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Geometrik modelleme ve sentetik eğrilerin analizi

Geometric modelling and synthetic curves

  1. Tez No: 66646
  2. Yazar: ENGİN ÇİFTÇİ
  3. Danışmanlar: PROF. DR. TEOMAN KURTAY
  4. Tez Türü: Yüksek Lisans
  5. Konular: Makine Mühendisliği, Mechanical Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1997
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Makine Mühendisliği Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 112

Özet

ÖZET Geometrik model bir planlamanın,dizayn ve üretimin başlangıç noktasıdır. Bu yüzden parça dizaynının, bir üretimde 1. kısmı olması ve parçanın onsuz tanımlanamaması nedeniyle başlangıç noktasıdır. Günümüzde artık bilgisayarlar her türlü alanda kullanılmaktadırlar bunlardan bir taneside imalattan önceki safha olan,tasarım alanıdır, imalatı yapılacak parçanın şeklinin bilgisayarda tanımlanabilmesi için çeşitli eğri metodları geliştirilmiştir. Bu eğrilerin arasında bildiğimiz basit eğriler daire, çizgi gibi analitik eğrilerde kullanılmaktadır. Yanlız bu eğriler çoğu zaman imalatı yapılacak parçanın çiziminde yeterli olmamaktadır. Bu yüzden daha kompleks eğri yüzeylerini tanıtmamak için sentetik eğriler oluşturulmuştur. Sentetik eğriler hem parçaya istenilen formun verilmesinde kolaylık hemde istenilen kısmın değiştirilmesinde bir kolaylık sağlamaktadır. îleriki bölümlerde sentetik eğrilerin bu avantajlarına detaylı olarak değinilmiştir. Bahsedilen başlıca sentetik eğriler ; Hermit, bezier, B-spline ve oransal eğrilerdir. Bilgisayarlı tasarımda en çok kullanılan eğri türü günümüzde oransal eğrilerdir. İlerideki bölümlerde bu eğrilerin avantajları, dezavantajları, birbirlerine karşı üstün tarafları ve matematiksel eşitlikleri örnekler ile detaylı olarak anlatılmıştır.

Özet (Çeviri)

SUMMARY GEOMETRIC MODELING and SYNTHETIC CURVES This work discusses geometric modeling and its relevance to CAD / CAM. Early CAD /CAM systems focused on modeling enginering objects. As a results geometric models, that once were more than adequate for drafting purposes are not acceptable for engineering applications. A basic requiriment.therefore. Is that a geometric model should be an unambiguous representation of its corresponding object. That is to say, the model should be unique and complete to all engineering functions from documantation (drafting and shading ) to engineering analysis to manufacturing. i A geometric model of an object and its related database have 3 types of geometrict models, wireframes »surfaces and solids. Users usually have to decide on the type of modeling technique based on the ease of the using the technique during the construction phase and on the expected utilization of the resulting database later in the design and manufacturing processes. Regardles of the chosen technique, the usuer constructs a geometric model of an object on a CAD/CAM system. To software data into a mathematical representation which it stores in the model database for later use. The user may retrieve and modify the model during the design and manufacturing processes. To convey the importance of geometric modeling to the CAD/ CAM proces, one may refer to other engineering disciplines and make the following anology. Geometric modeling to CAD/ CAM is as important as governing equilibrium equations to classical engineering fields as machanics and thermal fluids. From an engineering point of view modeling of objects is by its self unimportant. Rather,it is ameans (tool) to enable useful engineering analysis and judgment. As a mater of fact, the amount of time and effort a designer spends in creating a geometric model cannot be justified unless the resulting database is utilized by the aplication module. The need to study the mathematical basis of geometric modeling is many fold. XIFrom a strictly modeling point of view, it provides a good understanding of a terminology encountered in the CAD/CAM field as well as CAD / CAM system documantations. From an engineering and design point of view.studying geometric modeling provides engineers and designers with new sets of tools and capabalities that they can use in their daily engineering assigments. This is an important issue because,,historically,engineers cannot think in terms of tools they have not learned to use or been exposed to. The tools are powerful if utilized innovativelyin engineering applications. It is usually left to the individual imagination to apply these tools usefully to applications in a new contex. Having established the need for geometric modeling, what is the most useful geometric model to engineering applications? Unfortunately, ther is no direct answer to this question. Newrtheless, the follpwing answer has two levels. At one level »engineers may agree that some sort of geometry is required to carry enginering analysis. the degree of geometric details depend on the analysis procedure that utilizes the geometry. Engineers may also agree that there is no model that is sufficent to study all behavioural aspects of an engineering component or asystem.Aa machine part, for example,can be modeled as a lumped mass rigid body on one occasion or a distributed mass continuum on other occasion. At the second level, the adequace of geometry or a geometric model to an analysis procudure is decided by its related useful atributes to that procedure. Atributes of geometry is never an issue for manual procedures because the engineer's mind coordinates all the related facts and information. This work covers the avaliable types and most useful mathematical represantations of curves. There are two categories of curves ; analytic and synthetic. Analytic curves are defined as those that can be described by analytic equations such as lines,circles,and conies. synthetic curves are the ones that are described by a set of data points (control points) such as splines and bezier curves. Parametric polynomials usually fit the control points. while analytic curves provide very compact forms to represent shapes and simplify the computation of related properties such as areas and volumes,they are not attractive to deal with interactively.Alternatively, synthetic curves provide designer with greater flexibility and control of a curve shape by changing the positions of the control points. Analytic curves are usually not sufficent to meet geometric design requirements of mechanical parts. Products such as car bodies. ship hulls, airplane fuselage and wings, propeller blades and bottles are a few examples that require free-form,or synthetic »curves and surfaces. The xnneed for synthetic curves in design arises on two occasions : When a curve is represented by a collection of measured data points and when an existing curve must change to meet new design requirements. In the later occasion, the designer would need a curve represantation that is directly related to the data points and flexible enough to bend,twist,or change the curve shape by changing one more data points. Data points are usually called control points and the curve itself called an interpoland if it passes through all the data points. Mathematically, synthetic curves represent a curve-fitting problem to constuct a smoot curve that passes through given data points. Therefore, polynomials are the typical form of these curves. Various continuity requirements can be specified at the data points to impose various degrees of smoothness of the resulting curve. The order of contiunity becomes important when a complex curve is modeled by several curve segments pieced together end to end. Zero order continuity yiels a position continuous curve. First and second order continuityies imply slope and curvature continuous curves respectively curve is the minumum acceptable curve for engineering design. Figure 1 shows a geometrical interpretion of these order of continuity. - J anccnts Center of curvature \ + Cimtinl point ((«) = £ ^4», on(u) are the Berstein polynomials. Thus the bezier curve has a berstein basis. The Berstein polynomial serves as the blending or basis funtion for the bezier curve and is given by ; xvBu(u) = C(n,i) U^ı-u)01 (4) C(n,i); binomial coefficient c = TîÖ^Ö! (5) While bezier cuve seems superior to a cubic spline curve, it steel has some disadvantages.First the curve does not pass through the control points which may be inconvenient to some designers. Second, the curve lacks local control. It only has the global control nature. If one control point is changed.the whole curve changes. Therefore,the designer cannot selectively change parts of the curve. B-Spline Curves : B spline curves provide another effective method,besides that of bezier, of generating curves defined by polygons. In fact, B-spline curves are the proper and powerful generalization of bezier curves. In additin to sharing most of the characteristics of bezier curves they enjoy some other unique advanteges. They provide local control of the curve shape as opposed to global control by using a special set of blending functions that provide local influence. They also provide the ability to add control points without increasing the degree of the curve. In contrast to bezier curves,the theory of B-spline curves separates the degree of the resulting curve from the number of the given control points. Similar to bezier curves, the B-spline curve defined by n+1 kontrol points Pi is given by; P(u)=S W,*(«) 0n 0 < j < n+k (9) (10) and the range of U is ; 0 < U < n-k+2 (11) While the degree of the resulting b-spline curve is controlled by k, the range of the parameter u as given by Eq 11 implies that there is alimit on k that is determined by the number of the given control points. The local control of the curve can be achieved by changing the position of a control point 5using multiple control points by placing several points at the same location,or by choosing a different degree (k-1).As mentioned earlier changing one control point affects only k segments.Figure 3 shows the local control for a cubic B-spline curve by moving P3 to P3* and P3* *. The four curve segments surrounding P3 change only. Figure 3 Local control of B-spline curves. The closed B-spline curve of degree (k-1) or order k defined by (n+1) kontrol points is given by Eq.6 as the open curve.However,for closed curves : xvi iNi>k (u) = N0>k [(u-i+n+1) mod(n+l)] Uj=j, 0

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