Error Analysis of Form and Position Error Measurement-EEWORLD

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I. Introduction

As the main measurement means of form and position errors, the existing roundness instruments at home and abroad and the form and position error measuring instruments developed on the basis of roundness instruments (such as shape error measuring instruments, shape measurement systems, etc.) can obtain more accurate form and position error measurement results than traditional measuring instruments and measurement methods. However, in the use of these instruments, due to improper adjustment or inappropriate expansion of their scope of use, large measurement errors may occur. In the development of new form and position error measuring instruments, there are also problems such as failure to ensure the manufacturing accuracy and adjustment accuracy of the corresponding components of the instrument according to the design function requirements, or blindly pursuing high manufacturing accuracy, thereby excessively increasing manufacturing costs. This paper analyzes the system error and workpiece installation error of the form and position error measuring instrument, and studies the influence of these error factors on the measurement accuracy of the form and position error, thereby providing a theoretical basis for reducing the measurement cost while ensuring the measurement accuracy and the development of the form and position error measuring instrument.

II. System Error Analysis of Form and Position Error Measuring Instrument

The existing form and position error measurement system is mainly composed of mechanical part, microcomputer hardware part and measurement software part.

To ensure the accuracy of data processing, the measurement software mostly uses double-byte fixed-point number operation or three-byte floating-point number operation method, so the accuracy of the software part is generally not less than 0.01%. The accuracy of the microcomputer hardware part mainly depends on the accuracy of the forward channel. By analyzing the technical characteristics of each part such as the sensor device, amplification and filtering circuit, sampling and holding circuit and A/D analog-to-digital conversion circuit, the limit error of each circuit is obtained, and the Gaussian method is used to synthesize it. It can be seen that the total error value of the hardware device does not exceed 0.2%. Therefore, the microcomputer hardware and software errors of the measuring instrument (excluding the principle error of data processing) are very small and can be ignored. The measurement accuracy of the measuring instrument mainly depends on the accuracy of the mechanical part.

1. The rotation accuracy of the measuring instrument

During the rotation process, the relative displacement of the rotation axis to the average position of the axis is the rotation error motion. The error motion causes the rotation axis to move parallel or perpendicular to the axis at each instant. The former is called the end face error motion, and the latter is called the radial rotation error motion. The

end face error motion makes the sampling points of the measured workpiece within one rotation not all in one cross section, thereby reducing the correlation between the sampling points. However, since the end face error motion is generally very small, and the actual workpiece surface to be measured is smooth, the probe cannot be in pure point contact when sampling the measured surface, but in small area contact. Therefore, the impact of the end face error motion on the measurement accuracy can be ignored. The

rotation accuracy of the rotary table measuring instrument is mainly determined by the rotation accuracy of the measuring instrument spindle, while the rotation accuracy of the center clamping measuring instrument is jointly determined by the center accuracy of the measuring instrument and the shape accuracy of the center hole of the measured workpiece. The

radial rotation error δr will be directly transmitted to the sampling data Δri (i = 1, 2, 3...n), thereby affecting the calculation accuracy of the least squares center coordinates. The expression of the least squares center coordinates is [1]

newmaker.com (1)

Where (a, b) is the least squares coordinate of the center of the measured cross section, θi is the sampling angle, R is the average circle radius, and n is the number of sampling points.

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Therefore, radial rotation accuracy is the most important accuracy indicator of the form and position error measuring instrument.

2. Axial guide rail straightness error

(1) Axial guide rail straightness error in the plane where the axis of rotation is located
This error will be reflected in the measurement result at a ratio of 1:1. However, for the sampling data of the same section, it is equivalent to only a fixed error ΔSr. If the sampling method of even-numbered points with equal intervals is adopted, it can be known from formula (1) that

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Similarly, b′j = bj, so this error will not change the position of the center of the section. Therefore, for a form and position error measuring instrument that uses data processing software that conforms to the corresponding error definition, the guide rail straightness error in this direction will affect the measurement accuracy of the cylindrical error and element straightness error of the measured workpiece, but will not affect the measurement accuracy of the roundness, coaxiality, and axis straightness errors [2, 3].

(2) Guide rail straightness error perpendicular to the plane where the axial guide rail and the rotation axis are located

This error ΔShj will cause the probe to deviate from the radial direction, as shown in Figure 1, so that the measured radius increment is

newmaker.com (4)

Obviously, ΔShj is very small compared to r0, and the effect of this error on the measurement accuracy can be ignored.

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Figure 1 Error caused by the probe deviating from the radial direction

3. Parallelism error between the axial guide rail and the rotary axis of the measuring instrument

As analyzed in the previous section, the parallelism error perpendicular to the plane where the axial guide rail and the rotary axis are located is a non-sensitive error and can be ignored. The following only analyzes the influence of the parallelism error in the plane where the axial guide rail and the rotary

axis are located. When the rotary axis of the measuring instrument is not parallel to the axial movement direction of the probe, its parallelism error will be repeatedly reflected in the sampling data at a ratio of 1:1. As shown in Figure 2, assuming that the angle between the guide rail and the rotary axis is α, and the section spacing is Z, then the compression of the probe on the kth section is Δrz＝kZtgα. Obviously, this parallelism error is a linear system error, and for the sampling data of the same section, it is equivalent to a fixed error ΔSk, which will not change the position of the section center coordinates. Therefore, for measuring instruments that use data processing software that meets the corresponding error definition, this error only affects the cylindrical error evaluation result, and will not affect the measurement accuracy of roundness, coaxiality, axis and element line straightness errors［2,3］.

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Figure 2: The axial guide rail is not parallel to the rotation axis

4. Sampling angle error

If the actual deviation of each sampling point from the least square circle is εij, then [1]

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Therefore, there is

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In formula (7), daj, dbj are the effects of sampling angle error on the least squares center coordinates of the cross section. From formula (1), we can get

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Since the measured surface is smooth and the probe contacts the measured surface with a small area, when the error of the sampling angle θij is small, the influence d(Δrij) on the sampling data can be ignored. Therefore,

newmaker.com (9)

Take the number of sampling points n = 128, when the maximum sampling angle error dθjmax = 0.5° = 0.009rad, daj = dbj < 0.001Δrjmax. It can be seen that the influence of the sampling angle error on the least squares circle center coordinates can be ignored.

Similarly, ignoring d(Δrij), substituting equation (9) into equation (7) yields

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It can be seen that the sampling angle error has little effect on the measurement result and can be ignored.

3. Workpiece installation error analysis

The installation error of the workpiece includes the installation eccentricity error and the installation tilt error.

1. Workpiece installation eccentricity error

When the analytical evaluation method is used to solve the form and position error, the error caused by the installation eccentricity e on the polar diameter at each sampling point is [1]

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Where R is the radius of the workpiece being measured. As can be seen, as long as a certain installation accuracy is guaranteed and the measuring range is not exceeded during the measurement process, this error is very small and can be ignored.

When a traditional roundness meter that records contour graphs is used for measurement, the image distortion error caused by the installation eccentricity e is [4]

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Where M is the magnification of the recorded contour figure. At this time, the measurement error caused by the installation eccentricity e is large and cannot be ignored. Usually, e should be ≤ 7% (R/M).

2. Workpiece installation tilt error

The verticality error of the workpiece installation base surface to the axis or the presence of foreign matter between the workpiece installation base surface and the worktable surface will cause the workpiece installation tilt error.

For the convenience of analysis, it is assumed that the workpiece is an ideal cylinder with a diameter of 2R, and the inclination angle of its axis to the rotation axis is γ, as shown in Figure 3. Due to the inclination of the workpiece, the profile of its measured cross-section is an ellipse, and the major and minor axes of the ellipse are 2Rsecγ and 2R respectively. The measurement error caused by the inclination of the workpiece installation is δt＝R（secγ－1）. If the installation inclination height difference t＝0.1mm, R＝25mm, then γ＝0.115°, δt＝0.05μm. Therefore, under the condition of ensuring that the inclination of the workpiece is small, δt can be ignored.

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Figure 3: Workpiece installation tilted

When the workpiece is installed at an angle, a second harmonic component is superimposed on the sampling data of each sampling point, which has no effect on the calculation accuracy of the least squares center coordinates of the measured cross-sectional profile.

Therefore, for measuring instruments that use analytical evaluation data processing software, the workpiece installation inclination error has little effect on the form and position error evaluation results and can be ignored; but for traditional roundness instruments that use the recorded contour image evaluation method, the recorded contour image is F = M2R (secγ-1), which shows that the workpiece installation inclination error has a greater impact on the measurement results, especially this inclination error will also affect the center position of the recorded contour of each sampling section, so it has a greater impact on the measurement results of various form and position errors.

4. Conclusion

The accuracy of form and position error measuring instruments mainly depends on the accuracy of the mechanical part, among which the rotation accuracy is the most important accuracy indicator; the straightness error of the axial guide rail will affect the evaluation results of the cylindrical error and the element line straightness error of the measured workpiece; the parallelism error of the axial guide rail to the rotation axis mainly affects the measurement accuracy of the cylindrical error. (end)

Reference address：Error Analysis of Form and Position Error Measurement

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