59. Gear Ratios.-The same principles as are applied to pulleys can be applied to gears. If we have two gears running together as shown in Fig. 9, the product of the diameter and R. P. M. of one gear will be equal to the product of the diameter and R. P. M. of the other. In studying gearing, we do not deal with the diameters so much as we do with the numbers of teeth. We find that gears are generally designated by the numbers of teeth. For example, we talk of 16 tooth gears and 24 tooth gears, etc., but we seldom talk about gears of certain diameters. In making these calculations for gears, we can use the numbers of teeth instead of the diameters. When a gear is revolving, the number of teeth that pass a certain point in one minute will be the product of the number of teeth times the R. P. M. of the gear. FIG. 9. If this gear is driving another one, as in Fig. 9, each tooth on the one gear will shove along one tooth on the other one. Consequently, the product of the number of teeth times R. P. M. of the second gear will be the same as for the first gear. This gives us our rule for the relation of the speeds and numbers of teeth of gears. Rule for Finding the Speeds or Numbers of Teeth of Gears.Take the gear of which we know both the R. P. M. and the number of teeth and multiply these two numbers together. Divide their product by the number that is known about the other gear. The quotient will be the unknown number. Example: A 38 tooth gear running 360 R. P. M. is to drive another gear at 190 R. P. M. What must be the number of teeth on the other gear? 38×360=13,680, the product of the number of teeth and revolutions of one gear. 190 Answer = 13,680 111. (a) If you draw $33.00 on pay day and another man draws $22.00, what is the ratio of your pay to his? (b) What is the ratio of his pay to yours? 112. The speeds of two pulleys are in the ratio of 1:4. If the faster one goes 260 R. P. M., how fast does the slower one go? 113. Two castings are weighed and the ratio of their weights is 5:2. If the lighter one weighs 80 lb., what does the heavier one weigh? 114. Find the unknown number in each of the following proportions: (a) 2:10 5:Answer (b) 6:42=5:Answer (c) 7:35 10:X (d) 6:72=8:X 115. If it takes 72 lb. of metal to make 14 castings, how many pounds are required to make 9 castings? 116. A 14 tooth gear is driving a 26 tooth gear. If the 14 tooth gear runs 225 revolutions per minute, what is the speed of the 26 tooth gear? 117. A 12 in. lineshaft pulley runs 280 revolutions and is belted to a machine running 70 revolutions. What must be the size of the pulley on the machine? 118. A lineshaft runs 250 R. P. M. A grinder with a 6 in. pulley is to run 1550 R. P. M. Determine size of pulley to put on the lineshaft to run the grinder at the desired speed. 119. An apprentice was given 100 bolts to thread. He completed threefifths of this number in 45 minutes and then the order was increased so that it took him 2 hours for the entire lot. How many bolts did he thread? 120. A 42 in. planer has a cutting speed of 30 ft. per minute and the ratio of cutting speed to return speed of the table is 1:2.8. What is the return speed in feet per minute? CHAPTER IX PULLEY AND GEAR TRAINS-CHANGE GEARS 60. Direct and Inverse Proportions.—A proportion formed of numbers of castings and the weights of metal required to make them is a direct proportion, because the amount of metal required increases directly as the number of castings increases. When two pulleys (or gears) are running together, one driving the other, the larger of the two is the one that runs the slower. The proportion formed from their diameters and revolutions is, therefore, called an Inverse Proportion, because the larger pulley runs at the slower speed. The number of revolutions of one pulley is said to vary inversely as its diameter, since the greater the diameter, the less the number of revolutions it will make. In every pair of gears one of them is driving the other, so the one can be called the driving gear, or the driver, and the other the driven gear, or the follower. These names are in quite general use to designate the gears and to assist in keeping the proportions in the right order. Accordingly, we have the proportion: This is an inverse proportion because the driver and the driven are in the reverse order in the second ratio from what they are in the first ratio. Perhaps this can be seen better if the ratios are written as fractions. R. P. M. of driven No. of teeth on driver R. P. M. of driver = seen. Here the reason for the name "inverse proportion" is easily The second fraction has the driver and the driven inverted from what they are in the first fraction. This method of writing proportions as fractions is much used in solving problems in gears or pulleys. 61. Gear Trains.-A gear train consists of any number of gears used to transmit motion from one point to another. Fig. 10 shows the simplest form of gear train, having but two gears. Fig. 11 shows the same gears A and B, as in Fig. 10, but with a third gear, usually called an intermediate gear, between them. The intermediate gear C can be used for either of two reasons: 1. To connect A and B and thus permit of a greater distance between the centers of A and B without increasing the size of the gears; or 2. To reverse the direction of rotation of either A or B. If A turns in a clockwise direction, as shown in both Figs. 10 and 11, B in Fig. 10 will turn in the opposite, or counter-clockwise direction, but in Fig. 11, B will turn in the same direction as A. Hence, the speed ratio of A to B is 2 to 1. In the case shown in Fig. 11 when A moves a distance of one tooth, the same amount of motion will be given to C, and C must at the same time move В one tooth. To move B 96 teeth, or one revolution, will require a motion of 96 teeth on A, or two revolutions of A. Hence, A will turn twice to each one turn of B, or the speed ratio of A to B is 2 to 1, just as in the case of Fig. 10. 62. Compound Gear and Pulley Trains.-Quite often it is. desired to make such a great change in speed that it is practically necessary to use two or more pairs of gears or pulleys to accomplish it. If a great increase or reduction of speed is made by a single pair of gears or pulleys, it means that the difference in the diameters will have to be very great. The belt drive of a lathe is an example of a compound train of pulleys, though here the train is used chiefly for other reasons. In the first step, the pulley on the lineshaft drives a pulley on the countershaft; then another pulley on the countershaft drives the lathe. The back gearing on a lathe is an example of compound gearing, two pairs of gears being used to make the speed reduction from the cone pulley to the spindle and face plate. Fig. 12 shows a common arrangement of compound gearing. Here A drives B and causes a certain reduction of speed. B and C are fastened together and therefore travel at the same speed. A further reduction in speed is made by the two gears C and D. A and C are the driving gears of the two pairs and B and D are the driven gears. In making calculations dealing with compound gear or pulley trains, we might make the calculations for each pair as explained in Chapter VIII and then proceed to the next pair, etc., but this can be shortened to form a much simpler process. The speed ratio for a pulley or gear train is equal to the product of the ratios of all the separate pairs of pulleys or gears making up the train. |