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A new and generalized methodology to design multi-stage gear drives by integrating the dimensional and the confi guration design process Tae Hyong Chong *, Inho Bae, Gyung-Jin Park Department of Mechanical Engineering, Hanyang University, 17 HaengDang-Dong, SeongDong-Gu, Seoul 133-791, South Korea Received 24 July 2000; accepted 3 August 2001 Abstract This paper proposes a new and generalized design methodology to support the designer at the preliminary design phase of multi-stage gear drives. The proposed design methodology automates the design process by integrating the dimensional design and the confi guration design processes in a formalized algorithm. The algorithmconsistsoffour steps. Inthe fi rst step, theuserprovisionally sets thenumberof reductionstages. In the second step, gear ratios of every stage are chosen by using the random search method within the specifi ed ratiorange,andtheratiosareusedasthebasicinputforthedimensionaldesignofgearsinthenextstep.Inthe thirdstep,thevaluesofthebasicdesignparametersofageararechosenbyusingthegenerateandtestmethod. Then the values of other design parameters, such as pitch diameter and outer diameter, are calculated for the confi guration design in the fi nal step. In the fi nal step, the confi guration design is carried out by using the simulated annealing algorithm. The positions of gears and shafts are determined to minimize the geometrical volume of a gearbox while satisfying spatial constraints. These steps are carried out iteratively until a de- sirable solution is acquired. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Gear; Multi-stage gear drives; Dimensional design; Confi guration design; Simulated annealing; Generalized new design algorithm 1. Introduction Until now, research on the design of gear drives has focused on the dimensional design of single-stage gear drives. In recent years, however, the need for designing multi-stage gear drives has been increased with more applications of the gear drives in high speed and small space. A Mechanism and Machine Theory 37 (2002) 295310 *Corresponding author. Fax: +82-2-2296-4799. E-mail address: thchonghanyang.ac.kr (T.H. Chong). 0094-114X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0094-114X(01)00078-7 number of complicated problems should be considered in the design of multi-stage gear drives, which do not arise in the design of single-stage gear drives. Firstly, the number of reduction (increasing) stages and the gear ratio of each stage should be determined properly in consideration of total gear ratio, available space, and other design requirements. No defi nite rule has been proposed to determine them. Secondly, the problem of confi guring gear drive elements (gears, shafts, bearings,.) into available space is also one of the major design problems. Pinions and gears should mesh properly with each other while satisfying spatial constraints between gears, and between other machine elements. The volume or weight of a gearbox is also considerably aff ected by the confi guration and the arrangement. There are several conventional methods to estimate gear sizes recommended by gear standards organizations or researchers. However, these methods do not take the confi guration and the arrangement of the gear drive elements into consideration, although the dimensions of them are directly aff ected by the confi guration. The designer should have solved the above problems only by using the trial and error method, largely depended on his intuitional sense. Thus, the design practices are time-consuming even for expert designers, and often end up with unsatisfactory design solutions. The objective of this paper is to develop a new and generalized design methodology for multi- stage cylindrical gear drives (spur and helical gear drives). The proposed design algorithm sup- ports a designer eff ectively at the preliminary design phase by integrating and automating the dimensional and the confi guration design processes. The algorithm consists of four steps. In the fi rst step, the designer provisionally sets the number of reduction stages in consideration of total gear ratio and other design requirements. In the second step, the gear ratios of every stage are determined by using the random search method, and the ratios are used as basic input for the dimensional design of gears in the third step. In the third step, the three basic design parameters of module, number of teeth, and face width are determined by using the generate and test method. In the preliminary design phase, it is possible to consider only the three basic design parameters, which have dominant eff ects on the overall size of a gear, and consequently on the confi guration design. Other design parameters, such as pressure angle, helix angle, addendum modifi cation coeffi cient, cause relatively small change in the overall size of a gear, and are generally determined later in the detail design phase of a gear design process. Strength and durability of the designed gear is guaranteed by bending strength and pitting re- sistance rating practices. In the fi nal step, the positions of the gears are determined to minimize the geometrical volume (size) of a gearbox by using the simulated annealing algorithm, while meshing properly between pinions and gears, and avoiding interferences between gears and shafts. The above four steps are carried out iteratively until a desirable design solution is acquired. The algorithm automates the preliminary design of multi-stage gear drives by effi ciently inte- grating the dimensional design and the confi guration design processes. The availability of the algorithm will be validated by design examples of four-stage gear drives. 2. The proposed design algorithm Fig. 1 shows the proposed algorithm for automating the preliminary design phase of multi- stage gear drives. As mentioned earlier, the algorithm consists of four design steps, and the steps are carried out iteratively until a desirable solution is acquired. 296T.H. Chong et al. / Mechanism and Machine Theory 37 (2002) 295310 In Step 1, the designer provisionally sets the number of reduction stages in consideration of total gear ratio, available space, and other design specifi cations. Several simple guides have been proposed to determine the number of stages. In gear design texts, it is recommended to handle gear ratios from 1:1 to 8:1 (or 10:1) in a single reduction for ordinary spur and helical design practices 1. AGMA recommends adding another stage to the gear train if the gear ratio of a stage is greater than 5:1 2. Thus, the designer can make a sensible choice for the number of reduction stages from the recommended ratio range. When the fi nal design solution is not sat- isfactory or the iteration exceeds the maximum number, i.e. the design is regarded provisionally as having no feasible solution, the designer can decide optionally whether or not to proceed with another number of reduction stages. It is rather ineffi cient to automate this step into the algo- rithm, since the number of stages can be selected in a relatively small range. Moreover, the au- tomation unnecessarily increases computation time in most cases. In Step 2, the gear ratios of each reduction stage are determined using the random search method within the specifi ed ratio range. No defi nite rule has been proposed to determine the gear ratios. The guide proposed by Niemann et al. 3 might be a practical one, in which gear ratios are determined based on the Hertz contact stress formula. However, this method is limited to the design of two- and three-stage gear drives, and the designer should previously determine the number of teeth or module in order to calculate the gear ratios of each stage 4,5. We have proposed two types of premises in order to employ the random search method. Firstly, gear ratios can be limited to a reasonable range. As mentioned earlier, it is reasonable to handle gear ratios from 1:1 to 8:1 in a single reduction in ordinary spur and helical design Fig. 1. Flowchart for the design of multi-stage gear drives. T.H. Chong et al. / Mechanism and Machine Theory 37 (2002) 295310297 practices. Ratios of even 10:1 are possible 1. Thus, the upper and the lower limits of gear ratios can be set to generally acceptable values according to the above guides, although the defi nite values of them are not known. Secondly, it is general to choose a greater value for the gear ratio of the fi rst reduction stage than that of the second stage. In the same way, the ratio of the second reduction stage should have a greater value than that of the third stage, and so forth. From these premises, a random value is generated for the gear ratio of the fi rst reduction stage u1 between the lower and upper limits previously specifi ed by the designer. The ratio range from 1:1 to 9:1 has been used for the fi rst reduction stage of the design examples in Section 4 (see Table 2). Then, the gear ratio of the second reduction stage u2can be selected by setting the gear ratio of the fi rst stage as the new upper limit. In other words, another random value is generated for the gear ratio of the second stage between the lower limit and the gear ratio of the fi rst stage previously determined. The gear ratios for every reduction stage uican be determined by the same way described above. Although the method randomly selects the gear ratios, the gear ratios of every stage shall eventually have proper values. This may be validated from the fact that there are direct corre- lations among gear ratios, the dimensions and the confi guration of gears, and the volume of a gearbox. That is to say, gear ratios aff ect the dimensions of gears, and the dimensions of gears do the confi guration of them. It is obvious that the confi guration of gears have a direct eff ect on the volume of a gearbox. This fact will be clearly shown by the design examples in Section 4. In Step 3, basic design parameters (module m, number of teeth z, and face width b) of gears are determined by using the generate and test method. There are several conventional methods to estimate gear sizes recommended by gear standards organizations or researchers. For example, AGMA 2 presents a complete guide for the preliminary design process of spur and helical gears, and Dudley 6 gives a general way of estimating gear sizes. However, these methods do not take into consideration of the confi guration and the arrangement of the gear drive elements, although the confi guration of the gears directly aff ects the dimensions of them. On the contrary, the pro- posed algorithm integrates the confi guration and the dimensional design of gears to consider the relation between them. Once the values for the basic design parameters are determined, pitch diameter and outer di- ameter are calculated from module and number of teeth for the confi guration design. Since the purpose of this paper is to automate the preliminary phase of the gear design process, the de- termination of other design variables, such as pressure angle, helix angle, and addendum modi- fi cation coeffi cient has not been considered. These design variables cause relatively small change in the overall size of a gear and are generally determined in the detail design phase. Thus, the variables have fi xed values in the design process. This is one of the key points of enabling the use of the generate and test method for the dimensional design, although the effi ciency of the method is not good in most cases. Another key point is that the search time of the method can be reduced considerably by limiting the search space of the design variables. Firstly, it is recommended by standards organizations 7 to use standard values for module, and thus it may be treated as a discrete variable. Moreover, the upper and the lower limits of it can be given according to the application of the gear drive. Secondly, the number of teeth is obviously an integer variable. The minimum value of the number of teeth in pinion can be specifi ed according to pressure angle, and the maximum value of it can be limited to a conventional value. Finally, supposing that the face width is specifi ed by an integer multiple of module (face width factor), as is in common practice, it 298T.H. Chong et al. / Mechanism and Machine Theory 37 (2002) 295310 can be treated as a discrete variable. It is also possible to specify the upper and the lower limits of it to conventional values in accordance with the application. Once the design variables are determined, then strength rating practice is carried out using the AGMA rating formulas 8 for bending strength and pitting resistance rating to test the validity of the dimensional design solution. If the gear does not satisfy the rating practice, the design restarts with increasing values of the basic design parameters from Step 2. Thus, the confi guration design in Step 4 is carried out only for the gears satisfying strength and durability criteria. In Step 4, the confi guration design is carried out to minimize the volume of a gearbox by using the simulated annealing algorithm. Since the outer diameter and the face width of a gear have been determined from the previous design steps, although the values are provisional, the con- fi guration design might be considered as a problem of packing gears of fi xed size in three-di- mensional space. There have been several researches to solve three-dimensional packing problems using optimization techniques 911. In particular, Szykman and Cagan 9,10 have reported signifi cantly good results for the optimal packing problems of three-dimensional elements of fi xed Fig. 2. Flowchart of simulated annealing algorithm for the confi guration design of a gear drive. T.H. Chong et al. / Mechanism and Machine Theory 37 (2002) 295310299 size using a simulated annealing algorithm. The problem of packing gears in a three-dimensional space is problematic to conventional gradient-based optimization methods due to discontinuities and severe nonlinearities in its objective function space. Simulated annealing is well suited to the problem, because it is zero-order algorithm requiring no derivative information, and thus dis- continuities can be easily dealt with 12,13. Fig. 2 shows the fl owchart of the simulated annealing algorithm used in this paper for the confi guration design of a gear drive. Starting from an initial random point, the algorithm takes a step and the function is evaluated. When minimizing a function, any downhill step is accepted and the process repeats from this new point. An uphill step may be accepted. Thus, it can escape from the local optima. This uphill decision is made by the Metropolis criteria. As the optimization process proceeds, the length of the step declines and the algorithm closes in on the global optimum. 3. Objective function formulation for confi guration design The objective function F for the confi guration design is formulated simply as the linear sum- mation of the volume of a virtual gearbox, i.e. a box completely bounding the gears, and the spatial constraints, as shown in Eq. (1) F WboxPboxVbox X nc i1 WiPiCijj;1 where Wbox, Pboxare the weighting factor and the normalizing factor for the volume Vboxof a virtual gearbox, respectively. Wiand Piare the weighting factors and the normalizing factors for ith constraint, Ci. The total number of constraints is nc. The values of normalizing factors Pboxand Piare one divided by the maximum values of Vboxand Ciat the current position, respectively. The spatial constraints Ci should be satisfi ed to confi gure the gear drive elements properly. The constraints consist of four types of spatial constraints; the center distance constraints for proper meshing of pinion and gear, the face distance constraints for mating of co-axis gears, the gear interference constraints to avoid the interference between gears, and the shaft interference con- straints to avoid the interference between gear and shaft. As the objective function F minimizes, the values of the constraints approach zero. In order to confi rm the validity of the confi guration design algorithm using Eq. (1), the con- fi guration design of six cylinders has been carried out. The cylinders consist of three cylinders with the same diameter of 10 mm, and three cylinders of 20 mm. The height of every cylinder is 10 mm. This confi guration is for the analogy of gear meshing of a three-stage reduction gear drive. Fig. 3 shows the optimal confi gurations of the six cylinders. The global optimal confi guration is in Fig. 3(a) with its bounding box having a volume of 24000 mm3. Fig. 3(b) shows another possible confi guration, i.e. a local optimum, having a volume of 24500 mm3 . This confi guration also might be regarded as a good design, though it is not a global optimum. The constraints to locate proper positions of cylinders are shown in Eqs. (2)(13). C1 d1 d2=2 ? ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi x1? x22 y1? y22 q 0;2 300T.H. Chong et al. / Mechanism and Machine Theory 37 (2002) 295310 C2 z1? z2 0;3 C3 d3 d4=2 ? ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi x3? x42 y3? y42 q 0;4 C4 z3? z4 0;5 C5 d5 d6=2 ? ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi x5? x62 y5? y62 q 0;6 C6 z5? z6 0;7 C7 z2j? z3j ? b2 b3=2 0;8 C8 x2? x3 0;9 C9 y2? y3 0;10 C10 z4j? z5j ? b4 b5=2 0;