Mathematical modelling and analysis of profile meter

The profiler measures the workpiece to reflect the contour curve shape of the surface of the workpiece. When measuring the same workpiece, the drawn contour curve will shift due to the different placement angle and horizontal position. In order to improve the measurement accuracy of the profiler, the paper proposes a new modelling and calculation method for the profiler. This method uses mathematical methods such as vectors and geometric relations to calculate parameters; then completes level correction by changing the coordinate system; finally, uses geometric methods and analogy inference methods to obtain the deviation between the calculated values of each parameter. The research results show that this method can realize the measurement of the profiler more accurately.

For the first problem, we use Excel to process and integrate the data in Annex I, and draw a scatter diagram. We use Pythagorean theorem to calculate the radius of arc and the length of oblique line segment, and use vector to calculate the relevant angle. We also use mathematical methods such as geometric relationship to calculate the width of notch, the distance between centers, the length of circular arc, the length of horizontal line segment, the angle between oblique line and horizontal line The value of the herringbone height parameter.
For the second problem, we process, analyze and integrate the data from Annex I down, draw the scatter diagram, complete the horizontal correction by changing the coordinate system, and obtain the deviation between the calculated values of parameters through data operation and analogy reasoning.
For the the third problem, we analyze and integrate the data of Annex II to obtain the scatter diagram of each group. According to the data and the drawn image, we can calculate the tilt angle, the parameter values, and draw the complete contour line of workpiece 2.

Assumptions of the model
Symbol description is shown in Table 1. In order to make the problem easier to understand, we make the following assumptions: 1. It is assumed that the contact profilometer has no problems such as probe contamination, probe defect and inaccurate scanning position.
2. It is assumed that the test frequency of the profilometer remains unchanged before and after the profilometer.
3. It is assumed that the worktable is flat and even. 4. Assume that the center angle of the arc in the figure is 180 degrees.   According to figure A2, the height of herringbone line is as follows:  Fig. A2 and Pythagorean theorem, the length of oblique line segment is as follows: According to figure A2 and vector angle formula: The angle between the straight line and the oblique line can be obtained as follows: According to Figure A2 and the model hypothesis,by using the formula L r   , the length of the circle can be obtained as follows: Figure A2 , it can be concluded that the distance between the centers of circles is:

4.2.Establishment and solution of the second problem model
Directly use Excel to convert the down data of Annex I table into scatter plot, as follow the figure 3: According to the imaging principle of profilometer and the analysis of Annex I, the radius of the arc of the contour line obtained by the same workpiece under different angles and different displacements is constant, and then the length of the arc is constant, and the height of the herringbone line is constant.
To sum up, the distance between the centers of the circle after correction follows:

Establishment and solution of the third problem model
Through the analysis and integration of the data in Annex II by Excel, we can get the following results from the geometric relationship and Pythagorean theorem: After processing each group of data in Annex II, the complete data of workpiece II are summarized according to the tilt angle, and then the complete contour line is drawn, as shown in Figure 4:

Summary
Based on the principle of profilometer, this paper analyzes the parameters of the image. In the process of solving and analyzing the model, we use a lot of mathematical formulas and geometric relations to support the established mathematical model. Using Excel to simplify the data processing, the data information and formula are verified each other, which makes the results more meticulous. However, there are some problems in the contact profilometer, such as probe contamination, probe defects, inaccurate scanning position and so on, which can not be ignored, and the center angle of circular arc in the figure is also uncertain. We should carefully select the instruments needed for measurement, so as to minimize the influence of the instrument itself on the error, so as to make the trajectory curve closer to the ideal model. We should use more mathematical software to provide more powerful data support for the established mathematical model, and make full use of the data information provided by its attachments to make the results more accurate, so as to achieve the purpose of reducing errors.