CN105242546B - A kind of High Speed Milling Force modeling method based on material property - Google Patents
A kind of High Speed Milling Force modeling method based on material property Download PDFInfo
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- 239000000463 material Substances 0.000 title claims abstract description 79
- 238000000034 method Methods 0.000 title claims abstract description 49
- 238000003801 milling Methods 0.000 claims abstract description 101
- 238000005520 cutting process Methods 0.000 claims abstract description 48
- 239000011159 matrix material Substances 0.000 claims description 9
- 230000002153 concerted effect Effects 0.000 claims description 3
- 230000000694 effects Effects 0.000 claims description 3
- 238000012545 processing Methods 0.000 claims description 3
- 230000007704 transition Effects 0.000 claims description 3
- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical compound C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 claims description 2
- 238000004519 manufacturing process Methods 0.000 abstract description 3
- PXHVJJICTQNCMI-UHFFFAOYSA-N nickel Substances [Ni] PXHVJJICTQNCMI-UHFFFAOYSA-N 0.000 description 5
- 238000002474 experimental method Methods 0.000 description 4
- 229910001069 Ti alloy Inorganic materials 0.000 description 3
- 238000005516 engineering process Methods 0.000 description 3
- 238000003754 machining Methods 0.000 description 3
- 229910052759 nickel Inorganic materials 0.000 description 3
- 229910000601 superalloy Inorganic materials 0.000 description 3
- 241000220317 Rosa Species 0.000 description 2
- 238000013473 artificial intelligence Methods 0.000 description 2
- 229910052751 metal Inorganic materials 0.000 description 2
- 239000002184 metal Substances 0.000 description 2
- 238000005457 optimization Methods 0.000 description 2
- 238000012360 testing method Methods 0.000 description 2
- 229910045601 alloy Inorganic materials 0.000 description 1
- 239000000956 alloy Substances 0.000 description 1
- 238000004458 analytical method Methods 0.000 description 1
- 238000004364 calculation method Methods 0.000 description 1
- 238000006243 chemical reaction Methods 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 239000006185 dispersion Substances 0.000 description 1
- 238000002386 leaching Methods 0.000 description 1
- 238000012821 model calculation Methods 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 238000010200 validation analysis Methods 0.000 description 1
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Abstract
A kind of High Speed Milling Force modeling method based on material property of the present invention belongs to difficult-to-machine material curved surface part precise high-efficiency manufacture field, is related to a kind of difficult-to-machine material High Speed Milling Force modeling method based on material property.This method is primarily based on infinitesimal thought and inclined cutting principle, by working angles infinitesimal Milling Force be expressed as normal pressure and chip caused by rake face extruding workpiece material along rake face slide caused by frictional force；During material property is introduced into milling force modeling based on this, the computation model of normal pressure and frictional force based on material property is established；Then the Milling Force Model established under workpiece coordinate system；The Converse solved method of individual layer mean force is finally based on to recognize the coefficient in model.The present invention realizes a kind of accurate prediction of model to different difficult-to-machine material High Speed Milling Forces, improves the robustness of model, has a wide range of application, and improves difficult-to-machine material curved surface part crudy and efficiency.
Description
Technical field
It is more particularly to a kind of to be based on material property the invention belongs to difficult-to-machine material curved surface part precise high-efficiency manufacture field
Difficult-to-machine material High Speed Milling Force modeling method.
Background technology
Difficult-to-machine material curved surface part high-speed milling technology is studied, there is important meaning to improving its machining accuracy and efficiency
Justice.During difficult-to-machine material curved surface part high-speed milling, Milling Force is joined as a significant process physics in milling process
Number, research milling force modeling method is to optimization difficult-to-machine material curved surface part high-rate wireless LAN technique, raising crudy tool
There is important guiding effect.
At present, milling force modeling method mainly has mechanical modelling, unified mechanical model method and artificial intelligence modeling etc..
Mechanical model method is combined as modeling basis with fixed cutting tool workpiece, and after changing workpiece, model will be no longer applicable, and need to test school again
Quasi- coefficient.Though unified mechanical model method considers the influence of cutter parameters, workpiece material yield strength and machining condition etc., power
Simplified formula has been used to assume make it that Milling Force precision of prediction is low and modeling process is complicated with some during credit analysis.Artificial intelligence
Energy modeling needs enough milling experiment samples, and can not consider actual milling situation, causes Milling Force prediction error larger.
Although milling force modeling method is more, lack more blanket for difficult-to-machine material curved surface part high-speed milling force modeling
Method, it is difficult to utilize a kind of accurate prediction of a variety of difficult-to-machine material High Speed Milling Forces of model realization.
King protects the " Instantaneous Milling Force during variable curvature curved surface side milling that liter et al. patent announcement number is CN104794305A
Forecasting Methodology ", for variable curvature part Flank machining, calculated according to tool position point and instantaneously cut out angle, so as to calculate
Actual instantaneous cutting-in, and then instantaneous not deformed thickness of cutting is obtained, the size of Milling Force is obtained based on not deformed thickness of cutting.So
And the influence that material property is brought to Milling Force is not considered in this method, do not have versatility to different workpiece materials.Text
Offer " Modeling and experimental validation of cutting forces in five-axis ball-
End milling based on true tooth trajectory ", Gaiyun He etc., International Journal
Of Advanced Manufacturing Technology, 2015,78 (1-4), 189-197, it is proposed that one kind is based on cutter tooth
The rose cutter five-axis milling power forecast model of actual motion track, it will predict that obtained Milling Force surveys Milling Force pair with experiment
Than, the results showed that prediction result and the experimental result goodness of fit are higher.However, the milling force modeling process involved by this method does not relate to
And the influence that material property is brought to Milling Force, thus it is accurately pre- to different difficult-to-machine material curved surface part process Milling Forces
Larger limitation be present in survey.
The content of the invention
It is contemplated that the defects of overcoming prior art, invents a kind of difficult-to-machine material high-speed milling based on material property
Force modeling method, material property is introduced into High Speed Milling Force modeling process, realizes that a kind of model is high to different difficult-to-machine materials
The accurate prediction of fast Milling Force, the robustness of model is improved, cutting process control for difficult-to-machine material Surfaces for High Speed Milling Operation provides theory
With technical support, difficult-to-machine material curved surface part crudy and processing efficiency are improved.
A kind of High Speed Milling Force modeling method based on material property of the technical solution adopted in the present invention, its feature exist
In being primarily based on infinitesimal thought and inclined cutting principle, working angles be equivalent to a series of line of small inclined cuttings processing
Property superposition, and by infinitesimal Milling Force be expressed as rake face extruding workpiece material caused by normal pressure and chip slide and produce along rake face
Raw frictional force；During material property is introduced into milling force modeling based on this, the normal pressure based on material property is established
With the computation model of frictional force, applicability of the model to workpiece material and milling condition is improved；Then, establish under workpiece coordinate system
Milling Force Model；It is finally based on the Converse solved method of individual layer mean force to recognize the coefficient in model, foundation is applied to
The High Speed Milling Force forecast model of difficult-to-machine material not of the same race.Modeling method comprises the following steps that：
1) instantaneous infinitesimal Milling Force calculates
To establish Milling Force Model, tool coordinate system Xc-Yc-Zc, local coordinate system t-r-a and workpiece coordinate are initially set up
It is Xw-Yw-Zw, and determines local coordinate system to the transition matrix A and tool coordinate system of tool coordinate system to workpiece coordinate system
Transition matrix B.
Based on infinitesimal thought and inclined cutting principle, infinitesimal Milling Force (dFt, dFa, dFr) can represent under local coordinate system
For：
In formula, η is chip flow angle, λsFor current inclined cutting inclination angle, Fn is that rake face extrudes workpiece material generation
Normal pressure, Fs be chip along rake face slide caused by frictional force.
SeparatelyThen have
2) rake face normal pressure calculates with frictional force
With reference to theory of metal cutting and difficult-to-machine material high-speed milling feature, rake face normal pressure Fn is expressed as：
In formula, c is coefficient, and σ is material yield strength, and ε is the thermal conductivity factor of difficult-to-machine material, and v is cutting speed, ADFor
Working angles material is squeezed area.Wherein, v=2 π Rsin κ S/60, R is tool radius, and κ is cutting infinitesimal axial direction
Position angle, S are the speed of mainshaft.
Working angles material is squeezed area ADIt is represented by：
AD=tn·dL (4)
In formula, tnFor instantaneous not deformed thickness of cutting corresponding to cutting edge infinitesimal, dL is tool in cutting sword infinitesimal length.
For difficult-to-machine material curved surface, judge whether blade infinitesimal participates in cutting using Z-map methods, obtain：
In formula, fzr,fzaRespectively feed engagement fzHorizontal component and vertical component, ψ be cutting infinitesimal radial direction leaching
Entering angle, κ is cutting infinitesimal axial location angle,For axial coordinate of the cutting infinitesimal under workpiece coordinate system, Z is work pieces process
Axial coordinate of the face under workpiece coordinate system.
In view of the plasticity ability T of material influences the coefficient of friction of rake face, coefficient of friction and material plasticity positive correlation, then before
Knife face frictional force is represented by：
Fs=nTFn (6)
In formula, n is coefficient.
3) high-speed milling force modeling
Infinitesimal Milling Force (dFt, dFa, dFr) under local coordinate system is transformed to by coordinate transform micro- under workpiece coordinate system
First Milling ForceFor：
In formula,For infinitesimal Milling Force under local coordinate system.
Total Milling Force under workpiece coordinate systemThe superposition of infinitesimal Milling Force is expressed as, is：
In formula, NjFor blade number, m is that blade instantaneously participates in cutting infinitesimal number.
4) High Speed Milling Force identification of Model Parameters
By formula (8), can obtain：
In formula, i is cutting infinitesimal layer index, and j indexes for blade.
Formula (3) and (6) are substituted into (9) and can obtained：
For the coefficient c and n introduced in identification model, dynamometry at any one cutter corner k need to be only chosen in working angles
What instrument measuredIt can be solved by any two of three equations in formula (10).Adopted to reduce error
With average milling force method, the milling that cutter rotates a circle makes a concerted effort to be：
It will rotate a circle and be divided into NkPart, then averagely Milling Force is expressed as：
OrderTo avoid computing repeatedly matrix, meter is improved
Calculate efficiency, the method combined using hierarchical solving with mean force.If there was only the difference in height of a cutting infinitesimal between cutting-in twice,
Then the difference of empirical average Milling Force is twice：
In formula (13),For constant matrices.Order：
Average Milling Force is：
It can be obtained by formula (15)：
Coefficient c and n can be obtained by formula (16), complete milling force modeling.
The remarkable result and benefit of the present invention is based on differential thought and inclined cutting principle, and material property is introduced at a high speed
During milling force modeling, a kind of High Speed Milling Force modeling method suitable for different difficult-to-machine materials has been invented, and based on single
The layer Converse solved method of mean force recognizes to the coefficient in model, and the High Speed Milling Force of difficult-to-machine material not of the same race can be achieved
Prediction, has a wide range of application.The method of invention improves modeling efficiency without many experiments.
Brief description of the drawings
Fig. 1 --- the High Speed Milling Force modeling method overall flow figure based on material property
The Milling Force prediction of Fig. 2 --- nickel base superalloy 718 and Experimental Comparison；MEA- Milling Force measured values, CAL- millings
Cut power calculated value
The Milling Force prediction of Fig. 3 --- titanium alloy TC 4 and Experimental Comparison；MEA- Milling Force measured values, CAL- Milling Force meters
Calculation value
Specific embodiment
Describe the embodiment of the present invention in detail with reference to accompanying drawing 1 and technical scheme.
In view of during difficult-to-machine material curved surface part high-speed milling, Milling Force is a significant process in milling process
Physical parameter, therefore milling force modeling method is studied to optimization difficult-to-machine material curved surface part high-rate wireless LAN technique, raising
Crudy has important guiding effect.In addition, the difference of workpiece material characteristic also results in the change of Milling Force.Accordingly, pin
To a variety of difficult-to-machine material curved surface part milling force modeling problems, a kind of high-speed milling force modeling based on material property has been invented
Method, overall flow is referring to accompanying drawing 1.
By taking rose cutter vertical milling sphere as an example, calculate and emulate by MATLAB softwares, describe the present invention in detail and implemented
Journey.
First, initial parameter is determined.Given working process parameter is cutting-in aP=0.2mm, feed engagement fz=
0.05mm/z, speed of mainshaft S=8000rpm, using two-edged tool sharpening, tool radius R=3mm, cutter nominal pitch angle alpha=
30 °, cutting edge inclination λs=0.1 °.Based on infinitesimal thought and inclined cutting principle, it is 1000 to give the cutter bulb some discrete number of plies,
Cutter rotates a circle dispersion number as 200, and cutter axial direction layer scattering is at intervals of 1 °.Thus, one group (i, j, k) is given, according to formula
(2) relation between infinitesimal Milling Force and rake face normal pressure and frictional force, is obtained.
Secondly, with reference to theory of metal cutting and difficult-to-machine material high-speed milling feature, material parameter is given.Embodiment 1 is selected
It is σ=550Mpa, ε=11W (mK) with nickel base superalloy 718-1, T=22A5%；Embodiment 2 from titanium alloy TC 4 be σ=
825Mpa, ε=6.8W (mK)-1, T=10A5%.Rake face normal pressure Fn and frictional force Fs mathematics are obtained according to formula (3)-(6)
Expression formula is：
Then, the High Speed Milling Force model obtained according to formula (7) and (8) under workpiece coordinate system is：
Finally, to avoid computing repeatedly matrix, computational efficiency is improved, the method combined using hierarchical solving with mean force is distinguished
Know coefficient c and n.Trying to achieve the average milling relation between difference and coefficient c and n of making a concerted effort by formula (13) is：
The milling force data obtained based on experiment, coefficient c and n are tried to achieve according to formula (16), and the coefficient tried to achieve is substituted into formula
(18)-(19), milling force modeling is completed.
To verify the validity of Milling Force Model, all infinitesimal Milling Forces are carried out after calculating solution superposition using MATLAB
It can obtain Milling Force calculated value.By taking nickel-base alloy 718 as an example, i=100, k=50, j=1, d κ=0.1 °, N are takenk=200, root
DL=5.24 × 10 are obtained according to cutting geometrical relationship-6Mm, κ=45 °, ψ=89.5 °, not deformed thickness of cutting t is tried to achieve according to formula (5)n
=3.9 × 10-6mm.Coordinate conversion matrixMatrixC=0.42, n=4.3 are obtained based on cutting test and formula (16), according to formula (17)
Try to achieve rake face normal pressure Fn=0.3547N, frictional force Fs=0.3408N, can be in the hope of micro- under local coordinate system according to formula (1)
First Milling Force isCan be in the hope of infinitesimal Milling Force under workpiece coordinate system according to formula (7)Infinitesimal Milling Force is under workpiece coordinate systemWorkpiece can be obtained by formula (18)
High Speed Milling Force is under coordinate system
For different difficult-to-machine materials, the milling force value measured and model calculation value are contrasted, referring to accompanying drawing 2 and accompanying drawing 3.
Milling Force prediction and Experimental Comparison result of the accompanying drawing 2 for nickel base superalloy 718, accompanying drawing 3 are predicted for the Milling Force of titanium alloy TC 4
With Experimental Comparison result.As a result show, Milling Force predicted value and the measured value goodness of fit are higher, and effectively hardly possible not of the same race can be added
Work material high-speed milling power is precisely predicted.
The present invention realizes that a variety of difficult-to-machine material High Speed Milling Forces are accurately predicted for being difficult by a kind of Milling Force Model,
A kind of difficult-to-machine material High Speed Milling Force modeling method based on material property has been invented, material property is introduced into milling force modeling
During, a kind of accurate prediction of model to different difficult-to-machine material High Speed Milling Forces is realized, improves the robustness of model, is applied
Scope is wide, and cutting process control for difficult-to-machine material Surfaces for High Speed Milling Operation provides theoretical and technical support, improves difficult-to-machine material curved surface
Part crudy and efficiency.
Claims (1)
1. a kind of High Speed Milling Force modeling method based on material property, it is characterised in that be primarily based on infinitesimal thought and oblique angle
Cutting principle, working angles is equivalent to a series of linear superposition of small inclined cutting processing, and infinitesimal Milling Force is represented
For rake face extrude normal pressure and chip caused by workpiece material along rake face slide caused by frictional force；Based on this by material
Expect that characteristic is introduced during milling force modeling, establish the computation model of normal pressure and frictional force based on material property, improve mould
Applicability of the type to workpiece material and milling condition；Then, the Milling Force Model established under workpiece coordinate system；It is finally based on individual layer
The Converse solved method of mean force recognizes to the coefficient in model, establishes the high-speed milling suitable for difficult-to-machine material not of the same race
Power forecast model；Modeling method comprises the following steps that：
1) instantaneous infinitesimal Milling Force calculates
To establish Milling Force Model, tool coordinate system Xc-Yc-Zc, local coordinate system t-r-a and workpiece coordinate system are initially set up
Xw-Yw-Zw, and determine that the transition matrix A and tool coordinate system of local coordinate system to tool coordinate system turn to workpiece coordinate system
Change matrix B；
Based on infinitesimal thought and inclined cutting principle, infinitesimal Milling Force (dFt, dFa, dFr) is represented by under local coordinate system：
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In formula, η is chip flow angle, λsFor current inclined cutting inclination angle, Fn is that rake face extrudes malleation caused by workpiece material
Power, Fs be chip along rake face slide caused by frictional force；
SeparatelyThen have
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2) rake face normal pressure calculates with frictional force
Rake face normal pressure Fn is expressed as：
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In formula, c is coefficient, and σ is material yield strength, and ε is the thermal conductivity factor of difficult-to-machine material, and v is cutting speed, ADFor cutting
Process material is squeezed area；Wherein, v=2 π Rsin κ S/60, R is tool radius, and κ is cutting infinitesimal axial location
Angle, S are the speed of mainshaft；
Working angles material is squeezed area ADIt is represented by：
AD=tn·dL (4)
In formula, tnFor instantaneous not deformed thickness of cutting corresponding to cutting edge infinitesimal, dL is tool in cutting sword infinitesimal length；
For difficult-to-machine material curved surface, judge whether blade infinitesimal participates in cutting using Z-map methods, obtain：
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In formula, fzr,fzaRespectively feed engagement fzHorizontal component and vertical component, ψ be cutting infinitesimal radial direction immerse angle,
κ is cutting infinitesimal axial location angle,For axial coordinate of the cutting infinitesimal under workpiece coordinate system, Z is work pieces process face in work
Axial coordinate under part coordinate system；
In view of the plasticity ability T of material influences the coefficient of friction of rake face, coefficient of friction and material plasticity positive correlation, then rake face
Frictional force is represented by：
Fs=nTFn (6)
In formula, n is coefficient；
3) high-speed milling force modeling
Infinitesimal Milling Force (dFt, dFa, dFr) under local coordinate system is transformed to by infinitesimal milling under workpiece coordinate system by coordinate transform
Cut powerFor：
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<mtd>
<mi>d</mi>
<mi>F</mi>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
<mo>)</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,For infinitesimal Milling Force under local coordinate system；
Total Milling Force under workpiece coordinate systemThe superposition of infinitesimal Milling Force is expressed as, is：
<mrow>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>F</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mn>1</mn>
<mrow>
<mi>N</mi>
<mi>j</mi>
</mrow>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mn>1</mn>
<mi>m</mi>
</munderover>
<msubsup>
<mi>dF</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>F</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mn>1</mn>
<mrow>
<mi>N</mi>
<mi>j</mi>
</mrow>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mn>1</mn>
<mi>m</mi>
</munderover>
<msubsup>
<mi>dF</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>F</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mn>1</mn>
<mrow>
<mi>N</mi>
<mi>j</mi>
</mrow>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mn>1</mn>
<mi>m</mi>
</munderover>
<msubsup>
<mi>dF</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, NjFor blade number, m is that blade instantaneously participates in cutting infinitesimal number；
4) High Speed Milling Force identification of Model Parameters
By formula (8), can obtain：
<mrow>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<msubsup>
<mi>F</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi>F</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi>F</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>j</mi>
</munder>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mi>d</mi>
<msubsup>
<mi>F</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>dF</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>dF</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>j</mi>
</munder>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mi>F</mi>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>F</mi>
<mi>s</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, i is cutting infinitesimal layer index, and j indexes for blade；
Formula (3) and (6) are substituted into (9) and can obtained：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<msubsup>
<mi>F</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi>F</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
<mtr>
<mtd>
<msubsup>
<mi>F</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>j</mi>
</munder>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>F</mi>
<mi>n</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>j</mi>
</munder>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mfrac>
<mrow>
<mn>30</mn>
<mo>&CenterDot;</mo>
<mi>&sigma;</mi>
<mo>&CenterDot;</mo>
<mi>&epsiv;</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>t</mi>
<mi>n</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>d</mi>
<mi>L</mi>
</mrow>
<mrow>
<mi>&pi;</mi>
<mo>&CenterDot;</mo>
<mi>R</mi>
<mo>&CenterDot;</mo>
<mi>sin</mi>
<mi>&kappa;</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>S</mi>
</mrow>
</mfrac>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>c</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
</mrow>
For the coefficient c and n introduced in identification model, dynamometer survey at any one cutter corner k in working angles need to be only chosen
It can be solved by any two of three equations in formula (10)；To reduce error using flat
Equal milling force method, the milling that cutter rotates a circle make a concerted effort to be：
<mrow>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>&Sigma;</mo>
<msubsup>
<mi>F</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&Sigma;</mo>
<msubsup>
<mi>F</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&Sigma;</mo>
<msubsup>
<mi>F</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>j</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>k</mi>
</munder>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<msub>
<mi>F</mi>
<mi>n</mi>
</msub>
<mo>=</mo>
<munder>
<mo>&Sigma;</mo>
<mi>i</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>j</mi>
</munder>
<munder>
<mo>&Sigma;</mo>
<mi>k</mi>
</munder>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mfrac>
<mrow>
<mn>30</mn>
<mo>&CenterDot;</mo>
<mi>&sigma;</mi>
<mo>&CenterDot;</mo>
<mi>&epsiv;</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>t</mi>
<mi>n</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>d</mi>
<mi>L</mi>
</mrow>
<mrow>
<mi>&pi;</mi>
<mo>&CenterDot;</mo>
<mi>R</mi>
<mo>&CenterDot;</mo>
<mi>sin</mi>
<mi>&kappa;</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>S</mi>
</mrow>
</mfrac>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>c</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
It will rotate a circle and be divided into NkPart, then averagely Milling Force is expressed as：
<mrow>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</mfrac>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>&Sigma;</mo>
<msubsup>
<mi>F</mi>
<mi>x</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&Sigma;</mo>
<msubsup>
<mi>F</mi>
<mi>y</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>&Sigma;</mo>
<msubsup>
<mi>F</mi>
<mi>z</mi>
<mi>W</mi>
</msubsup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mi>i</mi>
<mi>m</mi>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mi>j</mi>
<msub>
<mi>N</mi>
<mi>j</mi>
</msub>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mi>k</mi>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mfrac>
<mrow>
<mn>30</mn>
<mo>&CenterDot;</mo>
<mi>&sigma;</mi>
<mo>&CenterDot;</mo>
<mi>&epsiv;</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>t</mi>
<mi>n</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>d</mi>
<mi>L</mi>
</mrow>
<mrow>
<mi>&pi;</mi>
<mo>&CenterDot;</mo>
<mi>R</mi>
<mo>&CenterDot;</mo>
<mi>sin</mi>
<mi>&kappa;</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>S</mi>
</mrow>
</mfrac>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>c</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
OrderTo avoid computing repeatedly matrix, improve and calculate effect
Rate, the method combined using hierarchical solving with mean force；If there was only the difference in height of a cutting infinitesimal between cutting-in twice, then two
The difference of secondary empirical average Milling Force is：
<mrow>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mi>x</mi>
<mi>W</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mi>y</mi>
<mi>W</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msubsup>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mi>z</mi>
<mi>W</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<msub>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mi>i</mi>
</msub>
<mo>-</mo>
<msub>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mi>i</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>N</mi>
<mi>K</mi>
</msub>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mi>j</mi>
<msub>
<mi>N</mi>
<mi>j</mi>
</msub>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mi>K</mi>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mfrac>
<mrow>
<mn>30</mn>
<mo>&CenterDot;</mo>
<mi>&sigma;</mi>
<mo>&CenterDot;</mo>
<mi>&epsiv;</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>t</mi>
<mi>n</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>d</mi>
<mi>L</mi>
</mrow>
<mrow>
<mi>&pi;</mi>
<mo>&CenterDot;</mo>
<mi>R</mi>
<mo>&CenterDot;</mo>
<mi>sin</mi>
<mi>&kappa;</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>S</mi>
</mrow>
</mfrac>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>c</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula (13),For constant matrices；Order：
<mrow>
<mi>M</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mi>j</mi>
<msub>
<mi>N</mi>
<mi>j</mi>
</msub>
</munderover>
<munderover>
<mo>&Sigma;</mo>
<mi>k</mi>
<msub>
<mi>N</mi>
<mi>k</mi>
</msub>
</munderover>
<mi>B</mi>
<mo>&CenterDot;</mo>
<mi>A</mi>
<mo>&CenterDot;</mo>
<mfrac>
<mrow>
<mn>30</mn>
<mo>&CenterDot;</mo>
<mi>&sigma;</mi>
<mo>&CenterDot;</mo>
<mi>&epsiv;</mi>
<mo>&CenterDot;</mo>
<msub>
<mi>t</mi>
<mi>n</mi>
</msub>
<mo>&CenterDot;</mo>
<mi>d</mi>
<mi>L</mi>
</mrow>
<mrow>
<mi>&pi;</mi>
<mo>&CenterDot;</mo>
<mi>R</mi>
<mo>&CenterDot;</mo>
<mi>sin</mi>
<mi>&kappa;</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>&CenterDot;</mo>
<mi>S</mi>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
Average Milling Force is：
<mrow>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>M</mi>
<mo>&CenterDot;</mo>
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>c</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
It can be obtained by formula (15)：
<mrow>
<mo>(</mo>
<mi>D</mi>
<mo>&CenterDot;</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>n</mi>
<mo>&CenterDot;</mo>
<mi>T</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>&CenterDot;</mo>
<mi>c</mi>
<mo>)</mo>
<mo>=</mo>
<msup>
<mi>M</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>&CenterDot;</mo>
<mover>
<mi>F</mi>
<mo>&OverBar;</mo>
</mover>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>16</mn>
<mo>)</mo>
</mrow>
</mrow>
Coefficient c and n can be obtained by formula (16), complete milling force modeling.
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