A Fuzzy Satisfactory Optimization Method Based on Stress Analysis for a Hybrid Composite Flywheel

Transcription

1 Wold Academy of Science, Engineeing and Technology Intenational Jounal of Mateials and Metallugical Engineeing A Fuzzy Satisfactoy Optimization Method Based on Stess Analysis fo a Hybid Composite Flywheel Liping Yang, Cuan Cawfod, J. Ren, Zhengyi Ren Intenational Science Index, Mateials and Metallugical Engineeing waset.og/publication/ Abstact Consideing the cost evaluation and the stess analysis, a fuzzy satisfactoy optimization (FSO) method has been developed fo a hybid composite flywheel. To evaluate the cost, the cost coefficients of the flywheel components ae obtained though calculating the weighted sum of the scoes of the mateial manufactuability, the stuctue chaacte, and the mateial pice. To expess the satisfactoy degee of the enegy, the cost, and the mass, the satisfactoy functions ae poposed by using the decline function and intoducing a satisfactoy coefficient. To imply the diffeent significance of the objectives, the object weight coefficients ae defined. Based on the stess analysis of composite mateial, the cicumfeential and adial stesses ae consideed into the optimization fomulation. The simulations of the FSO method with diffeent weight coefficients and stoage enegy density optimization (SEDO) method of a flywheel ae contasted. The analysis esults show that the FSO method can satisfy diffeent equiements of the designe and the FSO method with suitable weight coefficients can eplace the SEDO method. Keywods Flywheel enegy stoage, fuzzy, optimization, stess analysis. I. INTRODUCTION OMPARED with chemical enegy stoage, Flywheel C Enegy Stoage (FES) has the advantages of no pollution, no equiement of stict envionmental tempeatue, etc. Theefoe, along with the envionment potection being paid moe attention, FES technology has eceived moe and moe attention within the ecent yeas. As the coe components of a FES System (FESS), the flywheel stuctue is vey impotant not only fo stoage capacity, but also fo safety and manufactuing cost of the FESS. Since the accuate data of the cost ae not available in cost optimization design, Kack et al. poposed that the cost could be nomalized by the pice of composite mateial [1], []. Yan designed pe unit mass pice of glass fibe as 1 so that othe mateial pice could be calculated by efeing to the pice of glass fibe []. Howeve, when consideing the total manufactuing pocess of a flywheel, it can be found that the cost of a flywheel is also affected by the stuctue and manufactuing pocess. Hence, how to evaluate the cost of a flywheel should be deeply eseached. As the stuctue of the flywheel decides how much kinetic Liping Yang is with Reseach Institute of Enegy Stoage Technology and Application, the Habin Engineeing Univesity, Habin, China (coesponding autho, phone: , fax: , yanglipingylp0@ 16.com). Cuan Cawfod, J., was with Depatment of Mechanical Engineeing, Univesity of Victoia, Victoia, Bitish Columbia, Canada. Zhengyi Ren is with Reseach Institute of Enegy Stoage Technology and Application, the Habin Engineeing Univesity, Habin, China. enegy can be stoed by the whole system, many eseaches used the kinetic enegy as the optimization object to design a flywheel [], [5]. Howeve, too much mass will bing inconvenience to the assembly and tanspotation of the flywheel especially fo the application in mobile machine. Hence, Stoage enegy density (SED) is used as anothe optimization object [6]-[8]. Along with the development of the composite mateial and the magnetic suspended beaing technologies, the cost of flywheel becomes an impotant facto affecting the application of the FESS. Hence, how to obtain big SED and low cost simultaneously is becoming moe and moe impotant. On the othe hand, the safety of a high velocity flywheel is vey impotant. In ode to avoid the failue of the flywheel, Cutiss et al. eseached and deived an equation which expesses the elationship between the stesses and flywheel paametes, such as the oute adii, inne adii, Poisson s atio, imum disk im velocity, adial and cicumfeential elastic modulus, etc. [9]. Kack et al. used the equation of stesses to veify the numeical tess esults, and used as a constaint fo the optimization design to calculate the imum angula velocity [1]. Ha et al. also consideed the stess into the design optimization to calculate the optimum adii unde the condition of the imum angula velocity being fixed [10]. But, all the eseaches did not use the stess equation fo calculating the imum angula velocity and the imum oute adii simultaneously. In this study, a flywheel cost evaluation method is developed. Then, to expess the satisfactoy degees fo diffeent objectives easonably, the FSO method based on fuzzy satisfactoy function and the object weight coefficients is poposed. Moeove, to guaantee the safety of the high velocity flywheel, based on the analysis of the elationship between the imum angula velocity, imum oute adii and stesses, FSO method based on the calculation of stesses is poposed. At last, the compaison of simulation esults fom diffeent optimization design methods shows the effective of the poposed method. II. COST EVALUATION FOR FLYWHEELS The taditional cost is calculated only accoding to the pice of mateials, it is clea that this evaluation method has geat limitations. We know that although being made by same mateial, the flywheels with diffeent stuctue have diffeent costs. It is also obvious that some mateials ae easy to be machined, but some ae not. Hence, we poposed that the cost of a flywheel is influenced not only by the mateial pice, but also by the manufactuing pocess and the mateial machinability. Intenational Scholaly and Scientific Reseach & Innovation 11(8) schola.waset.og/ /

2 Wold Academy of Science, Engineeing and Technology Intenational Jounal of Mateials and Metallugical Engineeing Intenational Science Index, Mateials and Metallugical Engineeing waset.og/publication/ To evaluate the cost moe easonably, the flywheel is divided into m components accoding to the mateial chaacte and the flywheel stuctue. Fo evey component, thee ae some main indicatos being used to evaluate the flywheel cost, such as mateial manufactuability, stuctue chaacte, and mateial pice. Mateial manufactuability indicates that the mateial is easy o not to be fomed and machined. Stuctue chaacte means that the stuctue is easy o not to be manufactued. Theefoe, the evaluation indicato set of the i th (i=1,,,m) component of a flywheel is descibed as follows: ψ i ={f i 1, f i, f i }={mateial manufactuability, stuctue chaacte, mateial pice}. To evaluate the pefomance of each element f i j, the scoe of xij [0,100](i=1,,,m; j=1,,) is given to f i j. The i highe the scoe fo f 1 is, the moe difficulty the mateial manufactuing pocess is. On the othe hand, the highe the scoe fo f i is, the moe complicated the stuctue is. Similaly, the highe the scoe fo f i is, the moe expensive the mateial pice is. Accoding to the expet evaluation, the weight vecto is W ( w1, w, w), wj 1, 0 w j 1. Theefoe, the cost j1 coefficient of the i th (i=1,,,m ) component is: C w x j ij di (1) j1 100 Accoding to the cost coefficient and mass distibution, the flywheel equivalent cost can be calculated as: m D C V( d, h) () di i i i i i1 whee i is the density of the i th component, d i and h i ae the oute diamete and the height of the i th component, Vi( di, h i) is the volume of the i th component, which is the function about d i and h i. III. MULTI-OBJECTIVE OPTIMIZATION BASED ON FSO METHOD High stoage enegy and low cost usually ae the pupose of the flywheel optimization poblem. Moeove, less mass will make the tanspotation moe convenient and will decease the otational inetia which will makes the flywheel otate moe stable. Theefoe, the imum stoage enegy, the cost, and the mass ae the optimization objects of the flywheel. In ode to expess the optimization objects, M ( X ), D ( X ) and ( X ), espectively, ae defined as the imum kinetic enegy, the equivalent cost and the mass of the flywheel. X (, di, hi) ( i=1,,,m) is the decision vecto, whee is the imum angula velocity of the flywheel, d i and h i ae oute diamete and height of the i th component. Accoding to the flywheel woking pinciple, M ( X ), D ( X ), and ( X ) can be epesented as: m M ( X) V ( d, h) () i i i i i1 m D ( X) C V( d, h) () di i i i i i1 1 ( X) J (5) whee J is the otational inetia which is the function of d i and h i. Fo diffeent designes, thei satisfactoy degees fo the objective values ae diffeent. Fo this eason, the fuzzy satisfactoy degee functions ae intoduced. Fo multi-objective optimization poblem in fuzzy envionment, thee ae vaious kinds of membeship functions such as linea, hypebolic, and exponential functions [11]. To simplify the calculation and epesent the desiability of designes, the tiangle-like membeship functions ae used as the fuzzy satisfactoy degee functions fo the flywheel optimization objectives [1]. In ode to adjust the tiangle-like membeship functions easily, a satisfactoy coefficient of (0.5 1) is used to descibe membeship functions. Hence, the satisfactoy degee functions of the optimization objects ae epesented as: 1 M M ( X) ( 1) M M min min ( X ) M M ( X) ( 1) M M (6) min ( 1 ) M M M ( X) M min ( M M ) min 1 D D ( X) ( 1) D D ( X ) D D ( X) ( 1) D D min ( 1 ) D D D ( X) D ( D D ) min min min min (7) Intenational Scholaly and Scientific Reseach & Innovation 11(8) schola.waset.og/ /

3 Wold Academy of Science, Engineeing and Technology Intenational Jounal of Mateials and Metallugical Engineeing E ( X)-E k kmin E E ( X) E ( 1) E kmin k k kmin ( X ) ( - ) E E E k kmin 1 E ( 1 ) E E ( X) E k kmin k k (8) Intenational Science Index, Mateials and Metallugical Engineeing waset.og/publication/ whee M and M min, D and D min, E k and E k min ae uppe and lowe bounds of M ( X ), D ( X ) and ( X ), espectively. As an example, the fuzzy satisfactoy degee function of flywheel mass is shown in Fig. 1. To ealize the multi-objective optimization along with the diffeent significance of the objectives, the weight coefficients of ε (0<ε 1) and γ (0<ε+εγ 1) ae defined. ε is the coefficient of the enegy significance, and γ is the coefficient of the cost significance elative to the enegy. Accodingly, the multi-objective optimization fomulation based on the satisfactoy function is epesented as follows: Maximize: [ E( X ) D ( X)] [1 (1 ) ] M ( X) Subject To: X min X X (1 ) 1 Fig. 1 Fuzzy satisfactoy membeship function of mass IV. STRESS ANALYSIS FOR FLYWHEEL OPTIMIZATION DESIGN Compaed with metal mateial, the composite mateial has the chaacteistics of light weight and high stength, which is the pefeed mateial fo the high velocity flywheel. As the safety of the flywheel is vey impotant, the influence of the composite mateial stess should be fully consideed. Othewise, the flywheel failue will esult in disastous consequences. Since the stesses on the oute ing have the geatest influence on the flywheel safety, the elationship between the stess and the mateial paametes and the size of this component is analyzed. The oute ing is a kind of axisymmetic constant thickness spinning disk, so the adial stess σ and cicumfeential stess σ θ ae epesented as follows [9]: (9) v 1 v / R R / R R 1 1 o i o i 1 v ( e v R i Ro Ro Ri v / R / 1 v / i Ro Ri R o v R i Ro Ro Ri v R o v R R / R R 1 1 o i o i 1 v ( e Ri Ro Ro Ri 1 1 Ri Ro Ri / Ro 1 ) Ri Ro Ro Ri Ro v ) (10) (11) whee is the mateial density, is the Poisson s atio of contaction in adial diection due to cicumfeential θ tension, ve is the imum disk im velocity, Ri and R o ae the disk inne and oute adii, E0 E, E0 and E ae adial elastic moduli, and cicumfeential elastic modulus, ( Ri Ro) is the adius of select point. Defining Ri Ro and x Ro, the σ θ and σ can be deived as: whee v ( cx c x c x ) (1) 1 1 e 1 v ( c x c x c x ) (1) v 1 1 c, c 9 1 c v e 1 v, 9 1, c v. 9 Assume that in and in ae the cicumfeential stess coefficient and adial stess coefficient, espectively. As ve Ro and the stesses should be less than the allowable stesses, fom (1) and (1), the following fomulas ae deived as: Ro [ ]/ in (1) R o [ ]/ in (15) whee Intenational Scholaly and Scientific Reseach & Innovation 11(8) schola.waset.og/ /

4 Wold Academy of Science, Engineeing and Technology Intenational Jounal of Mateials and Metallugical Engineeing Intenational Science Index, Mateials and Metallugical Engineeing waset.og/publication/ in cx c x c x in cx cx cx The imum of in and in can be calculated fo a kind of composite mateial. Theefoe, fo a fixed value of α, the multi-objective optimization fomulation consideing stesses constaints is epesented as follows: Maximize: [ E( X) D ( X)] [1 (1 ) ] M( X) Xmin X X 0 1 (16) Subject To: 0 (1 ) 1 Ro min{ [ ] / in, [ ] / in } R / R = i o whee X min and X ae the minimum value and imum value of X, in and in ae the imum of in and, espectively. in V. COMPARISON OF SIMULATION RESULTS In ode to analyze the pesented multi-objective FSO, a composite mateial flywheel stuctue is designed which is shown in Fig.. Fig. A flywheel stuctue Accoding to the mateial and the stuctue, the flywheel is divided into fou components: shaft, im1, im, and im. The mateial of shaft is steel. The mateial of im1 and im is aluminium alloy. The mateial of im is cabon fibe. Because oute ing of this flywheel is im, thee ae Ri = d /, Ro = d /. The vaiables include diametes of the fou components d, d, d, d (m) and the imum angula velocity (ad/s). 1 Setting =0.5, thee ae X (, d, d, d ) 1. Accoding to the design equiement, thee ae X min (00,0.06, 0.15, 0.6) and X ( 000, 0.08, 0.5, 0.8).Othe stuctue paametes of the flywheel ae h1 0.5 m, h 0.1 m, h h = 0. m. The densities of steel, aluminium alloy and cabon fibe ae 7800 kg/m, 700 kg/ m, and 1780 kg/ m, espectively. Fo mateial manufactuability, the manufactuing pocesses of the steel and aluminium alloy ae obviously diffeent fom the pocess of the cabon fibe. Because the aluminum alloy mateial belongs to the non-feous metal, the hadness is infeio to the steel, and the pocessing difficulty must be smalle than the steel. Due to the high equiement of the winding technology of cabon fibe flywheel, the technology matuity is not as good as the metal manufactuing pocess. Theefoe, the mateial manufactuability scoes of the fou components ae: x 0 11, x 0 1, x 0 1, x 80 1 On the othe hand, the stuctue chaacte of the shaft is elatively complex. It not only equies stict dimensional accuacy, igoous suface oughness, and suface hadness but also equies high shock esistance. On contast, the stuctue chaacte of im1 and im mainly eflects in the complex shape. Additionally, the demands on dimensional accuacy and suface oughness of the inne hole suface of im1 ae moe igoous than the oute cicle suface of im. The im is winded by cabon fibe which has egulaly shape and lowe dimensional accuacy. Hence, the scoe of stuctue chaacte of im is lowest in the scoes of the fou components. Theefoe, the scoe of the stuctue chaacte of the fou components ae: x 90 1, x 50, x 0, x 10 Based on the maket pices of steel, aluminium alloy and cabon fibe, the scoes of the mateial pice ae: x 5, x 1, x 1, x Accoding to the impotance degee, the weight set of the evaluation indicatos is: W= (0., 0., 0.5). The cost coefficient of the i th (i=1,,,m) component is calculated espectively as: C d C d C d Intenational Scholaly and Scientific Reseach & Innovation 11(8) schola.waset.og/ /

5 Wold Academy of Science, Engineeing and Technology Intenational Jounal of Mateials and Metallugical Engineeing Intenational Science Index, Mateials and Metallugical Engineeing waset.og/publication/ C d Accoding to the mass distibution of each component, the equivalent cost of the flywheel is: D (.V 0.59V 0.5V 1.5 V ) 10 1 Accoding to the (), (), (5), the M ( X ), D ( X ) and ( X) can be epesented as follows: 10 ( X ) (.6d1 0.5d 0.8d 0.5 d) 10 D ( X) (1.091d 0.1d 0.d 0. d ) 1 10 M ( X) (.6d 0.5d 0.8d 0.5 d ) 1 Hence, when the β=0.8, the fuzzy satisfactoy degee functions ae as follows: 6 ( X) ( X) E( X) E ( X) D( X) 11.5 ( X) D D( X) D( X) M ( X) 5.6 ( X) M M ( X) M ( X) when β=1, the fuzzy satisfactoy degee functions ae as follows: 6 ( X) E( X). ( X) D ( X) 95. D( X) D( X) M ( X) 1.8 M( X) M( X) The paametes of the cabon fibe ae v =0., E GPa, E 56 GPa, 1780 kg/m [1]. Hence, can be deived as: k E0 E Accoding to (1) and (1), the values of c 1, c, c, and c can be calculated as follows: and c1-0.6, c , c -.085, c 0.6 Hence, using the MATLAB optimization tool box, in ae calculated as: 0.617, inamx in inamx Since the longitudinal and tansvese tensile stengths of cabon fibe ae [ ] =500MPa and [ ] =56MPa [11], we can obtain: in, in [ ]/( ) 1786 [ ] / ( ) 79.5 Theefoe, fom (16) the epesentation of the optimization fomula is ewitten as follows: Maximize: [ E( X) D ( X)] (1 ) M( X) Xmin X X 0 1 Subject To: 0 (1 ) 1 d/ 79.5 d / d =1/ (17) A. Compaison of FSO with Diffeent β, ε and γ To veify the FSO method, the optimum esults of diffeent β, ε, and γ ae listed in Table I. TABLE I THE OPTIMUM RESULTS OF FSO WITH DIFFERENT COEFFICIENTS ε =0.8, γ=0. ε=0., γ=1 ε=0., γ= Indicatos β=0.8 β=1 β=0.8 β=1 β=0.8 β=1 ω (ad/s) d 1 (mm) d (mm) D (mm) E k(mj) D M (kg) μ E μ D μ M In ode to analyze the optimization effect, the values of the object which has the imum weight among the thee objects ae compaed. In the case of ε =0.8 and γ=0., the optimum value of enegy of β=1 is 5.6% highe than the value of β=0.8. In the case of ε =0. and γ=1, the optimum value of mass of β=1 is 18.5% lowe than the value of β=0.8. Moeove, in the case of ε =0. and γ=, the optimum value of cost of β=1 is 15.6% lowe than the value of β=0.8. All these show that when β=1, Intenational Scholaly and Scientific Reseach & Innovation 11(8) schola.waset.og/ /

6 Wold Academy of Science, Engineeing and Technology Intenational Jounal of Mateials and Metallugical Engineeing Intenational Science Index, Mateials and Metallugical Engineeing waset.og/publication/ the design equiements of the satisfaction degee ae moe stingent. On the othe hand, in Table I, when ε=0. and β=1, the optimum esults with diffeent γ ae same. But when ε =0. and β=0.8, the optimum cost of γ=1 is 0.6% bigge than the cost of γ=, and μ D of γ=1 is 0.5% smalle than μ D of γ=. It is obvious that the impotance of cost inceases with the incease of β. B. Compaison of FSO with and without the Stess Constaints When β=1, the optimum esults of FSO with o without the stess constaints ae listed in Table II. Consideing the stess constaints, the values of ωd / unde diffeent ε and γ is smalle than the allowable value of 79.5 ad m/s, as shown in (17). On contast, the optimum esults of ωd / without stess constaints ae all bigge than 79.5 ad m/s. It is obvious that the optimization with the stess constaints can guaantee the safety of the flywheel. TABLE Ⅱ THE OPTIMUM RESULTS WITH OR WITHOUT STRESS CONSTRAINTS Indicatos ε =0.8, γ=0. ε =0.6, γ=0.5 With without with without ω(ad/s) d 1 (mm) d (mm) d (mm) E k(mj) D M (kg) ωd /(ad m/s) C. Simulation Contast of FSO and SEDO In ode to povide futhe insight on the FSO, the optimum esults of FSO of β=1 and SEDO with same constaints ae listed in Table III. Indicatos TABLE Ⅲ THE OPTIMUM RESULTS OF FSO AND SEDO. ε =0.8, γ=0. FSO ε =0.5, γ=0.5 SEDO ω(ad/s) d 1 (mm) d (mm) d (mm) E k(mj) D M (kg) SED(MJ/kg) Fo the optimum E k and M of FSO, the SED of ε =0.8 and γ=0. is 0.08% smalle than that of SEDO. But, on the contay, the SED of ε =0.5 and γ=0.5 is 0.% bigge than that of SEDO. It can be seen that the optimum ESD can be acquied by the FSO method when the values of the weight coefficients ae suitable. VI. CONCLUSION An equivalent cost evaluation method was poposed. To evaluate the cost moe easonably, the flywheel is divided into seveal components accoding to the mateial chaacte and the flywheel stuctue. Fo evey component, mateial manufactuability, stuctue chaacte and mateial pice wee consideed as the main indicatos evaluating the flywheel cost. Based on intoducing the fuzzy satisfactoy degee functions, the enegy coefficient and the cost coefficient, the FSO method was poposed. The tiangle-like membeship functions ae used as the fuzzy satisfactoy degee functions fo the flywheel optimization objectives. A satisfactoy coefficient was used to adjust the tiangle-like membeship functions. On the basis of the stess analysis, the elativity between the oute diamete, imum velocity and the flywheel stesses wee calculated. Using the imum of the cicumfeential stess and adial stess as the constaints, the FSO method consideing the stesses constaints was poposed. To veify the effect of FSO, the FSO methods with diffeent satisfactoy coefficient and weight coefficients wee simulated. The esult analysis demonstated that the FSO could ealize the diffeent design taget. The contast of FSO simulation with/without stess constaint showed that only the calculated adial stess with stess constaint was smalle than the allowance adial stess. Additionally, the contast of SEDO and the FSO indicated that the FSO could each a highe SED when the values of paametes ae suitable. REFERENCES [1] M. Kack, M. Secanell, P. Metiny, Cost Optimization of Hybid Composite Flywheel Rotos fo Enegy Stoage, Stuct Multidisc Optim, vol.1, 010, pp [] M. Kack, M. Secanell, P. Metiny, Cost Optimization of a Hybid Composite Flywheel Roto with a Split-type Hub Using Combined Analytical/numeical Models, Stuct Multidisc Optim, vol., 011, pp [] X. l. Yan, Cost Optimization Design of Hybid Composite Flywheel Roto, Jounal of mechanical engineeing, vol.. n1, 01, pp [] J. Huang, G. 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