Computer Aided Design and Manufacturing Systems

Computer Aided Design and Manufacturing Systems

1. ME8691 Computer Aided Design and Manufacturing Unit- II Geometric Modeling A.Eswaran , AP/Mech

2. Geometric Modeling It is a branch of computational geometry and applied mathematics under which we study different algorithm for the description of different shapes. Requirements of geometric modeling . At the time of parts inspection, inter changeable manufacturing tolerance is required . There should be an automatic assembly of the model for checking interference, modeling , etc. Kinematic analysis and finite element analysis is required. Properties and geometrical evaluation of area, volume, weight, density, etc are required . There should be a graphical visualization of cross-section, line that are hidden in the geometry. The boundary of the solid should be uniquely identified.

3. Methods of Geometric Modeling There are basically three different methods to represents the geometric modeling. 1. Wire frame modeling 2. Surface modeling 3. Solid modeling

4. Wire frame Modeling It is one of the methods used in geometric modeling system. Wireframe represents a solid shape in the form of lines, edges and points. It is used to represent mathematically in the computer.

5. Surface modeling It is used to represent the complex object that cannot be represented by the wireframe modeling. It provides more and less ambiguous representation. Surface representation can be done in both parametric and non-parametric form. Surfaces available in the CAD/CAM systems are Bezier surface, B-spline surface, plane surface , coons path, surface of revolution, etc.

6. Solid modeling It provides the complete information of the object as compared with the surface modeling. It stores the geometric data and topological information of the object.

7. Representation of Curves Analytical curves : The curves which are defined as those that can be described by analytic equation such as lines, circle and conics. Synthetic curves : The curves which are described by a set of data points or the control points such as splines, Bezier curve, B-spline curve, etc.

8. Order of Continuity of curves To achieve the smoothness of a function is measured by the number of derivative it has that are continuous and for that certain continuity conditions has to be imposed . Zero order continuity C : It ensures that the two curves meet at a point where the values remain same and such a curve is called as zero order continuity curve or C curve.

9. Order of Continuity of curves contd … First order continuity C 1 : It ensures that the slope at the end of the curve C1 is same as the slope at the starting of the curve C2 and thus we obtained smoother curve . The slope can be found by differentiating the parametric equation .

10. Order of Continuity of curves contd … Second order continuity C 2 : When the first order equation are differentiated further then the condition of second order continuity is obtained i.e. they satisfy both slope as well as curvature continuity.

11. Interpolation and Approximation of Curve When the curve passes through all the control points then such a curve is known as interpolated curve . Generally we use Lagrangian polynomial for interpolated curve but it is unsuitable as they tend to oscillate about control point and thus it become inconvinent for storing in the system. Whereas when it is not necessary to pass through all the control points then the resulting curve is known as approximated curve.

12. Difference between Interpolation Curve and Approximation Curve

13. Hermite Cubic Curve When the curve is defined by the two end points and their slope are termed as Hermite cubic curve. These types of curve are generally used to interpolate a curve for a given data points . It is commonly known as splines . Cubic equation of the spline is given as : ---- Equ . 1

14. To define the Hermite cubic spline we need the tangent vectors at the points which can be found by differentiating equation 1 w.r.t u } --------- Equ . 2

15. Substituting ( i ), (ii), (vii) and (viii) in equation 1,

16. Bezier Curve These are approximation curve as in Bezier curve it is not necessary that the curve should pass through all the data points, but the shape of the curve is influenced by the control points. Bezier curve does not use first order differential as used in case of cubic spline curve . The order of the curve depend on the number of control points. Characteristics of Bezier curve are : The Bezier curve passes through start point and the end point. The control points define the order, derivative and the shape of the curve. As the position of control point changes the shape of the curve would change . Bezier curve is always tangential to the first and the last control point as shown in Fig.

17. ---- Equ . 1 ---- Equ . 2

18. B-Spline Curve It is a generelised form of the Bezier curve. It is similar to the Bezier curve which is been defined by the number of control points. The main difference between the B-spline curve and the Bezier curve is that they have an ability to control the shape of the curve locally then the global control in the Bezier curve.

19. Difference between Hermite Cubic Spline, Bezier Wave and B-Spline Curve

20. Rational Curve The non-uniform rational B-spline curve is also known as NURB which include both the Bezier and B-spline curve. Hence it is a standard curve defintion for data exchange. Rational curve is a function in which one polynomial curve is divided by another polynomial curve, which is given as :

21. Surface Modeling It is a mathematical method used by the computer-aided design applications for displaying solid appearing objects. Surface modeling is a popular technique for architectural design and rendering. Surface models are preferred for the representation of complex objects such as car , ship, aircraft and casting. Surface models helps the designers to obtain good visualization of the entire surface . The advanced surface models can be used for generating NC tool path. It only stores the geometry of the object and not its topology. The surface is generated by connecting the points of the *wireframe. CAD uses the two basic method for the creation of surfaces. The first begins with the construction curves from which 3D surface is swept. The second method is direct creation of the surface with manipulation of the surface poles or control points.

22. Classification of Surfaces in Geometric Modeling Plane surface : It is one of the simplest form of the analytical surface which require three non-coincident points to define the plane as shown in the Fig . 2. Loafed surface : It is a linear surface which is formed by interpolating between the boundaries . It needs two boundaries to define ruled or a loafed surface as shown in the Fig.

23. 3. Edge surfaces : It is an extension to the ruled surface as in this the surface is been patched between the boundaries as shown in the Fig. Classification of Surfaces in Geometric Modeling

24. 4. Surface of revolution : It is used for axisymmetric object which can be revolved around the axis to form the surface. The revolution can be controlled by controlling the angle of revolution as shown in the Fig. Classification of Surfaces in Geometric Modeling

25. 5. Tabulated surface : It is used to generate the surface by extending the planar curve in either direction as required in the object. This method is suitable for identical curved cross-section as shown in the Fig. Classification of Surfaces in Geometric Modeling

26. 6. Coons patch : This surface is formed by curves which form the closed boundaries. Hence , we required different forms of curve to obtain Coons path. Classification of Surfaces in Geometric Modeling

27. Blending Function Blending functions are also known as Basis function which is an element from a particular basis for a function space. As every vector in a vector space can be represented as a linear combination of basic vector. Blending function determine how the control points influence the shape of the given curve for values of the parameters u over the range from 0 to 1. For e.g. : In Bezier spline curve we have two points and two tangents which are used to plot the graph whose parametric range varies from 0 to 1. The equation of Bezier curve is given by