KINETOSTATIC ANALYSIS OF SIX-BAR MECHANISM USING VECTOR LOOPS AND THE VERIFICATION OF RESULTS USING WORKING MODEL 2D

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1 International Journal of Mechanical Engineering an Technology (IJMET) Volue 8, Issue 8, ugust 017, pp , rticle I: IJMET_08_08_110 vailable online at ISSN Print: an ISSN Online: IEME Publication Scopus Inexe KINETOSTTI NLYSIS OF SIXBR MEHNISM USING VETOR LOOPS N THE VERIFITION OF RESULTS USING WORKING MOEL het Shala an Mirlin Bruqi* Faculty of Mechanical Engineering, University of Prishtina, Prishtina, Kosovo *orresponing author: irlin.bruci@unipr.eu BSTRT In this paper is escribe use kineatics analysis is base on vectorial algebra known as vector loops for the sixbar echanis. First is require to escribe vectorial & scalar equations for each siple planar echaniss, either closechain or openchain. ynaic equations of otions for kinetic analysis are use. Solution of vectorial/scalar equations is one using Math software. Results are verifie using Working Moel software. Keywors: Mechanis, Kineatics, Kinetic, ynaic, Vector loop ite this rticle: het Shala an Mirlin Bruqi, Kinetostatic nalysis of SixBar Mechanis Using Vector Loops an the Verification of Results Using Working Moel, International Journal of Mechanical Engineering an Technology 8(8), 017, pp INTROUTION Kineatics an ynaic stuy of echaniss using plans an graphic ethos is known that perissive error [1, 3]. To avert this error is require use of analytic ethos [], especially for ynaic stuy of sart echaniss [3, ], where error is not allowe. In espite of har work uring calculations which are require for analytic stuy, for sart echaniss this etho is require. Except that, using analytic etho can be achieve require exactness, which can t be arrive using other two ethos [1]. Usually, by using analytic stuy is eterine relations of kineatics paraeters of guie an guie links [6, 7]. nalytical ethos for analysis of echaniss are too any [, 6], in view is presente application of vectors loops [1] etho for kineatic analysis an use of ynaic equations of otions for kinetostatic analysis of echanis eitor@iaee.co

2 Kinetostatic nalysis of SixBar Mechanis Using Vector Loops an the Verification of Results Using Working Moel. KINEMTIS NLYSIS OF MEHNISM USING VETOR LOOPS This represents one general stuy etho, which is base on vectorial algebra. Base on this etho, start & en connection points of each link represent the vectors with length sae as link have, irection is eterine accoring in the otion of echanis. In view is represente exaple of usage of this etho for sixbar echanis. Basic ata of links of echanis (Figure 1): Lengths L 1 =O O = c; L =O 1 = c; L 3 =B=6 c; L a =O B= c; L b =B=1 c; L ==7 c; y =1 c Masses = kg; 3 =6 kg; = kg; = kg; 6 =1 kg Kineatic friction between slier an baseent µ= 0.1 onstant angular velocity of river link, ω = 1ra/s θ = t river otion: ra Before starting any analysis it is iportant to check continues otion of given echanis for iniu one circle of otion. Use of Grashof conition is very useful. In our exaple ain basic echanis is fourbar echanis O 1 BO. The Grashof conition for a fourbar linkage states: If the su of the shortest an longest link of a planar quarilateral linkage is less than or equal to the su of the reaining two links, then the shortest link can rotate fully with respect to a neighboring link. In other wors, the conition is satisfie if L +L 3 L 1 +L where L is the shortest link, L 3 is the longest, an L 1 an L are the other links. pplying our ata: ω L +L 3 L 1 +L => +6 + => 8 9 Which is thru an our given echanis is Grashof or tiecontinuously. Figure 1 Sixbar planar echanis eitor@iaee.co

3 het Shala an Mirlin Bruqi y L b B θ L L 3 O 1 L θ L 1 θ 3 O L a θ x y x Figure Schee of vectors loops Fro Figure can be seen two vector loops which represent two vectorial equations of basic echaniss, as in view: L +L 3 =L 1 + L a ; (a) L a + L b + L = x + y ; (b) To fin the solution of syste of equations, tie invariant, in Mathca it is enough to call function Given in the beginning. To avoi inverse solutions it is require to eclare initial positionvalues (approxiately) of unknown paraeters. Projections / coponents of equations (a) an (b) in horizontal (x) an vertical (y) irection are: L L ( L ( L cosθ + L3 cosθ3 = L1 + La cosθ sinθ + L3 sinθ3 = 0 + La sinθ a a + L + L b b )cosθ + L )sinθ + L cosθ = x sinθ = y (c) () Fro syste of equations (c) an () can be foun all require expressions for positional analysis of echanis, usually unknown are angles: θ3, θ, θ an istancecoorinate x : kin_sol( := Fin θ 3, θ, θ, x Using first an secon erivative of equations (c) an () can be foun require velocities an accelerations of characteristic points an links of echanis. 3. YNMI NLYSIS OF MEHNISM ynaic analysis of echaniss consists on writing of equations of otion base ynaics kinetostatic laws e.g. ynaic equations of planar otion for each link of the echanis. To fin the solution of syste of equations tie invariant in Mathca it is enough on top to call function Given. Inverse solutions are avoie uring kineatic analysis; here is enough to eclare initial forcesvalues starting fro zero (0) just to avoi iaginary solutions of unknown paraeters. ynaic equations of otion for riven link : eitor@iaee.co

4 Kinetostatic nalysis of SixBar Mechanis Using Vector Loops an the Verification of Results Using Working Moel O XO M tr a y ε YO θ X P ax Y ax = XO1 ( X (1) a y ( t = YO1 Y g () ) L L 0 = X O1 sin( θ ( YO1 cos( θ ( + X L Y cos( θ ( + M tr ynaic equations of otion for link 3: L sin( θ ( (3) B X B ε3 a y3 YB X Y θ 3 P 3 a 3 x3 a 3( = X + X 3 x B () a y3( t = Y + YB 3 g () 3 ) L3 L3 J3 ε3( = X sin( θ3( Y cos( θ3( L3 L3 X B sin( θ3( + YB cos( θ3( ynaic equations of otion for link : X (6) Y B X B ε B Y ax X O O θ YO P a y ax ( = X O X B X (7) a y ( t = YO YB Y g (8) ) eitor@iaee.co

5 het Shala an Mirlin Bruqi J Y ε( = X O L + Y O B L L a + X B L L + a Y L ynaic equations of otion for link : + X L cos( θ( L sin( θ( (9) X ay ε Y X θ 3 ax P Y a ( = X + X x (10) a y ( t = Y + Y g (11) ) L J ε ( = ( X X ) sin( θ ( L ( Y Y ) cos( θ( ynaic equations of otion for link 6: (1) a y Y X a x N P6 F µ 6 a ( = X + F µ (13) 0 Y + N g (1) = 6 oulob s law for frictionnoral force is: Fµ = sign( v ( µ N (1) In previous equations are 1 unknown forcesoents: yn_sol( := Fin X O1, Y O1, X, Y, X B, Y B, X O, Y O, X, Y, X, Y, F µ, N, M tr Where: X i an Y i are coponents of joint forces, F µ friction force an M tr transitte oent on river link joint O 1, or Torque of otor. In the equation (1) is use signu function to take in consieration change of oveent irection of slier eitor@iaee.co

6 Kinetostatic nalysis of SixBar Mechanis Using Vector Loops an the Verification of Results Using Working Moel. RESULTS OF KINEMTIS N YNMIS NLYSIS, USING SOFTWRE S WORKING MOEL N MTH FOR SIXBR MEHNISM.1. Graphical Results iagras fter is obtaine syste which escribe the otion of echanis, require kineatics paraeters are solve using Math software, e.g. position, velocity, acceleration, angular velocity an acceleration, reaction forces on joints an Torque of otor (transitte oen place on joint O 1. Results are verifie in Working Moel. Base on Figure position of point 3 (centre of ass of link 3) is eterine by coorinates: L 3 x 3 ( := L cos( θ ( ) + an cos ( θ 3 ( ) y 3 ( := L sin θ ( Where: θ ( t ω t is given an θ can be foun fro equations (a) an (b). ) = 3 t L 3 + sin θ 3 ( Velocity an acceleration is calculate using firs an secon erivatives of position: v 3 ( = x3( + y3(, a 3 3( ) + t = x t y 3( Trajectory of point 3 3 y 3 ( 1 Working Moel x 3 ( Math Figure 3 Trajectory of point 3 3 Velocity of point 3. v 3 ( Working Moel t Math Figure Velocity of point eitor@iaee.co

7 het Shala an Mirlin Bruqi cceleration of point 3 3 a 3 ( 1 Working Moel t Math Figure cceleration of point 3 Fro Figure position of point (joint between link an link ) is eterine by coorinates: x ( := L 1 + L cos θ ( an y ( := L sin θ ( Where; θ can be foun fro equations (a) an (b). t Velocity an acceleration is calculate using first an secon erivatives of position: v ( = x ( + y (, a t = x + t y t Fro solution of ynaic equations of otion (1) to (1) is represente iagra of reaction force on joint an Transitte oent Torque of otor on joint O 1. Figure 6 Reaction force on joint an Transitte oent Torque of otor on joint O 1... Nuerical Results For coparisons of use software s in Table 1 are represente nuerical results for e.g. tie 0. secons. Table 1 Results of kineatics paraeters in Math an Working Moel for tie t=0. s Tie x position of slier Velocity of point Velocity of point 3 Velocity of point Velocity of point Velocity of slier t x v x v y v x v y v x v y v x v y v WM Mathca eitor@iaee.co

8 Kinetostatic nalysis of SixBar Mechanis Using Vector Loops an the Verification of Results Using Working Moel WM Mat hca Tie ngular velocity of link 3 ngular velocity of link ngular velocity of link ccelerat ion of point 3 ccelerat ion of point cceleratio n of point ccelerati on of point t ω 3 [s 1 ] ω 3 [s 1 ] ω 3 [s 1 ] a x a y a x a y a x a y x ngular acceleration link 3 ngular acceleration link ngular acceleration link ε 3 [s ] ε [s ] ε [s ] WM Mathca Table Results of ynaic paraeters in Math an Working Moel for tie t=0. s Ti e Force Force B Force O Force Force WM 0. Mathca t X Y X B Y B X O Y O X Y X Y Tie Force O 1 Friction Force Reaction Force Torque of Motor O 1 t X O1 Y O1 F µ N M tr WM Mathca Figure 7 ifferent positions (9 trucking fraes) of echaniss uring the otion eitor@iaee.co

9 het Shala an Mirlin Bruqi. ONLUSIONS Base on results can be conclue that Working Moel in coparison with Math nee less tie an less theoretical knowlege s to have exact results of any engineering oels generally, especially any or 3 oel of echaniss. Fro Tables 1 an, can be see sall ifference on results which result of use of erivative an integral atheatical operations. ifference shows avantage of use of Working Moel, because results fro this software are ore realistic in coparison with results fro Math which are theoretical ieal, receive by strict calculations. In secon circle, otion of echanis is stabilize. Results fro both software s are ientically.. REFERENES [1] H.J. Soer: Vector loops facilitate siple kineatic analysis of planar echaniss, either closechain or openchain, Mechanical & Nuclear Engineering Faculty, US, [] J. M. Mcarthy an G. S. Soh, 010, Geoetric esign of Linkages, Springer, New York. [3] B. Paul, Kineatics an ynaics of Planar Machinery, PrenticeHall, NJ, 1979 [] L. W. Tsai, Robot nalysis: The echanics of serial an parallel anipulators, John Wiley, NY, [] K. J. Walron an G. L. Kinzel, Kineatics an ynaics, an esign of Machinery, n E., John Wiley an Sons, 00. [6] Torby, Bruce (198). "Energy Methos". vance ynaics for Engineers. HRW Series in Mechanical Engineering. Unite States of erica: BS ollege Publishing. ISBN [7] T. R. Kane an.. Levinson, ynaics, Theory an pplications, McGrawHill, NY, 00. [8] K. Sai Prasa, G. Prabhakar Rey, N hanra Sekhar Rey, K. Sai Teja, n utoate Mechanis For Sart Packaging Line Manageent, International Journal of Mechanical Engineering an Technology, 8(6), 017, pp [9] r. U. Sathish Rao an r. Lewlyn L.R. Rorigues. Enhancing the Machining Perforance of HSS rill in the rilling of GFRP oposite by Reucing Tool Wear through Wear Mechanis. International Journal of Mechanical Engineering an Technology, 8(1), 017, pp eitor@iaee.co