Available online at ScienceDirect. Procedia Technology 14 (2014 ) Kailash Chaudhary*, Himanshu Chaudhary

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1 Avalable onlne at ScenceDrect Proceda Technology 4 (4 ) 35 4 nd Internatonal Conference on Innovatons n Automaton and Mechatroncs Engneerng, ICIAME 4 Optmum balancng of slder-crank mechansm usng equmomental system of pont-masses Kalash Chaudhary*, Hmanshu Chaudhary Department of Mechancal Engneerng, Malavya Natonal Insttute of Technology Japur, Japur 37, Inda Abstract An optmzaton technque for dynamc balancng of planar mechansms s presented n ths paper. The shakng forces and shakng moments developed due to nerta forces n mechansms are mnmzed usng the genetc algorthm (GA). The nertal propertes of rgd lnks of mechansm are represented by dynamcally equvalent systems of pont-masses. The shakng force and shakng moment are then evaluated n terms of the pont-mass parameters and presented as the obectve functon for the proposed optmzaton problem. Usng the pont-mass parameters as desgn varables, the soluton of ths optmzaton problem optmzes the mass dstrbuton of each lnk. The results obtaned by usng genetc algorthm are found better than the conventonal optmzaton algorthm results. The masses and nertas of the optmzed lnks are computed from the optmzed desgn varables. The effectveness of the proposed methodology s shown by applyng t to a problem of slder-crank planar mechansm avalable n the lterature. 4 Elsever The Authors. Ltd. Ths Publshed an open by Elsever access artcle Ltd. under the CC BY-NC-ND lcense Selecton ( and/or peer-revew under responsblty of the Organzng Commttee of ICIAME 4. Peer-revew under responsblty of the Organzng Commttee of ICIAME 4. Keywords: Dynamc balancng; Shakng force and shakng moment; Equmomental system; Optmzaton; Genetc algorthm. Introducton If an unbalanced mechansm runs at hgh speed, t transmts forces and moments to the frame known as shakng forces and shakng moments. These forces and moments are the vector sum of the nerta forces and moments of all the movng lnks of the mechansm. The shakng forces and the shakng moments need to be elmnated to mprove the dynamc performance of the mechansm. Several methods are presented n the lterature for reducng these * Correspondng author. Tel.: ; fax: E-mal address: k.chaudhary.mech@gmal.com Elsever Ltd. Ths s an open access artcle under the CC BY-NC-ND lcense ( Peer-revew under responsblty of the Organzng Commttee of ICIAME 4. do:.6/.protcy.4.8.6

2 36 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) 35 4 shakng forces and shakng moments due to nerta. The complete force balancng can be acheved by makng the mass center of movng lnks of a mechansm statonary []. Ether by redstrbutng the mass or by addng the counterweghts to the movng lnks, the mass center s made statonary. Ths method was further extended for the mechansms havng prsmatc onts under certan condtons [, 3]. The complete force balancng alone ncreases other dynamc performance characterstcs such as shakng moment, drvng torque and bearng forces n onts [4]. Therefore, to balance the shakng moment along wth the full force balancng, several methods are proposed n the lterature [5-7]. The addton of duplcate mechansm, nerta or dsk counterweghts also elmnates the shakng force and shakng moment [8-]. However, t s generally not recommended due to complexty and practcal reasons. Ths leads that the complete balancng s not possble usng nternal mass redstrbuton wthout addng extra lnks. Consequently, the optmzaton methods are developed to mnmze the shakng force and the shakng moment. Several trade-off methods were developed to mnmze dfferent dynamc quanttes smultaneously [, 3]. As the shakng force and shakng moment depend on lnk masses, ther locatons of CGs and moment of nertas, these trade-off methods fnd the optmal dstrbuton of the lnk masses [4]. The evolutonary optmzaton technques lke partcle swarm optmzaton (PSO) and genetc algorthm (GA) can be appled to fnd the global optmum soluton for an optmzaton problem [7, 8]. Most of the optmzaton methods use the conventonal optmzaton algorthms that requre an ntal guess pont to start searchng the optmum soluton and lkely to produce local optmum soluton close to the start pont. In ths paper, the formulaton of optmzaton problem s smplfed by modellng the rgd lnks of mechansm as dynamcally equvalent system of pont-masses, known as equmomental system [5, 6]. Ths optmzaton problem s solved by usng genetc algorthm whch doesn t requre a start pont and searches the soluton n the entre desgn space. Therefore, t produces the global optmum soluton for the problem. The proposed method can be effectvely used to balance the mechansms havng revolute and prsmatc onts whle most of the methods avalable n the lterature are for the mechansms wth revolute onts only. A slder-crank mechansm s balanced n ths paper by optmally dstrbutng the lnk masses whle a cam mechansm wth counterweght was used to balance the same mechansm n [8]. Therefore, t s advantageous to use the method proposed n ths paper as compared to the method presented n [8] whch ncreases the overall mass and complexty of the mechansm. The structure of ths paper s as follows. Secton presents the equatons of moton for rgd body and the same n equmomental pont-masses. The problem of mnmzng shakng force and shakng moment smultaneously for a slder-crank mechansm s formulated n Secton 3. A numercal example s solved usng the proposed method and results are presented n Secton 4. Fnally, conclusons are gven n Secton 5.. Equmomental system of pont-masses In ths secton, the concept of equmomental system of pont-masses s dscussed and the dynamc equaton of moton for a rgd body s rewrtten n terms of the pont-masses... Equatons of moton of rgd body The lnks of a mechansm can be modeled as rgd bodes for smplfyng the knematc and dynamc analyses. Consder an th rgd lnk havng moton n XY plane for whch a local frame, X Y, s fxed at O (Fg. ). The Newton-Euler (NE) equatons of moton for the th rgd lnk n the fxed nertal frame, OXY, are wrtten as []: M t C t w ()

3 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) In Eq. (), 3-vectors, t, t and w are twst, twst-rate, and wrench vectors of the th lnk wth respect to O, respectvely,.e., t ; v t v and w n f () where, and v are the scalar angular velocty about the axs perpendcular to the plane of moton and the - vector of lnear velocty of the orgn O, respectvely. Accordngly, and v are tme dervatves of and v, respectvely. Also, the scalar, n, and the -vector, f, are the resultant moment about O and the resultant force at O, respectvely. In Eq. (), the 33 matrces, M and C are defned as: I M mdsn( ) mdcos( ) m d sn( ) m mdcos( ) ; C m dcos( ) (3) m sn( ) md To defne the lnk length, O to O + are fxed at the onts connectng precedng and succeedng lnks. The body fxed frame, O X Y, s then defned n such a way that the axs X s algned from O to O +. The locaton of the mass center, C, can be defned by the polar coordnates, d and... Modfed equatons of moton for equmomental system of pont-masses To formulate an optmzaton problem to mnmze shakng force and shakng moment, the rgd lnks are modeled as dynamcally equvalent systems of pont-masses referred to equmomental systems (Fg. ). The rgd lnk and the system of pont-masses wll be dynamcally equvalent (equmomental) f they have same mass, same center of mass and same nerta tensor wth respect to same coordnate frame [5]. Hence, a set of dynamcally equvalent system of rgdly connected n pont-masses, m, located at l,, must satsfy the followng condtons: m m (4) O + O + Y m C a X Y m l a X l m d O Y O Y m l O X O X Fg.. th rgd lnk Fg.. Equmomental system of pont-mass

4 38 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) 35 4 mlcos( ) mdcos( ) ; mlsn( ) md sn( ) (5-6) ml I (7) where m and I are the mass of the th lnk and ts mass moment of nerta about O. The frst subscrpt denotes the lnk number, and the second subscrpt represents the pont-mass. The NE equatons of moton, Eq. (), are now rewrtten for the equmomental system of pont-masses n the same local and global reference frames. It can be shown that ther form, Eq. (), does not change except the elements of matrces, M and C, whch are gven as: M m l m l S( m l C( ) ) m l S( m ) mlc( ) ; C m m l C( ) m l S( ) ; (8) In Eq. (8), C and S are abbrevatons for cosne and sne functons, respectvely. There are 3k parameters, m,, l for =,,,k f k pont-masses are defned for the th lnk. For a mechansm of n movng lnks, there wll be total 3kn pont-mass parameters. All or some of these can be taken as the desgn varables n optmzaton formulaton dscussed n the next secton. 3. Formulaton of optmzaton problem The problem for mnmzng the shakng force and shakng moment n a planar slder-crank mechansm s now formulated on the bass of the dynamcs presented n the prevous secton. The slder-crank mechansm under consderaton s shown n Fg. 3. The lnks are numbered as #, #, # and #3, where lnk # represents the frame. The onts are numbered as,, 3 and 4 whle a, a and a represent the lnk lengths. The fxed nertal frame, OXY, s located at ont, between lnk # and the frame #. Y # # C a C a C 3 n 3 -f f e - n e n 3 X a # 4 O4 f 3y -f 3y #3 -n 3 O Fg. 3. Planar slder-crank mechansm

5 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) Identfcaton of desgn varables A system of k equmomental pont-masses s used for each lnk and the correspondng pont-mass parameters are taken as the desgn varables. Therefore, the 3k-vector of desgn varables for the th lnk ncludes pont-mass and ther locatons and s defned as: DV m T l m l... mk lk k Hence, the desgn varable 3kn-vector, DV, for mechansm havng n movng lnks can be defned as: DV DV T T T T DV... DVn (9) () 3.. Obectve functon and constrants For a mechansm n moton, shakng force s the vector sum of the nerta forces, whereas the shakng moment about any pont s the sum of the nerta couples and the moment of the nerta forces about that pont [4]. In the current problem, the external forces lke gravty and dsspatve forces are not consdered. Once all the ont reactons are determned, the shakng force and shakng moment at and about ont are obtaned as: f sh and n n n a x f ) ( f f3) sh e ( 3 3 () In Eq. (), f and f 3 are the reacton forces of the frame on the lnks # and #3, respectvely. The drvng torque appled at ont # s represented by n e whle n 3 represents the reacton of the nerta couple about ont #3. a represents the vector from O to O 4. Consderng the RMS values of the shakng force, f sh,rms, and the shakng moment, n sh,rms, the optmzaton problem s proposed as: Mnmze Z w f sh,rms wn sh,rms () Subect to m,mn m m ; I m l for =,, 3 and =,,, k (3),max, mn where w and w are the weghtng factors used to assgn weghtage to the dfferent obectves ncluded n a sngle obectve functon. The values of these factors vary between and dependng upon the applcaton. For complete shakng force balance and complete shakng moment balance, values are taken as (w =, w =) and (w =, w =), respectvely. To balance force and moment smultaneously, values are taken as (w =.5, w =.5) for gvng equal weghtage to both the obectves. The mnmum mass and nerta, m,mn and I,mn, of th lnk can be defned accordng to ts force bearng capabltes and materal propertes. 4. Solutons and results After formulatng the balancng problem as an optmzaton problem, t can be solved by usng ether conventonal or evolutonary optmzaton algorthms. The gradent-based conventonal algorthms use the gradent nformaton of the obectve functon wth respect to the desgn varables. Startng wth an ntal guess pont, these methods converge on the optmum soluton near to the startng pont and thus produce local optmum soluton. Genetc algorthm s evolutonary search and optmzaton algorthm based on the mechancs of natural genetcs and natural selecton [9]. Ths algorthm evaluates only the obectve functon and genetc operators - selecton, crossover and mutaton are used for explorng the desgn space. After selecton operaton, crossover and mutaton operators are used to form the new populaton [].

6 4 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) 35 4 The proposed method s appled to solve the balancng problem of a slder-crank mechansm avalable n the lterature [8] for whch the parameters are gven n Table. As the shakng force and the shakng moment are of dfferent unts, these quanttes are made dmensonless wth respect to the parameters of the frst lnk for addng them n a sngle obectve functon. For ths example, the nput lnk,.e. lnk #, rotates wth a constant speed of rad/sec. Each lnk s represented by three equmomental pont-masses to reduce the dmenson of the problem. Out of nne varables, m, l,, for =,, 3, for the th lnk, fve parameters are assgned as: 3 and l =l 3 =l. The other four parameters, namely, m, m, m 3, and l are brought nto the optmzaton scheme as the desgn varables. A MATLAB program was developed usng the equmomental condtons, Eqs. (4)-(7), for fndng the dynamcally equvalent pont-masses for each lnk. Table. Parameters of standard mechansm. Lnk Length a (m) Mass m (kg) Moment of nerta Ic zz (kg-m ) CG dstance d (m) CG angle (deg) Consderng m,mn = m, m,max = m and I,mn =.5I for the th lnk, the optmzaton problem as explaned n Eqs. ()-(3) s frst solved usng fmncon functon n Optmzaton Toolbox of MATLAB []. The results correspondng to the dfferent combnatons of the weghtng factors are presented n Table and shown n Fg. 4. The case s complete shakng force balancng n whch the rms value of shakng moment ncreases to four tmes of that the unbalanced mechansm. Smlarly, n case 3, shakng force ncreases whle shakng moment reduces substantally. Reducton n both the quanttes can occur n case, n whch equal weghtng factors are assgned to them. Table. RMS values of dynamc quanttes for dfferent combnatons of weghtng factors Shakng force Shakng moment Standard value Case : (w =.;w =.).47x Case : (w =.5;w =.5) Case 3: (w =.;w =.) x -5 Normalsed Shakng Force Orgnal value Case Case Case Tme (sec) Normalsed Shakng Moment Orgnal value Case Case Case Tme (sec) (a) Normalsed shakng force (b) Normalsed shakng moment

7 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) Fg. 4. Comparson of dfferent cases for shakng force and shakng moment The same problem s then solved by usng ga functon n Genetc Algorthm and Drect Search Toolbox of MATLAB wth equal weghtng factors for both the quanttes. The comparson of the orgnal rms values wth the optmum rms values of the shakng force and shakng moment obtaned usng conventonal and genetc algorthm are presented n Table 3 and Fg. 5. The optmzed lnk parameters are found by usng the equmomental condtons presented n Eqs. (4)-(7) and shown n Table 4. Table 3. RMS values of dynamc quanttes of normalzed standard and optmzed mechansms. Balancng method Shakng force Shakng moment Standard mechansm fmncon.93 (-).883(-8) Genetc algorthm.5(-46).5 (-99) The values n the parenthess denote the percentage ncrement/decrement wth respect to the correspondng RMS values of the standard mechansm. Table 4. Parameters of balanced mechansm. Lnk Mass m (kg) Moment of nerta Ic zz (kg-m ) CG dstance d (m) CG angle (deg) By usng the conventonal optmzaton method, the reducton of % and 8% was found n the rms values of shakng force and shakng moment, respectvely. The applcaton of the genetc algorthm results n reducton of 46% and 99% n the values of shakng force and shakng moment, respectvely. It s observed that f the mass of slder s not at CG, shakng moment reduces n the mechansm. The moment of nerta of slder about CG doesn t affect the values of shakng force and shakng moment and hence not gven for orgnal and balanced mechansms. Moreover, the shakng force rses up to 7 n the orgnal unbalanced mechansm. In the balanced mechansm, t goes up to 5 as shown n Fg. 5(a). However, the peak value of normalsed shakng moment reduces from.8 to.5 as shown n Fg. 5(b). Normalsed Shakng Force Orgnal value fmncon value GA value Tme (sec) Normalsed Shakng Moment Orgnal value - fmncon value GA value Tme (sec) (a) Normalsed shakng force (b) Normalsed shakng moment Fg. 5. Varatons n shakng force and shakng moment for complete cycle

8 4 Kalash Chaudhary and Hmanshu Chaudhary / Proceda Technology 4 ( 4 ) Conclusons For the dynamc balancng of planar mechansms, an optmzaton method s developed usng the concept of the equmomental system of pont-masses. The dynamc equatons of moton are formulated n the parameters related to the equmomental pont-masses. Usng these equatons, the optmzaton problem s formulated for the mnmzaton of the shakng force and shakng moment n a planar slder-crank mechansm. The genetc algorthm produced better results as compared to the conventonal optmzaton algorthm. Usng the equal weghtng factors to both the shakng force and the shakng moment, about 46% and 99% reductons are acheved n shakng force and shakng moment, respectvely. The method proposed n ths paper s general and can be appled for complex planar mechansms. References [] Berkof RS, Lowen GG. A new method for completely force balancng smple mechansms. Journal of Engneerng for Industry 969; 9(): -6. [] Tepper FR, Lowen GG. General theorems concernng full force balancng of planar mechansms by nternal mass redstrbuton. Journal of Engneerng for Industry 97; 94 (3): [3] Walker MJ, Oldham K. A general theory of force balancng usng counterweghts. Mechansm and Machne Theory 978; 3: [4] Lowen GG, Tepper FR, Berkof RS. The quanttatve nfluence of complete force balancng on the forces and moments of certan famles of four-bar lnkages. Mechansm and Machne Theory 974; 9: [5] Carson WL, Stephenes JM. Feasble parameter desgn spaces for force and root-mean-square moment balancng an n-lne 4R 4-bar syntheszed for knematc crtera. Mechansm and Machne Theory 978; 3: [6] Hans RS. Mnmum RMS shakng moment or drvng torque of a force-balanced mechansm usng feasble counterweghts, Mechansm and Machne Theory 98; 6: [7] Ye Z, Smth MR. Complete balancng of planar lnkages by an equvalent method. Mechansm and Machne Theory 994; 9(5): 7-7. [8] Arakelan V, Brot S. Smultaneous nerta force/moment balancng and torque compensaton of slder - crank mechansms. Mechancs Research Communcatons ; 37: [9] Esat I, Baha H. A theory of complete force and moment balancng of planar lnkage mechansms. Mechansm and Machne Theory 999; 34: [] Feng G. Complete shakng force and shakng moment balancng of 7 types of eght-bar lnkages only wth revolute pars. Mechansm and Machne Theory 99; 6(): [] Chou ST, Ba GJ, Chang WK. Optmum balancng desgns of the drag-lnk drve of mechancal presses for precson cuttng. Internatonal Journal of Machne Tools and Manufacture 998; 38 (3): 3-4. [] Chaudhary H, Saha SK. Dynamcs and Balancng of Multbody Systems. Germany: Sprnger; 9. [3] Lee TW, Cheng C. Optmum balancng of combned shakng force, shakng moment, and torque fluctuatons n hgh speed mechansms. Journal of Mechansms, Transmssons, and Automaton n Desgn 984; 6(): 4-5. [4] Chaudhary K, Chaudhary H. Concept of equmomental system for dynamc balancng of mechansms. Internatonal Conference on Automaton and Mechancal Systems. Fardabad, Inda; 3. [5] Routh EJ. Treatse on the dynamcs of a system of rgd bodes, Elementary Part I. New York: Dover Publcaton Inc.; 95. [6] Chaudhary H, Saha SK. Balancng of four-bar mechansms usng maxmum recursve dynamc algorthm. Mechansm and Machne Theory 7; 4(): 6-3. [7] Farman MR. Multobectve optmzaton for force and moment balance of a four-bar mechansm usng evolutonary algorthms. Journal of Mechancal Scence and Technology ; 5 (): [8] Erkaya S. Investgaton of balancng problem for a planar mechansm usng genetc algorthm. Journal of Mechancal Scence and Technology 3; 7 (7): [9] Deb K. Optmzaton for Engneerng Desgn Algorthms and examples. New Delh, Inda: PHI Learnng Prvate Lmted;. [] Gao Y, Sh L, Yao P. Study on mult-obectve genetc algorthm. 3 rd World Congress on Intellgent Control and Automaton. Hefe, P R Chna;. [] MATLAB Optmzaton Toolbox, verson (R8b).