CN105867311B - Arc-blade diamond cutter precision turning highly steep asphere method - Google Patents

Arc-blade diamond cutter precision turning highly steep asphere method Download PDF

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CN105867311B
CN105867311B CN201610239408.3A CN201610239408A CN105867311B CN 105867311 B CN105867311 B CN 105867311B CN 201610239408 A CN201610239408 A CN 201610239408A CN 105867311 B CN105867311 B CN 105867311B
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CN105867311A (en
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孙涛
孔繁星
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Harbin Institute of Technology
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/41—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by interpolation, e.g. the computation of intermediate points between programmed end points to define the path to be followed and the rate of travel along that path
    • G05B19/4103—Digital interpolation
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/34—Director, elements to supervisory
    • G05B2219/34092—Polar interpolation

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Abstract

The invention discloses a kind of arc-blade diamond cutter precision turning highly steep asphere method, pass through given processing index, calculate interpolation segments and interpolated point number, optimize interpolation chord length, determine the movement locus of cutter heart, with ensure in turning process knife tool arc sword and workpiece meridian section curve at interpolated point it is tangent.The present invention using diamond cutter Circular Nose Cutting Edge with workpiece is tangent is processed, participate in completing the high accuracy processing of highly steep asphere without B axle, solve the problems, such as machining high-precision, big steepness aspheric curve under conditions of B axle is not increased, avoid due to increase motion B axle and the single shaft installation introduced and control error, and multiaxis coordinates error, the manufacturing cost of lathe is greatly reduced simultaneously, and then reduces the processing cost of highly steep asphere.

Description

Arc-blade diamond cutter precision turning highly steep asphere method
Technical field
The invention belongs to Mechanical Manufacturing and Automation technical field, is related to one kind and uses arc-blade diamond cutter turning side Method, for high steepness optical mirror turnery processing (conformal optics, reflection optics, space optics).
Background technology
High accuracy, big steepness axial symmetry aspheric curve are due to the radius of curvature value changes phase of each point on its meridian section curve To larger, cause processing and manufacturing very difficult.Row digital control processing meeting is directly programmed into using existing XZ two-axle interlockings CNC lathes Cutter interference is produced, as shown in Figure 1.Therefore, the numerical control workshop of this big steepness aspheric curve extends B frequently with band at present The landing tee office lathe of axle is completed, as shown in Fig. 2 the mode for being formed by XYB interpolations of its movement locus is realized, son The Curvature varying of each point is compensated by the B axle of extension on noon section curve, ensures cutter in process with the rotation of B axle It is tangent with workpiece meridian section curve, avoid cutter interference.
Increase a motion B axle on the basis of original XZ two-axle interlockings, single shaft installation will necessarily be introduced and control misses Poor and multi-shaft interlocked coordination error, this surface quality to workpieces processing is unfavorable.Simultaneously because add motion B Axle so that the manufacturing cost of lathe greatly increases, and then adds the processing cost of highly steep asphere.
The content of the invention
The purpose of the present invention is the diamond turning problem for high accuracy, big steepness axial symmetry aspheric curve, there is provided A kind of arc-blade diamond cutter precision turning highly steep asphere method, by given processing index, calculate interpolation segmentation Number and interpolated point number, optimize interpolation chord length, the movement locus of cutter heart is determined, to ensure knife tool arc sword and work in turning process Part meridian section curve is tangent at interpolated point, i.e., solves machining high-precision, big steepness aspheric under conditions of B axle is not increased The problem of curved surface.
The purpose of the present invention is achieved through the following technical solutions：
A kind of arc-blade diamond cutter precision turning highly steep asphere method, comprises the following steps：
First, interpolated point calculates
1st, workpiece meridian section curve terminal A, B coordinate (z are assumedA,xA)、(zB,xB) it is, it is known that first according to workpiece Surface configuration precision, the processing such as surface roughness index calculate the interpolation chord length for meeting error requirements, then willAlong Z Direction is divided into n-1 interpolation section, calculates n interpolation point coordinates:
Then, the calculation formula of interpolation chord length is:
2nd, in i-th of interpolation straightway, it is k to make slopeMi=(xi+1-xi)/(zi+1-zi) and with workpiece meridian section curve It is tangential on point MiStraight line, make kMi=x 'MiThe M of each interpolation straightway can be releasediPoint coordinates (zMi,xMi), then cross point Mi's Normal equation is：
3rd, M is crossediPoint normal and i-th section of Interpolation step-length Δ LiIntersect at point Ni, then NiPoint coordinates calculation formula is：
4th, by MiAnd NiThe coordinate of point can calculate the approximate error of the i-th interpolation section, and calculation formula is：
5th, by the approximate error δ of each interpolation straightwayiAfter draw maximum approximate error δmax, work as δmaxLess than requiring Surface shape error when, you can determine rough interpolation segmentation number n.
2nd, the optimization of interpolation chord length
On the premise of workpiece surface form error requirement is met, make each interpolation straightway that there is equal chord length δ L= mFT≤ΔLmin, wherein, Δ LminFor Δ LiIn minimum value, F (mm/min) be lathe feed speed, T (s) be interpolation week Phase, m are integer, and interpolation point coordinates calculation formula is：
3rd, cutting tool path calculates
Assuming that knife tool arc radius is rd, the coordinate of knife tool arc central point is (zdi,xdi), cross workpiece meridian section curveUpper interpolated point Pi(zi,xi) normal hand over Z axis in ξiPoint, makes ρi=Piξi, θi=∠ PiξiZ, then：
1st, P is crossedi(zi,xi) point intercept of the normal in z-axis calculation formula it is as follows：
ξi=zi+xix′i。
2、ρiAnd θiCalculation formula it is as follows：
3rd, cutter heart coordinate calculation formula is as follows：
4th, turnery processing is carried out to highly steep asphere according to the cutting tool path that step 3 is calculated.
The processing of highly steep asphere is completed with Digit Control Machine Tools more than 3 axles or 3 axles at present, and the present invention utilizes circle The very high arc-blade diamond cutter of radian precision is completed on the axle lathes of xz two, i.e.,：Utilize diamond cutter Circular Nose Cutting Edge and work Part is tangent to be processed, and is participated in completing the high accuracy processing of highly steep asphere without B axle, is not being increased the condition of B axle Under solve the problems, such as machining high-precision, big steepness aspheric curve, avoid due to increase motion B axle and introduce single shaft installation and Control error, and multiaxis to coordinate error, while greatly reduce the manufacturing cost of lathe, and then reduce highly steep asphere Processing cost.
Brief description of the drawings
Fig. 1 is produces cutter interference when directly programming turning highly steep asphere using XZ two-axle interlocking CNC lathes, a) circle Arc cutter turning sword, b) without cutter compensation when produce form error, c) produce cutter interference when having cutter compensation；
Fig. 2 is that the T-shaped layout lathe with B axle avoids cutter interference, a) the T-shaped layout lathe with B axle, b) cutter and workpiece It is tangent；
Fig. 3 is 0.2 μm of cutter of circular arc degree；
Fig. 4 calculates for interpolation chord length；
Fig. 5 is rough interpolation error calculation；
Fig. 6 is cutter heart trajectory calculation；
Fig. 7 is turning experiment.
Embodiment
Technical scheme is further described below in conjunction with the accompanying drawings, but is not limited thereto, it is every to this Inventive technique scheme is modified or equivalent substitution, without departing from the spirit and scope of technical solution of the present invention, all should cover In protection scope of the present invention.
Under conditions of rotation B axle is not increased, want to realize the meridian section of cutter and workpiece by the axle interpolation of X, Z two Curve is tangent at interpolated point, completes the aspherical high-precision turning of big steepness, can select the Circular Nose Cutting Edge gold that circular arc degree is higher Diamond cutter is completed.The circular arc degree of diamond cutter cutting edge can accomplish 0.1~0.2 μm at present, as shown in figure 3, working as tool blade When mouth circular arc and tangent workpiece, the form error as caused by circular arc degree is certainly less than 1 μm.It follows that by making circular arc angle of throat Tool with workpiece it is tangent come improve traditional numeric-control method for turning process highly steep asphere when caused cutter interference problem be feasible 's.Therefore, the invention provides a kind of arc-blade diamond cutter precision turning highly steep asphere method-interpolation track meter Calculation method, is comprised the following steps that：
First, interpolated point calculates
1st, as shown in Figure 4, it is known that aspheric noon section curve equation：
Assuming that workpiece meridian section curve terminal A, B coordinate (zA, xA), (zB, xB) are, it is known that first according to workpiece The processing such as surface configuration precision, surface roughness index calculates the interpolation chord length for meeting error requirements, then willAlong Z side To n-1 interpolation section is divided into, n interpolation point coordinates is calculated, formula is as follows:
Obviously, the calculation formula of interpolation chord length is:
2nd, as shown in figure 5, in i-th of interpolation straightway, it is k to make slopeMi=(xi+1-xi)/(zi+1-zi) and it is sub with workpiece Noon section curve is tangential on point MiStraight line, make kMi=x 'MiThe M of each interpolation straightway can be releasediPoint coordinates (zMi,xMi), Then cross point MiNormal equation be：
3rd, M is crossediPoint normal and i-th section of Interpolation step-length Δ LiIntersect at point Ni, then NiPoint coordinates calculation formula is：
4th, by MiAnd NiThe coordinate of point can calculate the approximate error of the i-th interpolation section, and calculation formula is：
5th, by the approximate error δ of each interpolation sectioniAfter draw maximum approximate error δmax, work as δmaxLess than desired table During the form error of face, you can determine the number n of rough interpolation segmentation.
2nd, the optimization of interpolation chord length
The interpolation chord length of each interpolation section calculated according to above method is unequal, it is assumed that F (mm/min) is machine The feed speed of bed, T (s) is interpolation cycle, then only works as Δ LiWhen=mFT, m are integer, the end points of interpolation chord length could be with Interpolated point overlaps, i.e. interpolation track does not produce error at interpolated point.Therefore, workpiece surface form error requirement can met On the premise of, make each interpolation straightway that there is equal chord length δ L=mFT≤Δ Lmin, wherein, Δ LminFor Δ LiIn most Small value, interpolation point coordinates calculation formula are：
Obviously, the interpolation chord length Δ L at last interpolation straightwayn-1≤ δ L, i.e., having at last interpolated point can Error can occur, so δ L and Δ Ln-1Depending on actual calculated case the error at last interpolated point can be made to the greatest extent may be used It can reduce.
3rd, cutting tool path calculates
It is determined that after meeting the rough interpolation point number n needed for workpiece surface form error requirement, you can according to it is selected then Tool radius rdThe movement locus of cutter is calculated etc. parameter.As shown in Figure 6, it is assumed that knife tool arc radius is rd, in knife tool arc The coordinate of heart point is (zdi,xdi), cross workpiece meridian section curveUpper interpolated point Pi(zi,xi) normal hand over Z axis in ξiPoint, order ρi=Piξi, θi=∠ PiξiZ。
1st, ξ is solvedi：
Any point P on meridian is released to z derivations by formula (1) both endsi(zi,xi) place derivative calculations formula:
Obviously, P is passed throughi(zi,xi) point normal equation be：
X=0 is made, tried to achieve Pi(zi,xi) point intercept of the normal in z-axis：
ξi=zi+xix′i (10)。
2nd, ρ is solvediAnd θi：
3rd, cutter heart coordinate calculates：
The cutter heart track calculated according to rough interpolation is controlled, and not only realizes cutter radius compensation, be ensure that and is being inserted It is tangent to mend knife tool arc sword and workpiece meridian section curve at point, and make it that transition track need not be introduced at interpolated point, It can ensure interpolated point on curve is interpolated.
4th, experimental verification
As shown in fig. 7, selection corner radius rd=0.4 diamond cutter, it is larger to steepness using the above method Ellipsoid and hyperboloid carry out turning experiment.The workpiece after processing is measured using Taylor Hobson contourgraphs, tied Fruit is that surface configuration precision PV values are less than 3 μm, and surface roughness RMS value is less than 7nm, with agreement with theoretical calculation.

Claims (2)

  1. A kind of 1. arc-blade diamond cutter precision turning highly steep asphere method, it is characterised in that methods described step is such as Under：
    First, interpolated point calculates
    (1) workpiece meridian section curve terminal A, B coordinate (z are assumedA,xA)、(zB,xB) it is, it is known that willAlong Z-direction decile Into n-1 interpolation section, n interpolation point coordinates is calculated:
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>z</mi> <mi>A</mi> </msub> <mo>-</mo> <mfrac> <mrow> <mo>|</mo> <mrow> <msub> <mi>z</mi> <mi>B</mi> </msub> <mo>-</mo> <msub> <mi>z</mi> <mi>A</mi> </msub> </mrow> <mo>|</mo> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>&amp;times;</mo> <mi>i</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>......</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
    Then, the calculation formula of interpolation chord length is:
    <mrow> <msub> <mi>&amp;Delta;L</mi> <mi>i</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>......</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    (2) in i-th of interpolation straightway, it is k to make slopeMi=(xi+1-xi)/(zi+1-zi) and it is tangent with workpiece meridian section curve In point MiStraight line, make kMi=x 'MiThe M of each interpolation straightway can be releasediPoint coordinates (zMi,xMi), then cross point MiNormal Equation is：
    <mrow> <mi>x</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> <mo>;</mo> </mrow>
    (3) M is crossediPoint normal and i-th section of Interpolation step-length Δ LiIntersect at point Ni, then NiPoint coordinates calculation formula is：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>x</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>k</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> </mfrac> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>x</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;DoubleRightArrow;</mo> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    (4) by MiAnd NiThe coordinate of point can calculate the approximate error of the i-th interpolation section, and calculation formula is：
    <mrow> <msub> <mi>&amp;delta;</mi> <mi>i</mi> </msub> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>z</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>M</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mrow> <mi>N</mi> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>,</mo> <mrow> <mo>(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>......</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
    (5) by the approximate error δ of each interpolation straightwayiAfter draw maximum approximate error δmax, work as δmaxLess than desired table During the form error of face, you can determine the number n of rough interpolation segmentation；
    2nd, the optimization of interpolation chord length
    On the premise of workpiece surface form error requirement is met, make each interpolation straightway that there is equal chord length δ L=mFT ≤ΔLmin, wherein, Δ LminFor Δ LiIn minimum value, F (mm/min) is the feed speed of lathe, and T (s) is interpolation cycle, m For integer, then interpolation point coordinates calculation formula is：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>-</mo> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <msup> <mi>&amp;delta;L</mi> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
    3rd, cutting tool path calculates
    Assuming that knife tool arc radius is rd, the coordinate of knife tool arc central point is (zdi,xdi), cross workpiece meridian section curve Upper interpolated point Pi(zi,xi) normal hand over Z axis in ξiPoint, then cutter heart coordinate calculation formula is as follows：
    The θiCalculation formula it is as follows：
    <mrow> <msub> <mi>&amp;theta;</mi> <mi>i</mi> </msub> <mo>=</mo> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>t</mi> <mi>g</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <msub> <mi>z</mi> <mi>i</mi> </msub> <mo>-</mo> <msub> <mi>&amp;xi;</mi> <mi>i</mi> </msub> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow>
    ξiTo cross Pi(zi,xi) point intercept of the normal in z-axis；
    4th, turnery processing is carried out to highly steep asphere according to the cutting tool path that step 3 is calculated.
  2. 2. arc-blade diamond cutter precision turning highly steep asphere method according to claim 1, it is characterised in that The ξiCalculation formula it is as follows：
    ξi=zi+xixi′
    Wherein, xi' representative function x=f (z) is in Pi(zi, xi) point derivative.
CN201610239408.3A 2016-04-18 2016-04-18 Arc-blade diamond cutter precision turning highly steep asphere method Active CN105867311B (en)

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CN109669395B (en) * 2018-12-04 2021-08-06 天津津航技术物理研究所 Variable-radius circular interpolation method for axisymmetric aspheric surface
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JP2005001100A (en) * 2003-02-21 2005-01-06 Seiko Epson Corp Method of working aspherical face and method of forming aspherical face
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