Program 60-151 Synchronic Index of In-line Geared Systems Introduction This TK Model calculates the synchronic index of an in-line geared transmission system. The synchronic index of a geared system is the minimum number of full turns made by the driver to bring the system back to the initial alignment of the gear teeth. After this number of turns all gears will be back to their initial angular alignment with respect to each other and the gear housing. When designing instrument and control gear systems it is desirable to know the synchronic index of a geared system. For example, a gear train that is to drive a process that must repeat, such as the drum drives in multi-color copiers or printing presses, a synchronic index which produces exactly the same tooth contact pattern each time the process repeats will produce the least possible register error due to inaccuracies in the gears. If position or torque tests are required for instrument gears it is necessary to turn the system through the entire meshing cycle to be sure that all positions of the gears have been tested. The synchronic index will tell us how many turns of the system are required. In power gearing, depending upon the materials being used, it is often desirable to have a high synchronic index to distribute the wear and load fluctuations as evenly as possible. Adjustment of the synchronic index can sometimes be useful in changing the vibration or noise frequency of a drive caused by gear meshing error repetition to avoid vibration resonance or move the frequency away from the frequencies to which the ear is most sensitive. Reference: Synchronic Index of Gear Trains by E.D. Knab, Bell Telephone Labs, Whippany, NJ in Gear Design and Application edited by Nicholas P. Chironis and published by McGraw-Hill Book Company NY, NY in 1967
UTS Integrated Gear Software Examples Example 1 Example 1 is a control gear system consisting of three stages with a total ratio of 30 to 1. To obtain the correct rotation direction, the second stage has an idler gear. The first stage pinion is connected directly to a stepping motor. It is desired that a drum driven by the last gear in the third stage repeat its motion as closely as possible to hold the register on multi-color printed sheets. The optimum design would then be to have the system return to the same tooth alignment for one revolution of the last gear in the third stage. The preliminary design has the following numbers of teeth: 1st Stage - A 12 tooth pinion driving a 36 tooth gear 2nd Stage - A 22 tooth pinion driving a 38 tooth idler meshed with a 110 tooth gear 3rd Stage - A 22 tooth pinion driving the final 44 tooth gear When you open a new analysis in 60-151, an interactive data entry table appears as shown below. You may add the total number of columns you need first one for each gear in the system or you may add columns as you fill them in. We must first find the synchronic index for the preliminary gear system. The first of the three trains is a simple train consisting of a 12 tooth pinion driving a 36 tooth gear. The image on the next page shows the interactive table with the inputs for the first train Element 3 is the connecting shaft between the first stage gear and the second stage pinion, so we enter Shaft. 2
60-151 Synchronic Index of In-line Geared Systems The second train is a simple train consisting of a 22 tooth pinion driving a 38 tooth idler gear which is meshed with a 110 tooth output gear. Element 7 is the connecting shaft between the last gear in the second stage and the third stage pinion; we enter Shaft. The third train is another simple train with a 22 tooth pinion driving a 44 tooth output gear. We make entries accordingly in the wizard screens. Element 10 is the shaft connected to the last gear in the third stage which is the output of the system. So we select Shaft. 3
UTS Integrated Gear Software Train 3 is the last train in the system. Solve the model by clicking the Solve button. The solution is shown in Figure 1-1. Fig. 1-1 We can see that it will take 19 revolutions of the third stage gear to return the system to its original alignment. This, of course, would require 570 revolutions of the stepping motor. The system is certainly not optimized for repeatability of the process. 4
60-151 Synchronic Index of In-line Geared Systems Under the listing Is, Train in the table we can see the synchronic index of each individual stage. It is apparent that the culprit is the second stage with a synchronic index of 95 while the first stage has an index of 3 and the third stage has an index of only 2. We will attempt to adjust the tooth numbers in the second stage to reduce the synchronic index. We must keep the ratio of the stage at 5 to 1 and it would be desirable to use numbers of teeth that would keep the center distance from the pinion to the gear the same, assuming that the idler is in line with the pinion and gear. First we will try a train consisting of a 23 tooth pinion, a 35 tooth gear and a 115 tooth gear. Move the cursor to the interactive table, change the tooth numbers for gears 4, 5 and 6 to 23, 35 and 115, and solve. The result is shown in Figure 1-2. Fig. 1-2 The synchronic index for the second train is 35. The final gear would turn 7 times to restore the alignment. We will keep trying. Change the tooth numbers for gears 4, 5 and 6 to 24, 32 and 120 and solve again. See Figure 1-3. 5
UTS Integrated Gear Software Fig. 1-3 Try 25, 29 and 125. See Figure 1-4. Fig. 1-4 No good. Try 26, 26 and 130. See Figure 1-5. 6
60-151 Synchronic Index of In-line Geared Systems Fig. 1-5 We are successful at last. If there are no strength problems with the smaller idler and no clearance problems with the change in gear diameters we have a successful solution. The drum driven by gear 9 will turn only once and all gears will be back to the original alignment positions. This should give the best repeatability possible. Of course, the angular errors in the drum position caused by the individual tooth errors in each gear must be accounted for within each drum revolution, but whatever this effect is it will repeat with every drum revolution. 7
UTS Integrated Gear Software Example 2 If you have not run Example 1 please do so before proceeding to Example 2. In this example it is assumed that you know how to enter data and solve the model. Example 2 is a two stage speed increaser driven by a motor and driving a turbocompressor. It is desired to keep the synchronic index high so that any cyclic disturbance from repeatability of gear errors will be at a low frequency where the likelihood of exciting a lateral or torsional resonance is low. The original design has a 120 tooth gear driving a 24 tooth pinion in the first stage and a 100 tooth gear driving a 25 tooth pinion in the second stage. The total ratio is 20 to 1. This design is shown in Figure 2-1. Fig. 2-1 The same teeth would go through mesh and disturbance from any runout and unbalance would repeat with each revolution of the motor shaft (every 20 revolutions of the compressor) with the number of teeth specified. Let's try 100 driving 24 teeth in the first stage and 120 driving 25 teeth in the second stage. See Figure 2-2. The ratio is still exactly 20 to 1. 8
60-151 Synchronic Index of In-line Geared Systems Fig. 2-2 With this tooth combination is would take 6 revolutions of the motor (and 120 revolutions of the compressor) to repeat the pattern. The repeat frequency is 6 times lower. If a slight change in ratio is acceptable we might try 100 driving 23 and 120 driving 26 (Figure 2-3). The change in ratio is about 0.3% high. Fig. 2-3 9
UTS Integrated Gear Software Now it would take 299 revolutions of the motor (6000 revolutions of the compressor) to repeat the meshing pattern for the system. The first train would repeat every 23 revolutions of the motor and the second train would repeat every 60 revolutions of the compressor. Of course, an acceptable repeat frequency would be different with each individual case. Any natural frequencies that are to be avoided would determine what the synchronic index should be. Example 3 If you have not run Example 1 please do so before proceeding to Example 3. In this example it is assumed that you know how to enter data and solve the model. Example 3 is a very accurate reduction unit used in a feedback loop. The total reduction is 186 to 1. The maximum angular error must be measured and certified using electronic angular position measurements. We must know how many revolutions of the input shaft must be checked to be sure that all possible conditions of gear alignment have been tested. The system consists of three stages of reduction. The first stage is a 25 tooth pinion driving a 155 tooth gear. The second stage is a planetary set with a 28 tooth sun, three 77 tooth planets and a fixed 182 tooth internal gear. The third stage is also a planetary with a 26 tooth sun, four 26 tooth planet gears and a fixed 78 tooth internal gear. Figure 3-1 shows the interactive table with the data entered and the model solved. 10
60-151 Synchronic Index of In-line Geared Systems Sheet 3-1 It would require 2046 revolutions of the input shaft (11 revolutions of the output planet carrier) while testing to ensure that all possible positions of gear alignment have passed the test. Note that even though the overall ratio is exactly 186 to 1, it takes many more than 186 revolutions of the input shaft to return the gears to initial alignment. This system could be changed to reduce the synchronic index if desired. For example: Fig. 3-2 11
UTS Integrated Gear Software Example 4 If you have not run Example 1 please do so before proceeding to Example 4. In this example it is assumed that you know how to enter data and solve the model. It is possible to alter the synchronic index of epicyclic gear sets by changing the number of teeth in the planet gears without affecting the gear set ratio or the multiple planet assembly conditions. (In addition to altering the synchronic index it is sometimes advantageous to change from odd to even or even to odd number of teeth in the planets to control phasing of torque pulsations at the sun and ring gear meshes due to tooth deflection and/or errors.) Another advantage to changing the number of teeth in the planet gears is control of the operating pressure angles at the sun/planet and planet/ring meshes. The best power to weight ratios are usually achieved when the sun/planet mesh has a high operating pressure angle and the planet/ring mesh has a low operating pressure angle. As an example we will check the synchronic index and operating pressure angles for a 1.26 to 1 ratio solar gear. This set has a fixed sun gear with 26 teeth and the input internal gear has 100 teeth. The nominal pressure angle of the gears is 20 degrees. The standard number of teeth for the planets is 37. (One half the difference between the number of ring and sun teeth.) We could use either 2 or 3 equally spaced planets in this set. Figure 4-1 is the synchronic index for 37 tooth planets. Fig. 4-1 12
60-151 Synchronic Index of In-line Geared Systems The synchronic index for the system is 2331 revolutions of the input gear. With the center distance at mid-point between the sun and ring reference pitch diameters the operating pressure angle of both the sun/planet mesh and the planet/ring mesh is 20 degrees. Figure 4-2 is the synchronic index for planets with 36 teeth. Fig. 4-2 The synchronic index has been reduced to 567 revolutions of the input gear. With the center distance still at mid-point between the sun and ring reference pitch diameters the operating pressure angle of the sun/planet mesh is 22.4 degrees and of the planet/ring mesh 17.3 degrees. Figure 4-3 is the synchronic index for planets with 35 teeth. 13
UTS Integrated Gear Software Fig. 4-3 The synchronic index is now down to 441 revolutions of the input gear. With midpoint center distance the operating pressure angle of the sun/planet mesh is 24.5 degrees and of the planet/ring mesh 14.2 degrees. We have gone about as far as possible as the plant/ring mesh is down to 14.2 degrees and a lower operating pressure angle may cause trouble with gear tooling or tooth strength. However, the operating pressure angles can be adjusted further by changing the center distance. This will cause both pressure angles to increase or decrease together where changing the number of planet teeth causes an increase in one while the other decreases. We have assumed that we wished to reduce the synchronic index, increase the sun/planet pressure angle and reduce the planet/ring pressure angle. In this case reducing the number of teeth in the planet gears accomplished all of these. Each case must be investigated as reducing the number of planet gear teeth does not always reduce the synchronic index. 14