1、90 FTM 9A Procedure That Accounts forManufacturing Errors in the Design Minimizationof Transmission Error in Helical Gearsby: Sivakumar Sundaresan, Kosuke Ishii and Donald R. Houser,The Ohio State University,!American Gear Manufacturers AssociationI II Ill! ITECHNICAL PAPERA Procedure That Accounts
2、for Manufacturing Errors in the DesignMinimization of Transmission Error in Helical GearsSivakumar Sundaresan, Kosuke Ishii and Donald R. Houser,The Ohio State UniversityThe Statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion o
3、f the American Gear Manufacturers Association.ABSTRACT:This paper deals with the design of helical gears that have minimum transmission error and, at the sametime, are less sensitive to manufacturing errors. The paper addresses two stages in design: 1) Designgeneration stage where feasible designs a
4、re generated for a given specification and 2) Design of profile andlead modifications that minimize transmission error and its sensitivity to manufacturing errors. The paperpresents a brief discussion on how one can effectively minimize transmission error in helical gears bycombining both lead and p
5、rofile modifications.Copyright 1990American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-561-1A PROCEDURE THAT ACCOUNTS FOR MANUFACTURING ERRORSIN THE DESIGN MINIMIZATION OF TRANSMISSION ERRORIN HELICAL GEARSSivakumar Sundaresan, Grad
6、uate Research AssociateKosuke Ishii, Assistant ProfessorDonald R. Houser, ProfessorDepartment of Mechanical Engineering,The Ohio State University, Columbus, Ohio 43210-11071 INTRODUCTION lead due to designed tooth modifications and1.1, _cl_ound manufacturing errors, 2) Elastic deflection of theThis
7、paper focuses on the noise minimization gear tooth due to the transmitted load.aspect of gear design and takes into account the The explanation for the variation in theeffect of manufacturing errors on noise and transmission error and mesh stiffness lies in thevibration. Gear design is a combination
8、 of two conjugate action of gears. The number of pairsprocesses, namely, synthesis and analysis. The of teeth in contact for a low contact ratio spurfirst step generates a set of feasible solutions gear pair oscillates between one and two. Thiswhich satisfy the design requirements and leads to a sud
9、den increase or decrease in thephysical constraints. This procedure specifies mesh stiffness and hence, results in the variationthe geometry of the gear and its strength of the transmission error. One way to reducecharacteristics, while the second step takes the this variation is to modify the profi
10、le of the gearfeasible solutions and evaluates their tooth (by relieving the tip/root of the gear tooth)performance for such issues as durability, such that one tooth starts unloading when thelubrication, noise, and vibration, second tooth makes initial contact. This processThe prediction of gear no
11、ise has always been (called profile modification) lessens the chancea major concern in gear design. In recent years, of premature gear tooth impact at the initiationresearchers have established a relationship of tooth contact.between static transmission error and gear Profile modification may be cha
12、racterized bynoise. Studies typically employ analytical tools to two parameters: 1) the start radius of profilepredict the static and dynamic transmission modification and 2) the amount of relief at theerrors and subsequently, gear noise reduction tip/root of the gear teeth. The amount of tipleads t
13、o minimizing both transmission error and relief is usually estimated as the deflection of themesh stff_ess variation, gear mesh at the maximum load. In our model,The static transmission error is the deviation the shape of the modification can be eitherin rotation from its ideal position when a gear
14、linear (linear profile modification) or quadraticpair is rotated with constant torque. Welbourn (parabolic profile modification). However, this(1979) defined it as “the difference between the design procedure may produce lower contactactual position of the output gear and the ratios or substantial s
15、hortening of the path ofposition it would occupy if the gear drive is contact when operated at lower loads.perfect (infinite stiffness and conjugate teeth).“ Premature gear tooth contacts in helical gearsThis error is usually expressed in terms of the are reduced by modifying the tooth in bothdispla
16、cement along the line of conjugate gear profile and lead directions. We characterize theaction. The two main causes of the transmission lead modification by two parameters: 1) theerror at the meshing frequency and its starting position along the face width and 2) theharmonics are: 1) Non-ideal tooth
17、 profile and amount of relief at the end faces of the gear1tooth. Lead modifications also known as meshes with different contact ratios. Theycrowning) are usually applied to reduce the stated that helical gears with face contact ratioadverse effects of shaft misalignment. (FCR greater than 1.0 gener
18、ally operate withThis paper deals with determining: 1 the low vibration levels and modifications are verybasic gear design numbers of teeth, pressure effective for gears with total contact ratio greaterangle, etc. and 2) the profile modification that than 2.0 and FCR less than 1.0. Though researchmi
19、nimizes the transmission error, and at the has been reported on the effect of lead andsame time, is least sensitive to manufacturing profile modifications on transmission error andvariance. We apply the concept of Statistical load distribution in helical gears, little work hasOptimization developed
20、by Taguchi (1978) and been reported on the interactions between leadadapt this methodology to gear design. An and profile modifications and how they can beexisting program called the Load Distribution combined effectively to reduce transmissionProgram (Houser, 1990), is used to evaluate the error.tr
21、ansmission error as a function of elastic In recent years, there have been a growingproperties of the gear drive and the initial number of statistical optimization techniquesseparation tooth modifications). The Load applied to manufacturing quality control.Distribution Program takes into account the
22、 Taguchi 1978 introduced the concept ofshaft bending, shaft torsion, gear tooth bending, parameter design as a cost-effective means oftransverse shear, Hertzian compression, and improving the quality of a product whosebase rotation/translation of the gear tooth, manufacturing process involves signif
23、icantvariability or “noise.“ He recognized that a1.2. Literature Review process/product can perform its intendedOptimization techniques have been used in function at many settings/values of designmany areas of gear design. Typical objective parameters. The parameter design conceptcriteria include we
24、ight, space and load capacity. “reduces variation in performance by reducingSome gear designs are limited by bearing size. the sensitivity of an engineering design toGear designers seek to find the optimum helix sources of variations rather than controlling theangle in helical gears such that the su
25、pporting sources.“ Kackar 1985 presented statisticalbearings can withstand the axial thrust without methods for conducting parameter designsubstantial decrease in the load sharing of the experiments.gear mesh. In cases where center-distance is Hunter 1985 discussed several statisticalnot a constrain
26、t, designers aim at the least tools that are used to support Taguchiscenter-distance to minimize space and pitch- approach to product design, dEntremont andllne velocity. The choice of the design variables Ragsdell 1988 developed a non-linearsuch as diametral pitch, pressure angle, and programming c
27、ode which applies Taguchisnumber of teeth is usually based on the strength quality-loss concepts to design optimization.characteristics of the gears. Little research has Sundaresan et al. 1989 introduced a procedurebeen reported on the effect of these variables on to design spur gears that have mini
28、mumtransmission error, transmission error and at the same time, areModeling of the transmission error started less sensitive to manufacturing errors. Thiswith simple spring models for spur gear teeth, paper applies the same methodology to helicalLaskin 1968 approximated a tooth of a spur gears.gear
29、as a cantilever beam with deflections due tobending, shear, and Hertzian stresses at the 1.3. The Approachcontact zone. Tavakoli and Houser 1986 used a Most gear design problems present a list ofprocedure similar to Boxs complex method requirements and constraints. In our case, we(Reklaitis, 1983 to
30、 determine the profile will use the following: 1 Requirements: centermodification that resulted in minimized distance, reduction ratio, and transmittedtransmission error. Conry and Seireg 1972 torque are all fixed, and 2) Constraints:developed a model for estimating transmission maximum outer diamet
31、er, maximum face widtherror in helical gears. Their model is now and maximum stress. Our proposed procedureincorporated in the Load Distribution Program. generates candidate gear designs which meetLin, et al., 1989 studied the effect of profile these requirements and, in addition, satisfiesmodificat
32、ion on low contact ratio spur gears. He geometrical constraints such as avoidingrecommended parabolic tooth profile undercut, having a minimum root clearance andmodification on gears that operate over a range working depth. Depending on the specificationsof loads and linear tip modification on gears
33、 that and the step size of change in design variables,operate at a constant load. Biggert and Houser there may be anywhere between 10 to 100(1988 presented a method for evaluating the feasible designs. The candidate designs are thensensitivity of transmission error to errors in analyzed and their to
34、oth profiles are modified tomanufacturing and to operating at off-design reduce transmission error.loads. They considered four design profiles and The optimization programs used in thestudied their sensitivity to deviations in profile analysis predict the profile modification thatand torque, results
35、 in minimized transmission error.Umezawa, et al., (1985 studied the effects of However, manufacturing variance associated withprofile and. lead modifications on helical gear the modification may result in a level of2transmission error far larger than that occurs at x* Peak optimum.the peak optimum.
36、The proposed procedure x* Statistical optimum.determines the profile modification thatminimizes transmission error and, at the same Y Performance sample mean.time, is least sensitive to manufacturing variance. _ Weighting constant in the objectiveThe procedure subjects every candidate design functio
37、n (F)to peak and statistical profile optimization. In _2 Varianceeffect, an exhaustive search strategy is taken for _ Generating pressure angle of the gearthe “best“ design. The procedure thus aids the FCR Face contact ratiodesigner in studying the effect of each design PCR Profile contact ratiovari
38、able on performance issues such as noise and LDP Load Distribution Program.vibration. The work presented in this paper will PPTE Peak to peak transmission error.be the basis of a more sophisticated statistical SAP Start of active profileoptimization scheme that encompasses not only SFTFH Sum of the
39、first three fourierthe profile modification but also the basic design amplitudes of the transmission errorgeneration stage that includes discrete design SI Sensitivity index.variables such as number of teeth, pressure TIF True involute formangle, etc.Section 2 gives the problem statement.Section 3 d
40、escribes the proposed procedure for 2. PROBLEM DEFINITIONthe robust design, starting with the basic design While gear design problems come in variousgeneration (section 3.1), followed by a discussion forms, this paper uses the following problemon the effect of tooth modifications on formulation to i
41、llustrate the proposed procedure.transmission error (3.2), peak optimization (3.3)and statistical optimization (3.4) of transmission I) Specificationserror. Section 4 illustrates the procedure with 1 ) Fixed center distance between shafts (Cd).two examples. Conclusions and future directions 2) Speed
42、 reduction ratio (n)of research comprise section 5. 3) Design torque (T)1.4 Nomenclature 2) Design variablesat Addendum coefficient of the tool (hob) 1) Discrete geometric variables (Number ofap Addendum coefficient of the pinion, teeth, etc.)ag Addendum coefficient of the gear. 2) Profile modificat
43、ion variablesCd Center distance, a) Start roll angle of modification (point P)Dop Outer diameter of the pinion, b) Amount of tip modification (distance BQ)Dog Outer diameter of the gearDp Search domain in peak optimization Fig. 1 shows the profile of a gear tooth withDs Search domain in statistical
44、radius scaled in terms of the tooths roll angle.Line AB represents the true involute form of aoptimization, gear tooth from its root to tooth tip.fc Transmission error at the centernode. B TIPfi Transmission error at the corner _ TRUE INVOLUT_F FORMnodes. ROOT p TIPf* Optimal performance in Ragsdell
45、 A(1988) criterion. ROLLANGLEF Objective function during statistical -_-optimization. ALmin Minimum theoretical total length of RELIEFAMOUNT Qline of contact.n Reduction ratio Fig.1 Profile ModificationN 1,N2 Number of teeth on the pinion and thegear respectively. 3) ConstrAintsPd Generating diametr
46、al pitch 1) Physical constraints (eg. minimumR Resultant vector used for the clearance)movement of point C. 2) Stress constraints (eg. maximumrt Tooltip radius, bending and contact stresses)S/N Taguchis signal to noise ratio.S: Sample variance of performance 4) InputsVariation in profile modificatio
47、n variables.tpp Pinion tooth thickness at theoperating pitch circle. 5) Objective functiontpg Gear tooth thickness at the operating Minimize transmission error and itspitch circle, variation due to errors in profileXl, Start roll angle of modification, modificationx2 Amount of tip relief3Fig. 2 show
48、s the steps in the procedure for I ) Avoiding undercut : This can be done bysolving the problem, either limiting the minimum number ofpinion teeth or by implementing hobprofile shift on the pinion.2) Minimum clearance at the root of thegear teeth, ie, addendum of one elementshould be less than the d
49、edendum of theother.3) Minimum working depth.4) Minimum top land thickness of 0.3/Pd).5 Minimum face and profile contact ratio.6) Minimum roll angle of 15 at the pitchcircle to avoid tight and close meshes.7) Bending strength and surface durability.When a range is specified for gear reductionratio (eg. 6.23 + 0.05), not all pinion toothFig. 2 Stages in the procedure numbers may be feasible. This limits thenumber of possible combinations of design3. STATISTICAL DESIGN FOR NOISE variables.3.1 Design Generation The bending stress is evaluated only for th
