CN107728576A - Multi-shaft numerical control machining knife axial vector optimization method based on cutter stress deformation - Google Patents

Multi-shaft numerical control machining knife axial vector optimization method based on cutter stress deformation Download PDF

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CN107728576A
CN107728576A CN201710861499.9A CN201710861499A CN107728576A CN 107728576 A CN107728576 A CN 107728576A CN 201710861499 A CN201710861499 A CN 201710861499A CN 107728576 A CN107728576 A CN 107728576A
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msub
cutter
mrow
mtd
msup
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CN107728576B (en
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段现银
彭芳瑜
闫蓉
蒋国璋
周敏
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Luzhou Hanfei Aerospace Technology Development Co ltd
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Wuhan University of Science and Engineering WUSE
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/404—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by control arrangements for compensation, e.g. for backlash, overshoot, tool offset, tool wear, temperature, machine construction errors, load, inertia
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/37—Measurements
    • G05B2219/37087—Cutting forces
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/37—Measurements
    • G05B2219/37355—Cutting, milling, machining force
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/37—Measurements
    • G05B2219/37397—Measuring gap between tool and workpiece

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  • Human Computer Interaction (AREA)
  • Manufacturing & Machinery (AREA)
  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Automation & Control Theory (AREA)
  • Numerical Control (AREA)

Abstract

The invention discloses a kind of multi-shaft numerical control machining knife axial vector optimization method based on cutter stress deformation, comprise the following steps：1) parameter acquisition；2) cutting force and multi-axis numerical control equipment process system rigidity establish cutter force deformation model during multiaxis NC maching according to suffered by cutter in process；3) determine model meets constraints；4) according to cutter force deformation model and constraints, target is minimised as with cutter force deformation, establishes optimal tool orientation mathematical modeling；5) above optimal tool orientation mathematical modeling is solved, determines optimal generating tool axis vector Vc,optThe cutter top rake and angle of heel at place；6) cutter top rake and angle of heel are converted into generating tool axis vector, the generating tool axis vector after being optimized.The present invention compensate for existing multi-shaft numerical control machining knife axial vector planning and only consider geometrical constraint, the weak point of process system rigidity characteristic, and new thinking is provided for the planning of complex-curved multi-shaft numerical control machining knife axial vector.

Description

Multi-shaft numerical control machining knife axial vector optimization method based on cutter stress deformation
Technical field
The present invention relates to numerical control machine bed technique, more particularly to a kind of multiaxis NC maching based on cutter stress deformation Optimal tool orientation method.
Background technology
Complex curved surface parts extensive application in the fields such as Aero-Space, energy source and power, generally using multi-axis numerical control Process equipment is processed.In complex-curved NC Machining Process, cutter with processed curved surface there is good contact state to be Ensure one of key factor of part crudy, ideally, generating tool axis vector direction should be with normal side at Machining of Curved Surface point To consistent.The appearance of multiaxis NC maching rotary axis of machine tool allows cutter opposite piece to realize motion, cutter shaft in three dimensions Opposite piece surface is continuously swung, by adjusting in local coordinate system followed by angle and side drift angle, from geometrically meet cutter with There is good contact state between processed curved surface.With curved surface face shape even more complex, dashed forward particularly with local curvature Becoming the complex-curved of feature, changing in cutter shaft relative to larger generating tool axis vector in piece surface swing process, be present, so as to draw Play the corner change of big lathe yaw or turntable.Existing multi-shaft numerical control machining knife axial vector planning only considers geometrical constraint and work Process system stiffness characteristics.
The content of the invention
The defects of the technical problem to be solved in the present invention is to be directed in the prior art, there is provided one kind is become based on cutter stress The multi-shaft numerical control machining knife axial vector optimization method of shape.
The technical solution adopted for the present invention to solve the technical problems is：A kind of multi-axis numerical control based on cutter stress deformation Optimal tool orientation method is processed, is comprised the following steps：
1) cutter relevant parameter gathers, and the parameter includes the cantilevered length L of cutter and cutter bar parts, the cutter tooth portion of cutter Divide length Lf, the effective diameter and diameter D of cutter cutter tooth parteAnd D, the elastic modulus E of cutter material, cutter and cutter bar part The moment of inertia I；The design curved surface data information (processing rear curved surface) of workpiece, the Cast blanks surface data message of workpiece are (bent before processing Face), cutter-contact point trace line data message, the speed of mainshaft, the cutting edge number N of cutterf, the radial direction of cutting edge infinitesimal, it is tangential and Axial cutting force COEFFICIENT Kr、KtAnd Ka, cutter radially contacts with angleThe axially contact angle κ of cutter；
2) cutting force and multi-axis numerical control equipment process system rigidity establish multi-axis numerical control according to suffered by cutter in process Cutter force deformation model e (α, β) in process；
E is cutter stress deformation synthetic quantity；α and β is respectively the top rake and angle of heel of cutter；
3) determine model meets constraints：Constraints is used as including reachable tree requirement and contact requirement；
4) according to cutter force deformation model and constraints during multiaxis NC maching, with cutter force deformation Target is minimised as, establishes optimal tool orientation mathematical modeling, the expression formula of the optimal tool orientation mathematical modeling is：
min e(α,β)
s.t.
1.v(Pi,j,α,β)∈VG(Pi,j)
2.v(Pi,j,α,β)∈Vc(Pi,j)
5) above optimal tool orientation mathematical modeling is solved, determines optimal generating tool axis vector Vc,optThe cutter top rake at place and Angle of heel；
6) cutter top rake and angle of heel are converted into generating tool axis vector, the generating tool axis vector after being optimized, conversion formula is：
V (α, β)=xpsinαcosβ+ypsinαsinβ+zpcosα. (1)
Wherein, v is generating tool axis vector, and α and β are respectively the top rake and angle of heel of cutter, xp、yp、zpRespectively cutter-contact point is sat Mark three change in coordinate axis direction vectors of system.
By such scheme, the expression formula of the cutter force deformation mathematical modeling is：
Wherein, e is cutter stress deformation synthetic quantity, ex、ey、ezRespectively cutter stress deformation is along the three of tool coordinate system The component of individual change in coordinate axis direction, ex、ey、ezExpression formula be：
Wherein, Sx、Sy、SzRespectively component of the cutter tips static compliance along three change in coordinate axis direction of tool coordinate system, Sx、Sy、SzExpression formula be：
Wherein, LsFor the cantilevered length of cutter and cutter bar part, LfFor the cutter tooth partial-length of cutter, LsfAlways overhang for cutter Length, and Lsf=Ls+Lf, μtFor cutter tooth part effective diameter coefficient, and μt=De/ D (wherein DeIt is respectively cutter cutter tooth portion with D The effective diameter and diameter divided), E is the modulus of elasticity of cutter material, and I is the moment of inertia of cutter and cutter bar part.
F in formula (3)x、Fy、FzStatic cutting force respectively suffered by cutter is along three reference axis sides of tool coordinate system To component, Fx、Fy、FzExpression formula be：
Wherein,It is time t function for angle of eccentricity, fx、fy、fzDynamic cutting force respectively suffered by cutter Along the component of three change in coordinate axis direction of tool coordinate system, fx、fy、fzExpression formula be：
Wherein, j be cutter cutting edge sequence number, NfFor the cutting edge number of cutter, z1,jWith z respectively2,jCut for j-th Cut the upper limit of integral and lower limit of integral of sword, hjFor the undeformed chip thickness of j-th of cutting edge, Kr、KtAnd KaRespectively cutting edge The radial direction of infinitesimal, tangential and axial cutting force coefficient,For the angle that radially contacts with of cutter, κ is the axially contact angle of cutter, db (z) it is the not deformed chip width at z to represent axial location；
z1,jWith z respectively2,jFor the upper limit of integral and lower limit of integral of j-th of cutting edge, upper limit of integral and lower limit of integral are by work Part processing front curve, work pieces process rear curved surface, tool in cutting sword envelope surface three determines, is by Tool in Cutting specifically What sword envelope surface was formed in the forward and backward curved surface of work pieces process scans the decision of body body, and when processing front curve (the workpiece hair of workpiece Base curved surface) and processing rear curved surface (workpiece design curved surface) have determined that after, scanning body is determined by the generating tool axis vector of cutter, also It is to say z1,jWith z respectively2,jTo be the function of generating tool axis vector, being determined by generating tool axis vector.And, it is necessary to import specific in specific calculate The Cast blanks surface model of workpiece, the design surface model of workpiece, the cutter-contact point trace line and tool geometrical parameter of cutter, then Judge whether it participates in cutting according to the position relationship before and after the point and work pieces process in tool in cutting sword using Z-map methods.
hjIt is the function of generating tool axis vector for the undeformed chip thickness of j-th of cutting edge.Not deformed chip is in workpiece material Be removed in material the two neighboring cutting edge cycle of moment cutter enveloping surface and between Partial Resection workpiece portion, it is thick Degree is projection of the cutting edge infinitesimal feeding vector outside cutter enveloping surface on direction of normal, and the outer direction of normal of cutter enveloping surface with Generating tool axis vector is related.
The beneficial effect comprise that：The present invention compensate for existing multi-shaft numerical control machining knife axial vector planning and only consider It the weak point of geometrical constraint, process system rigidity characteristic, can be achieved to be based on cutter stress deformation, and consider geometrical constraint and cut The multi-shaft numerical control machining knife axial vector optimization for requiring constraint is touched, is provided for the planning of complex-curved multi-shaft numerical control machining knife axial vector New thinking.
Brief description of the drawings
Below in conjunction with drawings and Examples, the invention will be further described, in accompanying drawing：
Fig. 1 is the method flow diagram of the embodiment of the present invention；
Fig. 2 is the constraints of the embodiment of the present invention and optimal generating tool axis vector formation figure.
Embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, with reference to embodiments, to the present invention It is further elaborated.It should be appreciated that specific embodiment described herein is not used to limit only to explain the present invention The fixed present invention.
As shown in figure 1, a kind of multi-shaft numerical control machining knife axial vector optimization method based on cutter stress deformation, including it is following Step：
1) cutter relevant parameter gathers, and the parameter includes the cantilevered length L of cutter and cutter bar parts, the cutter tooth portion of cutter Divide length Lf, the effective diameter and diameter D of cutter cutter tooth parteAnd D, the elastic modulus E of cutter material, cutter and cutter bar part The moment of inertia I；The design curved surface data information (processing rear curved surface) of workpiece, the Cast blanks surface data message of workpiece are (bent before processing Face), cutter-contact point trace line data message, the speed of mainshaft, the cutting edge number N of cutterf, the radial direction of cutting edge infinitesimal, it is tangential and Axial cutting force COEFFICIENT Kr、KtAnd Ka, cutter radially contacts with angleThe axially contact angle κ of cutter；
2) cutting force and multi-axis numerical control equipment process system rigidity establish multi-axis numerical control according to suffered by cutter in process Cutter force deformation model e (α, β) in process；
E is cutter stress deformation synthetic quantity；α and β is respectively the top rake and angle of heel of cutter；
3) determine model meets constraints：Constraints is used as including reachable tree requirement and contact requirement；
4) according to cutter force deformation model and constraints during multiaxis NC maching, with cutter force deformation Target is minimised as, establishes optimal tool orientation mathematical modeling, the expression formula of the optimal tool orientation mathematical modeling is：
min e(α,β)
s.t.
1.v(Pi,j,α,β)∈VG(Pi,j)
2.v(Pi,j,α,β)∈Vc(Pi,j)
5) above optimal tool orientation mathematical modeling is solved, determines optimal generating tool axis vector Vc,optThe cutter top rake at place and Angle of heel；
6) cutter top rake and angle of heel are converted into generating tool axis vector, the generating tool axis vector after being optimized, conversion formula is：
V (α, β)=xpsinαcosβ+ypsinαsinβ+zpcosα. (1)
Wherein, v is generating tool axis vector, and α and β are respectively the top rake and angle of heel of cutter, xp、yp、zpRespectively cutter-contact point is sat Mark three change in coordinate axis direction vectors of system.
The expression formula of cutter force deformation mathematical modeling is in the embodiment of the present invention：
Wherein, e is cutter stress deformation synthetic quantity, ex、ey、ezRespectively cutter stress deformation is along the three of tool coordinate system The component of individual change in coordinate axis direction, ex、ey、ezExpression formula be：
Wherein, Sx、Sy、SzRespectively component of the cutter tips static compliance along three change in coordinate axis direction of tool coordinate system, Sx、Sy、SzExpression formula be：
Wherein, LsFor the cantilevered length of cutter and cutter bar part, LfFor the cutter tooth partial-length of cutter, LsfAlways overhang for cutter Length, and Lsf=Ls+Lf, μtFor cutter tooth part effective diameter coefficient, and μt=De/ D (wherein DeIt is respectively cutter cutter tooth portion with D The effective diameter and diameter divided), E is the modulus of elasticity of cutter material, and I is the moment of inertia of cutter and cutter bar part.
F in formula (3)x、Fy、FzStatic cutting force respectively suffered by cutter is along three reference axis sides of tool coordinate system To component, Fx、Fy、FzExpression formula be：
Wherein,It is time t function for angle of eccentricity, fx、fy、fzDynamic cutting force respectively suffered by cutter Along the component of three change in coordinate axis direction of tool coordinate system, fx、fy、fzExpression formula be：
Wherein, j be cutter cutting edge sequence number, NfFor the cutting edge number of cutter, z1,jWith z respectively2,jCut for j-th Cut the upper limit of integral and lower limit of integral of sword, hjFor the undeformed chip thickness of j-th of cutting edge, Kr、KtAnd KaRespectively cutting edge The radial direction of infinitesimal, tangential and axial cutting force coefficient,For the angle that radially contacts with of cutter, κ is the axially contact angle of cutter, db (z) it is the not deformed chip width at z to represent axial location；
z1,jWith z respectively2,jFor the upper limit of integral and lower limit of integral of j-th of cutting edge, upper limit of integral and lower limit of integral are by work Part processing front curve, work pieces process rear curved surface, tool in cutting sword envelope surface three determines, is by Tool in Cutting specifically What sword envelope surface was formed in the forward and backward curved surface of work pieces process scans the decision of body body, and when processing front curve (the workpiece hair of workpiece Base curved surface) and processing rear curved surface (workpiece design curved surface) have determined that after, scanning body is determined by the generating tool axis vector of cutter, also It is to say z1,jWith z respectively2,jTo be the function of generating tool axis vector, being determined by generating tool axis vector.And, it is necessary to import specific in specific calculate The Cast blanks surface model of workpiece, the design surface model of workpiece, the cutter-contact point trace line and tool geometrical parameter of cutter, then Judge whether it participates in cutting according to the position relationship before and after the point and work pieces process in tool in cutting sword using Z-map methods.
hjIt is the function of generating tool axis vector for the undeformed chip thickness of j-th of cutting edge.Not deformed chip is in workpiece material Be removed in material the two neighboring cutting edge cycle of moment cutter enveloping surface and between Partial Resection workpiece portion, it is thick Degree is projection of the cutting edge infinitesimal feeding vector outside cutter enveloping surface on direction of normal, and the outer direction of normal of cutter enveloping surface with Generating tool axis vector is related.
Constraints has two in the embodiment of the present invention：Meet that generating tool axis vector of reachable tree requirement and contact requirement is empty Between be used as constraints, wherein, the satisfaction of reachable tree requirement is realized by determining without interference generating tool axis vector subspace, no interference Generating tool axis vector subspace be by without global interference generating tool axis vector subspace, without curvature interference generating tool axis vector subspace, without knife bottom Interference generating tool axis vector subspace and sought common ground determination without lathe interference generating tool axis vector subspace；Contact is required to meet by cutter curved surface Meet second order contact relational implementation between designing curved surface workpiece, by meeting reachable tree requirement and contact requirement at the same time Optimal generating tool axis vector is determined in constraint space, to cause the cutter force deformation in constraint space to minimize.
First, P at cutter-contact point is establishedi,jWithout interference generating tool axis vector subspace VG(Pi,j), as shown in Fig. 2 this is without interference knife The expression formula of axial vector subspace is：
VG(PI, j)=VTs(PI, j)∩Vκ(PI, j)∩VTb(PI, j)∩VM(PI, j).
Wherein, VG(Pi,j) it is P at cutter-contact pointi,jWithout interference generating tool axis vector subspace, VTs(Pi,j) it is P at cutter-contact pointi,j Be without global interference generating tool axis vector subspace, its expression formula：
VTs(Pi,j)={ v | dmin(STs(Pi,j,v),SW) ＞ δTs}.
Wherein, STsFor tool circumferential envelope surface, SWFor the real-time curved surface of workpiece, δTsFor avoid global interference set safety away from From.
Vκ(Pi,j) it is P at cutter-contact pointi,jBe without curvature interference generating tool axis vector subspace, its expression formula：
Vκ(Pi,j)={ v | κT(Pi,j, v) and ＞ κmax(Pi,j)}
Wherein, κTFor cutter envelope surface STIn cutter-contact point Pi,jThe curvature of place's effectively cutting curve, κmaxExist for workpiece Maximum curvature at cutter-contact point；
VTb(Pi,j) it is P at cutter-contact pointi,jBe without knife bottom interference generating tool axis vector subspace, its expression formula：
VTb(Pi,j)={ v | dmin(STb(Pi,j,v),SW) ＞ δTb}
Wherein, STbFor cutter bottom cutting edge envelope surface, δTbTo avoid the interference of knife bottom from setting safe distance.
VM(Pi,j) it is P at cutter-contact pointi,jBe without lathe interference generating tool axis vector subspace, its expression formula：
VM(Pi,j)={ v | dmin(STs(Pi,j,v),SM) ＞ δM}
SMFor the machine tool component external envelope curved surface such as jig, workbench, δMTo avoid lathe interference from setting safe distance.
2nd, P at cutter-contact point is establishedi,jMeet second order contact requirement generating tool axis vector subspace Vc, as shown in Fig. 2 the second order is cut Touch and require that the expression formula that the need of generating tool axis vector subspace meet is：
Wherein, k1And k2To design curved surface P at cutter-contact pointi,jTwo principal curvatures, k11、k12、k21、k22It is bent based on respectively Rate k1And k2For the partial derivative of line of curvature arc length, R is the radius of gyration of point of a knife.
Two constraint subspace V that can determine that cutter more thanGc, as shown in Fig. 2 it is excellent then to establish generating tool axis vector Change mathematical modeling, the expression formula of the optimal tool orientation mathematical modeling is：
mine(α,β)
s.t.
1.v(Pi,j,α,β)∈VG(Pi,j)
2.v(Pi,j,α,β)∈Vc(Pi,j)。
It should be appreciated that for those of ordinary skills, can according to the above description be improved or converted, And all these modifications and variations should all belong to the protection domain of appended claims of the present invention.

Claims (2)

1. a kind of multi-shaft numerical control machining knife axial vector optimization method based on cutter stress deformation, it is characterised in that including following Step：
1) cutter relevant parameter gathers, and the parameter includes the cantilevered length L of cutter and cutter bar parts, the cutter tooth partial-length of cutter Lf, the effective diameter and diameter D of cutter cutter tooth parteAnd D, the elastic modulus E of cutter material, the moment of inertia of cutter and cutter bar part I；The design curved surface data information of workpiece, the Cast blanks surface data message of workpiece, cutter-contact point trace line data message, main shaft turn Speed, the cutting edge number N of cutterf, the radial direction of cutting edge infinitesimal, tangential and axial cutting force COEFFICIENT Kr、KtAnd Ka, the footpath of cutter To contact angleThe axially contact angle κ of cutter；
2) cutting force and multi-axis numerical control equipment process system rigidity establish multiaxis NC maching according to suffered by cutter in process During cutter force deformation model e (α, β)；
E is cutter stress deformation synthetic quantity；α and β is respectively the top rake and angle of heel of cutter；
3) determine model meets constraints：Constraints is used as including reachable tree requirement and contact requirement；
4) it is minimum with cutter force deformation according to cutter force deformation model and constraints during multiaxis NC maching Target is turned to, establishes optimal tool orientation mathematical modeling, the expression formula of the optimal tool orientation mathematical modeling is：
mine(α,β)
s.t.
1.v(Pi,j,α,β)∈VG(Pi,j)
2.v(Pi,j,α,β)∈Vc(Pi,j)
5) above optimal tool orientation mathematical modeling is solved, determines optimal generating tool axis vector Vc,optThe cutter top rake at place and inclination Angle；
6) cutter top rake and angle of heel are converted into generating tool axis vector, the generating tool axis vector after being optimized, conversion formula is：
V (α, β)=xpsinαcosβ+ypsinαsinβ+zpcosα.
Wherein, v is generating tool axis vector, and α and β are respectively the top rake and angle of heel of cutter, xp、yp、zpRespectively cutter-contact point coordinate system Three change in coordinate axis direction vectors.
2. the multi-shaft numerical control machining knife axial vector optimization method according to claim 1 based on cutter stress deformation, it is special Sign is that the expression formula of cutter force deformation mathematical modeling is in the step 2)：
<mrow> <mi>e</mi> <mrow> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <msub> <mi>e</mi> <mi>x</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>e</mi> <mi>y</mi> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>e</mi> <mi>z</mi> </msub> <mn>2</mn> </msup> </mrow> </msqrt> <mo>.</mo> </mrow>
Wherein, e is cutter stress deformation synthetic quantity, ex、ey、ezRespectively three seats of the cutter stress deformation along tool coordinate system The component in parameter direction, ex、ey、ezExpression formula be：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>e</mi> <mi>x</mi> </msub> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> <mo>=</mo> <msub> <mi>S</mi> <mi>x</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>y</mi> </msub> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> <mo>=</mo> <msub> <mi>S</mi> <mi>y</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>e</mi> <mi>z</mi> </msub> <mo>(</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> <mo>=</mo> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>F</mi> <mi>z</mi> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mfenced>
Wherein, Sx、Sy、SzRespectively component of the cutter tips static compliance along three change in coordinate axis direction of tool coordinate system, Sx、Sy、 SzExpression formula be：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>S</mi> <mi>x</mi> </msub> <mo>=</mo> <mfrac> <mtable> <mtr> <mtd> <mrow> <mn>3</mn> <msup> <msub> <mi>L</mi> <mi>s</mi> </msub> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>r</mi> <mo>-</mo> <msup> <msub> <mi>L</mi> <mi>s</mi> </msub> <mn>3</mn> </msup> <mo>+</mo> <mn>2</mn> <msup> <msub> <mi>L</mi> <mi>f</mi> </msub> <mn>2</mn> </msup> <mo>/</mo> <msup> <msub> <mi>&amp;mu;</mi> <mi>t</mi> </msub> <mn>4</mn> </msup> <mo>+</mo> <mn>3</mn> </mrow> </mtd> <mtd> <mrow> <mn>2</mn> <msub> <mi>L</mi> <mi>s</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>r</mi> <mo>-</mo> <msup> <msub> <mi>L</mi> <mi>s</mi> </msub> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow> <msub> <mi>L</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>r</mi> </mrow> </mtd> </mtr> </mtable> <mrow> <mn>6</mn> <mi>E</mi> <mi>I</mi> </mrow> </mfrac> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mi>y</mi> </msub> <mo>=</mo> <mfrac> <mtable> <mtr> <mtd> <mrow> <mn>3</mn> <msup> <msub> <mi>L</mi> <mi>s</mi> </msub> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>r</mi> <mo>-</mo> <msup> <msub> <mi>L</mi> <mi>s</mi> </msub> <mn>3</mn> </msup> <mo>+</mo> <mn>2</mn> <msup> <msub> <mi>L</mi> <mi>f</mi> </msub> <mn>2</mn> </msup> <mo>/</mo> <msup> <msub> <mi>&amp;mu;</mi> <mi>t</mi> </msub> <mn>4</mn> </msup> <mo>+</mo> <mn>3</mn> </mrow> </mtd> <mtd> <mrow> <mn>2</mn> <msub> <mi>L</mi> <mi>s</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>L</mi> <mrow> <mi>s</mi> <mi>f</mi> </mrow> </msub> <mo>-</mo> <mi>r</mi> <mo>-</mo> <msup> <msub> <mi>L</mi> <mi>s</mi> </msub> <mn>2</mn> </msup> </mrow> </mtd> <mtd> <mrow> <msub> <mi>L</mi> <mi>f</mi> </msub> <mo>-</mo> <mi>r</mi> </mrow> </mtd> </mtr> </mtable> <mrow> <mn>6</mn> <mi>E</mi> <mi>I</mi> </mrow> </mfrac> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mi>z</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>L</mi> <mi>s</mi> </msub> <mo>+</mo> <msub> <mi>L</mi> <mi>f</mi> </msub> <mo>/</mo> <msup> <msub> <mi>&amp;mu;</mi> <mi>t</mi> </msub> <mn>2</mn> </msup> </mrow> <mrow> <msub> <mi>EA</mi> <mi>s</mi> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced>
Wherein, LsFor the cantilevered length of cutter and cutter bar part, LfFor the cutter tooth partial-length of cutter, LsfFor the total cantilevered length of cutter, And Lsf=Ls+Lf, μtFor cutter tooth part effective diameter coefficient, and μt=De/ D, wherein DeIt is respectively having for cutter cutter tooth part with D Diameter and diameter are imitated, E is the modulus of elasticity of cutter material, and I is the moment of inertia of cutter and cutter bar part；
F in formulax、Fy、FzComponent of the static cutting force along three change in coordinate axis direction of tool coordinate system respectively suffered by cutter, Fx、Fy、FzExpression formula be：
Wherein,It is time t function for angle of eccentricity, fx、fy、fzDynamic cutting force respectively suffered by cutter is along knife Have the component of three change in coordinate axis direction of coordinate system, fx、fy、fzExpression formula be：
Wherein, j be cutter cutting edge sequence number, NfFor the cutting edge number of cutter, hjCut for j-th of the not deformed of cutting edge Consider to be worth doing thickness, be generating tool axis vector function；Kr、KtAnd KaThe respectively radial direction of cutting edge infinitesimal, tangential and axial cutting force coefficient,For the angle that radially contacts with of cutter, κ is the axially contact angle of cutter, and db (z) represents that axial location is wide for the not deformed chip at z Degree；z1,jAnd z2,jThe upper limit of integral and lower limit of integral of respectively j-th cutting edge, z1,jAnd z2,jBy work pieces process front curve, work Part processing rear curved surface, tool in cutting sword envelope surface three decision, z1,jWith z respectively2,jFor be the function of generating tool axis vector, by cutter shaft Vector determines.
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