Cantilever Beam and Torsion Rod Design Optimization Using Genetic Algorithm

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1 ISSN (Onlne) : ISSN (Prnt) : Internatonal Journal of Innovatve Research n Scence, Engneerng and Technology Volume 3, Specal Issue 3, March Internatonal Conference on Innovatons n Engneerng and Technology (ICIET 1) On 21 st & 22 nd March Organzed by K.L.N. College of Engneerng and Technology, Madura, Taml Nadu, Inda Cantlever Beam and Torson Rod Desgn Optmzaton Usng Genetc Algorthm 1 V.Muthukumaran, 2 R.Rajmurugan, 3 V.K.Ram kumar Department of Mechancal Engneerng, Kumaraguru college of Technology, Combatore, Tamlnadu, Inda. ABSTRACT- Any desgn optmzaton problem would be formulated as a multvarable, mult objectve, constraned, non-lnear programmng problem. Varous conventonal optmzaton technques are avalable for solvng the above such problems. Due to the problem complextes and computatonal neffcences, better methods are needed. To acheve ths goal, new technques popularly known as ntellgent technques have been ntroduced. In ths work t s proposed to mplement Genetc Algorthm for solvng the desgn optmzaton problem of Cantlever beam and Torson rod. Cantlever beam and Torson rod are used as structural elements n many Mechancal Engneerng applcatons. The objectve here s to desgn a cantlever beam and a Torson rod for mnmum weght n order to wthstand the gven workng condton. KEYWORDS- Optmzaton, Cantlever Beam, Torson rod, Genetc Algorthm I. INTRODUCTION The goal of optmzaton s to mnmze or maxmze certan quanttes such as lfe, mass, etc. In the mathematcal model these goals are expressed as functons of certan varables. In a desgn problem those varables may be the varous dmensons of the component. There are many methods avalable for solvng these models towards mnmzaton or maxmzaton [1]. GA gves effcent result from mult-varable constraned non-lnear programmng problem. Genetc algorthms are global optmzaton methods based on several metaphors from bologcal evoluton. The GA search for an optmum soluton s from the populaton of canddate soluton accordng to an objectve functon, whch s used to establsh the ftness of each canddate, as a soluton. The cantlever beam and Torson rods are wdely used as engneerng components n varous forms of applcatons and optmsng the mass of these types of components becomes mportant. In ths study, a problem has been formulated to mnmze the mass of a cantlever beam and Torson rod. A genetc algorthm has been used n ths study to optmze the desgn parameters to acheve the mnmum mass whch satsfy varous constrants. Research has been carred out to develop a optmzaton algorthm based on both bnary-coded and real-coded genetc algorthms [2] to solve mechancal component desgn problems lke gear tran desgn, Bellevlle sprng desgn, welded beam desgn. A method s proposed to prove the effcency and ease of applcaton of genetc algorthms by solvng four dfferent mechancal desgn problems [3]. A genetc algorthm based approach s developed for prelmnary cam shape desgn and optmzaton based on predctve computer smulaton of cam mechansm [].The am of ths work s to descrbe a method that can be effcently used for automatc cam shape determned and systematc seekng for optmum cam shaped. New approach has been proposed to optmze the structure optmzaton wth frequency constrants [5]. Work has been done to optmze tolerance desgn for gear tran and over runnng clutch usng genetc Copyrght to IJIRSET

2 algorthm [6]. The objectve of ths work s to develop a genetc algorthm based optmzaton procedure. The machne element col sprng, hollow shaft, pulley drve desgn has been optmzed usng the other non-tradtonal optmzaton algorthm lke Smulated annealng [7]. The objectve of ths work s to mply that the smulated annealng algorthm technque can be appled effcently and effectvely for any type of desgn problem. II. GENETIC ALGORITHM Genetc algorthm smulates the survval of the fttest among ndvduals over consecutve generaton for solvng a problem. Each generaton conssts of a populaton of character.each ndvduals represents a pont n a search space and a possble soluton. The ndvdual n the populaton are then made to go through a process of evoluton [8].The basc concept of genetc algorthm s to encode a potental soluton to a problem as a seres of parameters. A sngle set of parameters value s termed as the genome of an ndvdual soluton. An ntal populaton of ndvduals s generated randomly or statcally. In every generaton the ndvduals n the current populaton are decoded accordng to a ftness functon. The chromosome wth the hghest populaton ftness s selected for matng. The genes of the two parameters are allowed to exchange to reproduce off sprngs. These chldren then replace ther parents n the next generaton. Thus, the old populaton s dscarded and the new populaton becomes the current populaton [9]. The current populaton s checked for acceptablty or soluton.the teraton s stopped after the completon of maxmal number of generatons or on the attanment of the best results. The varous steps nvolved are brefly descrbed as gven below ) Start: Random populaton of n chromes (sutable soluton for the problem) are generated. Tournament selecton s used. [Melane Mtchell,1998] v) Cross-Over: Two parents are crossed to form a new off sprng wth a cross over probablty. v) Chld Ness: If no cross over s performed offsprng s an exact copy of parents. v) Mutaton: New off sprngs are mutated wth a mutaton probablty. v) Acceptng: The new off sprng are placed n a new populaton. x) Replace: Newly generated populaton s used for a further run of algorthm that s ndvduals from old populaton are klled and replaced by the new ones. x) Test: The generaton s stopped, f the end condton s satsfed and returns the best soluton n the current populaton. x) Loop: If the termnaton crtera are not met, the loop s repeated from the ftness step agan as reported above. A. Problem defnton III. OPTIMIZATION MODEL Case 1: Cantlever Beam Mnmse the mass of a cantlever beam shown n Fg. (1). It shows a cantlever beam arrangement whch s subjected to a load P. Due to ths load P stress and deflecton are produced n the beam. The objectve of the problem s to fnd the optmal values of the cross secton parameters lke r and r o, such that the mass of the beam wll be mnmum wthn the gven range of r and r o [10]. P ) Ftness: The ftness functon of each chromosome n the populaton s evaluated A ro ) New Populaton: A new populaton s created by repeatng followng steps. A l (m) r v) Selecton: Two parent chromosomes are selected from the populaton accordng to ther ftness, bgger the chance to be selected. In ths work Secton A-A Outer radus - r o Inner radus - r Fg. 1 Cantlever Beam Copyrght to IJIRSET

3 1) Desgn varables-the chosen cross secton here s hollow crcular cross secton. In ths desgn problem the varables are nner and outer radus of the cross secton r, r o. 2) Objectve functon-the objectve of ths work s to mnmse the mass f(x) of the cantlever beam. Mnmse f(x) = ρ l П (r o 2 r 2 ) Takng the constrants n to account a penalty factor s ntroduced and the new objectve functon f(x) =ρ l П (r o 2 r 2 )+ Penalty factor The penalty factor s the value whch corresponds to any volaton of the gven constran. 3) Stress constrants-the maxmum allowable stress whch s developed n the beam due to the load P.Let σ b be the bendng stress and τ be the shear stress developed wth the gven load P Let σ a & τ a be the allowable bendng and shear stress respectvely. Then the stress constrants are σ b σ a & τ τ a 1.27* * l * ro b r r 0 0.2* P*( r o o r 2 r 0 σ a (165 MPa) 2 r r r ) τa (50MPa) ) Dmensonal Constrants-The cross sectonal area must be non-negatve (.e.) A 0. r < r o and r o r 0. The range of these r and r o should be selected accordngly..e. thckness t = r o - r Therefore, r = r o -t And t 0 and t max < r o mn B. Implementaton of GA Data for the Problem The upper and lower lmts of the desgn varables are selected as follows consderng the dmensonal constrans. 3.5 cm r o 6 cm 1 cm t 5 cm C. Selecton of GA parameters The GA parameters Mutaton probablty and cross over probablty values were selected by makng varous runs wth dfferent combnaton of these parameters as tabled n table-i. The results were plot n Fg-2 and from whch t s selected that the CP as 0.9 and the MP as 0.. Cantlever TABLE I- GA PARAMETER Total 20 dgts Sample Sze 0 Selecton Operator Cross-over Probablty (Pc) Mutaton Probablty (Pm) No. of generatons n each run Tournament Selecton 0.7, 0.8, , 0.005, 0.010, 0.020, 0.030, 0.00, No. of runs Mutaton Probablty CP 0.7 CP 0.8 CP 0.9 Fg. 2 Mnmum Objectve Functon of Cantlever Beam Allowable bendng stress σ a =165 MPa Allowable shear stress τ a = 50 MPa Densty ρ =7850 kg/m 3 Length l =10 mts Load p=1 KN Copyrght to IJIRSET 268

4 D. Problem defnton Case 2: Torson Rod ) Dmensonal Constrant- A A 1 (m) T 0 r o r Desgn the x 1 = Outsde dameter d 0 x 2 = Insde dameter / Outsde dameter 0.02m d 0.5m d 0.6 d o o Secton A-A, Outer radus - r o, Inner radus -r Fg. 3 Torson Rod hollow torson rod as shown n Fg 3 to satsfy the followng requrements [10]. 1) Desgn varables- The chosen cross secton here s hollow crcular cross secton. In ths desgn problem the varables are nner and outer dameter of the cross secton d, d 0. 2) Objectve functon-the objectve of ths work s to mnmse the mass f(x) of the cantlever beam. Mnmse f(x) = (П /)ρl(d 0 2 d 2 ) Takng the constrants n to account a penalty factor s ntroduced and the new objectve functon f(x) = (П /)ρl(d 0 2 d 2 )+Penolty factor The penalty factor s the value whch corresponds to any volaton of the gven constran. 3) Stress constrants-calculated shear stress shall not exceed the allowable shear stress under normal operatng torque T o (N m). E. Implementaton of GA Data for the Problem Fve dfferent materals were tred at three dfferent lengths as 0.5m, 0.75m and 1m.The rod requrements, materals and propertes are gven n the followng tables II and IV. Torson rod TABLE II- ROD REQUIREMENTS Length l (m) Normal torque T 0 (kn m) Max Tmax ( kn m) Allowabl e twst θa (degrees) The calculated angle of twst θ c shall not be exceedng the allowable twst θ a (radans). The member shall not buckle under a short duraton of T max (N m) cr 12 l c 0 a GJ C 0 a a J d E 0 d 1 v d m Max Copyrght to IJIRSET

5 Cantlever Beam and Torson Rod Desgn Optmzaton Usng Genetc Algorthm F. Selecton of GA parameters The GA parameters were selected as n the case 1 and the MP as 0.05 and CP as 0.8 from Fg-.The parameters are tabled n table -III Cantlever Case 1: Fg () Mnmum Objectve functon of Torson Rod TABLE III- GA PARAMETER Total 20 dgts Sample Sze 0 Selecton Operator Cross-over Probablty (CP) Mutaton Probablty (MP) No. of generatons n each run Mutaton Probablty CP-0.7 CP-0.8 CP-0.9 Tournament Selecton 0.7, 0.8, , 0.02, 0.03, 0.0, 0.5, 0.8, 0.1, IV. RESULTS AND DISCUSSION Fg 5 shows the soluton hstory of the GA run. The mnmum objectve functon s acheved durng the generaton number 6 as kg. Mnmum Objectve Functon(CP 0.9/MP 0.0) Materal alloy steel 2. Al alloy 2 ST 3. Mg alloy A261. Beryllum 5.Ttanu m Case 2: Fg (5) Soluton hstory of GA Run TABLE IV- MATERIALS AND PROPERTIES Densty Kg/m3 Allowabl e shear stress τ a (M Pa) Elastc Modulu s E (G pa) Shear modulu s G(G pa) Posso ns rato ν The optmum mass value for the torson rod wth dfferent materals and wth dfferent lengths were found through GA and lsted n table V. The fgures from 6 to 20 shows the soluton hstory of those GA runs and found the curves get converged. Copyrght to IJIRSET

6 TABLE V- OPTIMUM MASS VALUE OF TORSION ROD Mass 27 kg Fg (6) Mnmum Objectve Functon 10 Alloy Steel/ L 0.5m Materal Length m Mn 10 alloy steel Alumnum Alloy 2 ST Mg Alloy A Beryllum Ttanum Mass 80 kg Fg (7) Mnmum Objectve Functon10 Alloy Steel/ L 0.75 m Mass 13 kg Fg (9) Mnmum Objectve Functon Alumnum Alloy 2 ST/L 0.5 m Fg (8) Mnmum Objectve Functon 10 Alloy Steel/ L 1 m 3 32 Mass 30kg Fg (10) Mnmum Objectve Functon Alumnum Alloy 2 ST/L 0.75m Copyrght to IJIRSET

7 Fg (11) Mnmum Objectve Functon Alumnum Alloy 2 ST/L 1 m Fg (1) Mnmum Objectve Functon Magnesum Alloy A261 /L 1 m Mass 8.5kg Fg (12) Mnmum Objectve Functon Magnesum Alloy A261 /L 0.5 m Fg (13) Mnmum Objectve Functon Magnesum Alloy A261 / L 0.75 m 15 1 Mass 13kg Fg (15) Mnmum Objectve Functon Beryllum/L 0.5 m Mass 20 kg Fg (16) Mnmum Objectve Functon Beryllum/L 0.75 m Fg (17) Mnmum Objectve Functon Beryllum/L 1 m Copyrght to IJIRSET

8 Fg (18) Mnmum Objectve Functon Ttanum/L 0.5 m Mass 35kg Fg (19) Mnmum Objectve Functon Ttanum/ L 0.75 m Mass 75kg Fg (20) Mnmum Objectve Functon Ttanum/L 1 m V. CONCLUSION A standardsed computer program usng C language was developed that can be easly manpulated to synthess the optmal desgn for the components lke cantlever beam and torson rod. The proposed procedure usng GA has the capablty to acheve mnmum mass for the gven component. Consderable reducton n computatonal effort s acheved by ths proposed procedure. Dfferent tradtonal technques have been reported earler to optmse the above knd of components. In ths work, t s demonstrated that a GAbased procedure can be appled to a varety of desgn optmsaton problems. REFERENCES [1] Sngresu S. Rao, (2009) Engneerng Optmzaton Theory and Practce, John wley and sons. [2] Kalyanmoy Deb and Mayank Goyal, (1996) A combned genetc adaptve Search (GeneAS) for Engneerng desgn, Computer Scence and Informatcs 26(), [3] Kalyanmoy Deb and Mayank Goyal, (1998). A Flexble Optmzaton Procedure for Mechancal Component Desgn Based on genetc adaptve Search, Journal of Mechancal Desgn, [] lampnen.j. (2003) Cam shape optmsaton by genetc algorthm, Computer Aded Desgn 35: [5] We Lngyun, Zhao Me, Wu Guangnng, Meng Guang (2005), Truss optmzaton on shape and szng wth frequency constrants based on genetc algorthm, Computatonal Mechancs,35: [6] Noorul hag.a, K.Svakumar, R. Saravanan, V. Muthuah (2005).Tolerance desgn optmzaton of machne elements usng genetc algorthm, Internatonal Journal of Advanced Manufacturng Technology, 25: [7] Rathnam.Ch, V.B. Rajendra Prasad, KNS Prakasha Rao (2006) Optmal Desgn of Machne elements by usng smulated Annealng Algorthms, Industral Engneerng Journal, vol. XXXV No. 6: [8] Kalyanmoy Deb, Optmzaton for Engneerng Desgn Algorthms and examples, (2012), Prentce Hall of Inda prvate Lmted. [9] Goldberg, (2013) Genetc Algorthm, Pearson Educaton Inda. [10] Jasbr S. Arora, Introducton to Optmum Desgn, (2011) Elsever. APPENDIX NOMENCLEATURE: Symbols M = Mass of the rod (kg) r = Insde dameter of the beam r o = Outsde dameter of the beam d 0 = Outsde dameter of the rod (m) d = Insde dameter of the rod (m) Ρ = Mass densty of materal (kg/m 3 ) L = Length of the rod (m) T 0 = Normal operatng torque (N m) C = Dstance from rod axs to extreme fbre (m) J = Polar moment of nerta (m ) θ = Angle of twst (radans) G = Shear modulus (Pa) T cr = Crtcal bucklng torque (N m) E = Modulus of elastcty (pa) ν = Posson s rato T Max = Maxmum Torque (N m) Copyrght to IJIRSET

9 σ b σ a τ τ a = Bendng stress = Allowable bendng stress = Shear stress = Allowable shear stress Copyrght to IJIRSET