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Some problems on the separation method of roundness and shafting error by multi-point method

the project supported by the special scientific research fund of doctoral discipline points in Colleges and universities allows the product itself to give full play to the three variables of roundness and shafting motion error measurement: the roundness error of workpiece and the motion error of shafting in X and Y directions. Using three independent measurement equations obtained by three probes, roundness and shafting motion error can be measured at the same time [1], This is the idea of three-point method in the separation method of roundness and shafting measurement error [1]. However, the roundness measured by the three-point method sometimes has large shape distortion. In order to solve this problem, predecessors have devoted a lot of energy [2 ~ 5]. The author puts forward the four point method [6], that is, four measuring equations are obtained by using four probes, which not only realizes the separation of roundness and shafting measurement errors, but also can effectively suppress the influence of probe reading and positioning errors by using the redundant measurement information, so as to greatly improve the roundness measurement accuracy. The idea of 4-point method can be extended to 5-point method, 6-point method and even N-point method

1 error separation method and measurement error analysis

1.1 error separation method

n probes PI are arranged on the circumference of the workpiece (I = 1, which is very suitable for this kind of satellite manufacturing ~ n), and the included angles between each probe and the y-axis are I 1 = 0 respectively (as shown in Figure 1). The roundness of the workpiece is R (), and the X and y components of the radial motion error of the shafting are x (), y (), respectively. The output of each probe is (

Si () = ri-r (- I) cos I) sin I, (1) + X () - Y (

Fig. 1 principle of multipoint method

in which RI is the zero reading number of the probe PI. Construct the linear equation

(2)

to expand R () into Fourier series, with

(3)

in which AK and BK are the Fourier coefficients of each harmonic of R ()

If

(4)

, then

(5)

is separation x (), y (), so that x = y = 0, that is, when k = 1, tense (4) is zero

Expand s () into Fourier series, there are 11

(6)

where FK and GK are the Fourier coefficients of each harmonic of S (), and

(7)

can be obtained from equation (5) (6), and the roundness can be obtained from AK and Bk. The radial motion error x (), y () of shafting can be calculated by the least square method according to formula (1)

1.2 measurement error analysis

there are two main errors in the measurement process: ① probe reading error, which affects FK and GK; ② Probe angular position error, which affects XK YK. By differentiating equation (7),

(8)

in which the influence of reading error is dominant. Assuming that the variances of each probe are 2, the k-th Fourier coefficient AK of roundness introduced by the reading error of the probe is obtained through error analysis. The variance of the measurement error of BK is

2 (AK) = 2 (BK) = 2q2k2, (9)

Where is the error transfer factor, indicating the influence of the reading error of the probe on the accuracy of roundness measurement

error analysis also shows that the influence of probe angular position error on circle measurement accuracy is directly proportional to QK. Therefore, in order to reduce the influence of probe reading error and probe angular position error on the measurement results, QK should be minimized

2 characteristics of multi-point method

QK in three-point method is only related to harmonic level K and probe angle position {I. at this time, in order to determine the best probe angle position, calculate {I for the given harmonic upper limit km, so that the maximum value of QK of all harmonics is the minimum. {I after determination, each harmonic will get the same synthesis coefficient {AK}, and then determine AK and Bk

when n> 3, there are (n-3) redundant degrees of freedom, and whether the array {AI} satisfies the equations. At this time, QK is related not only to K, I, but also to the redundant variables in {AI}. Even if {I has been determined, there are still (n-3) degrees of freedom. In this way, these redundant degrees of freedom can be used to minimize the error transfer factor of each harmonic, and then determine a set of optimal angular position {I, so that the minimum qkmi of QK of all harmonics is the minimum of the maximum value of sliding friction n. that is, under the optimal angular position, a set of optimal composite coefficient {AIK} structural formula (2) can be determined for each harmonic, Thus, the error transfer factor of each harmonic is further reduced.}

3 Relationship between the number of probes and measurement accuracy

3.1 analysis of calculation results

① in the same method, when the upper harmonic limit km is different, the QK of each corresponding harmonic is different. With the increase of KM, the optimal value of the objective function becomes larger, and the optimal angular position obtained by optimization is also different, as shown in Figure 2

Figure 2 Relationship between error transfer factor and harmonic level of multi-point method

② after optimization with three-point method, the measurement error of each harmonic is still large (see figures 2a and 2e); Compared with the 3-point method, the 4-point method greatly reduces the objective function value and the error transfer factor of each harmonic (see Fig. 2B, 2e and 2f)

③ with the increase of the number of probes, when the same harmonic upper limit is taken, the value of the objective function and the error transfer factor of each harmonic will further decrease, but the reduction amplitude will slow down. Compared with the three-point method, the maximum measurement error of each harmonic is reduced by 46% ~ 67%; Compared with the 4-point method, the 5-point method decreased by 14% ~ 29%; Compared with the 5-point method, the decrease range of the 6-point method is 8% ~ 26%. Due to the influence of factors such as probe sensitivity and other inconsistent performance, it can be considered that four probes are the best choice

3.2 comparison of measured results

under the same measurement conditions, the author adopts the traditional 3-point method (the angle position of the probe is not optimized according to the method in this paper), the optimized 3-point method, 4-point method, 5-point method and 6-point method respectively, and compares them with the measurement results of roundness instrument. The results show that with the increase of the number of probes, the accuracy of roundness measurement results improves and the measurement uncertainty decreases. The results are shown in the table below

comparison table of measurement results of six methods

4 Conclusion

3-point method can not further improve the measurement accuracy, which is determined by the limitation of the principle of 3-point method. Compared with the 3-point method, the 4-point method adds a probe, and the roundness measurement accuracy is significantly improved. With the addition of probe on the basis of 4-point method, the measurement accuracy will not be significantly improved

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