1、90 FTM 15Optimal Design of Straight Bevel Gearsby: Rajiv Agrawal, Biplab Sarkar, Gary L. Kinzer and Donald R. Houser,The Ohio State UniversityAmerican Gear Manufacturers AssociationIlllTECHNICAL PAPEROptimal Design of Straight Bevel GearsRajiv Agrawal, Biplab Sarkar, Gary L. Kinzel and Donald R. Hou
2、ser,The Ohio State UniversityThe Statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.ABSTRACT:This paper describes the design of a straight bevel gearset with the objective of mini
3、mizing the enclosedvolume. The specifications for the design are the power requirements, the gear ratio, pinion speed andthe material properties. The design variables are the number of pinion teeth, the diametral pitch, and theface width. Constraints are set on facewidth, minimum number of pinion te
4、eth, and the safety factors forbending and pitting strength. The complete analysis for the gearset is based on the rating proceduredescribed in the ANSI/AGMA 2003-A86 standard. The optimization procedure is illustrated through anumerical example and the design is also compared with a spur gear optim
5、ization method using Tregoldsapproximation.Copyright 1990American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-567-0OPTIMAL DESIGN OF STRAIGHT BEVEL GEARSRajiv Agrawal, Research AssistantBiplab Sarkar, Research AssistantGary L. Kinzel
6、, ProfessorDonald R. Houser, ProfessorDepartment of Mechanical EngineeringThe Ohio State UniversityColumbus, Ohio 43210I. Introduction or the life of the gearing. In doing so, theunknowns such as the number of pinion teeth,Traditionally, the design of bevel gears has diametral pitch, and facewidth a
7、re selected asbeen based on charts and formulae from handbooks design variables. The given specifications include1, 2, 3. However, with the growth of Computer- the gear ratio, power input, pinion speed, and theAided Design (CAD), the designer can now look for materials properties. The constraints ba
8、sed onlight weight gears and compact configuration by bending strength, pitting failure, and undercuttingformulating the problem as a nonlinear optimization are also imposed.problem. A general non-linear programming problem Previous research in the field of bevel gearcan be stated as follows : optim
9、ization is scarce. Kaixiu 4 describesoptimization of straight bevel gears with themin f(x) , x E _n objective to minimize the volume of the gearset.He also considers maximum power transmitted as ansubject toobjective function when the cone pitch isgj(x) = 0 for j = I, 2, . m e specified. Vanderplaat
10、s et al. 5 have developedgj(x) _ 0 for j = me+l , . ,m a spiral bevel gear design program which uses ax I S x S xu commercial optimization package. This programhandles a wide variety of objective functions likeHere f(x) is the objective function, gj(x) the weight, dynamic load factor or life. But th
11、eare the constraint functions, the first me program can only handle shaft angle 90“, pressurefunctions defining the equality constraints, x I angle 20“ and spiral angle 35“ because the I and Jand x u are the upper and lower bounds on the factor information from the AGMA standards 6 hasvariables, and
12、 _n is the n-dimensional space of been curve fitted.real numbers. Agrawal 7 has developed a computer aid forIn the case of bevel gear optimization, the the analysis of straight bevel gears based ondesigner sets a goal to optimize an objective ANSI/AGMA 2003-A86 standards. This analysisfunction such
13、as the weight, the enclosed volume, program can generate I and J factors for arbitrarypressure angles and shaft angles. The optimization strength.procedure described in this paper uses this Td = design torque.analysis program repeatedly to iterate to the Tt = maximum allowable torque using tensileop
14、timum solution Thus there is flexibility in the strength.algorithm to accommodate different values of shaft V = total volume of the gearset.and pressure angles. There are two objective Ve = enclosed volume of the gearset.functions considered here, namely the total volume _ = pressure angle.and the e
15、nclosed volume of the gear set. The total _ = shaft angle.volume is used to design light weight gears, while _ = pinion pitch cone angle.the enclosed volume is to ensure a compact design. F = gear pitch cone angle.For specific cases, the bevel gearoptimization is quite similar to the spur gear 3. Pr
16、oblem formulationoptimization procedure 8, 9 adopted forminimizing the center distance. This similarity In the formulation used for this paper, thecan be derived using Tregolds approximation 10 specified quantities are _, _, ms, P, np, Sac , Sat ,to convert the bevel gear geometry into an rT. The de
17、sign variables are taken as Pd Np andequivalent spur gear. Section 8 of this paper F. The following assumptions are made in the beveldescribes how bevel and spur gearing are equivalent gear analysis:to each other with respect to this type ofoptimization problem (i) The pitch diameter of the heel (la
18、rge end) ofgears is taken to be the representative2. Nomenclature diameter in calculating the total andenclosed volume. The gears are treated asAg = gear pitch cone distance, frustra of cones having the apex at theAp = pinion pitch cone distance, crossing point of shafts.Cp = elastic coefficient. (i
19、i) All the AGMA correction factors, except theF = facewidth of gear and pinion, geometry factors I and J, are assumed to beI = geometry factor for durability, unity.J = geometry factor for bending. (iii) For the J-factor calculation, the load isKa = external dynamic factor, assumed to be at the high
20、est point of singleKm = load distribution factor, tooth contact.Ks = size factor. (iv) In calculating the geometry factors, theKv = internal dynamic factor, circular tooth thickness factor is assumed toma = facewidth to pitch diameter aspect ratio, be zero.mG = gear ratio. (v) rT, the cutter edge ra
21、dius, is normalizedNg = number of gear teeth, with respect to Pd i.e. rT = (actual cutterNp = number of pinion teeth, radius x Pd ). In doing the iterations forNvp = virtual number of spur gear teeth, optimization, Pd changes continuously, thusnp = pinion speed, changing rT for a given cutter edge r
22、adiusP = power transmitted. In the present formulation, rT is keptPd = diametral pitch at the large end. constant and it is assumed that the actualrT = cutter edge radius, cutter edge radius is selected from theRg = gear pitch radius at the large end. standard cutters once Pd is determined.Rp = pini
23、on pitch radius at the large end.Sac = allowable contact stress. 3.1 Objective functionSat = allowable bending stress.T = normal operating torque. The objective function for the presentTc = maximum allowable torque using contact optimization problem is either the total volume Vor the enclosed volume
24、 Ve of the gear set. The /equations for the total and enclosed volumes can beexpressed in terms of the design variables andspecified quantities using the following geometric Enclosed volumerelations 8:Ng = Npmgtan7= sinZm G + cos ZF=Z-7NpaPdRpAp =-sin7 Figure i. Enclosed volume for a bevel gearset f
25、orNg _ = 90“.Rg = 2PdRgAg = - 3.2 ConstraintssinFSeveral constraints that limit the feasibleThe total volume of the gear set is given by :design space are imposed on the variables. TheV=_F3 Ap J “ P + the gears 4:R gl: -F-0COSF 3Ag 2 - PdApg2: -F0Figure 1 shows the enclosed volume for this case 3whe
26、n the shaft angle is 90“. Mathematically,Constraints on the safety factors are basedI I(NP_3 on the design method used by Coleman i:Ve = m2 when _ = 90“_Pd)g3: T-c-10But for an arbitrary value of the shaft angle, the Tdenclosed volume is given by:TtCase i: Z 90“ g4: -10%(Ng gup )Ve=l-l- -cs_/-sin_/P
27、d _ Pd Pd J_ Pd J where,63000 P KaCase 2 : Y. 90“ Td =T Ka =np(V “V “_vdAPd Pd Avd ) %= 2 K,_ Pd2Case 3:Z+T 133I J KvSat Np np5. Design Space _ based on bending strength.12600 KsKmP KaThe formulation of the bevel gearoptimization problem has three design variables. The design space for a particular
28、problem canThe problem can be reduced to two dimensions if the be seen by drawing the constraints on a two-facewidth is expressed in te_s of number of pinion dimensional plot of Np versus Pd“ Figure 2 shows ateeth and diametral pitch. The following relation t_ical design space for the case with _ =
29、3.13,which takes into account constraints gl and g2 is = 20, Z = 90“, rT = 0.012, P = 29 hp, np = 1800used for this purpose, rpm, Sat = 30 ksi, Sac = 200 ksi, Cp = 2800, Kv = Ks= Km = Ka = I. In this case, F = Ap/3 and theF=Min , acceptable design space is the upper left handcorner of the plot, abov
30、e the shaded region. SinceNext, the safety factor constraints, g3 and _ = 90“, the locus of equally optimum designs cang4 can be converted into e_ations relating Np and be found from :Pd by substituting the expressions for Tc, Tt, andTd. Thus, the diametral pitch limit based on Np=_II-_ IltmG)contac
31、t strength is given by:where Ve is constant. Thus, the contours for theF I Kv Sac2 Np2 np_L_0_ Ks Cp2 P K a enclosed volume are straight lines through theorigin, and the smallest value of the enclosedSimilarly, the diametral pitch limit based on volume corresponds to the straight line of minimumbend
32、ing strength is given by: slope. Point A (Np = 18, Pd = 8.02) in Figure 2has the minimum slope and it0=0ifx_0.v=INp)3_maCOS_sin2_I 3 3me +ma2 +_ J L3 4sin2y 2sin 72 _.2_ 3 +ma211_mamG cosLsln iI 3ma4sin 2F 2sinF SFor the case, when F = 10/P d ,ma=_ 7Npand case the volume expression becomes :i0_F 2 r
33、 3Np2 15Np 1V= 3|cos7sin 7_ _ +i00_+3_ L 4sin _ sin_2 3m_Np24sinF _mcosFsin F_-_- 15mGNp _100sinFFrom the above expression, it can be seen _mthat there is no direct dependence of V on theratio Np/P d. However, for the case when F = Ap/3 , Figure 3: 3-D plot Of modified enclosed volume V_1 with Np an
34、d Pd as variables, showing thema=-6sinT acceptable region for design with # =20“, = 90“, _ = 3.13, P= 29 hp, np =the volume expression can be simplified to: 1800 rpm, rT = 0.012_ Np In Figure 3, the penaltycoefficients,_ andV- f(_,F, mG) 12, are taken as 104 . The utility of this plotlies in the fac
35、t that the acceptable region can beThus, for the specified values of the shaft isolated and a simple search scheme can be appliedangle and gear ratio, the direct dependence of v on for the points in this region to yield the optimalNp/P d is similar to that of the enclosed volume, solution. In this p
36、aper, however, this plot is notTherefore, the design space of the total volume5used as a method of identifying the feasible _ Design exampleregion, rather it is mentioned for betterunderstanding of the objective function and the The optimization method is illustratedconstraints. Figure 3 also shows
37、the position of through the following example. The inputpoints A, B, and C which lie at the boundary of the specifications for the gearset are taken fromfeasible and the infeasible region. Coleman I.6. Optimization procedure gear ratio, mq 3.13power transmitted r P 29 hpIn the numerical optimization
38、 routine, the pinion speed, np 1800 rpmdesign variables are treated as continuouspressure anqle r _ 20“variables. Thus, when an optimum number of teeth Xp shaft angle r _ 90is found, it must be rounded-off to obtain Np. normalized cutter edge radius, rT 0.012There are two choices for the discrete va
39、lue of Np. allowable bending stress, Sat 30 ksiOne is the next higher integer value and the otherallowable contact stress, Sac 200 ksiis the integer value just lower than Xp. Similarlyif a standard diametral pitch is sought, the numberof different combinations of Np and Pd increases to The lower bou
40、nds for the number of pinionteeth, facewidth, and diametral pitch are 13, 0.I,four. In general, for n discrete variables, thereand 1.0, respectively. The corresponding upperare 2n combinations around the continuous optimum,bounds are 45, 10.0, and 20.0, respectively. Inthat have to examined in order
41、 to locate theeach iteration of the optimization algorithm, thediscrete optimum. The selection procedure is basedbevel gear analysis program which calculates the Ion a search scheme which evaluates the volume atand J factors is called to evaluate g3 and g4“ Theeach possibility and selects the one wi
42、th minimumenclosed volume of the gearset, Ve, is taken as thevolume.objective function. The results from theoptimization are :The numerical optimization routine is basedon the successive quadratic programming method forthe solution of a general nonlinear programmingEnclosed Volume, Ve 125.46 in3prob
43、lem. The problem functions are all assumed toTotal Volume r V 11.90 in3be continuously differentiable. The method isNumber of Pinion Teeth, Np 17.25based on iterative formulation and solution ofDiametral Pitch, Pd 7.37 in -Iquadratic programming subproblems. The subproblemsare obtained by using a qu
44、adratic approximation to Facewidth r F 1.281 inthe Lagarangian and by linearizing the constraints, gl 0.0749Subroutine NCONF of the IMSL math library 9 was g2 3.516x10 -5used for this study, g_ 5.27xi0 -3In the present problem formulation, there are g4 0.1409no equality constraints. The minimum numb
45、er of CPU time on a VAX 8550 1.29 secondsteeth is given as the lower bound on thecorresponding variable and thus the constraint g5 From the optimal solution, it can be seenis is not explicitly specified. The inequality that the constraints corresponding to g2 and g3 areconstraints gl through g4 are
46、given by the active. Next the number of teeth is rounded offequations described in the problem formulation. By and a standard diametral pitch is selected. Thecomputing the values of the g-functions at the possible combinations of Np and Pd and theoptimal solution, the active constraints can be corre
47、sponding enclosed volumes are given as:identified.Np m_N, I Pd Ve IComment3 Ve= -_Pd)17 7.5 114.1Infeasible,17 8 94.0 violates pitting then, the quantity to be minimized is again Np/P dconstraint for specified mg. The total volume expression also18 7.5 135.4 takes an equivalent form when F = _/3. Si
48、nce the18 8 111.6 Discrete Optimum shaft angle is specified, 7 and F are constants fora particular case. Now if Tregolds approximationThus the optimal design is : Np = 18, Pd = 11 is applied, then,8.0, F = 1.23. This corresponds to point A inNpFigure 2. It was found that at this solution, the Nvp =
49、cos7total volume, V, also attains a minimum valuebecause the pitch cone distance constraint g2 is where Nvp = number of teeth in equivalent spuractive, and thus the two volume expressions become pinion.equivalent to each other.Thus, the enclosed volume expression can be re-8. Similarity with spur gear optimization written as :Nvp 2 3Savage et al. 10 describe the optimization _ = mGcos yof a spur gearset with the objective to minimizethe center distance. The center distance is given As is seen from
