\\ {\left(\mathrm{EI}\right)}_{\mathrm{vi}}=h^2\frac{M_i}{z_{i+1}-{2z}_i+z_{i-1}}& i=1,...,n-1\end{array}\right.---\left(6\right)$

In formula, z _rifor the linearity of the actual measurement Z-direction of crossbeam discrete segments i in deformation induced by gravity experiment;

(EI) _vifor the equivalent bendind rigidity of the discrete micro-section of i of crossbeam;

(2) by deformation induced by gravity being tested the moment M that the measurement spacing (i.e. step-length), Z-direction linearity data and the free beam mechanical model that obtain calculate _isubstitute into formula (6) and calculate crossbeam each discrete segments equivalent bendind rigidity (EI) _v; Table 2 gives the result of calculation of experiment crossbeam each discrete segments equivalent bendind rigidity;

Table 2 equivalent bendind rigidity result of calculation

Step 5, practical set condition by finite element method for simulating heavy machine tool crossbeam, calculate crossbeam finite element gravity deformation curve after crossbeam and rail head being assembled; Wherein, crossbeam finite element gravity deformation comprises crossbeam bend distortion and torsional deflection;

(1) ANSYS WORKBENCH is utilized to carry out finite element simulation to crossbeam distortion under gravity; Finite-element preprocessing process:

The model of heavy planer-type milling machine, primarily of parts compositions such as crossbeam, column, ram, knife rests, needs each parts definition material attribute, comprises the density p of elastic modulus E, Poisson ratio ν and material before emulation solves; Each parts simulation parameter information is as shown in table 3, and the material properties defined is all draw according to experience of engineering;

Table 3 heavy planer-type milling machine component materials parameter

(2) constraint condition of crossbeam in practical set is analyzed: lathe right side uprights is head tree, left column is auxiliary strut, when installing crossbeam, in assembling place of head tree guide rail and crossbeam, tip iron is set to eliminate fit-up gap, together with the effect of cylinder clamp, make crossbeam at head tree place except the translational degree of freedom of Z-direction all the other 5 degree of freedom be all limited, therefore the displacement constraint in coordinate system X and Y direction is added in assembling place on the right side of crossbeam, and displacement is restricted to 0mm; And crossbeam is when assembling at auxiliary strut place, coordinate system Y direction is due to the effect of clamp device, column and cross beam contacting surface clamp, and coordinate system X-direction is owing to leaving the gap of 5 ~ 10mm, then Y-direction translation, X, Z-direction rotational freedom are limited, and the degree of freedom in its excess-three direction is unrestricted; The friction force that machine beam clamping device produces is not enough to support whole crossbeam, and crossbeam mainly relies on leading screw to support, and shows that the degree of freedom of crossbeam in feed screw nut position in coordinate system Z-direction is restricted; Therefore cylinder constraint is applied on the face of cylinder at lead screw position place, limit its axial freedom, the constraint condition of simulation leading screw; Load defines overall gravity load; The constraint condition of crossbeam and load define as shown in Figure 4 and Figure 5;

(2) crossbeam FEM Numerical Simulation:

Crossbeam finite element gravity deformation curve is obtained by the crossbeam point of a knife point Z-direction gravity deformation under simulation calculation crossbeam and knife rest effect; Emulation is in conjunction with actual test case, crossbeam is divided into left and right two parts, crossbeam left-half is solved respectively in the distortion at right knife rest point of a knife point of the distortion of left knife rest point of a knife point and crossbeam right half part to it, the crossbeam emulated data that point of a knife point Z-direction is out of shape under the Z-direction distortion that 460mm calculates a point of a knife point obtains crossbeam and blade carrier component Action of Gravity Field; Fig. 9 is point of a knife point gravity deformation in z-direction under Action of Gravity Field;

According to the emulated data that point of a knife point Z-direction under crossbeam and blade carrier component Action of Gravity Field is out of shape, the Z-direction deformation curve of the crossbeam gravity of drafting, obtains crossbeam finite element gravity deformation curve as shown in Figure 10;

Step 6, utilize crossbeam gravity deformation finite element simulation separation method to be separated by finite element gravity deformation curve to obtain rail head assembling rear cross beam flexural deformation finite element curve and torsional deflection finite element curve;

(1) flexural deformation caused due to gravity and torsional deflection all have larger impact to the precision of planer-type milling machine, need to consider its impact on planer-type milling machine precision simultaneously; Crossbeam distortion is under gravity made up of bending and torsion; Because the finite difference calibration model of bending and torsion is different, when using the method to calculate crossbeam gravity deformation curve, need the bending and torsion data in finite element simulation to be separated, could correct respectively; There is a torsional centre by the known section of beam of theory of mechanics of materials, can think in this torsional centre position and only occur bending and deformation, and torsional centre position and external load have nothing to do, only relevant with section of beam own form; Experiment section of beam input ANSYS software is carried out analyzing and obtains crossbeam floor cross section and the crossbeam torsional centre position without floor cross section, as is illustrated by figs. 11 and 12;

(2) because crossbeam torsional centre position does not exist solid model, need under the prerequisite not affecting gravity deformation simulation result, manual interpolation is by the entity of crossbeam torsional centre, therefore the shaft-like entity that interpolation material parameter is identical with crossbeam is selected, material end is in torsional centre position, arrange observation point to emulate Z-direction distortion in torsional centre position, obtain the finite element curve of rail head assembling rear cross beam flexural deformation curve and crossbeam torsional deflection; The setting of torsional centre observation point as shown in figure 13; Obtain rail head assembling rear cross beam flexural deformation curve and rail head assembling rear cross beam torsional deflection curve data as table 4;

Table 4 crossbeam bend and torsional deflection finite element simulation mask data

Step 7, the equivalent bendind rigidity utilizing step 4 to calculate, based on method of finite difference, the finite element curve that step 6 calculates rail head assembling rear cross beam flexural deformation and crossbeam torsional deflection is corrected, obtain final crossbeam gravity deformation curve;

(1) rail head assembling rear cross beam flexural deformation curve is corrected

(1) for flexural deformation part, there is the relation such as formula (7) in the Z-direction deformation values of crossbeam coordinate system and bendind rigidity:

$\left\{\begin{array}{cc}{z}_i{|}_{i=0}=z_0,z_i{|}_{i=n}=z_n& i=0,n\\ z_{i+1}-{2z}_i+z_{i-1}=\frac{h^2{M}_i}{{\left(\mathrm{EI}\right)}_i}& i=1,...,n-1\end{array}\right.---\left(7\right)$

(2) because the material properties parameter of finite-element preprocessing process input is certain value, truly cannot reflect the material inhomogeneity of actual crossbeam, therefore need the crossbeam equivalent bendind rigidity utilizing step 4 to calculate to revise the bendind rigidity that finite element analysis inputs, calculate the crossbeam gravity deformation curve considering material;

(3) utilize finite difference method, the emulated data of finite element is carried out data processing according to the left side of equation (5), obtain the finite difference fraction (8) of finite element simulation;

$\left\{\begin{array}{cc}{z}_i^{\mathrm{bs}}{|}_{i=0}=z_0^{\mathrm{bs}},z_i^{\mathrm{bs}}{|}_{i=n}=z_n^{\mathrm{bs}}& i=0,n\\ z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{bs}}+z_{i+1}^{\mathrm{bs}}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}& i=1,...,n-1\end{array}\right.---\left(8\right)$

In formula, z ^bs _ifor crossbeam each section of flexural deformation Z-direction deformation values that finite element simulation obtains;

(EI) _inputfor the theoretical bendind rigidity of the theoretical bendind rigidity that inputs during FEM (finite element) calculation and the discrete micro-section of i of crossbeam;

(4) actual correction rear cross beam each section of flexural deformation Z-direction deformation values z ^br _imeet formula (9);

$\left\{\begin{array}{cc}{z}_0^{\mathrm{br}}=z_0^{\mathrm{bs}},z_n^{\mathrm{br}}=z_n^{\mathrm{bs}}& i=0,n\\ z_{i-1}^{\mathrm{br}}-{2z}_i^{\mathrm{br}}+z_{i+1}^{\mathrm{br}}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& \\ =h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}\·\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& \\ =(z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{bs}}+z_{i+1}^{\mathrm{bs}})\·\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& i=1,...,n-1\end{array}\right.---\left(9\right)$

(5) the theoretical bendind rigidity (EI) of the discrete micro-section of i of crossbeam is made _inputwith equivalent bendind rigidity (EI) _viratio be the correction factor k of this discrete segments ⁱ _i; I.e. formula (10):

$\left\{\begin{array}{cc}{k}_0^I=k_n^I=0& i=0,n\\ k_i^I=\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& i=1,...,n-1\end{array}\right.---\left(10\right)$

(6) because method of finite difference is based on the diastrophic approximate differential equation of mechanics of materials middle cross beam, near bearing, the calculating of equivalent bendind rigidity causes correction factor excessive due to the simplification of load in computation model, does not meet actual conditions; According to the constraint condition that crossbeam assembles in head tree position, 6 degree of freedom are all limited, and auxiliary strut place Z-direction is limited, then when practical distortion and simulation calculation, the amount of deflection of crossbeam both sides constraint portions is identical with deformation extent, and namely the starting condition of formula is:

Make z ^bs ₀=z ^br ₀, z ^bs _n=z ^br _n, k ⁱ ₁=k ⁱ _n-1=1;

Δz ^br _i＝z ^br _i-1-2z ^br _i+z ^br _i+1，Δz ^bs _i＝z ^bs _i-1-2z ^bs _i+z ^bs _i+1，

Then formula is;

$\left\{\begin{array}{cc}{z}_0^{\mathrm{br}}=z_0^{\mathrm{bs}},z_n^{\mathrm{br}}=z_n^{\mathrm{bs}}& i=0,n\\ {\mathrm{\Δz}}_i^{\mathrm{br}}={\mathrm{\Δz}}_i^{\mathrm{bs}}\·k_i^I& i=1,...,n-1\end{array}\right.---\left(11\right)$

Formula (11) is crossbeam gravity deformation curvature correction model;

(2) the finite element curve of rail head assembling rear cross beam torsional deflection is corrected:

(1) for crossbeam torsional deflection part, by the relational expression (12) of formula and shear modulus G and elastic modulus E;

$G=\frac{E}{2(1+v)}---\left(12\right)$

Wherein, ν is Poisson ratio;

Therefore crossbeam coordinate system Z-axis direction deformation values and bendind rigidity exist such as formula relation:

$z^{\′}\left(x\right)=z\left({\mathrm{\θ}}^{\′}\left(x\right)\right)=z\left(\mathrm{\α}\left(x\right)\right)=f\left(\frac{1}{\mathrm{EI}}\right)---\left(13\right);$

Wherein, θ (x) crossbeam twist distortion time each position torsion angle; α (x) is unit length torsion angle;

(2) utilize the thought of finite difference, the calculating data of finite element are carried out data processing according to the molecular moiety of first order difference form;

$z^{\′}\≈\frac{z_i-z_{i-1}}{h}---\left(14\right)$

(3) starting condition that torsional deflection corrects corrects the identical starting condition Z being torsional deflection and correcting with flexural deformation ^ts ₀=z ^tr ₀, z ^ts _n=z ^tr _n, need move left and right knife rest respectively in conjunction with during actual measurement G5 item precision, deformation curve, in the discontinuous situation of mid point, obtains the finite difference fraction (15) of finite element simulation;

$\left\{\begin{array}{cc}{z}_0^{\mathrm{ts}},z_n^{\mathrm{ts}}& i=0,n\\ z_i^{\mathrm{ts}}-z_{i-1}^{\mathrm{ts}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_i}\right)& i=1~\left[\frac{n}{2}\right]\\ z_{i+1}^{\mathrm{ts}}-z_i^{\mathrm{ts}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_i}\right)& i=\left[\frac{n}{2}\right]+1~n-1\end{array}\right.---\left(15\right)$

In formula, z ^ts _ifor crossbeam each section of torsional deflection Z-direction deformation values that finite element simulation obtains;

(4) according to the trend of crossbeam practical distortion, the correction factor k of reverse part ^iI _ifor:

$\left\{\begin{array}{cc}{k}_0^{\mathrm{II}}=k_n^{\mathrm{II}}=0& i=0,n\\ k_i^{\mathrm{II}}=\left|\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}\right|& i=1,...,n-1\end{array}\right.---\left(16\right)$

(5) then the result on the right of equation in formula (16) is corrected, if the Z-direction deformation values correcting each section of rear cross beam is z ^tr _i(i=1 ~ n-1); Make z ^tr _i-z ^tr _i-1=Δ z ^tr _i(i=1 ~ [n/2]) and z ^tr _i+1-z ^tr _i=Δ z ^tr _i(i=[n/2]+1 ~ n-1), then formula (15) is:

$\left\{\begin{array}{cc}{z}_0^{\mathrm{tr}}=z_0^{\mathrm{ts}},z_n^{\mathrm{tr}}=z_n^{\mathrm{ts}}& i=0,n\\ {\mathrm{\Δz}}_i^{\mathrm{tr}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}\right)=k_i^{\mathrm{II}}\·f\left(\frac{1}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}\right)& i=1,...,n-1\end{array}\right.---\left(17\right)$

Formula (17) is crossbeam torsional deflection calibration model; So far obtain based on the namely final crossbeam gravity deformation curve of crossbeam gravity deformation curve computation model (18) of method of finite difference:

$z_i^r=z_i^{\mathrm{br}}+z_i^{\mathrm{tr}}---\left(18\right)$

Z in formula ^r _ifor the crossbeam each section of Z-direction deformation values after finite element result correction;

(3) crossbeam G5 item precision is measured after crossbeam being installed, by test data curve plotting as shown in figure 15; Crossbeam initial manufacture curve is deducted the practical distortion curve that crossbeam G5 item precision curve is crossbeam;

(4) namely crossbeam gravity deformation curve is as shown in figure 14 accurately for crossbeam finite element gravity deformation curvature correction result;

The above-mentioned crossbeam gravity deformation curve computing method based on method of finite difference are utilized to calculate (mask data need carry out interpolation processing according to 230mm interval) the finite element simulation data after separation, obtain the crossbeam gravity deformation curve data after correcting, as shown in table 5:

Table 5 is based on the crossbeam gravity deformation curve result of calculation of method of finite difference

Crossbeam practical distortion, finite element simulation, flexural deformation correction and flexural deformation torsional deflection are separated the correlation data corrected, and comparing result Figure 16.

Step 8, eventually through the correctness of these computing method of G5 item precision test measuring lathe;

The final crossbeam gravity deformation curve that the crossbeam finite element gravity deformation curve, practical distortion curve and the step 7 that crossbeam gravity deformation simulation curve and step 5 are obtained obtain; Computing method correct result and contrast, as shown in figure 16; Wherein, practical distortion curve is tested middle cross beam and keep flat Z-direction linearity curve by measuring crossbeam G5 (i.e. Figure 15) the item precision curve that obtains and deformation induced by gravity and obtain at the mathematic interpolation of same coordinate position;

Curve after correcting as can be seen from Figure 16 compares the beam deformation situation of finite element simulation curve more closing to reality; As calculated, the error rate of former result of finite element and actual beam deformation is 26.86%, and the crossbeam Z-direction distortion obtained based on the crossbeam gravity deformation computing method of method of finite difference is 8.37% with the average error rate of actual beam deformation, the error amount of main machining area is 0.0564mm to the maximum; Demonstrate based on method of finite difference and the correctness reversing the finite element result bearing calibration calculated; As namely Fig. 1 completes a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference.

The present invention also can have other various embodiments; when not deviating from the present invention's spirit and essence thereof; those skilled in the art are when making various corresponding change and distortion according to the present invention, but these change accordingly and are out of shape the protection domain that all should belong to the claim appended by the present invention.

Claims (7)

1. based on a heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology for method of finite difference, it is characterized in that: a kind of heavy planer-type milling machine crossbeam gravity deformation curve computing method based on method of finite difference are specifically carried out according to the following steps:

Step one, by placing parallels at crossbeam and column assembling place, simulate practical set condition design heavy machine tool crossbeam deformation induced by gravity and test, obtain crossbeam deformation induced by gravity curve;

Step 2, utilize theory of mechanics of materials, according to the stressing conditions of crossbeam under Gravitative Loads, crossbeam is reduced to crossbeam deformation induced by gravity model and crossbeam torsional deflection model;

Crossbeam gravity deformation discretization model is set up in conjunction with method of finite difference after step 3, crossbeam deformation induced by gravity model discretize that step 2 is obtained:

Crossbeam gravity deformation discretization model described in heavy machine tool crossbeam deformation induced by gravity curve described in step 4, integrating step one and step 3, calculates the equivalent bendind rigidity of discrete micro-section of crossbeam; Wherein, equivalent bendind rigidity characterizes the material properties of crossbeam;

Step 5, practical set condition by finite element method for simulating heavy machine tool crossbeam, calculate crossbeam finite element gravity deformation curve after crossbeam and rail head being assembled; Wherein, crossbeam finite element gravity deformation comprises crossbeam bend distortion and torsional deflection;

Step 6, utilize crossbeam gravity deformation finite element simulation separation method to be separated by finite element gravity deformation curve to obtain rail head assembling rear cross beam flexural deformation finite element curve and torsional deflection finite element curve;

Step 7, the equivalent bendind rigidity utilizing step 4 to calculate, based on method of finite difference, the finite element curve that step 6 calculates rail head assembling rear cross beam flexural deformation and crossbeam torsional deflection is corrected, obtain final crossbeam gravity deformation curve; Namely a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference is completed.

2. a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference according to claim 1, it is characterized in that: by placing parallels at crossbeam and column assembling place in step one, the experiment of simulation practical set condition design heavy machine tool crossbeam deformation induced by gravity, obtaining crossbeam deformation induced by gravity curve detailed process is:

(1), according to heavy planer-type milling machine crossbeam vertical view, using the crossbeam mid point in the surface level of crossbeam lower guideway place as coordinate origin O, cartesian coordinate system is set up;

(2) the equivalent S-curve that the crossbeam Finite element analysis results, calculated using level measurement spacing obtains in conjunction with the actual processing experiential modification of crossbeam processes line style as crossbeam;

(3), by crossbeam keep flat, according to crossbeam processing line style, utilize planer-type milling machine to process the beam guideway after keeping flat, and adopt autocollimator to measure the Z-direction linearity data on flat condition sill lower guideway surface;

(4), crossbeam be sidelong and place parallels at crossbeam and assembling place of column leading screw, simulate actual machining state, adopt level meter or autocollimator to measure crossbeam lower guideway in the linearity data of Z-direction, obtain the lower guideway Z-axis direction linearity data that crossbeam is sidelong;

(5), calculate the difference of being sidelong the Z-direction linearity data recorded after the Z-direction linearity that records to stabilization and crossbeam keep flat, utilize this difference to obtain crossbeam deformation induced by gravity curve.

3. a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference according to claim 1, it is characterized in that: in step 2, utilize theory of mechanics of materials, according to the stressing conditions of crossbeam under Gravitative Loads crossbeam be reduced to crossbeam deformation induced by gravity model and crossbeam torsional deflection model process is:

(1) according to working environment and the assembly constraint condition of machine tool beam, crossbeam bend crushed element is reduced to free beam;

(2) gravity puts on free beam as uniformly distributed load, represents uniformly distributed load with crossbeam gravity load intensity; Obtain free beam mechanical model;

(3) according to free beam mechanical model, application straight beam distortion line of deflection approximate differential equation (1) calculates the Z-axis direction flexural deformation of crossbeam, obtains theoretical deformation induced by gravity curve and the crossbeam deformation induced by gravity model of crossbeam bend part;

$z^{\′\′}\left(x\right)=\frac{M\left(x\right)}{\mathrm{EI}\left(x\right)}---\left(1\right)$

In formula, x is the coordinate figure of crossbeam along guide rail direction;

The deformation curve that z (x) is crossbeam;

The moment of flexure of M (x) suffered by crossbeam bend distortion;

E is the elastic modulus of crossbeam material;

The distribution function that I (x) is cross sectional moment of inertia;

(4) by crossbeam torsional deflection simplified partial be the beam that two ends are fixed; Obtain clamped beam mechanical model;

(5) angle of twist per unit length α when section of beam twists under gravity is obtained according to clamped beam mechanical model by straight beam torsional deflection unit torsion angle computing formula (2):

$\stackrel{\·}{\mathrm{\θ}}\left(x\right)=\mathrm{\α}\left(x\right)=\frac{T\left(x\right)}{{\mathrm{GI}}_p\left(x\right)}---\left(2\right)$

In formula, the torsion angle of each position when θ (x) is out of shape for crossbeam twists;

I _px () is for section of beam is to the polar moment of inertia of its centre of form;

The modulus of shearing that G (x) is material;

The moment of torsion of each position when T (x) is out of shape for crossbeam twists;

α (x) is unit length torsion angle

Under utilizing formula (3) to calculate twisting action, namely the displacement of point of a knife point in coordinate system Z-direction obtain crossbeam torsional deflection model (3);

$z_t\left(x\right)=\mathrm{\θ}\left(x\right)\·y_c^t---\left(3\right)$

In formula, z _tx () is point of a knife point displacement in z-direction under torsional deflection effect;

Y _c ^tfor knife rest center of gravity is to section of beam centre of form place side-play amount in the Y direction.

4. a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference according to claim 1, is characterized in that: the process setting up crossbeam gravity deformation discretization model in conjunction with method of finite difference after the crossbeam deformation induced by gravity model discretize obtained step 2 in step 3 is:

(1) to crossbeam deformation induced by gravity model (1) discretize under single material, crossbeam gravity deformation discretization model is set up; Crossbeam is equidistantly divided into n section and discrete micro-section of n crossbeam, and the coordinate x of discrete micro-section of i-th section of crossbeam _imeet:

x _i＝x ₀+ih, i＝0,1,...,n (4)

In formula, h is step-length, h=2L/n;

L is the half of crossbeam total length;

X ₀for the starting point coordinate of crossbeam left end;

(2) for crossbeam bend crushed element, crossbeam gravity deformation discretization model (5) is obtained according to the difference formula of second derivative and crossbeam deflection differential equation (1);

$\left\{\begin{array}{cc}{z}_i{|}_{i=0}=z_0,z_i{|}_{i=n}=z_n& i=0,n\\ z_{i+1}-{2z}_i+z_{i-1}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_i}& i=1,...,n-1\end{array}\right.---\left(5\right)$

In formula,

Z ₀, z ₁..., z _i..., z _nfor the Z-direction Deformation Theory value of each discrete segments of crossbeam;

M _ifor the moment suffered by each discrete segments i of crossbeam;

(EI) _ifor the bendind rigidity of crossbeam discrete segments i.

5. a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference according to claim 1, it is characterized in that: crossbeam gravity deformation discretization model described in heavy machine tool crossbeam deformation induced by gravity curve described in integrating step one and step 3 in step 4, the equivalent bendind rigidity detailed process calculating discrete micro-section of crossbeam is:

(1) according to heavy machine tool crossbeam deformation induced by gravity experiment measuring obtain Z-direction linearity data and crossbeam gravity deformation discretization model by crossbeam gravity deformation discretization model (5) arrange after formula (6);

$\left\{\begin{array}{cc}{z}_i=z_{\mathrm{ri}}& i=0,...,n\\ {\left(\mathrm{EI}\right)}_{\mathrm{vi}}=h^2\frac{M_i}{z_{i+1}-{2z}_i+z_{i-1}}& i=1,...,n-1\end{array}\right.---\left(6\right)$

In formula, z _rifor the linearity of the actual measurement Z-direction of crossbeam discrete segments i in deformation induced by gravity experiment;

(EI) _vifor the equivalent bendind rigidity of the discrete micro-section of i of crossbeam;

(2) by deformation induced by gravity being tested the moment M that the measurement spacing, Z-direction linearity data and the free beam mechanical model that obtain calculate _isubstitute into formula (6) and calculate crossbeam each discrete segments equivalent bendind rigidity (EI) _v.

6. a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference according to claim 1, is characterized in that: utilize crossbeam gravity deformation finite element simulation separation method to be separated by finite element gravity deformation curve in step 6 to obtain rail head assembling rear cross beam flexural deformation finite element curve and torsional deflection finite element curve detailed process is:

(1) section of beam is inputted ANSYS software to carry out analyzing and obtain crossbeam floor cross section and the crossbeam torsional centre position without floor cross section;

(2) the shaft-like entity that interpolation material parameter is identical with crossbeam is selected, material end is in torsional centre position, arrange observation point to emulate Z-direction distortion in torsional centre position, obtain the finite element curve of rail head assembling rear cross beam flexural deformation curve and crossbeam torsional deflection.

7. a kind of heavy planer-type milling machine crossbeam gravity deformation Forecasting Methodology based on method of finite difference according to claim 1, it is characterized in that: the equivalent bendind rigidity utilizing step 4 to calculate in step 7, correct the finite element curve that step 6 calculates rail head assembling rear cross beam flexural deformation and crossbeam torsional deflection based on method of finite difference, obtaining final crossbeam gravity deformation curve detailed process is:

(1) rail head assembling rear cross beam flexural deformation curve is corrected

(1) for flexural deformation part, there is the relation such as formula (7) in the Z-direction deformation values of crossbeam coordinate system and bendind rigidity:

$\left\{\begin{array}{cc}{z}_i{|}_{i=0}=z_0,z_i{|}_{i=n}=z_n& i=0,n\\ z_{i+1}-{2z}_i+z_{i-1}=\frac{h^2{M}_i}{{\left(\mathrm{EI}\right)}_i}& i=1,...,n-1\end{array}\right.---\left(7\right)$

(2) the crossbeam equivalent bendind rigidity utilizing step 4 to calculate is revised the bendind rigidity that finite element analysis inputs, and calculates crossbeam gravity deformation curve;

(3) utilize finite difference method, the emulated data of finite element is carried out data processing according to the left side of equation (5), obtain the finite difference fraction (8) of finite element simulation;

$\left\{\begin{array}{cc}{z}_i^{\mathrm{bs}}{|}_{i=0}=z_0^{\mathrm{bs}},z_i^{\mathrm{bs}}{|}_{i=n}=z_n^{\mathrm{bs}}& i=0,n\\ z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{bs}}+z_{i+1}^{\mathrm{bs}}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}& i=1,...,n-1\end{array}\right.---\left(8\right)$

In formula, z ^bs _ifor crossbeam each section of flexural deformation Z-direction deformation values that finite element simulation obtains;

(EI) _inputfor the theoretical bendind rigidity of the theoretical bendind rigidity that inputs during FEM (finite element) calculation and the discrete micro-section of i of crossbeam;

(4) actual correction rear cross beam each section of flexural deformation Z-direction deformation values z ^br _imeet formula (9);

$\left\{\begin{array}{cc}{z}_0^{\mathrm{br}}=z_0^{\mathrm{bs}},z_n^{\mathrm{br}}=z_n^{\mathrm{bs}}& i=0,n\\ z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{br}}+z_{i+1}^{\mathrm{br}}=h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& \\ =h^2\frac{M_i}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}\·\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& \\ =(z_{i-1}^{\mathrm{bs}}-{2z}_i^{\mathrm{bs}}+z_{i+1}^{\mathrm{bs}})\·\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& i=1,...,n-1\end{array}\right.---\left(9\right)$

(5) the theoretical bendind rigidity (EI) of the discrete micro-section of i of crossbeam is made _inputwith equivalent bendind rigidity (EI) _viratio be the correction factor k of this discrete segments ⁱ _i; I.e. formula (10):

$\left\{\begin{array}{cc}{k}_0^I=k_n^I=0& i=0,n\\ k_i^I=\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}& i=1,...,n-1\end{array}\right.---\left(10\right)$

(6) amount of deflection of crossbeam both sides constraint portions is identical with deformation extent, and namely the starting condition of formula is:

Make z ^bs ₀=z ^br ₀, z ^bs _n=z ^br _n, k ⁱ ₁=k ⁱ _n-1=1;

Δz ^br _i＝z ^br _i-1-2z ^br _i+z ^br _i+1，Δz ^bs _i＝z ^bs _i-1-2z ^bs _i+z ^bs _i+1，

Then formula is;

$\left\{\begin{array}{cc}{z}_0^{\mathrm{br}}=z_0^{\mathrm{bs}},z_n^{\mathrm{br}}=z_n^{\mathrm{bs}}& i=0,n\\ {\mathrm{\Δz}}_i^{\mathrm{br}}={\mathrm{\Δz}}_i^{\mathrm{bs}}\·k_i^I& i=1,...,n-1\end{array}\right.---\left(11\right)$

Formula (11) is crossbeam gravity deformation curvature correction model;

(2) the finite element curve of rail head assembling rear cross beam torsional deflection is corrected:

(1) for crossbeam torsional deflection part, by the relational expression (12) of formula and shear modulus G and elastic modulus E;

$G=\frac{E}{2(1+v)}---\left(12\right)$

Wherein, ν is Poisson ratio;

Therefore crossbeam coordinate system Z-axis direction deformation values and bendind rigidity exist such as formula relation:

$z^{\′}\left(x\right)=z\left({\mathrm{\θ}}^{\′}\left(x\right)\right)=z\left(\mathrm{\α}\left(x\right)\right)=f\left(\frac{1}{\mathrm{EI}}\right)---\left(13\right);$

Wherein, θ (x) crossbeam twist distortion time each position torsion angle; α (x) is unit length torsion angle;

(2) the calculating data of finite element are carried out data processing according to the molecular moiety of first order difference form;

$z^{\′}\≈\frac{z_i-z_{i-1}}{h}---\left(14\right)$

(3) starting condition that torsional deflection corrects corrects the identical starting condition Z being torsional deflection and correcting with flexural deformation ^ts ₀=z ^tr ₀, z ^ts _n=z ^tr _n, need move left and right knife rest respectively in conjunction with during actual measurement G5 item precision, deformation curve, in the discontinuous situation of mid point, obtains the finite difference fraction (15) of finite element simulation;

$\left\{\begin{array}{cc}{z}_0^{\mathrm{ts}},z_n^{\mathrm{ts}}& i=0,n\\ z_i^{\mathrm{ts}}-z_{i-1}^{\mathrm{ts}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_i}\right)& i=1~\left[\frac{n}{2}\right]\\ z_{i+1}^{\mathrm{ts}}-z_i^{\mathrm{ts}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_i}\right)& i=\left[\frac{n}{2}\right]+1~n-1\end{array}\right.---\left(15\right)$

In formula, z ^ts _ifor crossbeam each section of torsional deflection Z-direction deformation values that finite element simulation obtains;

(4) according to the trend of crossbeam practical distortion, the correction factor k of reverse part ^iI _ifor:

$\left\{\begin{array}{cc}{k}_0^{\mathrm{II}}=k_n^{\mathrm{II}}=0& i=0,n\\ k_i^{\mathrm{II}}=\left|\frac{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}\right|& i=1,...,n-1\end{array}\right.---\left(16\right)$

(5) then the result on the right of equation in formula (16) is corrected, if the Z-direction deformation values correcting each section of rear cross beam is z ^tr _i(i=1 ~ n-1); Make z ^tr _i-z ^tr _i-1=Δ z ^tr _i(i=1 ~ [n/2]) and z ^tr _i+1-z ^tr _i=Δ z ^tr _i(i=[n/2]+1 ~ n-1), then formula (15) is:

$\left\{\begin{array}{cc}{z}_0^{\mathrm{tr}}=z_0^{\mathrm{ts}},z_n^{\mathrm{tr}}=z_n^{\mathrm{ts}}& i=0,n\\ {\mathrm{\Δz}}_i^{\mathrm{tr}}=f\left(\frac{1}{{\left(\mathrm{EI}\right)}_{\mathrm{vi}}}\right)=k_i^{\mathrm{II}}\·f\left(\frac{1}{{\left(\mathrm{EI}\right)}_{\mathrm{input}}}\right)& i=1,...,n-1\end{array}\right.---\left(17\right)$

Formula (17) is crossbeam torsional deflection calibration model; So far obtain based on the namely final crossbeam gravity deformation curve of crossbeam gravity deformation curve computation model (18) of method of finite difference:

$z_i^r=z_i^{\mathrm{br}}+z_i^{\mathrm{tr}}---\left(18\right)$

Z in formula ^r _ifor the crossbeam each section of Z-direction deformation values after finite element result correction.