Adjusting Gear Sizes
From: Rob Subject: Proper gear tooth size Hello, I am in need of some help, and since you were very helpful to me on a question I asked previously, I thought I'd come back for more help. Setup: Some material (foam tape on a white liner) is to be die-cut into 4" long pieces. To accomplish this, the material goes through a station that has two rollers positioned vertically. The top roller makes a 4" cut that goes through the foam tape and not the liner. The bottom roller is connected to the driveshaft of the machine and has a 94-tooth, 2.9375" pitch diameter. The material is then fed into two more rollers that are positioned vertically. The bottom roller of this set is 3.310" diameter. The bottom roll gear of this set and the bottom roll gear of the other set are connected via a free-spinning transfer gear, and all three gears have a 94-tooth, 2.9375" pitch diameter with a 1/8" circular pitch. Before the material gets to the second set of rollers, the liner is stripped off. A new liner wraps around the second bottom roller with about a 120-degree wrap angle. The idea is to have the new liner moving fast enough to create a gap between the parts. In this case the part size needs to be 4" and the gap 1/8". Question: What gear (how many teeth) size do I need to use for the second bottom roller to achieve the required gap between parts? I know that only the transfer gear size will have to change to accommodate the second bottom roller's new dimensions. Everything else should stay the same. Calculations: I have made several calculations, but I keep getting more confused. If I assume a web speed of 40 feet per minute, that gives me 52 rpm for the first bottom roller and 46.19 rpm for the second bottom roller. The second bottom roll gear can be calculated at 4342 teeth/min. by multiplying 46.19 rpm by 94 teeth per revolution. Then things get really screwy when I calculate 40 fpm as 480 inches per minute. 40 feet of material gives me ten 4" pieces, and since the pieces must be separated by 1/8", I need an extra 1.125 inches. I came up with 91.4 teeth, but I don't know how. Could you please help me with this? I will be forever indebted to you. Thanks, Rob
From: Doctor Rick Subject: Re: proper gear tooth size Hello Rob. I hope I can help you again. I'm not quite sure I understand the problem fully. The picture, I think, looks something like this: --- //-- --\\ / \ ----- | | / \ | | | | | | | | | | | | | | | | \ / \ / \\-- --// ======> ----- --- ======> --+-------+-------+-------+-------++-------++-------++-------++----- ----- --- / \ //-- --\\ | | / \ | | | | | | | | | | | | \ / | | ----- | | \ / \\-- --// \ --- \ \ \ \ I don't quite see how the liner is stripped off and the new one applied, but I think what you're saying is that the parts, which came every 4 inches on the first liner (moving with the first set of rollers), will be spaced one every 4 1/8 inches on the second liner (moving with the second set of rollers.) This produces a gap of 1/8" between the 4" parts. Is that right? What this means is that the second liner must be moving at a speed 4.125/4 = 1.03125 times the speed of the first liner. As it stands, with the same number of teeth on all three rollers, the speed of the second liner is 3.310"/2.9375" = 1.12681 times the speed of the first liner. This is because the two rollers have the same rotational speed, and the linear velocity of their surfaces is in proportion to their diameters. At this speed ratio, the gap between 4" parts will be 4"(1.12681-1) = 0.5074". Is this correct? If I'm with you so far, you want to reduce the speed ratio from 1.12681 to 1.03125. To do this without changing the diameter of the second roller, you must reduce its rotational speed by a factor of 1.03125/1.12681 = 0.91519. Since the original tooth ratio is 1:1, the tooth ratio that you want is 0.91519:1, or 1:1.09266. Having more teeth on the second roller gear (by this latter factor) will make it turn more slowly, reducing the gap from 1/2 inch to 1/8 inch. With 94 teeth on the transfer gear, this comes out to 94*1.09266 = 102.71 teeth on the second roller gear. Can you make do with the nearest integer number of teeth? Let's work out what we get with 103 teeth on the second roller gear. For each turn of the first roller, the second roller turns 94/103 = 0.91262 times. Its surface moves 1.12681 times faster than it would if its diameter were the same as the first, so its surface moves 0.91262 * 1.12681 = 1.02835 times as fast as the surface of the first roller. When the first liner moves 4", the second liner will move 4" * 1.02835 = 4.1134". The gap is 0.1134", rather than 0.125" - it's a little too small now. As in the last time we corresponded, you'll have to decide whether this is good enough, or whether you will have to change the number of teeth on the transfer gear as well in order to get closer to the correct ratio. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.