Kinematic design of a double wishbone type front suspension mechanism using multi-objective optimization

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1 5 th utralaian Congre on pplied Mehani, CM Deember 2007, Bribane, utralia Kinemati deign of a double wihbone tpe front upenion mehanim uing multi-objetive optimiation J. S. wang 1, S. R. Kim 1 and S. Y. an 2 1 Deptment of Mehanial Engineering, raduate Shool, anang Univerit, Korea 2 Shool of Mehanial Engineering, anang Univerit, Korea btrat: kinemati deign of a double wihbone tpe front upenion mehanim wa ued to determine the optimal hardpoint poition while onidering ontrollabilit and tabilit performane of a vehile imultaneoul. Variou performane parameter were laified into two objetive funtion related to ontrollabilit and tabilit performane. ditane funtion method wa implemented with multi-objetive optimiation. Multi-objetive optimiation wa performed b uing a geneti algorithm. When the multi-objetive optimiation onited of the performane parameter related to onl ontrollabilit or tabilit performane, variation of eah performane parameter were minimied b emphaiing the importane of eah performane parameter. It wa onluded that multi-objetive optimiation uing the ditane funtion method i ver effetive for obtaining the optimal hardpoint poition of a upenion mehanim. Keword: Controllabilit erformane, eneti lgorithm, erformane arameter, Stabilit erformane, Supenion Mehanim. 1 Introdution t the beginning of deigning a upenion mehanim, it i eential to determine the hardpoint poition through kinemati anali in order to atif the required ontrollabilit and tabilit performane of a vehile. The performane parameter uh a amber angle, toe angle, king pin angle, and ater angle are related to ontrollabilit performane, while the performane parameter of roll enter height and perent anti-dive are related to tabilit performane. ll are highl dependent on the hardpoint poition. Upon kinemati anale of a upenion mehanim, Suh [1] uggeted a nonlinear motion anali method of a upenion mehanim uing the diplaement matri. Kang [2] uggeted the ontraint equation of a pherial linder link and performed motion anali of a Mheron tpe upenion mehanim. For the optimum deign of a upenion mehanim, Simioneu [3] performed an optimum deign of a multi-link tpe upenion mehanim to minimie the hange of variou performane tpe Kim [4, 5] uggeted an approimate ompoition method of a multi-link tpe upenion mehanim uing an imaginar rew ai when a wheel i in troke and teering. lo, enitivit anali of the upenion mehanim harateriti due to hange in the hardpoint loation, kinemati anali, wa neear. Lee [6] laified the harateriti of wheel alignment into the funtion of ontrollabilit and traightne performane and performed multi-objetive optimiation uing a geneti algorithm. In thi tud, a double wihbone tpe upenion mehanim wa kinematiall analed. lo, the performane parameter oupled with eah other were laified into the performane funtion of kinemati ontrollabilit and tabilit. Multi-objetive funtion onited of the laified funtion. Sine the harateriti of the objetive funtion are nonlinear, a geneti algorithm wa ued to obtain a global olution. The poition of the hardpoint were etablihed a deign variable. The ditane funtion method wa implemented with multi-objetive optimiation in order to make the value of eah performane parameter approah the value at urb height and to minimie it variation when the wheel i in troke.

2 2 Kinemati anali of a upenion mehanim 2.1 Diplaement matri and ontraint Rigid bod motion an be epreed b the diplaement matri [ D α, β, γ ], whih onit of rotation angle and diplaement. Suh [1] performed a kinemati motion anali of a rigid bod uing the diplaement matri and ontraint. oint q on a rigid bod after rigid motion an be epreed b (1). q q1 [ Rα, β, γ ] ( p [ Rα, β, γ ] p1 ) q1 = [ Dα, β, γ ] = where, p and q are the point in a rigid bod after rigid motion, and p 1 and q 1 are the initial point in a rigid bod. [ R α, β, γ ] i the rotation matri. α, β and γ are the rotation angle with repet to,, and ae, repetivel. Contraint equation of a double wihbone tpe upenion mehanim are hown below. Sine the length of a link, whih i a omponent of tie rod, i ontant before and after moving in Figure 1, it an be written a (2). (1) ( ) + ( ) + ( ) = ( ) + ( ) + ( ) i 0 i 0 i 0 ( i = 2, 3,..., m) (2) uuuuur The length of vetor CiC0 of the upper arm hown in Figure 1 i ontant before and after it i uuuuur moved and an be epreed b (3). The vetor CiC0, whih proeed through point C 0 and i uuur perpendiular to the unit vetor, an be denoted b (4). ( C C ) + ( C C ) + ( C C ) = ( C C ) + ( C C ) + ( C C ) i 0 i 0 i 0 ( i = 2, 3,..., m) (3) uu ( Ci C0 ) + uu ( Ci C0 ) + uu ( Ci C0 ) = 0, uuur (4) ( : unit vetor of i = 2, 3,..., m) u u For the lower arm, the ame ontraint an be epreed b (5) and (6), repetivel. ( B B ) + ( B B ) + ( B B ) = ( B B ) + ( B B ) + ( B B ) i 0 i 0 i 0 ( i = 2, 3,..., m) (5) u ( B B ) + u ( B B ) + u ( B B ) = 0, l i 0 l i 0 l i 0 uuur (6) ( : unit vetor of i = 2,3,..., m) u l C 0 0 C 1 1 front diretion B 0 S 1 E 1 bump rebound B 1 Figure 1 Shemati diagram of a front wheel ued with a double wihbon upenion mehanim

3 3 Optimum deign of a upenion mehanim 3.1 Formulation The multi-objetive optimiation problem onidering kinemati ontrollabilit and tabilit performane of a vehile in thi tud an be formulated a hown in (7). Minimie F ( ) = w F ( ) + w F ( ) Subjet to g ( ) 0 i = 1,2,..., m i (7) where, F ( ) i the objetive funtion for ontrollabilit performane oniting of amber angle, toe angle, king pin angle, and ater angle. F ( ) i the objetive funtion for tabilit performane, whih inlude roll enter height and perent anti-dive. w and w are the weighting fator for the ontrollabilit and tabilit performane objetive funtion, repetivel. The ditane funtion method a an optimiation tehnique wa implemented with multi-objetive optimiation in order to make the value of eah performane parameter approah the value of urb height and minimie their hange during wheel troke. ale fator wa emploed to make eah performane parameter uniform, and then eah performane parameter wa evaluated b (8). If f ε, where, i ver loed to ero m 1/ 2 2 i i = 1 Ue f f ( ) f, m = 13 Ele, ue f m i = 1 i ( ) f f f 1/2 2 (8) where, f, f ( ) i, and f are the value, i th value, and a ale fator of eah performane parameter, repetivel. The value of eah performane parameter at urb height for a double wihbone tpe upenion mehanim are lited in Table 1 [7]. 3.2 Contraint and deign variable Etablihment of the aeptable range of eah performane parameter at urb height i ver helpful for ontrollabilit and tabilit performane during the deign of a upenion mehanim. From the reearh reult of Lee [6], and alderman [7], the aeptable range of eah performane parameter at urb height were referred. lo, ine the harmon of amber angle and toe angle when the wheel i in troke enhane traightne performane and prevention of tire wear, the tenden of toe in and poitive amber during rebound, a well a toe out and negative amber during bump, hould be required. The poition of the hardpoint attahed to the vehile bod and wheel aembl were etablihed a deign variable. The aeptable range of the deign variable are ummaried in Table Optimiation reult Optimiation reult of eah performane parameter after adjutment b objetive funtion weighting fator for kinemati ontrollabilit and tabilit performane are hown in Figure 3 and 4. The -ai denote the degree of rebound and bump, and the -ai indiate the variation of eah performane parameter during rebound and bump. The weighting fator, wi ( i = 1,2, 3, 4) for kinemati ontrollabilit performane were given a 0.25 for wi ( i = 1,2, 3, 4). The weighting fator, w j ( j = 5, 6), for kinemati tabilit performane were given a 0.5. Figure 3 how the optimiation reult for kinemati ontrollabilit performane, whih onited of amber angle, toe angle, king pin angle, and ater angle in the ae of w = 0.8, w = 0.2, and vie vera. The aeptable range of eah performane parameter related to ontrollabilit a well a ta-

4 Table 1 Idle value of eah performane parameter f Cam Toe Kin Ca Rh Fap value mm 27.9% Table 2 Range of deign variable Upper arm Lower arm Tie Rod Lower DV Upper Lower DV Upper Lower DV Upper C C C B B B Table 3 Optimum olution for w = 0.2, w = 0.8, and vie vera (unit: mm) OS C 1 C 1 w = 0.2, w = w = 0.8, w = OS w = 0.2, w = w = 0.8, w = OS w = 0.2, w = w = 0.8, w = OS: optimum olution Start 1 B 1 B 1 C 1 B 1 Input weighting fator of ontrollabilit, tabilit and eah performane parameter Input parameter of geneti algoritnm Vertial movement of wheel enter 60 mm Ei _ 60 mm, i = 1,2,...,13 Simulation Solve nonlinear equation Calulation diplaed joint oordinate Calulate performane parameter 1. amber angle 2. toe angle 3. king pin angle 4. ater angle 5. roll enter height 6. perent anti-dive erform multi o bjetive optimiation Minimie F( ) = w F ( ) + w F ( ) Optimiatoin Convergene? No Stop Ye Figure 2 Flow hart of the imulation and optimiation proe

5 Figure 3 Variation of eah performane parameter related to ontrollabilit performane Figure 4 Variation of eah performane parameter related to tabilit performane

6 bilit were atified at urb height. In the ae of the larger weight fator for ontrollabilit performane, the variation of eah performane parameter were minimied, and approahed eah point ompared with the larger weight fator for tabilit performane. Furthermore it wa hown that the variation of toe angle were ver mall and almot the ame value a the point. Figure 4 how the optimiation reult for kinemati tabilit performane that onited of roll enter height and perent anti-dive in the ae of w = 0.2, w = 0.8 and vie vera. The aeptable range of eah performane parameter related to ontrollabilit a well a tabilit performane were atified at urb height. In the ae of the larger weighting fator for tabilit performane, the variation of eah performane parameter related to it wa minimied, and approahed eah point ompared with the larger weight fator for ontrollabilit performane. The optimal olution for the two ae are ummaried in Table 3. The aeptable range of all performane parameter at urb height were atified for all ae. In the ae of the larger weighting fator for eah performane parameter related to tabilit performane, it variation were minimied, and approahed eah point ompared to the maller weighting fator for the other performane parameter. In partiular, it i hown that the variation of perent anti-dive were ver mall and remained loe to the point. 5 Conluion In thi tud, a double wihbone tpe upenion mehanim wa kinematiall analed, and the ditane funtion method wa implemented with multi-objetive optimiation. Optimal poition of the hardpoint were determined b a geneti algorithm through multi-objetive optimiation. The onluion derived from thi tud are a follow: (1) It wa verified that multi-objetive optimiation wa effetivel performed uing the ditane method in order to make the value of eah harateriti fator approah the value at urb height and minimie their variation throughout the wheel troke. (2) In the ae of the larger weighting fator for ontrollabilit performane, the variation of amber angle, toe angle, king pin angle, and ater angle were minimied. Converel, in the ae of the larger weighting fator for tabilit performane, the variation of roll enter height and perent antidive were minimied. Moreover, it wa hown that the variation of toe angle were ver mall, produing a imilar value to that of the point. (3) In the ae of the larger weighting fator for eah performane parameter related to ontrollabilit or tabilit performane, it variation were minimied and approahed eah point in omparion to the maller weighting fator for the other performane parameter. Moreover, it wa hown that the variation of the toe angle wa ver mall and remained near the point. knowledgement Thi work wa upported b the BK21 projet of the Korea Reearh Foundation. Referene [1] Suh, C.., 1989, Snthei and nali of Supenion Mehanim with Ue of Diplaement Matrie, SE paper , pp. 189~200. [2] Kang,. Y. and Suh, C.., 1994, Snthei and nali of Spherial-Clindrial (SC) Link in the Mpheron Strut Supenion Mehanim, SME J. of Mehanial Deign, Vol. 116, pp. 599~606. [3] Simioneu,.. and Beale, D., 2002, Snthei and nali of the Five-Link Rear Supenion Stem ued in utomobile, Mehanim and Mahine Theor, Vol. 32, pp. 815~232. [4] Kim, S.., Shim, J. K. and Lee, T. Y., 1999, pproimate Snthei of 5-SS Multi Link Supenion Stem Uing Intantaneou Srew i, KSME 99F173, pp. 1010~1015. [5] Kim, S.., Shim, J. K., hn. B. E. and Lee, U. K., 2001, pproimate Snthei of 5-SS Multi Link Supenion Stem for Steering Motion, KSME, Vol. 25, pp. 32~38. [6] Lee, D.., Kim, T. S. and Kim, J. J., 2000, Optimum Deign of Supenion Stem Uing eneti algorithm, Tranation of KSE, Vol. 8, pp.138~147. [7] alderman, J. D. and Mithell, Jr. C. D., 2000, utomotive Steering, Supenion, and lignment, rentie all.

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