Optimization of Two-Stage Cylindrical Gear Reducer with Adaptive Boundary Constraints

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1 5 JOURNAL OF SOFTWARE VOL. 8 NO. 8 AUGUST Optimization of Two-Stage Cylindrial Gear Reduer with Adaptive Boundary Constraints Xueyi Li College of Mehanial and Eletroni Engineering Shandong University of Siene and Tehnology Qingdao China lixueyi7@tsinghua.org.n Shoubo Jiang and Qingliang Zeng College of Mehanial and Eletroni Engineering Shandong University of Siene and Tehnology Qingdao China jiangshoubo@6.om qlzeng@6.om Abstrat An optimization mathematial model for designing two-stage ylindrial gear reduer is firstly onstruted based upon the drive priniples and design riteria of ylindrial gear transmission. Then some tehniques for alulating the key performane parameters suh as profile fator and dynami load oeffiient et. and revising the orresponding boundary onstraints dynamially during the optimization proess are researhed and implemented. Finally dediated interative optimization design software is developed based on VC++/MATLAB mixed programming. Compared with the onventional optimization methods with fixed performane parameter the proposed method an aurately alulate all the key parameters and revise the orresponding onstraint funtions adaptively in eah iteration step. Optimization examples illustrate that the proposed gear optimization method is more effetive and reasonable. Index Terms optimization design gear reduer adaptive boundary onstraints performane parameters I. INTRODUCTION With the advantages of large transmission power and high effiieny gear drives are widely applied in the fields of onstrution mahinery auto industry et. and play an important part in the modern industries. Gears are the ore omponents of the various kinds of transmission systems suh as speed inreaser and reduer and their reliabilities an diretly deide the performanes of the entire system. To make sure the strength servie life and reliability gears are usually designed with generous safety toleranes by initial means whih an produe a transmission system with large volume and heavy weight. And the aompanied strong inertial fores will bring heavy shoks and impats whih have an adverse impat on the running of the transmission system. Therefore the optimal designs of gear drives have been taken seriously Corresponding author: Xueyi Li. *Supports from Shandong Provinial Natural Siene Foundation of China (Grant No.ZREM) Siene & Tehnology Researh Guidane Projets of China National Coal Assoiation (Grant No.MTKJ-59 Grant No.MTKJ-8) Projet for Sientifi development plan of Shandong Provine (No. GGX) and Dotoral Fund of Ministry of Eduation of PRC(No: 786). by researhers and designers. Optimal researhes about shortening the enter distane reduing the volume enhaning the load apaity and lengthening the lifespan have vital eonomial signifianes. Deep researhes have been taken on the optimal problems of mehanial engineering with various optimization algorithms by many sholars at home and abroad [~]. A multi-objetive optimal design of gear transmission by GA (Geneti Algorithm) onsidering EHL (Elasto-Hydrodynami Lubriation) status is arried out by Yin et al.[] and BP artifiial neural network is used to approximate the numerous key oeffiients suh as the internal dynami fator load fators form fator stress orretion fator and so on during the optimization proess. A novel mesh adjustable finite-element algorithm is used by Niu et al. [5] in order to optimize the gear dimensions and maximize the torque output. Optimal designs are also arried out by Savage et al. [6] for single mesh spur gear redutions aiming at the optimal design of system life system volume and system weight and optimal results are obtained. And based on the optimization of double publi gear speed hange transmission system a gear optimization software is developed by Jin et al. [7] using VC++ to improve the optimization effiieny. The similar mathematial models are established with the objetive funtion aiming at the minimum enter distane or volume by the above sholars. But when the alulation of the numerous key oeffiients is involved the oeffiients are usually regarded as onstants or approximated by urves due to the omplex and umbersome alulation proess. In fat these oeffiients have lose links with teeth number module and helix angle et. and they also have great influenes on gear strength alulation. During the proess of iterative optimization the values of the oeffiients will hange depending on the design variables suh as teeth number module helix angle and so on. So the oeffiients need to update to aommodate the new onstraint onditions after every iterative step. Therefore the optimal results are not aurate by the previous methods whih regard the oeffiients as onstants or approximate with urves. ACADEMY PUBLISHER doi:./jsw

2 JOURNAL OF SOFTWARE VOL. 8 NO. 8 AUGUST 5 In this paper a new optimal method of gear drive based on adaptive onstraint onditions is presented to overome the limitations of the previous optimization methods. During the optimization proess in this method the key oeffiients related to the onstraint onditions an be updated with the hanges of the design variables. Hene the adaptive update of the onstraint onditions an be realized. And the implementation proedure an be desribed as the following steps. First of all the values of all the key oeffiients are alulated based on the design variables aording to the strength riterions and the onstraint onditions an be formed with the urrent oeffiients. Then the new design variables an be produed after an iterative step whih will ause the hanges of the oeffiients. These iterations will ontinue until the objetive funtion is ahieved and the onstraint onditions are satisfied. Using this dynami alulation method the oeffiients an be updated in every iterative step and the limitations of regarding the oeffiients as onstants or simulating with urves an be solved. Taking the omplex alulation proess into onsideration a alulation unit is developed to alulate the oeffiients. And the MATLAB optimization toolbox is also used to optimize a two-stage ylindrial gear reduer with the objetive funtion of minimum enter distane. Based on the unit the dynami alulation an be realized during the optimization iterations. And the VC++ platform is used to enapsulate the MATLAB optimization unit and the optimization software for ylindrial gear reduer is developed to improve the design and optimization effiieny. In the end an optimal design for two-stage ylindrial gear reduer is arried out and the optimization results show the feasibility and auray ompared with other optimization methods. II. OPTIMIZATION METHODOLOGY A. Mathematial Model a a Figure. Struture sketh of a two-stage reduer Fig. shows the struture sketh of a two-stage ylindrial gear reduer with external mesh. The highspeed stage onsists of gear and while the low-speed stage onsists of gear and. Center distanes of the two stages are a and a respetively. The total enter distane of two stages is the sum of a and a. Whatever the optimization algorithm is utilized to solve the problem the orret mathematial model should firstly be established whih inludes the determinations of appropriate design variables expeted objetive funtion and detailed onstraint onditions [89]. a) Design variables During the optimization proess the seletions of design variables have ruial influenes on the auray of the optimization results. Whether there are links among the design variables or not ombined with the determination of objetive funtion need to be onsidered as to the seletions of the variables []. For example in an optimal design whih aims at the minimum volume the faewidths of two stages should be regarded as two isolated optimization variables. While in an optimal design aiming at the minimum total enter distane the limitations of faewidths don t ontribute to the optimal objetive. Sine the objetive funtion is for the minimum total enter distane of two stages in this paper faewidths are not seleted as design variables. And width oeffiients of two stages are regarded as onstants aording to the design need. The teeth numbers are integer variables while modules are disrete. In order to solve the optimal problem aording to the general nonlinear programming method both the teeth numbers and the modules are regarded as ontinuous variables in this paper. After the optimization proess the teeth numbers and modules need to be rounded as integer and disrete variables respetively aording to the atual situations. And the strengths of two stages also need to be heked again in order to satisfy the strength riterions. Therefore based on the above disussions the design variables are defined as the following: pinion teeth number z of high-speed stage module m helix angle β transmission ratio u ; pinion teeth number z module m and helix angle β. The seven design variables are shown in (). X x x x x x x x T z m u z m T () b) Objetive funtion As to the optimal problem of gear transmission there are several objetive funtions whih aim at different targets suh as the minimum enter distane the minimum volume and the maximum load apaity. And in this paper the objetive funtion is seleted as the total minimum enter distane whih seeks the minimum sum of enter distanes. The objetive funtion an be expressed as shown in () using the design variables in (). f X x x x os 5 6 x osx x x u x Where u Σ stands for the total transmission ratio. ) Constraint onditions In this optimal design of two-stage ylindrial gear reduer the onstraint onditions mainly onsist of four 7 () ACADEMY PUBLISHER

3 5 JOURNAL OF SOFTWARE VOL. 8 NO. 8 AUGUST aspets inluding strength onditions size onditions lubriation onditions and limitations of all design variables. The strength onditions an be defined as: the alulated ontat fatigue strengths and bending fatigue strengths of two stages should be less than the allowable ontat and bending fatigue strengths respetively. The size onditions are to make sure that there is no interferene between the wheel s addendum irle of high-speed stage and the output shaft and at the same time there is also no interferene between the input shaft and the pinion s addendum irle of low-speed stage. The lubriation onditions are to guarantee the essential lubriation status for the two wheels of two stages to assure that four gears an be lubriated. The limitation onditions an be desribed as: there is no underut for the two pinions and the values of modules and helix angles need to be limited in an aepted range. Therefore the onstraint onditions of two-stage optimization problem are not simple superposition of those of single stage. When to determine the onditions in detail the four types of onstraint onditions an be obtained based on the gear meshing theory and strength alulation riterions. Taking the high-speed stage for example the onditions an be expressed as the followings. Strength onstraint onditions The alulated ontat stresses of high-speed stage need to be less than the allowable stresses respetively as shown in (). X i () Hi Hi The alulated bending stresses of high-speed stage should be less than the allowable values respetively as shown in (). X i () Fi Fi Size onstraint onditions To make sure there is no interferene between the wheel s addendum irle of high-speed stage and the output shaft as (5) shown. * x x x ha X x x 5 6 S osx osx7 Where S stands for the minimum spae between the wheel s addendum irle of high-speed stage and the axis of output shaft and h a * denotes the addendum oeffiient. Lubriation onditions All the gears at the two stages need to be lubriated in the running proess. In order to ahieve this demand the high-speed stage transmission ratio u and low-speed stage transmission ratio u should satisfy the following relation [] as shown in (6). u x. ~ u u. After a transformation the lubriation onditions an be expressed as (7) and (8) shown. X. u x (5) (6) (7) X x u. (8) 5 Limitations of design variables Make sure there is no underut for pinion as shown in (9). 6 X x * ha tan n sin artan os x Where α n denotes the normal pressure angle. Maximum teeth number of pinion should be limited to as shown in () X x 7 (9) () Module of high-speed stage should be limited from to as () and () shown. X x () 8 X x () 9 And helix angle should be limited from 8 to 5 degrees as shown in () and (). 8 X x 8 5 X x () () 8 B. Implementation of Adaptive Constraints In the previous methods the key oeffiients related to the stress alulations are usually regarded as onstants or alulated by some simplified approximate funtions. During the alulations many harts and urves are needed in order to obtain preise values and some oeffiients should be determined based on the estimations whih results in a barrier to the optimization proess. And the estimated oeffiients will also ause the wrong optimization results to a large extent. In order to improve the auray of the optimization results dynami alulations of the key oeffiients during the iterative steps are ahieved in this paper. As a result the adaptive onstraint onditions an be realized. In eah iterative step the oeffiients are alulated based on the teeth number module helix angle et. Then the onstraint onditions an be determined following the urrent oeffiients and the design variables an be produed under the urrent onditions. These steps will loop until the minimum total enter distane is obtained and all the onstraint onditions are satisfied. Sine the oeffiients are updated with hanges of the design variables the limitations in the previous methods an be overome. Therefore the optimization results are more aurate ompared with other methods. III. SOFTWARE IMPLEMENTATION PROCESS ACADEMY PUBLISHER

4 JOURNAL OF SOFTWARE VOL. 8 NO. 8 AUGUST 55 Many equality and non-equality onstraints are involved in the optimal design of two-stage ylindrial gear reduer and this problem is a typial onstraint nonlinear programming problem. To solve this type of problems many optimization algorithms an be used for example the exterior point method the interior point method omplex method and feasible diretion method et. And muh ommerial numerial software suh as MATLAB an support existing optimization algorithms []. One the objetive funtion onstraint onditions and the initial values are defined the optimal design an be preeded by invoking an optimization funtion whih will improve the effiieny greatly. Hene the optimization programs are oded in MATLAB and dynami exhange is ahieved using the data interation tehnique during the optimization proess. Start Initial design of stages Save initial parameters Invoke optimization unit Read initial design parameters Calulate the key oeffiients Read the key oeffiients optimization unit an be formed in this way. Then the parameters inluding the teeth number module helix angle and working parameters are written into disk files to be read by the optimization unit. In eah iterative step the unit will read the parameters from the files and form new onstraint onditions. After the urrent step design variable are saved to files again. And based on these variables the key oeffiients an be alulated in order to adjust with the variables whih an realize the dynami optimization. One the optimization result is obtained design variables are rounded aording to the atual situation and modifiation oeffiients an be distributed in order to fit a speified enter distane. Based on this proess the software an be developed. IV. EXAMPLE AND ANALYSIS A. Initial Conditions Taking a ertain two-stage helial ylindrial gear reduer as example an optimal design is arried out. The working parameters of this reduer are known as the following: input power P =KW the input rotate speed n =5r/min the output rotate speed n =87.5r/min and total transmission ratio u Σ =. B. Initial Design The initial design parameters suh as the preision lasses (PC) of high-speed stage (stage) and low-speed stage (stage) materials ontat fatigue strength σ Hlim and bending fatigue strength σ Flim of the gears faewidth oeffiient Φ d and the omprehensive oeffiient K an be tabulated in Table. Iterative optimization TABLE I. MATERIALS AND WORKING PARAMATERS Satisfy Y onditions? Y Save optimization results Round parameters N Save variables Item PC Materials σ Hlim (MPa) σ Flim (MPa) Φ d K stage 8 Carburized steel / 6/6..5 stage 8 Carburized steel / 6/6.8.5 Based on the parameters in Table initial designs of high and low speed stage are respetively performed using the onventional design method and the design results an also be obtained aordingly. Table shows the design results inluding teeth numbers modules helial angles et. of high and low speed stages. Distribute modifiation oeffiients End Figure. Flow hart of the optimization proess Fig. shows the implementation proess. At first the optimization funtion fminon in MATLAB optimization toolbox is used to perform an optimization of two-stage gear reduer aording to the existing programs. And the ommand m-build is used to ompile the odes into dynami link libraries in order to be invoked in other programming environments suh as C and C++. The TABLE II. INITIAL DESIGN RESULTS Item z z m β ( ) b (mm) b (mm) Value Item z z m β ( ) b (mm) b (mm) Value C. Optimal Results After the initial designs the enter distanes of two stages are aordingly optimized using the design and optimization software. In order to show the differenes between the previous optimization method and the ACADEMY PUBLISHER

5 56 JOURNAL OF SOFTWARE VOL. 8 NO. 8 AUGUST adaptive optimization method two experiments using the same parameters are preeded respetively. During the entire optimization proess all the design variables are regarded as ontinuous variables. Sine the teeth number is an integer variable while module is disrete essential round should be done after the optimization and the enter distane also need to be adjusted aording to the atual situation. Based on the speified enter distane the modifiation oeffiients an be distributed on the priniples of equal strength or equal sliding ratio. Then the ultimate optimization results an be obtained. Table has shown the different optimization results of two-stage gear reduer using the two optimization methods. TABLE III. COMPARISON BETWEEN THE TWO METHODS. Parameter Value z z m β ( ) b (mm) b (mm) z z m β ( ) b (mm) b (mm) a Σ (mm) Previous Method Adaptive Method After the optimizations using the two methods safety fators of tooth ontat strength and root bending strength are heked again. Combined with the initial safety fators these fators an be summarized as Table shown. TABLE IV. COMPARISON OF THE SAFETY FACTORS. Safety fators Initial Design Previous Method Adaptive Method Tooth ontat strength Root bending strength High-speed stage Low-speed stage High-speed stage Low-speed stage Pinion S H Wheel S H Pinion S H Wheel S H Pinion S F Wheel S F Pinion S F Wheel S F From the data in Table and Table we an see that ompared with the initial design the teeth numbers of pinions modules and faewidths have redued slightly using the previous method and the adaptive optimization method. As a result the total enter distane has fallen. Meanwhile the optimal effet of adaptive optimization method is more obvious than that of previous method. The total minimum enter distane using adaptive method is 5.5mm whih is less than 8.99% from that using the initial method and less than.8% from that of previous method. Values of Safety Fators Initial Design Adaptive Constraints Method 5 Previous Method SH SH SH' SH' SF SF SF' SF' Safety Fators Figure. Line graph of safety fators. And from Fig. it an be seen that all the safety fators exept for the bending strength safety fators of high-speed stage have redued using the adaptive optimization method. And after the optimizations the ontat strength safety fators are all greater than. and the bending strength safety fators are greater than. whih indiates that the optimization results an guarantee the minimum enter distane as well as satisfy all the onstraint onditions. Therefore the results have shown that the optimization results of gear reduer obtained by the previous methods are not suffiient and there is still large spae to be optimized. Only by using the method of variable oeffiients optimization and adjusting the key oeffiients based on the newly produed design variables an the optimization be arried out ompletely. As a result the optimization results are more aurate and the optimization effets of gear reduer are better. V. CONCLUSIONS The following onlusions an be drawn from this work: () Using the method of adaptive boundary onstraint onditions the key oeffiients in every iterative step an be updated with hanges of design variables. This an make sure the dynami alulations of onstrain onditions as well as the auray of the optimization results. () The design and optimization software for two-stage ylindrial gear reduer developed by VC++ and MATLAB an be applied in the areas of initial design strength heks and optimization. The faster design and ACADEMY PUBLISHER

6 JOURNAL OF SOFTWARE VOL. 8 NO. 8 AUGUST 57 optimization an be ahieved and the design effiieny is improved aordingly. () From the experimental results it an be seen obviously that the total enter distane obtained by the variable oeffiients optimization method is smaller than that obtained by the previous method whih regards the oeffiients as onstants or approximates with urves. The result indiates the feasibility and auray of this method. algorithm Comput. Aided. Des. Vol. 5() pp doi:.6/-85(9)955-s. [] K. G. MKenna J. Carey N. Y. Leon and A. S. Galiano- Roth High performane industrial gear lubriants for optimal reliability in Amerian Gear Manufatures Assoiation Fall Tehnial Meeting 9 pp [] F. Gao Appliations of matlab in mathematial analysis J. Softw vol.6(7) pp. 5-9 doi:./jsw ACKNOWLEDGMENT This work was supported in part by grants from Shandong Provinial Natural Siene Foundation of China (Grant No.ZREM) Siene & Tehnology Researh Guidane Projets of China National Coal Assoiation (Grant No.MTKJ-59 Grant No.MTKJ-8) Projet for Sientifi development plan of Shandong Provine (No. GGX) and Dotoral Fund of Ministry of Eduation of PRC(No: 786). Xueyi Li was born in Hubei Provine China in 97. He reeived his PhD. in Mehanial Engineering from Xi an Jiaotong University (XJTU) China in. He urrently serves as Assoiate Professor in Shandong University of Siene and Tehnology (SDUST) China. From 7-8 he worked as a post-dotor in the State Key Laboratory of Automotive Safety and Energy Tsinghua University Beijing. His researh interests inlude CAD/CAE mehanial transmission and virtual prototype tehnology. REFERENCES [] H. Dong Z. Zhu W. Zhou and Z. Chen Dynami simulation of harmoni gear drives onsidering tooth profiles parameters optimization J. Comput. vol.7(6) pp. 9-6 doi:./jp [] S. Wang Y. Ji and S. Yang A stohasti ombinatorial optimization model for test sequene optimization J. Comput. vol.5(9) pp. -5 doi:./jp [] Q. Chen X. Chen and Y. Wu Optimization algorithm with Kernel PCA to support vetor mahines for time series predition J. Comput. vol. 5() pp.8-87 doi:./jp [] J. Yin Q S. Xie Y. K. Luo and L. J. Chen Multiobjetive optimal design of gear transmission based on geneti algorithm with respet to EHL effet Jixie Gongheng Xuebao/Chinese Journal of Mehanial Engineering Vol.9(6) pp [5] S. X. Niu N. N. Chen S. L. Ho and W. N. Fu Design optimization of magneti gears using mesh adjustable finite-element algorithm for improved torque IEEE Trans. Magn. Vol. 8() pp doi:.9/tmag..9. [6] M. Savage S. B. Lattime J. A. Kimmel and H. H. Coe Optimal design of ompat spur gear redutions J. Meh. Des. Trans. ASME Vol. 6() pp [7] Y. G. Jin and L. B. Chen Optimal design of double publi gear speed hange transmission system in mahine tool in nd World Congress on Computer Siene and Information Engineering CSIE Vol. 6 pp doi:.7/ [8] X. Zhang Y. Rong J. Yu L. Zhang and L. Cui Development of optimization design software for bevel gear based on integer serial number enoding geneti algorithm J. Softw. vol.6(5) pp doi:./jsw [9] B. Peng L. Zhang R. Zhao H. Zhang and Y. Zheng Struture design of twin-spirals sroll ompressor based on C J. Softw. vol.7(9) pp. 9-7 doi:./jsw [] H. Zarefar and S. N. Muthukrishnan Computer-aided optimal design via modified adaptive random-searh Shoubo Jiang was born in Shandong Provine China in 988. He reeived his B.S. in Mehanial Engineering from Shandong University of Siene and Tehnology (SDUST) China in. He is urrently pursuing his Master of Engineering Siene in SDUST. His researh interests are in area of finite element analysis dynamis and appliation program design. Qingliang Zeng was born in Shandong Provine China. He reeived his PhD. in Mehanial Engineering from China University of Mining and Tehnology (CUMT) China in. He urrently serves as Professor in Shandong University of Siene and Tehnology (SDUST) China. And his major researhes over the areas of hydrauli drive and ontrol mehatronis CIMS CE and virtual prototype. ACADEMY PUBLISHER