Optmal synthess of a Path Generator Lnkage usng Non Conventonal Approach Mr. Monsh P. Wasnk 1, Prof. M. K. Sonpmple 2, Prof. S. K. Undrwade 3 1 M-Tech MED Pryadarshn College of Engneerng, Nagpur 2,3 Mechancal Engneerng Dept. Pryadarshn College of Engneerng, Nagpur ABSTRACT: Ths paper studes the soluton methods of optmal synthess of a Path Generator Lnkage usng Non Conventonal Approach. The method s defned by usng Harmony Search Method and a common knd of goal functon whch s used to fnd the approprate dmenson and to mnmze the error and fnd the best mechansm wth accurate Soluton. The possblty of extendng s the advantage of ths method. Unlke others we do not consder nput angle as desgn varable, because n those cases when many precson ponts are avalable, computaton wll ncrease wthout havng exact soluton. So we dvded the path to some secton and fnd mnmum error between desred ponts and desgn ponts. Usng ths method, we can easly decrease the path error and processng tme. I. INTRODUCTION Mechansms whch compose some connected rgd members are exclusvely used n the area of mechancal engneerng to transfer energy from one member to another. To mprove the atlast of mechansms wth a lot of curves to solve the mechansm problems. These methods are easy and fast to use but offer a low precson rate. Dmensonal synthess can be classfed as moton generaton, functon generaton and path or trajectory generaton. Both graphcal and analytcal methods have been used for dmensonal synthess. Usng precson ponts whch are traced by a mechansm s also classfed but such methods are relatvely restrctve because of ther low precson rates and cannot be used when we have varety of precson pont s mechansm By ncreasng the power of computers, numercal methods are used commonly to mnmze the goal functon. They used some optmzaton methods to optmze the goal functon, the error between the ponts trace by the coupler and ts desre trajectory. But all the solutons have a dsadvantage of fallng f the solutons appear n a local mnmum. In ths paper the approach presented to the synthess of mechansm deals wth Harmony Memory Search Method algorthm and we can compare t wth other soluton II. HARMONY MEMORY SEARCH METHOD The HS algorthm conceptualzes a behavoral phenomenon of muscans n the mprovsaton process, where each muscan contnues to experment and mprove hs or her contrbuton n order to search for a better state of harmony. It s frst gven by Geem & Km [17]. Ths secton descrbes the HS algorthm based on the heurstc algorthm that searches for a globally optmzed soluton. 2.1 Basc Algorthm The procedure for a harmony search, whch conssts of steps 1-5. Step 1. Intalze the optmzaton problem and algorthm parameters. Step 2. Intalze the harmony memory (HM). Step 3. Improvse a new harmony from the HM. Step 4. Update the harmony memory. Step 5. Repeat steps 3 and 4 untl the termnaton crteron s satsfed. These steps are explaned below: Step 1: Intalze the optmzaton problem and algorthm parameters. Frst, the optmzaton problem s specfed as follows: Mnmze f(x) subjected to x ε X, =1, 2...N Where f(x) s an objectve functon; X s the set of decson varables; N s the number of decson varable; X s the set of the possble range of values for each decson varable, that s x L x x U, and x L and x U are the lower and the upper bounds for each decson varables, respectvely. The algorthm requres several parameters: Harmony memory sze (HMS), Maxmum number of Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 38 Page
mprovsatons (NI) Harmony Memory Consderaton Rate (HMCR), ptch adjustng rate (PAR), Bandwdth vector used n (bm). Step 2: The HM matrs ntally flled wth as many randomly generated soluton vectors as the HMS, as well wth the correspondng functon values of each random vector, f(x). Ths s shown below: Step-3 New Harmony vector, X=(x 1, x 2 x 3,, x N ), s mprovsed based on the followng three mechansms: (1) random selecton, (2) memory consderaton, and (3) ptch adjustment. In the random selecton, the value of each decson varable, n the New Harmony vector s randomly chosen wthn the value ranges wth a probablty of (1-HMCR). The HMCR, whch vares between 0 to 1, s the rate of choosng one value from hstorcal values stored n the HM, and (1-HMCR) s the rate randomly selectng one value from the possble range of values. ε,. HMS f rand 1,1 < HMCR, wt Probablty HMCR ε X oterwse, wt probablty (1 HMCR) The value of each decson varable obtaned by the memory consderaton s examned to determne whether t should be ptch-adjusted. Ths operaton uses the PAR parameter, whch s the rate orgnal ptch obtaned n the memory consderaton s kept wth a probablty of HMCR. (1-PAR). If the ptch adjustment decson for s made wth the probablty of PAR, s replaced wth ± rand 1,1 bw, where bw s an arbtrary dstance bandwdth for the contnuous desgn varable,and adjustment s appled to each varable as follows: ± u( 1,1. bw Wt probablty PAR(tat f rand 1,1 < PAR wt probablty(1 PAR) Step 4. Update the HM. If the New Harmony vector s better than the worst harmony vector n the HM, based on the evaluaton of the objectve functon value, the New Harmony vector s ncluded n the HM, and the exstng worst harmony vector s excluded from the HM. Step 5. If the stoppng crteron (or maxmum number of mprovsaton) s satsfed, the computaton s termnated. Otherwse, steps 3 and 4 are repeated. 3.FORMULATION OF WORK 3.1. COUPLER POINT COORDINATES In the problem of four-bar lnkage synthess there s some number of precson ponts to be traced by the coupler pont P. To trace the coupler pont, the dmenson of the lnks (a, b, c, d, Lx, Ly) s to be determned along wth the nput crank angle θ2, so that the average error between these specfed precson ponts (Pxd, Pyd), (where =1,2, N wth N as number of precson ponts gven) and the actual ponts to be traced by the coupler pont P gets mnmzed. The objectve or error functon can be calculated when the actual traced ponts (Pxd, Pyd) s evaluated whch s traced by the coupler pont P wth respect to the man coordnate from X, Y as shown n Fg.3.1. Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 39 Page
Fg.3.1 Four-Bar Lnkage Wth ABP As Coupler Lnk The poston vector of the coupler pont P reference frame Xr,Yr can be expressed as a vector equaton: r p = a + L x + L y Ths can be represented n ts components accordng to: Px r = acosθ 2 + L x cosθ 3 + L y ( snθ 3 ) Py r = asnθ 2 + L x snθ 3 + L y (cosθ 3 ) Here, for calculaton the coupler pont coordnates (Px, Py), we have to frst compute the coupler lnk angle θ3 usng the followng vector loop equaton for the four-bar lnkage: a b c d 0 Ths equaton also can be expressed n ts components wth respect to relatve coordnates: acosθ 2 + bcosθ 3 ccosθ 4 d = 0 asnθ 2 + bsnθ 3 csnθ 4 = 0 For ths equaton followng two solutons are obtaned: θ 3 1 = 2tan 1 E+ E2 4DF 2D θ 3 2 = 2tan 1 E E2 4DF 2D where E = 2snθ 2 D = cosθ 2 k 1 + k 4 cosθ 2 + k 5 F = k 1 + k 4 1 cosθ 2 + k 5 These solutons may be () real and equal () real and unequal and () complex conjugates. If the dscrmnant E 2-4DF s negatve, then soluton s complex conjugate, whch smply means that the lnk lengths chosen are not capable of connecton for the chosen value of the nput angle θ2. Ths can occur ether when the lnk lengths are completely ncapable of connecton n any poston. Except ths there are always two values of θ3 correspondng to any one value of θ2. These are called, () crossed confguraton (plus soluton) and () Open confguraton of the lnkage (mnus soluton) and also known as the two crcuts of the lnkage. The other methods such as Newton-Raphson soluton technque can also be used to get approxmate soluton for θ3. The poston of coupler P, wth respect to world coordnate system XOY s fnally defned by: P x = x 0 + Px r cosθ 0 Py r snθ 0 P y = y 0 + Px r snθ 0 Py r cosθ 0 3.2poston Error as Objectve Functon The objectve functon s usually used to determne the optmal lnk lengths and the coupler lnk geometry. In path synthess problems, ths part s the sum squares whch computes the poston error of the dstance between each calculated precson pont P x, P y and the desred ponts P xd, P yd whch are the target ponts ndcated by the desgner. Ths s wrtten as: N =1 f X = [ Pxd Px 2 + Pyd Py 2 ] Where X s set of varables to be obtaned by mnmzng ths functon. Some authors have also consdered addtonal objectve functons such as the devaton of mnmum and maxmum transmsson angles mn and max from 90 o, for all the set of ntal solutons consdered. 4. The constrants of the lnkage The synthess of the four-bar mechansm greatly depends upon the choce of the objectve functon and the equalty or the nequalty constrants whch s mposed on the soluton to get the optmal dmensons. Generally the objectve functon s mnmzed under certan condtons so that the soluton s satsfed by a set of the gven constrants. The bounds for varables consdered n the analyss are treated as one set Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 40 Page
of constrants, whle the other constrants nclude: Grashof condton, nput lnk order constrant and the transmsson angle constrant. 4.1. Grashof crteron For Grashof crteron, t s requred that one of the lnks of mechansm, should revolve fully by 360 o angle. There are three possble Grashof lnkages for a four-bar crank chan: (a) Two crank-rocker mechansms (adjacent lnk to shortest s fxed) (b) One double crank mechans m (shortest lnk s fxed) and (c) One double rocker mechansm (opposte to shortest lnk s fxed). Of all these, n the present task, only crankrocker mechansm confguraton s consdered. Here, the nput lnk of the four-bar mechansm to be crank. Grashof crteron states that the sum (Ls+Ll) of the shortest and the longest lnks must be lesser than the sum (La+Lb) of the rest two lnks. That s: (Ls+Ll <= La+Lb) Or 2(Ls+Ll)<= a+b+c+d Or g1 = 2(Ls+Ll)<= (a+b+c+d) 1<= 0 In the present work volaton s defned as follows: Grashof s = 1 f g1>0 Or =0 f g1<= 0 4.2 Input lnk angle order constrant Usually a large combnaton of the mechansms exsts that generates the coupler curves passng through the desred ponts, but those solutons may not satsfy the desred order. To ensure that the fnal soluton honors the desred order, testng for any order volaton s mposed. Ths s acheved by requrng that the drecton of rotaton of the crank as defned by the sgn of ts angular ncrements θ 2 = θ 2 θ2 1,between the two poston and -1, where = 3,4,5...N, have same drecton as that between the 1 st and the 2 nd postons (θ 2 2 θ 2 1 ). That checks the followng: Is sgn Δθ 0 == sgn θ 2 2 θ 2 1 for all = 3 to N Where sgn(z) = 1 f Z>= 0 = -1 f Z<0 If ths condton s not satsfed the soluton s rejected. 4.3 Transmsson Angle Constrant For a crank-rocker mechansm generally the best results the desgners recognze when the transmsson angle s close to 90 degree as much as possble durng entre rotaton of the crank. Alternatvely, the transmsson angle durng entre rotaton of crank should le between the mnmum and maxmum values. Ths can be wrtten as one of the constrants as follows. Frst of all, the expressons for maxmum and mnmum transmsson angles for crank-rocker lnkage are defned. µ max = cos 1 b 2 (d+a) 2 +c 2 2bc µ mn = cos 1 b 2 (d a) 2 +c 2 2bc The actual value of transmsson angle at any crank angle 2 s gven by: µ = cos 1 b 2 a 2 d 2 +c 2 +2adcos θ 2 2bc The condton to be satsfed s: mn max The constrant gven by above equatons are handled by penalty method. That s the nondmensonal constrant devaton s drectly added to the objectve functon for mnmzaton. For example, constrant eq. f not satsfed, the penalty term s gven as follows: Trans = N =1 1 Trans mn (µ µ mn ) 2 + 1 Trans max (µ µ max ) 2 Where Transmn = sgn b 2 + c 2 d a 2 2bc cosµ mn Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 41 Page
Transmax = sgn 2bc cosµ max b 2 + c 2 + a + d 2 4.4. Varable Bounds All varables consdered n the desgn vector should be defned wthn prespecfed mnmum and maxmum values. Often, ths depends on the type of problem. For example, f we have 19 varables n a 10 pont optmzaton problem, all the varables may have dfferent values of mnmum and maxmum values. Generally, n non-conventonal optmzaton technques startng wth set of ntal vectors, ths constrant s handled at the begnnng tself, whle defnng the random varable values. That s we use the followng smple generaton rule: X=Xmn +rand (Xmax-Xmn) Where rand s a random number generator between 0 and 1. 4.5 Overall optmzaton problem The objectve functon s the sum of the error functon and the penaltes assessed to volaton the constrants as follows: F(k) = f(x) + W1 Grashof + W2 Tran, Whereas W1 s the weghtng factor of the Grashof s crtera and W2 s the weghtng factor of the Transmsson angle constrants.these addtonal terms acts as scalng factors to fx the order of magntude of the dfferent varables present n the problem or the objectve functon. III. RESULTS AND DISCUSSION 5. Path Synthess: The effcency and accuracy of the proposed are verfed by studyng three m e t h o d cases (for more than fve target ponts) from the lterature. Three cases are explaned : (1) 6 ponts (15 varables) (2) 10 ponts (19 varables) Dfferent parameters are used. It ncludes HS algorthm. Number of varable NVAR=15, maxmum no of teraton Maxtr =10000, harmony memory sze HMS=30, harmony memory consderaton rate HMCR=0.95, maxmum ptch adjustment rate PARmax=0.9, mnmum ptch adjustment rate PARmn=0.4, bandwdth mnmum = 0.0001, bandwdth maxmum=1. 5.1Sx Ponts Path Generaton and 15 desgn varables: The frst case s a path syntheszed problem wth gven sx target ponts arranged n a vertcal lne wthout prescrbed tmng. Desgn varables are: X= [a, b, c, d, ly, lx, θ1, θ2, θ3, θ4, θ5, θ6, θ0, x 0, y 0 ]Target Ponts:[(20,20),(20,25),(20,30),(20,35),(20,40),(20,45)] Lmts of the varable: a, b, c, d ε [5, 60] Lx, ly, x 0, y 0 ε [-60, 60] θ1, θ2, θ3, θ4, θ5, θ6 ε [0,2π] The syntheszed geometrc parameters and the correspondng values of the precson ponts (Pxd, Pyd) and the traced ponts by the coupler pont (Px,Py) and the dfference between them are shown n table 1 and table 2 respectvely. Although the constrant of the sequence of the nput angles durng the evoluton s gnored n ths case.the accuracy of the soluton n case 1 has been remarkably mproved usng the present method. Fg(5.2) shows the convergence graph of HS algorthm.fg(5.3) shows the sx target ponts and the coupler curve obtaned usng the harmony memory search method wth NVAR=15,Maxtr=10000, HMS=30,HMCR=0.95,PARmax=0.9,PARmn=0.4,bwmn=0.0001,bwmax=1. The out-put of the varables values as shown below. Table 5.1. Syntheszed Results For Sx Target Ponts Problem a b c d ly Lx θ1 θ2 θ3 θ4 θ5 θ6 θ0 x0 y0 10.565 2 46.0859 26.477 6 33.613 7-8.234 6 15.914 4 4.698 8 5.166 6 5.701 1 6.108 5 0.199 7 0.672 9 0.769 1 25.964 3 18.057 7 Table 5.2. Percentage Error Of The Coupler Lnk And The Precson Ponts px px d px-pxd (pxpxd)2 PY PY D PY- PYD (PY- PYD)2 18.7297-20.9235 0.9235 3 20 1.2703 1.61359 5 20 5 %Error n y %Error n x 0.85293 5 15.1392 39.951 Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 42 Page
20.9740 24.3770-0.38803 2 25-4.026 16.2085 8 25 0.6229 6 47.9811 26.946 21.3923 1.3923 29.8954-0.01093 8 20 8 1.93872 5 30 0.1046 1 16.594 4.524 34.9462-0.00288 20.5503 20 0.5503 0.30283 7 35 0.0537 7 6.5583 2.3230 19.8659-39.9980-3.65E- 4 20 0.1341 0.01797 9 40 0.0019 06 1.5981 0.0821-44.3950 0.36598 18.9822 20 1.0178 1.03592 4 45-0.605 1 12.13 26.17 Table 5.3. Actual Ponts Whch Is Traced By The Coupler Lnk And The Precson Ponts Px Pxd Py Pyd Px 18.72973 20 20.92355 20 18.72973 20.97402 25 24.37708 25 20.97402 21.39238 20 29.89545 30 21.39238 20.5503 20 34.94627 35 20.5503 19.86594 20 39.99809 40 19.86594 18.9822 20 44.39504 45 18.9822 Ftness Functon Value = 22.74 Ten Ponts Path Generaton and 19 desgn varables: Desgn varables are: X= [ a, b, c, d, ly, lx,θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8, θ9, θ10,θ0, x0,yo] Target Ponts:[(20,10),(17.66,15.142),(11.736,17.878),(5,16.928),(0.60307,12.736), (0.60307, 7.2638), (5, 3.0718), (11.736, 2.1215), (17.66, 4.8577), (20,10)] Lmts of the varable: a, b, c, d ε [5,80] lx,ly,x0,yo ε [-80,80] θ1, θ2, θ3, θ4, θ5, θ6, θ10 ε [0,2π] The syntheszed geometrc parameters and the correspondng values of the precson ponts (Pxd, Pyd) and the traced ponts by the coupler pont (Px,Py) and the dfference between them are shown n table 3 and table 4 respectvely. Although the constrant of the sequence of the nput angles durng the evoluton s gnored n ths case.the accuracy of the soluton n case 1 has been remarkably mproved usng the present method. Fg(5.5) shows the convergence graph of HS algorthm, fg (5.6) shows the ten target ponts and the coupler curve obtaned usng the harmony memory search method wth NVAR=18, Maxtr=10000, HMS=30, HMCR=0.95, PARmax=0.9, PARmn=0.4,bwmn=0.0001,bwmax=1. Table 5.4 Syntheszed Results For Ten Target Ponts Problem Table 5.5. Actual Ponts Whch Is Traced By The Coupler Ponts And The Precson Ponts Px Pxd Py Pyd 19.356 20 10.1177 10 17.676 17.66 16.0281 15.14 11.538 11.736 19.493 17.87 5.0531 5 17.9153 16.92 1.254 0.603 12.9509 12.73 0.8418 0.603 8.62267 7.26 4.5868 5 2.6323 3.07 Ftness Functon Value = 11.43 IV. CONCLUSIONS Even ths work has concentrated on path synthess part wth some mportant constrants, some more constrants lke mechancal advantage of the lnkage, and flexblty effects can be also consdered to get the accuracy. Also as n hybrd synthess approach, the same lnkage may be adopted both for path synthess applcatons as well as moton synthess applcatons. The objectve functon should be modfed so as to Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 43 Page
get a dfferent optmum lnk dmensons. Fnally fabrcaton of the proto-type of ths lnkage may be done to know the dfference between theoretcally obtaned coupler coordnates and actual values acheved. V. ACKNOWLEDGEMENT I aval ths opportunty to extend my hearty ndebtedness to my gude PROF.M.K. Sonpmple, Prof. S.K. Undrwade Mechancal Engneerng Department, for hs valuable gudance, constant encouragement and knd help at dfferent stages for the executon of ths dssertaton work. I also express my sncere grattude to hm for extendng ther help n completng ths project. I take ths opportunty to express my sncere thanks to my project gude and co-gude for co-operaton and to reach a satsfactory concluson REFERENCES Optmal Synthess of Planar Mechansms wth PSO Algorthms Author: 1)S.Alna 2)M.Ghadm 3) H.D Kalj Journal: World Appled scences Journal18 (2):268-273, 2012 Objectve: Ths Paper studes the soluton methods of optmal synthess of planer mechansm. The method defned by usng PSO Technque and a common knd of goal functon whch s used to fnd the approprate dmensons and to mnmze the error and fnd the best mechansm wth accurate soluton. Optmzaton of Watt s Sx-Bar Lnkage to Generates Straght and Parallel Leg Moton Author:Hamd Mehdghol and Saeed Akbarnejad Journal: Journal of Humanods, Vol.1, No.1, (2008), ISSN 1006-7290, pp.11-16 Objectve: Ths paper consders optmal synthess of a specal type of four-bar lnkages. Combnaton of ths optmal four-bar lnkage wth on of ts cognates and elmnaton of two redundant cognates wll result n a Watts s sx-bar mechansm, whch generates straght and parallel moton. Ths mechansm can be utlzed for legged machnes. The advantage of ths mechansm s that the leg remans straght durng t s contact perod and because of ts parallel moton, the legs can be wde as desred to ncrease contact area and decrease the number of legs requred to keep body s stablty statcally and dynamcally. Optmal Synthess of Crank Rocker Mechansm for Pont to Pont Path Generaton Author:1)Subhash N Waghmare, 2) Roshan B. Mendhule, 3) M.K.Sonpmple Journal:Internatonal Journal of Engneerng Inventons ISSN:2278-7461,Vol 1,2Sept.(2012), PP:47-55 Objectve: The concept of orentaton structure error of the fxed lnk and present a new optmal synthess method of crank rocker lnkages for the path generaton. The orentaton structure error of the fxed lnk effectvely reflects the overall dfference between the desred and generated path. In ths paper Genetc Algorthm s used for the formulaton of work and the desred output s made by the same. Knematcs Desgn of a Planer Parallel Controllable Mechansm Based on Partcle Swarm Optmzaton Algorthm. Author:1) Ke Zhang 2) Shengze Wang Journal: Journal of Computers, Vol.6, No.6, Jue 2011 Objectve: Planer parallel controable mechansm, whch s a combnaton of two types of motor and mechansm, has better flexble transmsson behavor. In ths paper, the knematcs analyss for a planer parallel controllable fve-bar mechansm s ntroduced. In order to mprove knematc performance of the controllable mechansm an optmzaton desgn for the mechansm s performed wth reference to knematcs objectve functon. A hybrd optmzaton algorthm whch combnes Partcle Swarm Optmzaton (PSO) wth Matlab Optmzaton tool bos proposed to solve the optmal desgn problem wth constraned condton. Optmal synthess of mechansms wth genetc algorthms Author: 1) J.A. Cabrera,2) A. Smon, 3)M. Prado Journal: Mechansm and Machne Theory 37 (2002) 1165 1177 Objectve: Ths paper deals wth soluton methods of optmal synthess of planar mechansms. A searchng procedure s defned whch apples genetc algorthms based on evolutonary technques and the type of goal functon. Problems of synthess of four-bar planar mechansms are used to test the method, showng that solutons are accurate and vald for all cases. The possblty of extendng the method to other mechansm type s outlned. The man advantages of the method are ts smplcty of mplementaton and ts fast convergence to optmal soluton, wth no need of deep knowledge of the searchng space. Internatonal Conference on Advances n Engneerng & Technology 2014 (ICAET-2014) 44 Page