INVESTIGATION OF ELASTIC PENDULUM OSCILLATIONS BY SIMULATION TECHNIQUE


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1 P AM U K K A L E Ü N İ V E S İ T E S İ M Ü H E N D İ S L İ K F AK Ü L T E S İ P A M U K K A L E U N I V E S I T Y E N G I N E E I N G F A C U L T Y M Ü HE N D İ S Lİ K B İ Lİ M LE İ J O U N A L O F E N G I N E E I N G D E Gİ S İ S C I E N C E S YIL CİLT SAYI SAYFA : 9 : 5 : : 886 INVESTIGATION OF ELASTIC PENDULUM OSCILLATIONS BY SIMULATION TECHNIQUE Zekeriya GİGİN, Ersin DEMİ Paukkale University, Engineering Faculty, Mechanical Engineering Departent, 7, Denizli Geliş Tarihi :.7.8 Kabul Tarihi : ABSTACT In this study, elastic springass pendulu scillatins are investigated. In rder t slve a nnlinear differential equatin syste, Siulatin Technique based n Mdelica Language such as Dyla, SiulatinX etc., is used. It's assued that the spring cefficient in this syste is linear and spring ass is neglected. Under these cnditins, kineatic behavir f the pendulu was investigated. The gverning equatin f the syste pssessing tw nnlinear differential equatins which interacts each ther are slved siultaneusly. The btained results are cpared with previus wrks and seeed gd agreeents with thers. Keywrds : Elastic pendulu, Nnlinear scillatin, Siulatin technique, Mdelica, Dyla. SİMÜLASYON TEKNİĞİ İLE ELASTİK KÜTLEYAY SALINIMINLAININ İNCELENMESİ ÖZET Bu çalışada, elastik kütleyay sarkaç salınıları inceleniştir. Sistein lineer layan diferansiyel denklelerini çözek için Dyla, SiulatinX gibi Mdelica dili tabanlı Siülasyn Tekniği kullanılıştır. Sistedeki yayın direngenliği lineer ve kütlesi ihal ediliştir. Bu şartlar altındaki sarkacın kineatik davranışı inceleniştir. Sistei ifade eden genel denkle iki tane lineer layan ve birbirini etkileyen diferansiyel denkleden luşaktadır. Bu denkleler Siülasyn Tekniği ile çözülüştür. Elde edilen snuçlar önceki çalışalarla kıyaslanış ve uyulu lduğu görülüştür. Anahtar Kelieler : Elastik sarkaç, Lineer layan salını, Siülasyn Tekniği, Mdelica, Dyla.. INTODUCTION Earlier, Nayfeh btained the slutin f the pendulu scillatins analytically and nuerically by using Perturbatin Techniques (Nayfeh, 987). Mst f nnlinear differential equatins can be slved by different nuerical ethds, such as Finite Eleent Methd (FEM), Finite Difference Methd (FDM), Variatinal Iteratin Methd (VIM), Htpy Perturbatin Methd (HPM) and etc. Mst f the use re girds t slve syste nuerically ecept Differential Quadrature Methd (DQM). Liu and Wu (Liu and Wu, ) slved Duffing Equatin by using DQM and used Frechet Derivative in rder t ake syste linear. He (He, 999; 3) was prpsed VIM and HPM t slve nnlinear differential equatin (NDE) nuerically. In VIM, the prbles are initially appriated with pssible unknwns. Than a crrectin functin is cnstructed by a general Lagrange ultiplier, which can be identified via the variatinal thery. In cntrast t the traditinal perturbatin ethd, HPM des nt require a sall paraeter in an equatin. 8
2 Authr used a Duffing equatin with high rder f nnlinearity t illustrate it s effectiveness. The differential quadrature ethd is etended t slve secndrder initial value prbles by Fung (Fung, ). NDE systes, which interacts each ther, are nt able t be slved easily abve ethds. Fr this reasn, new nuerical techniques such as, Genetic Algriths (GAs), Artificial Neural Netwrk (ANN), Fuzzy Lgic (FL) etc. have been used recently. ST is anther alternative ethd in rder t slve NDE. Althugh the prble cnstructin is very easy, results are re crrect when cpared with thers (Gergiu, 999). Elastic pendulu scillatin behavirs were als investigated by Chang and Lee, wh applied GAs t investigate duble pendulu scillatins (Chang and Lee, 4). Girgin ipsed Cbining Methd t study nnlinear pendulu scillatins (Girgin, 8). Later, Lynch investigated three diensinal tin f the elastic pendulu assuing that aplitude is sall (Lynch, ). Lynch and Hugtn investigated three diensinal tin f the elastic pendulu in the case f resnance and derived suitable initial cnditins using envelpe dynaics (Lynch and Hugtn, 4). In additin t, Vetyukv et al. derived the nnlinear equatin f tin f the D flating rectangular frae and the defratin was interplated by eans f plynial shape functins (Vetyukv et al., 4).. GOVENING EQUATIONS OF THE SYSTEM Gverning equatin f the elastic pendulu is btained by writing, equilibriu equatin f the pendulu shwn in Figure at psitin. Crilis acceleratin is added t the syste because pendulu ass is cnnected t spring instead f crd which causes t crilis acceleratin. A reference line A reference line k(r) (Lr) (Lr) g r r Figure. Frce equilibriu f the elastic pendulu scillating abut pint A. Writing Newtn s principle in r and directin give us the gverning equatins; d r k dt dt dr d dt dt gsin dt L r L r L r r g cs () () where; L L : static stable length f the spring with pendulu. L : free spring length. : deflectin because f ass. Equatin () and () have tw variables (r and ) and calculated fr Figure. Mühendislik Bilileri Dergisi 9 5 () Jurnal f Engineering Sciences 9 5 () 886
3 The syste is cnservative because there is n daping. Therefre ttal energy (kinetic energy and ptential energy) f the syste is always cnstant and tie invariant (hlnic). Equatin (3) and (4) depict ttal energy f the pendulu at psitin and as shwn in Figure. dr E k r g L r dt (3) E k r g L r cs (4) where E dentes ttal energy at psitin i. i In rder t investigate behavirs f the elastic pendulu, se paraeters ust be given. Fr this reasn, natural frequency f spring and natural frequency f pendulu are given in Equatin (5) respectively. k g s ve p L (5) Furtherre, diensinless paraeter is given by, p (6) s Spring cnstant (k) and the free length f the spring L values are calculated by, L g, L L, k (7) L d cs d d d d sin (9) () Equatin (9) and () were slved siultaneusly in Dyla and their siulatin schee is depicted in Figure. d d d Figure. Diensinless siulatin schee f and given in Equatin (9) and (). Fr stable scillatins, the suitable value f, depends n initial angle, is btained with siulatin by aking feedback. Stable and instable scillatins are given in Figure 3 with cntinuus and dashed lines respectively..5 sin cs 3. NOMALIZED SOLUTION.5 esults are given in nralized fr; hence they can be cpared with ther wrks. Fllwing transfratins are used fr nralizatin in Equatin () and (). y psitin [] .5 r L ve p t (8)  where and are nralized psitin and tie paraeters. is given Equatin (5). p When the abve transfratins are applied, fllwing equatins are btained in diensinless fr psitin [] Figure 3. Stable and instable scillatin paths. Mühendislik Bilileri Dergisi 9 5 () Jurnal f Engineering Sciences 9 5 () 886
4 Initial angles taking are taken fr t 8 by step size. Initial angular velcity d and relative velcity are taken and crrespnding initial relative psitin depend n initial angle Table. values are taken fr fr Table. Cputed initial radial psitin values depend n initial angle diensinless stable scillatins fr,. () () ,6 ,3684 ,639 ,5 3 ,48 3 ,66 4 , , , , ,59 6 , , , , , 9 ,99 9 ,9856 ,3 , Figure 6. Diensinless linear psitin () diagra versus angular psitin () fr,. This present wrk shwn in Figure 4 was cpared with (Gergiu, 999) shwn in Figure 5 and gd agreeent was seeed. As shwn in Figure 4, ne can realize that althugh it is syetric abut ais, it is asyetric abut ais. The shape changes fr circle t the ne sided stretched ellipse like a balln. Siilarly, in Figure 6 is syetric abut ais and it is asyetric abut ais. In cntrast with Figure 7 and 8 are syetric in all aes. 6 4 d/ . / Figure 4. Diensinless linear velcity diagra versus linear psitin () fr, (present wrk) Figure 7. Diensinless angular velcity diagra versus angular psitin () fr,..5 d/ d/ Figure 5. Diensinless linear velcity diagra versus linear psitin () fr, (Gergiu s wrk) / Figure 8. Diensinless linear velcity diagra versus angular velcity fr,. Mühendislik Bilileri Dergisi 9 5 () Jurnal f Engineering Sciences 9 5 () 886
5 It s nte that pendulu scillatin,. If values eceed this interval, these figures changes rapidly because pendulu is rtating abut A in Figure instead f scillatins EFFECT OF ON THE PENDULUM OSCILLATION Oscillatin paths in y plane are shwn in Figures 9, and fr.,.3 and.4 values respectively. Althugh Figure 9 shws sthness, Figure shws different behavir scillatins fr 5 because f weak stiffness f the spring. This stage can be investigated later fr scientists. Initial angles 7. are taken y psitin [] psitin [] Figure. y psitin f the elastic pendulu in Cartesian crdinate syste fr, CONCLUSION y psitin [] In this paper, ST is used t analyze the elastic pendulu tin. It can be clearly seen that initial value NDEs can be easily slved by ST and the re values cause t instability in the syste. Advantages f ST can be arranged as fllws:. Prble cnstructin is easy.. esults are crrect. 3. There is n analytical prcedure fr a new prble cnstructin psitin [] Figure 9. y psitin f the elastic pendulu in Cartesian crdinate syste fr,. Disadvantage:. It is suitable fr nly initial value prbles. Fr nw ne, it can be etended t tw and three diensinal prbles fr scientists. Als it gives a new technique t slve NDE systes and pens a new area fr pendulu scillatins..5 y psitin [] EFEENCES Chang, C. L. and Lee, Z. Y. 4. Applying the duble side ethd t slutin nnlinear pendulu prble, Appl. Math.Cput. 49, psitin [] Figure. y psitin f the elastic pendulu in Cartesian crdinate syste fr, 3. Fung, T. C.. Slving initial value prbles by differential quadrature ethd. part : secnd and higherrder equatins, Int. J. Nuer. Meth. Eng. 5, Mühendislik Bilileri Dergisi 9 5 () Jurnal f Engineering Sciences 9 5 () 886
6 Gergiu, I. T On the glbal geetric structure f the dynaics f the elastic pendulu, Nnlinear Dyna. 8, Girgin, Z. 8. Cbining differential quadrature ethd with siulatin technique t slve nnlinear differential equatins, Int. J. Nuer. Meth. Eng. 75 (6), He, J. H Variatinal iteratin ethd a kind f nnlinear analytical technique: se eaples, Int. J. Nnlin. Mech., 34 (4), He, J. H. 3. Hpty perturbatin ethd: a new nnlinear analytical technique, Appl. Math. Cput. 35 (), Liu, G.. and Wu, T. Y.. Nuerical slutin fr differential equatins f duffingtype nnlinearity using the generalized differential quadrature rule, J. Sund. Vib. 37 (5), Lynch, P.. esnant tins f the threediensinal elastic pendulu, Int. J. Nnlinear Mech. 37, Lynch, P. and Hughtn, C. 4. Pulsatin and precessin f the resnant swinging spring, Physica D, 9, Nayfeh, A. H Nnlinear scillatins 7s. A WileyInterscience Publicatin. Vetyukv, Y., Gerstayr, H. and Irschik, H. 4. The Cperative Analysis f the fully nnlinear, the linear elastic and the cnsistently linearized equatins f tin f the d elastic pendulu, Cput. Struct. 8, Mühendislik Bilileri Dergisi 9 5 () Jurnal f Engineering Sciences 9 5 () 886
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