giesecke_revolutions

giesecke_revolutions - CHAPTER 9 REVOLUTIONS * 239....

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Unformatted text preview: CHAPTER 9 REVOLUTIONS * 239. Revolutions Compared with Auxiliary Views. — To obtain an auxiliary View, the observer shifts his position with respect to the object, whereas in revolutions, the object is turned with respect to the observer until the desired views may be obtained upon the ordinary planes of pro- jection. * If the regular views of an object are given, the object may be revolved about an axis perpendicular to the top, front, or side planes, and the views of the object in the new position may be‘drawn. This is called a simple revolution. If this drawing is then used as a basis for another revolution about an axis perpendicular to another plane of projection, the process is known as successive revolutions. Obviously, this procedure may be con- tinued indefinitely.T N etc that in each revolution the dimension of the object, parallel to the axis, is not changed. V ‘ 240. Revolution about an Axis Perpendicular to the Front Plane. —— A simple revolution is illustrated in Fig. 410. An imaginary axis X Y is assumed, about which the object is to revolve to the desired position. In this case, the axis is selected perpendicular to the front plane of projection. NOR AL PLSITION ’ REVOLVElD FIG. 410. A REVOLUTION ABOUT AN AXIs PERPENDICULAR TO THE FRONT PLANE. * A special aspect of multi-View projection. 1' Each view is an axonometric projection (§ 262); hence it is possible to obtain an isometric projection (§ 264) by the method of revolution. - - 220 “ §242] REVOLUTION ABOUT AN AXIS 221 During the revolution all points, of the object describe circular arcs parallel to that plane. The axis may pierce the object at any point or may be eXa terior to it. The front view is drawn revolved through the angle desired (30°, for example), and the top View and the side View are obtained by pro~ jecting from the front view. The depth of the top view and the side view is found by projecting from the top view of the initial unrevolved position because the depth, in this revolution, remains unchanged. If the front View of the revolved position is drawn directly without first drawing the normal or initial position, the depth of the object, as shown in the revolved top and side views, may be drawn to known dimensions. No difficulty should be encountered by the student who understands how to obtain projections of a‘ point and of a line. See §§ 203 to 207. Note the similarity between the top and side views in Fig. 410 (II), and some of the auxiliary views of Fig. 385 (c). Y ' Y NO MAL POSlTlON FIG. 411. A REVOLUTION ABOUT AN Axrs PERPENDICULAR TO THE TOP PLANE. 241. Revolution about an Axis Perpendicular to the ' Top Plane. — A revolution/about an axis perpendicular to the top plane of projection is shown in Fig. 411. _An imaginary axis X Y is assumed perpendicular to the top plane of projection, the top view is drawn revolved to the desired posi— tion (say 30°), and the other views are obtained by projecting from this VleW. . During the revolution all points of the object describe circular arcs paral- lel to the top plane. Hence the heights of all points in the front View and in the side view of the object in the revolved position remain unchanged and may be drawn by projecting from the initial front and side views. Note the similarity between the front and side views in Fig. 411 (II) and some of the auxiliary views of 386 (c). 242. Revolution about an Axis Perpendicular to the Side Plane. —— A revolution about an axis X Y perpendicular to the side plane of projection 222 REVOLUTIONS [Ch. 9 is illustrated in Fig. 412. During the revolution, all points of the object describe circular arcs parallel to the side plane of projection. Hence the width of the top view and 0f the front view of the Object in the revolved positions remains unchanged and may be obtained by projection from the top view and from the front View of the Object in its initial unrevolved position, or may be set off by direct, measure- ment. Note the similarity between the top and front views in Fig. 412 (II) and some of the auxiliary Views of Fig. 387 (c). 243. Successive Revolutions. —— It is possible to draw an object in an in— finite number of oblique positions by making successive revolutions. Such a procedure (Fig. 413) limited to three or four stages, offers excellent practice FIG- 412- AREVOLUTION ABOUT AN in orthographic projection. While it ESEPERPENDICUI‘AR To THE SIDE is possible to makeseveral revolutions . . of a simple object without the aid of a system of numbers, it is absolutely necessary in complex revolutions to assign a number or a letter to every corner of the object. “ The numbering or lettering must be consistent in the various views of the several stages of revolution. Figure 413 shows four sets of multi—view projections, or four drawings numbered, respectively, I, II, III, and IV. These represent the same object in different positions with reference to the planes of projection. In space I, the object is represented in its normal position, with its faces parallel to the planes of projection. In space II, the object is repre— sented after it has been revolved clockwise through an angle of 30° about an axis perpendicular to the front plane. ' During the revolution, all points of the object describe circular arcs parallel to the front plane of projection and remain at the same distance from that plane. The side View, therefore, may be projected from the side View of space I' and the front view of space 11. The top View may be pro- jected from the front View and side View of space II. REVOLVED § 244] COUNTER—REVOLUTION 223 In space III, the object is taken as represented in space II and is revolved counter—clockwise through an angle of 30° about an axis perpendicular to the top plane of projection. During the revolution, all points describe horizontal circular arcs and remain at the same distance from the top plane of projection. The top view FIG. 413. SUCCESSIVE REVOLUTIONS OF A PRISM. is copied from space II but is revolved through 30°; the front and side Views are obtained by projecting from the front and side views of space II and from the tOp view of space III. In space IV the object is taken as represented in space III and revolved clockwise through 15° about an axis perpendicular to the side plane of projection. During the revolution, all points of the object describe circu— lar arcs parallel to the side plane of projection and remain at the same dis— tance from that plane. The side view is copied (§ 97), from the side view of space III but revolved through 15°. The front and top views are projected from the side view of space IV and from the top and front views of space III. A similar successive revolution applied to a pyramid is shown in Fig. 414. 244. Counter—Revolution. —— It is sometimes necessary to draw the Views of a given object which is located with reference to a given inclined 224 REVOLUTIONS [Ch. 9 FIG. 414. SUCCESSIVE REVOLUTIONS OF A PYRAMID. surface. In such case, it is necessary to revolve the inclined surface until it is perpendicular to one or two of the planes of projection and to draw the Views of the given object in this revolved position. Having done this, the FIG. 415. COUNTER-REVOLUTION or A PRISM. inclined surface and the object are counter—revolved until the inclined surface occupies its original position, when the given object will be repre— sented in its true position. This is illustrated in Fig. 415. Assume that the given inclined surface 8—4—3—7 (I) is similar to the surface 8—4—3—7 shown in space IV of Fig. 413 and that it is required to find the Views of a prism one-half inch high, hav- ing the given surface as its base. Revolve the surface about any hori— zontal axis XX perpendicular to the side plane, until the edges 8—4 and 3—7 are horizontal, as shown in space H, Then revolve the surface about §245] TO FIND THE TRUE LENGTH OF A LINE. 225 any vertical axis YY until the edges 8—7 and 4—3 are parallel to the front plane, as shown in space III. In this position the given surface is perpen— dicular to the front plane, and the front and top views of therequired prism can be drawn, as shown by dashed lines in the figure, because the edges 4*1, 3—2, etc., are parallel to the front plane and are, therefore, shown in their true lengths (§ 245), one-half inch. Having drawn the two Views in space III, counter—revolve the object from III to II and then from II to I to find the required views of the given object in space I. 245. To Find the True Length of a Line.* —Fig. 416. If a line is parallel to one of the planes of projection, its projection on that plane is equal in length to the line. See §§ 204 and 205. In Fig. 416 (a), the element AB of the cone is oblique to the planes of projection; hence its projections are fore- shortened (§ 205). If AB is revolved about the axis of the cone until it I , 02 _E s F E TRUE TRUE LENGTH LENGTH ,s (d) F- F’ (a) (b) (C) FIG. 416. TRUE LENGTH OF A LINE. coincides with either of the contour elements, for example AB’, it will be shown in its true length in the front view because it ‘will then be parallel to the front plane of projection. Likewise, in Fig. 416 (b), the edge of the pyramid CD is shown in its true length CD’ when it has been revolved about the axis of the pyramid until it is parallel to the front plane of projection. In Fig. 416 (c), the line EF is shown in its true length when it has been revolved about a vertical line until it is parallel to the front plane of projection._ The true length of a line may also be found by constructing a right triangle * Note that here the true length is found by revolving the line until parallel to a plane of projection, while by the method of auxiliary views it is found by placing the observer in the necessary position to view the line in its true length. See § 236. ' See Prob. 5, p. 227. Layout 30. FIG. 418. Draw revolved positions of prism as indicated. See Prob. 6, p. 227. Layout 3C. ' 226 / § 246] PROBLEMS , 227 (Fig. 416, (1) whose base is equal to the top view of the line, and whose altitude is the difierence in elevation of the ends. The hypotenuse of the triangle is equal to the true length of the line. PROBLEMS * 246. Revolution Problems. —- Problems in revolution are of con- siderable value in training the constructive imagination, and it is recom— mended that at least a few be included in the student’s schedule of drawings. All corners should be numbered, especially when successive revolutions are to be drawn. The student is urged to compare carefully the method of auxiliary Views with that of revolutions (§§ 224—237) and to apply the principles of §§ 203*210 relative to points, lines, and planes. 7 "I i E11 :2; Lag/+1 Lag»! Hz“ was I“ Hm 324:7' :37 —l ‘7 “1 \ if} j if :\iV I II DI m ‘ I III FIG. 419. Draw revolved positions of assigned block as in Fig. 418, p. 226. Layout 30. Prob. 1, Figs. 410, 411, and 419.—-—Using Layout BC, divide the sheet into four equal rectangles. Draw a simple revolution as in Figs. 410 and 411, but use an object selected from Fig. 419. ‘ r Prob. 2, Figs. 412 and 419.-— Using Layout 30, divide the sheet into four equal rectangles. In the two left—hand spaces draw a simple revolution as in Fig. 412, using an object selected from Fig. 419. In the two right-hand spaces, draw another simple revolution as in Fig. 412, but using a different object from Fig. 419, and revolving through 45° instead of 30°. Prob. 3. ——-Using Layout 30, divided into four equal parts, draw the revolutions shown in Fig. 413. Use a block of dimensions %” X 1%” X 1%”. Prob. 4.———Using Layout 3C, divided into four equal parts, draw the revolutions shown in Fig. 413. Use a pyramid having a base 1%” X 1%” and an altitude of 1%”. Prob. 5, Fig. 417. —— Using Layout 30, divided into four equal rectangles, draw in the upper left space three views of a plane rectangle as shown, including co—ordinate axes. See Fig. 270, p. 162. Number the corners as shown and draw the succeeding revolu- tions in the order indicated. Prob. 6, Fig. 418.-——Using Layout 30, divided into four equal rectangles, draw the views of the object in the several positions indicated. * Problems in revolutions in convenient form for solution may be found in Technical Drawing Problems, designed to accompany this text, published by The Macmillan Company. 228 REVOLUTIONS [Ch. 9 Prob. 7, Fig. 419.—Using Layout 3C, divided into four equal rectangles, draw the object assigned from Fig. 419 in the same manner as shownvin Fig. 418. FIG. 420. (See Probs. 8 and 9, below) - Prob. 8, Fig. 420. —— Draw three views of a right prism 1%” high that has as its lower base the triangle shown in Fig. 420 (I). See § 244, Fig. 415. ' Prob. 9, Fig. 420. — Draw three views of a right pyramid 2’ ’ high, having as its lower base the parallelogram shown in Fig. 420 (II). See § 244, Fig. 415. ...
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giesecke_revolutions - CHAPTER 9 REVOLUTIONS * 239....

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