Topological Design of Compliant Mechanisms with Multiple Materials

Transcription

1 Advancs in Enginring, volum 87 nd Intrnational Confrnc on Automation, Mchanical and Elctrical Enginring (AMEE 7) Topological Dsign of Compliant Mchanisms with Multipl Matrials Jinqing Zhan,,3, Liangming Long, Chunhui Lin and Xiangji Yang,3* Jiangi Xiongying Aluminum Co., Ltd, Nanchang 335, China School of Mchanotronics and Vhicl Enginring, East China Jiaotong nivrsity, Nanchang 333, China 3 School of Mchanical and Elctrical Enginring, Nanchang nivrsity, Nanchang 333, China * Corrsponding author Abstract In ordr to obtain bttr mchanical proprtis by ploiting th advantags of diffrnt matrials, a topological dsign mthod of compliant mchanisms multipl matrials was proposd. Th multipl matrials intrpolation modl for topological dsign of compliant mchanisms was stablishd using th gnralizd Solid Isotropic Matrial with Pnalization mthod. Th objctiv function was dvlopd maimizing th output displacmnt, and th structural volum was rstrictd. Th adjoint approach is applid to prform th snsitivity analysis. Th improvd filtr tchniqu was applid to modify th snsitivitis in ordr to avoid th avoid th numrical instability phnomnon. Th mthod of moving asymptots was usd to solv th topological optimization problm. Th numrical ampls wr givn to illustrat that th proposd mthod is corrct and ffctiv. Th rsult shows that multipl matrials compliant mchanisms has bttr mchanical prformanc than singl-phas matrial compliant mchanisms. Kywords-topology optimization; compliant multipl matrials; th improvd filtr tchniqu I. mchanisms; INTRODCTION Compliant mchanisms ar flibl monolithic structurs that can achiv forc and motion transmission through lastic dformation of rlativ flibility of its mmbrs[]. Compard with rigid-body mchanisms, compliant mchanisms hav many advantags such as high prcision, as of fabrication, asy intgration, no friction losss and without any assmbly cost. Thus, it has bn applid widly in th micro-lctromchanical systms, prcision positioning, micro/nano manufacturing, fibr alignmnt and arospac[]. Topological dsign of compliant mchanisms has drawn mor and mor attntions bcaus it only nds to dsignat a dsign domain and th positions of th inputs and outputs. In rcnt yars, th rsarch on th topology optimization of compliant mchanisms has bn gratly dvlopd[3-5]. Topological dsign of compliant mchanisms with a singl matrial has bn unabl to mt crtain rquirmnts. Howvr, multipl matrials compliant mchanisms has bttr mchanical prformanc ploiting th advantags of diffrnt matrials. Sigmund[6] proposd a topology optimization mthod to dsign lctro-thrmo-mchanical actuators using two diffrnt matrials. Yin t al.[7] usd a pak function mthod to dsign multi-matrial compliant mchanisms. Alonso t al.[8] adoptd th squntial lmnt rjction and admission mthod to produc compliant mchanism with multipl matrials. Gaynor t al.[9] proposd a topological dsign and fabrication procss for multipl matrials compliant structurs. Zhang t al.[] dvlopd a nw paralll stratgy mthod for multi-objctiv topology optimization of multipl matrials compliant mchanisms. Compard to th vast amount of litratur on a singl matrial compliant mchanisms, study on topological dsign of compliant mchanisms with multipl matrials ar rlativly fw. In this study, w proposd a topological dsign mthod of multipl matrials compliant mchanisms using th improvd filtr tchniqu. Th multipl matrials intrpolation modl for topological dsign of compliant mchanisms was stablishd using th gnralizd Solid Isotropic Matrial with Pnalization modl. Th objctiv function was dvlopd maimizing th output displacmnt, and th structural volum was rstrictd. Th adjoint approach is applid to prform th dsign snsitivity analysis. Th modifid filtr tchniqu was applid to modify th snsitivitis in ordr to avoid th avoid th phnomnon of numrical instability. Th mthod of moving asymptots was usd to solv th topological optimization problm. II. O PTIMIZATION ORMLATIONS O MLTIPLE MATERIALS COMPLIANT MECHANISMS A Multipl Matrials Intrpolation Schm Th matrial intrpolation modl for topology optimization is mainly homognization mthod[] and Solid isotropic matrial with pnalization (SIMP) mthod []. Compard with th homognization mthod, th Solid isotropic matrial with pnalization mthod has fwr dsign variabls, and is asy to calculat. In this papr, SIMP mthod is adoptd as matrial intrpolation mthod to stablish th topology optimization modl with multiphas matrial. Basd on th gnralizd SIMP intrpolation modl[], th thr-phas matrial intrpolation modl of lastic modulus for ach lmnt can b prssd as Ei, j (, ) ( ) p [( ) p E ( ( ) p ) E ] ( ( ) p ) Evoid Copyright 7, th Authors. Publishd by Atlantis Prss. This is an opn accss articl undr th CC BY-NC licns ( () 89

2 Advancs in Enginring, volum 87 Whr, th first st of dsign variabls ar usd in dtrmining th optimal topology, and th scond st of dsign variabls ar usd for slcting th matrial at ach location within th topology. E, E and Evoid ar th lasticity modulus of th solid matrial, th solid matrial and th void matrial, rspctivly. p and p ar th pnalty cofficints, and ar takn as 3. Th objctiv of th optimization problm is to maimiz th output displacmnt, and ach phas matrial volum was rstrictd is usd as th constraints. Th optimization modl can b stablishd as Maimiz : (, ) K Each phas matrial is dcidd by th combination of two st of dsign variabls. N V V (, ) V* (4) N (,) solid phas (, ) ( )V Subjct to : V (, ) V* () (,) solid phas (, ) void (,,,N) (,,,N) Th distribution of thr-phas matrial is shown in ig. Whr, rfrs to th input load vctor, K is th global stiffnss matri, N rprsnts th total numbr of finit lmnts. V* and V* ar th allowabl volum of th solid matrial and thos of th solid matrial, rspctivly. V is th solid lmnt volum. C Snsitivity Analysis and Solution Tchniqu To apply th gradint-basd algorithm, th snsitivitis of th objctiv and constraint with rspct to th dsign variabls ar nd to solvd. Th drivativ of th output displacmnt with rspct to th dsign variabls is statd as IGRE I. DISTRIBTION O EACH PHASE MATERIAL B Optimization Modl for Multipl Matrials Compliant Mchanisms Th flibility rquirmnt of compliant mchanisms is dtrmind by maimizing th output displacmnt at th output point. Th output displacmnt can b dfind using th unit dummy load mthod. A unit dummy load ar applid at th output point in th dsird dirction, as shown in ig.. Th output displacmnt can b dfind as (3) (5) Th adjoint mthod is adoptd to solv th trm in "(5)". It is obtaind form th global quilibrium quation K Thus, th trm K (6) can b calculatd whr is th nodal displacmnt vctor. (7) Considring th unit dummy load, w hav IGRE II. DESIGN DOMAIN OR MLTIPLE MATERIALS COMPLIANT MECHANISM (8) whr is th nodal displacmnt vctor undr th unit dummy load. 9

3 Advancs in Enginring, volum 87 Substituting "(7)" and "(8)" into "(5)", w finally obtain th snsitivity as T K T K ˆ out rprsnts th modifid snsitivity of th output i displacmnt with rspct to i th ( i =,) dsign variabl for (9) Similarly, w can gt Whr, () lmnt, is th snsitivity of th output displacmnt i with rspct to i th dsign variabl for lmnt f. N is th st of lmnt for which th distanc to lmnt is smallr than th filtr radius rmin, H f is th wight factor. Th wight factor is dfind as Whr H f K p ( ) p [( ) p E p ( ( ) ) E K () Evoid ] K g ( ) p p ( ) p ( E E ) K g () In "(7)", th snsitivity of th objctiv function is modifid by th wightd avrag in th sub-domain whr th cntr-tocntr distanc is smallr than th filtr radius. It can not liminat thoroughly th chckrboard pattrns. Thrfor, th dnsity filtr is adoptd to transforms th original dnsitis as follows ˆi Th snsitivity of th objctiv function is modifid as N ˆ out i H f N H f if (3) N H f H f i f N (5) Whr, th filtrd dnsity ˆi for lmnt rfrs to th physical dnsity. if is i th dsign variabl dsign variabl for lmnt f. Whn dnsity filtr is applid, th snsitivitis of th objctiv and th constraint with rspct to th dsign variabls ar obtaind by N D Th Improvd iltring Tchnology Th dsigns of multipl matrials compliant mchanisms using topology optimization typically lad to chckrboard pattrns. In this papr, th improvd filtr tchnology[5] is adoptd to modify th snsitivitis of th objctiv and th constraints for liminating th numrical instability phnomna. (4) Whr, dist (, f ) rfrs to th cntr-to-cntr distant btwn lmnt and lmnt f. Whr K is th lmnt stiffnss matri, K g rfrs to th product of th lmnt stiffnss matri and th rciprocal of th lmnt lasticity modulus. Th optimization problms ar solvd usually by two approachs: Optimality Critria (OC) mthod [3] and mathmatical programming mthod. Th OC mthod is simpl and asy to b implmntd in topological dsign. But, it is not suitabl for multi-constraind topology optimization problm. In mathmatical programming mthod, squntial conv programming approachs includ th Mthod of Moving Asymptots (MMA) [4], Squntial linar programming (SLP)[] and othrs. MMA is mor flibl to dal with multi-constraind optimization problm. MMA is adoptd to updat th dsign variabls in th optimization problm. ma(, rmin dist (, f )) i N H f H f ˆi (6) Whr rprsnts th objctiv or th constraint. Whil th filtring tchniqu is adoptd, th valu of th filtr radius has th influnc on th configuration of compliant mchanisms. In this papr, th optimization stratgy of filtr radius is usd to choos th optimal valu during th optimization procss[5].th larg valu of filtr radius is applid in th initial stag of th optimization. If th filtr radius is largr than th st valu, th valu of th filtr radius for th currnt itration stp is changd to.99 tims th valu for th last itration stp. Th valu of th filtr radius is rducd gradually up to convrgnc. 9

4 Advancs in Enginring, volum 87 III. NMERICAL EXAMPLES In this sction, w prsnt som numrical ampls of topological dsign of multipl matrials compliant grippr. Th dsign domain, boundary conditions, and th prscribd output motion ar shown in ig. 3. Th siz of th dsign domain is 8 mm by 8 mm. w us Young's Modulus for th solid matrial = MPa, Young's Modulus for th solid matrial = MPa, allowabl volum = %, allowabl volum = %, Poisson's Ratio =.3. Th output springs stiffnss is st to k out = N/mm. Applying symmtry, th lowr half of th dsign domain contains 3 lmnts. bcaus th stiff matrial is largly distributd in th rgion of small dformation, and th soft matrial is largly distributd in th rgion of larg dformation. In addition, th configuration of multipl matrials compliant grippr is similar to that of singl-phas matrial compliant grippr. It can shows that th proposd mthod is corrct. In ordr to invstigat th influnc of th ratio of th lastic modulus on th obtaind mchanical proprtis. th ratio of th lastic modulus is st to.5:, :,.5:, and 3: whil othr paramtrs kp th sam. Th Young's Modulus for th solid matrial is st to MPa whil th Young's Modulus is changd. ig.5 shows topological configuration of multipl matrials mchanism grippr for diffrnt ratio of th lastic modulus. rom Tabl, w can find that th output displacmnt of th obtaind mchanism grippr is largr by incrasing th ratio of th lastic modulus. Thus, multipl matrials compliant grippr has bttr mchanical prformanc incrasing th ratio of th lasticity modulus within a crtain rang. IGRE III. DESIGN DOMAIN O COMPLIANT GRIPPER In ordr to compar topological rsult of singl-phas matrial with that of multipl matrials, topological dsign of compliant mchanisms with singl-phas matrial is carrid out. w us Young's modulus is E = MPa, Poisson's ratio =.3, allowabl volum =3% whil othr paramtrs kp th sam. (A) (B) IGRE IV. COMPLIANT GRIPPERS OBTAINED BY DIERENT METHODS: (A) SINGLE-PHASE MATERIAL TOPOLOGY OPTIMIZATION (B) MLTIPLE MATERIALS TOPOLOGY OPTIMIZATION Singl-phas matrial and multipl matrials compliant grippr ar obtaind by topology optimization as shown in ig. 4(a) and ig. 4(b), rspctivly. In ig. 4(b), th rd ara rprsnts th stiff matrial, th blu rgion rfrs to th soft matrial, and th whit ara is void. Th output displacmnt of th optimal mchanism obtaind by singl-phas matrial is.5mm, and that of th optimal mchanism obtaind by multipl matrials is 4.mm. Th output displacmnt is incrasd by 3.4% using multipl matrials topology optimization. It shows that th multipl matrials topology optimization has bttr mchanical prformanc. That is (A) (B) (C) (D) IGRE V. COMPLIANT GRIPPERS OR DIERENT RATIO O ELASTIC MODLS: (A) E : E =.5: (B) E : E = : (C) E : E =.5: (D) E : E = 3: TABLE I. OTPT DISPLACEMENT O COMPLIANT GRIPPER OR DIERENT RATIO O ELASTIC MODLS Ratio of Elastic Modulus E : E Output Displacmnt ( m ).5: 3.6 : 4..5: 6. 3:.9 9

5 Advancs in Enginring, volum 87 IV. CONCLSIONS A nw mthod for topology optimization of multipl matrials compliant mchanisms by th improvd filtr tchniqu is prsntd. Th topology optimization modl of multipl matrials compliant mchanisms is stablishd using th gnralizd Solid Isotropic Matrial with Pnalization modl. Th improvd filtr tchniqu is applid to modify th snsitivitis of th objctiv and th constraint. Compard with singl-phas matrial topology optimization, th multipl matrials compliant mchanisms has bttr mchanical prformanc bcaus th stiff matrial is largly distributd in th rgion of small dformation and th soft matrial is largly distributd in th rgion of larg dformation. In addition, multipl matrials compliant grippr has bttr mchanical prformanc incrasing th ratio of th lasticity modulus within a crtain rang. [] [3] [4] [5] Intrnational Journal for Numrical Mthods in Enginring, vol. 4, pp , Jun 998 M. P. Bndso, O. Sigmund, Matrial intrpolation schms in topology optimization, Archiv of Mchanics, vol. 69, pp , May 999 M. Zhou, G. Rozvany, Th COC algorithm, part II: topological, gomtry and gnralizd shap optimization, Computr Mthods in Applid Mchanics & Enginring, vol. 89, pp , August 99 K. Svanbrg, Th mthod of moving asymptots: a nw mthod for structural optimization, Intrnational Journal for Numrical Mthods in Enginring, vol. 4, pp , bruary 987 K. Long, Z. X. Zuo, Topology Optimization for Compliant Mchanism sing Matrial Point Topological Variabl, Journal of Computr-Aidd Dsign & Computr Graphics, vol., pp , January. ACKNOWLEDGMENT This rsarch was supportd by th National Natural Scinc oundation of China (53536), th Natural Scinc oundation of Jiangi, China (5BAB63, 6BAB65) and th Scinc and Tchnology Plan Projcts of Jiangi Provincial Education Dpartmnt (GJJ5547). This supports ar gratly acknowldgd. REERENCES [] L.L.Howll, Compliant mchanisms, John Wily & Sons: Nw York,, [] O. Sigmund, On th dsign of compliant mchanisms using topology optimization, Mchanics of Structurs and Machins, vol. 5, pp , April 997. [3] J.Z. Lin, Z. Luo, L. Y. Tong, A nw multi-objctiv programming schm for topology optimization of compliant mchanisms, Structural Multidisciplinary Optimization, vol. 4, pp. 4-55, January. [4] Y. X. Du, Z. Luo, Q. H. Tian, t al. Topology optimization for thrmomchanical compliant actuators using msh-fr mthods, Enginring Optimization, vol. 4, pp , July 9. [5] L. Mllo, R. Salas, E. Silva. On rspons tim rduction of lctrothrmomchanical MEMS using topology optimization, Computr Mthods in Applid Mchanics and Enginring, vol , pp. 93-, Novmbr. [6] O.Sigmund, Dsign of multiphysics actuators using topology optimization part II: two-matrial structurs, Computr Mthods in Applid Mchanics and Enginring, vol. 9, pp , Octobr. [7] L. Yin, G. K.Ananthasursh, Topology optimization of compliant mchanisms with multipl matrials using a pak function matrial intrpolation schm, Structural & Multidisciplinary Optimization, vol. 3, pp. 49-6, Dcmbr. [8] C. Alonso, R. Ansola, O. M. Qurin, Topology synthsis of multimatrial compliant mchanisms with a Squntial Elmnt Rjction and Admission mthod, init Elmnts in Analysis & Dsign, vol. 85, pp. 9, August 4. [9] A. T. Gaynor, N. A. Misl, C. B. Williams, t al. Multipl-matrial topology optimization of compliant mchanisms cratd via PolyJt thr-dimnsional printing, Journal of Manufacturing Scinc and Enginring, vol. 36, pp. 65(-), Octobr 4. [] X. M. Zhang, K. Hu, N.. Wang, t al. Multi-objctiv topology optimization of multipl matrials compliant mchanisms basd on paralll stratgy, Journal of Mchanical Enginring, vol. 5, pp. -8, Octobr 6.(In Chins) [] S. Nishiwaki, M. I. rchr, S. MIN, N. Kikichi, Topology optimization of compliant mchanisms using th homognization mthod, 93