CN114580090A - Dynamic characteristic resolving method for friction coefficient of rear cutter face of cutter tooth pair of square shoulder milling cutter - Google Patents

Dynamic characteristic resolving method for friction coefficient of rear cutter face of cutter tooth pair of square shoulder milling cutter Download PDF

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CN114580090A
CN114580090A CN202111622966.5A CN202111622966A CN114580090A CN 114580090 A CN114580090 A CN 114580090A CN 202111622966 A CN202111622966 A CN 202111622966A CN 114580090 A CN114580090 A CN 114580090A
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cutter
tooth
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姜彬
李伟恒
赵培轶
王成基
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Harbin University of Science and Technology
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Abstract

A dynamic characteristic resolving method for friction coefficients of rear cutter faces of cutter tooth pairs of square shoulder milling cutters belongs to the technical field of milling cutter machining. The method comprises the steps of calculating the area unit of the rear cutter face of the milling cutter tooth pair; solving the unit friction speed of the area of the rear cutter face of the milling cutter tooth pair; a method for extracting the characteristic parameters of the thermal coupling field of the rear cutter face of the milling cutter tooth pair; constructing an instantaneous friction energy consumption distribution function of a rear cutter face of a milling cutter tooth pair; verifying the accumulated friction energy consumption of the rear cutter face of the milling cutter tooth pair; a method for calculating instantaneous friction coefficient and instantaneous friction stress of a rear cutter face of a milling cutter tooth pair. The method is used for solving the problem of providing a method for calculating the dynamic characteristic of the friction coefficient of the rear cutter face of the cutter tooth pair of the square shoulder milling cutter, and comprises the steps of constructing an area unit of the rear cutter face of the cutter tooth pair, providing a specific calculation method, providing a calculation method capable of effectively representing instantaneous friction characteristic parameters based on an atomic interface theory, and constructing an accurate friction speed distribution function and a normal stress distribution function at a friction characteristic point on the area unit.

Description

Dynamic characteristic resolving method for friction coefficient of rear cutter face of cutter tooth pair of square shoulder milling cutter
Technical Field
The invention relates to a method for calculating the area unit of the rear cutter face of a milling cutter tooth pair, belonging to the technical field of milling cutter processing.
Background
In the intermittent milling process of the square shoulder milling cutter, under the influence of factors such as cutter tooth errors and milling vibration of the milling cutter, the rear cutter face of the cutter tooth pair and the machining transition surface of a workpiece show an unstable contact relation, so that the friction coefficient of a characteristic point of a friction contact area between the rear cutter face of the cutter tooth pair and the machining transition surface of the workpiece is dynamically changed, the existing friction coefficient is generally considered as a constant, the interaction of the friction coefficient, which is influenced by factors such as instantaneous contact infinitesimal area, friction speed and normal stress of the rear cutter face of the cutter tooth pair, is ignored, and as a result, the error of the friction coefficient, friction stress and friction energy consumption calculation result and an actual result is large. Therefore, an accurate method for calculating the friction coefficient of the friction contact area of the cutter tooth pair flank is provided, which is necessary for representing the distribution of the tribological characteristic parameters of the cutter tooth pair flank, such as friction stress and friction energy consumption, on the cutter tooth pair flank.
Disclosure of Invention
The present invention has been developed in order to solve the problems of providing a method for calculating the dynamic coefficient of friction of the flank side of a tooth pair of a square shoulder milling cutter, and a brief summary of the invention is provided below in order to provide a basic understanding of some aspects of the invention. It should be understood that this summary is not an exhaustive overview of the invention. It is not intended to determine the key or critical elements of the present invention, nor is it intended to limit the scope of the present invention.
The technical scheme of the invention is as follows:
a dynamic characteristic resolving method for friction coefficients of rear cutter faces of cutter tooth pairs of square shoulder milling cutters comprises the following steps: step 1, calculating a method for the area of the rear cutter face of a milling cutter tooth pair; step 2, based on the step 1, solving the unit friction speed of the area of the rear cutter face of the milling cutter tooth pair; step 3, extracting the characteristic parameters of the thermal coupling field of the rear cutter face of the milling cutter tooth pair; step 4, constructing an instantaneous friction energy consumption distribution function of the rear cutter face of the milling cutter tooth pair; step 5, verifying the accumulated friction energy consumption of the rear cutter face of the milling cutter tooth pair; and 6, calculating the instantaneous friction coefficient and the instantaneous friction stress of the rear cutter face of the milling cutter tooth pair.
Further, the step 1 comprises: the method for measuring the axial and radial errors of the milling cutter comprises the following steps:
Δri＝rmax-ri(i＝1,2···m) (1)
wherein, Δ riRadial error of the ith tooth, riRadius of gyration of ith cutter tooth tip point of the milling cutter, wherein i is 1,2,3, rmaxThe maximum radius of gyration of the three cutter tooth tip points of the milling cutter.
The axial error calculation method is as follows:
Δzi＝l1-li(i＝1,2···m) (2)
wherein, Δ ziAxial error of the ith tooth,/1Distance from the lowest point of the milling cutter to the end face,/iThe distance from the lowest point of the ith cutter tooth to the end face.
Knife tooth i coordinate system oi-aibiciAnd milling cutter structure coordinate system Os-a rotation matrix I of XYZ1Translation matrix M1Respectively as follows:
Figure BDA0003438786420000011
wherein
Figure BDA0003438786420000013
Is the ith cutter tooth instantaneous position angle;
milling cutter structure coordinate system Os-XYZ and the cutting coordinate system O of the milling cutter under the action of vibrationscInstantaneous rotation matrix I of UVW2Comprises the following steps:
Figure BDA0003438786420000012
wherein the content of the first and second substances,
Figure BDA0003438786420000014
is the instantaneous included angle of the V axis and the Y axis in the UVW plane:
Figure BDA0003438786420000021
in the formula (I), the compound is shown in the specification,
Figure BDA0003438786420000028
for initial cutting-in t of milling cutter0At time t, which is equal to 0, the angle between the V axis and the Y axis in the UVW plane is:
Figure BDA0003438786420000022
wherein a iseIs the cutting width.
Milling cutter cutting coordinate system O under vibration actioncUVW and milling cutter cutting coordinate system o without vibration0Instantaneous rotation matrix I of uvw3、I4Instantaneous translation matrix M2Respectively as follows:
Figure BDA0003438786420000023
Figure BDA0003438786420000024
wherein, theta1(t) is the W axis at vo0Instantaneous angle of projection on the w plane with the w axis, θ2(t) is the W axis at uo0Instantaneous angle of projection of w plane to w axis, Ax(t)、Ay(t)、AzAnd (t) is the vibration displacement of the milling cutter along the directions of the x axis, the y axis and the z axis respectively.
Milling cutter cutting coordinate system o without vibration0The instantaneous translation matrix M of uvw with the object coordinate system o-xyz3Comprises the following steps:
Figure BDA0003438786420000025
in the formula, xo0(t),yo0(t),zo0(t) is the instantaneous position coordinate of the milling cutter cutting coordinate system coordinate origin oo in the workpiece coordinate system o-xyz under the vibration-free action;
Figure BDA0003438786420000026
wherein v isfFor milling cutter feed speed, L0Is the length of the workpiece, W0Is the width of the workpiece, H0Is the height of the workpiece, apThe depth of cut of the workpiece;
milling cutter center point track O under vibration actioncThe specific solving method of the variables such as (x, y, z), the attitude angle theta (t) of the milling cutter, the direction angle theta' (t) of the milling cutter and the like is as follows:
(1) milling cutter cutting coordinate system O under vibration actionc-locus of movement O of UVW origin of coordinatesc(x, y, z) is:
Oc(x,y,z)＝[x,y,z,1]T＝M3·[Ax(t),Ay(t),Az(t),1]T (11)
(2) the instantaneous included angle theta (t) between the W axis and the W axis is as follows:
Figure BDA0003438786420000027
wherein theta (t) is the instantaneous attitude angle of the milling cutter; l is the overhanging length of the milling cutter.
(3) W axes are respectively at vo0w face, uo0The projection of the w surface and the w axis form an included angle:
Figure BDA0003438786420000031
(4) theta' (t) is the W axis at uo0And the included angle between the projection on the w plane and the v axis is calculated as follows:
Figure BDA0003438786420000032
(5) coordinate system o of cutter teethi-aibiciO-xyz conversion relation matrix with workpiece coordinate system
Figure BDA00034387864200000311
The following were used:
Figure BDA00034387864200000312
(6) in the work coordinate system, the minor cutting edge l'fThe equation of (a) is:
Figure BDA0003438786420000033
wherein the content of the first and second substances,
Figure BDA0003438786420000034
the length of any point on the secondary cutting edge of the cutter tooth from the point of the cutter point,
Figure BDA0003438786420000035
is total length of secondary cutting edge of the cutter tooth, k'rIs a secondary deflection angle of a secondary back face of the cutter tooth, lambda'sThe angle of the secondary edge of the rear cutter face of the cutter tooth pair is the inclination angle of the secondary edge of the cutter tooth pair;
(7) order to
Figure BDA0003438786420000036
Then the knife tooth i knife point oiMovement locus o in the object coordinate systemi(x, y, z) is as shown in formula (17):
Figure BDA00034387864200000310
(8) the rear tool face A of the tool tooth pair in the workpiece coordinate systemi' transition surface B with workpiece machiningiFriction pair ofmAs shown in equation (18):
Figure BDA0003438786420000037
wherein, the upper and lower boundaries of the instantaneous frictional wear formed by the rear cutter face of the cutter tooth pair in the cutting process are as shown in a formula (19);
Figure BDA0003438786420000038
wherein luFor the upper friction boundary of the rear face friction pair of the cutter tooth pair, /)dThe lower boundary of the knife tooth pair rear knife face friction pair friction is shown.
(9) In a workpiece coordinate system, the equation of the tool tooth pair flank is shown as the formula (20).
Figure BDA0003438786420000039
Figure BDA0003438786420000041
Is a rear knife face of the knife tooth pair,
Figure BDA0003438786420000042
each side of the rear cutter face of the cutter tooth pair.
Workpiece machining transition surface B in workpiece coordinate systemiThe calculation method is shown in formula (21).
Figure BDA0003438786420000043
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003438786420000044
the moment when the tool instantaneously cuts into the workpiece,
Figure BDA0003438786420000045
the moment when the cutting tool instantly cuts the workpiece is finished.
Through material mechanics analysis, in the instantaneous cutting process of the cutter tooth, when the equivalent stress sigma on the secondary cutting edge of the cutter tooth is larger than the yield strength sigma of the materialsWhen the cutter is used, the material in the cutting edge area falls off, so that the wear upper boundary of the cutter tooth can use the equivalent stress as a criterion as shown in the formula (22);
σ≥σs (22)
where σ is the equivalent stress, σsIs the yield strength.
Identifying the equivalent stress at each section as yield strength sigmasAnd connecting the characteristic points on all the sections in a cutter tooth coordinate system to construct an instantaneous friction upper boundary curve of the rear cutter face of the milling cutter tooth.
In the whole milling process, equivalent strain exists in a contact area and a non-contact area of a rear cutter face of a cutter tooth of a milling cutter from cutting in to cutting out, the effect variation value of a friction contact area and the like is large, the effect variation value is gradually reduced along the normal vector direction of a cutting edge, and the instantaneous wear lower boundary is subjected to sudden change, so that the wear lower boundary node of the rear cutter face of the cutter tooth can be identified through the equivalent strain variation rate epsilon', and the formula (23) shows;
Figure BDA0003438786420000046
in the formula, epsilon is equivalent strain, epsilon' is equivalent strain rate, YiThe ordinate of the coordinate system is measured for the cutter teeth.
And identifying the position of the maximum mutation of the equivalent strain change rate at each section, connecting the maximum mutation position points of each section, and constructing an instantaneous friction lower boundary curve of the rear cutter face of the cutter tooth in a cutter tooth coordinate system.
By representing the instantaneous contact relation of the friction pair, selecting characteristic points in an instantaneous contact friction area of a rear cutter face of the cutter tooth pair, and calculating an instantaneous contact area unit, wherein the calculating method comprises the following steps;
Figure BDA00034387864200000410
ds is the instantaneous contact area unit of the rear cutter face of the cutter tooth pair; daiIs the area unit length in the plane aioibiProjection on a plane; dbiArea unit width in plane aioibiProjection on a plane; gamma is the normal vector direction of the area unit and ciThe angle between the axes.
In the formula (24), γ (a)i，bi，ci) Constructing a function as an equation (25);
Figure BDA0003438786420000047
further, the step 2 comprises: the equation (17) is solved to obtain a knife tooth arbitrary point track parameter equation as follows:
Figure BDA0003438786420000048
and (3) calculating the time partial derivative of any point track of the cutter teeth in the formula (26) to obtain the following arbitrary component speeds along the x, y and z axis directions of the workpiece coordinate system:
Figure BDA0003438786420000049
in the formula (27), vnxIs the component velocity along the x-axis direction; v. ofnyIs the component velocity along the y-axis direction; v. ofnzIs the component velocity in the z-axis direction.
Relative motion velocity v of any pointnThe calculation method is as follows:
Figure BDA0003438786420000051
relative speed of motion vnUnit vector in the object coordinate system
Figure BDA0003438786420000052
The following were used:
Figure BDA0003438786420000053
intersection o of rear tool face of cutter tooth and transition surface of workpiecerThe following:
Figure BDA0003438786420000054
perorPoint and common tangent plane P of the rear tool face of the cutter tooth pair and the processing transition surface of the workpieceiThe following were used:
Figure BDA0003438786420000055
do vnIn the common tangent plane PiProjection on and passing through point orUnit vector in projection direction
Figure BDA0003438786420000056
The following:
Figure BDA0003438786420000057
wherein lpxIs a unit vector
Figure BDA0003438786420000058
A component vector in the x-axis direction; lpyIs a unit vector
Figure BDA0003438786420000059
A component vector in the y-axis direction; lpzIs a unit vector
Figure BDA00034387864200000510
Component vector in z-axis direction.
The friction speed v of any point on the flank face in the workpiece coordinate systemmComprises the following steps:
vm＝vn·cosθm (33)
wherein, thetamIs a relative movement velocity vnAnd unit vector
Figure BDA00034387864200000511
The calculation method of the included angle is as follows:
θm＝π-θc (34)
Figure BDA00034387864200000512
in the formula (35), θcIs a relative movement velocity vnWith a friction speed vmThe included angle therebetween.
In the workpiece coordinate system, the friction speed direction vector of any point on the secondary flank surface
Figure BDA00034387864200000513
The equation (36) is solved.
Figure BDA00034387864200000514
Wherein v ismx、vmy、vmzThe friction speed of a point on the rear face of the cutter tooth in a workpiece coordinate system is converted into the cutter tooth coordinate system through coordinate transformation, and the formula (37) is shown as follows:
Figure BDA00034387864200000517
wherein v ismaiIs the point on the flank of the cutter tooth along the edge a in the coordinate system of the cutter toothiA friction speed in the axial direction; v. ofmbiThe point on the flank of the knife tooth pair is along the b in the knife tooth coordinate systemiA friction speed in the axial direction; v. ofmciFor the point on the flank of the tooth of the knifeMiddle edge c of tooth coordinate systemiThe friction speed in the axial direction.
The frictional velocity v of any point in the tool tooth coordinate systemmComprises the following steps:
Figure BDA00034387864200000515
in order to represent the dynamic change of the friction characteristic parameters required to be solved from cutting-in to cutting-out of the area unit of the friction contact area of the rear cutter face of the cutter tooth, a calculation model for instantly cutting-in to cutting-out of a single-rotation cutter tooth of the milling cutter is provided;
Figure BDA00034387864200000516
in the formula (39), TiFor the i-th cutting tooth cut-in to cut-out time period, Ti sFor the ith tooth cutting-in time, Ti eCutting time of ith cutter tooth, k is cutting time of ith cutter tooth, phiiThe included angle of the ith cutter tooth from cut-in to cut-out, omega is the angular speed of the milling cutter, tiThe instant the ith tooth cuts into the workpiece.
Figure BDA0003438786420000061
In a similar manner, in formula (40), Ti-1For the i-1 th tooth cut-in to cut-out time period, Ti-1 sFor the ith tooth cutting-in time, Ti-1 eFor the i-1 th tooth cutting time, etaii-1Is the included angle between the ith cutter tooth and the (i-1) th cutter tooth, wherein phiiIs shown as formula (41)
Figure BDA0003438786420000062
Further, the step 3 comprises: in order to extract temperature and equivalent stress based on a thermal coupling field, performing thermal coupling field simulation on the square shoulder milling cutter, performing finite element simulation by using a Johnson-Cook constitutive model mode and using a vibration signal, a milling cutter track and a cutter tooth track measured by an experiment as finite element simulation boundary conditions, and performing finite element tool and workpiece simulation models;
a point-taking method for the friction area of the rear face of a cutter tooth is used for measuring a coordinate system Y parallel to the cutter toothiAxial cross-sections with a spacing Δ l between cross-sections, each cross-section being m1，…mi…，mnTaking points at equal intervals from the upper and lower margins of the wear of the cutter teeth on each section, wherein the interval between the two points is delta k, lmiIs the distance between the ith section and the coordinate origin of the cutter tooth, and is an arbitrary point k on the sectioniThe coordinate solving method of (2) is shown as formula (42);
selecting any point k on the flank faceiCoordinate k in the tool tooth coordinate systemi(aki，bki，cki) As shown in (42):
Figure BDA0003438786420000063
wherein the content of the first and second substances,
Figure BDA0003438786420000064
the distance between two adjacent points in the same cross section is selected.
Will select the point kiConverting the motion trajectory into a workpiece coordinate system to obtain a motion trajectory in the workpiece coordinate system according to the formula (43):
Figure BDA00034387864200000610
Fpthe normal stress on the micro-element of the rear cutter face of the cutter tooth pair passes through the node of the rear cutter face of the cutter tooth, is vertical to the rear cutter face of the cutter tooth and is vertical to the ciAngle of axis thetaci(ii) a Tau is equivalent stress of a cutter tooth rear cutter face at a network node under a thermal coupling field; f1、F2、F3Respectively equivalent stress in the tetrahedral infinitesimal direction; thetak1、θk2、θk3The included angles between the equivalent stress and the normal stress on the three sides are respectively formed; vector for component force cutter tooth coordinate system of equivalent stress on tetrahedral infinitesimal element
Figure BDA0003438786420000065
Expressed as shown in equation (44):
Figure BDA0003438786420000066
Figure BDA0003438786420000067
in the formula (45), the reaction mixture is,
Figure BDA0003438786420000068
is the unit vector of normal stress in the normal direction on the area micro element of the rear cutter face of the cutter tooth pair.
Therefore, θ can be obtained from the expressions (44) and (45)k1、θk2、θk3As shown in formula (46);
Figure BDA0003438786420000069
solving the normal stress as shown in the formula (47);
Fp(ai,bi,ci)＝F1(ai,bi,ci)·cosθk1+F2(ai,bi,ci)·cosθk2+F3(ai,bi,ci)·cosθk3 (47)
further, the step 4 comprises: obtaining the rate of change E of absorbed energy from the interface atom theoryvComprises the following steps:
Figure BDA0003438786420000071
in the formula (48), E is the instantaneous absorption energy at the time t, and upsilon is the atom forced vibration frequency as shown in the formula (49):
Figure BDA0003438786420000072
the instantaneous energy distribution function of absorption is as follows (50):
Figure BDA0003438786420000073
wherein: psi is lattice constant (2.9506X 10)-10m); h is Planck constant (h-6.62607015 × 10)-34J · s); delta is boltzmann constant (delta-1.380649 × 10)-23J/K)；vmRelative friction speed; t is tes1And tes2Respectively the instantaneous initial time and the termination time of the absorbed energy; the calculation method for the temperature rise of the atomic interface is shown as the formula (51):
Figure BDA0003438786420000074
wherein: m is atomic relative to atomic mass of 4.34X 10-26kg，ωnThe natural frequency of the atoms is 4.39 multiplied by 1011rad/s, chi is 1X 10 of the excitation force pair of the interface potential energy field-9N。
Further, the accumulated friction energy consumption in the step 5 is accumulated by an instantaneous energy consumption boundary, and in order to verify the correctness of the calculation model of the distribution function of the accumulated energy consumption, a verification method of the accumulated friction energy consumption of the secondary flank surface is provided as follows; the method for identifying the accumulated wear boundary G at the section of the rear cutter face of the cutter tooth pair comprises the following steps:
G(Xi,Yimax)＝0 (52)
Yimax＝max(Yi(t)) (53)
the whole cutting process of the cutter tooth from cutting in to cutting out the workpiece is obtained by utilizing the instantaneous boundary of the cutter tooth, and the cutting process is different from XiPosition ofAt the instantaneous boundary YiMaximum value YimaxFinally obtaining a compound of YimaxThe formed accumulated friction energy consumption boundary G;
selecting characteristic point energy consumption reconstruction distribution curved surfaces for cutter tooth rear cutter surfaces at the moment when the cutter teeth cut workpieces instantly in a workpiece cutting stage, extracting an energy consumption boundary curve and an abrasion boundary of the cutter tooth rear cutter surface of an experimental cutter tooth pair by using the characteristic point selection method provided in the step 3, and carrying out correlation analysis by using correlation coefficients as follows:
Figure BDA0003438786420000075
in the formula, Co upsilon is a covariance calculation formula, Xi，YiThe measured values of the characteristic points of the boundary curve in the cutter tooth measuring coordinate system,
Figure BDA0003438786420000076
and
Figure BDA0003438786420000077
the average value of the values of the boundary curve at each point is taken; the correlation coefficient ρ is calculated as follows:
Figure BDA0003438786420000078
wherein
Figure BDA0003438786420000079
And
Figure BDA00034387864200000710
as standard deviation, as follows:
Figure BDA00034387864200000711
Figure BDA00034387864200000712
the closer ρ is to 1, the stronger the correlation between the two correlation variables, and the closer to 0, the weaker the correlation between the two correlation variables.
Further, the step 6 comprises: normal pressure of frictional contact area unit is FnAnd the friction coefficient is mu, the work dS performed by the unit infinitesimal friction force in dt time is as follows:
dS＝μ(ai,bi,ci,t)·Fn(ai,bi,ci,t)·vm(ai,bi,ci,t)·dt (58)
assuming that the friction work of the cutter interface is completely converted into the heat energy of the system in the friction motion process
dS＝dE (59)
The friction coefficient distribution function is solved by the formula (59) as shown in the formula (60):
Figure BDA0003438786420000081
the friction stress distribution function obtained by resolving the formulas (58) to (60) is shown as a formula (61)
fp(ai,bi,ci,t)＝μ(ai,bi,ci,t)·Fp(ai,bi,ci,t) (61)
The invention has the following beneficial effects:
1. according to the method, a cutter tooth pair rear cutter face area unit is constructed, a specific calculation method is provided, a calculation method capable of effectively representing instantaneous friction characteristic parameters is provided based on an atomic interface theory, and an accurate friction speed distribution function and a normal stress distribution function at a friction characteristic point are constructed on the area unit;
2. the invention establishes the mapping relation among parameters such as a friction speed distribution function, a normal stress distribution function and the like, and solves the problem that the deviation between friction stress and friction energy consumption calculation and an actual value is larger because the friction coefficient is taken as a constant in the prior art;
3. the friction coefficient is always considered to be a constant in the prior research on the friction coefficient, the dynamic change characteristic of the friction coefficient in the whole cutting process cannot be revealed, and further, the solved characteristic parameters such as friction force, friction energy and the like are not accurate enough and have larger errors;
drawings
FIG. 1 is a schematic flow diagram of the present invention;
FIG. 2 is a model diagram of the rear face-workpiece friction pair of the cutter tooth of the present invention, wherein (1) in FIG. 2 is a square shoulder milling cutter structure, and (2) is the instantaneous contact state of the rear face of the cutter tooth with the workpiece;
FIG. 3 is a graph of the equivalent stress distribution of the rear face of tooth 1 of the present invention;
FIG. 4 is a graph showing equivalent stress at different positions on each cross section of the flank of tooth 1 of the present invention, wherein (a) in FIG. 4 is Xi0.8mm node equivalent stress, and (b) is Xi2.5mm node equivalent stress, (c) is XiEquivalent stress of each node of the section C is 4.2 mm;
FIG. 5 is a diagram of the instantaneous upper flank wear of tooth 1 of the present invention;
FIG. 6 is a graph showing equivalent strain curves of the flank of tooth 1 of the present invention at different positions of the cross-section thereof, wherein (a) in FIG. 6 is XiEquivalent stress change of each node at 0.8mm, and (b) is XiEquivalent strain at each node of 2.5mm, and (c) is XiEquivalent strain of each node is 4.2 mm;
FIG. 7 is a lower boundary view of the instant wear of the flank surface of tooth 1 of the present invention;
FIG. 8 is a unit representation of the flank area of the tooth pair of the milling cutter of the present invention;
FIG. 9 is a pictorial view of a square shoulder milling cutter of the present invention;
FIG. 10 is a diagram of a milling test machining and vibration test system of the present invention;
FIG. 11 is a time domain signal diagram of workpiece cutting vibrations in accordance with the present invention;
FIG. 12 is a model diagram of the friction speed calculation of the rear cutter area unit of the milling cutter tooth pair of the invention;
FIG. 13 is a diagram of a model for solving the cutting-in and cutting-out periods of the milling cutter of the present invention;
FIG. 14 is a selected plot of flank feature points for a tooth set of the present invention;
FIG. 15 is a graph of the friction speeds of the three cutter tooth feature points in the cutting-in stage, and in FIG. 15, (1) is the friction speed of the three cutter tooth feature points in the cutting-in stage, (2) is the friction speed of the three cutter tooth feature points in the two cutting stages, (3) is the friction speed of the three cutter tooth feature points in the middle cutting stage, (4) is the friction speed of the three cutter tooth feature points in the two cutting stages, and (5) is the friction speed of the three cutter tooth feature points in the cutting-out stage;
FIG. 16 is a diagram of a finite element simulation model of the present invention, in FIG. 16, (1) is a workpiece model and (2) is a square shoulder milling cutter tool model;
FIG. 17 is a graph of temperature simulation under a thermodynamic coupling field in accordance with the present invention;
FIG. 18 is a simulation diagram of equivalent stress in a thermodynamic coupling field according to the present invention;
FIG. 19 is a diagram of a method for taking points in the frictional contact area of the flank face of a tooth set in accordance with the present invention;
FIG. 20 is a diagram of an equivalent stress resolution method of the present invention;
FIG. 21 is a graph of normal stresses at the cutting-in to cutting-out stages of three cutter tooth feature points of the present invention, and in FIG. 21, (1) the normal stresses at the cutting-in stages of the three cutter tooth feature points, (2) the normal stresses at the cutting-in two stages of the three cutter tooth feature points, (3) the normal stresses at the cutting middle stages of the three cutter tooth feature points, (4) the normal stresses at the cutting-out two stages of the three cutter tooth feature points, and (5) the normal stresses at the cutting-out stages of the three cutter tooth feature points;
FIG. 22 is a diagram of the friction energy consumption change from the cut-in stage to the cut-out stage of three cutter tooth characteristic points of the present invention, and in FIG. 22, (1) is the friction energy consumption change from the cut-in stage of three cutter tooth characteristic points, (2) is the friction energy consumption change from the cut-in stage to the cut-in stage of three cutter tooth characteristic points, (3) is the friction energy consumption change from the cutting middle section of three cutter tooth characteristic points, (4) is the friction energy consumption change from the cut-out stage of three cutter tooth characteristic points, and (5) is the friction energy consumption change from the cut-out stage of three cutter tooth characteristic points;
fig. 23 is a coordinate diagram of the method for extracting the accumulated boundary of the rear face of the cutter tooth according to the present invention, where (1) in fig. 23 is the instantaneous friction energy consumption (t is 24.55s) of the rear face of the cutter tooth pair, (2) is the instantaneous energy consumption of each section node of the rear face of the cutter tooth pair, and (3) is the accumulated friction energy consumption boundary of the rear face of the cutter tooth pair;
FIG. 24 is a graph of instantaneous cutting energy consumption for the flank of a tooth in accordance with the present invention, where (1) is the tooth energy consumption for the initial plunge condition and (2) is the cumulative friction energy consumption for the tooth for the plunge condition;
FIG. 25 is a point diagram of the energy consumption boundary of the rear face of the cutter tooth according to the present invention, where (1) is the boundary of the accumulated frictional energy consumption of the rear face of the simulated cutter tooth and (2) is the boundary of the wear of the rear face of the experimental cutter tooth;
FIG. 26 is a graph of the flank energy consumption and wear boundary for a tooth in a simulated and experimental state of the present invention;
FIG. 27 is a graph of instantaneous friction coefficients of three tooth features of the present invention, where (1) in FIG. 27 is the instantaneous friction coefficient of tooth 1 feature, (2) is the instantaneous friction coefficient of tooth 2 feature, and (3) is the instantaneous friction coefficient of tooth 3 feature;
fig. 28 is a friction coefficient change diagram of the three cutter tooth feature points in-out stage of the present invention, where (1) in fig. 28 is a friction coefficient change of the three cutter tooth feature points in-out stage, (2) is a friction coefficient change of the three cutter tooth feature points in-out stage, (3) is a friction coefficient change of the three cutter tooth feature points in-out stage of cutting, (4) is a friction coefficient change of the three cutter tooth feature points in-out stage of cutting, (5) is a friction coefficient change of the three cutter tooth feature points in-out stage (Xi ═ 0.8 mm);
FIG. 29 is a graph of instantaneous frictional stress at characteristic points of three teeth according to the present invention, where (1) in FIG. 29 is the instantaneous frictional stress at characteristic points of tooth 1, (2) is the instantaneous frictional stress at characteristic points of tooth 2, and (3) is the instantaneous frictional stress at characteristic points of tooth 3;
fig. 30 is a graph of a change of a cutting-in friction stress to a cutting-out friction stress of three cutter tooth feature points (Xi ═ 0.8mm), in fig. 30, (1) is a change of a friction stress of a cutting-in stage of the three cutter tooth feature points, (2) is a change of a cutting-in two-stage friction stress of the three cutter tooth feature points, (3) is a change of a cutting middle-stage friction stress of the three cutter tooth feature points, (4) is a change of a cutting-out two-stage friction stress of the three cutter tooth feature points, and (5) is a change of a cutting-out stage friction stress of the three cutter tooth feature points;
Detailed Description
In order that the objects, aspects and advantages of the invention will become more apparent, the invention will be described by way of example only, and in connection with the accompanying drawings. It is to be understood that such description is merely illustrative and not intended to limit the scope of the present invention. Moreover, in the following description, descriptions of well-known structures and techniques are omitted so as to not unnecessarily obscure the concepts of the present invention.
The specific implementation mode is as follows: the embodiment is described with reference to fig. 1 to 30, and the method for calculating the dynamic friction coefficient of the flank face of the tooth pair of the square shoulder milling cutter according to the embodiment includes: step 1, calculating a method for the area unit of the rear cutter face of a milling cutter tooth pair; step 2, based on the step 1, solving the unit friction speed of the area of the rear cutter face of the milling cutter tooth pair; step 3, extracting the characteristic parameters of the thermal coupling field of the rear cutter face of the milling cutter tooth pair; step 4, constructing an instantaneous friction energy consumption distribution function of the rear cutter face of the milling cutter tooth pair; step 5, verifying the accumulated friction energy consumption of the rear cutter face of the milling cutter tooth pair; and 6, calculating the instantaneous friction coefficient and the instantaneous friction stress of the rear cutter face of the milling cutter tooth pair. Representing an instantaneous contact friction area of a cutter worker by constructing an instantaneous friction pair contact model of a rear cutter face of the cutter tooth pair, selecting an area unit in the friction contact area, giving an area unit resolving method and constructing a friction speed distribution function and a normal stress distribution function on the area unit; and further constructing instantaneous friction energy consumption, instantaneous friction coefficient and instantaneous friction stress distribution function by adopting a friction speed distribution function and a normal stress distribution function and based on an atomic interface theory. And verifying the correctness of the calculation method by utilizing a characteristic point selection method of a friction contact area of the rear cutter face of the cutter tooth pair, selecting boundary characteristic points of the energy consumption distribution curved surface, constructing an energy consumption boundary curve and performing correlation analysis on the energy consumption boundary curve and the experimental cutter tooth wear boundary curve.
The method comprises the following steps of 1, calculating the area unit of the rear cutter face of the milling cutter tooth pair:
in the cutting process of the milling cutter, due to the influence of the structure and vibration of the milling cutter, the contact time of a cutter worker is changed, so that the research on the contact relation between the rear cutter face of the cutter tooth and the machined transition surface is of great significance, and the instantaneous contact relation between the milling cutter structure and the cutter worker is shown in fig. 2.
TABLE 1 Square shoulder milling cutter Structure and variable interpretation of instantaneous cutting State of the cutter teeth
Figure BDA0003438786420000101
In fig. 2(1), the method for measuring the axial and radial errors of the milling cutter comprises the following steps:
Δri＝rmax-ri(i＝1,2···m) (1)
wherein, Δ riRadial error of the ith tooth, riRadius of gyration of ith cutter tooth tip point of the milling cutter, wherein i is 1,2,3, rmaxThe maximum radius of gyration of the three cutter tooth tip points of the milling cutter.
The axial error calculation method is as follows:
Δzi＝l1-li(i＝1,2···m) (2)
wherein, Δ ziAxial error of the ith tooth,/1Distance from lowest point of milling cutter to end face,/iThe distance from the lowest point of the ith cutter tooth to the end face.
In FIG. 2(2), the tooth i coordinate system oi-aibiciAnd milling cutter structure coordinate system Os-rotation matrix of XYZ I1Translation matrix M1Respectively as follows:
Figure BDA0003438786420000111
milling cutter structure coordinate system Os-XYZ and the cutting coordinate system O of the milling cutter under the action of vibrationscInstantaneous rotation matrix I of UVW2Comprises the following steps:
Figure BDA0003438786420000112
wherein the content of the first and second substances,
Figure BDA0003438786420000119
is the instantaneous included angle of the V axis and the Y axis in the UVW plane:
Figure BDA0003438786420000113
in the formula (I), the compound is shown in the specification,
Figure BDA0003438786420000114
for initial cutting-in t of milling cutter0At time t, which is equal to 0, the angle between the V axis and the Y axis in the UVW plane is:
Figure BDA0003438786420000115
wherein a iseIs the cutting width.
Milling cutter cutting coordinate system O under vibration actioncUVW and milling cutter cutting coordinate system o without vibration0Instantaneous rotation matrix I of uvw3、I4Instantaneous translation matrix M2Respectively as follows:
Figure BDA0003438786420000116
Figure BDA0003438786420000117
wherein, theta1(t) is the W axis at vo0Instantaneous angle of projection on the w plane with the w axis, θ2(t) is the W axis at uo0Instantaneous angle of projection of w plane to w axis, Ax(t)、Ay(t)、AzAnd (t) is the vibration displacement of the milling cutter along the directions of the x axis, the y axis and the z axis respectively. (ii) a
Milling cutter cutting coordinate system o without vibration0Instantaneous translation moments of uvw with the object coordinate system o-xyzMatrix M3Comprises the following steps:
Figure BDA0003438786420000118
in the formula, xo0(t),yo0(t),zo0(t) is the coordinate origin o of the milling cutter cutting coordinate system under the condition of no vibrationoInstantaneous position coordinates in the workpiece coordinate system o-xyz;
Figure BDA0003438786420000121
wherein v isfFor milling cutter feed speed, L0Is the length of the workpiece, W0Is the width of the workpiece, H0Is the height of the workpiece, apThe depth of cut of the workpiece;
milling cutter center point track O under vibration actioncThe specific solving method of the variables such as (x, y, z), the milling cutter attitude angle theta (t), the milling cutter direction angle theta' (t) and the like is as follows:
(1) milling cutter cutting coordinate system O under vibration actionc-locus of motion O of UVW origin of coordinatesc(x, y, z) is:
Oc(x,y,z)＝[x,y,z,1]T＝M3·[Ax(t),Ay(t),Az(t),1]T (11)
(2) the instantaneous included angle theta (t) between the W axis and the W axis is as follows:
Figure BDA0003438786420000122
wherein theta (t) is the instantaneous attitude angle of the milling cutter; l is the overhanging length of the milling cutter.
(3) W axes are respectively at vo0w face, uo0The projection of the w surface and the w axis form an included angle:
Figure BDA0003438786420000123
(4) theta' (t) is the W axis at uo0And the included angle between the projection on the w plane and the v axis is calculated as follows:
Figure BDA0003438786420000124
(5) coordinate system o of cutter teethi-aibiciO-xyz conversion relation matrix with workpiece coordinate system
Figure BDA0003438786420000129
The following were used:
Figure BDA00034387864200001210
(6) in the work coordinate system, the minor cutting edge l'fThe equation of (a) is:
Figure BDA0003438786420000125
wherein the content of the first and second substances,
Figure BDA0003438786420000126
the length of any point on the secondary cutting edge of the cutter tooth from the point of the cutter point,
Figure BDA0003438786420000127
is total length of secondary cutting edge of the cutter tooth, k'rIs a secondary deflection angle of a secondary back face of the cutter tooth, lambda'sThe angle of the secondary edge of the rear cutter face of the cutter tooth pair is the inclination angle of the secondary edge of the cutter tooth pair;
(7) order to
Figure BDA0003438786420000128
Then the knife tooth i knife point oiMovement locus o in the object coordinate systemi(x, y, z) is as shown in formula (17):
Figure BDA00034387864200001211
(8) tooth secondary flank face A 'in a workpiece coordinate system'iMachining transition surface B with workpieceiFriction pair ofmAs shown in equation (18):
Figure BDA0003438786420000131
wherein, the upper and lower boundaries of the instantaneous frictional wear formed by the rear cutter face of the cutter tooth pair in the cutting process are as shown in a formula (19);
Figure BDA0003438786420000132
wherein luFor the upper friction boundary of the rear face friction pair of the cutter tooth pair, /)dThe lower boundary of the knife tooth pair rear knife face friction pair friction is shown.
In a workpiece coordinate system, the equation of the rear tool face of the tool tooth pair is as follows
Figure BDA0003438786420000133
Figure BDA0003438786420000134
Is a rear knife face of the knife tooth pair,
Figure BDA0003438786420000135
each side of the rear cutter face of the cutter tooth pair.
Workpiece machining transition surface B in workpiece coordinate systemiThe calculation method is shown in formula (21).
Figure BDA0003438786420000136
Wherein the content of the first and second substances,
Figure BDA0003438786420000137
the moment when the tool instantaneously cuts into the workpiece,
Figure BDA0003438786420000138
the moment when the cutting tool instantly cuts the workpiece is finished.
In FIG. 2(2), fuFor the upper boundary of the instantaneous wear of the cutter teeth of the milling cutter, through material mechanics analysis, in the instantaneous cutting process of the cutter teeth, when the equivalent stress sigma on the secondary cutting edge of the cutter teeth is greater than the yield strength sigma of the materialsIn the process, the material of the cutting edge region may fall off. The wear upper boundary of the cutter tooth can use the equivalent stress as a criterion as shown in equation (22).
σ≥σs (22)
In FIG. 3, three are taken parallel to YiSection C of the shaft1、C2、C3And points are taken at equal intervals on the three sections, and an equivalent stress curve of the points on each section is drawn as shown in fig. 3.
In FIG. 2(2), fdIn the whole milling process, equivalent strain exists in a contact area and a non-contact area from the cutting-in of the cutter tooth to the cutting-out of the rear cutter face of the cutter tooth of the milling cutter, and the effect variable values of a friction contact area and the like are larger, gradually decrease along the normal vector direction of a cutting edge and are suddenly changed in the instantaneous wear lower boundary, so that the wear lower boundary node of the rear cutter face of the cutter tooth can be identified according to the equivalent strain change rate. As shown in equation (23).
Figure BDA0003438786420000139
In the formula, epsilon is equivalent strain, epsilon' is equivalent strain rate, YiThe ordinate of the coordinate system is measured for the cutter teeth.
By representing the instantaneous contact relation of the friction pair, characteristic points are selected in the instantaneous contact friction area of the rear cutter face of the cutter tooth pair, and the instantaneous contact area unit is calculated, wherein the calculation method is as follows.
In fig. 8, ds is the unit of instantaneous contact area of the flank of the tooth set; daiIs the area unit length in the plane aioibiProjection on a plane; db (b)iArea unit width in plane aioibiProjection on a plane; gamma is the normal vector direction of the area unit and ciThe angle between the axes, ds, is calculated as follows.
Figure BDA00034387864200001310
In the formula (24), γ (a)i,bi,ci) The function is constructed as equation (25).
Figure BDA0003438786420000141
Step 2, a friction speed calculating method of the area unit of the rear cutter face of the milling cutter tooth pair comprises the following steps:
the cutter tooth error of the square shoulder milling cutter is selected through experimental measurement, and a vibration signal of the whole machining process is measured, the cutting experiment is carried out on a machining center of a large continuous machine tool with the model (VDL-1000E), the diameter of the milling cutter is 25mm, the number of teeth is three cutter teeth, and the size of a workpiece is 200mm multiplied by 100mm multiplied by 20 mm. The milling mode is forward milling, the geometric characteristics of the milling cutter in the machining process are linear side walls, and the cutting-in end and the cutting-out end of the cutter in the milling machining process are determined. The square shoulder milling cutter and milling parameters are as follows.
TABLE 2 milling parameters
Figure BDA0003438786420000142
Before a milling experiment, the method for measuring the cutter tooth error of the square shoulder milling cutter is adopted, and the cutter setting gauge is used for measuring the axial and radial cutter tooth errors of the upper shoulder milling cutter, as shown in table 3.
TABLE 3 cutter tooth radial error and axial error
Figure BDA0003438786420000143
In the actual cutting process, the milling cutter generates vibration due to the action of impact force in the cutting process, so that a vibration acceleration signal generated by a workpiece under the action of the cutting force needs to be detected by adopting a transient signal test analysis system. Therefore, in order to avoid contact interference with the workpiece when the acceleration sensor is mounted on the spindle, the sensor is placed on the upper surface of the workpiece and a vibration signal is collected. The experimental process site and vibration test operation are shown in fig. 10.
The contact state of the rear cutter face of the cutter tooth pair and the machining transition surface of a workpiece is dynamically changed in the cutting process, so that the friction speed of the center point of the area unit of the rear cutter face of the cutter tooth pair is dynamically changed at any moment, and the calculation method of the center point of the area unit is as follows.
The equation (17) is solved to obtain a knife tooth arbitrary point track parameter equation as follows:
Figure BDA0003438786420000144
and (3) calculating the time partial derivative of any point track of the cutter teeth in the formula (26) to obtain the following arbitrary component speeds along the x, y and z axis directions of the workpiece coordinate system:
Figure BDA0003438786420000145
in the formula (27), vnxIs the component velocity along the x-axis direction; v. ofnyIs the component velocity along the y-axis direction; v. ofnzIs the component velocity in the z-axis direction.
Relative motion velocity v of any pointnThe calculation method is as follows:
Figure BDA0003438786420000146
relative movementDynamic velocity vnThe unit vectors in the object coordinate system are as follows:
Figure BDA0003438786420000147
intersection o of rear tool face of cutter tooth and transition surface of workpiecerThe following were used:
Figure BDA0003438786420000151
perorPoint and common tangent plane P of rear tool face of tool tooth pair and workpiece processing transition surfaceiThe following were used:
Figure BDA0003438786420000152
do vnIn the common tangent plane PiProjection on and passing through point orUnit vector in projection direction
Figure BDA0003438786420000153
The following were used:
Figure BDA0003438786420000154
wherein lpxIs a unit vector
Figure BDA0003438786420000155
A component vector in the x-axis direction; lpyIs a unit vector
Figure BDA0003438786420000156
A component vector in the y-axis direction; lpzIs a unit vector
Figure BDA0003438786420000157
Component vector in z-axis direction.
In the coordinate system of the workpiece, the flank face is arbitraryFrictional velocity v of a pointmComprises the following steps:
vm＝vn·cosθm (33)
wherein, thetamIs a relative movement velocity vnAnd unit vector
Figure BDA0003438786420000158
The calculation method of the included angle is as follows:
θm＝π-θc (34)
Figure BDA0003438786420000159
in the formula (35), θcIs a relative movement velocity vnWith a friction speed vmThe included angle therebetween.
In the workpiece coordinate system, the friction speed direction vector of any point on the secondary flank surface
Figure BDA00034387864200001510
The equation (36) is solved.
Figure BDA00034387864200001511
Wherein v ismx、vmy、vmzThe friction speed of a point on the rear face of the cutter tooth in a workpiece coordinate system is converted into the cutter tooth coordinate system through coordinate transformation, and the formula (37) is shown as follows:
Figure BDA00034387864200001515
wherein v ismaiIs the point on the flank of the cutter tooth along the edge a in the coordinate system of the cutter toothiA friction speed in the axial direction; v. ofmbiThe point on the flank of the knife tooth pair is along the b in the knife tooth coordinate systemiA friction speed in the axial direction; v. ofmciThe point on the rear face of the cutter tooth is on the cutter tooth seatMark system middle edge ciThe friction speed in the axial direction.
The frictional velocity v of any point in the tool tooth coordinate systemmComprises the following steps:
Figure BDA00034387864200001512
in order to represent the dynamic change of the friction characteristic parameters required to be solved from cutting-in to cutting-out of the area unit of the friction contact area of the flank of the cutter tooth, a calculation model for the instantaneous cutting-in to cutting-out of the single-rotation cutter tooth of the milling cutter is provided as follows.
In FIG. 13,. eta.ii-1Is the angle between the ith cutter tooth and the (i-1) th cutter tooth, phiiAnd (4) calculating the included angle of the ith cutter tooth from cutting-in to cutting-out according to formulas (39) to (41).
Figure BDA00034387864200001513
In formula (39), TiFor the i-th cutting tooth cut-in to cut-out time period, Ti sFor the ith tooth cutting-in time, Ti eCutting time of ith cutter tooth, k is cutting time of ith cutter tooth, phiiIs the milling contact angle of the milling cutter and omega is the angular velocity of the milling cutter. t is tiThe instant the ith tooth cuts into the workpiece.
Figure BDA00034387864200001514
The same is true. In the formula (40), Ti-1For the i-1 th tooth cut-in to cut-out time period, Ti-1 sFor the ith tooth cutting-in time, Ti-1 eFor the i-1 th tooth cutting time, etaii-1Is the included angle between the ith cutter tooth and the (i-1) th cutter tooth, wherein phiiThe calculation method of (2) is shown in the formula (41).
Figure BDA0003438786420000161
A point taking method through technical characteristic list is adopted to enable the rear tool faces of the three cutter teeth to be parallel to the cutter tooth coordinate systemiThe axis is divided into 10 section planes, the 3 rd section plane is taken, and three points k are taken at equal intervals on the upper and lower boundaries in the section plane0、k1、k2、k3And with k2The points are calculated as the arithmetic points. Selecting five time periods from cutting-in to cutting-out in the whole cutting process of three cutter teeth of the milling cutter, calculating the effective cutting time period of single-rotation cutting of the three cutter teeth, and calculating three cutter teeth k2Five cutting periods T from cutting-in to cutting-out of a point11-T15。
TABLE 4 selected cutting sessions for three teeth of a milling cutter
Figure BDA0003438786420000162
The friction speed of the cutter teeth is simulated by two pairs of three cutter teeth and five cutting time periods according to technical characteristics, and the friction speed obtained by calculation is shown in figure 15.
And 3, extracting the characteristic parameters of the thermal coupling field of the rear cutter face of the milling cutter tooth pair:
in order to extract the temperature and the equivalent stress based on the thermal coupling field, the thermal coupling field simulation of the square shoulder milling cutter is performed, a Johnson-Cook constitutive model mode is adopted, the vibration signal, the milling cutter track and the cutter tooth track measured by the experiment are used as finite element simulation boundary conditions for finite element simulation, and a finite element cutter and workpiece simulation model is shown in FIG. 18.
The point-taking method for the friction area of the rear face of the cutter tooth is shown in FIG. 19, and a measurement coordinate system Y parallel to the cutter tooth is madeiAxial cross-sections with a spacing Δ l between cross-sections, each cross-section being m1，…mi…，mn. Taking points at equal intervals from the upper and lower side distances worn by the cutter teeth on each section, wherein the interval between the two points is delta k, lmiIs the distance between the ith section and the coordinate origin of the cutter tooth, and is an arbitrary point k on the sectioniThe coordinate solving method is as follows(42) As shown.
In fig. 19, an arbitrary point k is selected on the flank faceiCoordinate k in the tool tooth coordinate systemi(aki，bki，cki) As shown in (42):
Figure BDA0003438786420000163
wherein l is the distance between two selected adjacent points in the same cross section.
Will select the point kiConverting the motion trajectory into a workpiece coordinate system to obtain a motion trajectory in the workpiece coordinate system according to the formula (43):
Figure BDA0003438786420000166
in FIG. 20, FpThe normal stress on the micro-element of the rear cutter face of the cutter tooth pair passes through the node of the rear cutter face of the cutter tooth, is vertical to the rear cutter face of the cutter tooth and is vertical to the ciAngle of axis thetaci(ii) a Tau is equivalent stress of a cutter tooth rear cutter face at a network node under a thermal coupling field; f1、F2、F3Respectively equivalent stress in the tetrahedral infinitesimal direction; thetak1、θk2、θk3The included angles between the equivalent stress and the normal stress on the three sides are respectively formed; vector for component force cutter tooth coordinate system of equivalent stress on tetrahedral infinitesimal element
Figure BDA0003438786420000164
Expressed as shown in equation (44):
Figure BDA0003438786420000165
Figure BDA0003438786420000171
therefore, θ can be obtained from the expressions (44) and (45)k1、θk2、θk3As shown in equation (46).
Figure BDA0003438786420000172
Solve to obtain normal stress FpAs shown in equation (47).
Fp(ai,bi,ci)＝F1(ai,bi,ci)·cosθk1+F2(ai,bi,ci)·cosθk2+F3(ai,bi,ci)·cosθk3 (47)
Step 4, constructing a distribution function of instantaneous friction energy consumption of the rear cutter face of the milling cutter tooth pair;
Figure BDA0003438786420000173
in the formula (48), E is the instantaneous absorption energy at the time t, and upsilon is the atom forced vibration frequency as shown in the formula (49):
Figure BDA0003438786420000174
the instantaneous energy distribution function of absorption is as follows (50):
Figure BDA0003438786420000175
wherein: psi is the lattice constant (2.9506X 10)-10m); h is Planck constant (h-6.62607015 × 10)-34J · s); delta is boltzmann constant (delta-1.380649 × 10)-23J/K)；vmRelative friction speed; t is tes1And tes2Respectively the instantaneous initial time and the termination time of the absorbed energy;Ωthe calculation method of the temperature rise of the atomic interface is as shown in a formula (51):
Figure BDA0003438786420000176
wherein: m is atomic relative to atomic mass of 4.34X 10-26kg，ωnThe natural frequency of the atoms is 4.39 multiplied by 1011rad/s, chi is 1X 10 of the excitation force pair of the interface potential energy field-9N。
The friction energy consumption obtained by simulating five time periods of three cutter teeth in the step 3 is shown in fig. 22.
Step 5, verifying the accumulated friction energy consumption of the rear cutter face of the milling cutter tooth pair;
the accumulated energy consumption is accumulated by an instantaneous energy consumption boundary, and in order to verify the correctness of a calculation model of an accumulated energy consumption distribution function, the method for verifying the accumulated friction energy consumption of the secondary flank is provided as follows. The method for identifying the abrupt change node at the section of the rear cutter face of the cutter tooth pair comprises the following steps:
G(Xi,Yimax)＝0 (52)
Yimax＝max(Yi(t)) (53)
the whole cutting process of the cutter tooth from cutting in to cutting out the workpiece is obtained by utilizing the instantaneous boundary of the cutter tooth, and the cutting process is different from XiInstantaneous boundary at position YiMaximum value YimaxFinally obtaining a compound of YimaxThe formed cumulative friction energy consumption boundary.
Selecting characteristic point energy consumption reconstruction distribution curved surfaces for the rear cutter face of the cutter tooth at the moment when the cutter tooth instantaneously cuts a workpiece in the workpiece cutting stage, extracting an energy consumption boundary curve and the abrasion boundary of the rear cutter face of the experimental cutter tooth pair by using the characteristic point selection method provided in the step 3, and carrying out correlation analysis by using correlation coefficients as follows.
Figure BDA0003438786420000177
In the formula, Co upsilon is a covariance calculation formula, Xi，YiThe measured values of the characteristic points of the boundary curve in the cutter tooth measuring coordinate system,
Figure BDA0003438786420000181
and
Figure BDA0003438786420000182
the average value of the values of the boundary curve at each point is taken; the correlation coefficient ρ is calculated as follows:
Figure BDA0003438786420000183
wherein
Figure BDA0003438786420000187
And with
Figure BDA0003438786420000188
As standard deviation, as follows:
Figure BDA0003438786420000184
Figure BDA0003438786420000185
the more rho is close to 1, the stronger the correlation degree of the two correlation variables is shown, the closer rho is to 0, the weaker the correlation degree of the two correlation variables is shown, and the low correlation degree is generally considered to be below 0.40; 0.40-0.69 are moderately correlated; 0.70-0.89 are highly correlated; 0.90-1.00 is extremely high correlation
In order to verify the correctness of the cutter tooth rear cutter surface energy consumption calculation method, the correlation degree analysis is carried out by using an energy consumption surface curve obtained by simulation and a frictional wear boundary curve measured by experiments.
The energy consumption versus cumulative energy consumption profile for a workpiece as the cutter tooth initially cuts into the workpiece is illustrated in fig. 24-26.
The correlation coefficient of the two boundary curves is 0.9977 by solving the calculation method of the equations (59) to (61), so that the two boundaries have strong correlation, and further the accuracy of the wear of the rear face of the cutter tooth is verified by using the energy consumption boundary.
Step 6, constructing the instantaneous friction coefficient and the instantaneous friction distribution function of the rear cutter face of the milling cutter tooth pair
Normal pressure of frictional contact area unit is FnAnd the friction coefficient is mu, the work dS performed by the unit infinitesimal friction force in dt time is as follows:
dS＝μ(ai,bi,ci,t)·Fn(ai,bi,ci,t)·vm(ai,bi,ci,t)·dt (58)
assuming that the friction work of the cutter interface is completely converted into the heat energy of the system in the friction motion process
dS＝dE (59)
The friction coefficient distribution function is solved by the formula (59) as shown in the formula (60):
Figure BDA0003438786420000186
the friction stress distribution function obtained by resolving the formulas (58) to (60) is shown as a formula (61)
fp(ai,bi,ci,t)＝μ(ai,bi,ci,t)·Fp(ai,bi,ci,t) (61)
The friction coefficients obtained by simulating five time periods of three cutter teeth in the step 3 are shown in fig. 27 and 28.
The frictional stress obtained by simulating the characteristic points selected in five time periods of the three cutter teeth in the step 3 is shown in fig. 29 and 30.
It is noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of example embodiments according to the present application. As used herein, the singular forms "a", "an" and "the" are intended to include the plural forms as well, and it should be understood that when the terms "comprises" and/or "comprising" are used in this specification, they specify the presence of stated features, steps, operations, devices, components, and/or combinations thereof, unless the context clearly indicates otherwise.
It should be noted that, in the above embodiments, as long as the technical solutions can be aligned and combined without contradiction, those skilled in the art can exhaust all possibilities according to the mathematical knowledge of the alignment and combination, and therefore, the present invention does not describe the technical solutions after alignment and combination one by one, but it should be understood that the technical solutions after alignment and combination have been disclosed by the present invention.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (7)

1. The dynamic characteristic resolving method for the friction coefficient of the rear cutter face of the cutter tooth pair of the square shoulder milling cutter is characterized by comprising the following steps of:
step 1, calculating a method for the area unit of the rear cutter face of a milling cutter tooth pair;
step 2, based on the step 1, solving the unit friction speed of the area of the rear cutter face of the milling cutter tooth pair;
step 3, extracting the characteristic parameters of the thermal coupling field of the rear cutter face of the milling cutter tooth pair;
step 4, constructing an instantaneous friction energy consumption distribution function of the rear cutter face of the milling cutter tooth pair;
step 5, verifying the accumulated friction energy consumption of the rear cutter face of the milling cutter tooth pair;
and 6, calculating the instantaneous friction coefficient and the instantaneous friction stress of the rear cutter face of the milling cutter tooth pair.
2. The method for resolving the dynamic friction coefficient of the flank face of the tooth pair of the square shoulder milling cutter according to claim 1, wherein the step 1 comprises:
the method for measuring the axial and radial errors of the milling cutter comprises the following steps:
Δri＝rmax-ri(i＝1,2···m) (1)
wherein, Δ riRadial error of the ith tooth, riRadius of gyration of ith cutter tooth tip point of the milling cutter, wherein i is 1,2,3, rmaxThe maximum radius of gyration of three cutter tooth tool points of the milling cutter is obtained;
the axial error calculation method is as follows:
Δzi＝l1-li(i＝1,2···m) (2)
wherein, Δ ziAxial error of the ith tooth,/1Distance from lowest point of milling cutter to end face,/iThe distance from the lowest point of the ith cutter tooth to the end face is;
knife tooth i coordinate system oi-aibiciAnd milling cutter structure coordinate system Os-a rotation matrix I of XYZ1Translation matrix M1Respectively as follows:
Figure FDA0003438786410000011
wherein
Figure FDA0003438786410000012
Is the ith cutter tooth instantaneous position angle;
milling cutter structure coordinate system Os-XYZ and the cutting coordinate system O of the milling cutter under the action of vibrationscInstantaneous rotation matrix I of UVW2Comprises the following steps:
Figure FDA0003438786410000013
wherein the content of the first and second substances,
Figure FDA0003438786410000021
is the instantaneous included angle of the V axis and the Y axis in the UVW plane:
Figure FDA0003438786410000022
in the formula (I), the compound is shown in the specification,
Figure FDA0003438786410000023
the included angle between the V axis and the Y axis in the UVW plane at the initial cutting-in time of the milling cutter, namely t is 0, namely:
Figure FDA0003438786410000024
wherein a iseIs the cutting width;
milling cutter cutting coordinate system O under vibration actioncUVW and milling cutter cutting coordinate system o without vibration0Instantaneous rotation matrix I of uvw3、I4Instantaneous translation matrix M2Respectively as follows:
Figure FDA0003438786410000025
Figure FDA0003438786410000026
wherein, theta1(t) is the W axis at vo0Instantaneous angle of projection on the w plane with the w axis, θ2(t) is the W axis at uo0Instantaneous angle between projection of the w plane and the w axis, Ax(t)、Ay(t)、Az(t) vibration displacements of the milling cutter along the directions of the x axis, the y axis and the z axis respectively;
milling cutter cutting coordinate system o without vibration0The instantaneous translation matrix M of uvw with the object coordinate system o-xyz3Comprises the following steps:
Figure FDA0003438786410000027
in the formula, xo0(t),yo0(t),zo0(t) is the coordinate origin o of the milling cutter cutting coordinate system under the condition of no vibrationoIn the workerInstantaneous position coordinates in the piece coordinate system o-xyz;
Figure FDA0003438786410000028
wherein v isfFor milling cutter feed speed, L0Is the length of the workpiece, W0Is the width of the workpiece, H0Is the height of the workpiece, apThe depth of cut of the workpiece;
milling cutter center point track O under vibration actioncThe specific solving method of the variables such as (x, y, z), the milling cutter attitude angle theta (t), the milling cutter direction angle theta' (t) and the like is as follows:
(1) milling cutter cutting coordinate system O under vibration actionc-locus of motion O of UVW origin of coordinatesc(x, y, z) is:
Oc(x,y,z)＝[x,y,z,1]T＝M3·[Ax(t),Ay(t),Az(t),1]T (11)
(2) the instantaneous included angle theta (t) between the W axis and the W axis is as follows:
Figure FDA0003438786410000031
wherein theta (t) is the instantaneous attitude angle of the milling cutter; l is the length of the milling cutter;
(3) wherein the W axes are respectively at vo0w face, uo0The projection of the w surface and the w axis form an included angle:
Figure FDA0003438786410000032
(4) theta' (t) is the W axis at uo0And the included angle between the projection on the w plane and the v axis is calculated as follows:
Figure FDA0003438786410000033
(5) coordinate system o of cutter teethi-aibiciThe o-xyz transformation relationship matrix with the workpiece coordinate system is as follows:
Figure FDA0003438786410000037
(6) in the work coordinate system, the minor cutting edge l'fThe equation of (a) is:
Figure FDA0003438786410000034
wherein the content of the first and second substances,
Figure FDA0003438786410000035
the length of any point on the secondary cutting edge of the cutter tooth from the point of the cutter point,
Figure FDA0003438786410000036
is total length of secondary cutting edge of the cutter tooth, k'rIs a secondary deflection angle of a secondary back face of the cutter tooth, lambda'sThe angle of the secondary edge of the rear cutter face of the cutter tooth pair is the inclination angle of the secondary edge of the cutter tooth pair;
(7) order to
Figure FDA0003438786410000041
Then the knife tooth i knife point oiMovement locus o in the object coordinate systemi(x, y, z) is as shown in formula (17):
Figure FDA00034387864100000410
(8) the rear tool face A of the tool tooth pair in the workpiece coordinate systemi' transition surface B with workpiece machiningiFriction pair ofmAs shown in equation (18):
Figure FDA0003438786410000042
wherein, the upper and lower boundaries of the instantaneous frictional wear formed by the rear cutter face of the cutter tooth pair in the cutting process are as shown in a formula (19);
Figure FDA0003438786410000043
wherein luFor the upper friction boundary of the rear face friction pair of the cutter tooth pair, /)dThe lower boundary of the knife tooth pair rear knife face friction pair friction;
(9) in a workpiece coordinate system, the equation of the flank face of the cutter tooth pair is
Figure FDA0003438786410000044
Figure FDA0003438786410000045
Is a rear knife face of the knife tooth pair,
Figure FDA0003438786410000046
each side of the rear cutter face of the cutter tooth pair;
workpiece machining transition surface B in workpiece coordinate systemiThe calculation method is shown as a formula (21);
Figure FDA0003438786410000047
wherein the content of the first and second substances,
Figure FDA0003438786410000048
the moment when the tool instantaneously cuts into the workpiece,
Figure FDA0003438786410000049
the moment when the cutting tool instantly cuts the workpiece is finished;
through material mechanics analysis, during the instantaneous cutting process of the cutter toothThe equivalent stress sigma on the secondary cutting edge of the cutter tooth is larger than the yield strength sigma of the materialsWhen the cutter is used, the material in the cutting edge area falls off, so that the wear upper boundary of the cutter tooth can use the equivalent stress as a criterion as shown in the formula (22);
σ≥σs (22)
where σ is the equivalent stress, σsIs the yield strength;
identifying the equivalent stress at each section as yield strength sigmasConnecting the characteristic points on all the sections in a cutter tooth coordinate system to construct an instantaneous friction upper boundary curve of the rear cutter face of the milling cutter tooth;
in the whole milling process, equivalent strain exists in a contact area and a non-contact area from the cutting-in of the cutter tooth to the cutting-out of the rear cutter face of the cutter tooth of the milling cutter, the effect variable values of a friction contact area and the like are large, the effect variable values are gradually reduced along the normal vector direction of a cutting edge, and the instantaneous wear lower boundary is subjected to mutation, so that the wear lower boundary node of the rear cutter face of the cutter tooth can be identified through the equivalent strain change rate, as shown in a formula (23);
Figure FDA0003438786410000051
in the formula, epsilon is equivalent strain, epsilon' is equivalent strain rate, YiMeasuring the ordinate of a coordinate system for the cutter teeth;
selecting characteristic points in an instantaneous contact friction area of a rear cutter face of the cutter tooth pair by representing an instantaneous contact relation of the friction pair, and resolving an instantaneous contact area unit, wherein the resolving method is as follows;
ds is the instantaneous contact area unit of the rear cutter face of the cutter tooth pair; daiIs the area unit length in the plane aioibiProjection on a plane; dbiArea unit width in plane aioibiProjection on a plane; gamma is the normal vector direction of the area unit and ciThe angle between the axes, ds, is calculated as follows:
Figure FDA0003438786410000052
in the formula (24), γ (a)i，bi，ci) Constructing a function as shown in equation (25);
Figure FDA0003438786410000053
3. the method for resolving the dynamic friction coefficient of the flank face of the tooth pair of the square shoulder milling cutter according to claim 2, wherein the step 2 comprises:
the equation (17) is solved to obtain a knife tooth arbitrary point track parameter equation as follows:
Figure FDA0003438786410000054
and (3) calculating the time partial derivative of any point track of the cutter teeth in the formula (26) to obtain the following arbitrary component speeds along the x, y and z axis directions of the workpiece coordinate system:
Figure FDA0003438786410000055
in the formula (27), vnxIs the component velocity along the x-axis direction; v. ofnyIs the component velocity along the y-axis direction; v. ofnzIs the component velocity along the z-axis direction;
relative motion velocity v of any pointnThe calculation method is as follows:
Figure FDA0003438786410000061
relative speed of motion vnUnit vector in the object coordinate system
Figure FDA0003438786410000062
The following were used:
Figure FDA0003438786410000063
intersection o of rear tool face of cutter tooth and transition surface of workpiecerThe following were used:
Figure FDA0003438786410000064
perorPoint and common tangent plane P of the rear cutter face of the cutter tooth and the processing transition surface of the workpieceiThe following were used:
Figure FDA0003438786410000065
do vnIn the common tangent plane PiProjection on and passing through point orUnit vector in projection direction
Figure FDA0003438786410000066
The following were used:
Figure FDA0003438786410000067
wherein lpxIs a unit vector
Figure FDA0003438786410000068
A component vector in the x-axis direction; lpyIs a unit vector
Figure FDA0003438786410000069
A component vector in the y-axis direction; lpzIs a unit vector
Figure FDA00034387864100000610
A component vector in the z-axis direction;
the friction speed v of any point on the flank face in the workpiece coordinate systemmComprises the following steps:
vm＝vn·cosθm (33)
wherein, thetamIs a relative movement velocity vnAnd unit vector
Figure FDA00034387864100000611
The calculation method of the included angle is as follows:
θm＝π-θc (34)
Figure FDA00034387864100000612
in the formula (35), θcIs a relative movement velocity vnWith a friction speed vmThe included angle between them;
in the workpiece coordinate system, the friction speed direction vector of any point on the secondary flank surface
Figure FDA00034387864100000613
Solving the equation (36);
Figure FDA00034387864100000614
wherein v ismx、vmy、vmzThe friction speed of a point on the rear face of the cutter tooth in a workpiece coordinate system is converted into the cutter tooth coordinate system through coordinate transformation, and the formula (37) is shown as follows:
Figure FDA00034387864100000615
wherein v ismaiIs the point on the flank of the cutter tooth along the edge a in the coordinate system of the cutter toothiA friction speed in the axial direction; v. ofmbiIs the rear face of the cutter tooth pairAlong b in the coordinate system of the cutter teethiA friction speed in the axial direction; v. ofmciIs the point on the flank of the cutter tooth along the c in the coordinate system of the cutter toothiA friction speed in the axial direction;
the frictional velocity v of any point in the tool tooth coordinate systemmComprises the following steps:
Figure FDA0003438786410000071
in order to represent the dynamic change of the friction characteristic parameters required to be solved from cutting-in to cutting-out of the area unit of the friction contact area of the rear cutter face of the cutter tooth, a calculation model for instantly cutting-in to cutting-out of a single-rotation cutter tooth of the milling cutter is provided;
Figure FDA0003438786410000072
in the formula (39), TiFor the i-th cutting tooth cut-in to cut-out time period, Ti sFor the ith tooth cutting-in time, Ti eCutting time of ith cutter tooth, k is cutting time of ith cutter tooth, phiiThe included angle of the ith cutter tooth from cut-in to cut-out, omega is the angular speed of the milling cutter, tiThe time for the ith cutter tooth to cut into the workpiece instantaneously;
Figure FDA0003438786410000073
in a similar manner, in formula (40), Ti-1For the i-1 th tooth cut-in to cut-out time period, Ti-1 sFor the ith tooth cutting-in time, Ti-1 eCutting time of the i-1 st tooth, etaii-1Is the included angle between the ith cutter tooth and the (i-1) th cutter tooth, wherein phiiIs shown as formula (41)
Figure FDA0003438786410000074
4. The method for resolving the dynamic friction coefficient of the flank face of the tooth pair of the square shoulder milling cutter according to claim 3, wherein the step 3 comprises: in order to extract temperature and equivalent stress based on a thermal coupling field, performing thermal coupling field simulation on the square shoulder milling cutter, performing finite element simulation by using a Johnson-Cook constitutive model mode and using a vibration signal, a milling cutter track and a cutter tooth track measured by an experiment as finite element simulation boundary conditions, and performing finite element tool and workpiece simulation models;
the point-taking method of the friction area of the rear cutter face of the cutter tooth is to make a measurement coordinate system Y parallel to the cutter toothiAxial cross-sections with a spacing Δ l between cross-sections, each cross-section being m1，…mi…，mnTaking points at equal intervals from the upper and lower margins of the wear of the cutter teeth on each section, wherein the interval between the two points is delta k, lmiIs the distance between the ith section and the coordinate origin of the cutter tooth, and is an arbitrary point k on the sectioniThe coordinate solving method of (2) is shown as formula (42);
selecting any point k on the flank faceiCoordinate k in the tool tooth coordinate systemi(aki，bki，cki) As shown in (42):
Figure FDA0003438786410000075
wherein l is the distance between two adjacent points selected in the same cross section;
will select the point kiThe transformation into the object coordinate system results in a movement trajectory in the object coordinate system as follows:
[x y z 1]T＝θ·[aki bki cki 1]T (43)
Fpthe normal stress on the micro-element of the rear cutter face of the cutter tooth pair passes through the node of the rear cutter face of the cutter tooth, is vertical to the rear cutter face of the cutter tooth and is vertical to the ciAngle of axis thetaci(ii) a Tau is the flank face of the cutter tooth under the thermal coupling fieldEquivalent stress at the network node; f1、F2、F3Respectively equivalent stress in the tetrahedral infinitesimal direction; thetak1、θk2、θk3The included angles between the equivalent stress and the normal stress on the three sides are respectively formed; vector for component force cutter tooth coordinate system of equivalent stress on tetrahedral infinitesimal element
Figure FDA0003438786410000081
Expressed as shown in equation (44):
Figure FDA0003438786410000082
Figure FDA0003438786410000083
in the formula (45), the reaction mixture is,
Figure FDA0003438786410000084
is a unit vector of normal stress in the normal direction on the area micro element of the rear cutter face of the cutter tooth pair;
therefore, θ can be obtained from the expressions (44) and (45)k1、θk2、θk3As shown in formula (46);
Figure FDA0003438786410000085
solving the normal stress as shown in the formula (47);
Fp(ai,bi,ci)＝F1(ai,bi,ci)·cosθk1+F2(ai,bi,ci)·cosθk2+F3(ai,bi,ci)·cosθk3 (47)。
5. the method for resolving the dynamic friction coefficient of the flank face of the tooth pair of the square shoulder milling cutter according to claim 4, wherein the step 4 comprises:
rate of change E of absorbed energy from interface atom theoryvComprises the following steps:
Figure FDA0003438786410000086
in the formula (48), E is the instantaneous absorption energy at the time t, and upsilon is the atom forced vibration frequency as shown in the formula (49):
Figure FDA0003438786410000087
the instantaneous energy distribution function of absorption is as follows (50):
Figure FDA0003438786410000091
wherein: psi is lattice constant (2.9506X 10)-10m); h is Planck constant (h-6.62607015 × 10)-34J · s); delta is boltzmann constant (delta-1.380649 × 10)-23J/K)；vmRelative friction speed; t is tes1And tes2Respectively the instantaneous initial time and the termination time of the absorbed energy;Ωthe calculation method of the temperature rise of the atomic interface is as shown in formula (51):
Figure FDA0003438786410000092
wherein: m is atomic relative to atomic mass of 4.34X 10-26kg，ωnThe natural frequency of the atoms is 4.39 multiplied by 1011rad/s, chi is 1X 10 of the excitation force pair of the interface potential energy field-9N。
6. The method for calculating the dynamic friction coefficient of the secondary flank of the tooth of the square shoulder milling cutter according to claim 5, wherein the step 5 accumulated friction energy consumption is accumulated by an instantaneous energy consumption boundary, and in order to verify the correctness of a calculation model of an accumulated energy consumption distribution function, the method for verifying the accumulated friction energy consumption of the secondary flank is proposed as follows;
the method for identifying the accumulated wear boundary G at the section of the rear cutter face of the cutter tooth pair comprises the following steps:
G(Xi,Yimax)＝0 (52)
Yimax＝max(Yi(t)) (53)
the whole cutting process of the cutter tooth from cutting in to cutting out the workpiece is obtained by utilizing the instantaneous boundary of the cutter tooth, and the cutting process is different from XiInstantaneous boundary at position YiMaximum value YimaxFinally obtaining a compound of YimaxThe formed accumulated friction energy consumption boundary G;
selecting a characteristic point energy consumption reconstruction distribution curved surface for a cutter tooth rear cutter surface at the moment when the cutter tooth instantaneously cuts a workpiece in a workpiece cutting stage, extracting an energy consumption boundary curve and an abrasion boundary of a cutter tooth rear cutter surface of an experimental cutter tooth pair by using the characteristic point selection method provided in example 3, and carrying out correlation analysis by using correlation coefficients as follows:
Figure FDA0003438786410000093
in the formula, Co upsilon is a covariance calculation formula, Xi，YiThe measured values of the characteristic points of the boundary curve in the cutter tooth measuring coordinate system,
Figure FDA0003438786410000094
and
Figure FDA0003438786410000095
the average value of the values of the boundary curve at each point is taken;
the correlation coefficient ρ is calculated as follows:
Figure FDA0003438786410000096
wherein
Figure FDA0003438786410000097
And
Figure FDA0003438786410000098
as standard deviation, as follows:
Figure FDA0003438786410000099
Figure FDA0003438786410000101
the more p is close to 1, the stronger the correlation between the two correlation variables is, and the more close p is to 0, the weaker the correlation between the two correlation variables is.
7. The method for resolving the dynamic friction coefficient of the flank face of the tooth pair of the square shoulder milling cutter according to claim 6, wherein the step 6 comprises:
normal pressure of frictional contact area unit is FnAnd the friction coefficient is mu, the work dS performed by the unit infinitesimal friction force in dt time is as follows:
dS＝μ(ai,bi,ci,t)·Fn(ai,bi,ci,t)·vm(ai,bi,ci,t)·dt (58)
assuming that the friction work of the cutter interface is completely converted into the heat energy of the system in the friction motion process
dS＝dE (59)
The friction coefficient distribution function is solved by the formula (59) as shown in the formula (60):
Figure FDA0003438786410000102
the friction stress distribution function obtained by resolving the formulas (58) to (60) is shown as a formula (65)
fp(ai,bi,ci,t)＝μ(ai,bi,ci,t)·Fp(ai,bi,ci,t) (61)。
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115647440A (en) * 2022-11-10 2023-01-31 哈尔滨理工大学 Method for solving milling infinitesimal energy consumption characteristic parameters of main and auxiliary cutting edges of square shoulder milling cutter
CN117034725A (en) * 2023-08-07 2023-11-10 哈尔滨理工大学 Thermodynamic entropy value resolving method for friction force and heat conduction of milling cutter rear cutter surface under vibration effect
CN117454659A (en) * 2023-06-17 2024-01-26 哈尔滨理工大学 Method for solving stress wave equation of rear cutter surface of cutter tooth of efficient milling cutter

Cited By (7)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN115647440A (en) * 2022-11-10 2023-01-31 哈尔滨理工大学 Method for solving milling infinitesimal energy consumption characteristic parameters of main and auxiliary cutting edges of square shoulder milling cutter
CN117454659A (en) * 2023-06-17 2024-01-26 哈尔滨理工大学 Method for solving stress wave equation of rear cutter surface of cutter tooth of efficient milling cutter
CN117521374A (en) * 2023-06-17 2024-02-06 哈尔滨理工大学 Method for calculating propagation and attenuation characteristics of friction stress wave
CN117454659B (en) * 2023-06-17 2024-04-05 哈尔滨理工大学 Method for solving stress wave equation of rear cutter surface of cutter tooth of efficient milling cutter
CN117521374B (en) * 2023-06-17 2024-04-09 哈尔滨理工大学 Method for calculating propagation and attenuation characteristics of friction stress wave
CN117034725A (en) * 2023-08-07 2023-11-10 哈尔滨理工大学 Thermodynamic entropy value resolving method for friction force and heat conduction of milling cutter rear cutter surface under vibration effect
CN117034725B (en) * 2023-08-07 2024-04-12 哈尔滨理工大学 Thermodynamic entropy value resolving method for friction force and heat conduction of milling cutter rear cutter surface under vibration effect

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