Multi-Objective Optimization of Impact Crusher Rotor Based on Response Surface Methodology

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1 Sed Orders for Reprits to The Ope Mechaical Egieerig Joural, 2014, 8, Ope Access Multi-Objective Optimizatio of Impact Crusher Rotor Based o Respose Surface Methodology Li-Mei Zhao *, Lu-Ju Che, Feg He ad Yu Luo College of Mechaical Egieerig, Guizhou Uiversity, Gui yag, , Chia Abstract: I this study, a method of multi-objective optimizatio is proposed to improve the quality of crushed materials ad vibratio performace of the rotor. This method is drive by the first order atural frequecy ad the radius of the rotor. The Cetral Composite Desig (CCD) experimet method was used to guide the selectio of appropriate structure fiite elemet aalysis samples i desig space. The quadratic polyomials were employed to costruct respose surface (RS) model based o the respose outputs of these samples obtaied by aalyzig the first order atural frequecy, the harmoic ad mass with the software ANSYS. Well-distributed samples were geerated i the desig space by shifted Hamersley samplig method. The promiet poits were selected by the weighig method as iitial samples. The multiobjective geetic algorithm was used to obtai the Pareto optimal solutio set. Through optimizatio, the first order atural frequecy was icreased by 5.5%; the radius of the rotor was elarged by 2.5% ad the amplitude of the vibratio was decreased by 11% at the positio of bearig. At the same time, the rotor mass did ot chage much. The results show strog egieerig practicability of the proposed method. Keywords: Fiite-elemet aalysis, multi-objective optimizatio, optimizatio desig, respose surface methodology. 1. INTRODUCTION Impact crusher is a ew style, high efficiecy crushig equipmet ad is widely used i miig, metallurgy, buildig idustry, ad so o. Because of fast rotatig rotor, these kids of crushers have the problems of vibratios ad loud oise i crushig operatios. Structure parameters of the rotor ifluece ot oly the vibratio of the machie, but also make a impact o the crusher s structure size ad the crushig product quality. I recet years, optimizatio desig of the crusher rotor has bee paid more attetio. Oe study [1] discussed the fiite elemet modal aalysis of the rotor. Natural frequecy ad the mode shapes were calculated through modal aalysis. But the study was limited to a prelimiary aalysis of the vibratio characteristics of a rotor body. Aother study [2] aalyzed stress o the rotor, ad performed optimizatio to reduce the stress cocetratio by chagig the rotor size. This research achieved some results, but it was oly a simple optimizatio based o fiite elemet aalysis, limited to sigle desig goal, ad did ot cosider the effect of rotor structure o crushig performace i optimizatio process. I this work, the domestic CF250 impact crusher was cosidered as the research object. A multi-objective optimizatio method was preseted. I order to improve the vibratio characteristics ad the crushig product quality, the cetral composite desig (CCD) experimet method, the respose surface (RS) model, shifted Hamersley samplig method ad geetic algorithm were all adopted to carry out multi-objective optimizatio for the rotor. This method *Address correspodece to this author at the College of Mechaical Egieerig, Guizhou Uiversity, Gui yag, , Chia; Tel: ; zlm0226@163.com X/14 avoids premature pheomeo ad low local searchig ability of the Multi - objective evolutioary algorithm (MOEA), ad provides a referece for the optimizatio desig of other mechaical structure. 2. OPTIMIZATION MODEL The rotor compoet of CF250 impact crusher is show i Fig. (1). The parametric model was established with ANSYS code as show i Fig. (2). Tetrahedro elemets were used i the fiite aalysis of the rotor base ad shaft. The physical parameter type of the mesh was set to mechacal. Fig. (1). Assembly drawig of rotary sectio Determiatio of the Objective Fuctio Oe of the optimizatio goals is to improve the dyamic characteristics of the rotor ad to reduce the vibratio of the rotor system by optimizig the structure parameters of the rotor ad shaft. To improve the uit mass, stiffess of the 2014 Betham Ope

2 520 The Ope Mechaical Egieerig Joural, 2014, Volume 8 Zhao et al. rotor ca reduce the vibratio of the rotor ad the crusher uder workig coditio to make the rotor vibratio suitable for the iteded purpose. Therefore, oe of the optimizatio objective fuctios is the first atural frequecy, which is closely related with the vibratio of the rotor system. Fig. (2). Fiite elemet model of rotor. I order to make the hammer obtai larger kietic eergy before it collides with materials ad obtai better crushig effect, the rotor radius is icreased appropriately. The bigger the rotor radius, the larger the average kietic eergy per uit mass. By makig the rotor radius bigger, crushig product should be fier ad the crushig product quality should be improved [3-5]. Therefore, rotor radius is cosidered as aother optimizatio objective fuctio Selectio of Desig Variables Geerally, parameters of the rotor such as diameter, spa ad mass distributio may be chaged to attai the goal of optimizatio. However, i some cases, rotor structure parameters are ot simply determied by dyamics requiremets of the rotor. The diameter of the rotor axis d ad axis legth l are restricted by performace ad desig requiremets of the rotor ad caot be arbitrarily chaged. The two rotor bearigs are fixed compoets ad their size caot be chaged. Therefore, d 2 ad l 3 are restricted by rotor bearigs ad caot be arbitrarily chaged too. Therefore, four shaft diameters ad six axis legth show i Fig. (3). were selected as desig variables. Cosiderig the fact that the rotor structure should be symmetrical, the two bilaterally symmetrical axes have a uiform size. Therefore, l 1, l 2, l 4 ad d 1, d 3 were selected as desig parameters. Total 7 desig parameters were selected Multi-Objective Optimal Model Accordig to the above aalysis, the optimizatio problem has two objectives; oe is to icrease the first-order atural frequecy of the rotor to reduce the rotor vibratio i the processio of work ad the secod is to icrease the rotor radius based o the costraits fulfilled i order to icrease the rotor s impact o kietic eergy ad improve crushig (a) Optimizig parameters of shaft (b)optimizig parameters of rotor base Fig. (3). Optimizig parameters of rotor.

3 Multi-Objective Optimizatio of Impact Crusher Rotor The Ope Mechaical Egieerig Joural, 2014, Volume effect. Seve dimesio parameters of the rotor were evaluated as desig variables. The maximum ubalace respose of rotor bearig ad the rotor mass are limitatios i this optimizatio. The multi-objective optimizatio mathematical model of the rotor is as follows. max f 1 X max R X s.t. RESP(X)! RESP 0 (X) m( X)! m 0 X = ( l 1,l 2,l 4, d 1, d 3, h 1, h 2 ) T X " d i! x i! d i i = 1, 2,! 30k (! + 2d) 15k" 1 $ R $ 2 " H 30k (! + 2d) 15k" 1 2 " H Where X is the desig variable; f 1 (X) deotes the first-order atural frequecy; R (X) is the rotor radius; m (X) is the rotor body mass; M 0 is the rotor body mass before optimizig; Ω is the desig domai; d i deotes the lower limit of a desig variable; d i deotes the upper limit of a desig variable; is the umber of desig variables, =7;! ad! deote the upper ad lower limit of the hammer thickess, respectively ad H ad H deote the upper ad lower limit of the hammer height respectively. 3. PROCEDURES OF OPTIMIZATION DESIGN The optimizatio procedures of the rotor are show i Fig. (4). First, the mathematical model of desig parameters ad target parameters were established with the rage of desig parameters referrig to the origial size. The test poits were selected usig cetral composite experimetal desig method to obtai the respose set of the test poits o sample by ANSYS fiite elemet aalysis. Followig this,, ANSYS Workbech Desig Exploratio (AWB DX) optimizatio module was used to build the respose surface model with the uiform sample i the -dimesioal feasible solutio regio obtaied by Shifted Hamersley samplig method. Moreover, sample poits were sorted through weighig fuctio to obtai the iitial populatio of geetic algorithm. Multi- objective geetic algorithm wasemployed to obtai the optimized results of the respose surface to judge whether the optimized result ca meet the desig requiremet. Whe the optimal solutio was obtaied, the result was geerated, otherwise, the optimizatio objective fuctio model was updated ad the geetic algorithm was used to obtai optimal results Cetral Composite Experimetal Desig Test poits selectio affects the respose surface accuracy. The respose surface caot be costructed if the test poit is ot ideal, therefore test poits should be selected by Fig. (4). Procedures of optimizatio desig. the experimetal desig theory [6-8]. Cetral composite desig (CCD) method ca provide much iformatio ad the test error by umerical experimets i the ceter with its extesio poits with miimal work cycle. Whe solvig RS problem, the ceter poit is evaluated equal to the structure fiite elemet aalysis result ad other desig poits are impartially evaluated by least squares method. The cetral composite face (CCF) method is the most simple ad quickest, i which each test variable oly has three levels. CCF method is also ot easy to detect error sources [9, 10]. CCF method was used to choose test poits i this work. Referrig to the rotor specificatios ad desig experiece, the level ad rage of the desig variables are show i Table 1. Eighty ie experimetal desig poits were costructed though CCF method. Table 1. Value rage of optimizatio desig variables. X(mm) Level x 1 x 2 x 3 x 4 x 5 x The test poits were calculated By AWB DX optimizatio module ad the respose vector Y at differet desig poits was obtaied. = ( f 1 ( X), RESP-LEFT ( X), m( X) ) T Y X

4 522 The Ope Mechaical Egieerig Joural, 2014, Volume 8 Zhao et al. Where f 1 ( X) is the first atural frequecy of the rotor; RESP-LEFT X bearig; ad m X is the ubalace respose of the left rotor is the rotor body mass Create Respose Surface Model RSM is the regressio method searchig the relatio betwee several desig variables ad respose variables. The basic fuctio of RSM is to replace a complex model with a approximate oe based o results obtaied at various poits. I this way, the computatioal burde of evaluatig umerous desigs is reduced. Quadratic RSM used i this work is formulated as the followig polyomial fuctio [9]: =! 0 +! i y X " x i + "! ij x 2 j + " "! ij x i x j (1) j=1 1 j=i+1 Where y is the respose; x i are desig variables; β 0, β I ad β ij represet ukow coefficiets, ad is the umber of variables. Cosiderig that respose surface model is usually used as secod - order model, the respose surface is expressed as: Y = X! + " (2) where Y = (y 1, y 2,!, y ) T ;! = (! 1,! 2,!,! k ) T ;! is a error term,! = (! 1,! 2,!,! k ) T ;! X = " 1 x 11! x 1k 1 x 21! x 2k " " " 1 x 1! x k $ ; % k is the umber of data. The method of least squares is typically used to estimate the ukow coefficiets i a multiple liear regressio model i the followig form:! = (X T X) "1 X T Y (3) Eq. (3) is substituted i equatio (2), the respose surface is defied by Eq. (4) Y! X" (4) Whe respose surface is costructed, R 2 (multiple coefficiet of determiatio) ad Ra 2 (multiple adjusted coefficiet of determiatio) are geerally used to evaluate the predictive ability of the respose surface [6]. R 2 = 1! S SE S ST R 2 a = S /! k!1 SE S ST /!1 S SE = "( Y i! y i ) 2 S ST =! Y i "! y i $ % ' ( 2 2 Respose surface fittig evaluatio values are show i Table 2. It ca be see that the predictive ability of the respose surface model met the requiremets. Table 2. Objective Fuctio The First Natural Frequecy Fittig quality. Determiatio Coefficiet R 2 (%) Adjusted Coefficiet of Determiatio Ra 2 (%) Mass RESP-LEFT(X) MULTI OBJECTIVE OPTIMIZATION BASED ON GENETIC ALGORITHM Normally, there are two traditioal methods to solve multi-objective optimizatio problems. Trasformig multiobjective optimizatio ito a sigle objective optimizatio problem by weightig method or retaiig oly oe goal ad chagig the other objectives ito costraits. But these methods have may limitatios, as the optimal solutio highly depeds o the desiger's preferece [12]. At preset, geetic algorithm is ofte used to solve the multi-objective optimizatio problem. There are a large umber of solvig methods for the multi-objective optimizatio problem based o geetic algorithm. No-domiated Sortig Geetic Algorithm- II(NSGA-) method based o Pareto was used i this work. This method ca speed up o - domiated sortig solutios ad maitai the elitist ad the populatio's diversity [13, 14]. Iitial populatio of geetic algorithm has a great effect upo NSGA-covergece. I order to avoid early covergece ad esure global optimum, Shifted Hamersley samplig techique ad weighig fuctio were adopted to produce iitial populatio [15]. Hamersley samplig techique is a kid of quasi radomsamplig techique based o Hamersley algorithm. Evely distributed sample poits ca be produced i the - dimesioal feasible solutio space through this techique [11]. I this work Shifted Hamersley samplig techique was adopted to overcome the shortcomigs that Hamersley sample poits showed i the regio of startig poit i K - dimesio cube. Hamersley sample poits were offset Δ=N/2, ad the sample poits were more eve, ad smaller low-biased, which esured that the multi-objective geetic algorithm ca be quickly coverged to the global optimal solutio. I this work, 500 sample poits were evely extracted i the feasible solutio regio Ω usig Shifted Hamersley techique. Sortig 500 samples by weighig fuctio as show i Eq. (5).! = " M i (5) " y M i = t! y % $ y max! y mi ' i Where is the total umber of objective fuctio ad costrait. M i is defied by Eq. (6), where y max is the (6)

5 Multi-Objective Optimizatio of Impact Crusher Rotor The Ope Mechaical Egieerig Joural, 2014, Volume Table 3. Multi-objective optimizatio solutio set. Ordial The First Natural Frequecy (Hz) Radius Related Parameters (mm) h 1 h 2 Rotor Mass (10 3 kg) RESP-LEFT (10-2 mm) maximum value of y i (X), y mi is y i (X) miimum, y t is the ideal solutio of objective fuctio y i (X), y is the curret value of objective fuctio y i (X). The smaller the weight fuctio value, the better the sample poit. The first 300 Shifted Hamersley sample poits were selected as the iitial populatio of geetic algorithm. The objective fuctio was evaluated by geetic algorithm i AWB DX. The umber of each iteratio idividual was100, ad the maximum operatig algebra was100 geeratios. The Pareto-optimal solutios obtaied are as show i Fig. (5). The abscissa ad ordiate respectively represet a objective fuctio i Fig. (5). Accordig to the desig requiremets, 5 groups of solutios were selected from the Pareto optimal solutio, as show i Table 3. Cosiderig the desig goal, the rotor first order atural frequecy ad the rotor radius were the most importat, therefore, the fourth group of solutios was selected. I order to modify the optimal results ad make the optimizatio results suitable for egieerig applicatios, the chage i rotor structural respose with respect to desig variables was aalyzed through sesitivity aalysis. The sesitivity aalysis results are show i Fig. (6). Accordig to Fig. (6), the rotor ubalace respose was proportioal to l 1, l 4, ad d 1, ad d 1 were observed to be the biggest effect factors. The rotor mass ubalace respose was iversely proportioal to d 3, h 1, h 2, ad d 3, with h 1 beig the biggest effect factor. By cotrast, h 2 had little effect. I additio, the first-order atural frequecy of the rotor was proportioal to l 1, l 4, d 1, d 3, h 1, ad l 2, while h 1 had little effect. Therefore, it ca be cosidered to properly icrease the h 1 to reduce the Fig. (5). Pareto optimal solutio set.

524 The Ope Mechaical Egieerig Joural, 2014, Volume 8 Zhao et al. (a) The rotor ubalace mass respose sesitivity results (b) The first-order atural frequecy sesitivity results Fig. (6).

Icreasig l 1, l 4, d 1 ca improve the first-order atural frequecy of the rotor, but icreasig l 1, l 4 ad d 1 will icrease the rotor mass ubalace respose, therefore the icrease i l 1, l 4 ad d 1 was

6 524 The Ope Mechaical Egieerig Joural, 2014, Volume 8 Zhao et al. (a) The rotor ubalace mass respose sesitivity results (b) The first-order atural frequecy sesitivity results Fig. (6). Sesitivity aalysis results. rotor mass ubalace respose. Icreasig l 1, l 4, d 1 ca improve the first-order atural frequecy of the rotor, but icreasig l 1, l 4 ad d 1 will icrease the rotor mass ubalace respose, therefore the icrease i l 1, l 4 ad d 1 was kept limited. Icreasig d 3 ca elarge the first-order atural frequecy of the rotor body; at the same time, it ca reduce mass ubalace respose of the rotor ad icrease the mass of the rotor body, therefore, the icrease of d 3 was kept restricted. The first-order atural frequecy is iversely proportioal to h 2, ad it was strogly affected by h 2, therefore, the first order atural frequecy of the rotor was icreased by reducig h 2. Optimized desig variables of the rotor were modified accordig to the results of sesitivity aalysis as show i Table 4. Table 5 idicates the first-order atural frequecy of the rotor which icreased from HZ to HZ-a 5.5- percet. The rotor radius icreased from 600mm to 615mm highlightig a 2.5-percet icrease. The ubalace respose of the rotor bearig decreased from e-2mm to 4.637e- 2mm showig a 11-percet. The optimized rotor mass was kg, which remaied largely uchaged. At preset, the optimizatio results have bee applied i CF250 crusher rotor productio. Table 4. Compariso of for optimum ad iitial /(mm). Desig Variables Iitial Desig Variables Optimum Desig Variables Revised Desig Variables l l d d h h

7 Multi-Objective Optimizatio of Impact Crusher Rotor The Ope Mechaical Egieerig Joural, 2014, Volume Table 5. Compariso of performaces for optimum ad iitial. Optimizatio Variables Parameter Iitial Performaces Optimum Performaces Objective Fuctio State Variables The first atural frequecy f 1(x) (Hz) Rotor radius R (mm) RESP-LEFT (mm) e e -2 Mass (10 3 kg) CONCLUSION (1) The parametric fiite elemet model of the rotor body was built as the referece model for the multiobjective optimizatio of the rotor body. (2) The quadratic respose surface model, low-biased SHS ad weighig fuctio were employed to esure quick covergece of the multi-objective geetic algorithm method. The global Pareto-optimal solutio was obtaied. This method is efficiet ad feasible ad also applicable to other multi-objective optimizatio problems. (3) The method of combiig the experimetal desig, Shifted Hamersley samplig techique, multiobjective geetic algorithm ad sesitivity aalysis was adopted to optimize the rotor structure. The rotor structure was aalyzed to reduce vibratio ad improve the optimizig crushig product quality. This work proposed a improved scheme for eterprise's practical productio. CONFLICT OF INTEREST The authors cofirm that this article cotet has o coflict of iterest. ACKNOWLEDGEMENTS We wish to ackowledge the Natioal Sciece ad Techology Support Program of Chia (2011BAF07B01). We also wish to ackowledge the Sciece ad Techology Foudatio of Guizhou Provice i Chia ([2012]2110). REFERENCES [1] L. M. Che, F. L. Qia ad F. Xu, Dyamic stiffess ad stregth aalysis of the rotor i impact-crusher, Miig Proces. Equip., vol. 36, pp , [2] Q. Wag, Y. D. Zhag ad H. Zhag, Applicatio of kietic simulatio i desig for impact-crusher, J. WuHa Uiv. Tech., vol. 28, pp , [3] J. Xie, S. D. Zhao ad J. T. Liag, Variable sequetial combiatio respose surface methodology for press rod system optimizatio, J. Xi a Jiao Tog Uiv., vol. 46, pp , [4] SNikolov, Modellig ad simulatio of particle breakage i impact crushers, Mieral Proces., vol. 74, pp , [5] SNikolov, A performace model for impact crushers, Mierals Eg., vol. 15, pp , [6] H. Jiag, Y. S. Gua ad Z. C. Qiu, Dyamic ad static multiobjective optimizatio of a vertical machiig ceter based o respose surface method, J. Mech. Eg., vol. 47, pp , [7] J. T. Xiog, Z. D. Qiao ad Z. H. Ha, Optimum aerodyamic desig of trasoic wig based o respose surface methodology, Acta Aeroauticaet Astroauteca Si., vol. 27, pp , [8] J. H. Liu, Z.D. Qiao ad X. D. Yag. Research of aerodyamicstructure itegrative optimizatio desig of wig based o respose surface methodology, Acta Aerodyamica Si., vol. 24, pp , [9] V. Abbas, Optimizatio of composite pressure vessels with metal lier by adaptive respose surface method, J. Mech. Sci. Tech., vol. 25, pp , [10] Z. H. Zhag, Z. He ad W. Guo, A comparative study of three cetral composite desigs i respose surface methodology, J. Sheyag Isti. Aeroaut. Eg., vol. 24, pp , [11] E.Y. Li, G. Y. Li ad H. Wag, Key techology of mixed respose method for absorbig member of auto, Appl. Res. Comput., vol. 25, pp , [12] Q. H. Feg, Q. K. Liu ad L. f. Hu, Structural desig of flat extrusio cotaier based o multi-objective optimizatio, Chia Mech. Eg., vol. 17, pp , [13] S. Poles, Y. Fu ad E. Rigoi, The effect of iitial populatio samplig o the covergece of multi-objective geetic algorithms multi-objective programmig ad goal programmig, Theoret. Results ad Pract. Appl., vol. 618, pp , [14] Y. Fu, ad U. M. Diwekar, A efficiet samplig approach to multi-objective optimizatio, A. Operat. Res., vol.132, pp , [15] U. M. Diwekar, ad J. R. Kalagaam, Robust desig usig a efficiet samplig techique, Comput. Chem. Eg., vol. 20, pp , Received: September 10, 2014 Revised: November 5, 2014 Accepted: November 5, 2014 Zhao et al.; Licesee Betham Ope This is a ope access article licesed uder the terms of the Creative Commos Attributio No-Commercial Licese ( by-c/4.0/) which permits urestricted, o-commercial use, distributio ad reproductio i ay medium, provided the work is properly cited.

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