21st Century Manufacturing Episode 1 Part 6 pptx

• In Step 1, Figure 3.15, the ordinary Boolean intersection,AnB, is computed to yield the volume of the object plus any dangling or lower dimensional faces, edges, or vertices.. • In Ste

Figure 3.9 Outline of the base exterior before the trimming process.

F1gure3.10 Outline of the base exterior after the trimming process

A good place to start is the shallow rounded pocket in the bottom face, because many consumer electronics products, including telephones, have a cover plate on the into The cover plate will close over the larger internal cavity that houses the printed circuit boards and wires The plate generally follows the contour of the bottom sur-face of the object For the joystick, the base plate itself is a flat trapezoid with roundedcorners

A shallow rounded pocket will thus be drawn in the next few images to accom-modate the plate This task begins in Figure 3.11 First, the straight lines of a trape-will then be connected, next rounded, and then copied and repeated for the depth of the cavity

In the first construction step, the lineiis drawn parallel to the base, at a small

Ilemporaryoonstructionlines(ceanddf)

Figure 3.11 Initial formation of the cavity in the base exterior

of the object on each side Lineiibegins by placing a tiny line of lengthw perpendi-cular to the other edge The full extent of linefiis then constructed from the end point

of this short construction line w It goes past the front of the object, parallel to the

angled bottom edge The line is completed by extending it with the extend command

such that it intersects linei,The extend command finds the intersection automati-cally Line iii is constructed in a similar manner from the short construction line w2 from the front face Once one of the lines marked "il" is constructed on the right side

of the object it is mirrored to the left, across the y axis, to form the image.

The next step is to trim lines I,ii,andiiiso that only the trapezoid remains.The

cor-ners of the trapezoid are then rounded with the fillet command The fillet command in

AutoCAD can be quite confusing to a new user because the default fillet radius is always set to zero The procedure is to enter the fillet command,hit the"r"key (for radius) when the dialog line appears in the command area, and then enter the desired fillet radius conunand has been used on each comer of the trapezoid, selecting both intersecting lines created for the curved region of each fillet The trapezoid has thus been transformed from four individual lines to eight segments: four lines plus four fillets

At this point it is convenient to join all eight segments comprising the trape-zoidal pocket into a single line This is done with the pedit (polyedit) command This

procedure begins by typing pedit and hitting enter When prompted, a segment of the designer would like to make it one To make it one, the designer can simply enter y(es) A set of polyline options is then presented, and the best choice is "[" for join Each segment to be included in the polyline that caps the top of the handle is then selected, remembering there are eight segments in this case Pressing "enter" joins all the segments into one entity Joined polylines can be separated into their original

<Line of lengthw

Figure3.12 Cornpletion of the cavity in

the hMe extenor

To create the depth of the cavity on the bottom, the copy command is used The existing polyline is copied and repeated a short distance along the positive z axis.This

cavity depth The preceding procedures are used to generate Figure 3.12

The outline of the base cavity could have been drawn with the rectang and/or polygon commands These commands automatically make polylines; however, they

are not preferred because individual segments might be edited out from the original polylines when the final smoothing command is applied The only way to avoid this

it is a simpler process to start by drawing line segments and then joining them into polylines

There are additional cavities within the object that house the accelerometers and circuitry, as shown in Figure 3.13

The process begins by drawing a series of lines in the shape of a large rectangle,

a much smaller rectangle, and a trapezoid These are shown in Figure 3.13 going in a

"northeast" direction First constructions are best done on the lower level with later extrusions in+z.

The corners of the large rectangle and trapezoid are then filleted (the radius needs to be reset to a smaller value) The common edge of the large rectangle and common edge between the trapezoid and the smaller rectangle is also removed, leaving the open space between b andII,and the segments are joined as a polyline The outline of the rectangles is then copied to a certain distance in the z direction, thus creating a pocket Next, lines are drawn at the vertical intersecting edges-a,b,

Coand d-to show depth The trapezoid is also copied, but to a greater distance in the

z direction Progress thus far is illustrated in Figure 3.13, which shows the main block from earlier drawings, the shallow recess for the base plate, the medium depth pocket

on the left, and the deeper trapezoidal pocket to the right Wires can be run between the electronic components in each pocket through the small slotabedin between the rectangular and trapezoidal pockets

A few more details remain The gripping handle is attached to the base with

Figure3.13 VRjoystick base with some interior cavities

in the bottom of the handle Therefore, so as not to interfere with the electronics inside the inner cavities, it is necessary to provide large countersinks for the bolt heads

The counter-bore for the large bolt head is represented by a large circle, and the hole for the screw thread part by a smaller concentric one Thus, originally, two one for the through-hole that will take the screw thread part

Circles are created with thecircle command The user is prompted for the center and radius of the circle The circles are then copied to the appropriate posi-from the distance ab in the earlier figures.Thus in Figure 3.14,the upper smaller cir-cles are coincident with the top surface, meaning they are through-holes for the screw threads

Figure 3.14 also illustrates that circleyintersects one of the vertical surfaces of the rectangular pocket Circleywas modified by trimming the entire right-side half Such manual operations are a major disadvantage of wire frame modeling Ambiguous drawings can result if they are not attended to, which may result in costly mistakes during manufacture

Figure 3.14 is now completed to the extent required for this illustration Screw holes for the base plate and a hole for the parallel port cable have been omitted here, but are eventually required

To provide greater clarity, internal lines can be changed to hidden lines With hidden lines the object meets all the criteria required of typical multiview drawings Each of the six possible orthogonal viewpoints can be obtained with thevpoim

command, but some hidden lines may have to be manually adjusted as the viewpoint what difficult to read The cylinders (i.e., through-holes) removed hom the base are cult to see depth and surfaces without concentrating

Development of the wire frame model has made no reference to manufacture-bility or even physical feasimanufacture-bility The drawing is simply a set of lines and curves

Flgure 3.14 Complete 3-D drawing of theVR joystick base

point to the wire frame model to render the surfaces.This rendering makes the object

representation However, since we are still working only with a wire frame, this ver-sion of rendering does not make the computer understand the surfaces Just to emphasize this last point: all the understanding so far is in the eye and

the brain of the beholder-the human CAD designer Actually, the human designer could have drawn an "Escher-like art image" that would be impossible to

manufac-ture, and the wire frame model would have happily accepted it!

3.9 SOLID MODELING OVERVIEW

3.9.'Introduction

In contrast to the previous wire frame methods, solid modeling creates objects that

devoted to the formal aspects of these topics For example, the details are

compre-hensively described in Geometric and Solid Modeling by Hoffmann (1989) and in Computer Graphics: Principles and Practices by Foley,van Dam, Feiner, and Hughes

(1992).The notes below on Boolean set operations, b-rep, and CSG are arranged in form is gratefully acknowledged

In the wire frame tutorial (Section 3.8), the person sitting at the CAD system's user interface mostly clicks on points and connects them with lines to create the joy-the CAD system's user interface constructs joy-the joystick by adding, subtracting, or intersecting individual bodies (Section 3.10) or destructs the joystick by starting with

a large block and subtracting smaller bodies (Section 3.11) Such construction or disassembling "Lego blocks" of different shapes and sizes.The operations are done

3.9.2 Regularized Bool8an Set Operations

The goal is to carry out the constructions without creating superfluous or missing material For example, careful examination of the second illustration of Figure 3.15

shows an extra dangling plane that is common to both solids after ordinary Boolean intersection This is the undesirable dangling tab sticking up in the second illustration.

solid in all cases: they can create dangling lower dimensional objects

To avoid such dangling features, it is necessary to use the regularized Boolean set operators for construction (Requicha, 1977) These are mathematically defined

so that operations on solids always create closed solids with no dangling points, lines,

or planes The regularized operators are written with superscript stars- as:

• Regularized union operator =U*

• Regularized intersection operator =n*

• Regularized difference operator= _ •.

Regularized operations are defined as follows:

In the above equation, op is one of U, n, or - For Figure 3.15 the op'" is the inter-section operator (n"')

What does the expression mean in practice when computingAn*B?

• In Step 1, Figure 3.15, the ordinary Boolean intersection,AnB, is computed to yield the volume of the object plus any dangling or lower dimensional faces, edges, or vertices

• In Step 2, the set of points that is the interior space of (AnB) =interior (A)n interior(B)is found

• In Step 3, the boundary points of (A n B) are added (these will also be faces,

edges, and vertices, but just those adjacent to any interior points of the

inter-section of A and B) The last cube in Figure 3.15 exhibits closure.

• Regularization eliminates any dangling lower dimensional objects that are not adjacent to any of the interior points of the new volume object, but keeps the

set of points that are on the boundary.

Flpre J.15 Intersection of two blocks A and B (from Geometric and Solid Modeling:

Figure 3.15 illustrates the problem with ordinary operators: the intersection of the two objectsAandBcontains the intersection of the interior and all the boundary

of one object, with the interior and all the boundary of the other one In Figure 3.15,

object A is L-shaped, whereas object B is a smaller rectangular block The ordinary face By contrast the regularized operator contains (a) the intersection of the inte-other, but (c) only a subset of the intersection of their boundaries.

Hoffman's example shows an external dangling face For further clarification, consider Foley and associates' two-dimensional example in Figure 3.16, which shows,

in cross section, an internal dangling edge CD that is removed when the regulariza-tion is done In this figure, going from (a) to (b), the lighter object on the left is inter-intersection of their boundaries (betweenAandDin Figure 3.16c) are in the

regu-larized intersection operation?

The criterion is that boundary-boundary intersections are included in the reg-ularized Boolean intersection if and only if the interiors of both objects lie on the objects have their interior regions on the same side of that part of the boundary (Aft words, when both objects have their interior regions on the same side of that part of the boundary, the interior of their intersection will include the same region with the same piece of boundary

Next, those parts of one object's boundary that intersect with the other object's interior must be included Thus, the small section BC was already part of the interior

of the darker object Now it is merged with the boundary of the lighter object and is included in the regularization as a part of the new object By contrast, if the interiors

of the original objects are on opposite sides of the shared boundary such as CD, that points adjacent to the boundary are included in the intersection Therefore, CD is not adjacent to any interior points of the resulting object and is not included after regu-larization

Boundary points are defined as those points whose distance from the object and the object's complement is zero However, boundary points need not necessarily set contains none The union of a set with the set of its boundary points is known as

interior consists of all the set's other points The regularization of a set is defined as

the closure of the set's interior points (Poley et aI., 1992)

In summary, a regular set contains no boundary points that are not immediately

adjacent to some interior point Considering the closed cube of Step 3 in Figure 3.15, its top-left-back edge is adjacent to the interior and is included But any of the points that were vertically above that edge and part of the dangling tab sticking up in Step 2 are not induded since those points are not adjacent to any interior regions of the new object

3.9 Solid Modeling Overview

,

(a)

101

(d)

Npre 3.16The regularized Boolean inlersectkm of two blocks includes the

following parts of the boundary; (1) section AB-because the interiors of both objects

lie on the same side ofit; (2) section BC-because itW8lJalready in the interior of one

of the objects and interior-boundary intersection are included; but (3) Dot section CD-because the obj«ts lie on opposite sides (from Foleylvan DamlFeiner/Hughes,

Computn Grophics: Principles and Practice.@I996,I990Addison-Wes1eyPublillhing

Company Reprinted by permission of Addison Wesley Longman,Inc.)

3.9,3 The Boundery Representation Method

Several methods are available to represent solid objects; primitive instancing, sweeps, boundary representations, spatial partitioning, and constructive solid geom-wood"; however, so far, no physical dimensions for the cube or its global position in space have been specified, Boundary representations, orb-reps, describe such an object in terms of itssurface boundaries. The b-rep could be a list of the cube's faces, each represented by a list of vertex coordinates The desirable properties needed to represent solids are described by Requlcha (1980), Resolvingambiguity is one example in Requicha's list Thus even for the simple cube it is important to list these vertices in such a way as to distinguish the outside and inside of the cube To do this

it has become customary to use the right-hand rule and list the vertices in a counter-clockwise (ccw) order as seen from the outside of the object

Also it is usual to only support solids whose boundaries are2-manifolds. The neighborhood of every point of a 2-manifold ishomeomorphic (or topologically equiv-alent) to a two-dimensional disc Figure 3.17a shows a point and its surrounding neigh-borhood on the surface of a triangular block Whether the point is on a face, diagram

an adjoined block as shown in Figure 3.17c, the neighborhood at the joint could be interpreted as two discs rather than one Figure 3.17c is therefore not a 2-manifo1d There are several specific ways to store a b-rep, As mentioned above, the sim-plest possibility is to list all of the faces with their vertices One drawback is that it is

Figure 3.17 B-rep systems usually support only 2-manifolds Every point on the surface is surrounded by a disc as shown in (a) and (b) The joined edge in (e) creates ambiguity for a topological disc Object (c) is therefore not 2-manifold because the neighborhood of the joined edge in (c) is topologically equivalent to

twu <.liM;' (frum Fuley/v"Il Dll,nlFeillerlHughe., Cumpu.ter Graphics: Principles and

Practice © 1996, 1990Addison-WesLey Publishing Company Reprinted by permission

of Addison Wesley Longman, inc.)

difficult to deriveadjacency relationships from such a representation For example, if

it is desirable to know all the faces that are incident to a particular vertex, then it would rep methods have been introduced to reduce the cost of such adjacency computation Baumgart's (1972, 1975)winged-edge data structure is one example of a b-rep aimed at compact representation and minimized computation costs.The winged-edge structure is shown in Figure 3.18 It is used to show thatedge eis used by both facesA

andBin the pyramid of Figure 3.19 Aright-hand rule(ccw) is always used in CAD

to keep track of the vertices, edges, and faces For faceAthis givesedge aas the pre-ceding edge andedge das the succeeding edge The right-hand rule for faceBgives

edge cas the preceding edge andedge bas the succeeding one Table 3.1 gives other data for the pyramid This is how the data are stored in the computer for compactness and accessibility It also captures the geometric relationships in a brief manner 3.9.4Constructive Solid Geometry (CSGI

The general techniques for solid modeling in CAD/CAM were developed during the wire frame methods

Flgure3.18 Wmged-edgedata structure

Preceding

Succeeding

Face A

FaceB

Succeeding

Preceding

F1gme 3.19 Pyramid example (from

Geometric and Solid Modeling: An Introduction by Christoph M Hoffman,

©1989 Morgan Kaufmann Publlshers.)

TABLE3.1 Defining the Pyramid with a Winged-Edge Data Structure

Shah and Mantyla (1995) describe the two "camps" that evolved with the fol-lowing quote "Ian Braid and his colleagues at the University of Cambridge worked

on boundary representations, models consisting of facets that were subsets of planar,

of Rochester introduced CSG models, consisting of a finite number of Boolean set

operations applied to half-spaces defined by algebraic inequalities (Requicha, 1977; Requicha and Voelcker, 1977)." Both methods resulted in commercial developments

by the early 1980s

In CSG, blocks can be added together, subtracted from each other, and inter-sected with each other to create more complex shapes An example of an Lehaped bracket is considered to illustrate the CSG method (Figure 3.20)

Two blocks and the hole are used to build up the more complicated solid The

two legs of the bracket are formed by a union The hole is taken out of the solid leg

by a difference Figure 3.21 shows the CSG tree for the bracket The shapes that made

which regularized Boolean operation should be carried out For the bracket, the two blocks and the hole are at the leaves

The second block is unioned with the first after being translated by (1) in the x

direction The bole is subtracted from the unioned pair after being translated by (5) in thexdirection and (2) in theydirection