. MECHANICAL DRAWING PROBLEMS BOOKS BY CHARLES WILLIAM WEICK, B. Sc. ELEMENTARY MECHANICAL DRAWING. 250 pages, 6x9, 324 Illustrations $1.75 MECHANICAL DRAWING PROBLEMS. 153 pages, 6x9, 48 Illustrations and 112 Plates $1.25 IN PREPARATION MACHINE DRAWING PROBLEMS MECHANICAL DRAWING PROBLEMS BY CHARLES WILLIAM WEICK, B. Sc. ASSISTANT PROFESSOR OF DRAWING AND DESIGN, TEACHERS COLLEGE COLUMBIA UNIVERSITY, IN THE CITY OP NEW YORK; AUTHOR OF "ELEMENTARY MECHANICAL DRAWING." FIRST EDITION McGRAW-HILL BOOK COMPANY, INC. 239 WEST 39TH STREET, NEW YORK LONDON: HILL PUBLISHING CO., LTD. 6 & 8 BOUVERIE ST., E. C. 1917 COPYRIGHT, 1917, BY THE MCGEAW-HILL BOOK COMPANY, INC. PREFACE This volume is a book of examples and problems for the study of mechanical drawing. It is intended to be used under the direc- tion of a teacher, although the ambitious and painstaking student can solve many of the problems without such aid. It aims to provide a large selection of typical drawings carefully worked out as examples, each one of which is accompanied by appropriate problems suitable for elementary and advanced work in mechan- ical drawing. Explanations and directions are, when needed, expressed in simple and direct language, and in the fewest possible words. The principles of the construction of a drawing are shown in the examples in the graphic language of the draftsman. The book is divided into three parts. The First Part gives in brief outline such matter as the student will need to know before beginning work. It contains, also, a limited number of geometrical constructions which will be found helpful in solving some of the problems given. The Second Part the practical work is divided into four sections, namely: Projections, Developments and Intersections, Isometric Drawings, and Machine Details. Nearly all the example-drawings are fully dimensioned and, where necessary, described by explanatory notes. They are not to be copied but are intended to serve as a guide to the student when making his own drawings for which directions are given in the form of problems. In each section there are many more problems than any one student can work in the amount of time which is usually allotted to the subject of mechanical drawing, and a judicious selection must necessarily be made by the teacher to meet the individual student's requirement. These problems are arranged in the order of their difficulty. The problems given in the begin- ning of each section are not too difficult for a beginner in mechan- ical drawing, while those toward the end of the sections will be found suitable for advanced students. Each section begins with problems that are suitable for Junior High Schools, High Schools, and Evening Schools, and advances gradually to problems that are more difficult of solution, suitable for Vocational Schools, Trade Schools, and Colleges. In the first three sections two vi PREFACE problems are given with each example-drawing, and in the fourth section, three problems are given. The problems on Developments and Intersections are suitable as an introductory course for students interested in sheet metal pattern drafting. The problems on Machine Details will form a logical introduction to machine design. The Third Part of the book contains tables and general infor- mation of use to draftsmen, and to which frequent reference should be made when working problems in the second and fourth sections of the Second Part. The author's not inconsiderable experience as draftsman and designer of machinery, and as teacher of drawing and design, leads him to believe that teachers will find sufficient material in the following pages to enable them to formulate suitable courses for classes in mechanical drawing for all schools where this subject is taught. The author is indebted to many books on mechanical drawing for helpful suggestions in the preparation of the drawings of this volume, especially to Professor Thomas E. French's "Engineering Drawing." In preparing the manuscript and drawings he gratefully acknowledges much valuable help he has received from Mr. Arthur F. Hopper, director of manual arts, Plainfield, New Jersey; Mr. Frank C. Panuska, instructor in mechanical drawing, and Mr. Ralph Breiling, of Teachers College. NEW YORK, October, 1917 CONTENTS PREFACE PART I. INTRODUCTION GENERAL INFORMATION. Instruments and Materials PAGE 1 Drawing Lrnes . 1 Lines Used Border Lines Lettering Dimensions -. . 2 2 ............. 4 . 6 Titles Sharpening Pencils 7 9 Penciling Sequence for Penciling Inking Sequence for Inking Use of Lines Location of Views Working Drawings 10 10 10 10 11 11 . . 12 GEOMETRIC CONSTRUCTIONS. To Bisect a Given Angle 13 To Bisect a Given Arc 13 To Set Off an Angle Equal to a Given Angle from a Point on a Given Line , . 13 To Divide a Given Line into Any Number of Equal Parts .... 14 To Find the Point of Tangency of a Given Circular Arc and a Given Straight Line 14 To Find the Point of Tangency of Two Given Circular Arcs. . . 14 To Draw an Arc of Given Radius Tangent to Two Straight Lines Meeting at Right Angles 15 To Draw an Arc of Given Radius Tangent to Two Intersecting Straight Lines 15 To Draw an Arc of Given Radius Tangent to a Given Straight Line and a Given Circular Arc 15 To Draw a Circular Arc Tangent to a Given Straight Line and Tangent at a Point on a Given Circular Arc 16 To Connect Two Given Parallel Lines with a Compound Curve aad Tangent at Given Points 16 To Draw a Circular Arc of Given Radius Tangent to Two Given Circular Arcs 16 To Construct a Regular Polygon of Any Number of Sides Within a Circle of Given Diameter. . 17 viii CONTENTS PAGE To Construct a Regular Polygon of Any Number of Sides on a Line of Given Length 18 To Draw an Ellipse, the Major and Minor Axes Being Given ... 18 To Draw an Approximate Ellipse, the Major and Minor Axes Being Given 18 CONVENTIONAL SCREW THREADS Screw Threads 19 INTERSECTION OF Two CYLINDERS. To Find the Line of Intersection of Two Cylinders with Axes in the Same Plane and at Right Angles to Each Other 20 PLANE INTERSECTION OF A SOLID. To Find the Lines of Intersection of a Surface of Revolution Cut by Two Planes at Right Angles to Each Other and Parallel to the Axis. 21 PART II. EXAMPLES AND PROBLEMS PROJECTIONS. Section I. Thirty Examples Sixty Problems 23-53 DEVELOPMENTS AND INTERSECTIONS. Section II. Thirty-six Examples Seventy-two Problems. . . 54-95 ISOMETRIC DRAWING. Section III. Twenty-two Examples Forty-four Problems. 96-117 MACHINE DETAILS. Section IV. Twenty-four Examples Seventy Problems . . 118-141 PART III. TABLES LTRT OF TABLES. I. Cap Screws 144 II. U.S. Standard Bolts and Nuts 145 III. Machine Screws 146 IV. Briggs Standard Pipe Threads 147 V. Set Screws 148 VI. Gib Keys ; , ' 148 VII. Feather Keys or Splines 148 VIII. Automobile Screws and Nuts 149 IX. Jarno Tapers 150 X. Morse Tapers !..... 150 XI. Decimal Equivalents 151 XII. Areas and Circumferences of Circles 152 Conventional Section Lines . 153 LIST OF PLATES SECTION I PROJECTIONS PAGE 1. Prism 25 2. Tapered Objects 25 3. Circular Objects 27 4. Geometric Objects 27 5. Geometric Object 29 6. Geometric Solid 29 7. Projection of Letter 31 8. Projection of Letter 31 9. Hollow Cylinder 33 10. Geometric Solid 33 11. Projection of Letter 35 12. Projection of Letter 35 13. Triangular Prism . . . , 37 14. Pentagonal Prism 37 15. Square Pyramid 39 16. Hexagonal Pyramid 39 17. Right Cone 41 18. Right Cylinder '41 19. Square Prisms 43 20. Cross and Prism 43 21. Letter and Prism ' 45 22. Letter and Prism 45 23. Cylinder and Prism 47 24. Cone and Cylinder 47 25. Projection Problems 49 26. Projection Problems 49 27. Projection Problems 51 28. Projection Problems 51 29. Projection Problems 53 30. Projection Problems 53 SECTION II DEVELOPMENTS AND INTERSECTIONS 31. Rectangular Prism 55 32. Truncated Rectangular Prism 55 33. Triangular Wedge 57 34. Truncated Square Prism 57 ix x LIST OF PLATES PAGE 35. Truncated Triangular Prism 59 36. Truncated Hexagonal Prism 59 37. Truncated Pentagonal Prism 61 38. Triangular Pyramid 61 39. Truncated Square Pyramid 63 40. Scalene Pyramid 63 41. Truncated Cylinder 65 42. Truncated Cone 65 43. Conic Section 67 44. Conic Section 67 45. Intersecting Cylinders 69 46. Intersecting Solids 69 47. Cone and Cylinder 71 48. Cone and Prism 71 49. Three-Piece Elbow 73 50. Vertical and Oblique Cylinders 73 51. Circular Offset Pipe 75 52. Three-Piece Conical Elbow 75 53. Intersecting Pipes 77 54. Intersecting Pipes 77 55. Transition Piece 79 56. Transition Piece 79 57. Scalene Cone 81 58. Transition Piece 81 59. Intersection of Two Prisms 83 60. Development of Two Intersecting Prisms 83 61. Intersection of Two Cones 85 62. Development of Two Intersecting Cones 85 63. Intersection of Two Cones 87 64. Development of Two Intersecting Cones 87 65. Intersection of Two Cones 89 66. Development of Two Intersecting Cones 89 Supplementary Problems 90-96 SECTION III ISOMETRIC DRAWING 67. Isometric Blocks 97 68. Joints 97 69. Joints 99 70. Mortise and Tenon Joints 99 71. Miter Box 101 72. Drawer and Table Joints 101 73. Box with Hinged Lid 103 74. Sawhorse ' 103 75. Isometric Prisms 105 76. Prisms . . 105 LIST OF PLATES xi PAGE 77. Isometric Circles 107 78. Isometric Arcs 107 79. Hollow Cylinder 109 80. Bearing Cap 109 81. Magnet Pole Pieces Ill 82. Milling Cutter and Face Plate Ill 83. Knife and Fork Box 113 84. Uniform Motion Cam 113 85. Bracket Shelf 115 86. Kitchen Table 115 87. Knuckle Joint 117 88. Small Bench 117 SECTION IV MACHINE DETAILS 89. Eccentric Sheave 119 90. Hand-Wheel 119 91. Engine Crank 121 92. Connecting-Rod End 121 93. Crane Hook 123 94. Clutch Couplings 123^ 95. Screw Threads 125" 96. Bolts and Nuts 125 97. Lathe Carrier 127 98. Clamp Coupling 127 99. Stuffing Box 129 100. Safety Coupling 129 101. Nut Coupling 131 102. Forked Rod 131 103. Shifting Gear 133 104. Planer Jack 133 105. Belt Pulley 135 106. Belt Pulley 135 107. Grease Cup 137 108. Lathe Chuck .... .~ 137 109. Pipe Union 139 110. Pipe Union 139 111. Screw Jack 141 112. Ball Bearing 141 MECHANICAL DRAWING PROBLEMS PART I INTRODUCTION GENERAL INFORMATION Instruments and Materials. To obtain good results in mechanical drawing a good set of drawing instruments is neces- sary. Economy in their purchase is unwise, because drawings made with inferior equipment will be of inferior quality in tech- nique. A good set of drawing instruments with proper care will serve a draftsman's needs almost a lifetime. The following list of instruments and materials comprises a minimum equipment consistent with good work: A compass with pencil leg, pen leg, and extension bar. Divid- ers. Bow pencil. Bow pen. Two ruling pens. 45 celluloid tri- angle. 30 X 60 celluloid triangle. Two celluloid curves. Pro- tractor. 12-inch architect's scale. Drawing board. T-square. Pencils. Sandpaper pencil pointer. Pencil and ink eraser. Two penholders and pens for lettering. Penwiper. Thumb tacks. Cleaning rubber. Drawing paper and drawing ink. Drawing Lines. Draw horizontal lines at the upper edge of the T-square blade; never draw lines at its lower edge. Draw verti- cal lines, and lines making angles 15, 30, 45, 60, and 75 with the triangles resting against the T-square blade. Lines making angles other than those mentioned are drawn by the aid of the T-square, or the triangle, placed in the desired position. Fig. 1 shows the method for drawing lines perpendicular to, and also lines making angles of 60, 30, and 45 with the T-square blade. The arrows shown on the lines indicate the direction in which the lines should be drawn. Fig. 2 shows methods for drawing lines making angles of 15 and 75 with the T-square blade. 1 2 MECHANICAL DRAWING PROBLEMS Circles and circular arcs are drawn with the compass, or the bow instruments. Irregular curves are drawn by aid of the cel- luloid curves, with pencil or ruling pen. 90- 6CJ\ 760' FIG. 1. Showing vertical and oblique lines. FIG. 2. Position of triangles for drawing 15 and 75 lines. Lines Used. The various lines shown in Fig. 3 are of the kind used by most draftsmen, and may be considered as standard. In drawings which are to be inked or traced, continuous pencil lines may be used where there is no likelihood of an error when inking. Border Lines. The object of a border line is to give the draw- ing a finished appearance. The border line and the trimming line should be drawn in pencil before the drawing itself is begun, and should have the dimensions shown in Fig. 4. When inking, the border line should be the last line drawn. INTRODUCTION VISIBLE OUTLINE HIDDEN OUTLINE CENTER LINE PPOUECTION LINE AUXILIARY LINE EXTENSION LINE DIMENSION LINE BORDER LINE FIG. 3. Conventional lines. - 1 4 "S * 3 : N /2 '2 /5 4 Q L 10 5 \ \\\fioflOR LINE DRAWING BOARD ^CUTTING LINE: X OGC or PAPCR FIG. 4. Layout of border and cutting lines. 4 MECHANICAL DRAWING PROBLEMS Lettering. Since the general effect in the appearance of a drawing depends in a large measure on the appearance of its title, notes and dimensions, lettering and figuring form an essential part of the draftsman's work. The prime requisite for good lettering and figuring is simplic- ity of style and uniformity of treatment. These are obtained by correctness of form of letters and figures, a uniform inclination and height, and proper spacing. These results must be obtained by accuracy of hand and eye, since no rules can be followed which will be practical for all combinations of letters and worde. Since it is generally difficult for a beginner to letter well, it will be necessary, in order to acquire proficiency and obtain good results, to devote some time to the practice of making letters singly and in combinations to form words. When placing a title, notes or figures on a drawing the begin- ner should always remember, no matter how good a drawing may be, or how much time was given to its execution, if the lettering or figuring is hurriedly or carelessly done, the completed drawing will not present a neat appearance. The "Gothic," or uniform line letters, shown in Fig. 5, find favor with draftsmen and are generally used for mechanical drawings. Limiting lines should always be drawn to serve as a guide for the height and proper alignment of letters. They are used by the most experienced of letterers. Since it is important that all letters have the same slant, slanting lines should be drawn to serve as a guide to the eye. These lines may be drawn about one-quar- ter inch apart. See Fig. 6. All limiting and guide lines should be drawn with a wedge-pointed pencil, and very fine so they may be easily erased. Letters and figures are made with a conical- pointed pencil. Small letters need not necessarily be drawn with the pencil, but may be put in directly with ink. Titles and large letters should always be drawn in pencil before inking. For titles and large letters a "Hunt's Extra Fine Shot Point Pen," Number 512, and for small letters and figures a "Hunt's Strand Pen," Number 54, will be found suitable. "Leonard's Ball Point Pen," Number 506F, for large letters, and "Gillott's Pen," Number 303, for small letters and figures, also find favor among many draftsmen. For practical work, the height of letters and figures should be INTRODUCTION -LETTERS AND FIGURES- ABCDEFGHIJKLMN OPQRSTUVWXYZ 12345676 9 O EXTENDED COMPRESSED ABCDEFGHIJKLMN OPQRSTUVWX Y Z 7234,567690 ETXTET/VOETO COMPRESSED obcdefgh i j k I m n O p q r s t u v w x y z Lower Case Letters =4-4 CAPITALS FOR TITLES*** CAPITALS FOR SUB -TITLES" ^> For Descriptive Matter" *ig For Dimensions ef 24^. M% FIG. 5. Inclined single-stroke letters. 6 MECHANICAL DRAWING PROBLEMS as shown in Fig. 5. The slant may be from 60 to 75, according to the judgment of the student; or, they may be vertical, if preferable, as shown in the last two lines of Fig. 6. FIG. 6. Showing use of guide lines. Dimensions. To dimension a drawing means to place upon it all measurements required for the construction of the object represented. Fig. 7 shows an object with all dimensions neces- sary for its construction. Dimensions are placed in dimension lines which terminate with arrow heads. A break, or space, should be left near but not necessarily in the center of the lines to receive the dimensions. See Fig. 7. Dimensions should not be placed on center lines. There are several ways for writing dimensions; for instance, four inches may be written as 4 inches, 4 in., 4", or simply 4, the accent sign (") being omitted when all dimensions of the object are in inches. The proper use of the sign (") is shown in Fig. 7. Feet and inches are written thus; 4 ft. 6 in., or, more commonly, 4'-6"; 3'-4|"; 5'-0", etc. The use or omission of the inch sign, which is omitted in the following drawings, is left to the judgment of the instructor. INTRODUCTION 7 Figures for whole numbers should be -& high, and fractions should be ^ high, over all. The division line of a fraction should be in line with the dimension line. The figures of a frac- tion should not touch the division line. This requires that each figure of the fraction be a trifle less in height than the whole num- ber. See last line of Fig. 5. "TT r wi* 1 FIG. 7. Showing a method for dimensioning. Titles. The title of a drawing may be placed at the top of the sheet centrally with respect to the vertical lines of the border, and the problem number in the upper left-hand corner, as shown in Fig. 8. The distance marked x should not exceed three-quarters of an inch in any drawing. In some drawings, depending on the problem and the arrangement of views, this distance will neces- sarily be somewhat less, but in no case should it be less than one- half inch. In working drawings, titles are always placed in the lower right- hand corner, as shown in Fig. 9, the problem number in the upper left-hand corner being omitted. Fig. 10 shows three forms of general titles. Any one of these forms may be adopted instead of the titles shown in the example drawings. It is customary in some schools to place titles of practice draw- ings at the top of the sheet, and titles of working drawings in the 8 MECHANICAL DRAWING PROBLEMS lower right-hand corner. A slight readjustment of views in the following drawings, which have their titles on top, will provide space for titles in the lower right-hand corner, if preferable. FIG. 8. Title-form when placed on top. TITLE DRAWINGS BORDER LINE FIG. 9. Title-form when placed at bottom. ^.RECTANGULAR PRISMS^ I DRAWN BY J.H.SMITH 6-12-17^ .THREE-PIECE -J* ___ HIHTJZI SCALE 6 - /' =ZZ__iIZI ~ H. SMITH UUNE 12 1917 DESIGN FOR ENGINE ECCENTRIC? ^JLH-L SIZE 6-12-17 ZTZL ^zf^^ DRAWN BY J. H. SMITH zzi f n? FIG. 10. Various title-forms. Before placing a title on a drawing, it would be well to make a trial title on a separate piece of paper, as several attempts may be necessary to produce a satisfactory result. When the trial title is found satisfactory, it can then be copied on the drawing. INTRODUCTION 9 Underscore lines drawn to a title, when placed at the top of the drawing, as shown in Fig. 8, are optional with the instructor. Their use, however, adds character and stability to lettering. FIG. 11. Wedge-shaped pencil point. FIG. 12. Cone-shaped pencil point. FIG. 13. Correct position of pencil. FIG. 14. Incorrect position of pencil. Sharpening Pencils. For wedge-shaped points, remove the wood as shown in Fig. 11, with a sharp knife, or a chisel, exposing the lead which is then sharpened with a sandpaper pencil pointer or a fine file. Fig. 12 shows a cone-pointed pencil. 10 MECHANICAL DRAWING PROBLEMS Fig. 13 shows the correct position of the pencil, relative to the T-square or triangle, for drawing straight and accurate lines. Fig. 14 shows an incorrect position of the pencil. Penciling. For good penciling, which is a prerequisite to good inking, two pencils should be used; one 5H for drawing lay-out lines, which are to be drawn fine, and one 4H for drawing lines whose positions and limitations have been determined. These pencils should be sharpened to long slender wedge-shaped points. For pointing off distances, making letters, figures, and arrow heads, a 3H pencil sharpened to a cone-point should be used. Pencils should be sharpened frequently to keep the points in good working condition. All lines should be drawn as fine as is con- sistent with clearness. Sequence for Penciling. A general sequence for penciling, when conditions permit, may be as follows: 1. Draw border and cutting lines. 2. Lay off space required for title. 3. Decide on number of views required. 4. Make a rough, free-hand sketch in note book of views decided upon. 5. Draw horizontal and vertical center lines. 6. Lay out with as few lines as possible the position of each view. 7. Draw limiting horizontal outlines of all views. 8. Draw limiting vertical outlines of all views. 9. Complete all views. 10. Draw dimension lines. 11. Fill in dimensions. 12. Add explanatory notes, if required. Inking. When preparing to ink a line do not overload the pen as the ink is likely to flow too freely and thereby cause a blot. Be sure, however, to have enough ink in the pen to finish the line about to be drawn, and always try the pen, after adjustment for width of line, on a piece of paper. When finished do not lay the pen aside without cleaning. Fig. 15 shows the correct position of the ruling pen for inking, relative to the T-square, triangle, or irregular curve. The ruling pen should not be used for drawing free-hand lines. Sequence for Inking. A sequence for inking is more necessary than one for penciling, due to the necessity of changing instru- ments and waiting for ink to dry. A good order for inking is as follows : INTRODUCTION 11 1. Small circles and circular arcs with the bow pen. 2. Large circles and circular arcs with the compass. 3. Irregular curves. 4. Horizontal lines beginning with the uppermost. 5. Vertical lines beginning with those at the left-hand end. 6. All 30, 60, and 45 lines. 7. Other oblique lines. 8. Horizontal and vertical center lines. 9. Extension and dimension lines. 10. Put in dimensions, arrow heads, and explanatory notes. 11. Section lines. 12. Border line. In some drawings it may be desirable to ink center lines first. Use of Lines. The diagram drawing shown in Fig. 16 illus- trates the correct use of the lines shown in Fig. 3. It also shows the correct and incorrect methods of forming junctions of full and hidden lines and of making arrow heads. The lengths of dashes and spaces of broken lines, and the size of arrow heads, are to be estimated by eye and should as closely resemble those shown in Fig. 3 as conditions permit. The drawing of projection lines is recommended for all practice drawings. In working drawings projection lines should be omitted. FIG. 15. Correct position of ruling pen. Location of Views. The amount of space to be occupied by the title, and the size and location of each view of an object about to be drawn, should be determined before the drawing is begun. This information can be obtained beforehand by making a free- 12 MECHANICAL DRAWING PROBLEMS hand sketch, preferably in a note book kept for the purpose, and to a scale if necessary, of the desired views, the amount of space between the views, and the distance of each view from the hori- zontal and the vertical lines comprising the border line. In many cases, if the drawing is quite simple, a few lines drawn very lightly will serve as a preliminary lay-out, and will permit of a readjustment of views should the first trial be unsuccessful. /I , EXTENSION LINE / M< y _4 vi-/-JJk INCORRECT DIMENSION LINE PROJECTION LINES +k H - x - \ S V N \ \ \ \ ^VISIBLE OUTLINE ^HIDDEN OUTLINE AUXILIARY LINE FIG. 16. Correct and incorrect junctures of lines. To obtain a pleasing appearance, the various views should be arranged and placed in such a way as to utilize the available space on the sheet to the best advantage. Working Drawings. A working drawing, also called a shop drawing, is one which will impart such definite and unmistakable information in the form of a graphic representation with all neces- sary dimensions, notes and explanatory matter, the kinds of materials to be used and methods of finishing, as is required by a workman for the construction of the object represented. INTRODUCTION GEOMETRIC CONSTRUCTIONS 13 To Bisect a Given Angle (Fig. 17). Let ABC be the given angle. With B as center and any suit- able radius describe arc ab. With a and 6 as centers and any suitable radius describe arcs intersecting at c. Draw Be, the bisector of the angle. FIG. 17. To Bisect a Given Arc (Fig. 18). Let AB be the given arc. With A and B as centers and any suitable radius describe arcs intersecting at a. With A and B as centers, and the same or other suitable radius, describe arcs intersecting at b. Draw ab intersecting the arc at c, the required point. I FN ! i rs FIG. 19. FIG. 20. To Set off an Angle Equal to a Given Angle from a Point on a Given Line (Fig. 19). Let ABC be the given angle and E the given point on line EF . With B as center and any suitable radius draw arc ab. From E with the same radius draw arc cd. With radius ab and c as cen- ter, intersect cd in e. Draw Ee. Then angle DEF equals angle ABC. 14 MECHANICAL DRAWING PROBLEMS To Divide a Given Line into Any Number of Equal Parts (Fig. 20). Let AB be the given line, and six the required number of parts. Draw Ba, at any angle to AB. With any convenient length lay off on Ba the required number of parts, giving points 6, c, d, e, f, g. From the points on Ba, and parallel to gA, draw lines intersecting A B at b', c', d', e', /', giving the required parts. FIG. 21. FIG. 22. To Find the Point of Tangency of a Given Circular Arc and a Given Straight Line (Fig. 21). Let AB be the given arc of center E; and CD, the given line. From E draw a perpendicular to CD. The intersection a will be the point of tangency. FIG. 23. To Find the Point of Tangency of Two Given Circular Arcs (Fig. 22). Let AB of center C and ED of center F be the given arcs. Draw CF. The intersection a will be the point of tangency. INTRODUCTION 15 To Draw an Arc of Given Radius Tangent to Two Straight Lines Meeting at Right Angles (Fig. 23). Let AB and AC be the given lines and R the given radius. With A as center and R as radius draw an arc intersecting the given lines in a and 6. With a and 6 as centers and the same radius draw arcs intersecting at c. With c as center and the same radius draw the required arc ab. Points a and b are the points of tangency of the arc and the given lines. To Draw an Arc of Given Radius Tangent to Two Intersecting Straight Lines (Fig. 24). Let AB and AC be the given lines, and R the given radius. At a distance R draw parallels to AB and AC, intersecting at c. With c as center and radius R draw the required arc. Points a and b are the points of tangency. FIG. 25. FIG. 26. To Draw an Arc of Given Radius Tangent to a Given Straight Line and a Given Circular Arc (Fig. 25). Let A B be the given line, CD the given circular arc of radius R', and R the given radius. With E as center and radius R + R' draw an arc; also draw a line parallel to AB at distance R, inter- secting the arc at a. With a as center and radius R draw the required arc. Points b and c are the points of tangency. NOTE. The point of tangency b lies on a straight line joining centers a and E, and the point of tangency c lies at the foot of a perpendicular drawn from a to AB. 16 MECHANICAL DRAWING PROBLEMS To Draw a Circular Arc Tangent to a Given Straight Line and Tangent at a Point on a Given Circular Arc (Fig. 26). Let AB be the given line and F the point on the given arc CD of radius R. Draw a line tangent at F intersecting line AB at A . With A as center and radius R' draw arc ab. With a and 6 as centers and radius R" draw arcs intersecting at c. Draw Ac; also draw a straight line from E through F giving point d. With d as center and radius dF draw the required arc. Points F and e are the points of tangericy. To Connect Two Given Parallel Lines with a Compound Curve and Tangent at Given Points (Fig. 27) . Let AB and CD be the given lines; E and F the given points. Connect E and F by a straight line. Assume any point, as G. Draw ab and cd, the bisectors of EG and CrF, respectively. At E and F erect perpendiculars giving points e and /. With center / and radius R, equal to /F, draw arc GF. With center e and radius 72', equal to eE, draw arc GE, completing the required curve. Points E and F are the points of tangency. Point G is the point of tangency of the two arcs. To Draw a Circular Arc of Given Radius Tangent to Two Given Circular Arcs (Fig. 28). Let R be the radius of the given arc; and R' and R" the radii of the given circular arcs. With E and F as centers, and radii R + R' and R + R", draw arcs intersecting at a. With a as center and radius R draw the required arc. Points 6 and c, on straight lines drawn from a to E and a to F, respectively, are the points of tangency. INTRODUCTION 17 To Construct a Regular Polygon of Any Number of Sides Within a Circle of Given Diameter (Fig. 29). Let ABCD be the given circle. Divide AC into as many equal FIG. 29. parts as the polygon is to have sides, in this case five. With A and C as centers, describe arcs of radius AC, intersecting at a. A line drawn through a2, intersecting the circle at 6, determines FIG. 30. the length bC, one side of the polygon. With bC as radius and b as center describe a small arc giving point d. The other points are found similarly. 18 MECHANICAL DRAWING PROBLEMS To Construct a Regular Polygon of Any Number of Sides on a Line of Given Length (Fig. 30). Let AB be the given length. With B as center and A B as radius draw the semi-circle A bo. Divide the semi-circle into as many equal parts as the polygon is to have sides, in this case five. Draw B2. Bisect AB and B2 to find the center of the circum- scribing circle. Draw lines from B through 3 and 4. The inter- section of these lines with the circumscribed circle determines the vertices of the polygon. A i To Draw an Ellipse, the Major and Minor Axes Being Given (Fig. 31). Let AB be the major axis and CD the minor axis. Lay off on the edge of a straight piece of paper the distance ac equal to Ao, the semi-major axis. Also lay off the distance ab equal to Co, the semi-minor axis. Place the paper so that b coincides with the major axis, and c coincides with the minor axis, or the minor axis produced; then a will give one point on the required ellipse. Locate as many points as are necessary to draw a smooth curve. To Draw an Approximate Ellipse, the Major and Minor Axes Being Given (Fig. 32). Let AB be the major axis and CD the minor axis. On the minor axis lay off oe and og equal to the difference between the major and the minor axes. On the major axis lay off of and oh equal to three-fourths of oe Draw ef, eh, gf, and gh, all produced. With center e draw arc mDn. With center g draw arc kCL With center/ draw arc kAm. With center h draw arc IBn, completing the required ellipse. INTRODUCTION 19 CONVENTIONAL SCREW THREADS Screw Threads. Fig. 33 shows a method for drawing conven- tional single thread screws. Draw two lines indicating the diam- eter. On one line lay off spaces equal to the pitch. Bisect one space and draw line ab. From b draw an inclined line of half b \ \ I \ \ \ t *l I j Pitch J( L_ / Pitch BUTTRESS THREAD M\^ Pitch a r SHARP V-THREAD FIG. 33. Showing steps for drawing conventional threads. the pitch. From the spaces laid off on one line draw parallels to the inclined line. Complete the thread, following the steps suggested in the figure. These conventions are used for large screw threads. FIG. 34. Common conventional thread. For screw threads under three-quarter inch diameter the con- vention shown in Fig. 34 may be used. The spacing for pitch should be estimated by the eye. 20 MECHANICAL DRAWING PROBLEMS INTERSECTION OF TWO CYLINDERS To Find the Line of Intersection of Two Cylinders with Axes in the Same Plane and at Right Angles to Each Other (Fig. 35). Let A, B, and C be the front, end, and top views, respectively. Points on the line of intersection may be found by the intersection of elements in the surface of one cylinder, with elements in the FIG. 35. Intersection by elements method. surface of the other. Let a", an assumed point, be the end view of an element, in the surface of the horizontal cylinder, which projected to the front and the top views gives a'b' and ab, the front and the top views of the element. Let b be the top view of an element, in the surface of the vertical cylinder, which pro- jected to the front view gives c'd', the front view of the element. The point of intersection of these elements gives b f , one point on INTRODUCTION 21 the required line of intersection. Additional points are found similarly. The method of finding the lines of intersection of the hollow vertical cylinder and the two holes cut through its thickness is evident from the figure. PLANE INTERSECTION OF A SOLID To Find the Lines of Intersection of a Surface of Revolution Cut by Two Planes at Right Angles to Each Other and Parallel to the Axis (Fig. 36). Let ab and a'b' be the projections of the axis, pp and p'p' the end view of the cutting planes, and cd the circular-arc outline of the surface. A transverse section at e, an assumed point on the it :tvt- FIG. 36. Intersection by cutting-plane method. curve, shown in the end view as a circle, is cut by plane pp. The intersecting points of the circle and plane projected back to the top view give //, points on the required line of intersection. A transverse section at g, another assumed point, shown in the end view by the arc of a circle, is cut by planes pp and p'p'. The intersecting points of the arc and planes projected back to the top and the front views give additional points on the line of intersection. Other points are found in a similar manner. Through these points, smooth curves are drawn. PART II EXAMPLES AND PROBLEMS DEFINITIONS PLANES OF PROJECTION For example see Plate 5 The Ground Line, designated in the drawing as GL, shows the division of two Planes. The surface above this line is called the Horizontal Plane, or H; the surface below the line is called the Vertical Plane, or V. The line marked G'U is called an Auxiliary Ground Line and shows the line of intersection of the vertical plane and of an Auxiliary Horizontal Plane. PROJECTIONS ON THREE PLANES For example see Plate 25 When drawing three views of an object, a Side, or Profile Plane is used in addition to the horizontal and vertical planes. The line abd is the ground line, while the line cbe is the Profile Plane Trace. The surface bounded by abc is the horizontal plane. The surface bounded by abe is the vertical plane and the surface bounded by ebd is the side, or profile plane, frequently designated as P. PROJECTIONS ON AUXILIARY PLANES For example see Plate 29 Projections on Auxiliary Planes are frequently made to show the true shape of oblique surfaces; that is, of surfaces which are not parallel to any one of the regular planes of projection. Auxiliary planes are generally perpendicular to H and inclined to V, or perpendicular to V and inclined to H. The surface bounded by ebd is the auxiliary plane, while the line be is the Auxiliary Plane Trace. 23 24 MECHANICAL DRAWING PROBLEMS SECTION I PROJECTIONS PRISMS This drawing shows the top views, also called plans, or hori- zontal projections, and the front views, also called front eleva- tions, or vertical projections, of four right prisms. The front views, since the prisms are all the same height and same width, are alike. The top views, which show their outline or shape, cannot be determined from the front views; therefore, two views are necessary to represent such objects completely. Problem 1. Make a drawing of four prisms similar to those shown. Let A = 3 inches, B = 2 inches, and C = 1 inch. Problem 2. Draw top and front views of four prisms similar to those shown. Let A 3J inches, B = 1| inches, and C = 11 inches. TAPERED OBJECTS This drawing shows the top and the front views of four tapered objects. The front views are alike while the top views differ. The first object is a wedge; the fourth, a cone. It will be observed that the front view of a wedge, and the front view of a cone may be exactly alike, although the objects are radically different, as shown by their top views. Problem 1. Draw top and front views of four objects similar to those shown. Let A = 2| inches, B = 1-J- inches, and C = f inch. Problem 2. Make a drawing showing top and front views of four objects similar to those shown. Let B = If inches, C = 1 inch, and A = 2, 2}, 2%, and 2f inches, respectively. PROJECTIONS 25 PLATE 1 PR/SMS Top Views or Plans or Horizontal Projections Front V/'ews or Front Elevations or Vertical Projections TAPERED OBJECTS PLATE 2 26 MECHANICAL DRAWING PROBLEMS CIRCULAR OBJECTS This drawing shows four circular objects whose top views are alike and whose front views vary considerably. Problem 1. Draw top and front views of objects similar to those shown and of the following dimensions: A B C D E First object Second object Third object Fourth object H H U H I 3 3i 3i 3f a H H 1 'j Problem 2. Draw top and front views of objects similar to those shown and of the following dimensions : A B C D E First object If 1 gJL Second object ill if 3& 7 Third object Fourth object H 1H 1 IjL 3A 3^ i& if H GEOMETRIC OBJECTS This drawing shows four objects whose top views are similar but whose front views differ. Problem 1. Make a drawing showing top and front views of objects similar to those shown and of the following dimensions: A B C D E F G H First object Second object.... Third object Fourth object.... H If H If 31 34 8| 3 1 I f 1 H U 11 H 1\ \ \ 1 2J 1 If U Problem 2. Make a drawing showing top and front views of objects similar to those shown and of the following dimensions : A B C D # F G H First object U 3$ 1 U 2i Second object.... U Si f M I 2 Third object If 31 f U Itt Fourth object.... 1J 84 i H i 1 2| 2H PROJECTIONS 27 PLATE 3 CIRCULAR OBJECTS UsJ GEOMETRIC OBJECTS 1 f 1 UJ f iJJ 1 O -4' Ul f i i PLATE 28 MECHANICAL DRAWING PROBLEMS GEOMETRIC OBJECT This drawing shows the top and the front views of an object in three positions in relation to GL. The top views show the object in contact with GL, therefore in contact with V. See page 23. Problem 1. Draw top and front views of the object in positions similar to those shown and of the following dimensions : A B c D B e 4> First position 2 If 2f t A Second position 2 H 3 ! A 30 | Third position 2 ii 31 I A 45 Problem 2. Make a drawing showing top and front views of the object in positions similar to those shown and of the following dimensions : A B C ., ... . D E e First position n H 2f A i Second position 2 U 3 1 A 15 Third position 2| if 31 A t 45 GEOMETRIC SOLID This drawing shows the top and the front views of a triangular solid in three positions. Problem 1. Draw top and front views of a similar solid having the fol- lowing dimensions and positions : A B C D E e First position 2J If 2* 11 * Second position 2| U 3 U 1 45 Third position 2| If 3| U H 30 Problem 2. Make a drawing showing top and front views of a similar solid having the following dimensions and positions : A B C D E e First position Second position Third position 21 2| 2} 2 11 If BJ 3 21 1! If 1 f f 15 75 PROJECTIONS 29 PLATE 6 GEOMETRIC OBJECT Position Second Position Third Position GEOMETRIC SOLID PLATE 6 30 MECHANICAL DRAWING PROBLEMS PROJECTION OF LETTER (V) This drawing shows the top and the front views of a letter con- sisting of horizontal, vertical, and oblique lines, in three positions relative to V. In solving the following problems draw the front view of the letter in its first position and project the top view. Transpose the top view to the required positions and project the front views. Problem 1. For the first position, draw the front view of the letter as shown; project the top view when removed one inch from V. For the second position, let one corner be in contact with V , and 6 = 30. For the third position show the nearest corner removed inch from V, and = 15. Problem 2. Draw front and top views, as explained in Problem 1, when the letter is inverted. Insert a cross-bar and change the letter to A. Show hidden lines in the front view of the third position. PROJECTION OF LETTER (K) This drawing shows the top and the front views of a letter con- sisting of horizontal, vertical, and oblique lines in three positions. For drawing the following problems read the instructions given for letter V. Problem 1. Draw front and top views in positions similar to those shown and of the following dimensions : A B C D E F G e 3f 3i 3 i 1 I 2| 30' 45 Show all hidden lines in the front view of the second position. Problem 2. Using the letter K as a guide, design the letter X and draw views in positions similar to those shown for K and of general dimensions, as follows: A B C D E F G e * 3i 3| n 1 \ - i 2J 15 30 Show all hidden lines in the front view of the third position. PROJECTIONS 31 PLATE 7 PROJECTION OF LETTER PROJECTION or LETTER PLATE 8 32 MECHANICAL DRAWING PROBLEMS HOLLOW CYLINDER This drawing shows the top and the front views of a hollow cylinder in three positions. For the first position, draw front and top views as shown. Divide the front view, giving points 1', 2', 3', etc. Project these points to the top view. For the second and the third positions, transpose the top view and locate points for the front views by projecting lines as shown. Problem 1. Draw the cylinder in positions similar to those shown and having the following dimensions : A B c e