International Journal of Advance Engineering and Research Development

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1 Scetfc Joural of mpact Factor (SJF): 4.4 teratoal Joural of Advace Egeerg ad Research Developmet Volume, ssue, March -0 Computer aded desg of lks plaar mechasms Kalash Chaudhary Departmet of Mechacal Egeerg Raj Egeerg College Jodhpur, da Dr. Hmashu Chaudhary Departmet of Mechacal Egeerg Malavya Natoal sttute of Techology Japur Japur, da e-ssn (O): p-ssn (P): Abstract The lk shapes satsfyg kematc ad dyamc requremets are very crucal for the desg of a mechasm ad ts performace. The shape sythess usg parametrc curves lke Hermte, Bezer ad B-sple curves leads to computer-aded desg (CAD) ad maufacturg of the mechasm lks. Through CAD modelg of the lks usg these curves; the desg, producto ad fuctoal detals ca be easly trasmtted betwee egeerg ad maufacturg operatos. The CAD modelg of the lks s also useful aalyzg the statc ad dyamc respose of the desged mechasm. The real-tme behavor of the mechasm s evaluated through computer smulato ad thus t elmates the eed of the expermetal tests for the actual mechasm. Therefore, the cost ad tme are saved to a great extet ad ay possble error s realzed before maufacturg of the mechasm lks. Key words Computer aded desg, Plaar mechasm, Lk shape, Parametrc curves, Gree s theorem, Optmzato NTRODUCTON The lks shapes for plaar mechasms are to be decded to carry varous loads actg o them. Several methods are suggested the lterature to fd the mechasm lk shapes for specfed ertal propertes. A Small Elemet Superposg Method (SESM) [] s developed to fd lk shapes whch dscretze the tal assumed shape to small mass elemets ad locate them systematcally alog the lk legth. the covex optmzato method [], couterweght desg s formulated as a covex olear optmzato problem. The Evolutoary Structural Optmzato (ESO) method [] s used to optmze the rotatg machery shaft s shape by gradually removg effectve materal from the desg doma. The lk shapes are sytheszed by maxmzg exteral work doe by a gve exteral force cosderg total volume of all lks as the costrat fucto [4]. The lk shapes are also foud through the topology optmzato based o parametrc curves [5] ad o-tersectg closed polygos []. The lk shapes for the terferece-free moto are foud by detfyg feasble materal doma assocated wth the lk geometres [7]. Smlarly, the mechasm s dmesoal sythess to geerate specfed path or moto based o graphcal ad aalytcal techques ca also be used for shape optmzato [8]. The lmtato of these methods for fdg lk shapes s that they requre a pre-defed desg doma to start wth. ths paper, the lks shapes are sytheszed through optmzato usg closed parametrc curve. The cotrol pots of cubc B-sple curve are take as the desg varables for lk shape formato cosderg proper costrats. The closed parametrc curve s used to represet the lk shape ad ts geometrc ad ertal propertes are calculated usg Gree s theorem. The proposed optmzato problem cludes the equalty costrats to keep the resultg ertal propertes same as the desred ertal propertes of the plaar mechasm. The proposed method of shape sythess ca be appled to ay plaar sgle ad multloop mechasm wth revolute as well as prsmatc jots. ts effectveess s demostrated by applyg t to plaar four-bar mechasm. LNK SHAPE The lk shape s represeted by the parametrc curve,.e., closed cubc B-sple curve as show Fg.. f the curve terpolates or approxmates a set of + cotrol pots, P 0, P,, P [9] the the posto of ay pot o the curve s defed as: P( u) PN ( u), 0 u u 0, k All rghts Reserved 9

2 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: P P P 0, P P - Fg.. Closed cubc B-sple curve ad ts cotrol pots For a curve of degree (k-), the B-sple fucto N, k ( u) s computed teratvely as: N, k- ( u) N, k- ( u) N, k ( u) ( u - u ) ( uk - u) () uk- - u uk - u where, u u u N, () 0, otherwse Eq. (), N,s a ut step fucto ad u are kow as parametrc kots or kot values. These values form a sequece of odecreasg tegers called the kot vector. The parametrc equato of th curve segmet of a cubc B-sple curve havg cotrol pots P -, P, P + ad P + for u u -, u s gve as: αp - 4 ( ) α P α P α P P u (4) where α -u u - u (5) u u ( - 9) u(- 9 - ) -u u (- 9) u( 9 ) 4 u u ( ) u( ) α () α 4 (7) α 4 (8) The cotrol pots form the vertces of the characterstc polygo of the B-sple curve as show Fg.. The cubc B-sple curve s a composte sequece of curve segmets coected wth C cotuty whch bleds two curve segmets wth same curvature. The coordates of ay pot o the th segmet of the curve are gve by Eq. (4) as: αx - αx αx α4x x ( u) (9) α y- α y α y α4 y y ( u) (0) where the terms α, α, α ad α 4 are defed Eqs. (5) (8), ad (x -, y - ), (x, y ), etc. are the coordates of pots P -, P, etc. respectvely. The mass ad erta of the lk that s sytheszed usg closed cubc B-sple curve ca be calculated usg Gree s theorem [0]. For two fuctos P(x, y) ad Q(x, y) over a closed rego D the plae wth boudary D, Gree's theorem presets: The area of closed rego D s calculated as: Ths area s calculated usg Gree's theorem by takg Px, y 0 ad Qx, y For a plae curve specfed parametrcally as ( ) ( ) x u, y u for u u, 0 u D Q P dxdy x y D D Pdx Qdy () A dxdy () x that gves: Q P A dxdy x y () D A xdy (4) D, Eq. (4) All rghts Reserved 70

3 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: Smlarly, the momet about x-axs ad y-axs of plae are computed as: usg P - y / ad Q 0 A u u0 xydu (5) u Mx ydxdy y dx y xdu u () 0 usg P 0 ad Q x / u M y xdxdy x dy x ydu u (7) 0 The geometrc cetrod ( x, y) of plae curve s gve by x M / y A ad y M x/ A. Fally, the area momets of erta ca be computed as: usg P y / ad Q 0 usg P 0 ad Q x / y dxdy y dx u xx u 0 y xdu u yy x dxdy x dy x ydu (9) u 0 The area, cetrod ad area momet of erta about cetrodal axes [ xx, yy, zz ] of the closed curve made of cubc B-sple segmets are calculated as: A u u - ' (8) x ( u) y ( u) du (0) x x ( u) y ( u ) du u ' A u- u ' A u- y y ( u) x ( u ) du xx yy u y u- u x u- zz ' ( u) x ( u) du ' ( u) y ( u) du xx yy () () () (4) (5) ' ' The frst dervatves x ( u) ad y ( u) of x ( u) ad y ( u) wth respect to u, respectvely, Eqs. (0) (4) are gve by: ' - ( ) β x β x x u β x β 4 x () ' - ( ) y u β y β y β y β 4 y (7) where β -u u - (8) β 9 u u (-9 ) (9) β -9 u u (- 9 ) 9 (0) β 4 u u ( ) () For geometrc propertes defed Eqs. (0) (5), the mass ad mass momet of erta of a lk wth shape represeted by the closed curve are calculated All rghts Reserved 7

4 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: wheret ad ρ represet thckess ad materal desty for the lk, respectvely. FORMULATON OF OPTMZATON PROBLEM m Atρ () tρ () ths secto, a optmzato problem s formulated to fd the optmum lk shapes correspodg to the specfed ertal parameters of the plaar mechasm. To formulate the optmzato problem, the Cartesa coordates of cotrol pots of cubc B- sple curve are take as desg varables as show Fg.. zz Fg.. Closed cubc B-sple curve represetg lk shape ad ts cotrol pots where P ad P j are two opposte pots about x- axs The lk legth,a, betwee jot orgs O to O + s dvded to equal parts. Hece, the x-coordates of the cotrol pots lyg betwee O ad O + are fxed accordg to the lk legth. Now, the y-coordates are take as the desg varables. Furthermore, the exteso of lk beyod O ad O + s cotrolled by pots P 0, P, P - at oe ed ad by pots P /-, P /, P /+ at other ed. Hece, x-coordate of P 0, y-coordates of P ad P - are chose as the desg varables at the rght ed ad same s doe at left ed. Fally, the desg vector s proposed as: T [ x0 y... y/- x/ y/... y-] x (4) The codtos for symmetrcal ad o-symmetrcal shapes are mposed by cotrollg coordates of the opposte pots as yj y ad y y, respectvely. addto to maufacturg beefts, the symmetrcal shapes have zero products of erta. j The ertal propertes of resultg shapes are costraed by the desred ertal propertes. These costrats esure that the lks wth optmum shapes have the same ertal propertes as that of the desred propertes of the plaar mechasm. The objectve fucto s formulated to mmze the percetage error resultg lks erta values as: y y j (for symmetrcal lk); y y (for o-symmetrcal lk) j m ; ( ) Mmze Z 00 Subject to m (5) x x ; y y for =,,, () Here parameters wth superscrpt represet desred parameters of plaar mechasm to be desged ad subscrpt s used for th lk of mechasm. The teachg-learg-based optmzato (TLBO) algorthm s used to solve ths optmzato problem. t s advatageous to use TLBO as compared to the other evolutoary optmzato algorthms, as () t does t requre ay algorthm specfc parameters to be defed to start the optmzato procedure ad () t coverges to the optmum soluto faster tha other evolutoary optmzato algorthms. Also, the tal values of the desg varables are ot requred to start searchg the optmum soluto ad hece o tal shape s requred.the thckess of mechasm lks s take as 0 percet of the drvg lk legth ad the lk materal s chose as the mld steel (desty = 7850 kg/m ) for decdg the desty ad maxmum permssble stress. Furthermore, the thckess of the lk s take uform ormal to the plae of moto ad ca be dfferet for dfferet lk the mechasm cosdered. The stress at the weakest secto each lk s calculated for the maxmum jot force occurred durg the complete cycle of operato. Moreover, the vo mses stresses for the peak load s cosdered to determe mmum cross-secto of each lk. The ertal propertes of lks are calculated usg Eqs. () () ad verfed by CAD models developed usg Autodesk vetor software. The flow chart show Fg. llustrates the optmzato method proposed for the optmum desg of the plaar mechasms. DESGN OF PLANAR All rghts Reserved 7

5 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: ths secto, the effectveess of the proposed optmzato method for lk shape sythess s show by applyg t to a plaar four-bar mechasm. Based o the desg varables defed Fg. 4, total 8 desg varables, amely, x 0, x, y y are ow cosdered for the optmum lk shape sythess. Desg varables: Coordates of cotrol pots of cubc B-sple curve T x x y... y x y... y [ 0 /- / / - ] Objectve fucto ad costrats: ( ) Mmze Z(x) x00 Subject to m m ; x x ; y y For =,,, Soluto usg TLBO MATLAB Optmzed values of cotrol pots for cubc B-sple curve Optmum lk shapes for mechasm Fg.. Optmzato scheme for shape sythess of mechasm lks Here, a represets the lk legth betwee jots O ad O +. The desg varables x 0 ad x are represetg lk legths beyod the jots O + ad O, respectvely. Fg. 4. Desg varables to fd optmum lk shape of plaar mechasms The legths a, x 0 ad x are dvded each to equal parts whch decde the x-coordates of cotrol pots. So, these x- coordates are gve as follows: x = a + x 0 ; x = a x 0 ; x = a x 0 ; x 4 = a + 0.5x 0 ; x 5 = a ;x = 0.75a ; x 7 = 0.50a ; x 8 = 0.5a ; x 9 =0; x 0 = -0.5x ; x =-0.50x ; x = -0.75x ; x = -x ; x 4 = -x ; x 5 = -0.75x ; x = -0.50x ; x 7 = -0.5x ; x 8 = 0; x 9 = 0.5a ; x 0 = 0.50a ; x = 0.75a ; x = a ; x = a + 0.5x 0 ; x 4 = a x 0 ; x 5 = a x 0 ; x = a + x 0 Moreover, the symmetrcal lk shapes ca be obtaed by cotrollg the y-coordates as: y 4 = -y y = All rghts Reserved 7

6 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: y 5 = -y y = -y 5 y = -y y = -y 4 y 7 = -y 0 y 4 = -y y 8 = -y 9 y 5 = -y y 9 = -y 8 y = -y y 0 = -y 7 Note that legths x 0 ad x are varables whle a s the legth of the th lk. The optmzato problem of lk shape sythess for a dyamcally balaced plaar four-bar mechasm (Fg. 5) [] s solved ad the resultg lk shapes are show Fg.. Y C m θ # a C # C d m # d θ a r #0 ' θ a m O 4 4 θ d a 0 X O Fg. 5. A plaar four-bar mechasm DESRED PARAMETERS OF PLANAR FOUR-BAR MECHANSM Lk Mass m (kg) Momet of erta (kg-m ) c d (m) θ (deg) Fg.. Orgal ad optmally desged plaar four-bar mechasm The values of the desg varables, defed Fg., for the optmally desged four-bar mechasm are gve Table whle the CAD model of the mechasm s show Fg. All rghts Reserved 74

7 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: Fg. 7. CAD model of optmally desged four-bar mechasm CONCLUSONS The physcally possble shapes are costructed for the desred ertal parameters of the mechasm lks ad the gve kematc structure. The percetage error of resultg lk erta values defed as the objectve fucto was foud wth ± 5 percet. The beeft assocated wth the proposed method s that the lks of the mechasm are of the uform thckess. The resultg stresses for lks of the balaced mechasm ca be calculated at the weakest sectos uder exteral loads. Lk DV DESGN VARABLES FOR OPTMALLY DESGNED FOUR-BAR MECHANSM (ALL PARAMETERS ARE N METERS) y y y y 4 y 5 y y 7 y 8 y 9 y 0 y y y x 0 x REFERENCES [] B. Feg, N. Morta, ad T. Tor, A New Optmzato Method for Dyamc Desg of Plaar Lkage wth Clearaces at Jots, ASME Joural of Mechacal Desg, 4, pp. 8-7, 00. [] B. Demeuleaere, M. Verschuure, J. Swevers, ad J.D. Schutter, A Geeral ad Numercally Effcet Framework to Desg Sector-Type ad Cyldrcal Couterweghts For Balacg of Plaar Lkages, ASME Joural of Mechacal Desg,, All rghts Reserved 75

8 teratoal Joural of Advace Egeerg ad Research Developmet (JAERD) Volume, ssue, March -0, e-ssn: , prt-ssn: (-0), 00. [] Y. Km, A. Ta, B. Yag, W. Km, B. Cho, ad Y. A, Optmum Shape Desg of Rotatg Shaft By ESO Method, Joural of Mechacal Scece ad Techology,, pp , 007. [4] H. Azegam, L. Zhou, K. Umemura, ad N. Kodo, Shape Optmzato for a Lk Mechasm, Structural Multdscple Optmzato, 48 (), pp. 5-5, 0. [5] D. Xu, ad G.K. Aathasuresh, Freeform Skeletal Shape Optmzato of Complat Mechasms, ASME Joural of Mechacal Desg, 5, pp. 5-, 00. [] G. Yogaad ad D. Se, Lk Geometry Sythess for Prescrbed erta, Proc. of 5 th Natoal Coferece o Maches ad Mechasms, Nov 0 - Dec 0, T Madras, da, 0. [7] D. Se, S. Chowdhury, ad S.R. Padey, Geometrc Desg of terferece-free Plaar Lkages, Mechasm ad Mache Theory, 9, pp , 004. [8] F. Freudeste, Approxmate Sythess of Four-bar Lkages, Resoace, 5 (8), pp , 00. [9]. Zed, ad R. Svasubramaa, CAD/CAM Theory ad Practce, Tata McGraw-Hll, New Delh, da, 009. [0] S. Brlek, G. labelle, ad A. Lacasse, The Dscrete Gree Theorem ad Some Applcatos Dscrete Geometry, Theoretcal Computer Scece, 4, pp. 0-5, 005. [] K. Chaudhary ad H. Chaudhary, Dyamc Balacg of Plaar Mechasms usg Geetc Algorthm, Joural of Mechacal Scece ad Techology, 8 (0), pp All rghts Reserved 7