Introduction This program is used to design a gearbox with a minimum weight. It is also useful in estimating the cost and weight of a gearbox before it is manufactured. Gearboxes are attached to prime movers such as electric motors, internal combustion engines, turbines, etc. In most cases, they are used to reduce the speed of the prime movers to suit a particular application. However, there are situations where they are used to increase the speeds provided by these prime movers. The cost of a gearbox depends to an extent upon its weight. That is one reason why it is important to design a gearbox that weighs the least amount. Typically, you know the input speed of a gearbox and what you want the output speed to be. What you would like to know is how this end result can be obtained with a gearbox that weighs the minimum. That is the purpose of UTS Program 580. During the use of this program, you will come across the following terms: Size of Search Window Number of Search Directions Explanation of Terms Size of Search Window Program 580 calculates the minimum volume (and thus minimum weight) of a gearbox using an iteration process. Therefore, you need to tell the program when to stop searching for an answer. The way to do that is to define the size of a search window. During a particular iteration, the program will find a temporary minimum volume of the gearbox. During the next iteration, another temporary minimum volume will be found. If the difference between these two temporary volumes is smaller than the size of the search window you entered, the program will stop making further calculations. If it is outside of this search window, it will continue with the iteration process. Typically, the size of the search window is a small number, such as 0.000001. The same number can be input to the computer in the scientific notation form as 1E 6-6 (meaning 1 times 10 raised to the power of 6).
UTS Integrated Gear Software Number of Search Directions There is one more factor in this program that controls the iteration process. It is the maximum number of times you will allow the program to change its search directions (in an effort to find the minimum volume). The program will stop searching if the size of the search window is met. As a rule, the accuracy of your answer will improve if the size of the search window is smaller and the number of search directions changes, which you will allow the program to go through, is larger. However, the amount of time it takes the program to run may increase as you search for more accurate answers to your problems. In fact, to give you an idea of this, the program keeps track of the run time and displays it for you. The program uses these two factors in estimating the weight of the gearbox, and you can change these factors to suit your own situation. After you have used Program 580 for some time, you will develop a feel for it. With some of this familiarity, you will be able to estimate the weight of a gearbox within 15% or so using this program. Of course, when you consider the actual design process, a computerized method is about the only way to determine the minimum weight gear ratios. Benefits You will find this program quite useful if you are involved in the following types of activities: 1. Designing gearboxes 2. Estimating the cost of gearboxes 3. Specifying the purchase of gearboxes 4. Learning gear technology as students or new employees If you are a designer of gearboxes, you can use this program to explore different design alternatives with a minimum of effort. The program will give you the minimum weight for each alternative and then you make your final choice. Gearboxes can be divided into three basic applications: Industrial/commercial applications Light weight Ultra light weight, such as those used in aircraft 2
Within each of these categories, you will find that the cost of manufacturing a gearbox can be estimated accurately if you know the weight. So, if you have received an inquiry from a customer for a gearbox, you can use this program to quickly and reliably estimate its weight and thus the cost. People who buy gearboxes can use this program to estimate the price they have to pay a vendor. They will also find that the cost of operating their equipment will be lower if they have a gearbox that is designed in an optimum way (higher efficiency yet lower cost). Overall, the benefits are in saving the time required to get good answers, producing designs that cost less to manufacture, and operating more efficiently in every day use. Results The results of this program will be in the following forms: 1. Ratios at each stage of speed change in a gearbox so that the total weight of the gearbox will be minimum. 2. Estimated weight of the gearbox. The accuracy of this answer will depend upon the two factors that you have to input into the program. You will easily develop a feel for these factors after you use the program for some time. In the meantime, you can use the factors that are suggested in this program. Typically, you will be able to estimate the weight without any problems. 3. Various alternatives plotted if you wish. Such plots are often useful to include in a quotation to a customer or at least to include in your own files. 3
UTS Integrated Gear Software Theory and References Assumptions This program calculates the ratios in a gearbox so that the gearbox has a minimum weight. The program is based upon the following assumptions: The weight of a gear train is proportional to the volume of the gears. The volume is proportional to the sum of the face width times the diameter squared, of all the gears in the system. Multiple input shafts carry equal torque. Multiple output shafts carry equal torque. References The minimum weight gearbox design, Program 580, uses the following references: American Gear Manufacturers Association. Standard Number 218.01. Dudley. Gear Design Handbook. New York: McGraw-Hill. Johnson, Ray C. Optimum Design of Mechanical Elements. 2nd Edition. New York: John Wiley & Sons Willis, R.J., Jr. Lightest Weight Gears Product Engineering Magazine. West Lynn, MA: General Electric Co., Advanced Gearing Group. 4
Starting Program 580 Choosing Program 580 from the Integrated Gear Software menu screen (see Introduction) produces the screen shown in Figure 1. Each gearset image in the upper right of the screen represents a gearset configuration. Each image is also a button that you use to select the gearset type. Fig. 1 5
UTS Integrated Gear Software Here are the icons/buttons and the types of gearsets they represent. Offset Planetary Offset with Idler Right-Angle Bevel Triple Reduction Multiple Branch Double Offset 2 in Input Set Only Double Reduction Multiple Branch Double Offset with Idler 2 in Input set Only Star Double Offset with Idler 1 in Output Set Only Split Double Reduction 1 in Output Set Only A Multiple Branch label means a gearset can be constructed with more than one power path internal to this section of the transmission system. This should not be confused with the number of stages of speed changes or the input/output shafts. Star and Planetary gearsets are both epicyclic gearsets. The two Double Offset 2 in gearsets are set up for dual input power sources. They can be used for input only. The Double Offset 1 in and Split Double Reduction 1 In are set up for dual output shafts. They can be used for output only. Size of Search Window and Maximum # of Search Directions fields contain default values supplied by the application. Entering a number for Total System Ratio activates the gearset buttons. 6
Examples Example 1 We will begin with a hypothetical two-stage reduction a pinion driving a gear for the first stage and a planetary gearset for the second. We will enter 47 for the total system ratio and leave the defaults for Size of Search Window (10-6 ) and Maximum Number of Search Directions (50). The K Factor value is an indicator of compressive stress and thus depends on how much load will be placed on the gears: the larger the K Factor value, the greater the allowable compressive stress. (See the references above, especially Dudley, for more on the K Factor.) Higher K Factor values will result in gearboxes with less weight, because the intensity of loading on the gear teeth will be increased. For this example we will use a K factor of 400. The application factor will vary between.6 and.8 for a commercial gearbox. (Lighter-weight gearboxes with webbed gears can be lower,.4 or.5, and ultra-light gearboxes, such as are used on aircraft, can be.3 or.4.) It is a judgment call. For this example, we will use.7. The input shaft of the double reduction gear is the high-speed shaft in this example, and will carry torque of 1,500 inch-pounds. Enter 1500 in the High Speed Shaft Torque field. When you enter this data, the screen should look like Figure 2. 7
UTS Integrated Gear Software Fig. 2 Click the double reduction button, shown at left, to place the first stage. The double reduction gearset will immediately appear in the large gear configuration window at left, and calculated data for it will immediately appear in the Stage 1 column of the data grid at the bottom of the screen. Typically you'll want to work with 20- or 25-degree pressure angle gears. The calculation default is 20 for this example. With this first gearset selected, the screen should look like Figure 3. 8
Fig. 3 Click the planetary button, at left, to place the second stage in the design. You can enter any integer up to 30 for the number of planets. The default in this example is 4, and we won t change it. The default number of effective planets, 3.7, will automatically appear. You may override this if you wish, but for this example we won't. The screen should look like Figure 4. 9
UTS Integrated Gear Software Fig. 4 The default pressure angle is 20. We will change it to 25. Click the Return key for the rest of the input form. Changes in the other fields are automatically calculated each time one field is changed. The screen should now look like Figure 5. 10
Fig. 5 Click the Report button to generate a printable, exportable report (Rich Text Format or Adobe Acrobat PDF format). A report for this run is shown in Report 1. 11
UTS Integrated Gear Software Report 1 Program 580 System Information Total System Ratio: 47.01 K Factor: 400 App. Factor: 0.7 High Speed Shaft Torque: 1500 Estimated Weight : Stage Information Stage 1 - DOUBLE REDUCTION GEARSET (Pin->Gear->C.S.->Pin->Gear) Spur Gears, Low Pressure Angle Capacity Factor for Tooth Type & Pressure Angle = 1.00 Branches = 2 Ratio 1 = 2.86 Ratio 2 = 4.04 Stage 2 - PLANETARY GEARSET (Pin->Planets->Carrier: Ring Gear Fixed) Spur Gears, Low Pressure Angle Capacity Factor for Tooth Type & Pressure Angle = 1.00 Actual Planets = 4 Efective Planets = 3.7 Ratio 1 = 4.07 12
Example 2 We want to design a high-speed gearbox for the following application: 4,000 horsepower Input: Turbine running at 16,000 RPM Output: 900 RPM, driving a generator K-factor: 360 Weight application factor: 0.6 The main objectives are high efficiency, low weight, low cost, and no problem with vibration, lubrication or other factors. It might seem that the simplest solution would achieve the desired reduction ratio most effectively. But weight, efficiency and cost are not optimized in such a design. And because the unit requires relatively large gear diameters, there will be high pitch line velocities and lower efficiency. So it is worth our while to test a few possibilities with Program 580: Single offset (single pinion and gear) Double reduction (two-stage pinion and gear) Single offset driving a star gear Two-stage star gear Opening Program 580 brings up the screen shown in Figure 1. We first enter the overall ratio (16000 / 900 = 17.778), the K factor, the weight application factor, and the high speed shaft torque 4,000 HP converted to inch-pounds, or 15,750. Single Offset Opening Program 580 brings up the Gear Configuration Design window shown in Figure 1 above. Click the button for a single offset gearset, shown at left. The gearset image appears in the configuration window and calculated values appear in the Stage 1 column of the data grid. The screen should look like Figure 6. 13
UTS Integrated Gear Software Fig. 6 The calculation form displays the ratio in this example there is only one and the estimated gearbox weight in pounds, 17,581.38. Obviously, we need to look at some of the other alternatives. Double Reduction (2 Stage Pinion and Gear) Click the same button as above to add a second offset stage. Use the default 20- degree normal pressure angle. Figure 7 shows the results. 14
Fig. 7 The much more favorable weight shows the worth of checking the other possibilities. Single Offset Driving a Star Gear To build this example, we will delete one of the offset gearsets in the previous analysis and add a star gearset using the star button shown at left. To delete a gearset from the gear configuration window, put the mouse pointer on the gearset image and click the right mouse button. In the popup menu that appears, click Delete. If it is the only image, deleting it will clear the configuration window. The Reset button clears the window, but it also clears the basic input data. If you are trying different approaches for the same problem, the popup Delete is more convenient. The star gear will have three planets. Make this change in the data grid. Figure 8 shows the results. As they show, there is even more of a weight advantage for this design. 15
UTS Integrated Gear Software Fig. 8 Two-Stage Star Gear Using the same procedures, evaluate a two-stage star gearset. As the results in Figure 9 show, this is the lowest weight of all the designs evaluated. Other design factors are optimal with this design. 16
Fig. 9 17
UTS Integrated Gear Software Example 3 In this example we will look at a gearbox for a heavy-duty hoist. Here are the specifications: Power source: Variable-speed hydraulic motor maximum input speed 2,100 RPM Speed reduction ratio: 42 to 1 Horsepower: 250 at 2,100 RPM Operation: 10 hours per day Direct drive into the hoist drum Maximum torque: 20,000 inch-pounds at 800 RPM Stall torque: 30,000 inch-pounds Design uses AGMA standard 420.04 An experienced engineer would probably recognize that an epicyclic (planetary) gearset would probably work best for a 42-to-1 speed reduction gearbox for a hoist. But lacking such insight, Program 580 can calculate minimum weight gear ratios for several designs: Double-Reduction Triple-Reduction Two-stage Planetary We will use helical gears in this design and a pressure angle of 25. We will use a K factor of 400 and an application factor of 2, consistent with the AGMA specification. We will compute the input torque using the standard formula P = TN 5250 18
Where P is horsepower, T the torque in foot-pounds, and N the shaft speed in RPM. Using this formula, high-speed shaft torque is 7,500 inch-pounds Double Reduction We use the double reduction button shown at left to build this gearset. Figure 10 shows the results for the inputs given. Fig. 10 19
UTS Integrated Gear Software Triple Reduction A triple-reduction gearset with the same inputs produces the results shown in Figure 11 Fig. 11 20
Two-Stage Planetary Finally, a two-stage planetary design gives the results shown in Figure 12 Fig. 12 If desired, a user could make a printable/exportable report of any of these analyses by clicking the Report button. 21