1、02FTM11A Gear Design Optimization Procedurethat Identifies Robust, Minimum Stressand Minimum Noise Gear Pair Designsby: D.R. Houser and J. Harianto, The Ohio State UniversityTECHNICAL PAPERAmerican Gear Manufacturers AssociationA Gear Design Optimization Procedure that IdentifiesRobust, Minimum Stre
2、ss and Minimum Noise GearPair DesignsD.R. Houser and J. Harianto, The Ohio State UniversityThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AbstractTypicalgeardesignproceduresarebasedonaniterat
3、iveprocessthatusesratherbasicformulastopredictstresses. Modi-ficationssuch astiprelief andlead crowningare basedon experienceand thesemodificationsareusually selectedafterthe design has been developed. In the traditional approach, the selection of the optimal designs is really sub-optimal,since ther
4、e are always compromises that are made between the many design performance evaluation variables.Thedesignprocedurethatisdescribedinthispaperstartswithmicro-topographiesintheformofprofileandleadmodifi-cationsthatareappropriatefor agiven design and thatareeasyto manufacture. Wethen evaluatetheloaddist
5、ribution,bending and contact stresses, transmission error, film thickness, flash temperature, etc., for a large number of designs(sometimesasmanyas1million)inanefforttofindthebestdesignforthepredeterminedtoothmicro-topographies.Thekeytotheanalysisistherapidevaluationoftheloaddistribution. Auniquegra
6、phicallyorientedselectionprocedureisthen used to determinedesignsthataregood in many of theabove mentioned design responsevariables. The bestofthese designs are then subjected to an evaluation of the effects of manufacturing errors on the performance evaluationvariables(stresses,noiseexcitations,etc
7、.). Designsthatareleastsensitivetomanufacturingerrorsarethendeemedmostrobust and hence best for the given application.Copyright 2002American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2002ISBN: 1-55589-811-41A GEAR DESIGN OPTIMIZATION PROCEDURE THAT
8、IDENTIFIES ROBUST, MINIMUM STRESS AND MINIMUM NOISE GEAR PAIR DESIGNS Dr. Donald R. Houser and Jonny Harianto Department of Mechanical Engineering The Ohio State University 206 West 18thAvenue, Columbus, OH 43210 Phone: (614) 292-5860 Fax: (614) 292-3163 email: houser.4osu.edu INTRODUCTION The desig
9、n of a gear pair is, indeed, a complicated process. Usually there are many measures of a good design including bending and contact stress balancing and minimization, minimizing scoring features such as flash temperature, and minimizing noise. Noise minimization is seldom considered in great detail a
10、t the initiation of a design, but is often considered only when noise becomes a problem. In addition, the designer seeks to find designs that are insensitive to manufacturing and mounting errors. In order to achieve good designs the designer may vary a large number of geometry and manufacturing fact
11、ors such as number of teeth, pressure angle, helix angle, tooth height, hob shift, etc. Usually designers apply their past experiences to select design parameters, but due to the tedium of the process may not consider very many alternative designs. Also, the tooth surface micro-geometry that is defi
12、ned by the tooth topography is rarely considered at the beginning of a design, but may be considered after the main geometric parameters are chosen. This paper presents an optimization methodology that allows the manipulation of a large number of design parameters and allows selecting designs that a
13、re superb in many of the measures of goodness that the designer chooses to use. Tooth micro-design and design for minimum noise are considered throughout the design process and finally, the procedure allows one to assess the sensitivity of the design to manufacturing and assembly deviations. DESIGN
14、OPTIMIZATION Design optimization has, over the past couple of decades, literally become an engineering specialty. Yet most demonstrations of optimization techniques vary few design variables and use relatively simple objective functions for the optimization of the designs. We will first discuss desi
15、gn optimizations issues encountered in gear design and then present a unique gear design optimization approach. Macro vs. Micro Gear Design By macro design we mean the varying of design parameters such as numbers of teeth, pressure angle, helix angle, etc. Changing these parameters can have a large
16、effect on evaluation parameters such as root stress or contact stress and allows the application of evaluation equations such as those used by AGMA 1 or that appear in design textbooks. Micro-design is the establishment of the specification of the shape of the gear tooths active profile. This proced
17、ure includes the application of lead crowning and lead modifications that are often necessary for compensating for misalignment and mounting inaccuracies and profile modifications that are used to reduce contact along the tooth tips and roots and for noise reduction. Micro design is usually applied
18、after one has selected the macro-design parameters. Although the shapes used in micro-design are unlimited, manufacturing considerations usually limit the complexity of the shapes that are practical. 2Optimization Background Most gear designers 2-5 who have used design optimization have applied thei
19、r methodology using conventional gear stress formulas in a macro-design approach. The approaches used usually have taken into consideration only one or two parameters and have rarely considered such conventional design approaches as the balancing of the strength and durability lives or the balancing
20、 of bending strength between the pinion and gear. Each of these criteria is best solved using cutter profile shift in order to adjust tooth thickness of the gear and pinion. The optimization approach applied by these authors requires the selection of an objective function that is usually minimized u
21、sing some type of steepest ascent method. We have observed several difficulties with conventional gear design optimization techniques: 1. The number of design variables in even the simplest gear design is of the nature of from 6-10. With this number of variables we find that there are many combinati
22、ons of design variables that provide near optimum solutions. These multiple optima lead to major shortcomings of hill climbing optimization methods that will reach the peak of one of these “hills” but may miss analyzing any of the other hills. 2. When multiple term objective functions are used, ther
23、e is no rational means of weighting each of the variables. For instance, a variable set might include minimizing bending stress, contact stress and flash temperature. If one chooses to weight them equally, just the definition of “equal” is difficult, since we are dealing with quantities with totally
24、 different units. 3. In addition, multiple term objective functions tend to provide a single optimum design solution. Simply changing one of the weighting factors by a small amount will likely provide a totally different “optimum” design. Since we are uncertain what weighting factors to use, we find
25、 that there are likely to be a huge number of satisfactory designs that fit within the range of weightings that we deem acceptable. 4. Since micro-topographies have rarely been used in gear design optimization schemes, it has been impossible to include such factors as noise or sensitivity to manufac
26、turing errors or assembly variations into the design procedure. Robustness Considerations In this paper, the word “robustness” refers to the insensitivity of a design to manufacturing and assembly errors. Assembly errors include such factors as center distance variability and shaft or gear mounting
27、misalignment. Manufacturing errors are best defined as deviations in the tooth profiles, leads, and spacing from their design goals. In this case both first order deviations id profile and lead (slope errors) and second order deviations (curvature errors) are used. Modern gear inspection machines ma
28、y be used to obtain statistics on these errors for different parts and manufacturing processes. Spacing errors 6 that are important in evaluating the effect of transverse load sharing on bending and contact stresses are not considered in this paper. DESIGN OPTIMIZATION APPROACH APPLIED IN THIS PAPER
29、 Our approach to gear design optimization that is applied in this paper addresses many of the issues presented in the forgoing paragraphs. First, tooth surface optimization is applied to the design, then an analysis of the load distribution is performed and the numerous design evaluation parameters
30、are calculated based on the predicted load distribution. The load distribution analysis is performed on a large number of designs (sometimes over one million) and two different schemes may be used for the selection of the best of these many designs. Finally, the procedure allows the evaluation of ro
31、bustness of the optimum designs. An alternative approach that is not shown in this paper would be to perform robustness analysis on each of the designs prior to selection of the best designs. Tooth Surface Optimization In an earlier work, Regalado, et. al. 7-8 showed that it was possible to control
32、tooth stresses while at the same time determining the tooth micro-topography for minimizing transmission error. Unfortunately, the topographical shape that results is very complex and would be literally impossible to manufacture in an accurate and economical manner. Therefore, our new approach is to
33、 start with easy to manufacture profile and lead modification shapes and then optimize the shape coefficients (amplitudes) as outlined below. The current approach to tooth surface optimization is two-fold: Provide adequate lead crowning for low misalignment sensitivity. Apply appropriate tip relief
34、to minimize transmission error and also to minimize peak contact stresses due to edge contact. In this analysis the starting roll angle and amplitude of tip relief are varied and the condition of minimum transmission error is found. Fig. 1 shows a typical result for the baseline helical gear pair to
35、 be analyzed later in this paper. In this case, parabolically shaped relief was applied and the best shape occurs when the relief starts near the center of the tooth height. Since this parabolic shape is nearly circular as a function of roll angle, all subsequent analyses will use circular profile m
36、odifications. 3Fig. 1: Effect of Starting Roll Angle and Parabolic Tip Modification Amplitude on Transmission Error Fig. 2: Effect of Starting Face Width Position and Parabolic Lead Modification on Transmission Error The next stage of topographical optimization is to weigh the amount of tip relief a
37、gainst the desirable lead modification. Fig. 2 shows the relative amplitude of circular tip relief and circular lead modification on transmission error. We see that a nearly linear relationship occurs where the sum of the profile modification and the lead modification are constant. The above informa
38、tion is used as a guide for choosing mean modifications that are then used for all of the gear design cases that are run. In essence, we are selecting an easy to manufacture tooth topography and then finding gear geometries for which these modifications provide optimal results for the evaluation par
39、ameters. Run-Many-Cases Approach In this methodology, we seek to run as many sets of design parameters as possible and then use a creative means to select the best designs. The methodology borrows from the Seeker Filter Viewer approach developed by Chandrasekaran, et. al. 9-10 and was applied to gea
40、rs by Houser, et. al. 11-12. A block diagram of the method is shown in Fig. 3. It consists of generating possible candidate designs based on the range of design parameters that are selected. Next, gear micro-geometries are selected based on the methods discussed earlier. These data are then input to
41、 a load distribution solver 13 that not only solves for the gear pairs load distribution, but also computes evaluation parameters such as root stresses, contact stresses, flash temperature, transmission error, etc. Two methods of reducing the number of design cases to a tractable number have been de
42、veloped (Range and Power methods). These methods are used in conjunction with a graphic processor that allows the designer to visualize the selected designs in perspective with the results of all designs that are evaluated. Finally, a robustness analysis may be performed to study the sensitivity of
43、the selected designs to manufacturing error. Fig. 3: Gear Design Optimization Analysis Schematic Table 1: NASA Baseline Gear Geometry Pinion Gear Number of teeth 25 31 Diametral pitch (1/in) 8.598 Pressure angle (degree) 23.45 Helix angle (degree) 21.50 Center distance (in) 3.5 Outside diameter (in)
44、 3.360 4.110 Root diameter (in) 2.727 3.457 Face width (in) 1.25 1.25 Torque (lb-in) 2000 lb-in Through the use of an example design problem whose base geometry is given in Table 1, each block of the schematic of Fig. 3 are explained in detail in the next section of this paper. The design data are t
45、aken from a set of gear pairs that were designed and built for noise testing at NASA-Glenn Research Center (Drago, et. al. 14). Generatingpossible candidates Configure / micro geometries Load Distribution Solver Viewing Range Reduction Method Power Reduction Method Robust AnalysisFinal Results 4Desi
46、gn Generation Table 2 shows the design parameters that were selected to be varied. The variables selected include basic geometrical parameters but also include the ability to use non-standard tooling and operation at extended and contracted centers using ranges of tool shift from full recess action
47、to full approach action. The number of multiple levels of each parameter may be selected and a special preprocessor eliminates all impossible designs or designs that have inadequate backlash, inadequate tip thickness or have other conditions such as hunting ratios, minimum starting roll angle for th
48、e start of active profile, amount of approach and recess action, etc. that may be specified by the designer. Table 2: Design Parameters Target %Tolerance Center Distance (in) 3.50 0.0 Gear Ratio (gear/pinion) 1.24 4.0 Min Max Level Number of pinion teeth 20 30 11 Helix angle (degree) 16 30 8 Pressur
49、e angle (degree) 15 25 5 Facewidth (in) 1.25 1.25 1 Tool Dedendum 1.0 1.2 3 Tool Addendum 1.35 1.55 3 Level Center Distance Ratio 5 Tool Hob Shift 5 Diametral Pitch / Module 1 Coefficient Coefficient of backlash 0.04 Limit for tip thickness 0.2 Limit for root clearance 0.2 Operating center distance: Many designs start with the center distance being fixed. However, some designs that start with a fresh sheet of paper are flexible in terms of both center distance and face width. In these situations, one may use the Run Many Cases approach to
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