of centres, and the radii of the two circles the proportiona. radii. These circles are called, by mill-wrights. in general, pitch-lines. The distances from the centres of two circles to the extremities of their respective teeth, are called the real radii, and the distances between the centres of two contiguous teeth measured upon the pitch-line, is called the pitch of the wheel. Two wheels acting upon one another in the same plane, are called spur geer. When they act at an angle, they are called bevel geer. Teeth of wheels and leaves of pinions require great care and judgment in their formation, so that they neither clog the machinery with unnecessary friction, nor act so irre gularly as to produce any inequalities in the motion, and a consequent wearing away of one part before another. Much has been written on this subject by mathematicians, who seem to agree that the epicycloid is the best of all curves for the teeth of wheels. The epicycloid is a curve differing from the cycloid formerly described, in this, that the generating circle instead of moving along a straight edge, moves on the circumference of another circle. The teeth of one wheel should press in a direction perpendicular to the radius of the wheel which it drives. As many teeth as possible should be in contact at the same time, in order to distribute the strain amongst them; by this means the chance of breaking the teeth will be diminished. During the action of one tooth upon another, the direction of the pressure should remain the same, so that the effect may be uniform. The surfaces of the teeth in working should not rub one against another, and should suffer no jolt either at the commencement or the termination of their mutual contact. The form of the epicycloid satisfies all these conditions; but it is intricate, and the involute of the circle is here substituted, as satisfying equally these conditions, and as being much more easily described. Take the circumference ABC of the wheel on which it is proposed to raise the teeth, and let a be a point from which one surface of one tooth is to spring, then fasten a string at B, such that when stretched and lying on the circumference shall reach to a; fix a ba pencil at a, and keeping the string equally tense, move the pencil outwards, and it will describe the involute of the circle which will form the curve for one side of the tooth. Fasten the string at B so that its end, to which the pencil is fixed, be at the point from which the other face of the tooth is to spring-and proceed as above; then the curve of the other side of the tooth will be formed; and the figure of one of the teeth being determined, the rest may be traced from it. The teeth of the pinion are formed in like manner. The observation of practical men has furnished us with a method of forming teeth of wheels, which is found to an swer fully as well in practice as any of the geometrical curves of the mathematician. We have the pattern here of the segment of a wheel with cogs fixed on in their rough state, and it is required to bring them to their proper figure: C they are consequently understood to be much larger than they are intended to be when dressed. The arc b, b, is the circumference of the wheel on which the bottoms of the teeth and cogs rest. Draw an arc a, a, on the face of the teeth for the pitch line of their point of action; draw also d, d, for their extremities or tops. When this is done, the pitch circle is correctly divided into as many equal parts as there are to be teeth. The compasses are then to be opened to an extent of one and a quarter of those divisions, and with this radius arcs are described on each side of every division on the pitch line a, a, from that line to the line d, d. One point of the compasses being set on c, the curve f, g, on one side of one tooth, and o, n, on the other sides of the other are described. Then the point of the compasses being set on the adjacent division k, the curve l, m, will be described this completes the curved portion of the tooth e. The remaining portion of the tooth within the circle a, a, is bounded by two straight lines drawn from g and m towards the centre. The same being done to the teeth all round, the mark is finished, and the cogs only require to be dressed down to the lines thus drawn. It will be easy to determine the diameter of any wheel having the pitch and number of teeth in that wheel given Thus, a wheel of 54 teeth having a pitch of 3 inches, we have 54 × 3 = 162 inches, the circumference, consequently, 162 3.1416 or about 4 feet 3 51.5 inches diameter, nearly. inches. In the following table we have given the radii of wheels of various numbers of teeth, the pitch being one inch. To find the radius for any other pitch, we have only to multiply the radius in the table by the pitch in inches, the product is the answer. Thus for 30 teeth at a pitch of 31⁄2 inches, 16.74 inches, the radius. we have 4.783 × 3.5 10 1.668 1-774 1.932 2.089 2.247 2.405 2.563 2-721 2-879 3-038 20 3.196 3 355 3.513 3-672 3-830 3.989 4-148 4.307 4-465 4.624 30 4.783 4-942 5-101 5-260 5-419 5-578 5.737 5.896 6.055 6.214 40 6-373 6.532 6.643 6-850 7.009 7.168 7-327 7-486 7-695 7.804 50 7.963 8.122 8-231 8.440 8.599 8-753 8.962 9-076 9-235 9-399 60 9.553 9.712 9.872 10-031 10-190 10-349 10-508 10-662 10-826 10.935) 70 11-144 11-303 11.463 11-622 11-731 11.940 12-099 12-758 12-417 12-676 80 12.735 12.895 13.054 13-213 13-370 13-531 13 690 13-849 14-008 14-168 90 14.327 14-436 14-645 14.804 14.963 15.122 15.281 15-441 15-600 15-759 100 15-918 16.072 16-236 16-395 16.554 16-713 16-873 17-032 17-191 17-350 110 17-504 17.668 17.987 17-827 18-146 18-305 18-464 18-623 18-782 18-941 120 19 101 19-260 19-419 19-578 19-737 19-896 20-055 20-214 20-374 20-533 130 20-692 20-851 21.010 21.169 21.328 21-488 21-647 21-806 21-460 22-124 140 22.283 22-442 22.602 22-761 22.920 23-074 23-238 23-397 23-556 23-716 150 23-875 24-034 24-193 24-352 24.511 24-620 24-830 24-989 25·148 25-307 160 25 466 25.625 25.784 25.944 26-103 26.262 26-421 26-580 26-739 26-894 170 27-058 27-217 27-376 27.535 27.694 27.853 27-931 28-172 28-331 28-490) 180 28-699 28.808 28.967 29-126 29-286 29-445 29-604 29-763 29.922 30-086 190 30-241 30-400 30-559 30-718 30-877 31-036 31.196 31-355 31.514 31-673 200 31-832 31.992 32-150 32-310 32-469 32.628 32-787 32-846 33-105 33-264) 210 33-424 33.583 33-742 33.901 34.060 34.219 34 278 34-537 34-697 34-856 220 35-015 35.174 35-333 35-492 35.652 35.811 35.970 36-129 36-288 36-447 230 36-607 36-766 36 925 37-084 37-243 37-402 37-561 37-720 37.880 38-039 240 38.198 38.357 38.516 38.725 38.835 38-994 39-153 39-312 39-471 39-631 250 39-790 39-949 40 108 40-262 40-426 40-585 40-744 40-904 41-063 41-222 260 41-381 41-541, 41-699 41.858 42 019 42-177 42-336 42-495 42-65442-813 270 42-973 43-132 43-291 43-450 43.609 43-768 43-927 44 087 44-231 44-405 280 44-564 44-723 44-882 45-042 45-201 45-369 45-519 45-678 45-837 45-996 290 46.156 46-315 46-474 46-633 46-792 46-751 47-111 47-270 47-429 47-588 This will be found a very useful table in abridging calculation, for instance, if we wish to find the radius of a wheel having 132 teeth, we look for 130 at the left-hand side column, and 2 at the top, and where these columns meet, we find the number 21.010, which, if the pitch of the wheel be 24 inches, we multiply by 24. 21.010 × 2.5 = 52.525 inches, radius of required wheel. An easy practical rule for the same purpose is the following: Take the pitch by a pair of compasses, and lay it off on a straight line, seven times, divide this line into eleven equal parts; each will be equal to four of the radius, which is supposed to consist of as many parts as the wheel has teeth. Let the pitch be two inches, and the number of teeth 60 then the diagram will show how to lay it down. The upper line is the pitch laid off seven times, and forming AB, which is divided into 11 equal parts, one of which, CD, being repeated for every four teeth in the wheel, that is, in this case, fifteen times, will give the radius. The same may be done by calculation, going by the principles of the rule, thus, parts of an inch. = Reversing the operation, let it be required to find the pitch, the radius of the wheel being 19.104, and number of teeth 60. We have 19.104 60 and 1.272 × 11 = and therefore, = 318, then 318 × 4 = 1.272, 13.992. Now this is the whole line AB, = = 1.998, which is so very nearly 7 13.992 two inches, the difference being 2 1.998 = ⚫002 of an inch, we ought in practice to take two as the pitch. A little reflection on the part of the reader will show 11 .636 ⚫636, and = 1-571, and ='159 7 (1) pitch 159 x number of teeth = radius. (1) 2 × 159 × 60 = 19.08 = radius, NOTE. The number 16 may be employed instead of 159, being easily remembered. These rules are approxinate, and the error diminishes as the number of teeth increases. The true pitch is a straight line, but these rules give it an arc of the circle, which passes through the centre of the teeth, whereas it should be the chord of the arc. An eminent writer on clock-work gives the following rules regarding wheels and pinions : (A) As the number of teeth in the wheel +2.25, Is to the diameter of the wheel, So is the number of teeth in the pinion + 1.5, A wheel being 12 inches diameter, having 120 teeth, drives a pinion of 20 leaves; wherefore, 120 +2.25 = 122.25 and 20 + 1·5 = 21.5, Then 122.25 : 12 :: 21·5 : 2·1104 = the diameter of the pinion. (B) As the number of teeth in the wheel + 2.25, Is to the wheel's diameter, So is (teeth in wheel + leaves in pinion) A wheel's diameter being 3.2 inches, number of teeth 96, the leaves in the pinion being 8, then, 96 +2.25, 98.25 and (96 +- 8) = 104 |