CN107346356A - Cell type thin-wall part milling stability Forecasting Methodology - Google Patents
Cell type thin-wall part milling stability Forecasting Methodology Download PDFInfo
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Abstract
The invention discloses a kind of cell type thin-wall part milling stability Forecasting Methodology, the technical problem low for solving existing thin-wall part milling stability Forecasting Methodology precision of prediction.Survey method；The workpiece dynamics for considering that material removes effect is calculated followed by component mode synthesis method；Then workpiece is extracted in different tool positions and the dynamic displacement of axially different height；The milling dynamics model of Multi-contact is finally established, the workpiece dynamics obtained before is substituted into and solves stability.The present invention considers change, its change at different tool positions and its change along cutter axial direction that workpiece kinetic parameter removes by material simultaneously, improves the precision of cell type thin-wall part milling stability prediction.
Description
Technical field
The present invention relates to a kind of thin-wall part milling stability Forecasting Methodology, more particularly to a kind of cell type thin-wall part milling is stable
Property Forecasting Methodology.
Background technology
It is a large amount of in aerospace field to use thin-walled parts, but the processing of thin-walled parts high-efficiency high-accuracy is one in manufacturing field
Individual problem.Such part is more than cell type unit of the 100, ratio of height to thickness more than 10 by wall thickness 1.5-4mm, slenderness ratio and formed, due to this
Class cell type cell-wall is thin, span is big, chamber groove depth, and unstable cutting easily occurs in milling process, has a strong impact on thin-wall part milling
Processing efficiency and part quality, while accelerate tool wear, even result in tool failure.Therefore, Accurate Prediction thin-wall part milling
Process stability is significant to improving its processing efficiency and crudy.Research shows, with conventional non-thin-walled parts
Milling process is different, in thin-wall part milling process, caused by tool-workpiece contact zone position is different and material removes effect
The change of workpiece dynamics has a great impact to its milling stability.And the kinetic parameter of cell type thin-walled parts is because of position
Different and material removes change clearly, therefore researcher has carried out to cell type thin-wall part milling process stability and largely ground
Study carefully work.
" J.X.Fei, B.Lin, S.Yan, X.F.Zhang, J.Lan, S.G.Dai, the Chatter prediction of document 1
for milling of flexible pocket-structure,International Journal of Advanced
It is pre- that Manufacturing Technology 89 (2017) 2721-2730. " discloses a kind of cell type thin-wall part week milling stability
Survey method；Consider cutter along direction of feed, perpendicular to the vibration of direction of feed and workpiece along cutter axis orientation, with workpiece centre position
The kinetic parameter put replaces the kinetic parameter of each position of workpiece, predicts cell type thin-wall part bottom milling stability.
" K.Ahmadi, Finite strip modeling of the varying dynamics of thin- of document 2
walled pocket structures during machining,International Journal of Advanced
It is pre- that Manufacturing Technology 89 (2017) 2691-2699. " disclose a kind of cell type thin-wall part milling stability
Survey method；The model via dynamical response of thin-wall part is established using finite strip method, and is integrated with Dynamic Model of Milling Process, in advance
Survey cell type thin-wall part milling stability.
The major defect of existing cell type thin-wall part milling stability Forecasting Methodology is not consider that workpiece dynamics is joined simultaneously
Change, its change at different tool positions and its change along cutter axial direction that number removes by material, make workpiece dynamics
The acquisition precision of parameter reduces, and ultimately results in the reduction of cell type thin-wall part stability prediction precision.
The content of the invention
In order to overcome the shortcomings of that existing thin-wall part milling stability Forecasting Methodology precision of prediction is low, the present invention provides a kind of box
Type thin-wall part milling stability Forecasting Methodology.This method hammers the modal parameter and workpiece of experiment measurement cutter first by mode
Damping ratios；The workpiece dynamics for considering that material removes effect is calculated followed by component mode synthesis method；So
Workpiece is extracted afterwards in different tool positions and the dynamic displacement of axially different height；Finally establish the milling dynamics of Multi-contact
Model, the workpiece dynamics obtained before is substituted into and solves stability.The present invention considers workpiece dynamics ginseng simultaneously
Change, its change at different tool positions and its change along cutter axial direction that number removes by material, it is thin to improve cell type
The precision of wall pieces milling stability prediction.
The technical solution adopted for the present invention to solve the technical problems：A kind of cell type thin-wall part milling stability prediction side
Method, it is characterized in comprising the following steps：
Step 1: by the milling cutter clamping used in milling process on machine tool chief axis, cutter-handle of a knife-axis system is entered
The hammering experiment of row mode, measurement obtains the frequency response function of cutter multiple points vertically, by multiple spot frequency response function to cutter-knife
Handle-axis system carries out experimental modal analysis, obtains the intrinsic frequency matrix ω of cuttert, damping ratio matrix ζtWith Mode Shape square
Battle array
Step 2: by clamping workpiece to be processed on platen, modal hammer is carried out to initial workpiece to be processed
Experiment is hit, obtains the damping ratio matrix ζ of workpiecew；
Step 3: by initial workpiece to be processed be considered as processing after workpiece and remove two minor structures of material；
Step 4: FEM model is established to workpiece minor structure after processing, the non-damping vibration dynamics of workpiece after processing
Equation is：
Mmwümw(t)+Kmwumw(t)=Fmw(t)
Wherein, t represents time, MmwAnd KmwRespectively be processing after workpiece mass matrix and stiffness matrix, umwAnd F (t)mw
(t) it is respectively the displacement of each node of workpiece and force vector after processing；
Step 5: the non-damping vibration kinetics equation of workpiece is written as after being processed in step 4
Wherein, subscript i represent processing after workpiece finite element node in not with remove material interior nodes, subscript
B represents the boundary node with removing material, umw,iAnd u (t)mw,b(t) the displacement arrow of interior nodes and boundary node is represented respectively
Amount, Fmw,b(t) force vector on boundary node, M are representedmw,ii, Mmw,ib, Mmw,biAnd Mmw,bbRespectively saved with interior nodes and border
Quality matrix in block form corresponding to point, Kmw,ii, Kmw,ib, Kmw,biAnd Kmw,bbRigidity respectively corresponding with interior nodes and boundary node
Matrix in block form；
Step 6: the FEM model of workpiece after processing is reduced using fixed boundary component mode synthesis method,
Obtain condensation matrix
Wherein, Φmw,ikRepresent the reservation mode collection of fixed boundary, Φmw,ibRepresent Constrained mode collection, Imw,bbExpression and side
Unit matrix corresponding to boundary's degree of freedom on a node basis；
Step 7: the removal material minor structure to step 3 carries out finite element modeling, the removal material before Milling Process is obtained
The non-damping vibration kinetics equation of material for making clothes structure
Wherein,WithIt is the mass matrix and stiffness matrix that material is removed before Milling Process respectively, um(t) andIt is displacement and the force vector that each node of material is removed before Milling Process respectively；
Step 8: it is discrete to cutter path progress, obtain nCSIndividual tool position point；
Step 9: according to cutter path calculate cutter from initial tool location point move to m-th of tool position point when
Cutter scans profile, and judges to be removed the unit that material includes during this；
Step 10: being multiplied by -0.999999 by the unit included to removing material, quality and stiffness variation matrix are obtained,
I.e.With
Step 11: by by before the Milling Process in step 7 removal material minor structure non-damping vibration dynamics
The quality and stiffness variation matrix of equation and m-th of tool position point, i.e.,WithCombination, obtains m-th of cutter position
The non-damping vibration kinetics equation of material minor structure is removed when putting
Wherein,The force vector of each node of material is removed when being m-th of tool position point；
Step 12: m-th of cutter of the non-damping vibration kinetics equation of workpiece and step 11 after step 4 is processed
The non-damping vibration kinetics equation assembling of material minor structure, the condensation matrix R obtained using step 6 are removed during location pointmwAnd
The displacement coordination condition and dynamic balance condition of two minor structures enter line translation to the equation group after assembling, obtain m-th of cutter position
Workpiece finite element reduced-order models when putting
Wherein,WithThe mass matrix of workpiece finite element reduced-order models when being m-th of tool position point respectively
And stiffness matrix, pwThe motion vector of each generalized node of workpiece when representing m-th of tool position point；
Step 13: carrying out computational modal analysis to step 12 finite element reduced-order models, m-th of tool position point is obtained
When workpiece intrinsic frequency matrixAnd modal transfer matrix
Step 14: according to the coordinate of m-th tool position point and axial cutting-in, extraction is located at tool-workpiece cutting region
In view of this, it is an object of the invention to provide user in a kind of non-orthogonal multiple air-ground coordination communication system to dispatchWhereinIt isSubmatrix；
Step 15: the cutter intrinsic frequency matrix ω obtained using step 1t, damping ratio matrix ζtWith Mode Shape square
Battle arrayThe damping ratio matrix ζ for the workpiece that step 2 obtainsw, the intrinsic frequency of the workpiece in the process that step 13 obtains
MatrixThe dynamic displacement matrix of workpiece in the process that step 14 obtainsTool motion is established to m cutters
Milling process kinetics equation during location point：
Wherein, Γt(t)、WithThe respectively displacement of the modal coordinate of cutter, speed and vector acceleration, Γw(t) be respectively workpiece modal coordinate displacement, speed and vector acceleration, F (t) is acts on knife
The milling force vector of tool-workpiece cutting zone；
Step 16: with promote half discrete time-domain method judgment step 15 in kinetics equation stability, and
Draw the stability lobes diagram.
The beneficial effects of the invention are as follows：This method hammers the modal parameter and workpiece of experiment measurement cutter first by mode
Damping ratios；The workpiece dynamics for considering that material removes effect is calculated followed by component mode synthesis method；So
Workpiece is extracted afterwards in different tool positions and the dynamic displacement of axially different height；Finally establish the milling dynamics of Multi-contact
Model, the workpiece dynamics obtained before is substituted into and solves stability.The present invention considers workpiece dynamics ginseng simultaneously
Change, its change at different tool positions and its change along cutter axial direction that number removes by material, it is thin to improve cell type
The precision of wall pieces milling stability prediction.
The present invention is elaborated with reference to the accompanying drawings and detailed description.
Brief description of the drawings
Fig. 1 be in the inventive method embodiment 1 side wall be plane cell type thin-wall part schematic diagram.
Fig. 2 is that the stability lobes diagram predicted when the speed of mainshaft is 5600rpm in the inventive method embodiment 1 is tied with experiment
The comparison diagram of fruit.In figure, solid line represents the predicted value of the present embodiment, and zero represents the stabilization result of experiment, × represent experiment not
Stabilization result.
Fig. 3 is the schematic diagram of the cell type thin-wall part of one curved surface sidewall of band in the inventive method embodiment 2.
Fig. 4 is the stability lobes diagram predicted when the speed of mainshaft is 7000rpm in the inventive method embodiment 2.
Fig. 5 be in the inventive method embodiment 2 cutter since cutting point along knife rail move at 174.8mm predict it is steady
Qualitative flap figure.
Embodiment
Following examples reference picture 1-5.
Embodiment 1：The prediction that side wall is the cell type thin-wall part week milling stability of plane, wherein box are carried out using the present invention
Molded dimension is 200mm × 100mm × 27mm, cassette bottom thickness 3mm, wall thickness 3mm on front side of milling, and radial cutting depth is 1mm, workpiece
Material is aluminium alloy 7050, and workpiece is by pressing plate clamping on platen.
(1) by diameter 16mm 3 tooth square end mills by spring clamping head knife handle clamping on machine tool chief axis, cutter stretch out
Length 50mm, cutter material are hard alloy；Mode hammering experiment is carried out to cutter-handle of a knife-axis system, measurement obtains cutter
The frequency response function of 3 points vertically, experimental modal analysis is carried out to cutter-handle of a knife-axis system by 3 frequency response functions, obtained
To the intrinsic frequency matrix ω of cuttert, damping ratio matrix ζtWith Mode Shape matrix
(2) by clamping workpiece to be processed on platen, it is real that mode hammering is carried out to initial workpiece to be processed
Test, obtain the damping ratio matrix ζ of workpiecew；
(3) initial workpiece to be processed is considered as to workpiece after processing and removes two minor structures of material；
(4) FEM model is established to workpiece minor structure after processing, the non-damping vibration kinetics equation of workpiece after processing
For：
Mmwümw(t)+Kmwumw(t)=Fmw(t)
Wherein, t represents time, MmwAnd KmwRespectively be processing after workpiece mass matrix and stiffness matrix, umwAnd F (t)mw
(t) it is respectively the displacement of each node of workpiece and force vector after processing；
(5) the non-damping vibration kinetics equation of workpiece is written as after being processed in (4) step
Wherein, subscript " i " represent processing after workpiece finite element node in not with remove material interior nodes, under
Mark the boundary node that " b " is represented and removed material, umw,iAnd u (t)mw,b(t) position of interior nodes and boundary node is represented respectively
Move vector, Fmw,b(t) force vector on boundary node, M are representedmw,ii, Mmw,ib, Mmw,biAnd Mmw,bbRespectively with interior nodes and side
Quality matrix in block form, K corresponding to boundary's nodemw,ii, Kmw,ib, Kmw,biAnd Kmw,bbIt is respectively corresponding with interior nodes and boundary node
Rigidity matrix in block form；
(6) FEM model of workpiece after processing is reduced using fixed boundary component mode synthesis method, obtained
Condensation matrix
Wherein, Φmw,ikRepresent the reservation mode collection of fixed boundary, Φmw,ibRepresent Constrained mode collection, Imw,bbExpression and side
Unit matrix corresponding to boundary's degree of freedom on a node basis；
(7) the removal material minor structure to (3) step carries out finite element modeling, obtains of the removal material before Milling Process
The non-damping vibration kinetics equation of structure
Wherein,WithIt is the mass matrix and stiffness matrix that material is removed before Milling Process respectively, um(t) andIt is displacement and the force vector that each node of material is removed before Milling Process respectively；
(8) it is discrete to cutter path progress, obtain nCSIndividual tool position point；
(9) according to cutter path calculate cutter from initial tool location point move to m-th of tool position point when cutter
Profile is scanned, and judges to be removed the unit that material includes during this；
(10) -0.999999 is multiplied by by the unit included to removing material, obtains quality and stiffness variation matrix, i.e.,With
(11) by by before the Milling Process in (7) step removal material minor structure non-damping vibration kinetics equation
With the quality and stiffness variation matrix of m-th of tool position point, i.e.,WithCombination, it can obtain m-th of tool position
The non-damping vibration kinetics equation of material minor structure is removed during point
Wherein,The force vector of each node of material is removed when being m-th of tool position point；
(12) by the non-damping vibration kinetics equation of workpiece after the processing of (4) step and m-th of cutter of (11) step
The non-damping vibration kinetics equation assembling of material minor structure, the condensation matrix R obtained using (6) step are removed during location pointmw
And the displacement coordination condition and dynamic balance condition of two minor structures enter line translation to the equation group after assembling, obtain m-th of cutter
Workpiece finite element reduced-order models during location point
Wherein,WithThe mass matrix of workpiece finite element reduced-order models when being m-th of tool position point respectively
And stiffness matrix, pwPatent of invention；
(13) computational modal analysis is carried out to the finite element reduced-order models of (12) step, you can obtain m-th of tool position
The intrinsic frequency matrix of workpiece during pointAnd modal transfer matrix
(14) it is multiple positioned at tool-workpiece cutting zone according to the coordinate of m-th tool position point and axial cutting-in, extraction
The dynamic displacement matrix of pointWhereinIt isSubmatrix；
(15) the cutter intrinsic frequency matrix ω obtained in step (1) is utilizedt, damping ratio matrix ζtWith Mode Shape matrixThe damping ratio matrix ζ of the workpiece obtained in step (2)wPatent of invention
Rate matrixThe dynamic displacement matrix of workpiece in the process obtained in step (14)Tool motion is established to m
Milling process kinetics equation during the point of tool position：
Wherein, Γt(t)、WithThe respectively displacement of the modal coordinate of cutter, speed and vector acceleration, Γw(t) be respectively workpiece modal coordinate displacement, speed and vector acceleration, F (t) is acts on knife
The milling force vector of tool-workpiece cutting zone；
(16) stability of the kinetics equation in the half discrete time-domain method judgment step (15) promoted is used, and is drawn steady
Qualitative flap figure.
By above step, predictable side wall is the cell type thin-wall part week milling the stability lobes diagram of plane, can from Fig. 2
To find out, prediction result of the invention is coincide with experiment, it was demonstrated that the accuracy of method.
Embodiment 2：The prediction of cell type thin-wall part week milling stability of the side wall with curved surface, the cell type part are carried out using the present invention
One side wall is curved surface, and surface equation is
X (u, v)=RWcos(θW+(u-1)(2θW-π))
Y (u, v)=RWsin(θW+(u-1)(2θW-π))+0.155-RW u,v∈[0,1]
Z (u, v)=28v
Wherein, θW=1.164705892287963rad, RW=0.43035714285714m, x, y, z units are m.Its
Its 3 side wall is plane.Cell type size is 340mm × 155mm × 28mm, cassette bottom thickness 2mm, milling anterior wall thickness 2.8mm, radially
Cutting depth is 1mm, and workpiece material is aluminium alloy 7050, and workpiece is by pressing plate clamping on platen.
(1) by diameter 16mm 3 tooth square end mills by spring clamping head knife handle clamping on machine tool chief axis, cutter stretch out
Length 50mm, cutter material are hard alloy；Mode hammering experiment is carried out to cutter-handle of a knife-axis system, measurement obtains cutter
The frequency response function of 3 points vertically, experimental modal analysis is carried out to cutter-handle of a knife-axis system by 3 frequency response functions, obtained
To the intrinsic frequency matrix ω of cuttert, damping ratio matrix ζtWith Mode Shape matrix
(2) by clamping workpiece to be processed on platen, it is real that mode hammering is carried out to initial workpiece to be processed
Test, obtain the damping ratio matrix ζ of workpiecew；
(3) initial workpiece to be processed is considered as to workpiece after processing and removes two minor structures of material；
(4) FEM model is established to workpiece minor structure after processing, the non-damping vibration kinetics equation of workpiece after processing
For：
Mmwümw(t)+Kmwumw(t)=Fmw(t)
Wherein, t represents time, MmwAnd KmwRespectively be processing after workpiece mass matrix and stiffness matrix, umwAnd F (t)mw
(t) it is respectively the displacement of each node of workpiece and force vector after processing；
(5) the non-damping vibration kinetics equation of workpiece is written as after being processed in (4) step
Wherein, subscript " i " represent processing after workpiece finite element node in not with remove material interior nodes, under
Mark the boundary node that " b " is represented and removed material, umw,iAnd u (t)mw,b(t) position of interior nodes and boundary node is represented respectively
Move vector, Fmw,b(t) force vector on boundary node, M are representedmw,ii, Mmw,ib, Mmw,biAnd Mmw,bbRespectively with interior nodes and side
Quality matrix in block form, K corresponding to boundary's nodemw,ii, Kmw,ib, Kmw,biAnd Kmw,bbIt is respectively corresponding with interior nodes and boundary node
Rigidity matrix in block form；
(6) FEM model of workpiece after processing is reduced using fixed boundary component mode synthesis method, obtained
Condensation matrix
Wherein, Φmw,ikRepresent the reservation mode collection of fixed boundary, Φmw,ibRepresent Constrained mode collection, Imw,bbExpression and side
Unit matrix corresponding to boundary's degree of freedom on a node basis；
(7) the removal material minor structure to (3) step carries out finite element modeling, obtains of the removal material before Milling Process
The non-damping vibration kinetics equation of structure
Wherein,WithIt is the mass matrix and stiffness matrix that material is removed before Milling Process respectively, um(t) andIt is displacement and the force vector that each node of material is removed before Milling Process respectively；
(8) it is discrete to cutter path progress, obtain nCSIndividual tool position point；
(9) according to cutter path calculate cutter from initial tool location point move to m-th of tool position point when cutter
Profile is scanned, and judges to be removed the unit that material includes during this；
(10) -0.999999 is multiplied by by the unit included to removing material, obtains quality and stiffness variation matrix, i.e.,With
(11) by by before the Milling Process in (7) step removal material minor structure non-damping vibration kinetics equation
With the quality and stiffness variation matrix of m-th of tool position point, i.e.,WithCombination, it can obtain m-th of tool position
The non-damping vibration kinetics equation of material minor structure is removed during point
Wherein,The force vector of each node of material is removed when being m-th of tool position point；
(12) by the non-damping vibration kinetics equation of workpiece after the processing of (4) step and m-th of cutter of (11) step
The non-damping vibration kinetics equation assembling of material minor structure, the condensation matrix R obtained using (6) step are removed during location pointmw
And the displacement coordination condition and dynamic balance condition of two minor structures enter line translation to the equation group after assembling, obtain m-th of cutter
Workpiece finite element reduced-order models during location point
Wherein,WithThe mass matrix of workpiece finite element reduced-order models when being m-th of tool position point respectively
And stiffness matrix, pwThe motion vector of each generalized node of workpiece when representing m-th of tool position point；
(13) computational modal analysis is carried out to the finite element reduced-order models of (12) step, you can obtain m-th of tool position
The intrinsic frequency matrix of workpiece during pointAnd modal transfer matrix
(14) it is multiple positioned at tool-workpiece cutting zone according to the coordinate of m-th tool position point and axial cutting-in, extraction
The dynamic displacement matrix of pointWhereinIt isSubmatrix；
(15) the cutter intrinsic frequency matrix ω obtained in step (1) is utilizedt, damping ratio matrix ζtWith Mode Shape matrixThe damping ratio matrix ζ of the workpiece obtained in step (2)w, the intrinsic frequency of the workpiece in the process obtained in step (13)
Rate matrixThe dynamic displacement matrix of workpiece in the process obtained in step (14)Tool motion is established to m
Milling process kinetics equation during the point of tool position：
Wherein, Γt(t)、WithThe respectively displacement of the modal coordinate of cutter, speed and vector acceleration, Γw(t) be respectively workpiece modal coordinate displacement, speed and vector acceleration, F (t) is acts on knife
The milling force vector of tool-workpiece cutting zone；
(16) stability of the kinetics equation in the half discrete time-domain method judgment step (15) promoted is used, and is drawn steady
Qualitative flap figure.
By above step, the band curved surface cell type thin-wall part week milling the stability lobes diagram in Fig. 3, Fig. 4 and Fig. 5 can be predicted
Illustrate that this method is applied to band curved surface cell type thin-wall part, the milling under such part difference tool position, different rotating speeds can be predicted
Cut stability.
Claims (1)
1. a kind of cell type thin-wall part milling stability Forecasting Methodology, it is characterised in that comprise the following steps：
Step 1: by the milling cutter clamping used in milling process on machine tool chief axis, mould is carried out to cutter-handle of a knife-axis system
State hammering experiment, measurement obtains the frequency response function of cutter multiple points vertically, by multiple spot frequency response function to cutter-handle of a knife-master
Axle system carries out experimental modal analysis, obtains the intrinsic frequency matrix ω of cuttert, damping ratio matrix ζtWith Mode Shape matrix
Step 2: by clamping workpiece to be processed on platen, it is real that mode hammering is carried out to initial workpiece to be processed
Test, obtain the damping ratio matrix ζ of workpiecew；
Step 3: by initial workpiece to be processed be considered as processing after workpiece and remove two minor structures of material；
Step 4: FEM model is established to workpiece minor structure after processing, the non-damping vibration kinetics equation of workpiece after processing
For：
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Step 5: the non-damping vibration kinetics equation of workpiece is written as after being processed in step 4
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<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>i</mi>
<mi>b</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>M</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>b</mi>
<mi>i</mi>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>M</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>b</mi>
<mi>b</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<msub>
<mi>u</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>i</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>u</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>b</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>F</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>b</mi>
</mrow>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein, subscript i represent processing after workpiece finite element node in not with remove material interior nodes, subscript b tables
Show the boundary node with removing material, umw,iAnd u (t)mw,b(t) interior nodes and the displacement vector of boundary node are represented respectively,
Fmw,b(t) force vector on boundary node, M are representedmw,ii, Mmw,ib, Mmw,biAnd Mmw,bbRespectively with interior nodes and boundary node pair
The quality matrix in block form answered, Kmw,ii, Kmw,ib, Kmw,biAnd Kmw,bbRigidity piecemeal respectively corresponding with interior nodes and boundary node
Matrix；
Step 6: being reduced using fixed boundary component mode synthesis method to the FEM model of workpiece after processing, obtain
Condensation matrix
<mrow>
<msub>
<mi>R</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
</mrow>
</msub>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&Phi;</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>i</mi>
<mi>k</mi>
</mrow>
</msub>
</mtd>
<mtd>
<msub>
<mi>&Phi;</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>i</mi>
<mi>b</mi>
</mrow>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>I</mi>
<mrow>
<mi>m</mi>
<mi>w</mi>
<mo>,</mo>
<mi>b</mi>
<mi>b</mi>
</mrow>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein, Φmw,ikRepresent the reservation mode collection of fixed boundary, Φmw,ibRepresent Constrained mode collection, Imw,bbRepresent to save with border
Unit matrix corresponding to the point free degree；
Step 7: the removal material minor structure to step 3 carries out finite element modeling, of the removal material before Milling Process is obtained
The non-damping vibration kinetics equation of structure
<mrow>
<msubsup>
<mi>M</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mn>0</mn>
<mo>></mo>
</mrow>
</msubsup>
<msub>
<mover>
<mi>u</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>m</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>K</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mn>0</mn>
<mo>></mo>
</mrow>
</msubsup>
<msub>
<mi>u</mi>
<mi>m</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mn>0</mn>
<mo>></mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
Wherein,WithIt is the mass matrix and stiffness matrix that material is removed before Milling Process respectively, um(t) andPoint
It is not displacement and the force vector that each node of material is removed before Milling Process；
Step 8: it is discrete to cutter path progress, obtain nCSIndividual tool position point；
Step 9: according to cutter path calculate cutter from initial tool location point move to m-th of tool position point when cutter
Profile is scanned, and judges to be removed the unit that material includes during this；
Step 10: being multiplied by -0.999999 by the unit included to removing material, quality and stiffness variation matrix are obtained, i.e.,With
Step 11: by by before the Milling Process in step 7 removal material minor structure non-damping vibration kinetics equation
With the quality and stiffness variation matrix of m-th of tool position point, i.e.,WithCombination, obtains m-th of tool position point
When remove material minor structure non-damping vibration kinetics equation
<mrow>
<mo>(</mo>
<msubsup>
<mi>M</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mn>0</mn>
<mo>></mo>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>&Delta;M</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
<mo>)</mo>
<msub>
<mover>
<mi>u</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>m</mi>
</msub>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
<mo>+</mo>
<mo>(</mo>
<msubsup>
<mi>K</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mn>0</mn>
<mo>></mo>
</mrow>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>&Delta;K</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
<mo>)</mo>
<msub>
<mi>u</mi>
<mi>m</mi>
</msub>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mi>m</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
Wherein,The force vector of each node of material is removed when being m-th of tool position point；
Step 12: m-th of tool position of the non-damping vibration kinetics equation of workpiece and step 11 after step 4 is processed
The non-damping vibration kinetics equation assembling of material minor structure, the condensation matrix R obtained using step 6 are removed during pointmwAnd two
The displacement coordination condition and dynamic balance condition of minor structure enter line translation to the equation group after assembling, obtain m-th of tool position point
When workpiece finite element reduced-order models
<mrow>
<msubsup>
<mover>
<mi>M</mi>
<mo>&OverBar;</mo>
</mover>
<mi>w</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
<msub>
<mover>
<mi>p</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>w</mi>
</msub>
<mo>+</mo>
<msubsup>
<mover>
<mi>K</mi>
<mo>&OverBar;</mo>
</mover>
<mi>w</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
<msub>
<mi>p</mi>
<mi>w</mi>
</msub>
<mo>=</mo>
<mn>0</mn>
</mrow>
Wherein,WithThe mass matrix and rigidity of workpiece finite element reduced-order models when being m-th of tool position point respectively
Matrix, pwThe motion vector of each generalized node of workpiece when representing m-th of tool position point；
Step 13: computational modal analysis is carried out to step 12 finite element reduced-order models, when obtaining m-th of tool position point
The intrinsic frequency matrix of workpieceAnd modal transfer matrix
It is Step 14: more positioned at tool-workpiece cutting zone according to the coordinate of m-th tool position point and axial cutting-in, extraction
The dynamic displacement matrix of individual pointWhereinIt isSubmatrix；
Step 15: the cutter intrinsic frequency matrix ω obtained using step 1t, damping ratio matrix ζtWith Mode Shape matrix
The damping ratio matrix ζ for the workpiece that step 2 obtainsw, the intrinsic frequency matrix of the workpiece in the process that step 13 obtainsThe dynamic displacement matrix of workpiece in the process that step 14 obtainsTool motion is established to m tool positions
Milling process kinetics equation during point：
<mrow>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>&Gamma;</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>t</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>&Gamma;</mi>
<mo>&CenterDot;&CenterDot;</mo>
</mover>
<mi>w</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>+</mo>
<mn>2</mn>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&zeta;</mi>
<mi>t</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msub>
<mi>&zeta;</mi>
<mi>w</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&omega;</mi>
<mi>t</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msubsup>
<mi>&omega;</mi>
<mi>w</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>&Gamma;</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>t</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mover>
<mi>&Gamma;</mi>
<mo>&CenterDot;</mo>
</mover>
<mi>w</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>+</mo>
<msup>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<msub>
<mi>&omega;</mi>
<mi>t</mi>
</msub>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<msubsup>
<mi>&omega;</mi>
<mi>w</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mn>2</mn>
</msup>
<mfenced open = "{" close = "}">
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&Gamma;</mi>
<mi>t</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>&Gamma;</mi>
<mi>w</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<msup>
<msub>
<mover>
<mi>U</mi>
<mo>^</mo>
</mover>
<mi>t</mi>
</msub>
<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>U</mi>
<mo>^</mo>
</mover>
<mi>w</mi>
<mrow>
<mo><</mo>
<mi>m</mi>
<mo>></mo>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mi>F</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
Wherein, Γt(t)、WithThe respectively displacement of the modal coordinate of cutter, speed and vector acceleration, Γw(t) be respectively workpiece modal coordinate displacement, speed and vector acceleration, F (t) is acts on tool-workpiece
The milling force vector of cutting zone；
Step 16: with the stability of the kinetics equation in the half discrete time-domain method judgment step 15 promoted, and draw
The stability lobes diagram.
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| CN108958167A (en) * | 2018-09-13 | 2018-12-07 | 大连理工大学 | It is a kind of towards cutting stability forecast across axis across mould measurement and parameter identification method |
| CN109840380A (en) * | 2019-02-16 | 2019-06-04 | 北京理工大学 | A kind of stability prediction method considering multiple modal vibrations and work pieces process response |
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| CN108958167A (en) * | 2018-09-13 | 2018-12-07 | 大连理工大学 | It is a kind of towards cutting stability forecast across axis across mould measurement and parameter identification method |
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