Harmonic Oscillations and Resonances in 3-D Nonlinear Dynamical System
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1 Intrnational Journal of Partial Diffrntial Equations and Applications 06 Vol. 4 No. 7-5 Availabl onlin at Scinc and Education Publishing DOI:0.69/ijpda-4-- Harmonic Oscillations and Rsonancs in -D Nonlinar Dynamical Systm Usama H. Hgazy * Mousa A. ALshawish Dpartmnt of Mathmatics Faculty of Scinc Al-Azhar Univrsity Gaza Palstin *Corrsponding author: uhijazy@yahoo.com u.hjazy@alazhar.du.ps Abstract his papr is concrnd with th thr dimnsional motion of a nonlinar dynamical systm. h motion is dscribd by nonlinar partial diffrntial quation which is convrtd by Galrkin mthod to thr dimnsional ordinary diffrntial quations. h thr dimnsional diffrntial quations undr th influnc of xtrnal forcs ar solvd analytically and numrically by th multipl tim scals prturbation tchniqu and th Rung-Kutta fourth ordr mthod. Phas plan tchniqu and frquncy rspons quations ar usd to invstigat th stability of th systm and th ffcts of th paramtrs of th systm rspctivly. Kywords: Galrkin mthod rsonancs nonlinaritis Cit his Articl: Usama H. Hgazy and Mousa A. ALshawish Harmonic Oscillations and Rsonancs in -D Nonlinar Dynamical Systm. Intrnational Journal of Partial Diffrntial Equations and Applications vol. 4 no. (06: 7-5. doi: 0.69/ijpda Introduction Problms involving nonlinar diffrntial quations ar xtrmly divrs and mthods of solutions or analysis ar problm dpndnt. Nonlinar systms ar intrsting for nginrs physicists and mathmaticians bcaus most physical systms ar nonlinar in natur. h subcombination intrnal rsonanc of a uniform cantilvr bam of varying orintation with a tip mass undr vrtical bas xcitation is studid. h Eulr Brnoulli thory slndr bam was usd to driv th govrning nonlinar partial diffrntial quation []. h dynamic stability of a moving string in thr-dimnsional vibration is invstigatd []. hr nonlinar intgro-diffrntial quations of motion ar studid and th analysis is focusd on th cas of primary rsonanc of th first in-plan flxural mod whn its frquncy is approximatly twic th frquncy of th first out-of-plan flxural-torsional mod []. h mthod of multipl tim scals is applid to invstigat th rspons of nonlinar mchanical systms with intrnal and xtrnal rsonancs. h stability of vibrating systms is invstigatd by applying both th frquncy rspons quation and th phas plan mthods. h numrical solutions ar focusd on both th ffcts of th diffrnt paramtrs and th bhavior of th systm at th considrd rsonanc cass [45]. h nonlinar charactristics in th larg amplitud thr-dimnsional fr vibrations of inclind saggd lastic cabls ar invstigatd [6]. h nonlinar forcd vibration of a plat-cavity systm is analytically studid. Galrkin mthod is usd to driv coupld nonlinar quations of th systm. In ordr to solv th nonlinar quations of plat-cavity systm multipl scals mthod is mployd. Closd form xprssions ar obtaind for th frquncy-amplitud rlationship in diffrnt rsonanc conditions [7]. h stady-stat priodic rspons of th forcd vibration for an axially moving viscolastic bam in th suprcritical spd rang is studid [8]. For this motion th modl is cast in th standard form of continuous gyroscopic systms. Intrnal Various approximat analytical mthods ar dvlopd for obtaining solutions for strongly nonlinar diffrntial quations in a complx function. h mthods of harmonic balanc Krylov-Bogoliubov and lliptic prturbation ar utilizd [9]. h problm of supprssing th vibrations of a hingd-hingd flxibl bam whn subjctd to xtrnal harmonic and paramtric xcitations is considrd and studid. h multipl scal prturbation mthod is applid to obtain a first-ordr approximat solution. h quilibrium curvs for various controllr paramtrs ar plottd. h stability of th stady stat solution is invstigatd using frquncyrspons quations. h approximat solution was numrically vrifid. It is found that all prdictions from analytical solutions wr in good agrmnt with th numrical simulations [0].. Equations of Motion h nonlinar partial diffrntial quation govrning th flxural dflction uxt ( of th bam subjct to harmonic axial xcitation p = p 0 p cos Ω t is givn by [] 4 u u u u ( 4 0 cos Ω m c EI p p t t t x x u u ( p0 p cos Ωt x x 7 u u u x x x EI = u u 9 u u x 4 4 x 4 x x (
2 8 Intrnational Journal of Partial Diffrntial Equations and Applications undr th following boundary conditions: u u( x = 0 and = 0 atx = 0 x = L. x Equation ( can b convrtd to a thr dimnsional nonlinar ordinary diffrntial quations applying th mthod of Galrkins and using th following xprssion πx πx πx uxt ( = gt ( sin ht ( sin kt ( sin L L L ( into quation (. hn w hav g g εα ( g ηh g ηgk ηhk 4 η4g η5g k ε ( η6gk η7g k η8h k ( η9h g η0h k ηh g ηg k ηg 4 η4h k g η5g k η6h g k = εfcos Ωt h wh ε( βh λhgk λh λhk λ4hg ε ( λ5hg λ6h k λ7h λ8hgk λ9hk (5 λ h g λ gkh λ hg k λ hkg = εf cos Ωt 0 εδ ( τ τ τ 4g 5k ( 5 6k 4 7g k 8g k 9k g 0h g 4 h g gh k k w k k h k gk gh τ τ ε τ τ τ τ τ τ τ τ khg τ g τ hk τ hk = εfcos Ωt.. Prturbation Solution ( (6 h mthod of multipl scals is applid to dtrmin an approximat solution for th diffrntial quations (4-6. Assuming that g h and k ar in th forms g( 0 = g0( 0 ε g( 0... h( 0 = h0( 0 ε h( 0... (7 k( 0 = k0( 0 ε k( 0... Whr 0 = t = ε0 = εt. h tim drivativs ar writtn as d = D0 ε D... dt (8 d = D 0 ε DD 0... dt Whr D0 = D =. (9 0 Substituting qs. (7-9 into qs. (4-6 and quating cofficints of sam powrs of ε yilds: 0 o( ε : ( D ( D ( D 0 g0 = 0 (0 0 h0 = 0. ( 0 k0 = 0 ( o( ε : ( D0 g= DDg 0 0αD0g0ηh0 g0 ( ηk0 g0 ηh0 k0 η4g0 η5g0 k0 Fcos( Ωt ( D0 h= D0Dh 0βD0h0λh0g0k0 (4 λh0 λhk 0 0 λ4hg 0 0 Fcos Ωt. ( D0 k= D0Dk 0δD0k0τh0 k0 (5 τk0 g0 τh0 g0 τ4g0 τ5k0 Fcos( Ωt. h solution of qs. (0- is xprssd as i0 i0 g0( 0 = A( A(. (6 i0 i0 h0( 0 = B( B(. (7 i0 i0 k0( 0 = C( C(. (8 Whr ABCar complx functions in Substituting qs. (6-8 into qs (-5w gt ia α i A ηbba i0 ( D0 g = ηcca η4a A i ( 0 i0 ηcbb η5caa η4a i0( i0( ηb A ηb A i0( i0( ηc A ηc A (9 i0( i0( ηcb ηcb i0( i0( η5ca η5ca iω F 0 cc. ( D0 h = ( ib β i BλB B i i λ CCB λ BAA λ B i0( i0( λbac λbac i0( i0( λbac λbac (0 i0( i0( λc B λc B i0( λ i ( 4BA 0 λ 4BA F iω 0 cc. ic δ i C i0 ( D0 k = τcbb τ5c C i0 ( τacc τbba τ4a A i0( i0( τb C τb C ( i0( i0( τac τac i0( i0( τb A τb A i τ 4 A 0 i0 F iω τ 0 5 C cc.
3 Intrnational Journal of Partial Diffrntial Equations and Applications 9 Whr cc dnots a complx conjugat of th prcding trm. h gnral solution of qs. (9-0 can b writtn in th following form ( ηcbb η5caa i0 ( 0 = ( ( η4a i0 ηb A i0( 8 4 ( ηb A i0( g 4 ( ηc A i0( 4 ( ηc A i0( 4 ( ηcb ( ( i0( ηcb i0( ( ( η5ca i0( ( ( η5ca i0( ( ( F iω 0 cc. ( Ω ( Ω λ B i0 h( 0 = 8 λ BAC i0( ( ( λ BAC i0 ( ( ( λ BAC i0 ( ( ( λ BAC i0( ( ( λc B i 0( 4 ( λc B i0( 4 ( λ4ba i0( 4 ( λ4ba i0( 4 ( F iω 0 cc. ( Ω ( Ω ( ( ( τacc τbba τ4a A i0 k( 0 = ( ( τbc i0( 4 ( τbc i0( 4 ( τ AC i0( ( ( τ AC i0( ( ( τb A i 0( ( ( τb A i0( ( ( τ4 A i0 ( ( τ 5 C i0 F iω 0 cc. (4 8 ( Ω ( Ω From qs. (-4 th following rsonanc cass ar xtractd: Intrnal Rsonanc:. =. = = Extrnal Rsonanc: a. Primary rsonanc. Ω=. Ω=. Ω= b. Simultanous rsonanc. Ω= = =. 4. Stability Analysis W shall considr th rsonanc cas Ω whn. Using th dtuning paramtr σ th rsonanc cas ar xprssd as Ω= εσ = εσ = εσ. (5 Substituting q. (5 into qs. (9- and liminating trms that produc scular trm thn prforming som algbraic manipulations w obtain th following modulation quations: ia α i A ηbba i0 ηcca η4a A i ( ( 0 i0 η CBB η5caa ηcb (6 i0( iω η5ca F = 0.
4 0 Intrnational Journal of Partial Diffrntial Equations and Applications i B iβ Bλ B B λ CCB λ BAA λ i0 4 i0( i0( BAC λbac i0( iω 4BA F 0. λ = Ltting ( δ τ τ5 i ( τacc τbba τ4a A i0 i C i C CBB C C 0 i0( iω τ 0 B C F = 0. i A a θ i = B a θ i = C= a θ. (7 (8 whr a a a θ θ θ ar functions of. Sparating ral and imaginary parts givs th following six quations govrning th amplitud and phas modulations aν5 = a( σσ ηaa ηaa 4 4 η4a ηaacosν 8 4 η5aacos ν ηaacosν 8 η5aacos ν4 Fcosν5 8 a a a a = α η sinν η5aasinν ηaasinν η5aasinν4 Fsinν5 ν 4 ν ν a 4 a = σ λ5aa λ6a a λ7a λ9a a λ0aa λa a a 6 4 cos cos λ8aa a cosν λaaa cosν λ6aa ν4 λ8aa a ν5 a = βa λ 8 aa a sin ν sin λ aaa ν λ 6 a a sinν 4 λ 8 aa a sin ν 5 (9 (0 ( ( aν = a ( σ σ τaa τ5a 4 8 τaa τaa τ4a cosν0( τa a cosν Fcos ν 8 τaa τaa 4 4 a = δa sinν0 τ4a (4 8 τa a sinν Fsin ν. 8 whr ν= θ σ σ ν= θ θ σ σ ν= θθθσ σ ν4= ν ν5= θ σ σ ν6= θθθσ ν7= θ σ σ ν8= θθ θ σ ν9 = θθ σ ν0 = θ θ σ σ ν = θ θ σ ν = θ σ σ. h stady-stat solutions of qs. (9-4 ar obtaind by stting a = a = a = ν 5 = ν 8 = ν = 0. into qs. (9-4. his rsults in th following nonlinar algbraic quations which ar calld th frquncy rspons quations: 6 4 a a a 4 0. Λ Λ Λ Λ = ( a 6a 7a 8a 9 0. Λ Λ Λ Λ Λ = ( a a a a 4a 5 0. Λ Λ Λ Λ Λ Λ = (7 h cofficints Λ i i = (...5 ar givn in Appndix. 5. Numrical Rsults and Discussions In this sction th Rung-Kutta fourth ordr mthod is applid to dtrmin th numrical tim sris solutions (t g (t h and (t kand th phas plans (g v (h v (k v rspctivly for th thr mods of th nonlinar systm (4-6. Morovr th fixd points of th modl is obtaind by solving th frquncy rspons quations (5-7 numrically. 5.. im-rspons Solution A non-rsonant tim rspons and th phas plan of th thr mods of vibration of th systm is shown in Figur. In Figur diffrnt rsonanc cass ar invstigatd and an approximat prcntag of incras if
5 Intrnational Journal of Partial Diffrntial Equations and Applications xists in maximum stady-stat amplitud compard to that in th non-rsonant cas is indicatd. Figur. Non-rsonant tim solution of th -D modl to xtrnal xcitation Figur (a. h intrnal rsonanc condition = =.6 Figur (b. h intrnal rsonanc condition = = =.6
6 Intrnational Journal of Partial Diffrntial Equations and Applications Figur (c. h primary rsonanc condition Ω= =.6 (a Intrnal rsonanc cass ( = ( Figur (d. h Simultanous rsonanc condition Ω= = = =. 50% Non Non Figur (a ( = = (50% Non NonFigur (b (b Extrnal rsonanc cass ( primary rsonanc: ( Ω= ( 50%50%60% Figur (c. ( Simultanous rsonancs ( Ω= = = ( 50%50%60% Figur (d. 5.. hortical Frquncy Rspons Solution h numrical rsults ar prsntd graphically in Figs. (-5 as th amplituds a a a against th dtuning paramtrs σ σ σ for diffrnt valus of othr paramtrs. Each curv in ths figurs consists of two branchs. Considring Figur (a as basic cas to compar with it can b sn from Figur (b (c that th stady-stat amplitud a dcrass as ach of αη ar incrasd but in Figur ( th stady- stat amplitud a incrass as ach of F incrass.whras th frquncy rspons curvs in Figur (h ar shiftd to th right as σ incrass. Considring Figur 4(a as basic cas to compar with it can b sn from Figur 4(c that th stady-stat amplitud a incrass as ach of F ar incrasing. But in Figurs 4(b(c (d and(i th stady- stat amplitud a dcrass as ach of a a β ar incrasd. In Figur 4(h th curvs ar shiftd to th right as σ incrass. Whras th frquncy rspons curv ar bnt to right as λ varis from ngativ to positiv valus showing hardning nonlinarity ffct Figur 4(f (g. Considring Figur 5(a as basic cas to compar with it can b sn From Figurs 5(b(c that th stady-stat amplitud a incrass as ach of a τ F ar incrasd. h nonlinarity ffct of τ 5 is shown in Figur 5(f (g whras th curvs ar bing shiftd in Figur 5(d.
7 Intrnational Journal of Partial Diffrntial Equations and Applications Figur. Frquncy rspons curvs of th first mod of th systm at rsonanc η = 0.5 η = 0.6 η = 0.4 η4 = 0. η5 = 0.8 a = 0.0 a = 0.05 σ = 0.0 α = 0.08 F = 0.8 =.9 Figur 4. Frquncy rspons curvs of th scond mod of th systm at rsonanc a = 0.0 σ = 0.0 β = 0.08 F = 0.8 =. λ = 0.5 λ = 0.6 λ = 0.4 λ 4 = 0.8 a = 0.04
8 4 Intrnational Journal of Partial Diffrntial Equations and Applications Figur 5. Frquncy rspons curvs of th third mod of th systm at rsonanc a = 0.04 a = 0.0 σ = 0.0 δ = 0.08 F = 0.8 =.6 τ = 0.55 τ = 0.6 τ = 0.44 τ4 = 0. τ5 = 0.88 (5 h stady-stat amplitud of th third mod incrass as ach of th xtrnal forc amplitud 6. Conclusions F and th first mod amplitud a ar incrasd. W hav studid th analytic and numrical solutions of thr dimnsional nonlinar diffrntial quations that dscrib th oscillations of abam subjctd to xtrnal forcs. h multipl scals mthod and Rung-Kutta fourth ordr numrical mthod ar utilizd to invstigat th systm bhavior and its stability.all possibl rsonanc cass wr b xtractd and ffct of diffrnt paramtrs on systm bhavior at rsonant condition wr studid. W may conclud th following: ( h stady-stat amplitud of th first mod incrass as ach of th xtrnal forc amplitud F and nonlinar cofficints η η 5 ar incrasd. ( h stady-stat amplitud of th first mod dcrass as ach of th linar damping cofficint α and th nonlinar cofficint η 4 and th natural frquncy ar incrasd. ( h stady-stat amplitud of th scond and third mod incras as th xtrnal forc amplitud F incrass. (4 h stady-stat amplitud of th scond mod dcrass as ach of th linar damping cofficint β and nonlinar cofficint λ and th scond mod amplitud a and th natural frquncy ar incrasd. Nomnclatur Natural frquncis of th systm ε Small dimnsionlss prturbation paramtr αβδ Linar damping cofficints ηi λi τ i Nonlinar paramtrs F Excitation forc amplitud Ω Excitation frquncy D0 D Diffrntial oprators r Radius of gyration of cross-sction ara m Mass pr unit lngth of bam E Young's modulus I Momnt of inrtia L Lngth of bam t im p0 p Ral cofficints 0 Fast tim scal Slow tim scal * * * * * t g h k Ω Non- dimnsional quantitis gi hi ki( i = 0 Prturbation variabls xpansion
9 Intrnational Journal of Partial Diffrntial Equations and Applications 5 θ θ θ Phas angls of th polar forms A ( B ( C ( Complx valud quantitis cc Complx conjugat for prcding trms at th sam quation σ i i = Dtuning paramtr a a a Stady-stat amplituds Rfrncs [] Yaman M Analysis of subcombination intrnal rsonancs in a non- linar cantilvr bam of varying orintation with tip mass Int J Non-Linar Mch [] Huang J. Fung R. and Lin C Dynamic stability of a moving string undrgoing thr-dimnsional vibration Int J Mch Sci 7 ( [] Pai P.F. and Nayfh A.H hr-dimnsional nonlinar vibrations of composit bams-iii. Chordwis xcitations Nonlinar Dyn [4] Hgazy U.H. : Intrnal rsonanc of a string-bam coupld systm with cubic nonlinaritis Commun Nonlinar Sci Numr Simul [5] Hgazy U.H Nonlinar vibrations of a thin plat undr simultanous intrnal and xtrnal rsonancs J Vib Acoust pags. 00. [6] Srinil N. Rga R. and Chuchpsaku S Larg amplitud thrdimnsional fr vibrations of inclind saggd lastic cabls Nonlinar Dyn [7] Sadri M. and Younsian D Nonlinar harmonic vibration analysis of a plat-cavity systm Nonlinar Dyn [8] Zhang G. Ding H. Chn L. and Yang S Galrkin mthod for stady-stat rspons of nonlinar forcd vibration of axially moving bams at suprcritical spds J Sound Vibr ( [9] Cvticanin L Analytical mthods for solving strongly non-linar diffrntial quations J Sound Vibr 4 ( [0] Hgazy U.H Singl-Mod rspons and control of a hingdhingd flxibl bam Arch Appl Mch Appndix h cofficints prsntd in (5 ar as follows : 0 Λ = η4 Λ = η4σ ηη 4a η η4a ησ 4 η a η 5 a ηη 5a Λ = η a η a ηη a a ησ a ησ a ησ a ησa σσ σ σ α ηη 5a a η a a η af η 5aF Λ 4 = η a a η aa F F. 6 4 h cofficints prsntd in (6 ar as follows: Λ 5 = 9 λ 6 λσ λλ a λσ λλa Λ = Λ 7= λλ 4aa σλ a σσ λ a λ 4a σλ 4a σ σ σλa σλ4a β λ a a λ a a Λ 8= λ aa F λ 4aF Λ 9= F h cofficints prsntd in (7 ar as follows : Λ 0 = 9 τ5 στ 5 στ 5 ττ 5a 4 τa τ a 64 Λ = Λ = ττ aa Λ = σ σσ στ a τ a σ στ a τ af ττ 4a ττ aa Λ 4= τaf ττ 4aa ττ aa Λ 5= τaa F τ 4aF F τ 4a ττ 4aa τ aa
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