: This slow relative motion we can utilise as follows:In the boss of B cut a thread, and cut on the spindle of A a thread to fit it. B is then a nut and A the Let B be prevented by bearings from rising or screw. falling. Then as A slowly gains on B the spindle gradually screws out of B and descends. A also would descend with it, and thus the motion would soon stop, for A would leave C. But we can prevent this by connecting A to its spindle, not by a tight key, but by a key which can slide in a long slot in the spindle, and now the spindle can descend leaving A behind in its bearings. Fig. 102 shows an end view of A and its spindle, k the key can slide in the slot, but is fixed to A. A Fig. 102. as it turns. In the practical case the spindle of A is that of a drilling machine, and the mechanism becomes a self-feeding arrangement, giving the spindle a slow descending motion Crabs.-The two practical cases we have last considered have been mechanisms, motion being the object aimed at. As an example in which modification of effort is required we may take the common crab. This is used in two forms. In each case the moment is applied to A either by manual effort on a handle or handles, or by an engine. The resistance is a weight to be lifted, the rope or chain which lifts it passing round a barrel attached to B, thus applying the resisting moment. If A drive B direct we have a single purchase. If intermediate wheels C, C' be used we have a double purchase. A and C' are small, while C and B are large wheels, hence we obtain a large mechanical advantage. Bevel Wheels. When two turning pairs whose axes are not parallel but meet in a point are to be connected, the connection can To prove this, suppose the contact at b is pure rolling, then we will prove it is so also at a. Let V the common velocity of the peripheries at b, Velocity of periphery of A at the point a=AA × am, am =V. nb' and similarly Velocity of periphery of B at the point a=V. ap bq so that at a the peripheral velocities of A and B are identical, and the motion is there pure rolling. K The same proof applies to any point on ab, the line of contact. So there is pure rolling everywhere. By cutting teeth as in the preceding we produce the same relative motion as pure rolling would produce (see page 123), and we then treat the bevel wheels as if they were simply the smooth wheels from which we have derived them. For the velocity ratio we have To construct a pair of bevel wheels, or at least the smooth wheels equivalent to them, the faces of which are called the pitch surfaces (compare pitch circle, page 124), we proceed thus— OC and OD are given, and we can select the maximum or minimum or mean diameter of one wheel. Let us take then ab as the maximum diameter of A, and set it off perpendicular to OC. Let n be the given velocity ratio, i.e. AB: AA. Take cd equal to ab/n, and set it off perpendicular to OD. Draw be, de parallel to OC, OD respectively, meeting Drop ef, eg perpendicular to OC, OD, and these are the maximum diameters of the two wheels. We then complete them as shown, taking any convenient thickness. Then The two cones are called the pitch cones, and the angles eOC, eOD their angles (Fig. 104). When the angle between the two shafts is 90° the wheels are called Mitre Wheels. If the angle of one cone be 180° the pitch surface is a plane, and the teeth are on the flat side of the wheel; such a wheel is called a Crown Wheel, A crown wheel is usually of large size compared to the bevel wheel which gears with it, and this wheel is called a Pinion. When a small common toothed wheel gears with a large one, the small one is called the Pinion and the large the Spur Wheel. EXAMPLES. 1. A pair of blocks have three sheaves in the upper and two in the lower block. Find the pull required to raise ton, assuming of the energy wasted. Ans. 186 lbs. 2. Two sheaves 8 ins. and 7 ins. diameter respectively are fastened together, and turn on one axle, forming the upper block of a Weston differential pulley. The lower block contains one sheave only. The left-hand ply of chain supporting the lower pulley passes over the 8 in. diameter to the right, and the right-hand ply over the 7 in. diameter to the left; the two loose ends are then connected, so the chain is endless, hanging in two loops, one loose and one supporting the movable block and weight. Find the weight which can be lifted by a pull of 10 lbs., of the energy being wasted. Ans. The pull being applied to the loose part hanging from the 8 in. sheave, let it turn the upper pulley once. Then space traversed by pull is 8 ins. The 8 in. sheave winds on from the loop supporting W 8π ins., but the 7 in. sheave must also turn once, being fixed to the 8 in., and this inwinds 77 ins. The tight loop is thus shortened ins. (and the loose one lengthened Whence weight raised = × 16 × 10=64 lbs. 3. A shaft is to be driven at 400 revolutions per minute; a pulley on it is 8 ins. diameter. The shaft from which it is to be driven makes 70 revolutions per minute. Find the size of driving pulley necessary. Ans. 3 ft. 9 ins. diameter. 4. In the preceding the belt is in. thick, and the slip is I per cent. Allowing for these obtain the correct diameter. Ans. 4 ft. 5. A pulley 4 ft. diameter is driven by two belts running over each other, each in. thick. The speed of the middle plane of the inner belt is 1800 ft. per minute. the outer gain on the inner per minute? How much does 6. Calculate the revolutions per minute of a dynamo driven by a belt in. thick, the diameter of the pulley of the dynamo being 6 inches, and the angular velocity of the driving pulley, 4 feet 6 inches diameter, being 10 radians per second. 2 per cent slip. Allow Ans. 818. 7. A lathe is to be driven so as, without the use of back gear, to cut brass from in. diameter to 3 in. diameter (see question 7, page 38). The main shop shaft runs at 60 revolutions, and the driving pulley on it is 2 ft. 6 in. diameter; the driving pulley of the lathe is to turn at 40 revolutions. Find the sizes of the overhead driven pulley and of the speed pulleys, the least diameter of the latter being 5 ins., and four speeds are required, the extreme as stated and two convenient intermediate speeds. Ans. Diameter of driven pulley, 18 ins. Speed ratios, 10/3, 5/3, 4/5, 10/36. Diameters of upper pulley, 54, 89, 123, 18 ins. lower pulley, 173, 143, 10, 5 8. The usual back gear is used in the preceding, viz. a wheel A on the speed pulley drives a wheel C; on the same shaft as C is a second wheel C' which drives B, which is connected to the mandril and drives the work (see Fig. 101). The speed pulley and mandril are disconnected when the back gear is in use, at other times they are bolted together. A has 18 teeth, and C and B are equal; find the number of teeth in C, C', and B re |