Fig. 208.
These half-circles are divided into any convenient number of equal divisions: thus in Figure 208, each has eight divisions, asa,b,c, etc., for the outer, andi,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 linesa,b,c, etc., too. Of these, the seven froma, toh, correspond to the seven froma'tog', and are for the top of the thread, and the seven fromitoocorrespond to the seven on the inner half-circle, asi,j,k, etc. Horizontal lines are then drawn from thepoints of the division to meet the vertical lines of division; thus the horizontal dotted line froma'meets the vertical linea, and where they meet, as at A, a dot is made. Where the dotted line fromb'meets vertical lineb, 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 fromi'to meeti, where dot I is marked. J is got fromj'andj, while K is got from the intersection ofk'withk, 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 fromatoh) 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 thesame 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.
Fig. 208 a.
Fig. 209.
The curves for a United States standard form of thread are obtained in the same manner as from theVthread 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 commonVthread 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 commonVthread and dividing its depth into eight equal divisions, as atx, in Figure 208a, 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 toolarge in diameter. Thus, in Figure 208a, 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 ata,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 cornerfon 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 arcsdande, 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 frombtoiare used, while for the tops the eight fromgtooare used. Henceg,handiare used for getting both curves, the divisions fromatoband fromotopbeing 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 Figure208 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.
Fig. 210.
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.
Fig. 211.
If the curves of the thread are to be drawn in, theymay be obtained as in Figure 211, which is substantially the same as described for a V thread. The curvesf, representing the sides of the thread, terminate at the centre lineg, and the curveseare equidistant with the curvescfrom the vertical linesd. Asthe curvesfabove the line are the same asfbelow the line, the template forfneed not be made to extend the whole distance across, but one-half only; as is shown by the dotted curveg, in the construction for finding the curve for square-threaded nuts in Figure 212.
Fig. 212.
Fig. 213.
A specimen of the form of template for drawing these curves is shown in Figure 213;gg, is the centre line parallel to the edges R, S; linesm,n, represent the diameter of the thread at the top, ando,p, that at the bottom or root; edgeais formed to the points (found by the constructions in the figures as already explained) for the tops of the thread, and edgefis 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 atf, where it is made fromotog, only being turned upside down to draw the other half of the curve; the notches cut out atx,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.
Fig. 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 linesa b,c d, representing the pitch of the spring. Divide the distance betweenaandbinto four equal divisions, as by lines 1, 2, 3, letting line 3 meet line B. Lineemeeting the centre line at linea, 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 ate f, etc. These lines give at their intersections with the lines C and B the centres for the half circlesg, which being drawn, the sidesh,i,j,k, etc., of the spring, may all be marked in. By the linesm,n,o,p, the other sides of the spring may be marked in.
Fig. 215.
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 upona diagonal line, as F, Figure 216, which will greatly facilitate the operation.
Fig. 216.
Before going into projections it may be as well to give some 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.
Fig. 217.
Figure 218 is an example of a stuffing box and gland, supposed to stand vertical, hence the gland has an oil cup or receptacle.
Fig. 218.
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.
Fig. 219. (Page 169.)
Fig. 220.
Fig. 221.
Fig. 222.
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 ofthe 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.
Fig. 223
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 ofpieces 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.
Fig. 224.
Fig. 225.
Fig. 225a.
It is sometimes necessary to reduce a drawing to a smaller scale, or to find a minute fraction of a givendimension, 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 ata,b,c, etc. Divide the first spacee dinto 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 linee. Point 2 will be 2/24 inch from linee, 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,a2, 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 intersectshorizontalline 1, is 99/100 from the line B D, and 1/100 from the line A C, while where it intersectshorizontalline 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 ofn, and the figure 3 on line B D, because from that 3 to the vertical lined4 is 6/10 or 60/100 inch, and from that vertical line to the diagonal atnis 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.
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 225awe 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 thecylinder. In the right-hand view, however, a different portion of the spiral or thread is hidden, namely fromlines 1 to 13 inclusive, being an equal proportion to that hidden in the left-hand view.
Fig. 226.
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, orvice 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 markeda,b,c,d,e, etc.; a quarter of the top view is similarly divided off, as atf,g,h,i,j; from these points of division lines are projected on to the side view, as shown by the dottedlinesk,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.
Fig. 227.
Fig. 228.
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 Figure227. 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 facea, in theend view, will be even with the facea'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 lineb, therefore linecprojected down frombwill at its junction with lineb', which corresponds to lineb, give the extreme depth to whichb'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 linesd,e,f, and from the points of division we draw linesj,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 lineg, and from pointh, in the end view, as a centre, the arcg'is marked. With the compasses set to the length of thickened linei, and fromhas a centre, arci'is marked, and with the length of thickened linejas a radius and fromhas a centre arcj'is marked; from these arcs linesk,l,mare drawn, and from the intersection ofk,l,m, with the circle of A, linesn,o,pare let fall. From the lines of division,d,e,f, the linesq,r,sare drawn, and where linesn,o,pjoin linesq,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.
Fig. 229.
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 mostclearly seen in the top view. In this case the process is the same except in the following points: In the side view the linew, corresponding to the linewin the end view, passes within the linexbefore the curve of intersection begins, and in transferring the lengths ofthe full linesb,c,d,e,fto the end view, and marking the arcsb',c',d',e',f', they are marked from the pointw(the point where the centre line of B intersects the outline of A), instead of from the pointx. In all other respects the construction is the same as that in Figure 227.
Fig. 230.
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-angleto 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 linesd,e,f,g,h, their respective lengths being transferred from W as a centre, and marked by the arcsd',e',f', which are marked on a vertical line and carried by horizontal lines, to the arc of A as ati,j,k. From these points,i,j,k, the perpendicular linesl,m,n,o, are dropped, and where these lines meet linesp,q,r,s,t, are points in the curve of intersection of B with A. It will be observed that each of the linesm,n,o, serves for two of the points in the curve; thus,mmeetsqands, whilenmeetspandt, andomeets the outline on each side of B, in the side view, and asi,j,kare obtained fromdande, the linesgandhmight 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, asa,b,c, etc., drawn at a right-angle to its axis A A, and from one end of these lines are let fall the perpendicularsf,g,h,i,j; from the ends of these (where they meet the centre line of A in the top view), half-circlesk,l,m,n,o, are drawn to meet the circle of B in the top view, and from their points of intersection with B, linesp,q,r,s,t, are drawn, and where these meet linesa,b,c,dande, which is atu,v,w,x,y, are points in the curve.
Fig. 231.
Fig. 231 a.
It will be observed, on referring again to Figure 229, that the branch or cylinder B appears to be of ellipticalsection 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 isas 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 onefrom 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 231a, 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 arcsa,b,c,d,e,f. From these points of division, linesg,h,i,j,k,lare drawn, and also linesm,n,o,p,q,r. From the intersection of these last lines with the face in the side view, liness,t,u,t,w,x,y,zare 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.
Fig. 232.
Fig. 233.
It is obvious also that the liness,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 obtainthe remaining points, linest,u,v,w,x,y(which start from the point on the face F where the linesm,n,o,p,q,r, respectively meet it) are drawn across the face of B as shown. The compasses are then set to the radiusg; that is, from centre line D to divisionaon 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 linet, 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 lineh(that is, from centre line D to point of divisionb), and this distance is transferred, setting the compasses on centre line G where it is intersectedby lineu, 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 linei, 11 and 12 from that ofj; points 13 and 14 from the length ofk, and 15 and 16 froml, 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,lare unnecessary, sincelandgare of equal length, as are alsohandkandiandj. 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, radiusagives points 1, 2, 3 and 4; radiusbgives points 5, 6, 7 and 8, and radiuscgives 9, 10, 11 and 12, the extreme diameter being obtained from lines S, Z, and H, H.
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.
Fig. 234.
Fig. 235.
It is obvious that the points of the teeth and thebottoms 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 asthe 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,3/10of thepitch.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) thewhole depth of the tooth for the coarsest pitch, being 7/10 of the length of A B.
Fig. 236.
The other diagonal lines are for the proportion of the dimensions marked on the figure. Thus thedepth 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.
Fig. 237.
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 ata,b,c,d,e, etc., as the respective wheels are to have teeth, and the thickness of tooth having been obtained from the scale, thisthickness is marked from the points of division on the pitch circles, as atfin 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.