Applying Axiomatic Design Theory to the Multi-objective Optimization of Disk Brake *

Transcription

1 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake * Zhiqian Wu Xianfu Chen an Junpin Yuan School of Mechanical an Electronical Enineerin East China Jiaoton University Nanchan China chf_n@sina.com Abstract. The multi-objective optimization involves multiple competin functionality requirements which is mainly limite to ownstream etaile esin. Aiomatic esin provies the theory to esin a comple system top own an eals with multiple functional requirements FR. It has emonstrate its strenth in various types of esin tasks. In fact the objective function is a FR an those variables affectin the objective function are the esin parameters Ps. This paper presents an application of aiomatic esin to multi-objective optimization. First ientify the relationship between FRs an Ps in terms of contribution of each P to each FR by usin orthoonal eperiment an analysis of variance ANOVA; then ientify important esin parameters to a FR an classify esin variables into roups base on uncouple esin philosophy; an then establish the function epenence table an sequentially optimize every objective function. An application in a isk brake esin is use to emonstrate the use of the propose metho in ealin with real-worl esin problems. The results show that the propose metho provies a promisin approach to optimize multiple competin esin objectives. Keywors: Aiomatic esin Multi-objective optimization ANOVA isk Brake. Introuction Multi-objective optimization is enerally more ifficult to achieve each objective optimum because of traeoff between the various objectives so the absolutely optimal solution may not eist. Aiomatic esin approach has emonstrate its strenth in various types of lare-scale system esin incluin vehicles aircrafts manufacturin facilities an so on []. Liu [] Hwan [] an Jeff [] apply inepenent aiom of aiomatic esin theory to improve an optimize multiobjective an lare-scale enineerin systems. In this paper a systematic metho is presente for applyin inepenent aiom to multi-objective optimization problems. Firstly ientify the relationship between FRs an Ps in terms of contribution of * Founation: Project supporte by the National Natural Science Founation of China 7.. Li an Y. Chen Es.: CCTA Part III IFIP AICT 7 pp. 7. IFIP International Feeration for Information Processin

2 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake each P to each FR by usin orthoonal eperiment an analysis of variance; seconly important esin parameters to a specific objective are ientifie an coul be roupe into one set of parameters by means of the uncouple philosophy of aiomatic esin; then establish the function epenence table an optimize every objective function in sequence. Multi-objective Optimization Formulation The mathematical moel of multi-objective optimization is enerally epresse as Min: F { f f f } T m s.t. hk j j q k p where n T is n-imensional esin variables; F is the vector of objective functions; f i is the ith sub-objective function an m is the number of subobjective function; i is the jth inequality constraint an q is the number of inequality constraint functions h k is the kth equality constraints an k is the number of equality constraint functions. In the above equation achievin the optimum of each objective is enerally ifficult. Specially when there are traeoffs between the various objectives namely the optimization problem has conflictin oals it is impossible to make each subobjective simultaneously attain optimum. Thus it shoul be require that the optimal solutions of all sub-objectives are balance so that the totally satisfactory solutions can be obtaine. Compare to sinle objective optimization the theory an computational methos on multi-objective optimization are not perfect. Therefore it is necessary to introuce other relative esin theories such as aiomatic esin to further promote multi-objective optimization theory an methooloy. Optimization esin Base on Uncouple Philosophy In fact the objective function is a FR an those variables affectin the objective function are the Ps alreay mentione. But in this case the FR must be able to be epresse by the Ps in the form of a mathematical equation. It is obvious that optimization esin is the metho mappin between one FR an a set of Ps at a lower or the lowest level of FR an P hierarchies in the process of aiomatic esin []. For the esin with only a FR FR is clearly inepenent an the optimal value of FR can be easily obtaine by ajustin the corresponin Ps. Here optimal esin is very effective. For those with multiple objective functions it is ifficult to tune the corresponin Ps to simultaneously obtain the optimal solutions of FRs. For eample in orer to obtain the optimum of FR the value of Ps shoul be tune. In orer to improve the FR the value of Ps shoul be further tune but the

3 Z. Wu X. Chen an J. Yuan FR will be chane. Therefore it is neee to return to tune the Ps to optimize FR; on the contrary the FR will be chane. So aain this is a trae-off amon multiple objectives. In the aiomatic esin uncouple or ecouple esin is a oo esin []. The functional forms of the relations with two functional requirements are as follows: For uncouple esin: f f f f For ecouple esin: f f f f where f an f are functional requirements an an are corresponin esin parameters. For uncouple esin the esin parameters can be etermine separately. For ecouple esin an shoul be etermine sequentially. In the first step of the ecouple esin the value of can be et by optimizin f an is fie in the net step as *. In the secon step the value of can be obtaine by optimizin f. The solution obtaine from the above process coul not be the optimum when is not fie in the secon step. Therefore the ecouple esin may not be oo one when the esin solution is calculate by a mathematical optimization. It is necessary to make the esin to be uncouple esin. In the multi-objective optimization problems there are usually two ifferent solvin methos. The one is that the most important FR is selecte as an objective function an the other FRs are inore. Thus multiple FRs are transforme into sinle FR an optimization is effective to etermine the optimal value of FR an the value of corresponin Ps but the other FRs can't be optimize. The other is that every FR is weihte to form the areate objective function. Althouh it is also sinle FR optimization it can't optimize FRs irectly. Aitionally it is ifficult to ientify the weiht. Therefore in orer to satisfy the inepenence aiom FRs shoul be reefine or Ps shoul be reselecte an the ifferent FRs is controlle by corresponin Ps inepenently. For uncouple esin each sub-objective of multiobjective optimization is inepenent mutually which is equivalent to sinle objective optimization. For couple esin problem: f f f f. If the influence of on f is far reater than that of on f an the influence of on f is far reater than that of on f the problem can be consiere as a nearly uncouple problem. Thus the aiomatic esin process can be applie. When the influence of esin parameters on the rest of esin objectives can't be inore it is a stron couplin problem that can t be ealt with by aiomatic esin. Generally there are more esin variables than objective functions in optimization esin. However accorin to aiomatic esin theory the number of esin variables esin parameters must be the same as that of the objective functions functional requirements in aiomatic esin. Therefore these esins are enerally reunant or may be couple an they violate the inepenence aiom an are not oo esins. When the number of esin variables is lare the variables can be roupe to have similar characteristics via the sensitivity information. That is important variables to a specific objective function can be roupe into one set of parameters an the number of the roups is the same as that of the objective functions. Thus the esin is rearrane to be a nearly uncouple esin. The pertinence is backe up by the Theorem 8 in aiomatic esin theory.

4 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake Theorem 8: Inepenence an Tolerance. A esin is an uncouple esin when the esiner-specifie tolerance is reater than ΔR i > n j j i Ri Pj ΔPj i n. in which case the non-iaonal elements of the esin matri can be nelecte from esin consieration. A multi-objective optimization problem will be transforme into sinle objective optimization problem in sequence by utilizin the metho of lare-scale optimization system or ecomposition an coorination metho of multi-isciplinary optimization. As the optimal solution of each sub-objective function may be not the optimum of multi-objective optimization amon of sub-objective function shoul be coorinate. Accorin to optimization metho on ecomposition an coorination the esin variable of the previous objective function is fie in the net objective function in the optimization process; the net objective function only optimize those esin variables that is classifie into this roup an the optimal variables will be returne to the previous objective function. By an iterative manner aain an aain the optimization process continues until it satisfies the converence conition. The key question of the "ecompositioncoorination" optimization is how to ientify the corresponin esin variables an constraints of each objective function. In larescale multi-isciplinary optimization the esin variables are roupe accorin to the locations of parts in structural esin or eperts-base omain knowlee. In this research the ecomposition is base on the relationship between Ps an FRs which is efine as "loical ecomposition". Optimization esin Metho Usin Inepenent Aiom The steps of multi-objective optimization esin usin inepenent aiom in aiomatic esin can be iven as in the followin: Step. Select esin variables an ientify objective functions an constraints. Consier an optimization esin problem that consists of si esin variables three objective functions an three constraints functions which is iven by the followin esin equation. T Min F X Min { f f f } s.t. The vector of esin variables is { } T.

5 Z. Wu X. Chen an J. Yuan Step. Put the optimization objectives an esin variables into a esin matri. The relationship between esin variables an optimization objectives is epresse as FR FR FR where FRi is the ith functional requirement namely esin objective. The sin means a relation eists an means there is no relation between FR an esin variable. If the esin matri is rearrane into a new matri by makin it as iaonal matri as possible it shows the optimization objectives are inepenent then turn to step. Step. Use orthoonal eperiment an ANOVA to ientify the contribution of each esin variables to objective functions an etermine the principal esin variables an inore those esin variables with small influence. In esin matri the sin means esin variable has sinificant effect on FR an means esin variable has little effect on FR. The equation is rewritten as FR FR FR Rearrane the esin matri to make it as iaonal matri as possible. FR FR FR The esin variables are roupe an the number of the roups is the same as that of optimization objectives. Thus the esin will be nearly uncouple esin. Accorin

6 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake 7 to equation the esin variables are classifie into three roups. The esin parameters P inclue an P inclue an an P inclue an. Then the formula can be rewritten as FR FR FR P P P 7 Step. Suppose p an q lie in the same roup f i if one constraint is only relative to p an q this constraint will be classifie into this roup. All other constraints are classifie into ifferent roups by analoy an then o to step. Otherwise o to the net step. Step. Accorin to the relationship between the esin functions incluin the objective function an constraint functions an esin variables establish the epenence matri on the esin functions an esin variables which is name as the function epenence table FT. If a constraint function relates to one esin variable the corresponin position in FT is marke as otherwise marke as. Here the contribution values of esin variables on the constraint function are not consiere. The relationship between the objective function an esin variables is still marke as "" or "". Table. Function epenence Table f f f In any case the constraints must be met so they can be classifie into the ifferent roups. Namely if a constraint relates to esin variables of multiple roups it can be respectively classifie into those roups. As shown in table only relate to {f ; } it will be classifie into this roup; relate to esin variables of three roups { ; ; } it will be respectively classifie into these roups. Step. If the esin functions of each roup are inepenent the multi-objective optimization problem can be solve by sinle objective optimization manner. Then o to step. However in many cases even if the objective functions are inepenent the constraint functions are often closely linke to each other. So it can not be solve in sinle objective optimization manner an shoul be optimize in sequence.

7 8 Z. Wu X. Chen an J. Yuan Step 7. etermine the orer of each roup an optimize them in sequence. Accorin to the orer of optimization if the optimization objective an constraints in one roup not only relate to the esin variables in this roup but also they relate to that in other roup the esin variables in this roup is only optimize an that in other roup will be fie. In the net roup the values of esin variables obtaine from the previous roup are fie. Each roup is optimize in sequence an set k. Step 8. Start from the first roup continuously the esin variables in other roups are fie. The other roups will be optimize in sequence an set k k. Step 9. etermine the optimization results whether meet the converence criteria. < k k k If ε an is usually set net cycle. ε then return to step where ε is the converence precision ~ * * Step. Output the optimal solution F. Otherwise return to step 8 an continue for the. The alorithm flow of optimization esin usin uncouple iea of aiomatic esin is shown in fiure. Case Stuy In the optimization esin for caliper isc brake use in automobile the short brakin time an small nave parameter uner controllable temperature are require. When the vehicle is full-loae the loa of a sinle wheel W is N; the vehicle s runnin spee v is kmh; the raius of a wheel r is mm; the number of brakes i.e. number of wheels m is ; the allowe maimum iameter for brake isk [ ma ] is mm; the allowe maimum temperature [T ma ] is ºC; the material of brake isc is steel an the oriinal temperature T i is ºC; the iameter of nave h is 7mm; the friction coefficient between pa an brake isc μ is an the ahesion coefficient between tire an roa surface φ is ; the allowe maimum pressure of pa [p ma ] is MPa; the allowe maimum oil pressure of hyro-cyliner [p ma ] is 7MPa; the thickness of hyro-cyliner wall t c is.mm. The structural relationship between caliper an brake isc is shown in fi.. The circular friction surface of pa is iscretize as an arc concentric with the isc as shown in fi.. Consierin that asymmetrical wear process lea to pv unit-pressure Rotate-spee become balance raually on the whole fiction plane so we suppose that pv Cconst. The force of pa actin on isc: F R R C lr r 8 where l r β r cos R r Rr

8 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake 9 Fi.. The flow iaram of optimization esin usin uncouple iea of aiomatic esin. piston. pa. isc. pa. pa isc Fi.. The structure iaram of isk brake Fi.. Calculation iaram of brake

9 7 Z. Wu X. Chen an J. Yuan The friction torque of brakin: R R f FI pra T μ μ 9 where R R r I l I R R r r l I urin the process of breakin the power issipation of pa an brake isc is as much as kinetic enery of auto vehicle. Then the brakin time can be erive: m n I F v W t a μ π. where W a is total weiht of a car N; n is the rotation spee of brake iscs or wheels when start brakin rmin; F is thrust force of oil cyliner p F p π where p is iameter of piston mm; p is oil pressure MPa. Mathematical Moel esin variables { } T { } T p a R p. The esin variables are shown in fi. an fi.. Constraint functions π ] [ π ] [ ] [ - - ma 8 7 ma ma ma h h T T c J E Wr I p p I p t i p c ρ π ϕ μ. where J is mechanical equivalent of heat an equals to N mj; c is specific heat capacity an equals to 7.8 JkºC; ρ is ensity an equals to 7.8-kmm ; E is power issipation of friction torque an equals to m v W a. Objective functions To assure the safety of vehicle it is essential to improve the work efficiency of brakes an to shorten the brakin time so minimizin brakin time shoul be

10 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake 7 consiere as optimization objective. In aition minimizin thickness an temperature of the brake isc can be also consiere as other two objectives. i.e.: Min f. F Min { f f } T E where f t f a f Jπcρ.. Computin Process an Results Ientify the relationship between optimization objectives an esin variables in terms of esin matri an then rearrane the esin matri to make it into a nearly iaonal matri or trianular matri. FR FR FR. As FR is inepenent of FR an FR an FR is only relate to so we only nee to etermine the contribution of an to FR. Since there are three subobjectives in optimization esin the esin variables can be roupe an the number of the roups is three. Table. Level of factor unit: mm Factor Level Ientify the contribution of an to FR usin orthoonal eperiment an ANOVA. Three levels are etermine for both an as shown in table an the orthoonal table L 9 is selecte. The results of ANOVA show that has reat influence on FR an has small influence on FR. Then the equation can be rewritten as FR FR FR P P P.

11 7 Z. Wu X. Chen an J. Yuan Establish the function epenence table. The FT of isc brake is shown in Table. Table. FT of isc brake f 7 f 8 f 8 Accorin to table f is inepenent of f an the relation between f an f is nearly uncouple. Thus f an f can be optimize simultaneously. But both the constraint in the roup {f } an 8 in the roup {f } relate to in the roup {f } shoul be fie in the optimization process of f an f. When f is optimize an shoul be fie. Continue the net iteration until converence criterion is satisfie. Then the optimal solutions are obtaine. The initial sizes of isc brake an the results of the weihte optimization metho an the propose metho in this paper are summarize in table here weihtin factors w w an w are.9.9 an. respectively. Accorin to the results an process of calculation it shows that results obtaine from the weihte optimization metho will chane with ifferent weihtin factors so it is ifficult to balance every objective. The optimal results base on the propose metho are more satisfactory which consier the influence of esin variables to esin objectives an make the esin to be nearly uncouple esin. The values of the first two objectives are smaller ecept the thir objective which is consistent with the importance of the first two objectives. Table. The optimal results ft fa ft mm mm mm mm mm MPa s mm ºC Initial value Weihte optimization Propose metho

12 Applyin Aiomatic esin Theory to the Multi-objective Optimization of isk Brake 7 Conclusions In this paper a new metho of multi-objective optimization is presente. The concept of contribution of esin variables to esin objectives is use to ientify the relation between them. Followin the uncouple esin iea of aiomatic esin the esin matri is rearrane to be as iaonal matri as possible. If the esin is a nearly uncouple esin an iterative metho is sueste. The important esin variables to a specific objective are ientifie an can be roupe into one set of parameters an then establish the function epenence table an optimize every objective function in sequence. The optimization esin of isc brake has been solve to show the valiity of the propose metho. References. Suh N.P.: Aiomatic esin: avances an applications. Ofor University Press New York. Liu X.P. Soerbor N.: Improvin an eistin esin base on aiomatic esin principles. In: First International Conference on Aiomatic esin Cambrie U.S. June - pp. 99. Hwan K.H. Lee K.W. Park G.J. et al.: Robust esin of the vibratory yroscope with unbalance inner torsion imbal usin aiomatic esin. In: Secon International Conference on Aiomatic esin pp. 7. Massachusetts Institution of Technoloy Boston. Jeff T. Ge P.: Applyin aiomatic esin theory to the evaluation an optimization of lare-scale enineerin systems. Journal of Enineerin esin 7. Chen K.Z.: Ientifyin the relationship amon esin methos-key to successful applications an evelopments of esin methos. Journal of Enineerin esin 999