1、O 99FTM16 The Multi-Objective Optimization of Nonstandard Gears Including Robustness by: D.R. Houser, A.F. Luscher, The Ohio State University and I.C. Regalado, CIATEQ k I TECHNICAL PAPER COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicese The Multi-Obje
2、ctive Optimization of Nonstandard Gears Including Robustness Donald R. Houser and Anthony E Luscher, The Ohio State University and Isaias C. Regalado, CIATEQ The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the Ameri
3、can Gear Manufacturers Association. Abstract The design of gears involves the optimization of different objectives, and the presence of errorsin the manufacturing and operating conditions affects the performance of the gears; therefore, a robust optimization procedure applying Tagushi methods was us
4、ed as a tool in the design of nonstandard cylindrical gears. This paper presents an outline of the procedure and discusses some of the results. Copyright O 1999 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 1999 ISBN: 1-55589-754-1 COPYRIGHT
5、 American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesTHE MULTI-OBJECTIVE OPTIMIZATION OF NONSTANDARD GEARS INCLUDING ROBUSTNESS Houser Donald, R. Mechanical Engineering Department, The Ohio State University Luscher Anthony, F. Mechanical Engineering Department, The
6、Ohio State University Regalado Isaias C. Mechanical Transmissions Department, CIATEQ A. C. INTRODUCTION: U, 4 Normal pressure angle Weighting for the objective i Y Helix angle Design outimization methods have often been applied togear design 4 161 11 12 Most of these attempts were one or two variabl
7、e optimizations and did not always take into account “real“ design considerations such as hob shifting and only one has included noise and/or transmission error in the design criteria ll. In this paper, an eight variable optimization procedure is presented. The procedure not only varies conventional
8、 design variables such as number of teeth, pressure angle, face width, center distance, and helix angle, but also allows for hob shift, different sized hobs and uses actual load distributions for stress computations. In addition robustness techniques are applied so that the optimum design is one tha
9、t is least sensitive to manufacturing errors. NOMENCLATURE: F Face width (mm) m Normal module (mm) x, hP /s/A Signal to noise ratio T Torque u, Number of teeth in the gear Number of teeth in the pinion Utility function for the objective i : where: -Balanced pitting and bending life -Balanced bending
10、 life pinion and gear -Minimum transmission error (noise) i = 1 i = 2 i = 3 i = 4 -Maximum efficiency i = 5 -Minimum volume I; Profile tolerance I;, Lead tolerance ,Y# Coefficient of tool shift in the gear ,Y? Coefficient of tool shift in the pinion e BENP, BENG Ratio between actual and design bendi
11、ng stresses CDR Ratio operating vs theoretical center Distance CRP Profile contact ratio CRF Axial contact ratio HERTZ Ratio between actual and design Hertzian stress LOSS Factor of power loss PPTE Peak to peak transmission error (pm) SI Sensitivity index (Criterion for robust optimum) VOL Volume (i
12、t) PPTE Peak to peak transmission error without modification (pm) PROBLEM OUTLINE: At the beginning of this study, sponsors of Ohio States Gear Dynamics and Gear Noise Research Laboratory http:/gearlab.eng.ohio-state.edu/l were polled regarding both geometric variable that they vary in gears design
13、and the factors that they would like considered in comparing designs. Based on their responses, the following design variables and objective functions were included in the analysis. XCI) Pinion teeth number (2) Pressure angle $3) Helix angle (4) Face width (5) Gear teeth number (6) Ratio between cen
14、ter distances (7) Pinion tool shifting 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesxt8) Normal module Geometric The range used for each variable is an input and tolerances are placed on such parameters as center distance so that that variable has
15、 a nominal value and a percentage range about that value. Gear tool shifting also occurs in a manner such that the working depth always has the same parameter. At this time only a cutting rack definition is applied, but in the future it is anticipated that rack addenda and dedenda could be changed.
16、The analysis also allows the use of lead crowning and profile modifications in the gear tooth design. Mechanical Five objective functions were considered, namely: (4) Balanced pitting and bending life (F) Balanced bending life in pinion and gear (F,) Minimize transmission error (F,) Improve efficien
17、cy IF,) Minimize volume Each of the above variables is evaluated at the design load for the gears. However, when minimizing transmission error, the load for minimization is often considerably lower than the design load. Hence, the procedure allows the designer to select a load other than the design
18、load for evaluation of transmission error. For each of the objective functions, a Single Attribute Utility Function (U) was defined lo, and these utility functions were combined into a Multi-Attribute Utility function given as: From a preliminary analysis lo, the following manufacturing errors and v
19、ariations in operating conditions were found to be relevant in the performance of the gears: Profile error (Pressure angle error) Misalignment (Lead error) Torque variations It was also observed that the sensitivity of the response to these errors may be non-linear. Therefore, during this investigat
20、ion, three levels in the noise factors were used, allowing a nonlinear (second degree) approximation of the response. Under the requirement of minimization in the attributes, the sensitivity analysis was based on the signal to noise ratio type s (Smaller is better) given The design for robustness mu
21、st include the optimization of the performance and the minimization of the sensitivity to parametric noise factors. Using the sensitivity index (SI) Only the argument in the logarithm of equation (2) proposed by Sundaresan l 11, equations (1) and (2) give the objective function as the minimization o
22、f: Where A is the number of replications at each candidate point for the robustness analysis. The constraints are of two types, geometrical and mechanical as shown in Table 1. In order to guarantee the performance of the gears regardless of the existence of parametric noise factors, all of these con
23、straints must be inactive or statistically active ll. Gear ratio Root clearance I Allowable pitting stress pinion I Allowable pitting stress gear interference. Limits of full recess 8 full approach action Defined only in assembled gears Table 1 Constraints involved in the optimization of The perform
24、ance of the candidate points as well as the mechanical constraints are evaluated using LDP instead of the traditional AGMA procedure l. In this way, the bending and pitting stresses are calculated using the actual load and point of application for all the points in the mesh cycle, and the stresses a
25、re far more realistic predictions that can take into account profile and lead modifications. However, in order to use LDP, the gear-set must meet all of the geometrical constraints, and there is the need of using these constraints as a filter for the cnadidate points. gears. Load Distribution Progra
26、m 2 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesSELECTION OF CANDIDATE POINTS O (v AND REPLICATIONS: The candidate points may be selected applying different optimization techniques using the objective function in equation (3). During this investiga
27、tion, a grid search method using Orthogonal Arrays (OA) was used to select the candidate points. This approach was used because it was observed (101 that the valid range in the independent variables is different for each combination of the parameters. Besides, the evaluation of constraints like poin
28、ted teeth and interference requires an iterative analysis, making it difficult to implement a higher order method for the selection of the candidate points. 4 I 2 Orthogonal array for the design variables The requirement for a second degree approximation is rotatability. Montgomery 8 recommends usin
29、g full factorials or Central Composite Designs (CCD). A full factorial with eight variables and three levels implies performing 3 = 6561 experiments per iteration. This number of experiments is prohibitive due to the computing time required to perform each experiment; therefore, a fractional factori
30、al was used. The analvsis of the geometrically feasible region Il01 showed that this region is not suitable-for -the application of OAs or CCDs. For instance, Fig. 1 shows a typical geometrically feasible region in the Space Of Ihp, hp .CDR) . In order to handle this odd-shaped design region, and ta
31、king into account the requirement of making the variance-covariance matrix (xTxrd as close to diagonal as possible, the following approach was used: A - Using an orthogonal array, generate a geometrically feasible point in the space of the four independent variables (hrr.+,v,F). During the implement
32、ation, standard (3) and ,(3”) OAs were used allocating the variables in the way shown in Table 2. With this designation of variables, the effects of 4 and w are confounded with the interactions of h, and the effect of + is confounded with the interactions of N, , F. r I Pr ofle error 5 - , - 2 2 Lea
33、d error ( 5) %T - - I Lead error 5 2 2 And the variation in torque is specified by the limits: T*,“ I T 5 T. ( 6) 2 It was assumed that the noise factors have a uniform distribution within these limits; therefore, an OA is applicable. The smaller OA for three noise factors at three levels is the ,(3
34、), and the assignation of the columns was as follows: Variable I Column No. Profile error 1 1 Final procedure With the approach used, each of the candidate points was geometrically feasible and should be analyzed for its level of performance and violation of the mechanical constraints. Fig. 3 shows
35、the flow diagram of the outlined design process. Two examples were performed giving a broad range in the independent variables, under the assumption that the designer has no previous experience about what the optimum for each particular objective will be. In this way, it is possible to see how the p
36、rogram locates the optimum in the design space depending on the aimed objective. The convergence criterion used the total range in the independent variables in order to assure that all the design space is searched for candidate points. The range in the independent variables was reduced in each itera
37、tion using: (S,-S J)=o.8(s,-s ,*,“ -i until the total range of each variable were within the following limits: hTP- - Npmin I 3 The value of 0.8 may be increased in order to perform a more refined analysis in the design space. However, this implies an increase in the number of iterations and calcula
38、tion time. for better results, it is recommended to run a first approximation with a big reduction of the range (0.6) to approximate the optimum, and then, use a smaller range of reduction (.e. 0.9) to refine the solution starting with smaller ranges. e In order to simulate what a real manufacturing
39、 procedure would produce, the profile modifications were performed using parabolic approximations of the tip, root and lead modifications. These approximations were based on the maximum deflection during the meshing of the gears as is shown in Fig. 4, where the magnitude of profile modification at t
40、he tip of the tooth was given as: r, =I.X.TE, for high contact ratio gears (HCR) T, =I 40.- for low contact ratio gears (LCR) and the start of profile modification (pointA) was located at the highest point of single tooth contact for LCR gears and the middle point between the pitch point and the hig
41、hest point of double tooth contact for HCR gears. The parabolic lead modification used was: L, = O.S. TE, 4 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesdesign vanables -+ - for minimum Peak to Peak No Generate points in the space of independent Mod
42、ify limits in the design vanables I t- - Generate a set of geometrically feasible points using me points of previous step and the pseudo-independent vanables 1 and lead modification using ,zero PPTE( Profile ) parabolic , modifications, approximation Transmission Errcr J L tooth deflections 1 Unfeas
43、ible / Yesy A feasible Fig. 3 Flow diagram for the robust optimization of gears. 5 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesNOISE FACTORS AGMA quality level 9 Minimum torque (relative to nominal) Maximum torque (relative to nominal) 0.90 1.10 In
44、 the case of the pressure angle and misalignment errors, the nominal values specified in the AGMA Handbook 2 for different quality numbers are used to obtain the level of error that is used. ROW+ monccr LOW mmncmn Fig. 4 Approximation of the profile and lead modification for the reduction of transmi
45、ssion error. Example: A 1:l and a 1:1.84 gear ratio were used as examples: however, due to space limitation, only the 1 : 1 ratio is presented here: Target center distance % tolerance Target gear ratio % tolerance Teeth in pinion (Min) Teeth in pinion (Max) Min. pressure angle Max. pressure angle Mi
46、nimum helix angle Maximum helix angle Minimum face width Maximum face width 190.5 mm 1 .o0 fi% 20 40 15.0 25.0 0.0 30.0 25.40 mm 76.20 mm s!% TRANSMITTED POWER Torque 949.41 X. m Speed pinion 1500 rpm Bending expected life 30000 Hr Pitting expected life 25 000 Hr MATERIAL Hardness pinion 400 BHN(Thr
47、ough hardened steel) Hardness gear 400 BHN (Through hardened steel) Allowable bending stress 289.7 MPO Allowable pitting stress 1 069.1 MPU TOOTH PROPORTIONS Addendum coefficient 1.25 Dedendum coefficient Tip radius coefficient 0.3 Coefficient of minimum tip thickness Coefficient of minimum root cle
48、arance Coefficient of backlash 0.035 1 .O (Full topping hob) 0.2 0.2 Computation of evaluation parameters Root stresses: These are calculated by computing the AGMA geometry factor for each design and then using the Load distribution factor to “weight“ the classic AGMA stress computation. Allowable s
49、tresses are calculated using the AGMA fatigue curves. Contact stresses: These are calculated using the load distributions predicted by LDP and then applying the classic Hertzian contact theory. Efficiency: Since it is always assumed a constant speed and load, it is actually computed the energy loss and then it is intended to minimize this loss as opposed to maximizing efficiency. The equation of Merrit 7 is used for this loss prediction. Transmission Error: The Load Distribution Program is a comprehensive pr
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