A PROPOSAL OF ROBUST DESIGN METHOD APPLICABLE TO DIVERSE DESIGN PROBLEMS

Transcription

1 A PROPOSAL OF ROBUST DESIGN METHOD APPLICABLE TO DIVERSE DESIGN PROBLEMS Satoshi Nakatsuka Takeo Kato Yoshiki Ujiie Yoshiuki Matsuoka Graduate School of Science and Technolog Keio Universit Yokohama Japan Facult of Science and Technolog Keio Universit Yokohama Japan ABSTRACT: The need for robust design that enables the products to ensure robust performance in diverse surroundings has been focused. Robust design guarantees robust performance in manufacturing variation of products and diverse use environment. In this stud a robust design method applicable to diverse design problems was proposed. In the proposed method to use the robust inde R the weighted robust inde R W and the adjusted robust inde R A as indees of robustness the proposed method is applicable to design problems in which distribution pattern of objective characteristic is non-normal distribution with multi peak the control factor is adjustable. And to confirm the effectiveness the proposed method was applied to a public seat design. As a result of the verification the solution obtained b the proposed method was better than the eisting. It was confirmed that the proposed method was superior to the eisting methods. KEYWORDS: ROBUST DESIGN ROBUST INDEX SEAT DESIGN

2 . INTRODUCTION The environment surrounding design product has been diverse due to increased individuation of the user needs and globaliation of markets. Therefore the need for robust design that enables the products to ensure robust performance in diverse surroundings has been focused. Robust design guarantees robust performance in manufacturing variation of products and diverse use environment. In robust design man methods including Taguchi method (Taguchi G. 99 have been proposed. However the eisting robust design methods (RDMs are not applicable to all design problems as most RDMs are based on the premise that objective function which is a relation epression of factors and objective characteristic is linear. Therefore it is considered that there are design problems to which eisting RDMs are not applicable. The objective of this stud is to clarif the characteristics of design problems to which eisting RDMs are not applicable to propose a RDM which is applicable to these diverse design problems and to confirm the possible application and effectiveness of the proposed method b appling a design case.. RDMS AND THEIR PROBLEMS RDMs can be classified into the methods based on eperiment and the methods based on simulation... RDMS BASED ON EXPERIMENTS RDMs based on eperiments have their origin in the design of eperiment. RDMs evaluate and improve the robustness of function on objective characteristic which is a phsical value to epress the design objective b using the data obtained b full factorial designs eperiment or orthogonal arra eperiment. Specificall the first step is to get the average and standard deviation of s eperimental value b eperimenting with each level of control factor which is a factor that designer is able to control and noise factor which is a factor that designer is unable to control. Using these data the second step is to make optimiation of two aims: to minimie the difference between the objective characteristic value and the target value τ and to minimie the

3 variations of. (Fig. shows an eample oforthogonal arraand mathematical formula of concept based on eperiment In this stud four methods Taguchi s method Otto s method(otto K.N. and Antosson E.K. 99 Sundaresan s method(sundaresan S. Ishii K. and Houser D.R. 99 and Yu s method (Yu J.C. and Ishii K. 99 were introduced as RDMs based on eperiment... RDMS BASED ON SIMULATIONS RDMs based on simulation can be further classified into the methods for transforming objective functions and the methods for transforming constraint functions which epresses the relation between factors and constraint characteristics. Methods for transforming objective functions evaluate and improve the robustness of. The first step is to transform the objective functions into different function based on the fluctuation of fluctuant factors (control factors and noise factors. The second step is to make optimiation of two aims: to minimie the difference between the objective characteristic value and the target value and to minimie the variations of with the variations of fluctuant factors. As a result of these processes a robust design solution which has the variations of smaller than the eisting design solution is obtained(fig.. In this stud seven methods Ramakrishnan s method(ramakrishnan B. and Rao S.S. 996 Belegundu s method(belegundu A.D. and Zhang S. 99 Arakawa s method(arakawa M. and Yamakawa H. 995 Wilde s method(wilde D.J. 99 Zhu s method(zhu J. and Ting Number of eperiments Number of factors Minimie n ( i Figure : Concept of RDMs based on eperiment. i = n τ n n i ( i i = i = Objective characteristic (eperimental value nnumber of eperiment τtarget value

4 K.L. 00 Gunawan s method(gunawan S. and Aarm S. 004 and Eggert s method (Eggert R.J. 99 were introduced as methods for transforming objective function. Net methods for transforming constraint function evaluate and improve the robustness of constraint characteristic c which is a constrained phsical value b transforming the constraint function into different function based on the fluctuation of fluctuant factors and configuring the constraint condition into a stringer one. As a result of these processes a robust design solution which c does not depart from its feasible area with fluctuation of fluctuant factors is obtained (Fig.. In this stud si methods Parkinson s method ( Parkinson s method ((Parkinson A. 995 Arakawa s method Sundaresan s method Eggert s method and Yu s method were introduced as methodsfor transforming constraint function. Objective characteristic τ Robust optimum solution Optimum solution Minimie f { f ( τ C ( } Fluctuation of objective characteristic { f ( f ( * * C ( } : * ( QC = : * fobjective function Control factor Noise factor Fluctuation value * Standard value without fluctuation Control factor Figure : Concept of methods for transforming objective function. Control factor g Robust optimum solution Optimum solution Fluctuation of design solution g Subject to { ( 0 C ( } g j : * ( QC = : * gconstraint function Design solution running out of feasible area Control factor Figure : Concept of methods for transforming constraint function. 4

5 .. PROBLEMS ASSOCIATED WITH RDMS For assessment above methods seven characteristics of design problem were identified: linearit of function differentiabilit and monotonicit of function distribution pattern of fluctuant factors independence of fluctuant factors eistence of tuning factors units of fluctuant factors qualit of weight of objective characteristic value and eistence of adjusted factor which is the control factor having variable range. Then eisting methods were assessed if the were applicable to these problems. (Table shows the result of this assessment. B rating indicates that the robust method is applicable to the design problem and A rating indicates that the method is applicable to the problem effectivel with little calculation amount. In addition C rating indicates that the method is applicable to the problem ecept when is nominal-the-better characteristic.the result of assessment confirmed that these methods are not applicable to design problems in which a function is non-linear and fluctuant factors are dependent weight of objective characteristic value is nonuniform or control factor is adjustable. For eample in the case of considering probabilit distribution of eisting methods are applicable to problems in which function is non-liner and the sum of squares of is normal distribution or bilaterall-smmetric uniform as shown in (Fig. 4(a. Table : Assessment table of problems for RDMs. RDMs RDMs based on eperiment RDMs based on simulation RDMs for transforming objective function RDMs for transforming constraint function Characteristics of design problems Taguchi's method Otto's method Sundaresan's method Yu's method Ramakrishnan's method Belegundu's method Arakawa's method Wilde's method Zhu's method Gunawan's method Eggert's method Parkinson's method ( Parkinson's method ( Arakawa's method Sundaresan's method Eggert's method Yu's method Linearit of function Differentiabilit and monotonicit of function Distribution pattern of fluctuant factors Independence of fluctuant factors Tuning factors Units of fluctuant factors Weight of objective characteristic Adjusted control factor Possible to appl lineariation A A A A A A A A A A B A A A A B B Impossible to appl lineariation B B B Differentiable B B B B A A A B A B A A A A B A B Indifferentiable and monotone increase (decrease B B B B A B A B Indifferentiable and not monotone increase (decrease B B B B B B Bilaterall-smmetric uniform B B B B A B B A A A B B B Not bilaterall-smmetric uniform B B B B B B B B B Normal B B B B B B B Independent A B A B A A A A A A A A A A A A B Dependent B B B Eistent B B B B B B B B B B B B B B B B B Noneistent C C B B B B B B B B B B B B B B B Identical unit B B B B B B B B A A B B B B B B B Nonidentical unit B B B B B B B B B B B B B B B Uniform B B B B B B B B B B B Nonuniform Eistent B B B B B B Noneistent B B B B B B B B B B B B B B B B B 5

6 On the other hand it is conceivable that distribution pattern of is often non-normal in actual design problems. In the case where the function is strong non-liner distribution pattern of is non-normal as shown in (Fig. 4(b.In addition in the case where multi functions are fluctuant stochasticall distribution pattern of is non-normal distribution with multi peak as shown in (Fig. 4(c because of superposition of multi distribution. In the second case there area few errors between distribution pattern and normal distribution; moreover it is possible to appl properl with Eggert s method to consider chi-square distribution or gamma distribution. However in the third case there are big errors between distribution pattern and normal distribution and it is difficult to relate to other distribution. Moreover there is no method to proper evaluate the design problem with adjusted mechanism as shown in (Fig. 5. uniform distribution f nonnormal distribution f nonnormal distribution f with multi peak f stochastic fluctuation (a Linear Objective function (b Nonlinear objective function (c Fluctuant objective function stochasticall Figure 4: Relation between objective function and probabilit distribution of. Figure 5: Adjusted mechanism. 6

7 . A PROPOSAL OF ROBUST DESIGN METHOD APPLICABLE TO DIVERSE DESIGN PROBLEMS The new method applicable to diverse design problems named RDM was proposed which is applicable to all design problems including the problems to which eisting method was not applicable as indicated in the second paragraph. In the proposed method robustness was defined as the feasibilit of the objective characteristic value being within tolerance. B epressing this concept with probabilit densit function of the proposed method was applicable to above diverse design problems. Here are the evaluation indees of robustness used in the proposed method... The robust inde R The robust inde R was the feasibilit of the objective characteristic value being within tolerance and was calculated to integrate probabilit densit function of. The feasibilit is epressed as follows: u R = p( d l ( where p is probabilit densit p( is probabilit densit function of the l is the lower tolerance limit u is the upper tolerance limit and τ is the target value. (Fig. 6 shows the concept of R. R is the evaluation inde to minimie the sum of squares of because it evaluates robustness b using the feasibilit of the objective characteristic value being within tolerance. In this stud R was calculated using the Monte Carlo method. First a random number of fluctuant factors were generated based on the probabilit densit function of their fluctuation. Second objective characteristic value was calculated based on a random number of them. Finall R is calculated as follows: P R Tolerance p( l τ u Figure 6: Concept of the robust inde R. 7

8 R = s s M i i= M i = 0 ( l i u ( otherwise ( where s is the number of samples generated for a random number of fluctuant factors... The weighted robust inde R W Here since R is onl able to evaluate if the objective characteristic value is within tolerance or not R is not able to evaluate the weight of the objective characteristic value. However since a design problem has a target value in man cases the weight of the objective characteristic value increases as the approaches the target value. In this stud the weight of the objective characteristic value was epressed as a weighting function. R W that enables the weight of the objective characteristic value to be evaluated is epressed as follows: u R W = w( p( d l ( where w is the weight of the objective characteristic value and w( is the weighting function of the objective characteristic value. (Fig. 7 shows the concept of R W. R W is the evaluation measure to optimie two aims to minimie the sum of squares of and to maimie the weight of the objective characteristic value because it evaluates the feasibilit of the objective characteristic value being within tolerance and the weight of the objective characteristic value. In this stud R W was calculated using the Monte Carlo method as in the case of R. R W is calculated as follows: R = s w( ( l i u s M i = i= 0 ( otherwise W M i (4 P Tolerance p( R W w( w l τ u Figure 7: Concept of the weighted robust inde RW. 8

9 R and R W are used depending on the need to consider the weight of objective characteristic value... The adjusted robust inde R A The adjusted robust inde R A is the evaluation measure to add the concept of adjusted factor to R. Specificall in the case where control factor has variable range R A is the feasibilit of the objective characteristic value being within tolerance for each control factor value t i in the range and is epressed as the ratio of the sum of sets of combination of fluctuant factors in the case that objective characteristic value being within tolerance in entire set. R A [ { C( l f ( ti u} ] Ut i = P (5 where i is level of adjusted factor P[A] is the feasibilit to happen event A. (Fig. 8 shows the concept of R A. R A is able to evaluate robustness in case where control factor has variable range. In addition it can calculate variable range of control factor for additional design with adjusted mechanism to get more robustness when design solution calculated b using R do not have enough robustness. As some eamples of adjusted mechanism design of automobile seat back structure and seat slide structure follow. R A was calculated using the Monte Carlo method as in the case of R and R W. First a random number of fluctuant factors were generated. Second each combination of generated fluctuant factors and objective characteristic value in the case of each combination were calculated. Third the case an of adjusted factors eists where objective characteristic value being within tolerance was calculated otherwise was 0. Finall the sum of them was divided b a random number. R W is calculated as follows: R = A M i s i= s M i ( t S t t = t; l f ( i i u t = 0 ( otherwise (6 4. CASE APPLICATIONS Finall to confirm the effectiveness the proposed method was applied to a public seat design preventing hip-sliding force F HS which is a source of discomfort with respect to sitting posture. Here cushion angle θ C which is the main factor to control F HS was selected as design object (Fig. 9. In the process of calculating design solution to confirm the effectiveness of the proposed method the solution calculated b Yu s method with average and standard deviation (θ C(Yu and the solution calculated b average onl without considering sum of squares of (θ C(µ are 9

10 compared in addition to the design solution calculated b R and R W (θ C(R and θ C(RW (table. Moreover probabilit distribution of of each solution was compared (Fig. 0. According to (Table R and R W in case of θ C(R and θ C (RW were 5 to 7 percent higher than θ C(Yu and θ C(µ. This result confirmed that the proposed method advanced the feasibilit of the objective characteristic value being within tolerance and the design solutions calculated b the proposed method have higher robustness. This consideration was confirmed b the fact that probabilit distribution within tolerance of θ C(R and θ C(RW were more than that of θ C(Yu and θ C(µ. Moreover to satisf more users proper variable range was calculated using the R A. As a result of this calculation the solution satisfied with 95 percent of users with setting the variable range of cushion angle from 6.5 to 9.; therefore it was confirmed that the proposed method were superior to the eisting methods. 5. CONCLUSIONS In this stud first the characteristics of design problems which eisting RDMs are not applicable were confirmed. Second the robust design method applicable to diverse design problems was P p( Tolerance C t l f ( t u C t l f ( t 0 0 u C f t t l ( u t t t 0 t t l u f( t Figure 8: Concept of the adjusted robust inde RA. F HS θ C Figure 9: Design object. 0

11 proposed as the method applicable to these problems. In the proposed method the robust inde R the weighted robust inde R W and the adjusted robust inde R A were used. Moreover the proposed method was applied to a public seat design and its possible application and effectiveness were indicated. Future stud is to confirm the general versatilit of the proposed method b appling to a wide range of design problems. 6. ACKNOWLEDGMENTS This work is supported in part b Grant in Aid for the st Centur Center of Ecellence for "Sstem Design: Paradigm Shift from Intelligence to Life" from Ministr of Education Culture Sport and Technolog in Japan. Table : Design solution of cushion angle. Measure of robustness Optimum solution (deg θ C (R =7.9 θ C (RW =7.6 θ C (Yu =8. θ C (µ = R RW F HSτ = 0 Tolerance Probabilit densit θ C(R =7.9 θ C(RW =7.6 θ C(Yu θ C(µ = Hip-sliding force [N] Figure 0: Probabilit densit distribution of

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