CN1257386C - Single-transposition roundness fault separating method - Google Patents

Abstract

The present invention relates to the technical field of precision instrument manufacture and measuring technology, particularly to an ultraprecise roundness error separative method. The method selects proper transposition angles alpha during the separating period to make workpieces carry out single time small angle transposition at corresponding angles corresponding to a rotary main shaft. By measuring total composite errors A (n) and B (n) of the workpieces before and after the position transposition comprising a workpiece error g (n) and a main shaft turning error z (n), full harmonic separation of z (n) and g (n) in the harmonic ranges of 1 to 100 upr can be realized under the assistance of mathematical processing methods, such as quick Fourier series transposition, harmonic analysis, etc. Compared with the ' multi-stage process ' error separation technology generally adopted by the existing roundness measuring instrument, the method thoroughly eradicates the ' harmonics singular ' problem in the harmonics range of 1 to 100 upr, and simultaneously, the present invention makes an error separating system which takes the method as a base greatly simplified. The separating process is shortened, and separating time is reduced. The method can provide an ideal roundness error separative method for the establishment of nanometer precision revolution standards.

Description

Single-transposition roundness fault separating method

Technical field

The invention belongs to exact instrument manufacturing and field of measuring technique, particularly a kind of ultraprecise roundness fault separating method.

Background technology

Error separating technology has been widely used in improving in the accurate roundness measuring instrument its measuring accuracy as the basic means of separate apparatus rotary main shaft sum of errors measured workpiece deviation from circular from, the error separating technology that adopts in the accurate roundness measurement can be divided into two big classes, one class is to utilize the repeatedly transposition of measured workpiece to obtain " the multi-index method " of additional information, another kind ofly obtains additional information " many gauge heads method " by increasing the sensor number." multi-index method " comprises " multistep processes ", " reverse method " again.As a whole, adopt above-mentioned error separating technology to make the precision level of accurate roundness measuring instrument that bigger improvement arranged, but do not reach desired effect, its principal element concentrates on the following aspects:

1) exist harmonic wave to suppress problem on " multistep processes " principle in " multi-index method ", the forward and backward two measurement cross sections that " reverse method " then exists the sensor secondary installing to bring do not overlap the error that causes, it is confirmed among circularity international comparison result in European five states for the second time to the reduction of deviation from circular from separation accuracy;

" the multiple multistep processes " that grows up by " multistep processes ", " full harmonic error partition method " though etc. method can solve harmonic wave and suppress problem, it is a prerequisite to increase workpiece transposition number.Shortcomings such as it exists measuring period long, and measuring system drift influence is big, this is difficult to control with regard to the variation that makes the state of measuring in the error separating process, finally easily causes separating resulting dispersed big, reduces separation accuracy;

2) " many gauge heads method " can solve harmonic wave inhibition problem by the position that increases between sensor number and suitable layout sensor in principle, each sensor characteristic is inconsistent, sensor mounting arrangement difficulty but exist in engineering construction, be difficult to guarantee that gauge head is on same measurement profile, and the measured workpiece diameter also need not readjusted simultaneously, therefore is difficult to be applied in roundness measuring instrument.

As seen, to suppress, simplify its detachment process and reduce disengaging time be the key that further improves roundness measuring equipment error separating precision to the harmonic wave that how to reduce error separating technology.

Round this key problem of deviation from circular from isolation technics, recently some scholars have proposed many valuable separation methods, as doctor Sun Tong two-step approach deviation from circular from isolation technics (SunTong.Two-step technique without harmonics suppression in error separation.Meas.Sci.Technol., 1996 based on the Prony analysis of spectrum have been proposed; 7:1563), analyzed the validity of this method theoretically; Professor Cao Linxiang has proposed the two-step approach error separating technology based on positive and negative Fourier transform, these methods in-depth and developed error separating technology, but still have deficiency such as computing complexity.

Summary of the invention

In order to overcome the deficiency of above-mentioned existing roundness fault separating method, and further improve the rotating accuracy of ultraprecise roundness measuring instrument, the present invention proposes a kind of new single-transposition roundness fault separating method.This method passes through to select appropriate transposition angle α in detachment process, make the relative rotary main shaft of workpiece carry out the single low-angle transposition of respective angles, by comprising composition error A (n) and the B (n) of workpiece error g (n) and spindle rotation error z (n) before the measuring workpieces transposition and after the transposition, utilize mathematical processing methods such as the fast Fourier transformation of series and frequency analysis, realize z (n) and g (n) full harmonic separation in 1-100upr harmonic wave scope.

Technical solution of the present invention illustrates in conjunction with Fig. 1: a kind of ultraprecise deviation from circular from list transposition separation method, and this method may further comprise the steps:

1) selectes measured workpiece 2 certain measurement cross section and carry out separating and measuring, sensor 1 records the composition error A (n) of the turn error z (n) that comprises roundness error of workpiece g (n) and main shaft 3 that is in initial index position a place, n=0 wherein, 1, ..., N-1, N are all sampling numbers of workpiece 2 circle contours;

2) workpiece 2 relative main shafts 3 are turned over angle [alpha] and arrive index position b, the workpiece 2 that is in index position b is measured again, sensor 1 records the composition error value B (n) in the b transposition that comprises roundness error of workpiece g (n) and spindle rotation error z (n), n=0 wherein, 1 ..., N-1;

3) separate carrying out deviation from circular from the composition error A (n) that records on a, the b position before and after transposition angle α and the workpiece 2 single transpositions and B (n) the substitution single-transposition roundness fault separating method, concrete steps are as follows:

1. remove the DC quantity of sampled signal A (n), B (n);

2. calculated difference signal r (n);

$r\left(n\right)=\underset{k=2}{\overset{S-1}{\mathrm{\Σ}}}((a_k(1-\cos \mathrm{k\α})-b_k\sin \mathrm{k\α})\cos (2\mathrm{n\πk}/N)+(a_k\sin \mathrm{k\α}+b_k(1-\cos \mathrm{k\α}))\sin (2\mathrm{n\πk}/N))$

Wherein, a _k, b _kBe the fourier progression expanding method coefficient of roundness error of workpiece g (n), S is an overtone order.

3. r (n) is carried out frequency analysis, ask its harmonic constant e _kAnd f _k

$\left\{\begin{array}{c}{e}_k=a_k(1-\cos \mathrm{k\α})-b_k\sin \mathrm{k\α}\\ f_k=a_k\sin \mathrm{k\α}+b_k(1-\cos \mathrm{k\α})\end{array}\right.$

4. calculate g (n) harmonic constant a _kAnd b _k

$\left\{\begin{array}{c}{a}_k=\frac{1}{2}{e}_k+\frac{\sin \mathrm{k\α}}{2(1-\cos \mathrm{k\α})}{f}_k\\ b_k=-\frac{\sin \mathrm{k\α}}{2(1-\cos \mathrm{k\α})}{e}_k+\frac{1}{2}{f}_k\end{array}\right.$

5. remove fundametal compoment and first harmonic component among the A (n);

A″(n)＝A(n)-A ₀-(h ₁?cos(2nπ/N)+l ₁?sin(2nπ/N))

Wherein, A ₀Be fundamental component, h ₁And l ₁Be A (n) fourier progression expanding method coefficient.

6. calculate the synthetic g (n) of harmonic wave;

$g\left(n\right)=\underset{k=2}{\overset{S-1}{\mathrm{\Σ}}}\left((a_k\cos (2\mathrm{n\πk}/N)+b_k\sin (2\mathrm{n\πk}/N)\right)$

7. calculate axial system error z (n);

z(n)＝g(n)-A″(n)

4) g (n) and z (n) are carried out roundness evaluation in the substitution roundness measurement assessment system respectively, obtained rejecting the circularities of workpiece (2) g (n) of turn error and the circularities of main shaft (3) turn error z (n).

The present invention has following characteristics and good result:

Compare with other error separating technology, deviation from circular from list transposition method isolation technics is in detachment process, only need make the relative main shaft of workpiece finish the transposition of single low-angle, just measured workpiece circularity and spindle rotation error can be carried out full harmonic separation in the 1-100upr scope through simple algorithm, this is that the present invention distinguishes one of innovative point of prior art;

Error separating method adopts the single position shifter error separation method based on frequency analysis, separates turntable during error separating and only need finish the specific low-angle transposition of single, and the deviation from circular from that can finish measured workpiece separates, this be the difference prior art innovative point two.

When single position shifter error separation method can avoid existing error separating method such as multistep processes, reverse method etc. to carry out the deviation from circular from separation, the detachment process complexity, disengaging time is long, and the deficiency that all kinds of drifts of introducing are bigger etc. can have been simplified error separating device and error separating process simultaneously greatly.

Description of drawings

The synoptic diagram of Fig. 1 initial transposition a for measured workpiece is in

Fig. 2 is in the synoptic diagram of index position b for measured workpiece

Fig. 3 is in the first inversion synoptic diagram for embodiment of the invention measured workpiece

Fig. 4 is in the second transposition synoptic diagram for embodiment of the invention measured workpiece

Fig. 5 is the graph of relation between Pi (k) and the k

Fig. 6 measured data g (n) raw data profile stretch-out view

The data profile stretch-out view of Fig. 7 measured data g (n) 1-100upr filter range

Fig. 8 data z (n) is in raw data profile stretch-out view

Fig. 9 data z (n) is in the data profile stretch-out view of 1-100upr filter range

During Figure 10 Δ α=0.00 °, single-transposition roundness fault separating method separates Δ s (n) curve of measured data

During Figure 11 Δ α=0.00 °, single-transposition roundness fault separating method separates Δ e (n) curve of measured data

During Figure 12 Δ α=0.01 °, single-transposition roundness fault separating method separates Δ s (n) curve map of measured data

During Figure 13 Δ α=0.01 °, single-transposition roundness fault separating method separates Δ e (n) curve map of measured data

Among the figure, 1 sensor, 2 workpiece, 3 main shafts, 4 workpiece monumented points, 5 main shafts are measured starting point.

Embodiment

Single-transposition roundness fault separating method of the present invention reaches accompanying drawing in conjunction with the embodiments and is described in detail as follows: with sensor swinging roundness measuring equipment is example, and the method and the process of separating spindle rotation error z (n) and roundness error of workpiece g (n) are described.

With a certain fixed position a is as the apparatus measures starting point on the roundness measuring equipment rotary main shaft, measured workpiece places on the error separating turntable, and the initial reference point of establishing workpiece is b.As shown in Figure 3, workpiece the first inversion position is measured finish after, rotation error separates turntable to be made workpiece reference point b measure starting point with respect to the roundness measuring equipment main shaft to turn over the α angle counterclockwise, make workpiece be in second index position and measure, as shown in Figure 4.If g (θ) is the deviation from circular from of workpiece, z (θ) is the turn error of main shaft, and A (θ), B (θ) are respectively the composition error that same sensor records on Fig. 3 measuring position and Fig. 4 measuring position.

If the direction of compression sensor gauge head is the sensor forward, measuring workpieces shape and instrument main shaft shape are respectively the forward of g (θ) and z (θ) to the direction of outer lug, because sensor is positioned on the instrument main shaft, then z (θ) is reverse all the time with the direction of A (θ) and B (θ) value.When then workpiece was in the first measuring position figure, the composition error data A (θ) that sensor records was:

A(θ)＝g(θ)-z(θ) (1)

With g (θ), it is the function g (θ) of first-harmonic that z (θ) expanded in time domain with one week of main shaft gyration, and then its fourier progression expanding method formula is:

$g\left(\mathrm{\&theta;}\right)={\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">g</figure-callout>}}_0+\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}(a_k\cos \mathrm{k\&theta;}+b_k\sin \mathrm{k\&theta;})---\left(2\right)$

$z\left(\mathrm{\&theta;}\right)={\mathrm{<figure-callout\; id="0"\; label="z"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">z</figure-callout>}}_0+\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}(c_k\cos \mathrm{k\&theta;}+d_k\sin \mathrm{k\&theta;})---\left(3\right)$

In the formula: g ₀, z ₀, a _k, b _k, c _kAnd d _kBe the fourier progression expanding method coefficient.

After the relative main shaft of workpiece turned over the α transposition, variation had taken place in the phase place of its all subharmonic, its value for g (θ+α), and the phase invariant of axial system error composition, its value still is z (θ), then the composition error data B (θ) that records of this second transposition upper sensor is:

B(θ)＝g(θ+α)-z(θ) (4)

In like manner, with g (θ+α) in time domain, expand into one week of main shaft gyration be first-harmonic function g (θ+α), its fourier progression expanding method formula is:

$g(\mathrm{\&theta;}+\mathrm{\&alpha;})={\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">g</figure-callout>}}_0^{\&prime;}+\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}(a_k\cos k(\mathrm{\&theta;}+\mathrm{\&alpha;})+b_k\sin k(\mathrm{\&theta;}+\mathrm{\&alpha;}))---\left(5\right)$

$={\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">g</figure-callout>}}_0^{\&prime;}+\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}((a_k\cos \mathrm{k\&alpha;}+b_k\sin \mathrm{k\&alpha;})\cos \mathrm{k\&theta;}+(b_k\cos \mathrm{k\&alpha;}-a_k\sin \mathrm{k\&alpha;})\sin \mathrm{k\&theta;})$

(1) and (4) two formulas are subtracted each other:

r(θ)＝A(θ)-B(θ)＝g(θ)-g(θ+α)

$={\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">g</figure-callout>}}_0^{\&prime;}\underset{k=1}{\overset{S-1}{\mathrm{\&Sigma;}}}((a_k(1-\cos \mathrm{k\&alpha;})-b_k\sin \mathrm{k\&alpha;})\cos \mathrm{k\&theta;}+(a_k\sin \mathrm{k\&alpha;}+b_k(1-\cos \mathrm{k\&alpha;}))\sin \mathrm{k\&theta;})---\left(6\right)$

In the formula: g ₀", a _kAnd b _kBe the fourier progression expanding method coefficient

Following formula is handled through N point sampling discretize, and only got 0 to the S-1 subharmonic.The angle of the n time sampled point is 2n π/N, and then its discrete form is as follows:

$r\left(n\right)={\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">g</figure-callout>}}_0^{\&prime;}+\underset{k=1}{\overset{S-1}{\mathrm{\&Sigma;}}}((a_k(1-\cos \mathrm{k\&alpha;})-b_k\sin \mathrm{k\&alpha;})\cos (2\mathrm{n\&pi;k}/N)+(a_k\sin \mathrm{k\&alpha;}+b_k(1-\cos \mathrm{k\&alpha;}))\sin (2\mathrm{n\&pi;k}/N))---\left(7\right)$

In like manner, r (n) is also deployable is the fourier series form:

$r\left(n\right)={\mathrm{<figure-callout\; id="0"\; label="r"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">r</figure-callout>}}_0+\underset{k=1}{\overset{S-1}{\mathrm{\&Sigma;}}}(e_k\cos (2\mathrm{n\&pi;k}/N)+f_k\sin (2\mathrm{n\&pi;k}/N))---\left(8\right)$

In the formula $r_{}$ ${}_0=\frac{1}{N}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}r\left(n\right)$

$e_k=\frac{2}{N}\underset{n=0}{\overset{n-1}{\mathrm{\&Sigma;}}}r\left(n\right)\cos (2\mathrm{n\&pi;k}/N)$

$f_k=\frac{2}{N}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}r\left(n\right)\sin (2\mathrm{n\&pi;k}/N)$

The harmonic components of k in comparison expression (7) and (8) 〉=2:

$\left\{\begin{array}{c}{e}_k=a_k(1-\cos \mathrm{k\&alpha;})-b_k\sin \mathrm{k\&alpha;}\\ f_k=a_k\sin \mathrm{k\&alpha;}+b_k(1-\cos \mathrm{k\&alpha;})\end{array}\right.---\left(9\right)$

Find the solution following formula:

$\left\{\begin{array}{c}{a}_k=\frac{1}{2}{e}_k+\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{f}_k\\ b_k=-\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{e}_k+\frac{1}{2}{f}_k\end{array}\right.---\left(10\right)$

Obtain the harmonic constant a of any k subharmonic _k, b _kAfter, again according to formula (2) and carry out discretize and handle the time domain discrete value g (n) can obtain roundness error of workpiece and be:

$g\left(n\right)={\mathrm{<figure-callout\; id="0"\; label="g"\; filenames="C20051000228700152.png,C20051000228700155.png"\; state="\{\{state\}\}">g</figure-callout>}}_0+\underset{k=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}((\frac{1}{2}{e}_k+\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{f}_k)\cos (2\mathrm{n\&pi;k}/N)+(-\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{e}_k+\frac{1}{2}{f}_k)\sin (2\mathrm{n\&pi;k}/N))$

Got by formula (1), the time domain discrete value z (n) of axial system error is:

z(n)＝g(n)-A(n) (11)

During roundness measurement, fundametal compoment in the harmonic component and first harmonic component (k=0,1) do not belong to the category of circularity.Therefore, g (n) can revise as follows:

$g^{\&prime;}\left(n\right)=\underset{k=2}{\overset{S-1}{\mathrm{\&Sigma;}}}(\left(a_k\cos \right)2\mathrm{n\&pi;k}/N)+b_k\sin (2\mathrm{n\&pi;k}/N)---\left(12\right)$

In like manner, when obtaining z (n), remove A (n), fundametal compoment among the B (n) and first harmonic component according to formula (11).A (n) is expanded into the fourier series form:

$A\left(n\right)=\frac{A_0}{2}+\underset{k=1}{\overset{S-1}{\mathrm{\&Sigma;}}}(h_k\cos (2\mathrm{n\&pi;k}/N)+l_k\sin (2\mathrm{n\&pi;k}/N))---\left(13\right)$

In the formula: $A_0=\frac{1}{N}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}A\left(n\right)$

$h_k=\frac{2}{N}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}A\left(n\right)\cos \left(2\mathrm{n\&pi;k}\right)$

$l_k=\frac{2}{N}\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}A\left(n\right)\sin (2\mathrm{n\&pi;k}/N)$

Even remove first-harmonic and first harmonic component k 〉=2, then have:

A″(n)＝A(n)-A ₀-(h ₁?cos(2nπ/N)+l ₁sin(2nπ/N))

Then the axial system error data are obtained by following formula:

z(n)＝g*(n)-A″(n) (14)

Transposition angle α's is preferred

Select any k (k 〉=2) subharmonic to discuss, get by formula (12):

C _k(n)＝a _kcos(2nπk/N)+b _k?sin(2nπk/N) (15)

In the following formula

$\left\{\begin{array}{c}{a}_k=\frac{1}{2}{e}_k+\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{f}_k\\ b_k=-\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{e}_k+\frac{1}{2}{f}_k\end{array}\right.---\left(16\right)$

From formula (16) as can be seen: when α=2n π/k (n=0,1 ... N-1) time, have

$\left\{\begin{array}{c}1-\cos \mathrm{k\&alpha;}=0\\ \frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}\&RightArrow;\&infin;\end{array}\right.---\left(17\right)$

A is arranged then _Ik→ ∞, b _Ik→ ∞, this subharmonic coefficient are infinitely great.

Order $P_i\left(k\right)=\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}---\left(18\right)$

P then _i(k) coefficient is infinitely great, causes this k time harmonic wave to be infinitely enlarged, and therefore this harmonic wave must be avoided when harmonic wave is synthetic.

The way that solves is to select suitable transposition angle α by numerical evaluation, makes P _iK value when (k) infinity appears in coefficient is greater than more than 50 times of the required harmonic wave of error separating.

Analyze for convenient, establishing transposition angle α is the integral multiple of the corresponding corner of adjacent two sampled points of circle contour measurement data, that is:

$\mathrm{\&alpha;}=\frac{2\mathrm{\&pi;}}{N}{n}_1---\left(19\right)$

N wherein ₁Be open position angle α corresponding sampling points number.

By formula (16) as seen, when k α ≠ 2 π J, the P of k subharmonic correspondence _i(k) just infinity can not occur, get by formula 19:

${\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="C20051000228700153.png"\; state="\{\{state\}\}">n</figure-callout>}}_1\&NotEqual;J\frac{N}{k}---\left(20\right)$

Wherein J is a positive integer.

In the roundness measuring equipment measuring system, N=1024, n is selected in k≤100 ₁=9, can satisfy above-mentioned requirements, this moment, corresponding transposition angle was:

When getting transposition angle α=3.164 °, P _i(k) and the relation between the k as shown in Figure 5.

At this moment, P _i(k) occur correctly to isolate this subharmonic and near harmonic wave thereof unusually at k=114,228,342,456,569,683,797 and 911 subharmonic places.

In the high precision roundness measuring instrument, the common contained harmonic component of measurement standard cylindrical workpiece mainly concentrates on the low frequency composition, and during roundness measurement, overtone order is selected for use to synthesize with interior harmonic wave for 50 times and got final product.And in single position shifter error separated, as long as suitable selection α, harmonic wave is unusual will to be appeared in the error separating beyond the needed 100upr harmonic wave.In the reality, when the transposition angle is chosen,, make k α ≠ 2 π J, all can realize the accurate separation in the 100upr harmonic wave scope as long as by suitable selection.

Emulation and measured data checking

For verifying the validity and the correctness of single transposition method error separating method separating effect in the 1-100upr filter range, take following verification method:

Select two groups of data of overtone order k≤100, one group as workpiece data g (n) (n=0,1 ... k ... N-1), another group is as main shaft data z (n) (n=0,1, k ... N-1), at first above-mentioned two groups of data correspondences are subtracted each other, obtain composition error data A (the n) (n=0 of first survey time, 1 ... k ... N-1).Then, preferred n ₁, with workpiece data g (n) translation n ₁Individual data point obtains g (n+n ₁), g (n+n ₁) subtract each other with main shaft data z (n) again, obtain second survey time composition error data B (n) (n=0,1 ... k ... N-1).With A (n), B (n) and n ₁In the substitution single-transposition roundness fault separating method, obtain workpiece data s (n) and main shaft data e (n) after the error separating.

Calculating through s (n) and g (n) pointwise subtracting each other difference DELTA s (n) is:

Δs(n)＝s(n)-g(n) (21)

Root-mean-square deviation RMS (s) between s (n) and g (n):

$\mathrm{RMS}\left(s\right)=\sqrt{\left[\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{(s\left(n\right)-g\left(n\right))}^2\right]/N}---\left(22\right)$

Calculate workpiece and separate forward and backward relative error δ (s):

$\mathrm{\&delta;}\left(s\right)=\frac{|{\left[\mathrm{\&Delta;s}\left(n\right)\right]}_{\mathrm{Max}}-{[\mathrm{\&Delta;s}\left(n\right)}_{\mathrm{Min}}|}{|{[g\left(n\right)}_{\mathrm{Max}}-{\left[g\left(n\right)\right]}_{\mathrm{Min}}|}\&times;100\%---\left(23\right)$

Calculate e (n) and z (n) pointwise subtracting each other difference DELTA e (n):

Δe(n)＝e(n)-z(n) (24)

Root-mean-square deviation RMS (e) between e (n) and z (n):

$\mathrm{RMS}\left(e\right)=\sqrt{\left[\underset{i=1}{\overset{N}{\mathrm{\&Sigma;}}}{(e\left(n\right)-z\left(n\right))}^2\right]/N}---\left(25\right)$

Calculate main shaft and separate forward and backward relative error δ (e):

$\mathrm{\&delta;}\left(e\right)=\frac{|{\left[\mathrm{\&Delta;e}\left(n\right)\right]}_{\mathrm{Max}}-{[\mathrm{\&Delta;e}\left(n\right)}_{\mathrm{Min}}|}{|{[z\left(n\right)}_{\mathrm{Max}}-{\left[z\left(n\right)\right]}_{\mathrm{Min}}|}\&times;100\%---\left(26\right)$

By comparing the size of Δ s (n), δ (s), RMS (s), Δ e (n), RMS (e) and δ (e) value, estimate the validity of single-transposition roundness fault separating method error separating method.

Measured data emulation

Select for use actual measurement as Fig. 6 and Fig. 7, Fig. 8 and the abundant standard ball circle contour data of two groups of harmonic waves shown in Figure 9, carry out the separate authentication experiment.With Fig. 6 and data shown in Figure 7 as g (n), wherein, the raw data profile stretch-out view of Fig. 6 for not carrying out any processing, Fig. 7 is its circle contour figure in the 1-100upr filter range, its Minimum Area circularity value is 24nm; As z (n), Fig. 8 is the raw data profile stretch-out view for not carrying out any processing also with Fig. 8 and data shown in Figure 9, and Fig. 9 is its circle contour figure in the 1-100upr filter range, and its Minimum Area circularity value is 17nm.

Preferred n ₁=9 by z (n) and g (n) formation A (n), B (n), and A (n), B (n) are separated with single-transposition roundness fault separating method.

Suppose Δ α=0.00 °, utilize single-transposition roundness fault separating method to isolate s (n) and e (n), calculate Δ s (n) and Δ e (n), corresponding data and curves as shown in Figure 10 and Figure 11, its corresponding RMS (s) ≈ 0.000nm, δ (s) ≈ 0.000%, RMS (e) ≈ 0.000nm, δ (e) ≈ 0.000%, in the 1-100upr filter range, s (n) is respectively 24nm and 17nm with e (n) Minimum Area circularity value.

When supposing Δ α=0.01 °, utilize single-transposition roundness fault separating method to isolate s (n) and e (n), calculate Δ s (n) and Δ e (n), corresponding data and curves such as Figure 12 and shown in Figure 13, its corresponding RMS (s) ≈ 0.158nm, δ (s) ≈ 1.480%, RMS (e) ≈ 0.158nm, δ (e) ≈ 1.060%.In the 1-100upr filter range, s (n) is respectively 24nm and 17nm with e (n) Minimum Area circularity value.

Compare Figure 10 and Figure 11 and Figure 12 and Figure 13, to the measured data in the 1-100upr filter range, because the influence of transposition angle error Δ α, cause the single-transposition roundness fault separating method separation accuracy to reduce, Δ s (n) and Δ e (n) value become big, but RMS (s) and RMS (e) are still less than 0.158nm, and δ (s) and δ (e) are still less than 1.480%.Workpiece is still equal with the circularities of axle before and after separating and equal 24nm and 17nm respectively.

Show by the experiment of measured data separate authentication: transposition angle α and Δ α thereof are the keys that influences the single-transposition roundness fault separating method separation accuracy.When Δ α=0 °,, can realize the separation fully of signal by preferred α value; When Δ α＜0.01 °, to the abundant measured signal of harmonic wave, single-transposition roundness fault separating method also can accurately separate the harmonic signal in its 1-100upr scope.

Claims (1)

1. single-transposition roundness fault separating method is characterized in that this method may further comprise the steps:

1) separating and measuring is carried out in certain measurement cross section of selected measured workpiece (2), sensor (1) records the composition error A (n) of the turn error z (n) that comprises roundness error of workpiece g (n) and main shaft (3) that is in initial index position a place, n=0 wherein, 1, ..., N-1, N are all sampling numbers of workpiece (2) circle contour;

2) the relative main shaft (3) of workpiece (2) is turned over angle [alpha] and arrive index position b, again the workpiece (2) that is in index position b is measured, same sensor (1) records the composition error value B (n) in the b transposition that comprises roundness error of workpiece g (n) and spindle rotation error z (n), n=0 wherein, 1, ..., N-1;

3) separate carrying out deviation from circular from the composition error A (n) that records on a, the b position before and after transposition angle α and workpiece (2) the single transposition and B (n) the substitution single-transposition roundness fault separating method, concrete steps are as follows:

1. remove the DC quantity of sampled signal A (n), B (n);

2. calculated difference signal r (n);

$r\left(n\right)=\underset{k=2}{\overset{S-1}{\mathrm{\&Sigma;}}}((a_k(1-\cos \mathrm{k\&alpha;})-b_k\sin \mathrm{k\&alpha;})\cos (2\mathrm{n\&pi;k}/N)+(a_k\sin \mathrm{k\&alpha;}+b_k(1-\cos \mathrm{k\&alpha;}))\sin (2\mathrm{n\&pi;k}/N))$

Wherein, a _k, b _kBe the fourier progression expanding method coefficient of roundness error of workpiece g (n), S is an overtone order.

3. r (n) is carried out frequency analysis, ask its harmonic constant e _kAnd f _k

$\left\{\begin{array}{c}{e}_k=a_k(1-\cos \mathrm{k\&alpha;})-b_k\sin \mathrm{k\&alpha;}\\ f_k=a_k\sin \mathrm{k\&alpha;}+b_k(1-\cos \mathrm{k\&alpha;})\end{array}\right.$

4. calculate g (n) harmonic constant a _kAnd b _k

$\left\{\begin{array}{c}{a}_k=\frac{1}{2}{e}_k+\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{f}_k\\ b_k=-\frac{\sin \mathrm{k\&alpha;}}{2(1-\cos \mathrm{k\&alpha;})}{e}_k+\frac{1}{2}{f}_k\end{array}\right.$

5. remove fundametal compoment and first harmonic component among the A (n);

A″(n)＝A(n)-A ₀-(h ₁cos(2nπ/N)+l ₁sin(2nπ/N))

Wherein, A ₀Be fundamental component, h ₁And l ₁Be A (n) fourier progression expanding method coefficient.

6. calculate the synthetic g (n) of harmonic wave;

$g\left(n\right)=\underset{k=2}{\overset{S-1}{\mathrm{\&Sigma;}}}\left((a_k\cos (2\mathrm{n\&pi;k}/N)+b_k\sin (2\mathrm{n\&pi;k}/N)\right)$

7. calculate axial system error z (n);

z(n)＝g(n)-A″(n)

4) g (n) and z (n) are carried out roundness evaluation in the substitution roundness measurement assessment system respectively, obtained rejecting the circularities of workpiece (2) g (n) of turn error and the circularities of main shaft (3) turn error z (n).

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