CAD/CAM Principles and Applications Ch 4 Geometric Modelling
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
1
Objectives • • • • • • • •
Understand the various requirements for the information that is generated during the geometric modeling stage. Study various types of geometric models possible and their applications Develop various methodologies used for geometric construction such as sweep, surface models, solid models, etc. Recognize the various types of surfaces and their application as used in geometric modelling Appreciate the concept of parametric modeling which is the current mainstay of most of the 3D modeling systems Develop the various mathematical representations of the curves used in the geometric construction Discuss the various CAD system requirements that need to be considered while selecting a system for a given application Understand the concept of rapid prototyping and the various methods available for the purpose. CAD/CAM Principles and Applications by P N Rao, 2nd Ed
2
4.1 Requirements of Geometric Modelling
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
3
Fig. 4.1
Total product cycle in a
manufacturing environment Geometric Modelling
Ideas
Design Analysis
Production
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
4
Functions of Geometric Modelling • Design analysis: – – – – – –
Evaluation of areas and volumes. Evaluation of mass and inertia properties. Interference checking in assemblies. Analysis of tolerance build-up in assemblies. Analysis of kinematics — mechanics, robotics. Automatic mesh generation for finite element analysis.
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
5
Functions of Geometric Modelling • Drafting – Automatic planar cross sectioning. – Automatic hidden line and surface removal. – Automatic production of shaded images. – Automatic dimensioning. – Automatic creation of exploded views for technical illustrations. CAD/CAM Principles and Applications by P N Rao, 2nd Ed
6
Functions of Geometric Modelling • Manufacturing – Parts classification. – Process planning. – Numerical control data generation and verification. – Robot program generation.
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
7
Functions of Geometric Modelling • Production Engineering – – – –
Bill of materials. Material requirement. Manufacturing resource requirement. Scheduling.
• Inspection and Quality Control: – Program generation for inspection machines. – Comparison of produced part with design. CAD/CAM Principles and Applications by P N Rao, 2nd Ed
8
Requicha and Voelker [1981] specified the following properties to be desired of in any geometric modelling (solids) system. • • • • •
•
The configuration of solid (geometric model) must stay invariant with regard to its location and orientation. The solid must have an interior and must not have isolated parts. The solid must be finite and occupy only a finite shape. The application of a transformation or other operation that adds or removes parts must produce another solid. The model of the solid in E3 (Euler space) may contain infinite number of points. However, it must have a finite number of surfaces, which can be described. The boundary of the solid must uniquely identify which part of the solid is exterior and which is interior. CAD/CAM Principles and Applications by P N Rao, 2nd Ed
9
4.2
Geometric Models
• Two-dimensional, and • Three-dimensional. • The three principal classifications can be – The line model, – The surface model, and – The solid or volume model CAD/CAM Principles and Applications by P N Rao, 2nd Ed
10
Fig. 4.2 3D geometric representation techniques P2 P3
P1
S8
P10 S6
S5
P4 P9 P11
P12
S4 P5
P8
P6
S3 S1 S7
S2
P7 (a) LINE MODEL
(b) SURFACE MODEL
V1 V2
(c) VOLUME MODEL
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
11
Fig. 4.3 A geometric model represented in wire-frame model
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
12
Fig. 4.4 Ambiguities present in the wire-frame model
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
13
Fig. 4.5 Impossible objects that can be modelled using a wire-frame model
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
14
Fig. 4.6 Generation of 3D geometry using planar surfaces S5
S3 S6 S6
S5
S8
S8 S4 S3
S4
S1 S7
S2
S1
S2 S7
(b) SURFACE MODEL
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
15
4.3 Geometric Construction Methods • The three-dimensional geometric construction methods which extend from the 2D that is normally used are: – Linear extrusion or translational sweep, and – Rotational sweep.
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
16
Fig. 4.8
Component model produced using
translational (linear) sweep (extrusion)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
17
Fig. 4.9
Component model produced using
translational (linear) sweep with taper in sweep direction
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
18
Fig. 4.10
Component model produced using linear
sweep with the sweep direction along a 3D curve
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
19
Fig. 4.11
Component model produced using
translational (linear) sweep with an overhanging edge
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
20
Fig. 4.12 Component produced by the rotational sweep technique
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
21
Fig. 4.13 Various solid modelling primitives
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
22
Fig. 4.14 The Boolean operators and their effect on model construction
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
23
Fig. 4.15 The Boolean operators and their effect on model construction
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
24
Fig. 4.16 Creating a solid with the 3D primitives in solid modelling and the model shown in the form of Constructive Solid Geometry (CSG)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
25
Fig. 4.17 Model generated using the sculptured surfaces (Image appears with the permission of IBM World Trade Corporation/Dassault Systems - Model generated using CATIA)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
26
Fig. 4.18 The various types of surfaces used in geometric modelling Classification of Surfaces Planar surfaces Plane
Polygon Polyhedra
Curved surfaces
Free form surfaces
Single curved
Double curved
Cylinders Cones
Spheres Ellipsoids Paraboloid Torus
Ruled surfaces
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
Coons surface
B-spline Bezier surface NURBS Fractals
Lofted surfaces
27
Fig. 4.19
Ruled surface on the left is shown the curves
from which the ruled surface on the right is formed.
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
28
Fig. 4.20 Coons surface generation
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
29
Fig. 4.21 The Bézier curve and the associated control polygon
Y
Control Polygon
Control points
Curve
X
O CAD/CAM Principles and Applications by P N Rao, 2nd Ed
30
Fig. 4.22 The various examples of Bézier curves depending on the associated control polygons p2
p2 p1 p3
p3 p0
p0
p1 p1
p3 p0 p2
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
31
Fig. 4.23 The modification of Bezier curve by tweaking the control points
Y
Control Polygon
Control points
Y
Curve
Curve
O
Control Polygon
Control points
X
X
O
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
32
Fig. 4.24 The spline curve
Y Control points
Control Polygon Curve
X
O CAD/CAM Principles and Applications by P N Rao, 2nd Ed
33
Fig. 4.25 The lofted surface
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
34
Fig. 4.26 Example of filleting or blend method for model generation
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
35
Fig. 4.27 Example of tweaking method for surface modification ((Image appears with the permission of IBM World Trade Corporation/Dassault Systems - Model generated using CATIA))
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
36
4.4 Constraint Based Modelling
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
37
Fig. 4.28 Example of initial sketch without any dimensions
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
38
Fig. 4.29 The sketch shown above which is fully constrained and dimensioned
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
39
Fig. 4.30 The sketch in Fig. 4.29 when swept along a linear path produces the solid
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
40
Fig. 4.31 The sketch for the new feature (a cut)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
41
Fig. 4.32 The solid after executing an extruded cut of the geometry in Fig. 4.31
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
42
Fig. 4.33 The final solid
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
43
Fig. 4.34 The model tree of the part showing the modelling process
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
44
Fig. 4.35 A geometric model created following the sequence of features as Box → Hole → Shell
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
45
Fig. 4.36 A geometric model created following the sequence of features as Box → Shell → Hole
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
46
Fig. 4.37 Feature based model and its modified form Base feature Holes - 3
Slots - 2
(A) Original model
Base feature
Slots - 2
Holes - 5
(B) Modified model CAD/CAM Principles and Applications by P N Rao, 2nd Ed
47
Fig. 4.38 Typical drawing for the variant method of modelling
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
48
Fig. 4.39 Part model produced using the symbolic programming T C R G N COMPOSED PART
SYMBOL KEYS G
C
C
KEY SEQUENCE
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
49
Fig. 4.40 Examples of form elements used for model generation in the case of axi-symmetric components Thread
Arc
Groove
Taper
Turn Fillet
Knurl Chamfer
Face
Blank
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
50
Fig. 4.41 Examples of form features for modelling axi-symmetric components with milled features
Taper
Turn
Chamfer
Groove
Step
Keyway
Splines
Concentric slot
Fillet
Face
Thread
Knurl
Axial hole
Radial hole
Axial slot
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
Radial slot
51
Fig. 4.42 Example component modelled using the features shown in Fig. 4.41 A
3.2
All chamfers 1x45
0
Straight Knurl Pitch 1mm M36x1
1.6
45
90
60
42
-0.015 -0.040
0.01 A
R1.5
42
32 75 187
+ -
0.50 0.75
76
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
52
Fig. 4.43 Example component modelled using the features shown in Fig. 4.41 Chamfer 2X2 @45 10 dia 4 holes
2X2 Groove
M8X1 LHT 12.5
75
50
Chamfer angle 45
25
2
12.5 25
80
Sectional Elevation CAD/CAM Principles and Applications by P N Rao, 2nd Ed
4 holes on pcd 37.5
End view 53
4.6 Curve representation • Implicit form, and • Parametric form. • In parametric form, the curve is represented as • X = x(t) • Y = y(t) • Z = z(t) CAD/CAM Principles and Applications by P N Rao, 2nd Ed
54
Fig. 4.44 Circle Y
(X, Y)
θ X
O
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
55
Fig. 4.45 Ellipse Y
2
2
x y + = 1 2 2 a b (X, Y) b
θ
O
a
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
X
56
Fig. 4.46 Parametric curve representation in Cartesian space p3 z p2
y u x
p1
p0 CAD/CAM Principles and Applications by P N Rao, 2nd Ed
57
Fig. 4.47 Two cubic Bézier curves joined at p3 p2 p3
p1 u
p4
p0 z p5
y p6 x
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
58
4.7 Surface Representation Methods
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
59
Fig. 4.48 Typical surface display with the parametric variables u and v
z
v u y
x CAD/CAM Principles and Applications by P N Rao, 2nd Ed
60
Fig. 4.49 A bi-cubic Bézier surface patch p(u,1), v=1 curve
p(0,v), u=0 curve
p23
p13
p24 p21
p12 p22
p44=p(1,1)
v p21
z p11=p(0,0)
p33
p14=p(0,1)
u
p32
p43
p31 p42
y
p(1,v), u=1 curve
p(u,0), v=0 curve
x
p41=p(1,0) CAD/CAM Principles and Applications by P N Rao, 2nd Ed
61
4.8 Modelling Facilities Desired • • • • • • •
The geometric modelling features. The editing or manipulation features. The display control facilities. The drafting features. The programming facility. The analysis features. The connecting features. CAD/CAM Principles and Applications by P N Rao, 2nd Ed
62
Fig. 4.50 Elimination of hidden lines in display
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
63
Fig. 4.51
Shaded image of a CAD geometric model ((Image
appears with the permission of IBM World Trade Corporation/Dassault Systems Model generated using CATIA))
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
64
Fig. 4.52 Orthographic views from a geometric model (Image appears with the permission of IBM World Trade Corporation/Dassault Systems Model generated using CATIA)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
65
Fig. 4.53 Section view generation from a geometric model
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
66
Fig. 4.54 Exploded view and bill of materials of an assembly modelled
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
67
4.9 Rapid Prototyping (RP)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
68
Figure 4.55 Schematic of Stereolithography device Scanning mirror
Cured resin (to form model) Liquid resin
Laser
Recoating bar
Platform
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
69
Figure 4.56 Schematic of selective laser sintering device Scanning mirror
Laser
Powder feed roller
Platform Build powder Sintered powder (to form parts)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
70
Figure 4.57 Schematic of Three-dimensional printing device Binder solution Powder feed roller
Printing head Nozzle Platform
Build powder Glued powder (to form parts)
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
71
Fig. 4.58 Schematic of Fused deposition modelling device Filament from a coil
Feeder
Melter
Extrusion nozzle Solidified plaster (to form model)
Platform
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
72
Fig. 4.59 Schematic of Laminated Object Manufacturing device
Top view
Splits in excess material (for ease of removal)
Band of build material
Contour of actual cross section of the model
Laminating roller
Scanning mirror Laser
Band of build material
Laminate model
Laminating roller
Excess laminate material
Take-up roll
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
Platform
Material supply roll
73
Summary • •
• •
Information entered through geometric modeling is utilized in a number of downstream applications such as drafting, manufacturing, inspection and planning. Geometric models are three types, viz line model, surface model and solid model. Line model though simple is rarely used because of the ambiguity present. Surface and solid models are extensively used in industrial applications. Among the geometric construction methods sweep or extrusion is most widely used, because of its simplicity and elegance in developing 3D models. Solid modeling provides the most unambiguous representation of the solid model, but is more computing intensive. However to get the correct geometric model, it is essential to utilize solid modeling approach. CAD/CAM Principles and Applications by P N Rao, 2nd Ed
74
Summary • •
• •
Surfaces are more widely used and it is necessary to use different types of surfaces such as b-splines, Bezier, NURB, lofted, to get the user requirements fulfilled. Constraint or parametric based modeling is the main methodology used by most of the 3D CAD systems. This system helps in grasping the designer’s intent and would greatly facilitate the modification and reuse of the existing designs. Some variant modeling systems are used based on tabular data for specific applications. Form features is another form of modeling system that helps in designing CAD systems with more intelligence built into the geometric entities that is possible by purely geometric systems discussed thus far.
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
75
Summary •
• • •
The mathematical representation of the geometric entities can be in implicit or parametric form, the latter being the preferred method used in CAD systems because of its easier adaptation in software development. The curve representation methods can be extended for surface representations such as used in free form surfaces. A number of modeling facilities need to be considered while selecting a CAD/CAM system for any given application. Rapid prototyping is used to generate the product directly from the 3D CAD model data. A number of different processes such as stereo lithography, selective laser sintering, 3D printing, fused deposition modeling, laminated object manufacturing, are used for this purpose.
CAD/CAM Principles and Applications by P N Rao, 2nd Ed
76
