235 Proceedings of 20th International Conference. Mechanika. 2015
Graphics Programming for Designing Conical-Helix Surface A. Sokas Vilnius Gediminas Technical University, Sauletekio al.11, 10223 Vilnius, Lithuania, E-mail:
[email protected] Abstract This article analyzes graphical methods for designing conical-helix surface in engineering drawings, presents design system of a surface model and its developments. The creation methods of automated design system of a conicalhelix surface are discovered with graphics programming. Graphical system can design 3D logarithmic spiral with using information of program. Then calculated node coordinates and designed mesh of a conical-helix surface. Graphical system can draw development of surface. Example of a conical-helix surface model and its developments are presented. Creation problems of automated design system are discussed and conclusions are made. KEY WORDS: graphics programming, logarithmic spiral, conical-helix surface. 1. Introduction The radius of curvature of a logarithmic spiral is proportional to its arc length, a property that is desirable for the design of aesthetic curves [1]. The planar logarithmic spiral was discovered several times by different mathematicians. The first to discover the spiral was René Descartes (1596-1650), who in 1638 named it the equiangular spiral: a spiral whose radius vector cuts the curve at a constant angle. Evangelista Torricelli (1608-1647) named it the geometrical spiral after another property of the curve – its radius increases exponentially as a function of the angle in the year 1645 [2]. Edmond Halley (1656-1742) named it the proportional spiral: a spiral in which the lengths of the segments of the curve cut by a fixed radial ray are in continued proportion. Jacob Bernoulli (1655-1705) was the first to describe the curve without using a polar equation in 1711 and gave the curve its fourth name – the logarithmic spiral [3]. Leonhard Euler (1707-1783) rediscovered the curve’s equations, described their properties, and derived a series expansion to the curve’s integrals [4]. Of course curves studied nowadays, for example paper described a method for approximating logarithmic spiral segments by rational cubic spline curves. This approach provides the tools for the construction of planar spline curves whose curvature radius plot is continuous and close to piecewise linear [1]. Accurate mathematical modeling of the tooth surface geometry for logarithmic spiral bevel gears is developed in study, based on the basic gearing kinematics and spherical involute geometry along with the tangent planes geometry; actually, the tooth surface is a parametric surface defined on a parallelogrammic domain [5]. The logarithmic spiral in severe cases applied to form the conical-helix curve and surface. A conical-helix is a space curve on a cone which intersects all generators at the same angle. It was written in the vector form [6]. Conical involute gear represent the most general type of the involute gear, which can mesh conjugately with most involute type gears of spur gears, helical gears, conical involute gears, worms and racks to serve the motion transmission between parallel, intersected and crossed axes in any relative position [7]. For power transmission between two shafts with large angles, hypoid and worm gears are superior to involute helical gears. However, due to design and manufacturing difficulties, it is not practical to use hypoid and worm gears when the shafts are intersected with small angles in such compact gear boxes. In such applications, intersected beveloid gears provide a competitive design option because of their compactness, low manufacturing cost and flexibility in gear ratio [8]. The helical springs are the structural elements that have the ability to absorb energy while deforming. They are widely used in engineering structures. For example, in mechanical engineering helical springs are used as dampers to reduce the vibration in machines, whereas in civil engineering they are used as helical carrier systems such as stairs [9]. The geometric analysis of 3-D chip forms are identified: the geometric properties at every point on the tool–chip separation line are fully determined once those at any one point are known, all possible 3-D chip forms are confined to a relatively restricted parameter space defining the chip velocity direction and the orientation of the axis of the helical chip, 3-D helical chips are only approximately conical, and the radii of up-curl and side-curl can be determined from a set of simple measurements of the chip-in-hand. Unlike past analyses, the new analysis paves the way to the study of chip forms from empirical data obtained from practical 3-D chips [10]. Other paper covers a wide range of staircase configurations including: helical staircases; spiral staircases; and composite staircases [11]. Paper studied the spiral angle between successive leaves [12]. A developable surface is a surface which can be unfolded into a plane without stretching or tearing. Because of this property, developable surfaces possess a variety of applications in manufacturing with materials that are not amenable to stretching. These include the formation of aircraft skins, ship hulls, ducts and automobile parts such as upholstery, body panels and windshields. Developable surfaces can be isometrically mapped into the plane, at least locally. When sufficient differentiability is assumed, they are characterized by the vanishing of their Gaussian curvature. A non
236 flat developable surface is the envelope of its one parameter family of tangent planes. Such a developable surface locally is either a conical surface, a cylindrical surface, or the tangent surface of a twisted curve. Globally, of course, it can be a rather complicated composition of these three surface types. Thus, developable surfaces are ruled surfaces, but with the special property that they possess the same tangent plane at all points of the same generator [13]. A graphical environment and a working programming language in this environment are required for design of such systems. In the AutoCAD environment we can program with Visual Basic for Applications language (VBA) [1416]. This article analyzes graphical methods for designing conical-helix surface in engineering drawings. There are two types of curves: logarithmic spiral and conical-helix curve which presented in the drawing. The conical-helix surface development is followed by numerical parameters, which can be achieved by using graphical programming techniques. 2. Logarithmic spiral The logarithmic spiral (Fig. 1) is a spiral whose polar equation is given by (1) where r is the distance from the origin, is the angle from the x-axis, and a and k are arbitrary positive real constants. The logarithmic spiral constants are a equal one and k presented by (2). It can be expressed parametrically as (3, 4) in (Fig. 2). The 3D logarithmic spiral can presented conical helical curve (Fig. 3) with parametric equations (1-5) in (Fig. 2).
aek
(1)
k sin( ) tg ( )
(2)
x = cos( ) y = sin( ) z = /tan(a)
Fig. 1 The logarithmic spiral
(3) (4) (5)
Fig. 2 The logarithmic spiral formulas
The angle between the axis Z and some generator of the cone is a and the angle between the tangent to the helical curve and the corresponding generator of the cone is b (Fig. 3). The projection of the conical helical curve parallel to the axis of the cone onto the plane perpendicular to the axis of the cone is a logarithmic spiral. The program fragment that defines the conical helical curve presented in (Fig. 4). Procedure in cycle assigns an angle p/16 part (9), a radius counts (10), x, y, z coordinates founds (11-13), a short section of line draws (14) and line end points are found (15-17). The cycle is repeated 124 times.
t1(0) = 0 : t1(1) = 0 :t1(2) = 0 (6) ka = Sin(alfa * pi / 180) / Tan(beta * pi / 180) (7) For i = 0 To n (8) fi(i) = i * pi / 16 (9) ro(i) = Exp(ka * fi(i)) (10) t2(0) = ro(i) * Cos(fi(i)) (11) t2(1) = ro(i) * Sin(fi(i)) (12) t2(2) = ro(i) / Tan(alfa * pi / 180) (13) Set linija = ThisDrawing.ModelSpace.AddLine(t1, t2) (14) t1(0) = t2(0) (15) t1(1) = t2(1) (16) t1(2) = t2(2) (17) Next i (18)
Fig. 3 The 3D logarithmic spiral or conical helical curve
Fig. 4 The program fragment which form the 3D logarithmic spiral
237 3. Conical-helix surface 3D logarithmic spiral includes a plurality of blades mounted symmetrically and curve along the axis Z of rotation in a logarithmic spiral shape. The surface consists of a logarithmic curve pattern with a certain curve radius and placed around a rotary axis Z. The conical-helix surface is perpendicular to the axis Z from each point on the spiral. The coordinates are found (19, 20) (Fig.5) and the three-dimensional matrix [xyz] is formed (21-25), where the first index represents the number of radius, the second index represents the number of points on the radius, the third index represents the number of x, y, z coordinates. The conical-helix mesh can simulate using matrix parameters (2635) (Fig.6). x = (t3(0) - t2(0)) * 0.25 y = (t3(1) - t2(1)) * 0.25
(19) (20)
For j = 0 to 3 xyz(i, j, 1) = t2(0) + x * j xyz(i, j, 2) = t2(1) + y * j xyz(i, j, 3) = t2(2) Next j
(21) (22) (23) (24) (25)
Fig. 5 Forming matrix [xyz]
For j = 0 To 3 For i = 0 To n t1(0) = xyz(i, j, 1) t1(1) = xyz(i, j, 2) t1(2) = xyz(i, j, 3) t2(0) = xyz(i + 1, j, 1) t2(1) = xyz(i + 1, j, 2) t2(2) = xyz(i + 1, j, 3) Set linija = ThisDrawing.ModelSpace.AddLine(t1, t2) Next i : Next j
(26) (27) (28) (29) (30) (31) (32) (33) (34) (35)
Fig. 6 Modeling conical-helix mesh
Program’s procedure results for designing a conical-helix surface model is shown in Figure 7.
Fig. 7 A conical-helix model The lines (36-39) finds a logarithmic spiral forming line fragments length and coordinate changes (Fig. 8, 9).
ilg(i + 1, j) = linija.Length dx(i + 1, j) = (xyz(i+1, j, 1) - xyz(i, j, 1)) dy(i + 1, j) = (xyz(i + 1, j, 2) - xyz(i, j, 2)) dz(i + 1, j) = (xyz(i + 1, j, 3) - xyz(i, j, 3))
Fig. 8 A fragment of conical helix
Fig. 9 A line length and coordinate changes
(36) (37) (38) (39)
238 4. Development of conical-helix surface Program’s procedure for designing a conical-helix surface developments are shown in Figure 10. Program’s procedure results for designing a conical-helix surface development are shown in Figure 11 and Figure 12, where presented separate surface fragments. j=0 (40) For i = 1 To 3 (41) xy2(i, j, 1) = xy2(i - 1, j, 1) - (dx(i, j) * ilg(i, j) / _ (Sqr(dx(i, j) ^ 2 + dy(i, j) ^ 2))) – dz(i) * Sin(fi(i)) (42) xy2(i, j, 2) = xy2(i - 1, j, 2) + (dy(i, j) * ilg(i, j) / _ (Sqr(dx(i, j) ^ 2 + dy(i, j) ^ 2))) + dz(i) * Cos(fi(i)) (43) t1(0) = xy2(i - 1, 0, 1) (44) t1(1) = xy2(i - 1, 0, 2) (45) t1(2) = 0 (46) t2(0) = xy2(i, 0, 1) (47) t2(1) = xy2(i, 0, 2) (48) t2(2) = 0 (49) Set linija = ThisDrawing.ModelSpace.AddLine(t1, t2)(50) Next i (51)
j=4 (52) For i = 1 To 31 (53) xy2(i, j, 1) = xy2(i - 1, j, 1) – dz(i) * (Sin(fi(i))) (54) xy2(i, j, 2) = xy2(i - 1, j, 2) + dz(i) * (Cos(fi(i))) (55) t1(0) = xy2(i - 1, 4, 1) (56) t1(1) = xy2(i - 1, 4, 2) (57) t1(2) = 0 (58) t2(0) = xy2(i, 4, 1) (59) t2(1) = xy2(i, 4, 2) (60) t2(2) = 0 (61) Set linija = ThisDrawing.ModelSpace. _ AddLine(t1, t2) (62) Next i (63)
Fig. 10 Drawing interchange format
Fig. 11 A conical-helix surface development
Fig. 11 A conical-helix surface development with separate surface fragments
239 5. Conclusions Carrying out a study of this conical-helix complex surface do the following conclusions: First, the planar logarithmic spiral was discovered from 1638 until 1711 years by different mathematicians. Second, the logarithmic spiral at certain angles (Fig. 3) is composing the outer of conical-helix surface. Third, all conical-helix surface is perpendicular to the rotational axis. Forth, the conical-helix three dimensional of mesh nodes coordinates matrix formed, which information is used for development making. Fifth, because the conical-helix development is covered by one of the other, it provides a number of drawings, which can be connected in series. Sixth, a graphical environment and a working programming language in this environment are required for design of such systems. For example, Visual Basic for Application programming language works with the AutoCAD environment. References 1. Baumgarten, C., Farin, G. Approximation of logarithmic spirals. Computer Aided Geometric Design, 1997, 14, p. 515–532. 2. Davis, P. J. Spirals: from Theodorus to Chaos, AK Peters, Ltd., Wellesley, 1993. 3. Harary, G., Tal, A. The Natural 3D Spiral. Computer Graphics Forum, 2011, 30(2), p. 237–246. 4. Lucas, P., Ortega-Yagües, J. A. Bertrand curves in the three-dimensional sphere. Journal of Geometry and Physics, 2012, 62, p. 1903–1914. 5. Fan, S., Zou, J., Shi, M. Parametric surface and properties defined on parallelogrammic domain. Journal of Computational Design and Engineering, 2014, 1(1), p. 27–36. 6. Tan, R., Chen, B., Peng, C., Li, X. Study on spatial curve meshing and its application for logarithmic spiral bevel gears. Mechanism and Machine Theory, 2015, 86, p. 172–190. 7. Chen, Y-C., Liu, C.-C. Contact stress analysis of concave conical involute gear pairs with non-parallel axes. Finite Elements in Analysis and Design, 2011, 47, p. 443–452. 8. Zhang, Y., Fang, Z. Analysis of tooth contact and load distribution of helical gears with crossed axes, Mechanism and Machine Theory, 1999, 34, p. 41–57. 9. Eratli, N., Argeso, H., Çalim, F. F., Temeld, B., Omurtag, M. H. Dynamic analysis of linear viscoelastic cylindrical and conical helicoidal rods using the mixed FEM. Journal of Sound and Vibration, 2014, 333, p. 3671–3690. 10. Kharkevich, A., Venuvinod, P. K. Basic geometric analysis of 3-D chip forms in metal cutting. Part 2: implications. International Journal of Machine Tools & Manufacture, 1999, 39, p. 965–983. 11. Afifi, M., Parke, B., Al-Hussein, M. Integrated approach for older adult friendly home staircasen architectural design. Automation in Construction 39 (2014) 117–125. 12. Prusinkiewicz, P., Lindenmayer, A. 1990. The Algorithmic Beauty of Plants. Springer-Verlag, New York. 13. Pottmann, H., Wallner, J. Approximation algorithms for developable surfaces. Computer Aided Geometric Design, 1999, 16, p. 539–556. 14. Cottingham, M. Mastering AutoCAD VBA, Sybex, 2001, 656 p. 15. Sutphin, J. AutoCAD 2006 VBA: Programmer’s Reference. Apress, 2006, 777 p. 16. Clark, J. E. VBA for AutoCAD 2002: Writing AutoCAD Macros. Prentice Hall PTR, 2001, 752 p.
