CN108262648A - Axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method - Google Patents
Axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method Download PDFInfo
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- 238000000034 method Methods 0.000 title claims abstract description 48
- 239000006061 abrasive grain Substances 0.000 claims abstract description 133
- 238000009825 accumulation Methods 0.000 claims abstract description 34
- 238000005070 sampling Methods 0.000 claims abstract description 30
- 230000005489 elastic deformation Effects 0.000 claims abstract description 23
- 239000011159 matrix material Substances 0.000 claims description 30
- 239000000463 material Substances 0.000 claims description 26
- 238000013178 mathematical model Methods 0.000 claims description 21
- 238000004364 calculation method Methods 0.000 claims description 8
- 238000000926 separation method Methods 0.000 claims description 8
- 238000002715 modification method Methods 0.000 claims description 6
- 239000002245 particle Substances 0.000 claims description 5
- 238000005520 cutting process Methods 0.000 claims description 4
- 238000011084 recovery Methods 0.000 claims description 4
- 238000009795 derivation Methods 0.000 claims description 3
- 239000000428 dust Substances 0.000 claims description 3
- 238000005183 dynamical system Methods 0.000 claims description 3
- 230000008676 import Effects 0.000 claims description 3
- 239000004576 sand Substances 0.000 claims description 2
- 238000010189 synthetic method Methods 0.000 claims description 2
- 238000012876 topography Methods 0.000 claims description 2
- 238000004088 simulation Methods 0.000 abstract description 8
- 230000003068 static effect Effects 0.000 abstract description 4
- 238000003754 machining Methods 0.000 abstract description 3
- 238000010586 diagram Methods 0.000 description 9
- 238000005259 measurement Methods 0.000 description 9
- 238000005516 engineering process Methods 0.000 description 6
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- 230000000694 effects Effects 0.000 description 4
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- 238000002474 experimental method Methods 0.000 description 3
- 238000012545 processing Methods 0.000 description 3
- 238000012795 verification Methods 0.000 description 3
- 238000009826 distribution Methods 0.000 description 2
- 230000003746 surface roughness Effects 0.000 description 2
- NAWXUBYGYWOOIX-SFHVURJKSA-N (2s)-2-[[4-[2-(2,4-diaminoquinazolin-6-yl)ethyl]benzoyl]amino]-4-methylidenepentanedioic acid Chemical compound C1=CC2=NC(N)=NC(N)=C2C=C1CCC1=CC=C(C(=O)N[C@@H](CC(=C)C(O)=O)C(O)=O)C=C1 NAWXUBYGYWOOIX-SFHVURJKSA-N 0.000 description 1
- 230000009471 action Effects 0.000 description 1
- 238000004458 analytical method Methods 0.000 description 1
- 238000013459 approach Methods 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 238000000205 computational method Methods 0.000 description 1
- 229910001651 emery Inorganic materials 0.000 description 1
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Classifications
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B24—GRINDING; POLISHING
- B24B—MACHINES, DEVICES, OR PROCESSES FOR GRINDING OR POLISHING; DRESSING OR CONDITIONING OF ABRADING SURFACES; FEEDING OF GRINDING, POLISHING, OR LAPPING AGENTS
- B24B1/00—Processes of grinding or polishing; Use of auxiliary equipment in connection with such processes
- B24B1/04—Processes of grinding or polishing; Use of auxiliary equipment in connection with such processes subjecting the grinding or polishing tools, the abrading or polishing medium or work to vibration, e.g. grinding with ultrasonic frequency
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- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Abstract
The present invention relates to a kind of axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction methods, can follow ultrasonic vibration dynamically to choose abrasive grain profile sampled point.Grinding groove is established for the characteristics of axial ultrasonic vibration on this basis to broaden model, it is further introduced into grinding elastic deformation model and plastic accumulation model is modified the dynamic outline method of sampling, realize the combination of geometric simulation and physical simulation, workpiece surface appearance prediction model is ultimately generated, and exports result figure.According to the simulation and prediction of workpiece surface appearance as a result, machined parameters can be in optimized selection in advance, so as to improve Grinding Machining Quality.Existing emulation mode is overcome due to the shortcomings that cannot being emulated using Static Sampling method when there is axial ultrasonic vibration.It has good practical value in grinding field.
Description
Technical field
The present invention relates to a kind of machine-building processing technology, more particularly to a kind of axial direction based on dynamic outline sampling method surpasses
Acoustic vibration assistant grinding workpiece surface appearance simulated prediction method.
Background technology
Supersonic vibration assistant grinding is to answer one kind that ultrasonic vibrating machining technology and plain grinding processing technology are combined
Close processing technology.The experimental results show that axial ultrasonic vibration-assisted grinding can obtain the workpiece surface of high quality.In order to
Analyse in depth the influence of axial ultrasonic vibration and grinding parameter to workpiece surface appearance wound into process, it is necessary to shake to axial ultrasonic
The workpiece surface appearance of dynamic auxiliary grinding is predicted.
In general, use workpiece topological matrix g in workpiece surface appearance predictionmnTo represent workpiece surface appearance.I.e. in work
With x directions separation delta x and y directions separation delta y grid divisions on part surface.With the height value z (m, n) at grid lattice point P (m, n)
As workpiece topological matrix gmnIn element, as shown in Figure 1.Workpiece surface appearance emulation is actually to need to transport by mathematics
Calculate and calculate numerous abrasive grain workpiece topological matrixs after grinding on grinding wheel.
Conventional workpiece surface appearance computational methods need to carry out Configuration of Grinding-wheel Surface static discrete sampling, are opened up with grinding wheel
Flutter matrix hijIt represents.In Fig. 1, by h (i, j) of the height value of the sampled point H (i, j) on wheel face abrasive grain profile as grinding wheel
Topological matrix hijElement.When calculating the workpiece surface appearance of plain grinding, the sampled point H in grinding wheel topological matrix is taken out successively
(i, j) calculates its track, such as curve 1 in Fig. 1.Then be obtained curve 1 passed through each workpiece topological matrix lattice point P (m,
N) the height value z (m, n) at place.It calculates by every track at lattice point P (m, n), it is to grind to ask for minimum value min (z (m, n))
The workpiece surface final residual height of the point after cutting.After the final residual height that each lattice point of workpiece surface has been obtained, you can
Obtain workpiece surface appearance.
But the workpiece surface appearance that this method is not suitable for axial ultrasonic vibration-assisted grinding calculates.Because plain grinding
It is different with the grain motion trajectory of axial ultrasonic vibration-assisted grinding.Sampled point H (i, j) in plain grinding on abrasive grain profile
Movement locus (curve 1 in Fig. 1) exist only in one and be parallel in the section of Oxz planes, can all cover workpiece topology
On the lattice point of matrix.However in axial ultrasonic vibration-assisted grinding, the track of sampled point H (i, j) is space three-dimensional curve (figure
Curve 2 in 1), curve 3 is projected as on plane Oxy, it is observed that workpiece topological matrix cannot be completely covered in curve 3
Lattice point on, therefore can not at the lattice point P (m, n) of workpiece topological matrix the corresponding height value z (m, n) of calculated curve 2, from
And workpiece surface appearance can not be obtained.This is because represent that Configuration of Grinding-wheel Surface is substantially advance with grinding wheel topological matrix
Ground statically samples abrasive grain profile.In the presence of having axial ultrasonic vibration, sampled point on grinding wheel cannot always with
The lattice point alignment of workpiece surface, also can not just carry out surface topography emulation.
In addition, in current grinding workpiece surface appearance emulation mode, it is ground elastic deformation model and plastic accumulation
Model is all established under the conditions of plain grinding, and under axial ultrasonic vibration condition, due to there is grinding groove to become broad effect,
The two models also can be different.
Invention content
It cannot be when there is axial ultrasonic vibration due to using Static Sampling method the present invention be directed to existing emulation mode
The problem of being emulated, it is proposed that a kind of axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method, it can be with
Abrasive grain profile sampled point is dynamically chosen with ultrasonic vibration.On this basis grinding is established for the characteristics of axial ultrasonic vibration
Groove broadens model, is further introduced into grinding elastic deformation model and plastic accumulation model repaiies the dynamic outline method of sampling
Just, it realizes the combination of geometric simulation and physical simulation, ultimately generates workpiece surface appearance prediction model, and export result figure.
According to the simulation and prediction of workpiece surface appearance as a result, machined parameters can be in optimized selection in advance, add so as to improve grinding
Working medium amount.
The technical scheme is that：A kind of axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method,
Specifically comprise the following steps：
1) grinding parameter is inputted, imports Configuration of Grinding-wheel Surface mathematical model：
Model represents with matrix G, the information of one abrasive grain of element representation in matrix G per a line, for i-th abrasive grain
Information, including iting the coordinate (x ' in wheel face coordinate systemi, y 'i, z 'i) and abrasive grain shape simplification be spherical diameter dgi,
N abrasive grain is shared in grinding wheel numerical model, the form of matrix G is N × 4,
2) grain motion trajectory is calculated：
Using workpiece as stationary reference frame, workpiece coordinate system Oxyz is established, wherein x-axis feeds negative direction along workpiece, and y-axis is along sand
Wheel shaft is to the position of origin O is selected in the workpiece surface highest point before grinding, during axial ultrasonic vibration-assisted grinding, grinds
The movement of grain is made of three parts：Around grinding wheel spindle with angular velocity omegasCircular motion, along direction of feed relative to workpiece with linear speed
Spend vwLinear motion, along grinding wheel axially with respect to workpiece with amplitude A, ultrasonic vibration that frequency f is carried out；
T at the time of if grinding starts0=0s, at this time the origin O ' of wheel face coordinate system positioned at grinding wheel minimum point and be located at
The surface of workpiece coordinate system origin O, ultrasonic vibration initial phase are 0；
Arbitrary equation of locus of the abrasive grain i in workpiece coordinate system：
Wherein rsFor grinding wheel radius, t is from t0Moment starts the time of grain motion, LzFor grinding wheel axis to O it is vertical away from
From,
λ=1,2,3...., λ represent the number of abrasive grain i incision workpiece, αiFor abrasive grain i to grinding wheel
The vertical line and O ' of axis are to the angle between the vertical line of grinding wheel axis；
3) the dynamic outline method of sampling：
Workpiece topological matrix g is used in workpiece surface appearance predictionmnRepresent workpiece surface appearance, i.e., on the surface of the workpiece
With x directions separation delta x and y directions separation delta y grid divisions, using the height value z (m, n) at grid lattice point P (m, n) as workpiece
Topological matrix gmnIn element,
First, the series of parallel sampled cross-section in plane Oyz is set on the surface of the workpiece, these sampled cross-sections cross workpiece
The lattice point of topological matrix is followed successively by 1,2,3 ... since O points along the number of positive direction of the x-axis, during grinding, a certain mill
Grain is in C1Point incision workpiece, in C2Point leaves workpiece, C1The sampled cross-section n on point the right1It is that first sampling interfered is cut
Face, C2The sampled cross-section n on the point left side2It is the last one, abrasive grain has been sequentially passed through during workpiece is ground from n1To n2One
Series of samples section, therefore abrasive grain grinding can be obtained in the scallop-height value for calculating each lattice point in these sampled cross-sections
Workpiece surface appearance afterwards, n1And n2Value can be obtained by following formula
L in formula1For C1Point arrives the horizontal distance of workpiece coordinate system origin O, l2For C2Point arrives the horizontal distance of O, l1And l2's
Value can utilize abrasive grain equation of locus in step 2) to be obtained；
When calculating each lattice point scallop-height value in n-th of sampled cross-section, the sampled cross-section is calculated first and is sat to workpiece
The horizontal distance x of mark system origin On：
xn=(n-1) Δ x
By xnValue substitute into abrasive grain equation of locus that abrasive grain center can be obtained in sampled cross-section n is former to workpiece coordinate system
The horizontal distance y of point OnWith vertical range zn, then position of the abrasive grain profile in sampled cross-section n be assured that, abrasive grain profile
Equation is expressed as：
(y-yn)2+(z-zn)2=(dg/2)2
D in formulagFor abrasive grain diameter, in C1And C2Intermediate C3Point arrives C4Abrasive grain and workpiece interfere between point, C3Point is right
The lattice point P on side1(m1, n) and it is first lattice point interfered, C4The lattice point P on the point left side2(m2, n) and it is the last one, m1And m2
Value can be obtained by following formula,
L in formula3For C3Point arrives the horizontal distance of workpiece coordinate system origin O, l4For C4Point arrives the horizontal distance of O points, l3And l4
Value can be obtained using abrasive grain profile equation.
Then, from P1To P2A series of lattice points at abrasive grain profile is sampled, wherein a certain sampled point H (m, n) is arrived
The horizontal distance of workpiece coordinate system origin O is lm：
lm=(m-1) Δ y
By y=lmIt substitutes into and following formula is obtained in abrasive grain profile equation, you can the ordinate z (m, n) of sampled point H (m, n) is obtained,
Assuming that the workpiece material interfered is completely removed, then the value of z (m, n) is exactly the workpiece remnants of the point after grinding
Height value.The coordinate value of sampled point H (m, n) is assigned to lattice point P (m, n), completes the update at the lattice point, similarly, update sampling
From P in the n of section1Point arrives P2The possessive case point coordinates of point completes the calculating in sampled cross-section n；When from n1To n2All samplings
After the completion of section all updates, it is possible to obtain the workpiece surface appearance after single grain grinding, continue thereafter with and adjust on this basis
It is updated with other abrasive grains, finally obtains complete workpiece surface appearance；
4) to the amendment of the method for sampling：The characteristics of for axial ultrasonic vibration-assisted grinding, establishes grinding groove and broadens
Model, and grinding elastic deformation model and plastic accumulation model are introduced on this basis, the dynamic outline method of sampling is repaiied
Just.
The grinding groove model modification method that broadens is as follows in the step 4)：
Groove contour is oval, ellipse short shaft dg, long axis de, groove contour equation is expressed as：
dgAnd deRatio be：
θ is grain motion speed vgWith the angle between sampled cross-section n, at sampled cross-section n, vgIt can be decomposed into along x-axis
Velocity component vxWith the velocity component v along y-axisy, obtained according to the synthetic method of movement：
vxAnd vyValue the derivation of time t can be obtained by abrasive grain equation of locus：
T in formulanAt the time of for abrasive grain center movement at sampled cross-section n, vsFor grinding speed, d can be obtainedeValue：
Work as deValue determine after, groove contour equation determines therewith, when there are during axial ultrasonic vibration, using groove contour
Equation replaces the abrasive grain profile equation in step 3), and sampled point is chosen on groove contour.
Grinding elastic deformation model modification method is as follows in the step 4)：
Assume to establish grinding elastic deformation theory's model, abrasive grain stressing conditions based on spherical wear particles and ideal plane grinding
Similar during to test Brinell hardness, abrasive grain is R by normal pressure, and the depth of abrasive grain incision workpiece is dp, cut when abrasive grain moves
During workpiece, the direction of R has turned over angle, θ ', abrasive grain is friction coefficient by frictional force μ a R, μ in bottom, abrasive grain yielding value
δcWith workpiece elasticity recovery value δwCalculation formula it is as follows：
δc=C [R (cos θ '-μ sin θs ')]2/3
δw=R (cos θ '-μ sin θs ')/k
C is constant in formula, and value range is 0.08~0.25, and average value 0.15, k is workpiece stiffness coefficient；
The calculation formula of θ ' and R is respectively：
R=π b2B
B is the half that abrasive grain cuts workpiece portion chord length in formula, and B is the ball hardness number of workpiece material；
The calculation formula that geometrical relationship can obtain b is：
The depth of abrasive grain incision workpiece is d in desired elastic deformation modelpIt is in the hypothesis that workpiece surface is ideal plane
Lower measurement, practical work piece surface is not ideal plane, sampled cross-section n septal fossulas channel profiles and the situation of practical work piece Surface Interference
Under, practical work piece surface is less than preferable workpiece surface, from H1(m1, n) and to H2(m2, n) series of points really interfere
Sampled point, each sampled point has different penetraction depths, real in grinding although practical work piece surface is not ideal plane
Border workpiece surface is very smooth, and the difference in height between each neighboring lattice points is little, therefore, it is considered that desired elastic deformation model is still near
Like establishment, the penetraction depth average value d with each sampled point is only neededp' instead of the d in desired elastic deformation modelp,
Z in formulaw(m, n) is the actual height of workpiece surface lattice point P (m, n) before grinding, and z (m, n) is sampled point H (m, n)
Depth, the scallop-height of workpiece surface lattice point P (m, n) should be after grinding：Z'(m, n)=z (m, n)+δc+δw
When updating workpiece surface appearance model, z is replaced with the value of z 'wValue.
Plastic accumulation model modification method is as follows in the step 4)：
During practical grinding, the material that is interfered on workpiece with abrasive grain only some be removed, form abrasive dust, and
The material not being removed then is plastically deformed, and is deposited in groove both sides, and plastic accumulation model is based on spherical wear particles and ideal
Flat surface grinding is it is assumed that the sectional area of removal material is Ag, the profile of material stacking part is assumed to be parabola, is highly hp, width
For 2wp, sectional area Ap, abrasive grain profile and the inclination angle of accumulation profile intersection tangent line are αp, wpAnd hpValue be respectively：
ApValue determined by grinding efficiency β,
For axial ultrasonic vibration-assisted grinding, it is contemplated that groove broadens phenomenon, should use the elliptical grooves profile established
Replace the round abrasive grain profile in ideal plasticity Mathematical Model of heaped-up,
The bottom of accumulation profile assumes that the workpiece surface for ideal plane in ideal plasticity Mathematical Model of heaped-up, but in grinding
Practical work piece surface is not ideal plane, and position is less than the ideal plane, although practical work piece surface is not ideal plane,
But due in grinding workpiece surface it is very smooth, almost plane is set up, groove therefore, it is considered that ideal plasticity Mathematical Model of heaped-up is still approximate
Profile is from sampled point H1(m1, n) and to H2(m2, n) with practical work piece Surface Interference, remove the sectional area A of materialgIt should be according to practical dry
Relate to situation calculating：
In order to establish accumulation profile, with sampled point H1To H2Practical work piece apparent height average value establish one it is equivalent flat
Face replaces the ideal plane in ideal plasticity Mathematical Model of heaped-up, the equivalent level value zdFor：
The height that accumulation profile is higher by equivalent plane is still hp, but width increases as 2wp', broaden model similarly with groove,
The degree that accumulation profile broadens is identical with the degree that groove contour broadens, and can calculate wp' value：
H is being determinedp, wp', zdValue after, the location and shape for accumulating profile just entirely define, accumulation profile cover
The height value of accumulation profile is calculated at each lattice point covered, former lattice dynamical system value is substituted, completes in a sampled cross-section more
Newly.
The beneficial effects of the present invention are：Axial ultrasonic vibration-assisted grinding workpiece surface appearance simulation and prediction side of the present invention
Method overcomes existing emulation mode due to that cannot have what is emulated during axial ultrasonic vibration to lack using Static Sampling method
Point.Grinding groove is established for the characteristics of axial ultrasonic vibration to broaden model, be further introduced into grinding elasticity on this basis
Distorted pattern and plastic accumulation model are modified the dynamic outline method of sampling, have ultimately generated workpiece surface appearance prediction mould
Type, and prediction result has also obtained verification experimental verification.It has good practical value in grinding field.
Description of the drawings
Fig. 1 is workpiece topological matrix schematic diagram in the present invention；
Fig. 2 is Configuration of Grinding-wheel Surface mathematical model graphics used in the present invention；
Fig. 3 is supersonic vibration assistant grinding schematic diagram axial in the present invention；
Fig. 4 is sampled cross-section set-up mode schematic diagram in the present invention；
Fig. 5 is the sampled point set-up mode schematic diagram in sampled cross-section n in the present invention；
Fig. 6 broadens model schematic for groove in the present invention；
Fig. 7 is the groove contour schematic diagram in sampled cross-section n in the present invention；
Fig. 8 is desired elastic deformation model schematic in the present invention；
Fig. 9 is actual elastic distorted pattern schematic diagram in the present invention；
Figure 10 is ideal plasticity Mathematical Model of heaped-up schematic diagram in the present invention；
Figure 11 is practical plastic accumulation model schematic in the present invention；
Figure 12 is workpiece surface appearance Prediction program general flow chart in the present invention；
Figure 13 a are the prediction workpiece surface appearance of plain grinding in the present invention；
Figure 13 b are the prediction workpiece surface appearance of axial supersonic vibration assistant grinding in the present invention；
Figure 14 a are the actual measurement workpiece surface appearance of plain grinding in the present invention；
Figure 14 b are the actual measurement workpiece surface appearance of axial supersonic vibration assistant grinding in the present invention.
Specific embodiment
A kind of axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method based on dynamic outline sampling method,
This method is as follows：
Step 1：Import Configuration of Grinding-wheel Surface mathematical model
Grinding is to carry out multiple-cutting-edge, the cutting of micro- sword, the diameter of wheel face abrasive grain and distribution feelings using the abrasive grain of wheel face
Condition has vital influence to workpiece surface appearance after grinding.In general, the diameter of abrasive grain is a certain on grinding wheel
It is Gaussian distributed in section, is to obey random distribution in the position of wheel face, in this regard, has there is maturation now
Configuration of Grinding-wheel Surface emulation mode can use.In order to by Configuration of Grinding-wheel Surface the application of mathematical model in this method, with existing side
The Configuration of Grinding-wheel Surface model of method generation should meet following condition：The model represents with matrix G, the member in matrix G per a line
Element represents the information of an abrasive grain.For the information of i-th abrasive grain, including iting the coordinate (x ' in wheel face coordinate systemi,
y’i, z 'i) and abrasive grain shape simplification be spherical diameter dgi.N abrasive grain is shared in grinding wheel numerical model, the form of matrix G is N
×4.Configuration of Grinding-wheel Surface mathematical model graphics used as shown in Figure 2 after the model drawing output.
Step 2：Calculate grain motion trajectory
Axial ultrasonic vibration-assisted grinding is on the basis of plain grinding, and high frequency simple harmonic oscillation is axially applied along grinding wheel
A kind of Combined Machining Technology on to workpiece or grinding wheel.Its grain motion trajectory has very very much not with plain grinding grain motion trajectory
Together, it is also different to the effect of workpiece surface appearance forming process, it is therefore necessary to give expression to axial ultrasonic with mathematical formulae and shake
Dynamic auxiliary is ground the movement locus of the arbitrary abrasive grain in medium plain emery wheel surface.For ease of research, using workpiece as stationary reference frame, workpiece is established
Coordinate system Oxyz, wherein axial ultrasonic vibration-assisted grinding schematic diagram as shown in Figure 3, x-axis feed negative direction, y-axis edge along workpiece
Grinding wheel is axial, and the position of origin O is selected in the workpiece surface highest point before grinding.During axial ultrasonic vibration-assisted grinding,
The movement of abrasive grain is made of three parts：Around grinding wheel spindle with angular velocity omegasCircular motion, along direction of feed relative to workpiece with line
Speed vwLinear motion, along grinding wheel axially with respect to workpiece with amplitude A, ultrasonic vibration that frequency f is carried out.
T at the time of if grinding starts0=0s, at this time the origin O ' of wheel face coordinate system positioned at grinding wheel minimum point and be located at
The surface of workpiece coordinate system origin O, ultrasonic vibration initial phase are 0.O ' is easy to get in workpiece by primary condition and kinematic relation
Equation of locus in coordinate system is：
R in formulasFor grinding wheel radius, t is from t0Moment starts the time of grain motion, LzFor grinding wheel axis to O it is vertical away from
From LzIt can be calculated by formula (3)：
Lz=rs+hmax-ap (3)
H in formulamaxFor the maximum projecting height of wheel face abrasive grain, apFor grinding depth.Known arbitrary abrasive grain i is in grinding wheel
Coordinate in surface coordinate system is (x 'i, y 'i, z 'i).Remember abrasive grain i to grinding wheel axis vertical line and O ' to grinding wheel axis vertical line
Between angle be αi, can be calculated by formula (4):
L hereiniWhat is represented is arc length of i-th of abrasive grain to grinding wheel coordinate origin.
When grinding wheel is from t0Moment starts to turn over angle [alpha]iWhen, abrasive grain i is located exactly at grinding wheel minimum point.One during in view of grinding
Abrasive grain may repeatedly cut workpiece, when grinding wheel turns over angle as αiDuring+2 π (λ -1), it is minimum that abrasive grain i is also located exactly at grinding wheel
Point, λ represent the number of abrasive grain i incision workpiece.It is t to remember this momenti：
Fall behind at the time of moving to grinding wheel minimum point than O ' at the time of understanding that abrasive grain i moves to grinding wheel minimum point by formula (5)
TiSecond, in the y-axis direction, abrasive grain i is y ' relative to the distance that O ' is deviatedi, on grinding wheel radius direction, abrasive grain i is higher by grinding wheel
The distance on surface is z 'i, arbitrary abrasive grain i can be obtained on the basis of formula (2) in workpiece coordinate system according to these relationships
Equation of locus：
Formula (6) is the general formula of Movement Locus Equations of the arbitrary abrasive grain i of wheel face in workpiece coordinate system.
Step 3：The dynamic outline method of sampling
The generally use workpiece topological matrix g in workpiece surface appearance predictionmnRepresent workpiece surface appearance.I.e. in workpiece table
With x directions separation delta x and y directions separation delta y grid divisions on face.Using the height value z (m, n) at grid lattice point P (m, n) as
Workpiece topological matrix gmnIn element, as shown in Figure 1.
First, the series of parallel sampled cross-section in plane Oyz is set on the surface of the workpiece, these sampled cross-sections cross workpiece
The lattice point of topological matrix is followed successively by 1,2,3 ... as shown in Figure 4 since O points along the number of positive direction of the x-axis.Δ x and Δ y
Value determine the sizing grid of workpiece surface topological matrix, also just determine simulation accuracy, occurrence should be by we
The user of service of method is set according to demand.Below embodiment fall into a trap the result of nomogram 13 when the value that uses for：Δ x=Δs y=
0.004mm, dgiAverage value=0.069mm, can be as reference.
During grinding, a certain abrasive grain is in C1Point incision workpiece, in C2Point leaves workpiece.C1The sampled cross-section n on point the right1
It is first sampled cross-section interfered, C2The sampled cross-section n on the point left side2It is the last one.Abrasive grain is in the process of grinding workpiece
In sequentially passed through from n1To n2A series of sampled cross-sections, therefore the remnants for calculating each lattice point in these sampled cross-sections are high
Abrasive grain workpiece surface appearance after grinding can be obtained in angle value.n1And n2Value can be obtained by formula (7).
L in formula1For C1Point arrives the horizontal distance of workpiece coordinate system origin O, l2For C2Point arrives the horizontal distance of O, l1And l2's
Value can be obtained using abrasive grain equation of locus (6).
When calculating each lattice point scallop-height value in n-th of sampled cross-section, the sampled cross-section is calculated first and is sat to workpiece
The horizontal distance x of mark system origin On：
xn=(n-1) Δ x (8)
By xnValue substitute into abrasive grain equation of locus (6) abrasive grain center can be obtained in sampled cross-section n to workpiece coordinate
It is the horizontal distance y of origin OnWith vertical range zn.Then position of the abrasive grain profile in sampled cross-section n is assured that, is such as schemed
Shown in 5.Abrasive grain profile can use equation (9) to represent.
(y-yn)2+(z-zn)2=(dg/2)2 (9)
D in formulagFor abrasive grain diameter.In Fig. 5, in C3Point arrives C4Abrasive grain and workpiece interfere between point.C3The lattice on point the right
Point P1(m1, n) and it is first lattice point interfered, C4The lattice point P on the point left side2(m2, n) and it is the last one.m1And m2Value can
To be obtained by formula (10).
L in formula3For C3Point arrives the horizontal distance of workpiece coordinate system origin O, l4For C4Point arrives the horizontal distance of O points, l3And l4
Value can be obtained using abrasive grain profile equation (9).
Then, from P1To P2A series of lattice points at abrasive grain profile is sampled, wherein a certain sampled point H (m, n) is arrived
The horizontal distance of workpiece coordinate system origin O is lm：
lm=(m-1) Δ y (11)
By y=lmIt substitutes into and formula (12) is obtained in abrasive grain profile equation (9), you can the ordinate z of sampled point H (m, n) is obtained
(m, n).
Assuming that the workpiece material interfered is completely removed, then the value of z (m, n) is exactly the workpiece remnants of the point after grinding
Height value.The coordinate value of sampled point H (m, n) is assigned to lattice point P (m, n), completes the update at the lattice point.Similarly, update sampling
From P in the n of section1Point arrives P2The possessive case point coordinates of point completes the calculating in sampled cross-section n.When from n1To n2All samplings
After the completion of section all updates, it is possible to obtain the workpiece surface appearance after single grain grinding, continue thereafter with and adjust on this basis
It is updated with other abrasive grains, finally obtains complete workpiece surface appearance.
Due to the presence for thering is axial ultrasonic to vibrate, in different sampled cross-sections, abrasive grain center to workpiece coordinate system origin O
Horizontal distance ynIt is different, therefore sampled point is movement relative to abrasive grain profile, sampled point dative point alignment always, this
It is exactly the dynamic outline method of sampling.Compared with Configuration of Grinding-wheel Surface topologizes method, the dynamic outline method of sampling can be each
Accurately reflect the profile of abrasive grain in sampled cross-section so that the workpiece surface appearance of axial ultrasonic vibration-assisted grinding is measured in advance
To realize.
Step 4：Amendment to the method for sampling
The above-mentioned dynamic outline method of sampling is to remove hypothesis completely based on material, but in practical grinding, is considered
It removes to the workpiece material interfered and non-fully, but a series of elastic-plastic deformations has occurred.Therefore, for axial ultrasonic
The characteristics of vibration-assisted grinding, establishes grinding groove and broadens model, and introduce on this basis grinding elastic deformation model and
Plastic accumulation model is modified the dynamic outline method of sampling.
1st, grinding groove broadens the foundation of model
Many scholars are found by experiment that, in axial ultrasonic vibration-assisted grinding, due to the axial movement of abrasive grain, along
The grinding groove of grinding wheel axial direction broadens, and interference degrees enhance between the groove of different abrasive grains, improve workpiece surface quality.Usually
In the case of be, the above-mentioned dynamic outline method of sampling axially measured along grinding wheel to workpiece surface roughness value measurement after grinding
The sampled cross-section of middle setting is also parallel to grinding wheel axial direction.Therefore, it in order to make prediction result truer, needs to consider to be ground
Groove broadens the influence generated to workpiece surface appearance, establishes grinding groove and broadens model, as shown in Figure 6.Fig. 6 is to regard
Angle observation grinding groove, sampled cross-section n is actually abrasive grain space swept from section D-D to section E-E motion processes
An oblique section.Since the distance of section D-D to section E-E are very short, grain motion approximate straight line motion, therefore abrasive grain from cut
Face D-D can be reduced to an a diameter of d to space swept section E-EgCylinder, with the sampled cross-section n bevels circle
Cylinder, gained profile is oval for one, groove contour as shown in Fig. 7, which is dg, long axis de, groove contour can
To be represented with equation (13).
Geometrical relationship in Fig. 7, dgAnd deRatio be：
θ is grain motion speed vgWith the angle between sampled cross-section n.At sampled cross-section n, vgIt can be decomposed into along x-axis
Velocity component vxWith the velocity component v along y-axisy.It can be obtained according to the compositional rule of movement：
vxAnd vyValue the derivation of time t can be obtained by abrasive grain equation of locus (6)：
T in formulanAt the time of for abrasive grain center movement at sampled cross-section n, vsFor grinding speed.Formula (16) is substituted into formula
(15), formula (15), which substitutes into formula (14), can obtain deValue：
Work as deValue determine after, groove contour equation (13) just determines therewith.From figure 7 it can be seen that groove contour produces
Raw groove width w2The groove width w generated than abrasive grain profile1It is wider.When there are during axial ultrasonic vibration, it should using groove
Profile equation (13) chooses sampled point instead of abrasive grain profile equation (9) on groove contour.The sampling of different location is cut
Face, deValue be different, it means that different groove contour equations is corresponding in different sampled cross-sections, what groove broadened
Degree is also dynamic.
2nd, it is ground the introducing of elastic deformation model
During grinding, since abrasive grain is flexibly supported by grinding wheel bond, elastic yield can occur for abrasive grain during grinding,
In addition elastic recovery can also occur for workpiece material after being ground.Existing emulation mode is based on spherical wear particles and ideal plane grinding is false
If grinding elastic deformation theory's model is established, as shown in Figure 8.Abrasive grain stressing conditions and similar, abrasive grain during test Brinell hardness
It is R by normal pressure, the depth of abrasive grain incision workpiece is dp.When abrasive grain moves cutting workpiece, the direction of R has turned over angle
θ ', abrasive grain are friction coefficient by frictional force μ a R, μ in bottom.Abrasive grain yielding value δcWith workpiece elasticity recovery value δwCalculating
Formula is (18) and (19) respectively.
δc=C [R (cos θ '-μ sin θs ')]2/3 (18)
δw=R (cos θ '-μ sin θs ')/k (19)
C is constant in formula, and value range is 0.08~0.25, and average value 0.15, k is workpiece stiffness coefficient.θ's ' and R
Calculation formula is respectively：
R=π b2B (21)
B is the half that abrasive grain cuts workpiece portion chord length in formula, and B is the ball hardness number of workpiece material.By the geometry of Fig. 8
The calculation formula that relationship can obtain b is：
The depth of abrasive grain incision workpiece is d in desired elastic deformation model shown in Fig. 8pIt is to be put down in workpiece surface to be preferable
It is measured under the hypothesis in face.Under actual conditions, most abrasive grains are continued on the basis of having polishing scratch as follow-up abrasive grain
Grinding.Practical work piece surface is not ideal plane, is illustrated in figure 9 sampled cross-section n septal fossulas channel profiles and is done with practical work piece surface
Situation about relating to, practical work piece surface is less than preferable workpiece surface, from H1(m1, n) and to H2(m2, n) series of points be really to occur
The sampled point of interference, each sampled point have different penetraction depths.
Although practical work piece surface is not ideal plane, practical work piece surface is very smooth in grinding, each adjacent
Difference in height between lattice point is little, sets up, is only needed with each sampled point therefore, it is considered that desired elastic deformation model is still approximate
Penetraction depth average value dp' instead of the d in desired elastic deformation modelp, as shown in formula (23).
Z in formulaw(m, n) is the actual height of workpiece surface lattice point P (m, n) before grinding, and z (m, n) is sampled point H (m, n)
Depth.The scallop-height of workpiece surface lattice point P (m, n) should be after grinding：Z'(m, n)=z (m, n)+δc+δw (24)
When updating workpiece surface appearance model, z is replaced with the value of z 'wValue.
3rd, it is ground the introducing of plastic accumulation model
During practical grinding, the material that is interfered on workpiece with abrasive grain only some be removed, form abrasive dust, and
The material not being removed then is plastically deformed, and is deposited in groove both sides.In order to consider the workpiece material plasticity of groove both sides
Accumulation situation introduces the plain grinding plastic accumulation model that existing emulation mode is established.As shown in Figure 10, which is based on spherical shape
Abrasive grain and ideal plane grinding are assumed.The sectional area for removing material is Ag, the profile of material stacking part is assumed to be parabola, high
It spends for hp, width 2wp, sectional area Ap.Abrasive grain profile and the inclination angle of accumulation profile intersection tangent line are αp。wpAnd hpValue
Respectively：
ApValue determined by grinding efficiency β.
Ideal plasticity Mathematical Model of heaped-up shown in Fig. 10 is processed for plain grinding and is established, and axial ultrasonic is vibrated auxiliary
Grinding aid is cut, it is contemplated that groove broadens phenomenon, should replace ideal plasticity Mathematical Model of heaped-up using the elliptical grooves profile just established
In round abrasive grain profile.In addition, the bottom that profile is accumulated in ideal plasticity Mathematical Model of heaped-up assumes that the workpiece for ideal plane
Surface, but practical work piece surface is not ideal plane in grinding, and position is less than the ideal plane, as shown in figure 11：
Although practical work piece surface is not ideal plane, since workpiece surface is very smooth in grinding, almost plane, because
This thinks that ideal plasticity Mathematical Model of heaped-up is still approximate and sets up.Groove contour is from sampled point H1(m1, n) and to H2(m2, n) and practical work piece
Surface Interference removes the sectional area A of materialgIt should be calculated according to practical interference situation：
In order to establish accumulation profile, with sampled point H1To H2Practical work piece apparent height average value establish one it is equivalent flat
Face replaces the ideal plane in ideal plasticity Mathematical Model of heaped-up, the equivalent level value zdFor：
The height that accumulation profile is higher by equivalent plane is still hp, but width increases as 2wp’.Model is broadened with groove similarly,
The degree that accumulation profile broadens is identical with the degree that groove contour broadens, and w can be calculated with reference to formula (17)p' value：
H is being determinedp, wp', zdValue after, the location and shape for accumulating profile just entirely define.It is covered in accumulation profile
The height value of accumulation profile is calculated at each lattice point covered, former lattice dynamical system value is substituted, completes in a sampled cross-section more
Newly.
Workpiece surface appearance Prediction program flow chart as shown in figure 12, the pre- flow gauge after designing as stated above, by this
Flow can carry out workpiece surface appearance prediction.
According to the pre- flow gauge of this method, to plain grinding and axial ultrasonic vibration auxiliary under typical process Parameter Conditions
The workpiece surface appearance of grinding is predicted, using parameter：Grinding wheel outer diameter 300mm, ultrasonic vibration frequency f=20.45KHz,
Amplitude A=15 μm, grinding wheel speed vs=20m/s, feed-speed vw=1m/min, grinding depth ap=4 μm.Prediction obtains
Result as shown in Figure 13 a, 13b.In order to verify the accuracy of above-mentioned Forecasting Methodology, corresponding verification test has been carried out.Figure
14a, 14b are the workpiece surface appearance of experiment actual measurement.
The prediction result and Figure 14 a of comparison diagram 13a, 13b, the measured result of 14b are can be found that：Common mill in Figure 13 a
It is narrow, straight to cut groove, there are apparent protuberance in keeping parallelism between adjacent trenches, groove both sides, and residual altitude is higher, with Figure 14 a
In actual measurement workpiece surface appearance feature it is similar.The groove width of axial supersonic vibration assistant grinding is wider in Figure 13 b, and groove is walked
To slight curving, there is apparent interference between adjacent trenches, the protuberance of groove both sides is mutually cut off by adjacent trenches, residual altitude compared with
It is low.Actual measurement workpiece surface appearance feature in Figure 14 b is similar.
Further comparison prediction and actual measurement workpiece surface roughness value Ra：0.88 μm of plain grinding predicted value, plain grinding
0.91 μm of measured value, 0.76 μm of ultrasonic grinding predicted value, 0.82 μm of ultrasonic grinding measured value.Mean error is 5.3%, prediction with
Result of the test approaches.The result of prediction and experiment actual measurement all shows the workpiece surface matter obtained by axial ultrasonic vibration-assisted grinding
Amount is better than plain grinding.The reason is that abrasive grain track becomes more complicated under the action of axial ultrasonic vibration, lead to groove width
It broadens, effect enhancing is interfered between groove so that remaining material and protuberance are removed originally for groove both sides, reduce work
The microfluctuation on part surface, so as to obtain higher-quality surface.
The above results demonstrate the feasibility and accuracy of Forecasting Methodology.Dynamic outline sampling method can be accurately by groove
Profile is mapped on workpiece surface appearance model, and can be readily incorporated into groove and be broadened model, elastic deformation model and modeling
Property Mathematical Model of heaped-up improves precision of prediction, reaches good prediction effect.
Claims (4)
1. a kind of axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method, which is characterized in that specifically include as
Lower step：
1) grinding parameter is inputted, imports Configuration of Grinding-wheel Surface mathematical model：
Model represents with matrix G, the information of one abrasive grain of element representation in matrix G per a line, for the letter of i-th abrasive grain
Breath, including iting the coordinate (x ' in wheel face coordinate systemi, y 'i, z 'i) and abrasive grain shape simplification be spherical diameter dgi, sand
It takes turns and N abrasive grain is shared in numerical model, the form of matrix G is N × 4,
2) grain motion trajectory is calculated：
Using workpiece as stationary reference frame, workpiece coordinate system Oxyz is established, wherein x-axis feeds negative direction along workpiece, and y-axis is along grinding wheel spindle
To, the position of origin O is selected in the workpiece surface highest point before grinding, during axial ultrasonic vibration-assisted grinding, abrasive grain
Movement is made of three parts：Around grinding wheel spindle with angular velocity omegasCircular motion, along direction of feed relative to workpiece with linear velocity vw
Linear motion, along grinding wheel axially with respect to workpiece with amplitude A, ultrasonic vibration that frequency f is carried out；
T at the time of if grinding starts0=0s, the origin O ' of wheel face coordinate system is positioned at grinding wheel minimum point and positioned at workpiece at this time
The surface of coordinate origin O, ultrasonic vibration initial phase are 0；
Arbitrary equation of locus of the abrasive grain i in workpiece coordinate system：
Wherein rsFor grinding wheel radius, t is from t0Moment starts the time of grain motion, LzFor the vertical range of grinding wheel axis to O,
λ=1,2,3...., λ represent the number of abrasive grain i incision workpiece, αiFor abrasive grain i to grinding wheel axis
Vertical line and O ' to the angle between the vertical line of grinding wheel axis；
3) the dynamic outline method of sampling：
Workpiece topological matrix g is used in workpiece surface appearance predictionmnRepresent workpiece surface appearance, i.e., on the surface of the workpiece with x
Direction separation delta x and y directions separation delta y grid divisions, are opened up using the height value z (m, n) at grid lattice point P (m, n) as workpiece
Flutter matrix gmnIn element,
First, the series of parallel sampled cross-section in plane Oyz is set on the surface of the workpiece, these sampled cross-sections cross workpiece topology
The lattice point of matrix is followed successively by 1,2,3 ... since O points along the number of positive direction of the x-axis, and during grinding, a certain abrasive grain exists
C1Point incision workpiece, in C2Point leaves workpiece, C1The sampled cross-section n on point the right1It is first sampled cross-section interfered, C2Point
The sampled cross-section n on the left side2It is the last one, abrasive grain has been sequentially passed through during workpiece is ground from n1To n2A series of adopt
Sample section, therefore abrasive grain work after grinding can be obtained in the scallop-height value for calculating each lattice point in these sampled cross-sections
Part surface topography, n1And n2Value can be obtained by following formula
L in formula1For C1Point arrives the horizontal distance of workpiece coordinate system origin O, l2For C2Point arrives the horizontal distance of O, l1And l2Value can
To be obtained using abrasive grain equation of locus in step 2)；
When calculating each lattice point scallop-height value in n-th of sampled cross-section, the sampled cross-section is calculated first to workpiece coordinate system
The horizontal distance x of origin On：
xn=(n-1) Δ x
By xnValue substitute into abrasive grain equation of locus abrasive grain center can be obtained in sampled cross-section n to workpiece coordinate system origin O's
Horizontal distance ynWith vertical range zn, then position of the abrasive grain profile in sampled cross-section n be assured that, abrasive grain profile equation
It is expressed as：
(y-yn)2+(z-zn)2=(dg/2)2
D in formulagFor abrasive grain diameter, in C1And C2Intermediate C3Point arrives C4Abrasive grain and workpiece interfere between point, C3Point the right
Lattice point P1(m1, n) and it is first lattice point interfered, C4The lattice point P on the point left side2(m2, n) and it is the last one, m1And m2Value
It can be obtained by following formula,
L in formula3For C3Point arrives the horizontal distance of workpiece coordinate system origin O, l4For C4Point arrives the horizontal distance of O points, l3And l4Value
It can be obtained using abrasive grain profile equation.
Then, from P1To P2A series of lattice points at abrasive grain profile is sampled, wherein a certain sampled point H (m, n) is to workpiece
The horizontal distance of coordinate origin O is lm：
lm=(m-1) Δ y
By y=lmIt substitutes into and following formula is obtained in abrasive grain profile equation, you can the ordinate z (m, n) of sampled point H (m, n) is obtained,
Assuming that the workpiece material interfered is completely removed, then the value of z (m, n) is exactly the workpiece scallop-height of the point after grinding
Value.The coordinate value of sampled point H (m, n) is assigned to lattice point P (m, n), completes the update at the lattice point, similarly, update sampled cross-section n
It is interior from P1Point arrives P2The possessive case point coordinates of point completes the calculating in sampled cross-section n；When from n1To n2All sampled cross-sections
After the completion of all updating, it is possible to obtain the workpiece surface appearance after single grain grinding, continue thereafter with and call it on this basis
He is updated abrasive grain, finally obtains complete workpiece surface appearance；
4) to the amendment of the method for sampling：The characteristics of for axial ultrasonic vibration-assisted grinding, establishes grinding groove and broadens model,
And grinding elastic deformation model and plastic accumulation model are introduced on this basis, the dynamic outline method of sampling is modified.
2. axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method, feature exist according to claim 1
The grinding groove model modification method that broadens is as follows in, the step 4)：
Groove contour is oval, ellipse short shaft dg, long axis de, groove contour equation is expressed as：
dgAnd deRatio be：
θ is grain motion speed vgWith the angle between sampled cross-section n, at sampled cross-section n, vgThe speed along x-axis can be decomposed into
Spend component vxWith the velocity component v along y-axisy, obtained according to the synthetic method of movement：
vxAnd vyValue the derivation of time t can be obtained by abrasive grain equation of locus：
T in formulanAt the time of for abrasive grain center movement at sampled cross-section n, vsFor grinding speed, d can be obtainedeValue：
Work as deValue determine after, groove contour equation determines therewith, when there are during axial ultrasonic vibration, using groove contour equation
Instead of the abrasive grain profile equation in step 3), sampled point is chosen on groove contour.
3. axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method, feature exist according to claim 2
In grinding elastic deformation model modification method is as follows in the step 4)：
Assume to establish grinding elastic deformation theory's model based on spherical wear particles and ideal plane grinding, abrasive grain stressing conditions are with surveying
Similar during examination Brinell hardness, abrasive grain is R by normal pressure, and the depth of abrasive grain incision workpiece is dp, when abrasive grain moves cutting workpiece
When, the direction of R has turned over angle, θ ', abrasive grain is friction coefficient by frictional force μ a R, μ in bottom, abrasive grain yielding value δcWith
Workpiece elasticity recovery value δwCalculation formula it is as follows：
δc=C [R (cos θ '-μ sin θs ')]2/3
δw=R (cos θ '-μ sin θs ')/k
C is constant in formula, and value range is 0.08~0.25, and average value 0.15, k is workpiece stiffness coefficient；
The calculation formula of θ ' and R is respectively：
R=π b2B
B is the half that abrasive grain cuts workpiece portion chord length in formula, and B is the ball hardness number of workpiece material；
The calculation formula that geometrical relationship can obtain b is：
The depth of abrasive grain incision workpiece is d in desired elastic deformation modelpIt is to be measured under hypothesis of the workpiece surface for ideal plane
, practical work piece surface is not ideal plane, real in the case of sampled cross-section n septal fossulas channel profiles and practical work piece Surface Interference
Border workpiece surface is less than preferable workpiece surface, from H1(m1, n) and to H2(m2, n) series of points be the sampling really interfered
Point, each sampled point have different penetraction depths,
Although practical work piece surface is not ideal plane, practical work piece surface is very smooth, each neighboring lattice points in grinding
Between difference in height it is little, set up therefore, it is considered that desired elastic deformation model is still approximate, only need the incision with each sampled point
Depth-averaged value dp' instead of the d in desired elastic deformation modelp,
Z in formulaw(m, n) is the actual height of workpiece surface lattice point P (m, n) before grinding, and z (m, n) is the depth of sampled point H (m, n)
Degree, the scallop-height of workpiece surface lattice point P (m, n) should be after grinding：Z'(m, n)=z (m, n)+δc+δw
When updating workpiece surface appearance model, z is replaced with the value of z 'wValue.
4. axial ultrasonic vibration-assisted grinding workpiece surface appearance simulated prediction method, feature exist according to claim 2
In plastic accumulation model modification method is as follows in the step 4)：
During practical grinding, the material that is interfered on workpiece with abrasive grain only some be removed, form abrasive dust, without
The material being removed then is plastically deformed, and is deposited in groove both sides, and plastic accumulation model is based on spherical wear particles and ideal plane
Grinding is it is assumed that the sectional area of removal material is Ag, the profile of material stacking part is assumed to be parabola, is highly hp, width is
2wp, sectional area Ap, abrasive grain profile and the inclination angle of accumulation profile intersection tangent line are αp, wpAnd hpValue be respectively：
ApValue determined by grinding efficiency β,
For axial ultrasonic vibration-assisted grinding, it is contemplated that groove broadens phenomenon, should be using the elliptical grooves profile established come generation
For the round abrasive grain profile in ideal plasticity Mathematical Model of heaped-up,
The bottom of accumulation profile assumes that the workpiece surface for ideal plane in ideal plasticity Mathematical Model of heaped-up, but practical in grinding
Workpiece surface is not ideal plane, and position is less than the ideal plane, although practical work piece surface is not ideal plane,
Since workpiece surface is very smooth in grinding, almost plane is set up, groove contour therefore, it is considered that ideal plasticity Mathematical Model of heaped-up is still approximate
From sampled point H1(m1, n) and to H2(m2, n) with practical work piece Surface Interference, remove the sectional area A of materialgFeelings should be interfered according to practical
Condition calculates：
In order to establish accumulation profile, with sampled point H1To H2Practical work piece apparent height average value establish an equivalent plane generation
For the ideal plane in ideal plasticity Mathematical Model of heaped-up, the equivalent level value zdFor：
The height that accumulation profile is higher by equivalent plane is still hp, but width increases as 2wp', it broadens model similarly with groove, accumulates
The degree that profile broadens is identical with the degree that groove contour broadens, and can calculate wp' value：
H is being determinedp, wp', zdValue after, the location and shape for accumulating profile just entirely define, accumulation profile cover
Each lattice point at calculate the height value of accumulation profile, substitute former lattice dynamical system value, complete the update in a sampled cross-section.
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