The Analysis of the Non-linear Deflection of Non-straight Ludwick type Beams Using Lie Symmetry Groups

Transcription

1 Proceedigs of the 3 rd Iteratioal Coferece o Cotrol, Dyamic Systems, ad Robotics CDSR 16 Ottawa, Caada May 9 10, 016 Paper No. 107 DOI: /cdsr The Aalysis of the No-liear Deflectio of No-straight Ludwick type Beams Usig Lie Symmetry Groups M. Ami Chagizi 1, Davut Erdem Sahi, Io Stiharu 3 1 Kowledge Egieerig, Itelliquip Co. 3 W Broad St, Bethlehem, PA 18018, USA achagizi@itelliquip.com Bozok Uiversity, Departmet of Mechaical Egieerig 6600, Yozgat, Turkey davut.sahi@bozok.edu.tr 3 Cocordia Uiversity, Departmet of Mechaical ad Idustrial Egieerig, 1455 De Maisoeuve Blvd. W., Motreal, Quebec, Caada, H3G 1M8 istih@alcor.cocordia.ca Abstract - Noliear deflectio of beams uder the various forces ad boudary coditios has bee widely studied. The predictio of deflectio of beams has bee of great iterest to geeratios of researchers. This seems to be a mudae problem as it is subject of textbooks o elemetary mechaics of materials. Although both aalytical ad umerical solutio have bee foud for specific type of loads, the geeral problem for beams that are ot geometrically perfectly straight has ot bee approached so far of a systematic fashio. The preset work preseted a geeral method based o Lie symmetry groups that yields a exact solutio to the geeral problem ivolvig ay arbitrary loadig o-straight Ludwick type micro-beams. Lie symmetry method is used to reduce the order of the ODE describig the large deflectio of the beam. The solutio is validated agaist the particular cases of loadig for which the large deflectio problem has bee solved ad preseted i the ope literature. I this work a approach i solvig the geeral problem based o Lie symmetry groups ad a geeral aalytical solutio of the problem is preseted below. The objective of the preset work is to ivestigate a versatile mathematical method ito the deflectio of geometrically o-straight catilever beams subjected poit loads ad momets applied at the free ed while experiecig o-liear deflectio. Lie symmetry method preseted below ca be used to ay geometry of the beded beams uder to coditio that there is o residual stress i the uloaded beam. The deflectio equatio was calculated based o Ludwick experimetal strai-stress curve. The itegral equatio was solved umerically ad the ed beam deflectio ad rotatio were calculated. The same problem of large deflectio catilever beams made from materials behavig of oliear fashio uder the tip poit force was solved by fiite differece methods. Keywords: No-Liear Deflectio, Ludwick Beams, Lie Symmetry 1. Itroductio Noliear deflectio of beams uder the various forces ad boudary coditios has bee widely studied. The predictio of deflectio of beams has bee of great iterest to geeratios of researchers. This seems to be a mudae problem as it is subject of textbooks o elemetary mechaics of materials. Although both aalytical ad umerical solutio have bee foud for specific type of loads, the geeral problem for beams that are ot geometrically perfectly straight has ot bee approached so far of a systematic fashio. Despite the iterest i the subject, so far there is o geeral solutio to describe the geeral case of loadig. This research presets a approach i solvig the geeral problem based o Lie symmetry groups ad a geeral aalytical solutio of the problem is preseted below. The objective of the preset work is to ivestigate a versatile mathematical method ito the deflectio of geometrically o-straight catilever beams subjected poit loads ad momets applied at the free ed while experiecig o-liear deflectio for Ludwick type Beams. Lie symmetry method preseted below ca be used to ay geometry of the beded beams uder to coditio that there is o residual stress i the uloaded beam. Noliear deflectio of beams subjected to various types of forces ad boudary coditios have bee extesively studied. The differetial equatio of large deflectio of catilever beam uder a poit force at the tip was solved i

2 [1]. I that approach the differetial equatio of the slope of the beam versus the legth of the deflected curve was formulated ad solved based o complete secod ad first kid elliptic itegrals. The differetial equatio of slope versus legth of the deflected curve based o cosideratio of shear force was umerically solved []. The authors used fiite differece methods to solve ordiary differetial equatio ODE for distributed force o catilever ad simple supported beams. They also used the same method to solve the ODE of the simple supported beam uder a poit force. A umerical solutio for the tapered catilever beam uder a poit force at tip was preseted i 1968 [3]. The author coverted ODE to o-dimesioal ODE ad used a computer to solve it. A catilever beam, made from materials exhibitig oliear properties subjected to a poit force was also studied [4]. The deflectio equatio was calculated based o Ludwick experimetal strai-stress curve. The itegral equatio was solved umerically ad the ed beam deflectio ad rotatio were calculated. The same problem of large deflectio catilever beams made from materials behavig of o-liear fashio uder the tip poit force was solved by fiite differece methods [5]. The authors solved the oliear ODE of curvature for a catilever made from oliear characteristics a material ad subjected to poit force at the tip by umerical methods. Power series ad eural etwork were used to solve large deflectio of a catilever beam uder tip force [6]. Noliear ODE were decomposed to a system of ODEs ad solved by eural etworks. Large deflectios of catilever beams made from oliear elastic materials uder uiform distributed forces ad a poit force at tip were also studied [7]. I this work a system of oliear ODEs was developed to model the system which was further solved by Ruge-Kutta method. Researchers [8] i used almost a similar method that was used i [1] to solve the large deflectio of a catilever uder the poit force at the tip ad they validated their results with experimets. Also they used o-dimesioal formulatio to simplify the oliear deflectio to liear aalysis. They showed that oliear small deflectio is same as those foud through the liear aalysis. Two dimesioal loadig of catilever beams with poit forces loads at the free ed was studied for o-prismatic ad prismatic beams [9]. Authors formulated the model for the geeral loadig coditios i beams. The result is a oliear PDE which is preseted i this paper. Further, the authors umerically solved the o-dimesioal equatio usig a polyomial to defie the rotatig agle of the beam. They preseted some examples applied to their methods. A catilever beam subjected to a tip momet with oliear bimorph material was theoretically ad umerically studied [10]. The authors used a exact solutio for the deflectio of a catilever with a momet applied at the tip. Catilever beam uder uiform ad tip poit force was umerically ad experimetally studied [11]. I this study, the authors used a system of ODEs to solve umerically this problem. Fiite differece methods for aalysis of large deflectio of a o-prismatic catilever beam subjected to differet types of cotiuous ad discotiuous loadigs was studied [1]. Authors formulated the problem based o [9] ad further used quasi-liearizatio cetral fiite differeces method to solve the problem. A explicit solutio for large deflectio of catilever beams subjected to poit force at the tip was obtaied by usig the homotopy aalysis method HAM preseted i [13]. Large deflectio of a o-uiform sprig-higed catilever beam uder a follower poit force at the tip was formulated ad solved umerically [14].. Ludwick Materials Geerally the relatio betwee stress ad strai uder small stress coditios is liear as stated by Hook Law. By icreasig stress oliear behavior of material is expressed. For most of materials the o-liear behaviour ca be defied as a stress-strai relatio like: σ = Eε 1 I = y +1 da = 1 +1 bh This kid of materials are called Ludwick materials. Here, B ad represet costats related to material properties. For a rectagular cross sectio of beam subjected to a momet at the free ed. 107-

3 . Formulatio of the Large Deflectio of Not-Straight Beam Problem Fig. 1: A o-straight catilever uder tip forces ad momet. A o-straight catilever beam as show i figure 1 is subjected a poit forces. Iteral reactio momet i ay cross sectio o the left had side of catilever ca be writte as: Mx, y = Vx 0 x + Hy 0 y + M 0 1 where: V = P si θ 0 H = P cos θ 0 M 0 = P. u u is distace from ed of tip to eutral axis of catilever beam. Euler Beroulli momet curvature relatioship gives: dθ dρ = Mx, y + 1 bh+1 Where: dx = cos θ 3 dρ dy = siθ 4 dρ θ is agle of curved beam. Derivative of gives: d θ dρ = V cosθ + Hsiθ + 1 bh+1 5 The boudary coditios for this ODE are: 107-3

4 L is legth of catilever. The equatio 5 ca be writte as dθ dρ = ρ=l θ ρ=0 = 0 6 M bh+1 7 d θ dρ = V cos θ Hsiθ + 1 bh+1 8 Oe ca show that for a secod order ODE ifiitesimal trasformatio η ca be calculated as i [1, 13] η = D η j! j! θ j+1d j ξ j=1 = η ρρ + η ρθ ξ ρρ θ + η θθ ξ ρθ θ ξ θθ θ 3 + η θ ξ ρ 3ξ θ θ θ = ξω p + ηω θ + η θ ξ ρ θ ξ θ θ ω θ 9 Where: ω = Vco sθ+hsiθ +1 bh +1 By decomposig 9 ito a system of PDEs, ξ ad η ca be calculated. Most of Lie symmetries icludig rotatio traslatio ad scalig could be foud with the help of the below trasformatios: ξ = C 1 + C ρ + C 3 θ η = C 4 + C 5 ρ + C 6 θ 10 where: C 1, C, C 3, C 4, C 5, C 6 are costat umbers. Substitutio 9 i 10 yields: Vco sθ + Hsiθ Vco sθ Hsiθ C 6 C 3C 3 θ = + 1 bh+1 C 4 C ρ + C 6 θ + 1 bh+1 11 By comparig terms, oe ca show that Therefore: C = C 3 = C 4 = C 5 = C 6 =

5 ξ = C 1 η = 0 13 Equatios 13 give the geeral trasformatio for 8. It is possible to cosider ξ ad η as: ξρ, θ = 1 14 ηρ, θ = 0 15 Hece, X = ξx, y + ηx, y, is called a poit symmetry, ca be simplified as a operator: x y X = ρ 16 caoical coordiates, sρ, θ ad rρ, θ ca be derived from the solutio of below ODE for rρ, θ: dθ ηρ, θ = dρ ξρ, θ 17 ad sρ, θ will be: dρ sρ, θ = ξρ, θr, ρ r=rρ,θ 18 Substitutig 14 ad 15 i 17 ad 18 gives By defiig ur as: rρ, θ = θ 19 sρ, θ = ρ 0 ur= 1 dθ dρ 1 Oe ca show that: dur dr = d θ dρ dθ dρ So: d θ dur = ur 3 dρ dr 3 Substitutio of 1 ad i 3 gives: 107-5

6 dur dr Vco sθ + Hsiθ = ur 3 4 Substitutio of 19 ad 0 i 4 will yield: duθ dθ = Vco sθ + Hsiθ uθ 3 5 For a first order ODE like 5 η 1 ca be writte as: η θ + η u ξ θ ω ξ u ω = ξω θ + ηω u 6 Where: ω = Vco sθ+hsiθ +1 bh +1 uθ 3 Substitutig ω ad its derivatives i 6 yields 1 η θ + η u ξ θ Vco sθ + Hsiθu 3 1 ξ Vco sθ + Hsiθu E 1 +1 ξ u Vco sθ + Hsiθ u 6 = + 1 bh+1 30 ηvco sθ + Hsiθu 7 From 7, equatig u 6 ad free term, it is foud that η θ = ξ u = 0 8 By comparig the coefficiets of u 3, oe ca write η u ξ θ Vco sθ + Hsiθ = ξ Vco sθ + Hsiθ 9 To satisfy 9, ξ must be assumed as zero ξ = 0 30 hece 7 becomes η u u = 3η

7 The itegral of this ODE is: η = u 3 3 So X = ξx, y x + ηx, y y becomes: X = u 3 u 33 Based o 33, caoical coordiates ca be calculated from 17 ad 18 as: Substitutig 34 ad 35 i ds = s x+s y fx,y = Fr [13]yields: dr r x +r y fx,y rθ, u = θ 34 sθ, u = 1 u 35 ds dr = Vco sr + Hsir 36 So: S = Vco sr + Hsir Further, by substitutig 1, 34 ad 35 i 37 oe gets: + C bh+1 dθ dρ = Vco sr + Hsir + C 1 38 By cosiderig boudary coditio 6, C 1 becomes: C 1 = M bh+1 Vco sθ f + Hsiθ f 39 where θ f is fial agle i the tip. Solutio of 38 yields: 107-7

8 ρ = dθ + C Vco sθ f + Hsiθ f + C 1 40 From equatios 3 ad 4 oe ca write equatio 38 as: a dx 0 b dy 0 = 0 = 0 θ f θ f dθ cos θ Vco sθ f + Hsiθ f + C 1 dθ si θ Vco sθ f + Hsiθ f + C which relatioship, after the itegratio will yield the deflectio at the tip of the ot-straight catilever beam. 4. Coclusio The deflectio of geometrically o-straight catilever beams subjected poit loads ad momets applied at the free ed while experiecig o-liear deflectio for Ludwick type Beams was obtaied based o Lie symmetry method to reduce the order of the ODE describig the large deflectio of the beam. Hereby geeral aalytical solutio of the deflectio for the ot-straight beam was systematically preseted. This method is very geeral ad powerful method for solvig oliear differetial equatios. Refereces: [1] K. E. Bisshopp ad D. Drucker, Large deflectio of catilever beam, Notes, vol.101, o. 3. [] T. M. Wag, S. L. Lee, ad O. C. Ziekiewicz, A umerical aalysis of large deflectios of beams, Iteratioal Joural of Mechaical Scieces, vol. 3, o. 3, pp. 19-8, [3] J. D. Kemper, Large deflectios of tapered catilever beams, Iteratioal Joural of Mechaical Scieces, vol. 10 o. 6, pp , [4] G. Lewis ad F. Moasa, Large deflectios of catilever beams of o-liear materials of the Ludwick type subjected to a ed momet, Iteratioal Joural of No-Liear Mechaics, vol. 17, o. 1, pp. 1-6, 198. [5] F. Moasa ad G. Lewis, Large deflectios of poit loaded catilevers with oliear behaviour, Zeitschrift für Agewadte Mathematik ud Physik ZAMP, vol. 34, o. 1, pp , [6] M. H. Ag Jr, W. Wei, ad L. Teck-Seg, O the estimatio of the large deflectio of a catilever beam, [7] K. Lee, Large deflectios of catilever beams of o-liear elastic material uder a combied loadig, Iteratioal Joural of No-Liear Mechaics, vol. 37, o. 3, pp , 00. [8] T. Belédez, C. Neipp, ad A. Belédez, Large ad small deflectios of a catilever beam, Europea joural of physics, vol. 3, pp. 371, 00. [9] M. Dado ad S. Al-Sadder, A ew techique for large deflectio aalysis of o-prismatic catilever beam, Mechaics Research Commuicatios, vol. 3, o. 6, pp , 005. [10] C. Baykara, U. Guve, ad I. Bayer, Large deflectios of a catilever beam of oliear bimodulus material subjected to a ed momet, Joural of Reiforced Plastics ad Composites, vol. 4, o. 1, pp , 005. [11] T. Beledez, et al., Numerical ad experimetal aalysis of large deflectios of catilever beams uder a combied load, Physica scripta, pp. 61,

9 [1] S. Al-Sadder ad R. A. O. Al-Rawi, Fiite differece scheme for large-deflectio aalysis of o-prismatic catilever beams subjected to differet types of cotiuous ad discotiuous loadigs, Archive of Applied Mechaics, vol. 75, o. 8, pp , 006. [13] J. Wag, J. K. Che, ad S. Liao, A explicit solutio of the large deformatio of a catilever beam uder poit load at the free tip Joural of Computatioal ad Applied Mathematics, vol. 1, o., pp , 008. [14] B. S. Shvartsma, Large deflectios of a catilever beam subjected to a follower force, Joural of Soud ad Vibratio, vol. 304, o. 3-5, pp ,