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In inking in horizontal lines begin at the top and mark in each line as the square comes to it; and in inking the vertical ones begin always at the left hand line and mark the lines as they are come to, moving the square or the triangle to the right, and great care should be taken not to let the lines cross where they meet, as at the corners, since this would greatly impair the appearance of the drawing.
These figures have been drawn without the aid of a centre line, because from their shapes it was easy to dispense with it, but in most cases a centre line is necessary; thus in Figure 134 we have a body having a number of steps. The diameters of these steps are marked by arcs, as in the previous examples, and their lengths may be marked by applying the measuring rule direct to the drawing paper and making the necessary pencil mark.
But it would be tedious to mark the successive steps true one with the other by measuring each step, because one step would require to be pencilled in before the next could be marked. To avoid this the centre line 1, Figure 134, is first marked, and the arcs for the steps are then marked as shown. Centre lines are also necessary to show the alignment of one part to another; thus in Figure 135 is a cube with a hole passing through it. The dotted lines in the side view show that the hole passes clear through the piece and is a parallel one, while the centre line, being central to the outline throughout the piece, shows that the hole is equidistant, all through, from the walls of the piece.
The pencil lines for this piece would be marked as in Figure 136, line 1 representing the centre line from which all the arcs are marked. It will be noted that the length of the piece is marked by arcs which occur, because being a cube the set of the compasses for arcs 2, 3, 4 and 5 will answer without altering to mark arcs 6 and 7.
If the hole in the piece were a taper or conical one, it would be denoted by the dotted lines, as in Figure 137, and that the taper is central to the body is shown by these dotted lines being equidistant from the centre line.
Suppose one of the sides to be tapered, as is the side A, in Figure 138, and that the hole is not central, and both facts will be shown by the centre lines 1 and 2 in the figure. The measurement of face A would be marked from A to line B at each end, but the distance the hole was out of the centre would be marked by the distance between the centre line 2 and the edge C of the piece.
If the hole did not pass entirely through the piece, the dotted lines would show it, as in Figure 139.
The designations of the views of a piece of work depend upon the position in which the piece stands, when in place upon the machine of which it forms a part. Thus in Figure 140 is a lever, and if its shaft stood horizontal when the piece is in place in the machine, the view given is an end one, but suppose that the shaft stood vertical, and the same view becomes a plan or top view.
In Figure 142 is a view of a lever which is a side view if the lever stands horizontal, and lever B hangs down, or a plan view if the shaft stands horizontal, but lever B stands also horizontal. We may take the same drawing and turn it around on the paper as in Figure 143, and it becomes a side view if the shaft stands vertical, and a plan view if the shaft stands horizontal and arm D vertical above it.
In a side or an end view, the piece that projects highest in the drawing is highest when upon the machine; also in a side elevation the piece that is at the highest point in the drawing extends farthest upward when the piece is on the machine. But in a plan or top view the height of vertical pieces is not shown, as appears in the case of arm D in Figure 143.
In either of the levers, Figures 142 or 143, all the dimensions could be marked if an additional view were given, but this will not be the case if an eye have a slot in it, as at E, in Figure 144, or a jaw have a tongue in it, as at F: hence, end views of the eye and the jaw must be given, which may be most conveniently done by showing them projected from the ends of those parts as in the figure.
This naturally brings us to a consideration as to the best method of projecting one view from another. As a general rule, the side elevation or side view is the most important, because it shows more of the parts and details of the work; hence it should be drawn first, because it affords more assistance in drawing the other views.
There are two systems of placing the different views of a piece. In the first the views are presented as the piece would present itself if it were laid upon the paper for the side view, and then turned or rolled upon the paper for the other views, as shown in Figure 145, in which the piece consists of five sections or members, marked respectively A, B, C, D, and E. Now if the piece were turned or rolled so that the end face of B were uppermost, and the member E was beneath, it will, by the operation of turning it, have assumed the position in the lower view marked position 2; while if it were turned over upon the paper in the opposite direction it would assume the position marked 3. This gives to the mind a clear idea of the various views and positions; but it possesses some disadvantages: thus, if position 1 is a side elevation or view of the piece, as it stands when in place of the machine, then E is naturally the bottom member; but it is shown in the top view of the drawing, hence what is actually the bottom view of the piece (position 3) becomes the top view in the drawing. A second disadvantage is that if we desire to put in dotted lines, to show how one view is derived from the other, and denote corresponding parts, then these dotted lines must be drawn across the face of the drawing, making it less distinct; thus the dotted lines connecting stem E in position 1 to E in position 3, pass across the faces of both A and B of position 1.
In a large drawing, or one composed of many members or parts, it would, therefore, be out of the question to mark in the dotted lines. A further disadvantage in a large drawing is that it is necessary to go from one side of the drawing to the other to see the construction of the same part.
To obviate these difficulties, a modern method is to suppose the piece, instead of rolling upon the paper, to be lifted from it, turned around to present the required view, and then moved upwards on the paper for a top view, sideways for a side view, and below for a bottom view. Thus the three views of the piece in Figure 145 would be as in Figure 146, where position 2 is obtained by supposing the piece to be lifted from position 1, the bottom face turned uppermost, and the piece moved down the paper to position 2, which is a bottom view of the piece, and the bottom view in the drawing. Similarly, if the piece be lifted from position 1, and the top face in that figure is turned uppermost, and the piece is then slid upwards on the paper, view 3 is obtained, being a top view of the piece as it lies in position 1, and the top view in the drawing. Now suppose we require to find the shape of member B, then in Figure 145 we require to look at the top of position 1, and then down below to position 2.
But in Figure 146 we have the side view and end view both together, while the dotted lines do not require to cross the face of the side view. Now suppose we take a similar piece, and suppose its end faces, as F, G, to have holes in them, which require to be shown in both views, and under the one system the drawing would, if the dotted lines were drawn across, appear as in Figure 147, whereas under the other system the drawing would appear as in Figure 148. And it follows that in cases where it is necessary to draw dotted lines from one view to the other, it is best to adopt the new system.
CHAPTER VII.
EXAMPLES IN BOLTS, NUTS, AND POLYGONS.
Let it be required to draw a machine screw, and it is not necessary, and therefore not usual in small screws to draw the full outline of the thread, but to represent it by thick and thin lines running diagonally across the bolt, as in Figure 149, the thick ones representing the bottom, and the thin ones the top of the thread. The pencil lines would be drawn in the order shown in Figure 150. Line 1 is the centre line, and line 2 a line to represent the lower side of the head; from the intersection of these two lines as a centre (as at A) short arcs 3 and 6, showing the diameter of the thread, are marked, and the arcs 5 and 6, representing the depth of the thread, are marked. The arc 7, representing the head, is then marked. The vertical lines 8, 9, 10, and 11 are then marked, and the outline of the screw is complete. The thick lines representing the bottom of the thread are next marked in, as in Figure 151, extending from line 9 to line 10. Midway between these lines fine ones are made for the tops of the thread. All the lines being pencilled in, they may be inked in with the drawing instruments, taking care that they do not overrun one another. When the pencil lines are rubbed out, the sketch will appear as in Figure 149.
For a bolt with a hexagon head the lines would be drawn in the order shown in Figure 152. At a right-angle to centre line 1, line two is drawn. The pencil-compasses are then set to half the diameter of the bolt, and from point A arcs 3 and 6 are pencilled, thus showing the width of the front flat of the head, as well as the diameter of the stem. From the point where these arcs meet line 2, and with the same radius, arcs 5 and 6 are marked, showing the widths of the other two flats of the head. The thickness of the head and the length of the bolt head may then be marked either by placing a rule on line 1 and marking the short lines (such as line 7) a cross line 1, or the pencil-compasses may be set to the rule and the lengths marked from point A. In the United States standard for bolt heads and nuts the thickness of the head is made equal to the diameter of the bolt. With the compasses set for the arcs 3 and 4, we may in two steps, from A along the centre line, mark off the thickness of the head without using the rule. But as the rule has to be applied along line 1 to mark line 7 for the length of the bolt, it is just as easy to mark the head thickness at the same time. The line 8 showing the length of the thread may be marked at the same time as the other lengths are marked, and the outlines 9, 10, 11, 12, 13 may be drawn in the order named. We have now to mark the arcs at the top of the flats of the head to show the chamfer, and to explain how these arcs are obtained we have in Figure 153 an enlarged view of the head. It is evident that the smallest diameter of the chamfer is represented by the circle A, and therefore the length of the line B must equal A. It is also evident that the outer edge of the chamfer will meet the corners at an equal depth (from the face of the nut), as represented by the line C C, and it is obvious that the curves that represent the outline of the chamfer on each side of the head or nut will approach the face of the head or nut at an equal distance, as denoted by the line D D. It follows that the curve must in each case be such as will, at each of its ends, meet the line C, and at its centre meet the line D D, the centres of the respective curves being marked in the figure by X.
It is sufficiently accurate, therefore, for all practical purposes to set the pencil on the centre-line at the point A in Figure 152 and mark the curve 14, and to then set the compasses by trial to mark the other two curves of the chamfer, so that they shall be an equal distance with arc 14 from line 9, and join lines 10 and 13 at the same distance from line 9 that 14 joins lines 3 and 4, so that as in Figure 153 all three of the arcs would touch a line as C, and another line as D.
The United States standard sizes for forged or unfinished bolts and nuts are given in the following table, Figure 154 showing the dimensions referred to in the table.
UNITED STATES STANDARD DIMENSIONS OF BOLTS AND NUTS.
KEY: A: Nominal. D. B: Effective.[*] C: Standard Number of threads per inch.
- BOLT. BOLT HEAD AND NUT. - Diameter. Long diameter, Short - I, or diameter diameter across corners. of hexagon A B C Hexa- Square. and gon. square, Depth of Depth of or width nut, bolt across J. H. head, K. - - 1/4 .185 20 9/16 23/32 1/2 1/4 1/4 5/16 .240 18 11/16 27/32 19/32 5/16 19/64 3/8 .294 16 25/32 31/32 11/16 3/8 11/32 7/16 .345 14 29/32 1-3/32 25/32 7/16 25/64 1/2 .400 13 1 1-1/4 7/8 1/2 7/16 9/16 .454 12 1-1/8 1-3/8 31/32 9/16 31/64 5/8 .507 11 1-7/32 1-1/2 1-1/16 5/8 17/32 3/4 .620 10 1-7/16 1-3/4 1-1/4 3/4 5/8 7/8 .731 9 1-21/32 2-1/32 1-7/16 7/8 23/32 1 .837 8 1-7/8 2-5/16 1-5/8 1 13/16 1-1/8 .940 7 2-3/32 2-9/16 1-13/16 1-1/8 29/32 1-1/4 1.065 7 2-5/16 2-27/32 2 1-1/4 1 1-3/8 1.160 6 2-17/32 3-3/32 2-3/16 1-3/8 1-3/32 1-1/2 1.284 6 2-3/4 3-11/32 2-3/8 1-1/2 1-3/16 1-5/8 1.389 5-1/2 2-31/32 3-5/8 2-9/16 1-5/8 1-9/32 1-3/4 1.491 5 3-3/16 3-7/8 2-3/4 1-3/4 1-3/8 1-7/8 1.616 5 3-13/32 4-5/32 2-15/16 1-7/8 1-15/32 2 1.712 4-1/2 3-19/32 4-13/32 3-1/8 2 1-9/16 2-1/4 1.962 4-1/2 4-1/32 4-15/16 3-1/2 2-1/4 1-3/4 2-1/2 2.176 4 4-15/32 5-15/32 3-7/8 2-1/2 1-15/16 2-3/4 2.426 4 4-29/32 6 4-1/4 2-3/4 2-1/8 3 2.629 3-1/2 5-11/32 6-17/32 4-5/8 3 2-5/16 3-1/4 2.879 3-1/2 5-25/32 7-1/16 5 3-1/4 2-1/2 3-1/2 3.100 3-1/4 6-7/32 7-19/32 5-3/8 3-1/2 2-11/16 3-3/4 3.317 3 6-5/8 8-1/8 5-3/4 3-3/4 2-7/8 ... 3.567 3 7-1/16 8-21/32 6-1/8 4 3-1/16 4-1/4 3.798 2-7/8 7-1/2 9-3/16 6-1/2 4-1/4 3-1/4 4-1/2 4.028 2-3/4 7-15/16 9-23/32 6-7/8 4-1/2 3-7/16 4-3/4 4.256 2-5/8 8-3/8 10-1/4 7-1/4 4-3/4 3-5/8 5 4.480 2-1/2 8-13/16 10-25/32 7-5/8 5 3-13/16 5-1/4 4.730 2-1/2 9-1/4 11-5/16 8 5-1/4 4 5-1/2 4.953 2-3/8 9-11/16 11-27/32 8-3/8 5-1/2 4-3/16 5-3/4 5.203 2-3/8 10-3/32 12-3/8 8-3/4 5-3/4 4-3/8 6 5.423 2-1/4 10-17/32 12-29/32 9-1/8 6 4-9/16 - - * Diameter at the root of the thread.
The basis of the Franklin Institute or United States standard for the heads of bolts and for nuts is as follows:
The short diameter or width across the flats is equal to one and one-half times the diameter plus 1/8 inch for rough or unfinished bolts and nuts, and one and one-half times the bolt diameter plus, 1/16 inch for finished heads and nuts. The thickness is, for rough heads and nuts, equal to the diameter of the bolt, and for finished heads and nuts 1/16 inch less.
The hexagonal or hexagon (as they are termed in the shop) heads of bolts may be presented in two ways, as is shown in Figures 155 and 156.
The latter is preferable, inasmuch as it shows the width across the flats, which is the dimension that is worked to, because it is where the wrench fits, and therefore of most importance; whereas the latter gives the length of a flat, which is not worked to, except incidentally, as it were. There is the objection to the view of the head, given in Figure 156, however, that unless it is accompanied by an end view it somewhat resembles a similar view of a square head for a bolt. It may be distinguished therefrom, however, in the following points:
If the amount of chamfer is such as to leave the chamfer circle (as circle A, in Figure 153) of smaller diameter than the width across the flats of the bolt-head, the outline of the sides of the head will pass above the arcs at the top of the flats, and there will be two small flat places, as A and B, in Figure 156 (representing the angle of the chamfer), which will not meet the arcs at the top of the flats, but will join the sides above those arcs, as in the figure; which is also the case in a similar view of a square-headed bolt. It may be distinguished therefrom, however, in the following points:
If the amount of chamfer is such as to leave the chamfer circle (A, Figure 153) of smaller diameter than the width across the flats of the bolt-head, the outline of the sides will pass above the arc on the flats, as is shown in Figure 157, in which the chamfer A meets the side of the head at B, and does not, therefore, meet the arc C. The length of side lying between B and D in the side view corresponds with the part lying between E and F in the end view.
If we compare this head with similar views of a square head G, both being of equal widths, and having their chamfer circles at an equal distance from the sides of the flats, and at the same angle, we perceive at once that the amount of chamfer necessary to give the same distance between the chamfer circle and the side of the bolt (that is, the distance from J to K, being equal to that from L to M), the length of the chamfer N for the square head so greatly exceeds the length A for the hexagon head that the eye detects the difference at once, and is instinctively informed that G must be square, independently of the fact that in the case of the square head, N meets the arc O, while in the hexagon head, A, which corresponds to N, does not meet the arc C, which corresponds to O.
When, however, the chamfer is drawn, but just sufficient to meet the flats, as in the case of the hexagon H, and the square I, in Figure 157, the chamfer line passes from the chamfer circle to the side of the head, and the distinction is greater, as will be seen by comparing head H with head I, both being of equal width, having the same angle of chamfer, and an amount just sufficient to meet the sides of the flats. Here it will be seen that in the hexagon H, each side of the head, as P, meets the chamfer circle A. Whereas, in the square head these two lines are joined by the chamfer line Q, the figures being quite dissimilar.
It is obvious that whatever the degree or angle of the chamfer may be, the diameter of the chamfer circle will be the same in any view in which the head may be presented. Thus, in Figure 158, the line G in the side view is in length equal to the diameter of circle G, in the end view, and so long as the angle of the chamfer is forty-five degrees, as in all the views hitherto given, the width of the chamfer will be equal at corresponding points in the different views; thus in the figure the widths A and B in the two views are equal.
If the other view showing a corner of the head in front of the head be given, the same fact holds good, as is shown in Figure 159. That the two outside flats should appear in the drawing to be half the width of the middle flat is also shown in Figure 158, where D and E are each half the width of C. Let us now suppose, that the chamfer be given some other angle than that of 45 degrees, and we shall find that the effect is to alter the curves of the chamfer arcs on the flats, as is shown in Figure 160, where these arcs E, C, D are shown less curved, because the chamfer B has more angle to the flats. As a result, the width or distance between the arcs and line G is different in the two views. On this account it is better to draw the chamfer at 45 degrees, as correct results may be obtained with the least trouble.
If no chamfer at all is to be given, a hexagon head may still be distinguished from a square one, providing that the view giving three sides of the head, as in Figure 158, is shown, because the two sides D and E being half the width of the middle one C, imparts the information that it is a hexagon head. If, however, the view showing but two of the sides and a corner in front is given, and no chamfer is used, it could not be known whether the head was to be hexagon or square, unless an end view be given, as in Figure 161.
If the view showing a full side of the head of a square-headed bolt is given, then either an end view must be given, as in Figure 162, or else a single view with a cross on its head, as in Figure 163, may be given.
It is the better plan, both in square and hexagon heads, to give the view in which the full face of a flat is presented, that is, as in Figures 155 and 163; because, in the case of the square, the length of a side and the width across the head are both given in that view; whereas if two sides are shown, as in Figure 161, the width across flats is not given, and this is the dimension that is wanted to work to, and not the width across corners. In the case of a hexagon the middle of the three flats is equal in width to the diameter of the bolt, and the other two are one-half its width; all three, therefore, being marked with the same set of compasses as gives the diameter of the body of the bolt, were as shown in Figure 152. For the width across flats there is an accepted standard; hence there is no need to mark it upon the drawing, unless in cases where the standard is to be departed from, in which event an end view may be added, or the view showing two sides may be given.
To draw a square-headed bolt, the pencil lines are marked in the order shown by figures in Figure 164. The inking in is done in the order of the letters a, b, c, etc. It will be observed that pencil lines 2, 9, and 10 are not drawn to cross, but only to meet the lines at their ends, a point that, as before stated, should always be carefully attended to.
To draw the end view of a hexagon head, first draw a circle of the diameter across the flats, and then rest the triangle of 60 degrees on the blade s of the square, as at T 1, in Figure 165, and mark the lines a and b. Reverse the triangle, as at T 2, and draw lines c and d. Then place the triangle as in Figure 166, and draw the lines e and f.
If the other view of the head is to be drawn, then first draw the lines a and b in Figure 167 with the square, then with the 60 degree triangle, placed on the square S, as at T 1, draw the lines c, d, and turning the square over, as at T 2, mark lines e and f.
If the diameter across corners of a square head is given, and it be required to draw the head, the process is as follows: For a view showing one corner in front, as in Figure 168, a circle of the given diameter across corners is pencilled, and the horizontal centre-line a is marked, and the triangle of 45 degrees is rested against the square blade S, as in position T 1, and lines b and c marked, b being marked first; and the triangle is then slid along the square blade to position T 1, when line c is marked, these two lines just meeting the horizontal line a, where it meets the circle. The triangle is then moved to the left, and line d, joining the ends of b and c, is marked, and by moving it still farther to the left to position T 2, line e is marked. Lines b, c, d, and e are, of course, the only ones inked in.
If the flats are to lie in the other direction, the pencilling will be done as in Figure 169. The circle is marked as before, and with the triangle placed as shown at T 1, line a, passing through the centre of the circle, is drawn. By moving the triangle to the right its edge B will be brought into position to mark line b, also passing through the centre of the circle. All that remains is to join the ends of these two lines, using the square blade for lines c, d, and the triangle for e and f, its position on the square blade being denoted at T 3; lines c, d, e, f, are the ones inked in.
For a hexagon head we have the processes, Figures 170 and 171. The circle is struck, and across it line a, Figure 170, passing through its centre, the triangle of sixty degrees will mark the sides b, c, and d, e, as shown, and the square blade is used for f, g.
The chamfer circles are left out of these figures to reduce the number of lines and so keep the engraving clear. Figure 171 shows the method of drawing a hexagon head when the diameter across corners is given, the lines being drawn in the alphabetical order marked, and the triangle used as will now be understood.
It may now be pointed out that the triangle may be used to divide circles much more quickly than they could be divided by stepping around them with compasses. Suppose, for example, that we require to divide a circle into eight equal parts, and we may do so as in Figure 172, line a being marked from the square, and lines b, c and d from the triangle of forty-five degrees; the lines to be inked in to form an octagon need not be pencilled, as their location is clearly defined, being lines joining the ends of the lines crossing the circle, as for example, lines e, f.
Let it be required to draw a polygon having twelve equal sides, and the triangle of sixty is used, marking all the lines within the circle in Figure 173, except a, for which the square blade is used; the only lines to be inked in are such as b, c. In this example there is a corner at the top and bottom, but suppose it were required that a flat should fall there instead of a corner; then all we have to do is to set the square blade S at the required angle, as in Figure 174, and then proceed as before, bearing in mind that the point of the circle nearest to the square blade, straight-edge, or whatever the triangle is rested on, is always a corner of a polygon having twelve sides.
In both of these examples we have assumed that the diameter across corners of the polygon was given, but suppose the diameter across the flats were given, and the construction is a little more complicated. Circle a, a, in Figure 175, is drawn of the required diameter across the flats, and the lines of division are drawn across with the triangle of 60 as before; the triangle of 45 is then used to draw the four lines, b, c, d, e, joining the ends of lines i, j, k, l, and touching the inner circle, a, a. The outer circle is then pencilled in, touching the lines of division where they meet the lines b, c, d, e, and the rest of the lines for the sides of the polygon may then be drawn within the outer circle, as at g, h.
It is obvious, also, that the triangle may be used to draw slots radiating from a centre, as in Figure 176, where it is desired to draw a chuck-plate having 6 slots. The triangle of 60 is used to draw the centre lines, a, b, c, etc., for the slots. From the centre, the arcs e, f, g, h, etc., are marked, showing where the centres will fall for describing the half circles forming the ends of the slots. Then half circles, i, j, k, l, etc., being drawn, the sides of the slots may be drawn in with the triangle, and the outer circle and the slots inked in.
If the slots are not to radiate from the centre of the circle the process is as follows:
The outer circle a, Figure 177, being drawn, an inner one b is drawn, its radius equalling the amount; the centres of the slots are to point to one side of the centre of circle a. The triangle is then used to divide the circle into the requisite number of divisions c for the slots, and arcs i, j, are then drawn for the lengths of the slots. The centre lines e are then drawn, passing through the lines c, and the arcs i, j, etc., and touching the perimeter of the inner circle b; arcs f, g, are then marked in, and their sides joined with the triangle adjusted by hand. All that would be inked in black are the outer circle and the slots, but the inner circle b and a centre line of one of the slots should be marked in red ink to show how the inclination of the slot was obtained, and therefore its amount.
For a five-sided figure it is best to step around the circumference of the circle with the compasses, but for a three-sided one, or trigon, the construction is as follows: It will be found that the compasses set to the radius of a circle will accurately divide it into six equal divisions, as is shown in Figure 178; hence every other one of these divisions will be the location for a corner of a trigon.
The circle being drawn, a line A, 179, is drawn through its centre, and from its intersection with the circle as at b, here a step on each side is marked as c, d, then lines c to d, and c and d to e, where A meets, the circle will describe a trigon. If the figure is to stand vertical, all that is necessary is to draw the line a vertical, as in Figure 180. A ready method of getting the dimension across corners, across the flats, or the length of a side of a given polygon, is by means of diagrams, such as shown in the following figures, which form excellent examples for practice.
Draw the line O P, Figure 181, and at a right angle to it the line O B; divide these two lines into parts of one inch, as shown in the cut, which is divided into inches and quarter inches, and from these points of division draw lines crossing each other as shown.
From the point O, draw diagonal lines, at suitable angles to the line O P. As shown in the cut, these diagonal lines are marked:
40 degrees for 5 sided figures. 45 " " 6 " " 49 " " 7 " " 52-1/2 " " 8 " " 55-1/2 " " 9 " "
But still others could be added for figures having a greater number of sides.
1. Now it will be found as follows: Half the diameter, or the radius of a piece of cylindrical work being given, and the number of sides it is to have being stated, the length of one side will be the distance measured horizontally from the line O B to the diagonal line for that particular number of sides.
EXAMPLE.—A piece of work is 2-1/2 inches in diameter, and is required to have 9 sides: what will be the length of the sides or flats?
Now the half diameter or radius of 2-1/2 inches is 1-1/4 inches. Then look along the line O B for 1-1/4, which is denoted in the cut by figures and the arrow A; set one point of the compasses at A, and the other at the point of crossing of the diagonal line with the 1-1/4 horizontal line, as shown in the figure at a, and from A to a is the length of one side.
Again: A piece of work, 4 inches in diameter, is to have 9 sides: how long will each side be?
Now half of 4 is 2, hence from B to b is the length of each side.
But suppose that from the length of each side, and the number of sides, it is required to find the diameter to which to turn the piece; that is, its diameter across corners, and we simply reverse the process thus: A body has 9 sides, each side measures 27/32: what is its diameter across corners?
Take a rule, apply it horizontally on the figure, and pass it along till the distance from the line O B to the diagonal line marked 9 sides measures 27/32, which is from 1-1/4 on O B to a, and the 1-1/4 is the radius, which, multiplied by 2, gives 2-1/2 inches, which is the required diameter across corners.
For any other number of sides the process is just the same. Thus: A body is 3-1/2 inches in diameter, and is to have 5 sides: what will be the length of each side? Now half of 3-1/2 is 1-3/4; hence from 1-3/4 on the line O B to the point C, where the diagonal line crosses the 1-3/4 line, is the length of each of the sides.
2. It will be found that the length of a side of a square being given, the size of the square, measured across corners, will be the length of the diagonal line marked 45 degrees, from the point O to the figures indicating, on the line O B or on the line O P, the length of one side.
EXAMPLE.—A square body measures 1 inch on each side: what does it measure across the corners? Answer: From the point O, along diagonal line marked 45 degrees, to the point where it crosses the lines 1 (as denoted in the figure by a dot).
Again: A cylindrical piece of wood requires to be squared, and each side of the square must measure an inch: what diameter must the piece be turned to?
Now the diagonal line marked 45 degrees passes through the 1-inch line on O B, and the inch line on O P, at the point where these lines meet; hence all we have to do is to run the eye along either of the lines marked inch, and from its point of meeting the 45 degrees line, to the point O, is the diameter to turn the piece to.
There is another way, however, of getting this same measurement, which is to set a pair of compasses from the line 1 on O B, to line 1 on O P, as shown by the line D, which is the full diameter across corners. This is apparent, because from point O, along line O B, to 1, thence to the dot, thence down to line 1 on O P, and along that to O, encloses a square, of which either from O to the dot, or the length of the line D, is the measurement across corners, while the length of each side, or diameter across the flats, is from point O to either of the points 1, or from either of the points 1 to the dot.
After graphically demonstrating the correctness of the scale we may simplify it considerably. In Figure 182, therefore, we have applications shown. A is a hexagon, and if one of its sides be measured, it will be found that it measures the same as along line 1 from O B to the diagonal line 45 degrees, which distance is shown by a thickened line.
At 1-1/2 is shown a seven-sided figure, whose diameter is 3 inches, and radius 1-1/2 inches, and if from the point at 1-1/2 (along the thickened horizontal line), to the diagonal marked 49 degrees, be measured, it will be found exactly equal to the length of a side on the polygon.
At C is shown part of a nine-sided polygon, of 2-inch radius, and the length of one of its sides will be found to equal the distance from the diagonal line marked 52-1/2 degrees, and the line O B at 2.
Let it now be noted that if from the point O, as a centre, we describe arcs of circles from the points of division on O B to O P, one end of each arc will meet the same figure on O P as it started from at O B, as is shown in Figure 181, and it becomes apparent that in the length of diagonal line between O and the required arc we have the radius of the polygon.
EXAMPLE.—What is the radius across corners of a hexagon or six-sided figure, the length of a side being an inch?
Turning to our scale we find that the place where there is a horizontal distance of an inch between the diagonal 45 degrees, answering to six-sided figures, is along line 1 (Figure 182), and the radius of the circle enclosing the six-sided body is, therefore, an inch, as will be seen on referring to circle A. But it will be noted that the length of diagonal line 45 degrees, enclosed between the point O and the arc of circle from 1 on O B to one on O P, measures also an inch. Hence we may measure the radius along the diagonal lines if we choose. This, however, simply serves to demonstrate the correctness of the scale, which, being understood, we may dispense with most of the lines, arriving at a scale such as shown in Figure 183, in which the length of the side of the polygon is the distance from the line O B, measured horizontally to the diagonal, corresponding to the number of sides of the polygon. The radius across corners of the polygon is that of the distance from O along O B to the horizontal line, giving the length of the side of the polygon, and the width across corners for a given length of one side of the square, is measured by the length of the lines A, B, C, etc. Thus, dotted line 2 shows the length of the side of a nine-sided figure, of 2-inch radius, the radius across corners of the figure being 2 inches.
The dotted line 2-1/2 shows the length of the side of a nine-sided polygon, having a radius across corners of 2-1/2 inches. The dotted line 1 shows the diameter, across corners, of a square whose sides measure an inch, and so on.
This scale lacks, however, one element, in that the diameter across the flats of a regular polygon being given, it will not give the diameter across the corners. This, however, we may obtain by a somewhat similar construction. Thus, in Figure 184, draw the line O B, and divide it into inches and parts of an inch. From these points of division draw horizontal lines; from the point O draw the following lines and at the following angles from the horizontal line O P.
A line at 75 deg. for polygons having 12 sides. " 72 deg. " " 10 " " 67-1/2 deg. " " 8 " " 60 deg. " " 6 "
From the point O to the numerals denoting the radius of the polygon is the radius across the flats, while from point O to the horizontal line drawn from those numerals is the radius across corners of the polygon.
A hexagon measures two inches across the flats: what is its diameter measured across the corners? Now from point O to the horizontal line marked 1 inch, measured along the line of 60 degrees, is 1 5-32nds inches: hence the hexagon measures twice that, or 2 5-16ths inches across corners. The proof of the construction is shown in the figure, the hexagon and other polygons being marked simply for clearness of illustration.
Let it be required to draw the stud shown in Figure 185, and the construction would be, for the pencil lines, as shown in Figure 186; line 1 is the centre line, arcs, 2 and 3 give the large, and arcs 4 and 5 the small diameter, to touch which lines 6, 7, 8, and 9 may be drawn. Lines 10, 11, and 12 are then drawn for the lengths, and it remains to draw the curves in. In drawing these curves great exactitude is required to properly find their centres; nothing looks worse in a drawing than an unfair or uneven junction between curves and straight lines. To find the location for these centres, set the compasses to the required radius for the curve, and from the point or corner A draw the arcs b and c, from c mark the arc e, and from b the arc d, and where d and e cross is the centre for the curve f.
Similarly for the curve h, set the compasses on i and mark the arc g, and from the point where it crosses line 6, draw the curve h. In inking in it is best to draw in all curves or arcs of circles first, and the straight lines that join them afterward, because, if the straight lines are drawn first, it is a difficult matter to alter the centres of the curves to make them fall true, whereas, after the curves are drawn it is an easy matter, if it should be necessary, to vary the line a trifle, so as to make it join the curves correctly and fair. In inking in these curves also, care must be taken not to draw them too short or too long, as this would impair the appearance very much, as is shown in Figure 187.
To draw the piece shown in Figure 188, the lines are drawn in the order indicated by the letters in Figure 189, the example being given for practice. It is well for the beginner to draw examples of common objects, such as the hand hammer in Figure 190, or the chuck plate in Figure 191, which afford good examples in the drawing of arcs and circles.
In Figure 191 a is a cap nut, and the order in which the same would be pencilled in is indicated by the respective numerals. The circles 3 and 4 represent the thread.
In Figure 192 is shown the pencilling for a link having the hubs on one side only, so that a centre line is unnecessary on the edge view, as all the lengths are derived from the top view, while the thickness of the stem and height of the hubs may be measured from the line A. In Figure 193 there are hubs (on both sides of the link) of unequal height, hence a centre line is necessary in both views, and from this line all measurements should be marked.
In Figure 194 are represented the pencil lines for a double eye or knuckle joint, as it is sometimes termed, an example that it is desirable for the student to draw in various sizes, as it is representative of a large class of work.
These eyes often have an offset, and an example of this is given in Figure 195, in which A is the centre line for the stem distant from the centre line B of the eyes to the amount of offset required.
In Figure 196 is an example of a connecting rod end. From a point, as A, we draw arcs, as B C for the width, and E D for the length of the block, and through A we draw the centre line. It is obvious, however, that we may draw the centre line first, and apply the measuring rule direct to the paper, and mark lines in place of the arcs B, C, D, E, and F, G, which are for the stem. As the block joins the stem in a straight line, the latter is evidently rectangular, as will be seen by referring to Figure 197, which represents a rod end with a round stem, the fact that the stem is round being clearly shown by the curves A B. The radius of these curves is obtained as follows: It is obvious that they will join the rod stem at the same point as the shoulder curves do, as denoted by the dotted vertical line. So likewise they join the curves E F at the same point in the rod length as the shoulder curves, both curves in fact being formed by the same round corner or shoulder. The centre of the radius of A or B must therefore be the same distance from the centre of the rod as is the centre from which the shoulder curve is struck, and at the same time at such a distance from the corner (as E or F) that the curve will meet the centre line of the rod at the same point in its length as the shoulder curves do.
Figure 198 gives an example, in which the similar curved lines show that a part is square. The figure represents a bolt with a square under the head. As but one view is given, that fact alone tells us that it must be round or square. Now we might mark a cross on the square part, to denote that it is square; but this is unnecessary, because the curves F G show such to be the case. These curves are marked as follows: With the compasses set to the radius E, one point is rested at A, and arc B is drawn; then one point of the compass is rested at C, and arc D is drawn; giving the centre for the curve F by a similar process on the other side of the figure, curve G is drawn. Point C is obtained by drawing the dotted line across where the outline curve meets the stem. Suppose that the corner where the round stem meets the square under the head was a sharp one instead of a curve, then the traditional cross would require to be put on the square, as in Figure 199; or the cross will be necessary if the corner be a round one, if the stem is reduced in diameter, as in Figure 200.
Figure 201 represents a centre punch, giving an example, in which the flat sides gradually run out upon a circle, the edges forming curves, as at A, B, etc. The length of these curves is determined as follows: They must begin where the taper of the punch joins the parallel, or at C, C, and they must end on that part of the taper stem where the diameter is equal to the diameter across the flats of the octagon. All that is to be done then is to find the diameter across the flats on the end view, and mark it on the taper stem, as at D, D, which will show where the flats terminate on the taper stem. And the curved lines, as A, B, may be drawn in by a curve that must meet at the line C, and also in a rounded point at line D.
CHAPTER VIII.
SCREW THREADS AND SPIRALS.
The screw thread for small bolts is represented by thick and thin lines, such as was shown in Figure 152, but in larger sizes; the angles of the thread also are drawn in, as in Figure 202, and the method of doing this is shown in Figure 203. The centre line 1 and lines 2 and 3 for the full diameter of the thread being drawn, set the compasses to the required pitch of the thread, and stepping along line 2, mark the arcs 4, 5, 6, etc., for the full length the thread is to be marked. With the triangle resting against the $T$-square, the lines 7, 8, 9, etc. (for the full length of the thread), are drawn from the points 4, 5, 6, on line 2. These give one side of the thread. Reversing the drawing triangle, angles 10, 11, etc., are then drawn, which will complete the outline of the thread at the top of the bolt. We may now mark the depth of the thread by drawing line 12, and with the compasses set on the centre line transfer this depth to the other side of the bolt, as denoted by the arcs 13 and 14. Touching arc 14 we mark line 15 for the thread depth on that side. We have now to get the slant of the thread across the bolt. It is obvious that in passing once around the bolt the thread advances to the amount of the pitch as from a to b; hence, in passing half way around, it will advance from a to c; we therefore draw line 16 at a right-angle to the centre line, and a line that touches the top of the threads at a, where it meets line 2, and also meets line 16, where it touches line 3, is the angle or slope for the tops of the threads, which may be drawn across by lines, as 18, 19, 20, etc. From these lines the sides of the thread may be drawn at the bottom of the bolt, marking first the angle on one side, as by lines 21, 22, 23, etc., and then the angles on the other, as by lines 24, 25, etc.
There now remain the bottoms of the thread to draw, and this is done by drawing lines from the bottom of the thread on one side of the bolt to the bottom on the other, as shown in the cut by a dotted line; hence, we may set a square blade to that angle, and mark in these lines, as 26, 27, 28, etc., and the thread is pencilled in complete.
If the student will follow out this example upon paper, it will appear to him that after the thread had been marked out on one side of the bolt, the angle of the thread might be obtained, as shown by lines 16 and 17, and that the bottoms of the thread as well as the tops might be carried across the bolt to the other side. Figure 204 represents a case in which this has been done, and it will be observed that the lines denoting the bottom of the thread do not meet the bottoms of the thread, which occurs for the reason that the angle for the bottom is not the same as that for the top of the thread.
In inking in the thread, it enhances the appearance to give the bottom of the thread and the right-hand side of the same, heavy shade lines, as in Figure 202, a plan that is usually adopted for threads of large diameter and coarse pitch.
A double thread, such as in Figure 205, is drawn in the same way, except that the slant of the thread is doubled, and the square is to be set for the thread-pitch A, A, both for the tops and bottoms of the thread.
A round top and bottom thread, as the Whitworth thread, is drawn by single lines, as in Figure 206. A left-hand thread, Figure 207, is obviously drawn by the same process as a right-hand one, except that the slant of the thread is given in the opposite direction.
For screw threads of a large diameter it is not uncommon to draw in the thread curves as they appear to the eye, and the method of doing this is shown in Figure 208. The thread is first marked on both sides of the bolt, as explained, and instead of drawing, straight across the bolt, lines to represent the tops and bottoms of the thread, a template to draw the curves by is required. To get these curves, two half-circles, one equal in diameter to the top, and one equal to the bottom of the thread, are drawn, as in Figure 208.
These half-circles are divided into any convenient number of equal divisions: thus in Figure 208, each has eight divisions, as a, b, c, etc., for the outer, and i, j, k, etc., for the inner one. The pitch of the thread is then divided off by vertical lines into as many equal divisions as the half-circles are divided into, as by the lines a, b, c, etc., to o. Of these, the seven from a, to h, correspond to the seven from a' to g', and are for the top of the thread, and the seven from i to o correspond to the seven on the inner half-circle, as i, j, k, etc. Horizontal lines are then drawn from the points of the division to meet the vertical lines of division; thus the horizontal dotted line from a' meets the vertical line a, and where they meet, as at A, a dot is made. Where the dotted line from b' meets vertical line b, another dot is made, as at B, and so on until the point G is found. A curve drawn to pass from the top of the thread on one side of the bolt to the top of the other side, and passing through these points, as from A to G, will be the curve for the top of the thread, and from this curve a template may be made to mark all the other thread-tops from, because manifestly all the tops of the thread on the bolt will be alike.
For the bottoms of the thread, lines are similarly drawn, as from i' to meet i, where dot I is marked. J is got from j' and j, while K is got from the intersection of k' with k, and so on, the dots from I to O being those through which a curve is drawn for the bottom of the thread, and from this curve a template also may be made to mark all the thread bottoms. We have in our example used eight points of division in each half-circle, but either more or less points maybe used, the only requisite being that the pitch of the thread must be divided into as many divisions as the two half-circles are. But it is not absolutely necessary that both half-circles be divided into the same number of equal divisions. Thus, suppose the large half-circle were divided into ten divisions, then instead of the first half of the pitch being divided into eight (as from a to h) it would require to have ten lines. But the inner half-circle may have eight only, as in our example. It is more convenient, however, to use the same number of divisions for both circles, so that they may both be divided together by lines radiating from the centre. The more the points of division, the greater number of points to draw the curves through; hence it is desirable to have as many as possible, which is governed by the pitch of the thread, it being obvious that the finer the pitch the less the number of distinct and clear divisions it is practicable to divide it into. In our example the angles of the thread are spread out to cause these lines to be thrown further apart than they would be in a bolt of that diameter; hence it will be seen that in threads of but two or three inches in diameter the lines would fall very close together, and would require to be drawn finely and with care to keep them distinct.
The curves for a United States standard form of thread are obtained in the same manner as from the $V$ thread in Figure 208, but the thread itself is more difficult to draw. The construction of this thread is shown in Figure 208, it having a flat place at the top and at the bottom of the thread. A common $V$ thread has its sides at an angle of 60 degrees, one to the other, the top and bottom meeting in a point. The United States standard is obtained from drawing a common $V$ thread and dividing its depth into eight equal divisions, as at x, in Figure 208 a, and cutting off one of these divisions at the top and filling in one at the bottom to form flat places, as shown in the figure. But the thread cannot be sketched on a bolt by this means unless temporary lines are used to get the thread from, these temporary lines being drawn to represent a bolt one-fourth the depth of the thread too large in diameter. Thus, in Figure 208 a, it is seen that cutting off one-eighth the depth of the thread reduces the diameter of the thread. It is necessary, then, to draw the flat place on top of the thread first, the order of procedure being shown in Figure 209. The lines for the full diameter of the thread being drawn, the pitch is stepped off by arcs, as 1, 2, 3, etc.; and from these, arcs, as 4, 5, 6, etc., are marked for the width of the flat places at the tops of the threads. Then one side of the thread is marked off by lines, as 7, which meet the arcs 1, 2, 3, etc., as at a, c, etc. Similar lines, as 8 and 9, are marked for the other side of the thread, these lines, 7, 8 and 9, projecting until they cross each other. Line 10 is then drawn, making a flat place at the bottom of the thread equal in width to that at the top. Line 12 is then drawn square across the bolt, starting from the bottom of the thread, and line 13 is drawn starting from the corner f on one side of the thread and meeting line 12 on the other side of the thread, which gives the angle for the tops of the thread. The depth of the thread may then be marked on the other side of the bolt by the arcs d and e, and the line 14. The tops of all the threads may then be drawn in, as by lines 15, 16, 17 and 18, and by lines, as 19, etc., the thread sides may be drawn on the other side of the bolt. All that remains is to join the bottoms of the threads by lines across the bolt, and the pencil lines will be complete, ready to ink in. If the thread is to be shown curved instead of drawn straight across, the curve may be obtained by the construction in Figure 208, which is similar to that in Figure 207, except that while the pitch is divided off into 16 divisions, the whole of these 16 divisions are not used to get the curves, some of them being used twice over; thus for the bottom the eight divisions from b to i are used, while for the tops the eight from g to o are used. Hence g, h and i are used for getting both curves, the divisions from a to b and from o to p being taken up by the flat top and bottom of the thread. It will be noted that in Figure 207, the top of the thread is drawn first, while in Figure 208 the bottom is drawn first, and that in the latter (for the U.S. standard) the pitch is marked from centre to centre of the flats of the thread.
To draw a square thread the pencil lines are marked in the order shown in Figure 210, in which 1 represents the centre line and 2, 3, 4 and 5, the diameter and depth of the thread. The pitch of the thread is marked off by arcs, as 6, 7, etc., or by laying a rule directly on the centre line and marking with a lead pencil. To obtain the slant of the thread, lines 8 and 9 are drawn, and from these line 10, touching 8 and 9 where they meet lines 2 and 5; the threads may then be drawn in from the arcs as 6, 7, etc. The side of the thread will show at the top and the bottom as at A B, because of the coarse pitch and the thread on the other or unseen side of the bolt slants, as denoted by the lines 12, 13; and hence to draw the sides A B, the triangle must be set from one thread to the next on the opposite side of the bolt, as denoted by the dotted lines 12 and 13.
If the curves of the thread are to be drawn in, they may be obtained as in Figure 211, which is substantially the same as described for a V thread. The curves f, representing the sides of the thread, terminate at the centre line g, and the curves e are equidistant with the curves c from the vertical lines d. As the curves f above the line are the same as f below the line, the template for f need not be made to extend the whole distance across, but one-half only; as is shown by the dotted curve g, in the construction for finding the curve for square-threaded nuts in Figure 212.
A specimen of the form of template for drawing these curves is shown in Figure 213; g g, is the centre line parallel to the edges R, S; lines m, n, represent the diameter of the thread at the top, and o, p, that at the bottom or root; edge a is formed to the points (found by the constructions in the figures as already explained) for the tops of the thread, and edge f is so formed for the curve at the thread bottoms. The edge, as S or R, is laid against the square-blade to steady it while drawing in the curves. It may be noted, however, that since the curve is the same below the centre line as it is above, the template may be made to serve for one-half the thread diameter, as at f, where it is made from o to g, only being turned upside down to draw the other half of the curve; the notches cut out at x, x, are merely to let the pencil-lines in the drawing show plainly when setting the template.
When the thread of a nut is shown in section, it slants in the opposite direction to that which appears on the bolt-thread, because it shows the thread that fits to the opposite side of the bolt, which, therefore, slants in the opposite direction, as shown by the lines 12 and 13 in Figure 210.
In a top or end view of a nut the thread depth is usually shown by a simple circle, as in Figure 214.
To draw a spiral spring, draw the centre line A, and lines B, C, Figure 215, distant apart the diameter the spring is to be less the diameter of the wire of which it is to be made. On the centre line A mark two lines a b, c d, representing the pitch of the spring. Divide the distance between a and b into four equal divisions, as by lines 1, 2, 3, letting line 3 meet line B. Line e meeting the centre line at line a, and the line B at its intersection with line 3, is the angle of the coil on one side of the spring; hence it may be marked in at all the locations, as at e f, etc. These lines give at their intersections with the lines C and B the centres for the half circles g, which being drawn, the sides h, i, j, k, etc., of the spring, may all be marked in. By the lines m, n, o, p, the other sides of the spring may be marked in.
The end of the spring is usually marked straight across, as at L. If it is required to draw the coils curved instead of straight across, a template must be made, the curve being obtained as already described for threads. It may be pointed out, however, that to obtain as accurate a division as possible of the lines that divide the pitch, the pitch may be divided upon a diagonal line, as F, Figure 216, which will greatly facilitate the operation.
Before going into projections it may be as well to give some examples for practice.
CHAPTER IX.
EXAMPLES FOR PRACTICE.
Figure 217 represents a simple example for practice, which the student may draw the size of the engraving, or he may draw it twice the size. It is a locomotive spring, composed of leaves or plates, held together by a central band.
Figure 218 is an example of a stuffing box and gland, supposed to stand vertical, hence the gland has an oil cup or receptacle.
In Figure 219 are working drawings of a coupling rod, with the dimensions and directions marked in.
It may be remarked, however, that the drawings of a workshop are, where large quantities of the same kind of work is done, varied in character to suit some special departments—that is to say, special extra drawings are made for these departments. In Figures 220 and 221 is a drawing of a connecting rod drawn, put together as it would be for the lathe, vise or erecting shop.
To the two views shown there would be necessary detail sketches of the set screws, gibbs, and keys, all the rest being shown; the necessary dimensions being, of course, marked on the general drawing and on the details.
In so simple a thing as a connecting rod, however, there would be no question as to how the parts go together; hence detail drawings of each separate piece would answer for the lathe or vise bands.
But in many cases this would not be the case, and the drawing would require to show the parts put together, and be accompanied with such detail sketches as might be necessary to show parts that could not be clearly defined in the general views.
The blacksmith, for example, is only concerned with the making of the separate pieces, and has no concern as to how the parts go together. Furthermore, there are parts and dimensions in the general drawing with which the blacksmith has nothing to do.
Thus the location and dimensions of the keyways, the dimensions of the brasses, and the location of the bolt holes, are matters that have no reference to the blacksmith's work, because the keyways, bolt holes, and set-screw holes would be cut out of the solid in the machine shop. It is customary, therefore, to send to the blacksmith shop drawings containing separate views of each piece drawn to the shape it is to be forged; and drawn full size, or else on a scale sufficiently large to make each part show clearly without close inspection, marking thereon the full sizes, and stating beneath the number of pieces of each detail. (As in Figure 222, which represents the iron work of the connecting rod in Figure 220). In some cases the finished sizes are marked, and it is left to the blacksmith's judgment how much to leave for the finishing. This is undesirable, because either the blacksmith is left to judge what parts are to be finished, or else there must be on the drawing instructions on this point, or else signs or symbols that are understood to convey the information. It is better, therefore, to make for the blacksmith a special sketch, and mark thereon the full-forged sizes, stating on the drawing that such is the case.
As to the material of which the pieces are to be made, the greater part of blacksmith work is made of wrought iron, and it is, therefore, unnecessary to write "wrought iron" beneath each piece. When the pieces are to be of steel, however, it should be marked on the drawing and beneath the piece. In special cases, as where the greater part of the work of the shop is of steel, the rule may, of course, be reversed, and the parts made of iron may be the ones marked, whereas when parts are sometimes of iron, and at others of steel, each piece should be marked.
As a general rule the blacksmith knows, from the custom of the shop or the nature of the work, what the quality or kind of iron is to be, and it is, therefore, only in exceptional cases that they need to be mentioned on the drawing. Thus in a carriage manufactory, Norway or Swede iron will be found, as well as the better grades of refined iron, but the blacksmith will know what iron to use, for certain parts, or the shop may be so regulated that the selection of the iron is not left to him. In marking the number of pieces required, it is better to use the word "thus" than the words "of this," or "off this," because it is shorter and more correct, for the forging is not taken off the drawing, nor is it of the same; the drawing gives the shape and the size, and the word "thus" conveys that idea better than "of," "off," or "like this."
In shops where there are many of the same pieces forged, the blacksmith is furnished with sheet-iron templates that he can lay directly upon the forging and test its dimensions at once, which is an excellent plan in large work. Such templates are, of course, made from the drawings, and it becomes a question as to whether their dimensions should be the forged or the finished ones. If they are the forged, they may cause trouble, because a forging may have a scant place that it is difficult for the blacksmith to bring up to the size of the template, and he is in doubt whether there is enough metal in the scant place to allow the job to clean up. It is better, therefore, to make them to finished sizes, so that he can see at once if the work will clean up, notwithstanding the scant place. This will lead to no errors in large work, because such work is marked out by lines, and the scant part will therefore be discovered by the machinist, who will line out the piece accordingly.
Figure 223 is a drawing of a locomotive frame, which the student may as well draw three or four times as large as the engraving, which brings us to the subject of enlarging or reducing scales.
REDUCING SCALES.
It is sometimes necessary to reduce a drawing to a smaller scale, or to find a minute fraction of a given dimension, such fraction not being marked on the lineal measuring rules at hand. Figure 224 represents a scale for finding minute fractions. Draw seven lines parallel to each other, and equidistant draw vertical lines dividing the scale into half-inches, as at a, b, c, etc. Divide the first space e d into equal halves, draw diagonal lines, and number them as in the figure. The distance of point 1, which is at the intersection of diagonal with the second horizontal line, will be 1/24 inch from vertical line e. Point 2 will be 2/24 inch from line e, and so on. For tenths of inches there would require to be but six horizontal lines, the diagonals being drawn as before. A similar scale is shown in Figure 225. Draw the lines A B, B D, D C, C A, enclosing a square inch. Divide each of these lines into ten equal divisions, and number and letter them as shown. Draw also the diagonal lines A 1, a 2, B 3, and so on; then the distances from the line A C to the points of intersection of the diagonals with the horizontal lines represent hundredths of an inch.
Suppose, for example, we trace one diagonal line in its path across the figure, taking that which starts from A and ends at 1 on the top horizontal line; then where the diagonal intersects horizontal line 1, is 99/100 from the line B D, and 1/100 from the line A C, while where it intersects horizontal line 2, is 98/100 from line B D, and 2/100 from line A C, and so on. If we require to set the compasses to 67/100 inch, we set them to the radius of n, and the figure 3 on line B D, because from that 3 to the vertical line d 4 is 6/10 or 60/100 inch, and from that vertical line to the diagonal at n is seven divisions from the line C D of the figure.
In making a drawing to scale, however, it is an excellent plan to draw a line and divide it off to suit the required scale. Suppose, for example, that the given scale is one-quarter size, or three inches per foot; then a line three inches long may be divided into twelve equal divisions, representing twelve inches, and these may be subdivided into half or quarter inches and so on. It is recommended to the beginner, however, to spend all his time making simple drawings, without making them to scale, in order to become so familiar with the use of the instruments as to feel at home with them, avoiding the complication of early studies that would accompany drawing to scale.
CHAPTER X.
PROJECTIONS.
In projecting, the lines in one view are used to mark those in other views, and to find their shapes or curvature as they will appear in other views. Thus, in Figure 225a we have a spiral, wound around a cylinder whose end is cut off at an angle. The pitch of the spiral is the distance A B, and we may delineate the curve of the spiral looking at the cylinder from two positions (one at a right-angle to the other, as is shown in the figure), by means of a circle having a circumference equal to that of the cylinder.
The circumference of this circle we divide into any number of equidistant divisions, as from 1 to 24. The pitch A B of the spiral or thread is then divided off also into 24 equidistant divisions, as marked on the left hand of the figure; vertical lines are then drawn from the points of division on the circle to the points correspondingly numbered on the lines dividing the pitch; and where line 1 on the circle intersects line 1 on the pitch is one point in the curve. Similarly, where point 2 on the circle intersects line 2 on the pitch is another point in the curve, and so on for the whole 24 divisions on the circle and on the pitch. In this view, however, the path of the spiral from line 7 to line 19 lies on the other side of the cylinder, and is marked in dotted lines, because it is hidden by the cylinder. In the right-hand view, however, a different portion of the spiral or thread is hidden, namely from lines 1 to 13 inclusive, being an equal proportion to that hidden in the left-hand view.
The top of the cylinder is shown in the left-hand view to be cut off at an angle to the axis, and will therefore appear elliptical; in the right-hand view, to delineate this oval, the same vertical lines from the circle may be carried up as shown on the right hand, and horizontal lines may be drawn from the inclined face in one view across the end of the other view, as at P; the divisions on the circle may be carried up on the right-hand view by means of straight lines, as Q, and arcs of circle, as at R, and vertical lines drawn from these arcs, as line S, and where these vertical lines S intersect the horizontal lines as P, are points in the ellipse.
Let it be required to draw a cylindrical body joining another at a right-angle; as for example, a Tee, such as in Figure 226, and the outline can all be shown in one view, but it is required to find the line of junction of one piece, A, with the other, B; that is, find or mark the lines of junction C. Now when the diameters of A and B are equal, the line of junction C is a straight line, but it becomes a curved one when the diameter of A is less than that of B, or vice versa; hence it may be as well to project it in both cases. For this purpose the three views are necessary. One-quarter of the circle of B, in the end view, is divided off into any number of equal divisions; thus we have chosen the divisions marked a, b, c, d, e, etc.; a quarter of the top view is similarly divided off, as at f, g, h, i, j; from these points of division lines are projected on to the side view, as shown by the dotted lines k, l, m, n, o, p, etc., and where these lines meet, as denoted by the dots, is in each case a point in the line of junction of the two cylinders A, B.
Figure 227 represents a Tee, in which B is less in diameter than A; hence the two join in a curve, which is found in a similar manner, as is shown in Figure 227. Suppose that the end and top views are drawn, and that the side view is drawn in outline, but that the curve of junction or intersection is to be found. Now it is evident that since the centre line 1 passes through the side and end views, that the face a, in the end view, will be even with the face a' in the side view, both being the same face, and as the full length of the side of B in the end view is marked by line b, therefore line c projected down from b will at its junction with line b', which corresponds to line b, give the extreme depth to which b' extends into the body A, and therefore, the apex of the curve of intersection of B with A. To obtain other points, we divide one-quarter of the circumference of the circle B in the top view into four equal divisions, as by lines d, e, f, and from the points of division we draw lines j, i, g, to the centre line marked 2, these lines being thickened in the cut for clearness of illustration. The compasses are then set to the length of thickened line g, and from point h, in the end view, as a centre, the arc g' is marked. With the compasses set to the length of thickened line i, and from h as a centre, arc i' is marked, and with the length of thickened line j as a radius and from h as a centre arc j' is marked; from these arcs lines k, l, m are drawn, and from the intersection of k, l, m, with the circle of A, lines n, o, p are let fall. From the lines of division, d, e, f, the lines q, r, s are drawn, and where lines n, o, p join lines q, r, s, are points in the curve, as shown by the dots, and by drawing a line to intersect these dots the curve is obtained on one-half of B. Since the axis of B is in the same plane as that of A, the lower half of the curve is of the same curvature as the upper, as is shown by the dotted curve.
In Figure 228 the axis of piece B is not in the same plane as that of D, but to one side of it to the distance between the centre lines C, D, which is most clearly seen in the top view. In this case the process is the same except in the following points: In the side view the line w, corresponding to the line w in the end view, passes within the line x before the curve of intersection begins, and in transferring the lengths of the full lines b, c, d, e, f to the end view, and marking the arcs b', c', d', e', f', they are marked from the point w (the point where the centre line of B intersects the outline of A), instead of from the point x. In all other respects the construction is the same as that in Figure 227.
In these examples the axis of B stands at a right-angle to that of A. But in Figure 229 is shown the construction where the axis of B is not at a right-angle to A. In this case there is projected from B, in the side view, an end view of B as at B', and across this end at a right-angle to the centre line of B is marked a centre line C C of B', which is divided as before by lines d, e, f, g, h, their respective lengths being transferred from W as a centre, and marked by the arcs d', e', f', which are marked on a vertical line and carried by horizontal lines, to the arc of A as at i, j, k. From these points, i, j, k, the perpendicular lines l, m, n, o, are dropped, and where these lines meet lines p, q, r, s, t, are points in the curve of intersection of B with A. It will be observed that each of the lines m, n, o, serves for two of the points in the curve; thus, m meets q and s, while n meets p and t, and o meets the outline on each side of B, in the side view, and as i, j, k are obtained from d and e, the lines g and h might have been omitted, being inserted merely for the sake of illustration.
In Figure 230 is an example in which a cylinder intersects a cone, the axes being parallel. To obtain the curve of intersection in this case, the side view is divided by any convenient number of lines, as a, b, c, etc., drawn at a right-angle to its axis A A, and from one end of these lines are let fall the perpendiculars f, g, h, i, j; from the ends of these (where they meet the centre line of A in the top view), half-circles k, l, m, n, o, are drawn to meet the circle of B in the top view, and from their points of intersection with B, lines p, q, r, s, t, are drawn, and where these meet lines a, b, c, d and e, which is at u, v, w, x, y, are points in the curve.
It will be observed, on referring again to Figure 229, that the branch or cylinder B appears to be of elliptical section on its end face, which occurs because it is seen at an angle to its end surface; now the method of finding the ellipse for any given degree of angle is as in Figure 231, in which B represents a cylindrical body whose top face would, if viewed from point I, appear as a straight line, while if viewed from point J it would appear in outline a circle. Now if viewed from point E its apparent dimension in one direction will obviously be defined by the lines S, Z. So that if on a line G G at a right angle to the line of vision E, we mark points touching lines S, Z, we get points 1 and 2, representing the apparent dimension in that direction which is the width of the ellipse. The length of the ellipse will obviously be the full diameter of the cylinder B; hence from E as a centre we mark points 3 and 4, and of the remaining points we will speak presently. Suppose now the angle the top face of B is viewed from is denoted by the line L, and lines S', Z, parallel to L, will be the width for the ellipse whose length is marked by dots, equidistant on each side of centre line G' G', which equal in their widths one from the other the full diameter of B. In this construction the ellipse will be drawn away from the cylinder B, and the ellipse, after being found, would have to be transferred to the end of B. But since centre line G G is obviously at the same angle to A A that A A is to G G, we may start from the centre line of the body whose elliptical appearance is to be drawn, and draw a centre line A A at the same angle to G G as the end of B is supposed to be viewed from. This is done in Figure 231 a, in which the end face of B is to be drawn viewed from a point on the line G G, but at an angle of 45 degrees; hence line A A is drawn at an angle of 45 degrees to centre line G G, and centre line E is drawn from the centre of the end of B at a right angle to G G, and from where it cuts A A, as at F, a side view of B is drawn, or a single line of a length equal to the diameter of B may be drawn at a right angle to A A and equidistant on each side of F. A line, D D, at a right angle to A A, and at any convenient distance above F, is then drawn, and from its intersection with A A as a centre, a circle C equal to the diameter of B is drawn; one-half of the circumference of C is divided off into any number of equal divisions as by arcs a, b, c, d, e, f. From these points of division, lines g, h, i, j, k, l are drawn, and also lines m, n, o, p, q, r. From the intersection of these last lines with the face in the side view, lines s, t, u, t, w, x, y, z are drawn, and from point F line E is drawn. Now it is clear that the width of the end face of the cylinder will appear the same from any point of view it may be looked at, hence the sides H H are made to equal the diameter of the cylinder B and marked up to centre line E.
It is obvious also that the lines s, z, drawn from the extremes of the face to be projected will define the width of the ellipse, hence we have four of the points (marked respectively 1, 2, 3, 4) in the ellipse. To obtain the remaining points, lines t, u, v, w, x, y (which start from the point on the face F where the lines m, n, o, p, q, r, respectively meet it) are drawn across the face of B as shown. The compasses are then set to the radius g; that is, from centre line D to division a on the circle, and this radius is transferred to the face to be projected the compass-point being rested at the intersection of centre line G and line t, and two arcs as 5 and 6 drawn, giving two more points in the curve of the ellipse. The compasses are then set to the length of line h (that is, from centre line D to point of division b), and this distance is transferred, setting the compasses on centre line G where it is intersected by line u, and arcs 7, 8 are marked, giving two more points in the ellipse. In like manner points 9 and 10 are obtained from the length of line i, 11 and 12 from that of j; points 13 and 14 from the length of k, and 15 and 16 from l, and the ellipse may be drawn in from these points.
It may be pointed out, however, that since points 5 and 6 are the same distance from G that points 15 and 16 are, and since points 7 and 8 are the same distance from G that points 13 and 14 are, while points 9 and 10 are the same distance from G that 11 and 12 are, the lines, j, k, l are unnecessary, since l and g are of equal length, as are also h and k and i and j. In Figure 232 the cylinders are line shaded to make them show plainer to the eye, and but three lines (a, b, c) are used to get the radius wherefrom to mark the arcs where the points in the ellipse shall fall; thus, radius a gives points 1, 2, 3 and 4; radius b gives points 5, 6, 7 and 8, and radius c gives 9, 10, 11 and 12, the extreme diameter being obtained from lines S, Z, and H, H.
CHAPTER XI.
DRAWING GEAR WHEELS.
The names given to the various lines of a tooth on a gear-wheel are as follows:
In Figure 233, A is the face and B the flank of a tooth, while C is the point, and D the root of the tooth; E is the height or depth, and F the breadth. P P is the pitch circle, and the space between the two teeth, as H, is termed a space.
It is obvious that the points of the teeth and the bottoms of the spaces, as well as the pitch circle, are concentric to the axis of the wheel bore. And to pencil in the teeth these circles must be fully drawn, as in Figure 234, in which P P is the pitch circle. This circle is divided into as many equal divisions as the wheel is to have teeth, these divisions being denoted by the radial lines, A, B, C, etc. Where these divisions intersect the pitch circle are the centres from which all the teeth curves may be drawn. The compasses are set to a radius equal to the pitch, less one-half the thickness of the tooth, and from a centre, as R, two face curves, as F G, may be marked; from the next centre, as at S, the curves D E may be marked, and so on for all the faces; that is, the tooth curves lying between the outer circle X and the pitch circle P. For the flank curves, that is, the curve from P to Y, the compasses are set to a radius equal to the pitch; and from the sides of the teeth the flank curves are drawn. Thus from J, as a centre flank, K is drawn; from V, as a centre flank, H is drawn, and so on.
The proportions of the teeth for cast gears generally accepted in this country are those given by Professor Willis, as average practice, and are as follows:
Depth to pitch line, 3/10 of the pitch. Working depth, 6/10 " " Whole depth, 7/10 " " Thickness of tooth, 5/11 " " Breadth of space, 6/11 " "
Instead, however, of calculating the dimensions these proportions give for any particular pitch, a diagram or scale may be made from which they may be taken for any pitch by a direct application of the compasses. A scale of this kind is given in Figure 235, in which the line A B is divided into inches and parts to represent the pitches; its total length representing the coarsest pitch within the capacity of the scale; and, the line B C (at a right-angle to A B) the whole depth of the tooth for the coarsest pitch, being 7/10 of the length of A B.
The other diagonal lines are for the proportion of the dimensions marked on the figure. Thus the depth of face, or distance from the pitch line to the extremity or tooth point for a 4 inch pitch, would be measured along the line B C, from the vertical line B to the first diagonal. The thickness of the tooth would be for a 4 inch pitch along line B C from B to the second diagonal, and so on. For a 3 inch pitch the measurement would be taken along the horizontal line, starting from the 3 on the line A B, and so on. On the left of the diagram or scale is marked the lbs. strain each pitch will safely transmit per inch width of wheel face, according to Professor Marks.
The application of the scale as follows: The pitch circles P P and P' P', Figure 236, for the respective wheels, are drawn, and the height of the teeth is obtained from the scale and marked beyond the pitch circles, when circles Q and Q' may be drawn. Similarly, the depths of the teeth within the pitch circles are obtained from the scale or diagram and marked within the respective pitch circles, and circles R and R' are marked in. The pitch circles are divided off into as many points of equal division, as at a, b, c, d, e, etc., as the respective wheels are to have teeth, and the thickness of tooth having been obtained from the scale, this thickness is marked from the points of division on the pitch circles, as at f in the figure, and the tooth curves may then be drawn in. It may be observed, however, that the tooth thicknesses will not be strictly correct, because the scale gives the same chord pitch for the teeth on both wheels which will give different arc pitches to the teeth on the two wheels; whereas, it is the arc pitches, and not the chord pitches, that should be correct. This error obviously increases as there is a greater amount of difference between the two wheels.
The curves given to the teeth in Figure 234 are not the proper ones to transmit uniform motion, but are curves merely used by draughtsmen to save the trouble of finding the true curves, which if it be required, may be drawn with a very near approach to accuracy, as follows, which is a construction given by Rankine:
Draw the rolling circle D, Figure 237, and draw A D, the line of centres. From the point of contact at C, mark on D, a point distant from C one-half the amount of the pitch, as at P, and draw the line P C of indefinite length beyond C. Draw the line P E passing through the line of centres at E, which is equidistant between C and A. Then increase the length of line P F to the right of C by an amount equal to the radius A C, and then diminish it to an amount equal to the radius E D, thus obtaining the point F and the latter will be the location of centre for compasses to strike the face curve.
Another method of finding the face curve, with compasses, is as follows: In Figure 238 let P P represent the pitch circle of the wheel to be marked, and B C the path of the centre of the generating or describing circle as it rolls outside of P P. Let the point B represent the centre of the generating circle when it is in contact with the pitch circle at A. Then from B mark off, on B C, any number of equidistant points, as D, E, F, G, H, and from A mark on the pitch circle, with the same radius, an equal number of points of division, as 1, 2, 3, 4, 5. With the compasses set to the radius of the generating circle, that is, A B, from B, as a centre, mark the arc I, from D, the arc J, from E, the arc K, from F, and so on, marking as many arcs as there are points of division on B C. With the compasses set to the radius of divisions 1, 2, etc., step off on arc M the five divisions, N, O, S, T, V, and at V will be a point on the epicycloidal curve. From point of division 4, step off on L four points of division, as a, b, c, d; and d will be another point on the epicycloidal curve. From point 3, set off three divisions, and so on, and through the points so obtained draw by hand, or with a scroll, the curve.
Hypocycloids for the flanks of the teeth maybe traced in a similar manner. Thus in Figure 239, P P is the pitch circle, and B C the line of motion of the centre of the generating circle to be rolled within P P. From 1 to 6 are points of equal division on the pitch circle, and D to I are arc locations for the centre of the generating circle. Starting from A, which represents the location for the centre of the generating circle, the point of contact between the generating and base circles will be at B. Then from 1 to 6 are points of equal division on the pitch circle, and from D to I are the corresponding locations for the centres of the generating circle. From these centres the arcs J, K, L, M, N, O, are struck. The six divisions on O, from a to f, give at f a point in the curve. Five divisions on N, four on M, and so on, give, respectively, points in the curve.
There is this, however, to be noted concerning the construction of the last two figures. Since the circle described by the centre of the generating circle is of a different arc or curve to that of the pitch circle, the length of an arc having an equal radius on each will be different. The amount is so small as to be practically correct. The direction of the error is to give to the curves a less curvature, as though they had been produced by a generating circle of larger diameter. Suppose, for example, that the difference between the arc a, b, and its chord is .1, and that the difference between the arc 4, 5, and its chord is .01, then the error in one step is .09, and, as the point f is formed in five steps, it will contain this error multiplied five times. Point d would contain it multiplied three times, because it has three steps, and so on.
The error will increase in proportion as the diameter of the generating is less than that of the pitch circle, and though in large wheels, working with large wheels, so that the difference between the radius of the generating circle and that of the smallest wheel is not excessive, it is so small as to be practically inappreciable, yet in small wheels, working with large ones, it may form a sensible error.
For showing the dimensions through the arms and hub, a sectional view of a section of the wheel may be given, as in Figure 240, which represents a section of a wheel, and a pinion, and on these two views all the necessary dimensions may be marked.
If it is desired to draw an edge view of a wheel (which the student will find excellent practice), the lines for the teeth may be projected from the teeth in the side view, as in Figure 240 a. Thus tooth E is projected by drawing lines from the corners A, B, C, in the side view across the face in the edge view, as at A, B, C in the latter view, and similar lines may be obtained in the same way for all the teeth.
When the teeth of wheels are to be cut to form in a gear-cutting machine, the thickness of the teeth is nearly equal to the thickness of the spaces, there being just sufficient difference to prevent the teeth of one wheel from becoming locked in the spaces of the other; but when the teeth are to be cast upon the wheel, the tooth thickness is made less than the width of the space to an amount that is usually a certain proportion of the pitch, and is termed the side clearance. In all wheels, whether with cut or cast teeth, there is given a certain amount of top and bottom clearance; that is to say, the points of the teeth of one wheel do not reach to the bottom of the spaces in the other. Thus in the Pratt and Whitney system the top and bottom clearance is one-eighth of the pitch, while in the Brown and Sharpe system for involute teeth the clearance is equal to one-tenth the thickness of the tooth.
In drawing bevil gear wheels, the pitch line of each tooth on each wheel, and the surfaces of the points, as well as those at the bottom of the spaces, must all point to a centre, as E in Figure 241, which centre is where the axes of the shafts would meet. It is unnecessary to mark in the correct curves for the teeth, for reasons already stated, with reference to the curves for a spur wheel. But if it is required to do so, the construction to find the curves is as shown in Figure 242, in which let A A represent the axis of one shaft, and B that of the other of the pair of bevil wheels that are to work together, their axes meeting at W; draw the line E at a right angle to A A, and representing the pitch circle diameter of one wheel, and draw F at a right angle to B, and representing the pitch circle of the other wheel; draw the line G G, passing through the point W and the point T, where the pitch circles or lines E F meet, and G G will be the line of contact of the tooth of one wheel upon the tooth of the other wheel; or in other words, the pitch line of the tooth. |
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