Shape Optimization of Flexible Fixed Beam

Transcription

1 ISS (Onlin) : ISS (Print) : Intrnational Journal of Innoati Rsarc in Scinc, Enginring and Tcnology An ISO 97: 007 Crtifid Organization Volum, Spcial Issu 4, Marc 014 ational Confrnc on Rcnt Adancs in Ciil Enginring (CRACE-01) During ombr, 01 Organizd by Dpartmnt of Ciil Enginring, ort Eastrn Rgional Institut of Scinc and Tcnology, irjuli, Itanagar, Arunacal Prads, India. Sap Optimization of Flxibl Fixd Bam K.S. Sing 1, M.K. Miti and S. Mato B. Tc. Studnt, ME Dpt., ERIST, irjuli, Arunacal Prads, India 1, Assoc. Prof., ME Dpt., ERIST, irjuli, Arunacal Prads, India Abstract In tis work, sap optimization is carrid out of a flxibl fixd bam. Bam is considrd undr Eulr-Brnoulli tory and finit lmnt formulation is don for its dynamics analysis using wmark s scm. Squntial quadratic programming (SQP) mtod is usd to optimiz t orall prformanc of t bam. Optimizd fixd bam may b prfrrd in t ral world applications for spcific prformanc rquirmnt. Kywords Eulr-Brnoulli bam tory, flxibl fixd bam, finit lmnt mtod, sap optimization, squntial quadratic programming I. ITRODUCTIO Conntional bams ar comprisd of ig rigidity. Du to ig rigidity, bams ar gnrally ay and bulky. Flxibl bams a bn a topic of instigation in sral filds. Flxibility du to ligt wigt of bams a sral adantags (lik lss matrials, transportabl, tc). Howr, tr ar crtain disadantags associatd wit flxibl bams,.g. ibration du to low stiffnss. Static dflction and ibration ar t callnging tasks for flxibl bam applications. To rtain t adantags of flxibl bam, it nds its optimal dsign. Most of t rsarcrs optimizd t fundamntal frquncy of t cantilr bam or manipulator. Cranc and Adlr [1] prsntd t closd-form solutions in trm of Bssl s functions for t natural frquncis. Unconstraint non-uniform bams wit four kinds of rctangular cross-sctions is considrd for mod saps analysis. Hidbrct [] dtrmind t approximat natural frquncis and mod saps of a non-uniform simply supportd bam from frquncy quation. Baily [] sold t frquncy quation drid from Hamilton s principl to obtain natural frquncis of t non-uniform cantilr bams. Elwany and Barr [4] prsntd work wic maximizs t fundamntal frquncy for a gin bam wigt or quialnt bam wigt minimization for a spcific alu of fundamntal frquncy. Oloff and Prbry [5] dtrmind t optimal dsign of a transrsly ibrating tin lastic bam using cross-sctional ara function as t dsign ariabl tat maximizs t diffrnc btwn two adjacnt natural frquncis. Gupta and Murty [6] studid t optimal dsign of uniform non-omognous bams undr transrs ibration. Optimum tapring of cantilr bam carrying tip mass is dtrmind by Karialoo and iordson [6] to maximiz fundamntal frquncy. Lio [7] dlopd a gnralizd mtod for t dsign of a cantilr bam of circular cross-sction in flxural ibration. T bam is composd of two matrials along t lngt. Wang [9] addrssd optimum dsign of a singl link manipulator to maximiz its fundamntal frquncy. H formulatd t dsign problm as a nonlinar ignalu problm using ariational mtod. H dmonstratd t incras of fundamntal frquncy as a rsult of optimization. Wang and Mirowitc [10] xtndd t work of Karialoo and iordson [7] to find substantial impromnt in optimum sap troug simplifying original analysis. Wang and Russl [11] proposd minimax dsign mtod to construct t optimum sap undr a finit rang of tip loads. Xu and Anantasurs [1] mployd squntial quadratic programming (SQP) mtod aailabl in MATLAB for sap optimization of sgmnt of compliant mcanism. Gunjal and Dixit [1] addrssd t sap optimization of a rotating bam at diffrnt spds wit constraints on its mass and static tip dflction. Ty studid natural Copyrigt to IJIRSET 7

2 ISS (Onlin) : ISS (Print) : Intrnational Journal of Innoati Rsarc in Scinc, Enginring and Tcnology An ISO 97: 007 Crtifid Organization Volum, Spcial Issu 4, Marc 014 ational Confrnc on Rcnt Adancs in Ciil Enginring (CRACE-01) During ombr, 01 Organizd by Dpartmnt of Ciil Enginring, ort Eastrn Rgional Institut of Scinc and Tcnology, irjuli, Itanagar, Arunacal Prads, India. frquncis and dynamic rspons of t optimizd bam. Dixit t al. [14] prsntd FEM modl of singl link flxibl robotic manipulator for rolut and prismatic joint. Ty usd SQP for optimizing bam saps undr diffrnt optimization conditions and compard its dynamic rsponss and fundamntal frquncis. From t abo study, it is obsrd tat most of t rsarcrs contributd sap optimization for cantilr bam and rotating cantilr bam to impro crtain objctis. Tr is no muc rsarc contribution in sap optimization of flxibl fixd bam. In tis work, autors considrd tr diffrnt optimization problms for sap optimization for comparati study. II. MODELLIG AD SOLUTIO TECHIQUE Flxibl bams a significant transrs dflctions. Ty ba as a nonlinar lastic bams and xibit ibratory motions in transrs dirction. Formulations ar linarizd for small transrs dflction du to bnding motion undr linar bam tory as a two-dimnsional idalization. Tis simplifid modl is not suitabl for modling t dynamic baior of flxibl bam wit larg dflctions. T finit lmnt formulation as bn dscribd in Dixit [14]. It is dscribd r for t sak of compltnss. Figur 1(a) sows flxibl fixd bam in wic XOV rprsnts t stationary co-ordinat fram. F rprsnts t applid forc at t mid-position of a bam, q rprsnts t loading intnsity (load pr unit lngt) in t transrs plan and E, I, L, ρ, A and M rprsnts t Young s modulus, ara momnt of inrtia, lngt, mass dnsity, cross-sctional ara and payload mass (at t cntr) rspctily. Motion of t manipulator is rprsntd by fixd XOV co-ordinat fram. Bam is considrd slndr. So, transrs sar and rotary inrtia ffcts ar nglctd allowing it to b tratd as an Eulr- Brnoulli bam. Bam is assumd to ibrat dominantly in rtical plan (XOV), nglcting graity ffcts. Fig. 1 (a) Configuration of flxibl manipulator, (b) typical finit lmnt wit four dof Considr an infinitsimal link lmnt P on t manipulator at a distanc x from t fixd nd (lft sid). Position of t lmnt P wit rspct to inrtial fram of rfrnc (XOV) aftr tim ' t ' and transrs dflction ( x, t ) is gin by t position ctor P( x, ) wit rspct to t fixd fram. From basic mcanics, quation of motion of t flxibl bam may b writtn as EI m q 0 (1) x x t wr m (mass pr unit lngt) and I ar function of x and transrs load is t function of bot x and t. T following gomtry boundary conditions act at t fixd nd sids: ( x 0, t) 0 & 0 x x 0 and ( x L, t) 0 & 0. () x x L In t FEM formulation t bam is diidd into lmnts, ac lmnt aing four dgrs of frdom as sown in Figur 1(b). In t figur, 1,,, 4 ar t transrs dflction and slops at t first and scond nods of t lmnt. Transrs dflction is xprssd by t approximat function insid t lmnt at point P. Tn rsidual of Equation 6 is gin by R EI m q (4) x x t Bam transrs dflction is approximatd in finit lmnt as () Copyrigt to IJIRSET 8

3 ISS (Onlin) : ISS (Print) : Intrnational Journal of Innoati Rsarc in Scinc, Enginring and Tcnology An ISO 97: 007 Crtifid Organization Volum, Spcial Issu 4, Marc 014 ational Confrnc on Rcnt Adancs in Ciil Enginring (CRACE-01) During ombr, 01 Organizd by Dpartmnt of Ciil Enginring, ort Eastrn Rgional Institut of Scinc and Tcnology, irjuli, Itanagar, Arunacal Prads, India (5) wr x is t local coordinat, t lmnt lngt and 1,, and 4 ar known as t Hrmitian sap functions and Galrkin s wigt function W is approximatd in t sam fasion as is dfind as W 1 W 1 W W W 4. 4 (6) Using Galrkin s FEM approac, wak form of t diffrntial quation for an lmnt is gin by W R dx 0. (7) 0 W W W EI EI EI d x x x x x x x mw dx Wqdx 0. (8) 0 t 0 Using Equation 5 and 6, t Equation 8 bcoms 1 1 EI 1 4 dx m 1 4dx q dx Intrnal forc ctor. (9) 0 4 Effct payload mass is incorporatd in t global mass matrix and stiffnss matrix using Dirac-dlta function as dscribd by Dixit t al. [14]. Payload mass is dfind μ (ratio of tip mass to bam mass). Equation 9 can b xprssd in matrix form: M K F. (10) wr [ M ], [ K ] andf ar t lmnt mass matrix, stiffnss matrix and lmnt load ctor. Structural damping of t bam is not considrd for numrical study. Aftr assmbling lmnt quations, t global systm gorning quation can b xprssd as [ M ]{ V } [ K ]{ V} { F}, (11) wr [M] and [K] ar t global mass and stiffnss matrics rspctily. Global load ctor {F} and global nodal displacmnt ctor {X} ar gin by F F F F F F 1 m n1 n (1) and T V 1... n-1, n (1) wr n is numbr of nods takn (1) and m is midposition nod no (11). glcting load ctor, Equation 11 bcoms standard ignalu problm, wic is sold to obtain natural frquncis of t systm. wmark mtod is usd to sol t Equation 11 to prdict t dynamic baiour of t bam. T wmark intgration scm is basically t xtnsion of t linar acclration mtod. It is a constant arag acclration scm. Using wmark s mtod, transrs dflction ( ), slop ( ) and its driati ar obtaind. III. OPTIMIZATIO PROCEDURE Tr optimization problms ar considrd for t comparison of static and dynamic baior of t flxibl fixd bam systm. Minimization of maximum dynamic tip dflction is considrd as an objcti for ig spd opration of t robotic systm. Minimization of mass of t uniform bam manipulator is kpt constraints. Gnral form of an optimization problm is xprssd as Optimization Problm /Bam rfrrd TABLE 1 DIFFERET OPTIMIZATIO PROBLEMS Objcti T Constraints Copyrigt to IJIRSET 9

4 ISS (Onlin) : ISS (Print) : Intrnational Journal of Innoati Rsarc in Scinc, Enginring and Tcnology An ISO 97: 007 Crtifid Organization Volum, Spcial Issu 4, Marc 014 ational Confrnc on Rcnt Adancs in Ciil Enginring (CRACE-01) During ombr, 01 Organizd by Dpartmnt of Ciil Enginring, ort Eastrn Rgional Institut of Scinc and Tcnology, irjuli, Itanagar, Arunacal Prads, India. Minimization of I (Bam-I) static tip dflction Maximization of II (Bam-II) fundamntal bam frquncy Minimization of III (Bam-III) maximum dynamic tip dflction LB UB Prmissibl Bound : X X X wr... T X b 1 b b n M M 0 M M 0 M M 0 is a dsign ctor wit b i indicating widt of t i t finit lmnt, f(x) indicats L t diffrnt objcti function. Lowr bounds ( X ) and U uppr bound ( X ) ar t ctors of dsign ariabls rspctily. M is t mass of t optimizd manipulator, M is t prscribd mass of t uniform. T MATLAB function fmincon uss squntial quadratic programming (SQP) tcniqu for constraind optimization of nonlinar function. III. RESULTS AD DISCUSSIO A comparati dynamic analysis as bn carrid out for sap optimizd flxibl fixd bam. For t numrical study, a bam aing uniform widt 10 mm, ticknss 4 mm, lngt 1750 mm, mass gm, Young s modulus of lasticity 71 GPa is considrd. Optimizd bams ar subjctd to a sinusoidal forc of amplitud 0.5 -m (Figur d) at t mid-position of t bam. Fig. Excitation Forc (a) Bang-bang, (b) Triangular, (c) Trapzoidal and (d) Sinusoidal Fig. Optimizd saps for diffrnt payloads mass ( ) (a) Bam-I, (b) Bam-II, (c) Bam-III Fig. 4 All optimizd saps for payloads (µ=0) Optimizd saps of t fixd bam as pr t optimization problms dfind in Equation 11 ar plottd in Figur. Tr ar diffrnt optimal saps for diffrnt optimization problms. In optimization problm-i and II, tr is almost no ariations of optimal saps for diffrnt payload masss, owr tr is for optimization problm-ii. Ts saps gi t optimal prformanc for tat particular objcti. Diffrnt optimizd saps undr diffrnt optimization problms for payload mass ( 0) ar plottd in Fig. 4 for comparati obsration. Copyrigt to IJIRSET 10

5 ISS (Onlin) : ISS (Print) : Intrnational Journal of Innoati Rsarc in Scinc, Enginring and Tcnology An ISO 97: 007 Crtifid Organization Volum, Spcial Issu 4, Marc 014 ational Confrnc on Rcnt Adancs in Ciil Enginring (CRACE-01) During ombr, 01 Organizd by Dpartmnt of Ciil Enginring, ort Eastrn Rgional Institut of Scinc and Tcnology, irjuli, Itanagar, Arunacal Prads, India. Fig. 5 Static bam dflction optimizd at µ=0 du to 1 forc at cntr For comparati static bam dflctions, 1 static load is considrd at t mid-position of t bam. Static bam dflctions of uniform bam and optimizd bams ar plottd in Fig. 5. Optimizd Bam-I and III ar dflctd lssr static bam dflction tan tat of uniform bam. As xpctd, Bam-I is last dflctd. Bam-II is not improd for static bam dflction. Fig. 7 Static bam dflction optimizd at µ=0 du to 1 forc at cntr Dynamic bam dflctions du to sinusoidal forc (0.5 sin 4 t) ar sown in Fig. 7. All optimizd bams (at 0) a minimum bam dflction tan t uniform bam dflction for no payload mass cass. Bam-III is optimizd for minimization of maximum dynamic bam dflction. Hnc, it supprss t ibration maximum Dynamic bam dflctions of Bam-III (at 0) ar also plottd in Fig. 8. It is obsrd sam trnds of baiour of t optimizd bams undr diffrnt four diffrnt forc xcitations as sown in Fig. ar plottd in Fig. 7. Troug numrical xprimnts, it also obsrd tat Bam-III optimizd at igr payload mass always gi minimum of maximum dynamic bam dflction wit rspct to uniform fixd bam for lowr payload mass cass but rrs is not tru. Fig. 6 Static bam dflction optimizd at µ=0 du to 1 forc at cntr atural frquncis ar important caractristics of any structural systm. Fundamntal frquncis of uniform bam and optimiz bams ar plottd in Fig. 6. All t optimizd bams a igr fundamntal frquncy tan tat of uniform bam. Howr, optimizd Bam-II nancd t fundamntal frquncy of uniform bam maximum. Copyrigt to IJIRSET 11

6 ISS (Onlin) : ISS (Print) : Intrnational Journal of Innoati Rsarc in Scinc, Enginring and Tcnology An ISO 97: 007 Crtifid Organization Volum, Spcial Issu 4, Marc 014 ational Confrnc on Rcnt Adancs in Ciil Enginring (CRACE-01) During ombr, 01 Organizd by Dpartmnt of Ciil Enginring, ort Eastrn Rgional Institut of Scinc and Tcnology, irjuli, Itanagar, Arunacal Prads, India. Fig. 8 Comparison of dynamic tip dflction du to bang- input torqu, bam optimizd at (a) µ=0, (b) µ=0.1, (c) µ=0. & (d) µ=0.5 IV. COCLUSIO In tis work, sap optimization of flxibl fixd bam is carrid out troug linar modling. Toroug FE analysis as bn conductd and succssi SQP itration scm as bn usd to sol constraind optimum sap of t flxibl fixd bam to optimiz its static/dynamic prformancs. Tr ar diffrnt optimal saps for diffrnt optimization problms. Ts optimum saps gi optimizd orall prformanc for tat particular payload. Bam optimizd for igr payload always gis minimum of maximum dynamic bam dflction wit rspct to tat of uniform bam manipulator for rang of lowr payloads but rrs is not ncssarily tru. [10] F.Y. Wang and L. Miroitc, Optimum Dsign of Vibrating Cantilrs: A classical problm. Risitd, Journal of Optimization Tory and Applications, ol. 84(), pp , [11] F.Y. Wang, and J.L. Russl, Optimum Sap Construction of Flxibl Manipulators wit Total Wigt Constraint, Systms, Man and Cybrntics, ol. 5(4), pp , [1] D. Xu, and G.K. Anantasurs, Frform Skltal Sap Optimization of Compliant Mcanism, Journal of Mcanical Dsign, ol. 15, pp. 5 61, 00. [1] S.K. Gunjal, and U.S. Dixit, Vibration Analysis of Sap Optimizd Rotating Cantilr Bam, Enginring Optimization, ol. 9(1), pp , 007. [14] U.S. Dixit, R. Kumar and S.K. Dwidy, Sap Optimization of Flxibl Robotic Manipulator, ASME Journal of Mcanical Dsign, ol. 11(), pp , 006. REFERECES [1] E.T. Cranc and A.A. Adlr, Bnding ibration of ariabl crosssction bams, J. of Applid Mcanics, ASME, ol.(1), pp , [] A.C. Hidbct, Vibration of non-uniform simply supportd bams, J. of Enginring Mcanics Diision, procding of t ASCE, ol. 9(EM), pp. 1-15, [] C.D. Baily, Dirct analytical solution to non-uniform bam problm, J. of Sound of Vibration, ol. 56(4), pp , [4] M.H.S. Elwany and Barr, A.D.S. Barr, "Optimal dsign of bams undr flxural ibration", J. of Sound and Vibration, 88(), pp , 198. [5]. Oloff and R. Parbry, "Dsigning ibrating bams and rotating safts for maximum diffrnc btwn adjacnt natural frquncis", J of Solid Structur, 0(1), pp. 6-75, [6] V.K. Gupta and P.. Murty, "Optimal dsign of uniform nonomognous ibrating bams", J. of Sound and Vibration, 59(4), pp , [7] B.L Karialoo and F.I. iordson, Optimum dsign of ibrating cantilr, Journal of Optimization, Tory and Applications, ol. 11(6), pp , 197. [8] Y.S. Lio, A gnralizd mtod for t Optimal Dsign of Bams undr Flxural Vibration, Journal of Sound and Vibration, ol. 167(), pp. 19 0, 199. [9] Fi-Yu Wang, On t Extrnal Fundamntal Frquncis of on Link Flxibl Manipulators, T Intrnational Journal of Robotics Rsarc, ol. 1, pp , Copyrigt to IJIRSET 1