Optimization of the 2 ½ D Processing Method of Complex Parts, through a Predictive Algorithm for Controlling the Geometric Shape Deviations Resulting from Processing
Abstract
:1. Introduction
1.1. The Importance of Processing Methods with Numerical Control
- The new mathematical concept of optimal surface generation in 2 ½ D with control over the accuracy of the geometric shape resulting from the processing.
- A new algorithm for the mathematical concept for the optimal generation of the surface in 2 ½ D with a new program that was made for the optimal generation with preliminary control of the deviation of the geometric form when processing the complex surfaces in space.
- Applied experimental research on optimal modelling of surface generation in 2 ½ D on production technology systems equipped with numerical control;
- Determination based on the 2D modelling of the characteristic points of the profile piece and of the equidistant curve with an original soft;
- Determination based on optimal modelling in 2 ½ D of the characteristic coordinates of the level planes used and the points corresponding to the spatial contours of the equidistant curve so that the deviations from the resulting geometric shape should fall within the required limits;
- Experimental results.
1.2. General Context
2. The Mathematical Concept of Optimal Surface Generation in 2 ½ D
2.1. The Concept of Optimal Surface Generation in 2 ½ D with Predictive Control of the Geometric Shape Deviations Resulting from the Processing
- = straight segment (forms the angle α1 with the OX axis)
- = arc of a circle, of radius R2 (may be tangent or not to neighboring segments)
- = straight segment (forms the angle α3 with the OX axis)
- P4P5 = arc of a circle, of radius R4 (may be tangent or not to neighboring segments)
- P5P6 = arc of a circle, of radius R5, (may be tangent or not to P4P5 arc), with the observation that in the case of tangent P5 will be the inflexion point. The arc is tangent for technological reasons at the bottom surface of the piece.
2.2. Optimal Modelling of Concave Surface Generation
- Determine the law of dependence of the optimum distance between the planes of the profile geometry (function Δ2,i opt);
- Calculate the successive positions of the planes.
2.3. Optimal Modelling of Convex Surfaces and Marginal Surfaces
3. Application—The Result of Optimization for a Real Piece–Extract for New Surfaces
3.1. Case Study
- = 60-degree straight segment
- P2 P3 = 100 mm concave circle tangent to segment P1 P2 and P3 P4
- = 30-degree straight segment tangent to P2 P3
- P4 P5 = arc of 100 mm radius convex circle tangent to segment P3 P4 and
- P5 P6 = 100 mm radius concave arc tangent to the P4 P5 arc
- = horizontal straight line tangent to the P5 P6.
- P3, E3 = points defined by a circle and an angle (module 1 option 4, original algorithm and soft ([6] Rece)
- P4, E4 = points defined on the structure of a given straight line (Module 5 Option 3)
- P5, E5 = points defined by a circle and an angle (module 1 option 4)
- P5, E5 = points defined by a circle and an angle (module 1 option 4)
- P6, E6 = points defined by the right/circle intersection (module 1 option 3, all calculated with ([6])
3.2. Results of Optimisation Based on the Optimal Modelling in 2 ½ D of the Characteristic Coordinates of the Level Planes and the Points Corresponding to the Spatial Contours of the Equidistant–Extract
- P2P3 = concave arc of a circle
- P3P4 = straight segment
- P4P5 = convex arc of a circle
4. Conclusions
- Identifying the characteristic sections of the complex surface, A-A, A′-A′, A″-A″, and so on and for each other the set of the corresponding characteristic points {Pi}, {Pi′} etc., as well as the types of segments corresponding to Pi, Pi+1.
- Identify the defining dimensions for each segment (Pi, Pi+1) using the execution drawing (angles, rays, interpolation centers, lengths, etc.).
- The calculation for each segment of optimal processing distances Δi,opt using the specific relationships determined for each type, taking into account the following observations:
- (a)
- In the case of straight segments, the optimum distance between the planes is unique (depending on the required form tolerance, the radius at the tip of the tool and the segment angle with the OX axis). This value is used for positioning all processing planes in space, resulting in uniform formulas.
- (b)
- In the case of some concave or convex circle segments, two solutions with specific relations can be used, namely Case no. 1, and Case no.2:
- Case no. 1 involves the determination of a single value Δi,opt, for the entire segment, a value that results in the accuracy of the required form. This case is simpler in terms of solution. Case no. 1 leads to accuracy by using several processing planes greater than the strictly necessary.
- Case no. 2 takes into account the spatial position of the processing plane versus the generating curve, and leads to superior results by obtaining absolute values for the distances between the processing planes, according to their position regarding the analyzed segment. It ultimately presents the advantage of obtaining under the conditions of minimizing the number of processing planes, deviations from the given shape of the surface, uniforms and within the accepted limits.
- (c)
- In the case of particular circle segments and end segments tangent to frontal plane surfaces or other types of special surfaces, it is recommended to resort to case no. 2, the first case leading to an oversized plane number, which ensures the shape of the imposed surface, but with low productivity. This situation is also avoided in terms of the final stage of compiling numerical control programs, a stage which in this case would implicitly lead to the development of more complex programs of long duration.
- The calculus of the coordinates of the intermediate characteristic points Pi,j of the profile, on each segment Pi, Pi+1, obtained by processing in successive Pi,j+1 at the optimal distances determined previously. These coordinates are determined from almost using the recurrent relationships presented and the definition data of each segment.
- Determination of the coordinates of the intermediate characteristic points (Ei,j) of equidistance. At each characteristic intermediate point on the profile (Pi,j), generally corresponds to a single intermediate point characteristic on the equidistant (Ei,j) which is determined by the position of the characteristic section relative to the complex surface area of the piece, the level curve and tool parameters with specific mathematical relationships.
- Finally, the execution of N.C. l programs for each processing plan based on the sets {Ei,j} and {E′i,j}.
Author Contributions
Funding
Conflicts of Interest
References
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P3 (x,z) = P3 (151.851, −56.603) | |
KNOWING THE POINT OF A CIRCLE DEFINED BY ANGLE A (ENCODE P/CA) | |
ENTER THE COORDINATES OF THE CIRCLE | |
CENTRE | XC = 101.851 |
YC = 30 | |
RADIUS | R = 100 |
ANGLE (grade/horizontal) | A = −60 |
COORDINATES OF THE POINT: | X = 151.851 |
Y = −56.603 | |
E3 (x, z) = E3 (146.851, −47.942) | |
KNOWING THE POINT OF A CIRCLE DEFINED BY ANGLE A (ENCODE P/CA) | |
ENTER THE COORDINATES OF THE CIRCLE | |
CENTRE | XC = 101.851 |
YC = 30 | |
RADIUS | R = 90 |
ANGLE (grade/horizontal) | A = −60 |
COORDINATES OF THE POINT: | X = 146.851 |
Y = −47.942 | |
P4 (x, z) = P4 (134.531, −66.603) | E4 (x, z) = E4 (129.531, −57.942) |
P5 (x, z) = P5 (97.928, −103.205) | E5 (x, z) = E5 (89.268, −98.205) |
C4 (x, z) = C4 (184.531, −153.205) | C5 (x, z) = C5 (11.325, −53.205) |
P6 (x, z) = P6 (11.325, −153.205) | E6 (x, z) = E6 (11.325, −143.205) |
P7 (x, z) = P7 (1.325, −153.205) | E7 (x, z) = E7 (1.325, −143.205) |
SIGNATURE: Intermediate and Endpoint Coordinates on Segment P1P2 |
---|
ENTER THE COORDINATES OF THE POINT |
XC = 200 |
YC = 0 |
THE COORDINATES OF THE INTERMEDIATE POINTS ON THE LINE P1P2 |
NO = 1 X = 198.716 Y = −2.223 |
NO = 2 X = 197.434 Y = −4.445 |
NO = 3 X = 196.867 Y = −6.667 |
NO = 4 X = 194.867 Y = −8.889 |
NO = 5 X = 193.585 Y = −11.112 |
NO = 6 X = 192.302 Y = −13.334 |
NO = 7 X = 191.018 Y = −15.556 |
NO = 8 X = 189.735 Y = −17.778 |
NO = 9 X = 188.453 Y = −20.000 |
SIGNATURE: Intermediate and endpoint coordinates on segment E1E2 |
ENTER THE COORDINATES OF THE POINTS |
XC = 191.339 |
YC = 5 |
THE COORDINATES OF THE INTERMEDIATE POINTS ON THE LINE E1E2 |
NO = 1 X = 190.055 Y = 2.777 |
NO = 2 X = 188.773 Y = 2.555 |
NO = 3 X = 187.490 Y = −1.657 |
NO = 4 X = 186.207 Y = −3.989 |
NO = 5 X = 184.824 Y = −5.112 |
NO = 6 X = 183.641 Y = −8.334 |
NO = 7 X = 182.338 Y = −10.556 |
NO = 8 X = 181.074 Y = −12.778 |
NO = 9 X = 179.792 Y = −15.000 |
Angle α2,i | Optimal Distance D2iopt | Coordinates on the X axis Xei | Coordinates on the Z axis Zei |
---|---|---|---|
1.046 | 2.33388 | 179.7394 | −15.09331 |
1.015766 | 2.291968 | 178.3406 | −17.42713 |
0.9855136 | 2.47933 | 176.871 | −19.71917 |
0.9552408 | 2.29181 | 175.3318 | −21.9671 |
0.9249472 | 2.153634 | 173.724 | −24.1689 |
0.8946319 | 2.103446 | 172.0491 | −26.32254 |
0.8642936 | 2.051285 | 170.3083 | −28.42598 |
0.8339312 | 1.997192 | 168.5031 | −30.47727 |
0.8035431 | 1.94121 | 166.6348 | −32.47446 |
0.7731281 | 1.883384 | 164.705 | −34.41567 |
0.7426839 | 1.823758 | 162.7151 | −36.29907 |
0.7122089 | 1.762378 | 160.6667 | −38.1228 |
0.6017008 | 1.699293 | 158.5613 | −39.88519 |
0.6511568 | 1.634549 | 156.4006 | −41.58449 |
0.620571 | 1.568195 | 154.1862 | −43.21904 |
0.5899486 | 1.509279 | 151.9197 | −44.78723 |
0.5592769 | 1.43085 | 145.9197 | −46.28752 |
0.5285536 | 1.349955 | 145.6026 | −47.71837 |
ENTER THE COORDINATES OF THE POINTS E3E4 |
---|
XC = 146.851 |
YC = −47.942 |
THE VALUE OF THE ANGLE DEGREES = 210 |
THE LENGTH OF AN INTERVAL = 2.875 |
NUMBER OF INTERVALS = 7 |
SIGNATURE: The intermediate and the endpoint coordinates on segment E3E4 |
NO = 1 X = 44.376 Y = −49.371 |
NO = 2 X = 141.902 Y = −50.799 |
NO = 3 X = 139.428 Y = −52.228 |
NO = 4 X = 136.954 Y = −53.657 |
NO = 5 X = 134.479 Y = −55.085 |
NO = 6 X = 132.005 Y = −56.514 |
NO = 7 X = 129.531 Y = −57.942 |
Angle | Optimal Distance | Coordinates on the X Axis | Coordinates on the Z Axis |
---|---|---|---|
α4,i | Δ4ιοπτ | Xei | Zei |
0.523 | 1.489165 | 129.5881 | −57.90929 |
0.549498 | 1.408428 | 127.0825 | −59.39846 |
0.5735447 | 1.463318 | 124.8436 | −60.80688 |
0.5976748 | 1.517414 | 122.6371 | −62.27019 |
0.621707 | 1.570679 | 120.4644 | −63.7876 |
0.6458198 | 1.623078 | 118.3271 | −65.35827 |
0.6699521 | 1.674573 | 116.2267 | −66.98135 |
0.6941026 | 1.725132 | 114.1645 | −68.65591 |
0.7182705 | 1.774719 | 112.1418 | −70.38105 |
0.7424545 | 1.823302 | 110.1602 | −72.15576 |
0.7666542 | 1.870849 | 108.2208 | −73.97906 |
0.7908685 | 1.917328 | 106.3249 | −75.8499 |
0.8150969 | 1.962709 | 104.4739 | −77.76723 |
0.8393388 | 2.006962 | 102.6688 | −79.72995 |
0.8635934 | 2.050059 | 100.911 | −81.7369 |
0.8878606 | 2.09197 | 99.20143 | −83.78696 |
0.9121396 | 2.13267 | 97.54134 | −85.87893 |
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Rece, L.; Florescu, V.; Modrea, A.; Jeflea, V.; Harničárová, M.; Valíček, J.; Borzan, M. Optimization of the 2 ½ D Processing Method of Complex Parts, through a Predictive Algorithm for Controlling the Geometric Shape Deviations Resulting from Processing. Mathematics 2020, 8, 59. https://doi.org/10.3390/math8010059
Rece L, Florescu V, Modrea A, Jeflea V, Harničárová M, Valíček J, Borzan M. Optimization of the 2 ½ D Processing Method of Complex Parts, through a Predictive Algorithm for Controlling the Geometric Shape Deviations Resulting from Processing. Mathematics. 2020; 8(1):59. https://doi.org/10.3390/math8010059
Chicago/Turabian StyleRece, Laurentiu, Virgil Florescu, Arina Modrea, Victor Jeflea, Marta Harničárová, Jan Valíček, and Marian Borzan. 2020. "Optimization of the 2 ½ D Processing Method of Complex Parts, through a Predictive Algorithm for Controlling the Geometric Shape Deviations Resulting from Processing" Mathematics 8, no. 1: 59. https://doi.org/10.3390/math8010059
APA StyleRece, L., Florescu, V., Modrea, A., Jeflea, V., Harničárová, M., Valíček, J., & Borzan, M. (2020). Optimization of the 2 ½ D Processing Method of Complex Parts, through a Predictive Algorithm for Controlling the Geometric Shape Deviations Resulting from Processing. Mathematics, 8(1), 59. https://doi.org/10.3390/math8010059