Investigation of Excited Duffing s Oscillator Using Versions of Second Order Runge-Kutta Methods

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1 Internatinal Jurnal f Science and Technlgy Vlume 1 N. 1, December, 01 Investigatin f Excited Duffing s Oscillatr Using Versins f Secnd Order Runge-Kutta Methds Salau T. A.O. 1, Ajide O.O. Department f Mechanical Engineering, University f Ibadan, Nigeria. ABSTRACT This investigatin derived its strng mtivatin in the adptin f versins f secnd-rder Runge-Kutta methds where there is presently dearth f relevant literature t re-establish the cmplicated nature f slutin f Duffing scillatr dynamics. The chice f secnd-rder Runge-Kutta methds hinged n its simplest algebraic frmulatin f relevant cefficients based n Taylr series expansin cmparing with its higher rder cunterpart. Validatin f FORTRAN-90 cdes f algrithms was achieved by phase plts cmparisn reference t Dwell (1988) as standard. The nature f simulated slutins were visually determined with scatter plt f phase variables btained frm simultaneus implementatin f large number f versins f secnd-rder Runge-Kutta methds in cnjunctin with the crrespnding literature results. Validatin results are acceptable t within the accuracy limit f Runge-Kutta methds adpted. The scatter plts n phase plane fr cases investigated are well structured and bunded (strange) and cmpare crrespndingly well with literature Pincare sectins. This investigatin re-establishes the cmplex nature f slutin f Duffings scillatr dynamics. Its established prcedures prvide an alternative Pincare sectin methd and can be utilised fr preliminary verificatin f system dynamics behaviur subject t cnfirmatin by additinal dynamics tests. Keywrds: Excited Duffing Oscillatr, Secnd rder Runge-Kutta, Taylr Series and Pincare Sectin 1. INTRODUCTION Runge kutta technique is unarguably ne f the mst highly favured numerical tls in simulating chas dynamics. The furth-rder Runge-Kutta methd has been the preferred numerical integratin scheme fr slving chatic prblems in nn-linear systems (Mrthy et al, 1993). Accrding t the authrs wrk, the methd is cnsidered very accurate but requires very small timesteps and fur equatin slutins per time-step. These drawbacks hinder the slutin f chatic prblems in multi-degree-f-freedm (MDOF) systems. The paper presents the slutin f the chatic prblem f impacting single-degree-f-freedm (SDOF) scillatrs, using the Newmark methd which is cmputatinally efficient and uncnditinally stable. The results f their study are cmpared with thse btained frm the furth-rder Runge-Kutta methd. It is cncluded that the Newmark methd with an adequate check n the slutin accuracy culd give qualitatively the same results as the Runge- Kutta methd. The methd has the benefit f an extensin t MDOF real-life prblems f chas which culd be slved using numerical techniques. Chatic mtin f a virtual duble pendulum has been studied by Aan et al (011). Large scillatins f this pendulum were mdelled by the system f nnlinear differential equatins in the Hamiltn frm. This system was slved n the wrksheet f Cmputer Package Mathcad numerically, using the Runge-Kutta algrithm f furth rder. The results btained were demnstrated by sme frames f vide clips visualizing the chatic mtin f a duble physical pendulum. The paper cncluded that the results btained can serve as teaching aids in the prcess f analytical mechanics fr engineering students. A study which utilizes cmbinatin f phase plts, time steps distributin and adaptive time steps Runge-Kutta and fifth rder algrithms t investigate a harmnically Duffing scillatr has been carried ut by Salau and Ajide (01 (a) ). The bject f their study was t visually cmpare furth and fifth rder Runge-Kutta algrithms perfrmance as tls fr seeking the chatic slutins f a harmnically excited Duffing scillatr. It is deduced that althugh fifth rder algrithms favurs higher time steps and as such faster t execute than furth rder fr all studied cases, the reliability f results btained with furth rder wrth its higher recrded ttal cmputatin time steps perid. The bjective f Khatami et al (008) paper is the analytical investigatin f the dynamics f vibratin f parametrically excited scillatr with strng cubic negative nnlinearity based n Mathieu- Duffing equatin. The analytic investigatin was cnducted by using He s hmtpy-perturbatin methd (HPM).The authrs emplyed Runge-Kutta s algrithm t slve the gverning equatin via numerical slutin. The effects f variatin f the parameters n the accuracy f the hmtpy- perturbatin methd were equally studied. A IJST 01 IJST Publicatins UK. All rights reserved. 679

2 Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 mdified Runge Kutta methd with phase-lag f rder infinity fr the numerical slutin f the Schr dinger equatin and related prblems has been develped in Sims and Jesu s (001) paper. The mdified methd is based n the classical Runge Kutta methd f algebraic rder fur. The numerical results indicate that the new methd develped is mre efficient fr the numerical slutin f the Schr dinger equatin and related prblems than the well knwn classical Runge Kutta methd f algebraic rder fur. The behaviur f chatic dynamical systems is understd by calculating flws in phase space. Stable pints can emerge after iterating these flw many times, revealing significant infrmatin abut the system (Benjamin, 011). A rigrus integratr has been develped using Taylr mdels and implemented in the cde COSY INFINITY which integrates ODEs and PDEs rigrusly. The authr was able t illustrate these integratins f varius pint densities using an eighthrder Runge-Kutta with autmatic step size cntrl using reverse cmmunicatin. Optimum fractal disk dimensin algrithms has been used t characterize the evlved strange attractr when adaptive time steps Runge-Kutta furth and fifth rder algrithms are emplyed t cmpute simultaneusly multiple trajectries f a harmnically excited Duffing scillatr frm very clse initial cnditins ( Salau and Ajide, 01 (b) ). The bject f the study was t enable visual cmparisn f the chas diagrams in the excitatin amplitude versus frequency plane. The chas diagrams btained by furth rder algrithms is accepted t be mre reliable than its fifth rder cunterpart, its utility as tl fr searching pssible regins f parameter space where chatic behaviur/mtin exist may require additinal dynamic behaviur tests. Despite the availability f numerus literatures n the chice f Runge kutta as a numerical tl fr characterizing nnlinear dynamics, there is presently dearth f relevant literature which re-establishes the cmplicated nature f slutin f Duffing Oscillatr dynamics. The gal f this paper is t investigate this dynamics by adpting versins f secnd-rder Runge- Kutta methds. The fcus is what gemetrical shape will set f results btained when the same Duffing equatin is simulated frm the same initial cnditins (displacement and velcity) using several versins f secnd rder Runge-Kutta (1,,3,----n; fr very large n). The justificatin fr the chice f secnd rder Runge-Kutta is that it is the simplest methd where in the mathematical relatinships between cefficients can easily be btained withut much f rigrus derivatin.. METHODOLOGY.1 Harmnically Excited Duffing s Oscillatr The present study investigated nrmalized gverning equatin f harmnically excited Duffing system given by equatin (1) with reference t Mn (1987), Dwell (1988) and Narayanan and Jayaraman (1989b). x x x (1 x ) P Sin( t) (1) In equatin (1) x, x and x represents respectively displacement, velcity and acceleratin f the scillatr abut a set datum. The damp cefficient is. Amplitude strength f harmnic excitatin, excitatin frequency and time are respectively P, and t. Accrding t literature cmbinatin f = 0.168, P = 0.1, and = 1.0 r = , P = 0.09 and = 1.0 parameters leads t chatic behaviur. Hwever Dwell (1988) reprted perid ne, tw and fur at respective excitatin amplitude f 0.177, and when the damp and excitatin frequency are fixed respectively at and 1.0. The present investigatin utilised family f secnd rder Runge-Kutta methd driven at cnstant time step (0.01) t seek bth transient and steady slutins f equatin (1) ver a ttal f seventy (70) excitatin perids and initial cnditins (1, 0). Scatter diagrams were made in each studied case n phase plane with the last cmputed stable slutins.. Generalized Runge-Kutta Methds Accrding t Chapra and Canale (006) many variatins f explicit Runge-Kutta methd exist, hwever all can be cast in the generalized frm given by equatin (). x i 1 x i ( ti, xi, t) t () In equatin () ( t, x, t) is called an incremental i i functin and best interpreted as a representative slpe f multivariable functin f ( x, t) ver the time interval ( t ). The increment functin can be written in general frm 's 's as in equatin (3) fr cnstants and K. K K K (3) 1 n n 1.3 Secnd-Order Runge-Kutta Methds The secnd rder versin f equatin () is given by equatin (4). x i 1 x i ( K 1 K) t (4) 1 IJST 01 IJST Publicatins UK. All rights reserved. 680

3 Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 In equatin (4) K1 f ( t, x ) and K f ( t t, x K t). The cnstant i 1 i 11 1 cefficients are related as in equatins (5) t (7) with details equivalent relatinship between equatin (4) and Taylr series expansin t the secnd rder term prvided in Chapra and Canale (006). 1 (5) 1 1 (6) (7) The three simultaneus equatins (5) t (7) related fur unknwn cnstants and suggested nn availability f unique set f cnstants that satisfy these equatins. The secnd rder Runge-Kutta methds is a chice f this investigatin because f its simplest nature f the derivatin f equatins (5) t (7) cmpare with its higher rder cunterpart that invlve a large amunt f tedius, time cnsuming and errr prne algebraic manipulatin. Mre insights are prvided by Cartwright and Pir (199). The present study assumed nine hundred and ninety nine (999) distinct values fr within using cnstant step size f 1 and slved fr ther three cnstants using equatins (5) t (7). The cmbinatin f crrespnding 1, 1, 11 and the assumed i i tagged (V1, V V999) were used ne after the ther t seek the slutins f equatin (1). Mathematical arguments by Chapra and Canale (006) are that each f these versins (V1, V V999) wuld prduce same results prvided the rdinary differential equatins (ODE) have slutin that is quadratic, linear r cnstant and varied results therwise. It is here that lays the pivt f the present study with ne f the bjective being t bserve any structural pattern f assembly f slutin pints frm versins f secnd-rder Runge- Kutta methds n the phase plane. The last f cmputed displacement and velcity in the steady slutins realm prduced by V1, V V999 were used t plt scatter diagrams n the phase plane..4 Parameters f Cases Investigated A cnstant time step ( t = 0.01), initial cnditins (1, 0) and crrespnding cefficients cmbinatin (V1, V V999) are cmmn t all investigated cases. The unsteady slutins spanned the first twenty (0) simulatin perid reference t harmnic excitatin. Case-I: = 0.168, P = 0.1, and = 1. Case-II: = , P = 0.09, and = 1.0 Case-III: = 0.168, P = 0.197, and = 1. Case-IV: = 0.168, P = 0.178, and = 1. Case-V: = 0.168, P = 0.177, and = 1. Case-VI: = 0.168, P = 0.1, and = 1 in additin t table 1. The versin (V500) is ne f the three mst cmmnly used and preferred. Cefficients Table 1: Cefficients cmbinatin fr selected versin f secnd-rder Runge-Kutta methds Versins f secnd-rder Runge-Kutta methds V100 V00 V300 V400 V500 V600 V700 V800 V RESULTS AND DISCUSSIONS The secnd-rder Runge-Kutta methds apprpriately cded in FORTRAN-90 was used t cmpute results presented under this sectin. The phase plts in figures 1 and were used t validate Runge-Kutta algrithms by cmparing crrespnding figures with its literature cunterpart. Figure 1 cmpare very well its literature cunterpart which was cmputed by higher rder Runge- Kutta. The bserved significant variatin in figure with its literature cunterpart can be explained respect t higher rder Runge-Kutta difference. Dynamics cmputatins with higher rder Runge-Kutta methds are well knwn t be stable and mre reliable than lwer rder cunterpart. IJST 01 IJST Publicatins UK. All rights reserved. 681

4 Velcity Velcity Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 Phase plt (Case-I) Figure 1: Steady Phase plt (Case-I) cmputed by V500 ver 41 st t 70 th excitatin perid Phase plt (Case-V) Figure : Steady Phase plt (Case-V) cmputed by V500 ver 41 st t 70 th excitatin perid IJST 01 IJST Publicatins UK. All rights reserved. 68

5 Velcity Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, V100 V00 V300 V400 V500 V600 V700 V800 V Time Figure 3: Steady time displacement histry (Case-VI) ver 70 th excitatin perid V100 V00 V300 V400 V500 V600 V700 V800 V Time Figure 4: Steady time velcity histry (Case-VI) ver 70 th excitatin perid. It is amazing t nte that as wildly varied as time histry in figures 3 and 4 are the difference amng the series cmputatin is the versin f the Runge-Kutta methds used. The time displacement histry predicted ver a whle perid f excitatin by versin V800 was almst linear. The cmmn and preferred versin (V500) f secnd-rder Runge-Kutta methds is n exceptin in the bserved wild variatins. Furthermre the series in figures 3 and 4 impresses large number f bjects swarming at different rate with cmmn fcus being t get nt arbitrary pints n the strange attractr f Duffing s scillatr. Sme f these attractrs are reprted in figures 5 t 9. IJST 01 IJST Publicatins UK. All rights reserved. 683

6 Velcity Velciy Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 Scatter plt (Case-I) Figure 5: Steady scatter plt (Case-I) phase variables n phase plane cmputed at the end f 70 th excitatin perid Scatter plt (Case-II) Figure 6: Steady scatter plt (Case-II) phase variables n phase plane cmputed at the end f 70 th excitatin perid IJST 01 IJST Publicatins UK. All rights reserved. 684

7 Velcity Velcity Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 Scatter plt (Case-III) Figure 7: Steady scatter plt (Case-III) phase variables n phase plane cmputed at the end f 70 th excitatin perid Scatter plt (Case-IV) Figure 8: Steady scatter plt (Case-IV) phase variables n phase plane cmputed at the end f 70 th excitatin perid IJST 01 IJST Publicatins UK. All rights reserved. 685

8 Velcity Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 Scatter plt (Case-V) Figure 9: Steady scatter plt (Case-V) phase variables n phase plane cmputed at the end f 70 th excitatin perid The strange structured and bunded images f figures 5 and 6 are qualitatively acceptable within the accuracy limit f secnd-rder Runge-Kutta methds as equivalent t the Pincare sectins reprted by Dwell (1988) and Salau and Ajide (01 (c)) and fr respective crrespnding driven parameters cmbinatin. Therefre it can be cncluded that equivalent Pincare sectin can be btained by simulatin f equatin (1) either frm set f very clse multiple initial cnditins r by implementatin f large number f varied versin f Runge-Kutta methds. The standardised methd f generating Pincare sectins used by Dwell (1988) being the strbscpic reprt f predicted phase variables at interval f ne excitatin length perid perfrmed repeatedly ver infinite time length. Figure 7 match fairly the finite average dts number f perid fur respnse reprted by Dwell (1988). Hwever figures 8 and 9 deviated significantly frm respective perid tw and ne that Dwell (1988) predicted. The nticeable deviatin can be accunted t lwer instead f higher rder Runge- Kutta methds used. The phase plt in figure elucidates inaccuracy f adpted slutin methd and why figures 8 and 9 appeared disjinted unlike their figure 5 cunterpart. 4. CONCLUSIONS This study re-affirms cmplicated and varied nature f slutins t gverning equatin f Duffing s scillatr as change frm ne set f driven parameters cmbinatin t anther are effected. Thrugh use f scatter plt f the phase variables btained frm versins f Runge-Kutta methds emerge structured and bunded strange images n the phase plane that qualitatively replicate Pincare sectins reference t literature. The bservatin was cnsistent especially fr driven parameters cmbinatin that guarantees chatic respnse f the Duffing s Oscillatr. The utility f the prcedures f this study can be in the preliminary verificatin f chatic behaviur f dynamic systems subject t cnfirmatin by additinal acceptable dynamic tests. REFERENCES [1] Aan A., Aarend E. and Heinl M. (011), chatic mtin f a duble pendulum. Agrnmy Research Bisystem Engineering, Special Issue 1, Pg.5-1 [] Benjamin L. (011),Verificatin f a rigrus integratr using an eighth-rder Runge-Kutta, Michigan State University, Taylr Mdel Methds VII. [3] Dwell E.H. (1988), Chatic scillatins in mechanical systems, Cmputatinal Mechanics, 3, [4] Francis C. Mn (1987), Chatic Vibratins-An Intrductin fr Applied Scientists and Engineers, Jhn Wiley & Sns, New Yrk, ISBN [5] Julyan, H.E. Cartwright and Oreste Pir (199), The Dynamics f Runge-Kutta methds, Internatinal Jurnal f Bifurcatin and Chas,, pp IJST 01 IJST Publicatins UK. All rights reserved. 686

9 Internatinal Jurnal f Science and Technlgy (IJST) Vlume 1 N. 1, December, 01 [6] Khatami I., Pashai M.H., and Tlu N. (008), Cmparative Vibratin Analysis f a Parametrically Nnlinear Excited Oscillatr Using HPM and Numerical Methd.Hindawi Publishing Crpratin Mathematical Prblems in Engineering, Vlume 008, Article ID , 11 pages.di: /008/ [7] Mrthy R. I. K., Kakdkar A., Srirangarajan H. R. and Suryanarayan S. (1993), An assessment f the newmark methd fr slving chatic vibratins f impacting scillatrs Cmputers & Structures, 49 (4). pp ISSN [8] Narayanan S. and Jayaraman K. (1989). Cntrl f Chatic Oscillatins by Vibratin Absrber. ASME Design Technical Cnference, 1 th Biennial Cnference n Mechanical Vibratin and Nise. DE 18.5, [9] Salau T.A.O. and Ajide O.O. (01( a) ), Cmparative Analysis f Time Steps Distributin in Runge-Kutta Algrithms, Internatinal Jurnal f Scientific & Engineering Research Vlume 3, Issue 1, January ISSN [10] Salau T.A.O. and Ajide O.O.(01 (b) ), Cmparative Analysis f Numerically Cmputed Chas Diagrams in Duffing Oscillatr, Jurnal f Mechanical Engineering and Autmatin 01, Vl.(4): DOI: /j.jmea Scientific and Academic Publishing, USA. [11] Salau, T.A.O. and Ajide, O.O. (01 ( c) ), Fractal Characterizatin f evlving Trajectries f Duffing Oscillatr, Internatinal Jurnal f Advances in Engineering & Technlgy Research (IJAET), Vlume, Issue 1, Pg [1] Sims T.E., Jesu s V. A. (001), A mdified Runge Kutta methd with phase-lag f rder infinity fr the numerical slutin f the Schr dinger equatin and related prblems. Cmputers and Chemistry, Vl.5, Pg [13] Steven C. Chapra and Raymnd P. Canale (006), Numerical methds fr engineers, Fifth editin, McGraw-Hill (Internatinal editin), New Yrk, ISBN IJST 01 IJST Publicatins UK. All rights reserved. 687