CN107728576B - multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation - Google Patents
multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation Download PDFInfo
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- CN107728576B CN107728576B CN201710861499.9A CN201710861499A CN107728576B CN 107728576 B CN107728576 B CN 107728576B CN 201710861499 A CN201710861499 A CN 201710861499A CN 107728576 B CN107728576 B CN 107728576B
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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
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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
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/37—Measurements
- G05B2219/37087—Cutting forces
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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
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/37—Measurements
- G05B2219/37355—Cutting, milling, machining force
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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
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/37—Measurements
- G05B2219/37397—Measuring gap between tool and workpiece
Abstract
the invention discloses a multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation, which comprises the following steps of 1) parameter acquisition, 2) establishing a cutter stress deformation model in the multi-axis numerical control machining process according to cutting force borne by a cutter in the machining process and the rigidity of a multi-axis numerical control equipment process system, 3) determining that the model meets constraint conditions, 4) establishing a cutter shaft vector optimization mathematical model by aiming at minimizing the cutter stress deformation according to the cutter stress deformation model and the constraint conditions, 5) solving the cutter shaft vector optimization mathematical model to determine a cutter front inclination angle and a side inclination angle at an optimal cutter shaft vector V c,opt, and 6) converting the cutter front inclination angle and the side inclination angle into cutter shaft vectors to obtain optimized cutter shaft vectors.
Description
Technical Field
The invention relates to the technology of numerical control machine tools, in particular to a multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation.
Background
The complex curved surface part has wide application in the fields of aerospace, energy power and the like, and is generally processed by adopting multi-axis numerical control processing equipment. In the numerical control machining process of the complex curved surface, the cutter and the machined curved surface have good contact cutting state, which is one of the key factors for ensuring the machining quality of parts, and ideally, the vector direction of the cutter shaft is consistent with the normal direction of the machining point of the curved surface. The appearance of a rotating shaft of the multi-axis numerical control processing machine tool enables a cutter to move in a three-dimensional space relative to a workpiece, a cutter shaft continuously swings relative to the surface of the workpiece, and the cutter and a processed curved surface are in good contact state geometrically by adjusting a heel angle and a side slip angle in a local coordinate system. With the increasingly complex surface shape of the curved surface, particularly for a complex curved surface with local curvature abrupt change characteristics, a large cutter shaft vector change exists in the swinging process of the cutter shaft relative to the surface of a part, so that a large rotation angle change of a machine tool swing head or a rotary table is caused. The existing multi-axis numerical control machining cutter shaft vector planning only considers the characteristics of geometric constraint and process system rigidity.
Disclosure of Invention
The invention aims to solve the technical problem of providing a multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation aiming at the defects in the prior art.
The technical scheme adopted by the invention for solving the technical problems is as follows: a multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation comprises the following steps:
1) Collecting relevant parameters of the cutter, wherein the parameters comprise the overhanging length L of the cutter bar part of the cuttersLength L of tooth portion of toolfEffective diameter and diameter D of the tooth portion of the tooleD, the elastic modulus E of the cutter material and the inertia moment I of the cutter bar part of the cutter; design curved surface data information (curved surface after machining) of workpiece, blank curved surface data information (curved surface before machining) of workpiece, tool contact trajectory data information, spindle rotation speed, number N of cutting edges of toolfRadial, tangential and axial cutting force coefficients K of cutting edge infinitesimalr、KtAnd KaRadial contact angle of toolThe axial contact angle kappa of the cutter;
2) Establishing a tool stress deformation model e (alpha, beta) in the multi-axis numerical control machining process according to the cutting force borne by the tool in the machining process and the rigidity of a multi-axis numerical control equipment process system;
e is the resultant of the deformation of the cutter under stress; alpha and beta are respectively a front rake angle and a side rake angle of the cutter;
3) determining the satisfied constraint conditions of the model: the method comprises the following steps of taking reachable space requirements and contact requirements as constraint conditions;
4) establishing a cutter shaft vector optimization mathematical model according to a cutter stress deflection model and constraint conditions in a multi-axis numerical control machining process by taking the minimization of the cutter stress deflection as a target, wherein the expression of the cutter shaft vector optimization mathematical model is as follows:
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) solving the cutter axis vector optimization mathematical model to determine the optimal cutter axis vector Vc,optthe tool rake and roll angles;
6) Converting the front rake angle and the side rake angle of the cutter into a cutter axis vector to obtain an optimized cutter axis vector, wherein the conversion formula is as follows:
v(α,β)＝xpsinαcosβ+ypsinαsinβ+zpcosα， (1)
wherein v is a cutter axis vector, alpha and beta are respectively a front inclination angle and a side inclination angle of the cutter, and xp、yp、zpThree coordinate axis direction vectors of the knife contact point coordinate system are respectively.
According to the scheme, the expression of the mathematical model of the cutter stress deformation amount is as follows:
wherein e is the resultant of the deformation of the cutter under stress, ex、ey、ezAre the components of the tool force deformation along the three coordinate axis directions of the tool coordinate system, ex、ey、ezThe expression of (a) is:
Wherein S isx、Sy、SzAre respectively the components of the static flexibility of the end of the cutter along the three coordinate axis directions of the cutter coordinate system Sx、Sy、SzThe expression of (a) is:
wherein L issIs the overhang length of the shank portion of the tool, Lfis the length of the tooth part of the tool, Lsfis the total overhanging length of the cutter, and Lsf＝Ls+Lf，μtIs a coefficient of the effective diameter of the tooth portion, and mut＝DeD (wherein D)eand D are the effective diameter and diameter of the cutter tooth portion, respectively), E is the modulus of elasticity of the cutter material, and I is the moment of inertia of the cutter shank portion.
F in formula (3)x、Fy、FzThe components of the static cutting force on the tool along three coordinate axes of the tool coordinate system, Fx、Fy、FzThe expression of (a) is:
Wherein the content of the first and second substances,As spindle angle, as a function of time t, fx、fy、fzThe components of the dynamic cutting force on the tool along three coordinate axes of the tool coordinate system, fx、fy、fzThe expression of (a) is:
Wherein j is the number of the cutting edge of the tool, NfThe number of cutting edges of the tool, z1,jAnd respectively z2,jUpper and lower integral limits of the jth cutting edge, hjUndeformed chip thickness of the jth cutting edge, Kr、 KtAnd Karespectively the radial, tangential and axial cutting force coefficients of the cutting edge infinitesimal,Is the radial contact angle of the tool, κ is the axial contact angle of the tool, db (z) represents the undeformed chip width at axial position z;
z1,jAnd respectively z2,jThe integral upper limit and the integral lower limit of the jth cutting edge are determined by a workpiece processing front curved surface, a workpiece processing back curved surface and a cutter cutting edge envelope curved surface, in particular, by a swept body formed by the cutter cutting edge envelope curved surface in the workpiece processing front curved surface and the workpiece processing back curved surface, and when the processing front curved surface (workpiece blank curved surface) and the processing back curved surface (workpiece design curved surface) of the workpiece are determined, the swept body is determined by a cutter shaft vector of the cutter, namely z1,jAnd respectively z2,jIs determined from the arbor vector as a function of the arbor vector. In the specific calculation, a blank curved surface model of a specific workpiece, a designed curved surface model of the workpiece, a tool contact point trajectory line of a tool and geometric parameters of the tool need to be introduced, and then whether the workpiece participates in cutting is judged according to the position relationship between a point on a cutting edge of the tool and the workpiece before and after machining by adopting a Z-map method.
hjThe undeformed chip thickness, which is the jth cutting edge, is a function of the arbor vector. The undeformed chips are parts of the workpiece partially cut away between and on the tool envelope surface of two adjacent cutting edge periods at the moment the chips are cut away in the workpiece material, and the thickness of the undeformed chips is the projection of the cutting edge infinitesimal feed vector onto the outside normal vector direction of the tool envelope, which is related to the arbor vector.
The invention has the following beneficial effects: the method overcomes the defects that only geometric constraint and process system rigidity characteristics are considered in the conventional multi-axis numerical control machining cutter shaft vector planning, can realize multi-axis numerical control machining cutter shaft vector optimization based on cutter stress deformation and geometric constraint and contact requirement constraint, and provides a new idea for the multi-axis numerical control machining cutter shaft vector planning of the complex curved surface.
drawings
The invention will be further described with reference to the accompanying drawings and examples, in which:
FIG. 1 is a flow chart of a method of an embodiment of the present invention;
Fig. 2 is a diagram showing constraint conditions and an optimal arbor vector according to the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
As shown in fig. 1, a multi-axis numerical control machining cutter axis vector optimization method based on cutter stress deformation includes the following steps:
1) Collecting relevant parameters of the cutter, wherein the parameters comprise the overhanging length L of the cutter bar part of the cutterslength L of tooth portion of toolfEffective diameter and diameter D of the tooth portion of the tooled, the elastic modulus E of the cutter material and the inertia moment I of the cutter bar part of the cutter; design curved surface data information (curved surface after machining) of workpiece, blank curved surface data information (curved surface before machining) of workpiece, tool contact trajectory data information, spindle rotation speed, number N of cutting edges of toolfRadial, tangential and axial cutting force coefficients K of cutting edge infinitesimalr、KtAnd KaRadial contact angle of toolThe axial contact angle kappa of the cutter;
2) Establishing a tool stress deformation model e (alpha, beta) in the multi-axis numerical control machining process according to the cutting force borne by the tool in the machining process and the rigidity of a multi-axis numerical control equipment process system;
e is the resultant of the deformation of the cutter under stress; alpha and beta are respectively a front rake angle and a side rake angle of the cutter;
3) Determining the satisfied constraint conditions of the model: the method comprises the following steps of taking reachable space requirements and contact requirements as constraint conditions;
4) Establishing a cutter shaft vector optimization mathematical model according to a cutter stress deflection model and constraint conditions in a multi-axis numerical control machining process by taking the minimization of the cutter stress deflection as a target, wherein the expression of the cutter shaft vector optimization mathematical model is as follows:
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) solving the cutter axis vector optimization mathematical model to determine the optimal cutter axis vector Vc,optThe tool rake and roll angles;
6) Converting the front rake angle and the side rake angle of the cutter into a cutter axis vector to obtain an optimized cutter axis vector, wherein the conversion formula is as follows:
v(α,β)＝xpsinαcosβ+ypsinαsinβ+zpcosα， (1)
Wherein v is a cutter axis vector, alpha and beta are respectively a front inclination angle and a side inclination angle of the cutter, and xp、yp、zpThree coordinate axis direction vectors of the knife contact point coordinate system are respectively.
The expression of the mathematical model of the cutter stress deformation in the embodiment of the invention is as follows:
wherein e is the resultant of the deformation of the cutter under stress, ex、ey、ezare the components of the tool force deformation along the three coordinate axis directions of the tool coordinate system, ex、ey、ezThe expression of (a) is:
wherein S isx、Sy、Szare respectively the components of the static flexibility of the end of the cutter along the three coordinate axis directions of the cutter coordinate system Sx、Sy、SzThe expression of (a) is:
Wherein，LsIs the overhang length of the shank portion of the tool, LfIs the length of the tooth part of the tool, LsfIs the total overhanging length of the cutter, and Lsf＝Ls+Lf，μtIs a coefficient of the effective diameter of the tooth portion, and mut＝DeD (wherein D)eAnd D are the effective diameter and diameter of the cutter tooth portion, respectively), E is the modulus of elasticity of the cutter material, and I is the moment of inertia of the cutter shank portion.
F in formula (3)x、Fy、FzThe components of the static cutting force on the tool along three coordinate axes of the tool coordinate system, Fx、Fy、FzThe expression of (a) is:
Wherein the content of the first and second substances,As spindle angle, as a function of time t, fx、fy、fzThe components of the dynamic cutting force on the tool along three coordinate axes of the tool coordinate system, fx、fy、fzThe expression of (a) is:
wherein j is the number of the cutting edge of the tool, Nfthe number of cutting edges of the tool, z1,jAnd respectively z2,jUpper and lower integral limits of the jth cutting edge, hjUndeformed chip thickness of the jth cutting edge, Kr、 KtAnd KaRespectively the radial, tangential and axial cutting force coefficients of the cutting edge infinitesimal,is the radial contact angle of the tool, κ is the axial contact angle of the tool, db (z) represents the undeformed chip width at axial position z;
z1,jAnd respectively z2,jThe integral upper limit and the integral lower limit of the jth cutting edge are determined by a workpiece processing front curved surface, a workpiece processing back curved surface and a cutter cutting edge envelope curved surface, in particular, by a swept body formed by the cutter cutting edge envelope curved surface in the workpiece processing front curved surface and the workpiece processing back curved surface, and when the processing front curved surface (workpiece blank curved surface) and the processing back curved surface (workpiece design curved surface) of the workpiece are determined, the swept body is determined by a cutter shaft vector of the cutter, namely z1,jAnd respectively z2,jis determined from the arbor vector as a function of the arbor vector. In the specific calculation, a blank curved surface model of a specific workpiece, a designed curved surface model of the workpiece, a tool contact point trajectory line of a tool and geometric parameters of the tool need to be introduced, and then whether the workpiece participates in cutting is judged according to the position relationship between a point on a cutting edge of the tool and the workpiece before and after machining by adopting a Z-map method.
hjThe undeformed chip thickness, which is the jth cutting edge, is a function of the arbor vector. The undeformed chips are parts of the workpiece partially cut away between and on the tool envelope surface of two adjacent cutting edge periods at the moment the chips are cut away in the workpiece material, and the thickness of the undeformed chips is the projection of the cutting edge infinitesimal feed vector onto the outside normal vector direction of the tool envelope, which is related to the arbor vector.
in the embodiment of the invention, the constraint conditions are two: the method comprises the following steps that an arbor vector quantum space meeting reachable space requirements and contact requirements serves as constraint conditions, wherein the reachable space requirements are met by determining an interference-free arbor vector subspace, and the interference-free arbor vector subspace is determined by solving an intersection through a global interference-free arbor vector quantum space, a curvature-free interference arbor vector subspace, a bottom-of-tool interference arbor vector subspace and an inorganic tool interference arbor vector subspace; the contact requirement is met by meeting a second-order contact relation between the curved surface of the cutter and the designed curved surface of the workpiece, and the optimal cutter shaft vector is determined in a constraint space which meets the reachable space requirement and the contact requirement at the same time, so that the stress deformation of the cutter in the constraint space is minimized.
First, establishing P at the knife contacti,jNon-interference cutter axial vector quantum space VG(Pi,j) As shown in fig. 2, the expression of the non-interference arbor vector quantum space is:
VG(Pi，j)＝VTs(Pi，j)∩Vκ(Pi，j)∩VTb(Pi，j)∩VM(Pi，j)，
Wherein, VG(Pi,j) Is the point P of knife contacti,jwithout interference of the knife axis vector subspace, VTs(Pi,j) Is the point P of knife contacti,jThe global interference-free arbor vector subspace has the expression:
VTs(Pi,j)＝{v|dmin(STs(Pi,j,v),SW)＞δTs}，
Wherein S isTsIs a circumferential envelope surface of the tool, SWFor real-time curved surface of work, deltaTsA safety distance is set to avoid global interference.
Vκ(Pi,j) Is the point P of knife contacti,jThe non-curvature interference knife axis vector subspace has the expression as follows:
Vκ(Pi,j)＝{v|κT(Pi,j,v)＞κmax(Pi,j)}
Wherein, κTFor enveloping a curved surface S of the toolTat knife contact point Pi,jcurvature of the point of the effective cutting curve, κmaxmaximum curvature of the workpiece at the knife contact point;
VTb(Pi,j) Is the point P of knife contacti,jThe expression of the non-bottom interference cutter axis vector subspace is as follows:
VTb(Pi,j)＝{v|dmin(STb(Pi,j,v),SW)＞δTb}
Wherein S isTbIs a curved surface enveloped by a cutting edge at the bottom of the cutter, deltaTband a safety distance is set for avoiding the interference of the knife bottom.
VM(Pi,j) As a knife contactat Pi,jThe expression of the inorganic tool interference cutter axis vector subspace is as follows:
VM(Pi,j)＝{v|dmin(STs(Pi,j,v),SM)＞δM}
SMfor the outer envelope curve of machine tool parts such as machine tool clamps, work tables, etc., deltaMa safety distance is set for avoiding machine tool interference.
secondly, establishing a knife contact point Pi,jCutter shaft vector subspace V meeting second-order contact cutting requirementcAs shown in fig. 2, the second-order contact requirement is expressed by the following expression that the second-order contact requirement is satisfied by the arbor vector subspace:
Wherein k is1and k2For designing curved surface at knife contact point Pi,jTwo principal curvatures of, k11、k12、k21、k22Respectively, main curvature k1And k2For the partial derivative of the arc length of the curvature line, R is the radius of gyration of the nose.
The constraint subspace V of the cutter can be determined by the two partsGcAs shown in fig. 2, a mathematical model for arbor vector optimization may then be established, where the expression of the mathematical model for arbor vector optimization 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)。
it will be understood that modifications and variations can be made by persons skilled in the art in light of the above teachings and all such modifications and variations are intended to be included within the scope of the invention as defined in the appended claims.
Claims (2)
1. A multi-axis numerical control machining cutter shaft vector optimization method based on cutter stress deformation is characterized by comprising the following steps:
1) collecting relevant parameters of the cutter, wherein the parameters comprise the overhanging length L of the cutter bar part of the cuttersLength L of tooth portion of toolfEffective diameter and diameter D of the tooth portion of the tooleD, the elastic modulus E of the cutter material and the inertia moment I of the cutter bar part of the cutter; design curved surface data information of the workpiece, blank curved surface data information of the workpiece, tool contact point trajectory data information, main shaft rotating speed, and number N of cutting edges of the toolfradial, tangential and axial cutting force coefficients K of cutting edge infinitesimalr、KtAnd KaRadial contact angle of toolThe axial contact angle kappa of the cutter;
2) Establishing a tool stress deformation model e (alpha, beta) in the multi-axis numerical control machining process according to the cutting force borne by the tool in the machining process and the rigidity of a multi-axis numerical control equipment process system;
e is the resultant of the deformation of the cutter under stress; alpha and beta are respectively a front rake angle and a side rake angle of the cutter;
3) Determining the satisfied constraint conditions of the model: the method comprises the following steps of taking reachable space requirements and contact requirements as constraint conditions;
4) Establishing a cutter shaft vector optimization mathematical model according to a cutter stress deflection model and constraint conditions in a multi-axis numerical control machining process by taking the minimization of the cutter stress deflection as a target, wherein the expression of the cutter shaft vector optimization mathematical model is as follows:
min e(α,β)
s.t.
1.v(Pi,j,α,β)∈VG(Pi,j)
2.v(Pi,j,α,β)∈Vc(Pi,j)
Wherein, VG(Pi,j) Is the point P of knife contacti,jwithout interference of the knife axis vector subspace, Vc(Pi,j) Is the point P of knife contacti,jthe cutter shaft vector subspace meeting the second-order contact requirement is met;
5) solving the cutter shaft vector optimization mathematical model to determine the optimal cutter shaftvector Vc,optthe tool rake and roll angles;
6) converting the front rake angle and the side rake angle of the cutter into a cutter axis vector to obtain an optimized cutter axis vector, wherein the conversion formula is as follows:
v(α,β)＝xpsinαcosβ+ypsinαsinβ+zpcosα，
wherein v is a cutter axis vector, alpha and beta are respectively a front inclination angle and a side inclination angle of the cutter, and xp、yp、zpthree coordinate axis direction vectors of the knife contact point coordinate system are respectively.
2. The multi-axis numerical control machining cutter shaft vector optimization method based on cutter force deformation according to claim 1, wherein the expression of the mathematical model of the cutter force deformation in the step 2) is as follows:
Wherein e is the resultant of the deformation of the cutter under stress, ex、ey、ezare the components of the tool force deformation along the three coordinate axis directions of the tool coordinate system, ex、ey、ezThe expression of (a) is:
Wherein S isx、Sy、Szare respectively the components of the static flexibility of the end of the cutter along the three coordinate axis directions of the cutter coordinate system Sx、Sy、SzThe expression of (a) is:
Wherein L issis the overhang length of the shank portion of the tool, LfIs the length of the tooth part of the tool, Lsfis the total overhanging length of the cutter, and Lsf＝Ls+Lf，μtis a coefficient of the effective diameter of the tooth portion, and mut＝DeD, wherein Ded is the effective diameter and the diameter of the cutter tooth part of the cutter respectively, E is the elastic modulus of the cutter material, and I is the inertia moment of the cutter bar part of the cutter;
In the formula Fx、Fy、FzThe components of the static cutting force on the tool along three coordinate axes of the tool coordinate system, Fx、Fy、FzThe expression of (a) is:
wherein the content of the first and second substances,As spindle angle, as a function of time t, fx、fy、fzThe components of the dynamic cutting force on the tool along three coordinate axes of the tool coordinate system, fx、fy、fzthe expression of (a) is:
Wherein j is the number of the cutting edge of the tool, NfThe number of cutting edges of the tool, hjthe undeformed chip thickness, which is the jth cutting edge, is a function of the arbor vector; kr、KtAnd KaRespectively the radial, tangential and axial cutting force coefficients of the cutting edge infinitesimal,Is the radial contact angle of the tool, κ is the axial contact angle of the tool, db (z) represents the undeformed chip width at axial position z; z is a radical of1,jAnd z2,jUpper and lower integral limits, z, of the jth cutting edge, respectively1,jAnd z2,jIs determined by a workpiece processing front curved surface, a workpiece processing back curved surface and a cutter cutting edge enveloping curved surface, z1,jAnd respectively z2,jIs determined from the arbor vector as a function of the arbor vector.
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