Applied Mathematics, 2011, 2, doi: /am Published Online December 2011 (

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1 Applied Mthemtics,,, doi:.6/m..9 Pulished Online Decemer ( The Anlyticl nd Numericl Solutions of Differentil Equtions Descriing of n Inclined Cle Sujected to Externl nd Prmetric Excittion Forces Astrct Mohmed S. Ad Elkder, Deprtment of Mthemtics nd Sttistics, Fculty of Science, Tif University, El-Tif, Kingdom of Sudi Ari Deprtment of Engineering Mthemtics, Fculty of Electronic Engineering, Menoufi University, Menouf, Egypt E-mil: moh_6_@yhoo.com Received Octoer, ; revised Novemer, ; ccepted Novemer, The nlyticl nd numericl solutions of the response of n inclined cle sujected to externl nd prmetric excittion forces is studied. The method of perturtion technique re pplied to otined the periodic response eqution ner the simultneous principl prmetric resonnce in the presence of : internl resonnce of the system. All different resonnce cses re extrcted. The effects of different prmeters nd worst resonnce cse on the virting system re investigted. The stility of the system re studied y using frequency response equtions nd phse-plne method. Vrition of the prmeters α, α, β, γ, η, γ, η, f leds to multi-vlued mplitudes nd hence to jump phenomen. The simultion results re chieved using MATLAB 7.6 progrms. Keywords: Perturtion Method, Resonnce, Chotic Response, Stility. Introduction Cle structures ply n importnt role in mny engineering fields, such s civil, ocen nd electric engineering. Arft nd Nyfeh [] studied the motion of shllow suspended cles with primry resonnce excittion. The method of multiple scles is pplied to study nonliner response of this suspended cles nd its stility nd the dynmic solutions. Some interesting work on the nonliner dynmics of cles to the hrmonic excittions cn e found in the review rticles y Reg [,]. Nielsen nd Kierkegrd [] investigted simplified models of inclined cles under super nd comintoril hrmonic excittion nd gve nlyticl nd purely numericl results. Zheng, Ko nd Ni [5] considered the super-hrmonics nd internl resonnce of suspended cle with lmost commensurle nturl frequencies. Zhng nd Tng [6] investigted the chotic dynmics nd glol ifurctions of the suspended inclined cle under comined prmetric nd externl excittions. Nyfeh et l. [7] investigted the nonliner nonplnr responses of suspended cles to externl excittions. The equtions of motion governing such systems contin qudrtic nd cuic nonlinerities, which my result in : nd : internl resonnces. Chen nd Xu [8] investigted the glol ifurctions of the inclined cle sujected to hrmonic excittion leding to primry resonnces with the externl dmping y using verging method. Kmel nd Hmed [9], studied the nonliner ehvior of n inclined cle sujected to hrmonic excittion ner the simultneous primry nd : internl resonnce using multiple scle method. Ae [] investigted the ccurcy of nonliner virtion nlyses of suspended cle, which possesses qudrtic nd cuic nonlinerities, with : internl resonnce. The nonliner dynmics of suspend cle structures hve een studied with : internl resonnces y the uthors [,]. Experimentl studies of this prolem hve een conducted y Alggio nd Reg [] nd Reg nd Allgio [], however explicit stility regions for the semi-trivil solution hve not een clculted nlyticlly. Here, we use modl model to compute the instility oundry for rnge of excittion frequencies close to the : resonnce for n inclined cle, including nonliner modl interction. The out-of-plne

2 7 dynmic stility of inclined cles sujected to in-plne verticl support excittion is investigted y Gonzlez- Buelg et l. [5]. Perkins [6] exmined the effect of one support motion on the three-dimensionl nonliner response. Using the Glerkin method, he constructed two-degree-of-freedom model to nlyze the : internl resonnce. Lee nd Perkins [7] extended the work to include second-order perturtions nd multiple internl resonnces. Still, the focus ws on the : internl resonnce, wheres the excittion ws chnged to hrmoniclly vrying lod per unit length cting in the sttic equilirium plne. Lee nd Perkins [8] lso used three-degree-of-freedom model to simulte non-liner response of suspended, inclined cles driven y plnr excittion nd determined the existence nd stility of four clsses of periodic solutions. Eiss nd Syed [9-] nd Syed [], studied the effects of different ctive controllers on simple nd spring pendulum t the primry resonnce vi negtive velocity feedck or its squre or cuic. Syed nd Hmed [] studied the response of two-degree-of-freedom system with qudrtic coupling under prmetric nd hrmonic excittions. The method of multiple scle perturtion technique is pplied to solve the non-liner differentil equtions nd otin pproximte solutions up to nd including the second-order pproximtions. Syed nd Kmel [,5] investigted the effects of different controllers on the virting system nd the sturtion control to reduce virtions due to rotor lde flpping motion. The stility of the otined numericl solution is investigted using oth phse plne methods nd frequency response equtions. Amer nd Syed [6], studied the response of one-degree-of freedom, non-liner system under multi-prmetric nd externl excittion forces simulting the virtion of the cntilever em. Vrition of some prmeters leds to multi-vlued mplitudes nd hence to jump phenomen. Syed et l. [7], investigted the non-liner dynmics of two-degree-of freedom virtion system including qudrtic nd cuic non-linerities sujected to externl nd prmetric excittion forces. The stility of the system is investigted using oth frequency response curves nd phse-plne trjectories. The effects of different prmeters of the system re studied numericlly. This work dels with model hving two-degree-offreedom nonliner system sujected to externl nd prmetric excittion forces descries the virtions of n inclined cle. The method of multiple scles perturtion is pplied to otin modultion response equtions ner the simultneous principl prmetric resonnce in the presence of : internl resonnce ( nd ). The stility of the proposed nlytic nonliner solution ner the ove cse is studied nd the stility condition is determined. The effect of different prmeters on the stedy stte response of the virting system is studied nd discussed from the frequency response curves. The numericl solution nd chotic responses of the nonliner system of n inclined cle for some different prmeters re lso studied. A comprison with previously pulished work is included.. Mthemticl Anlysis Our ttention is focused on n elstic-sg hnging t fixed supports nd excited y hrmonic nd prmetric distriuted verticl forcing in plne. The two-degree-offreedom descriing the nonliner dynmics of cle shown in Figure, cn e written s: x c x x x y x xy () yc y yxy y x f cos ty f cos t where x nd y denote in-plne nd out-of-plne displcements, respectively, nd dots denote derivtives with respect to the time t. The prmeters c nd c re the viscous dmping coefficients, nd re the nturl frequencies ssocited with in-plne nd out-ofplne modes nd re the excittion frequencies, f nd f re the excittion forces mplitude,,,,,, nd re the coefficients of nonliner prmeters. The liner viscous dmping forces, the exciting forces nd nonliner prmeters re ssumed to e y c cˆ ˆ ˆ ˆ ˆ ˆ, c c, fn fn, s s, s s, ˆ ˆ, ˆ s s n, s, where is smll perturtion prmeter nd. For the convenience of the nlysis of Equtions ()-(), the non-dimensionl prmeter is introduced. We cn otin x cx ˆ ˆ ˆ x( x y ) () ( ˆ x ˆ xy ) y cˆ y yˆ xy ( ˆ y ˆ x y) ( fˆ ˆ cost y fcos t) The prmeters ˆ ˆ, ˆ, re of the order of nd the prmeters cˆ, cˆ, ˆ, ˆ, ˆ, ˆ, fˆ, fˆ re of the order of. The pproximte solution of Equtions ()-() cn e otined using the method of multiple scles [8]. Let x( t; ) x( T, T, T) x( T, T, T) (5) x ( T, T, T ) () ()

3 Figure. A schemtic of inclined cle under comined excittions. y( t; ) y ( T, T, T ) y ( T, T, T ) y( T, T, T) n where, Tn t (n =,, ) re the fst nd slow time scles respectively. In terms of T, T nd T, the time derivtives trnsform ccording to 7 (6) d D D D dt (7) d D DD ( D DD dt ) where cc denotes the complex conjugte of the preceding terms nd A, B re complex functions in T nd T which determined through the elimintion of seculr nd smll-divisor terms from the first nd second-order of pproximtions. In this cse, we nlyze the cse where nd. To descrie quntittively the nerness of the resonnces, we introduce the detuning prmeters nd ccording to ˆ, ˆ. Sustituting Equtions ()-(5) into Equtions ()-() nd eliminting the seculr terms leds to the solvility conditions for the first-order expnsion s: ˆ i D A B exp( i ˆ T ) (6) i D B ˆ ABexp( i ˆ T ) (7) After eliminting the seculr terms, the prticulr solutions of Equtions ()-() re given y: ˆ ˆ ˆ x A exp( i T) AA BB cc (8) ˆ y ABexpi T cc(9) where. Sustituting Equtions Now sustituting Equtions ()-(5) nd Equtions Dn Tn (5)-(6) nd (8)-(9) into Equtions ()-(), the following re otined (7) into Equtions ()-() nd equting the coefficients of similr powers of in oth sides, we otin the differentil equtions s follows: ( D ) x Order ( ): ( D Aicˆ A i DA ABB A A)exp( it ) ( D ) x (8) NST cc () ( D ) y (9) Order ( ): ( D ) y ( D ˆ ˆ ) x DDx x y () DB icb ˆiDB AABBBexp( it ) ( D ˆ ) y DDy xy () fb ˆ exp( i( ) T) NSTcc Order ( ) : () where ( D ) x D x DDxDDx cdx ˆ ˆ ˆ ˆ ˆ ˆ ˆ xx yy () ˆ, ( ) ˆ ˆ x xy ˆ ˆ, ( D ) y D y DDyDDy cdy ˆ ˆ ˆ ( xy yx ) y () ˆ ˆ ˆ ˆ ˆ ˆ yx fcos T y fcos T ˆ, ( ) The solution of Equtions (8)-(9) cn e expressed in the complex form: ˆ ˆ ˆ x AT (, T)exp( it) cc () y B( T, T )exp( i T ) cc (5) nd NST stnds for non-seculr terms. Eliminting the

4 7 seculr terms leds to the solvility conditions for the second-order expnsion i D A D A i ˆ c A ABB A A () i D B D B i c ˆ B AAB ˆ ˆ BB f B exp( i T) () Stility Anlysis of Nonliner Solutions From Eqution (7), multiplying oth sides e i, i we get da i ida ida () dt db i i DB idb (5) dt To nlyze the solutions of Equtions (6)-(7) nd Equtions ()-(), we express A nd B in the polr form i i A( T, T ) ( ) e, B( T, T ) ( ) e (6) where, nd s ( s,) re the stedy stte mplitudes nd phses of the motion respectively. Sustituting Equtions (6), (6)-(7) nd Equtions ()-() into Equtions ()-(5) nd equting the rel nd imginry prts we otin the following equtions descriing the modultion of the mplitudes nd phses: where c sin (7) cos (8) f c sin sin (9) 8 cos 8 9 () f cos 7 8 9, 8 8 5, 6, 7, 8,,, nd ˆ T, ˆ T () Form the system of Equtions (7)-() to hve sttionry solutions, the following conditions must e stisfied: () It follows from Eqution () tht, () Hence, the stedy stte solutions of Equtions (7)-() re given y c sin () 8 ( ) cos (5) f c sin sin (6) 8 cos 8 9 (7) f cos Solving the resulting lgeric equtions for the fixed points of the prcticl cse where,, tht is non-plnr motions, we otin the following frequency response equtions 6 ( ) c ( ) ( ) 6 c where f cos( ) f (8) (9) 5 6,, nd. 8 The stility of the otined fixed points for the simul-

5 7 tneous primry, principl prmetric nd : internl resonnce cse is determined nd studied s follows: one lets, () nd s s s where, nd s re the solutions of Equtions ()-(7) nd,, s re perturtions which re ssumed to e smll compred to, nd s. Sustituting Eqution () into Equtions (7)-(), using Equtions ()-(7) nd keeping only the liner terms in,, s we otin: c K cos () K sin ( ) 6 cos 9 8 sin sin cos cos 6 f cos f sin sin f c sin sin f cos cos cos sin 9 f () () 9 cos cos f sin () The system of Equtions ()-() re first order utonomous ordinry differentil equtions nd the stility of prticulr fixed point with respect to n infinitesiml disturnce proportionl to exp( t) is determined y eigenvlues of the Jcoin mtrix of the right hnd sides of Equtions ()-(). The zeros of the chrcteristic eqution re given y L L L L (5) L, L, L L (,,,, c, c,,,,,,,, f,,, where, nd re functions of the prmeters ). According to the Routh-Hurwitz criterion the necessry nd sufficient conditions for ll the roots of Eqution (5) to possess negtive rel prts re: L, LL L, L LL L LL, L (6) The system is stle if the eigenvlues hve negtive rel prts, otherwise is unstle. In the frequency response curves, solid/dotted lines denote stle/ unstle periodic responses, respectively.. Results nd Discussion The response of the two-degree-of-freedom nonliner system under oth prmetric nd externl excittions is studied. The solution of this system is determined up to nd including the second order pproximtion y pplying the multiple time scle perturtion. The stedy stte solution nd its stility re determined nd representtive numericl results re included. The stility zone nd effects of the different prmeters re discussed using frequency response curve. The stility of the numericl solution is studied lso using the phse-plne method. Some of the resulting resonnce cses re confirmed pplying well-known numericl techniques. The effects of the some different prmeters on the virting system ehvior re investigted nd discussed... Numericl Solution Figure shows tht the response of the inclined cle for the non-resonnt t the prcticl vlues of the prmeters c =., c =., α =., β =.5, γ =., η =.5, α =., η =.5, γ =., =, =., =.75, =., =., =.5. It cn e seen from this figure tht the stedy stte mplitude is out.5 with dynmic chotic ehvior for the inplne mode nd out.8 with multi-limit cycle for the out-of-plne mode. The mplitudes decresing with incresing time nd tend to stedy stte motion nd hve stle solution. The worst resonnce cse is lso confirmed numericlly s shown in Figure. From this figure, it cn e notice tht the mximum stedy stte mplitude of the in-plne mode is out times tht of sic cse with multi-limit cycle, while the mximum mplitude of out-of-plne mode is out times of the sic cse with chotic motion. Effects of externl nd prmetric excittion forces f nd f.

6 7 Amplitude(x) Amplitude(y) Time Time Figure. Non-resonnce system ehvior (sic cse) Ω ω ω. Amplitude(x) Amplitude(y) - Time - Time Figure. Simultneous principl prmetric resonnce in the presence of : internl resonnce ( nd ). For incresing the mplitude of the externl or prmetric excittion forces f or f, we oserve tht the modes of virtion hve incresing mgnitudes nd there exist chotic dynmic motion s shown in Figures nd 5... Frequency Response Curves The frequency response Equtions (8)-(9) re nonliner lgeric equtions in the mplitudes of the system (in-plne mode) nd (out-of-plne mode). The stility of fixed point solution is studied y exmintion of the eigenvlues of Eqution (5). The numericl results of Equtions (8) nd (9) re plotted in Figures 6-8. Figure 6, show the frequency response curves of the two modes of inclined cle ginst detuning prmeter. From the geometry of the figures we oserve tht the mplitudes hve two rnches nd these rnches re ent to the right, the ending leds to multi-vlued solutions nd hence the effective nonlinerity is hrdening type. In Figure 6(), there re two rnches of nontrivil solution such tht the left rnch stle nd the right rnch lose stility s.. Figure 6(), show tht the stedy stte mplitudes re incresing for incresing Amplitude(x) Amplitude(y) Time Time Figure. Effects of incresing vlue of externl excittion force f = 5. Amplitude(x) Amplitude(y) - - Time - - Time Figure 5. Effects of incresing vlue of prmetric excittion force f =. prmetric excittion force f. The region of instility for two modes is incresing for incresing f. For in cresing nonliner prmeter (i.e. ) s shown in Figure 6(c), we show tht the regions of definition re decresing nd the two rnches of the stedy stte mplitude curve re contrcted nd give one continuous curve which is stle nd response mplitude of the inplne mode is incresed. Figure 6(d) show tht the response mplitudes of the inclined cle re incresing for

7 () () (c) (d) (e) (f) Figure 6. (): Frequency response curves for mplitudes ginst σ ; (): Frequency response curve for incresing prmetric excittion force f =.; (c): Frequency response curve for incresing nonliner prmeter β =.; (d): Frequency response curve for decresing nonliner prmeter η =.; (e): Frequency response curve for incresing nonliner prmeter γ =.8; (f): Frequency response curve for negtive vlue of nonliner prmeter γ =.. decresing nonliner prmeter nd the regions of multi-vlued nd instility of two modes re incresing. The regions of instility solutions re incresing for incresing nonliner prmeter s shown in Figure 6(e). Figure 6(f) shows tht for negtive vlue of nonliner prmeter the response mplitudes re incresing nd the stility solution re decresing with incresing region of multi-vlued. Figure 7, represent the vrition of the mplitudes of the inclined cle ginst the detuning prmeter. In Figure 7(), we see tht ech mode of the inclined cle hs one continuous curve nd single vlued solution nd it is symmetric out the origin nd it is noticed tht the in-plne mode reches mximum vlue t nd the out-of-plne mode reches minimum vlue t the sme vlue of. Also, it intersects in two points nd these modes hve stle nd unstle solutions. From Figure 7(), we oserve tht for incresing prmetric excittion force f the symmetric rnch moves up with incresed mgnitudes nd the region of stility is incresed. For incresing nonliner prmeter, we note tht the mplitudes of the two modes of the inclined cle hve decresing mgnitudes nd incresing stle solutions, s shown in Figure 7(c). The stedy stte mplitudes of the two modes re incresing for decresing nonliner prmeter s shown in Figure 7(d). Also, the region of stility solutions is incresed. From Figure 7(e) we oserve tht the stedy stte mplitudes nd of the two modes re incresing for decresing vlue of nonliner prmeters respectively with incresing stle solutions. The stility solution is decresing s the nonliner prmeter is increse nd the curves re shifted to the right nd hs hrdening phenomen nd there exists jump phenomen, s shown in Figure 7(f). Figure 8 represent force-response curves for the nonliner solution of the cse of simultneous principl prmetric resonnce in the presence of : internl resonnces. In this figure the mplitudes of the inclined cle re plotted s function of the prmetric excittion force f. Figure 8 shows tht the response mplitudes of the inclined cle hve continuous curve nd the curve hs stle nd unstle solutions.. Comprison with Pulished Work In comprison with the previous work [8], we hve the glol ifurction of this inclined cle leding to primry resonnces nd : internl resonnce is investigted. A new glol perturtion technique is employed to nlyze Shilnikov type homoclinic orits nd chotic dynmics in the inclined cle. Kmel nd Hmed [9],

8 76 Amplitudes () f =.6 f =. f = () f =. f =.5 f = et=. et=.6 et=.6.8 et=..6. et= et= (c) (d) (e) (f) Figure 7. (): Frequency response curves for simultneous principl prmetric resonnce in the presence of : internl resonnce nd ; (): Frequency response curve for prmetric excittion force f ; (c): Frequency response curve for nonliner prmeter γ ; (d): Frequency response curve for nonliner prmeter η ; (e): Frequency response curve for nonliner prmeter α ; (f): Frequency response curve for nonliner prmeter α f 6 8 f Figure 8. Force response curves for (, ). studied the nonliner ehvior of n inclined cle sujected to hrmonic excittion ner the simultneous primry nd : internl resonnce y using multiple scle method. In this pper, periodic nd chotic response of discretiztion two-degree-of-freedom model of suspended inclined cle, contining : internl resonnce, suject to hrmonic externl nd prmetric excittion re otined. The stle/unstle periodic solutions re determined using the method of multiple scle nd re presented through frequency response plots. Chotic responses re determined y numericl integrtion of the governing ordinry differentil equtions of motion. Vrition of the prmeters,,,,,,, fleds

9 77 to multi-vlued mplitudes nd hence to jump phenomen. 5. Conclusions Cles re very efficient structurl memers nd hence hve een widely used in mny long-spn structures, including suspension, roofs nd guyed towers. The nonliner dynmic response of the nonliner system sujected to externl nd prmetric excittions is investigted. The method of multiple scles is pplied to otin the solution of the considered system up to second order pproximtion. The numericl solutions nd chotic response of this nonliner system re investigted. The stility of the proposed nlytic nonliner solution is studied t worst resonnce cse which is the simultneous principl prmetric resonnce in the presence of : internl resonnces. The modultion equtions of the mplitudes nd phses re otined nd stedy stte solutions re determined. The effects of some nonliner prmeters on the stedy stte response of the virting cle leding to multi-vlued solutions. From the nlysis the following my e concluded. ) For the resonnce cse, we note tht the stedy stte mplitude is incresed to out % compred to sic cse with multi-limit cycle, nd it is etter to void this resonnce cse s working conditions for the system. ) The stedy stte mplitude of the system re incresing for incresing externl or prmetric excittion force, nd for lrge vlues of the system ecome unstle. ) Vrition of α, α, β, γ, η, γ, η, f leds to multi-vlued mplitudes nd hence jump phenomen. ) For incresing prmetric excittion force f or negtive vlue of the nonliner prmeter γ we oserve tht the stedy stte mplitudes of the two modes re incresing with incresing instility solutions. 5) Incresing of the nonliner prmeters η or γ cn reduce the mplitude of the system nd otin the effect of reduction of the mplitude. 6) Vrition of the prmeter α leds to multi-vlued mplitudes nd hence to jump phenomen. 7) For incresing prmetric excittion force f or decresing nonliner prmeter α we show tht the stedy stte mplitudes of the two modes re incresing. For incresing nonliner prmeter η we note tht the stedy stte mplitudes of the two modes re decresing with decrese of the stility solutions. 6. References [] H. N. Arft nd A. H. Nyfeh, Non-Liner Responses of Suspended Cles to Primry Resonnce Excittions, Journl of Sound nd Virtion, Vol. 66, No.,, pp doi:.6/s-6x()9-7 [] G. Reg, Non-Liner Virtions of Suspended Cles; Prt I: Modeling nd Anlysis, Journl of Applied Mechnics Review, Vol. 57, No. 6,, pp doi:.5/.777 [] G. Reg, Non-Liner Virtions of Suspended Cles; Prt II: Deterministic Phenomen, Journl of Applied Mechnics Review, Vol. 57, No. 6,, pp doi:.5/.7775 [] S. R. Nielsen nd P. H. Kirkegrd, Super nd Comintoril Hrmonic Response of Flexile Inclined Cles with Smll Sg, Journl of Sound nd Virtion, Vol. 5, No.,, pp doi:.6/jsvi..979 [5] G. Zheng, J. M. Ko nd Y. O. Ni, Super-Hrmonic nd Internl Resonnces of Suspended Cle with Nerly Commensurle Nturl Frequencies, Nonliner Dynmics, Vol., No.,, pp doi:./a:9599 [6] W. Zhng nd Y. Tng, Glol Dynmics of the Cle under Comined Prmetricl nd Externl Excittions, Interntionl Journl of Non-Liner Mechnics, Vol. 7, No.,, pp doi:.6/s-76()6-9 [7] A. H. Nyfeh, H. Arft, C. M. Chin nd W. Lcronr, Multimode Interctions in Suspended Cles, Journl of Virtion nd Control, Vol. 8, No.,, pp doi:.77/ [8] H. Chen nd Q. Xu, Bifurction nd Chos of n Inclined Cle, Nonliner Dynmics, Vol. 57, No. -, 9, pp doi:.7/s [9] M. M. Kmel nd Y. S. Hmed, Non-Liner Anlysis of n Inclined Cle under Hrmonic Excittion, Act Mechnic, Vol., No. -,, pp doi:.7/s77--9-x [] A. Ae, Vlidity nd Accurcy of Solutions for Nonliner Virtion Anlyses of Suspended Cles with Oneto-One Internl Resonnce, Nonliner Anlysis: Rel World Applictions, Vol., No.,, pp doi:.6/j.nonrw [] N. Srinil, G. Reg nd S. Chucheepskul, Two-yo-One Resonnt Multi-Modl Dynmics of Horizontl/Inclined Cles. Prt I: Theoreticl Formultion nd Model Vlidtion, Nonliner Dynmics, Vol. 8, No., 7, pp. -5. doi:.7/s [] N. Srinil nd G. Reg, Two-To-One Resonnt Multi- Modl Dynmics of Horizontl/Inclined Cles. Prt II: Internl Resonnce Activtion Reduced-Order Models nd Nonliner Norml Modes, Nonliner Dynmics, Vol. 8, No., 7, pp doi:.7/s z [] R. Alggio nd G. Reg, Chrcterizing Bifurctions nd Clsses of Motion in the Trnsition to Chos through D-Tori of Continuous Experimentl System in Solid Mechnics, Physic D, Vol. 7, No.,, pp doi:.6/s67-789(99)69- [] G. Reg nd R. Alggio, Sptio-Temporl Dimensionl-

10 78 ity in the Overll Complex Dynmics of n Experimentl Cle/Mss System, Interntionl Journl of Solids nd Structures, Vol. 8, No. -,, pp doi:.6/s-768()5-9 [5] A. Gonzlez-Buelg, S. A. Neild, D. J. Wgg nd J. H. G. Mcdonld, Modl Stility of Inclined Cles Sujected to Verticl Support Excittion, Journl of Sound nd Virtion, Vol. 8, No., 8, pp doi:.6/j.jsv.8.. [6] N. C. Perkins, Modl Interctions in the Non-Liner Response of Inclined Cles under Prmetric/Externl Excittion, Interntionl Journl of Non-liner Mechnics, Vol. 7, No., 99, pp. -5. doi:.6/-76(9)98-j [7] C. L. Lee nd N. C. Perkins, Nonliner Oscilltions of Suspended Cles Contining Two-to-One Internl Resonnce, Nonliner Dynmics, Vol., 99, pp [8] C. L. Lee nd N. C. Perkins, Three-Dimensionl Oscilltions of Suspended Cles Involving Simultneous Internl Resonnce, Proceedings of ASME Winter Annul Meeting D-, 99, pp [9] M. Eiss nd M. Syed, A Comprison etween Pssive nd Active Control of Non-Liner Simple Pendulum Prt-I, Mthemticl nd Computtionl Applictions, Vol., No., 6, pp [] M. Eiss nd M. Syed, A Comprison etween Pssive nd Active Control of Non-Liner Simple Pendulum Prt-II, Mthemticl nd Computtionl Applictions, Vol., No., 6, pp [] M. Eiss nd M. Syed, Virtion Reduction of Three DOF Non-Liner Spring Pendulum, Communiction in Nonliner Science nd Numericl Simultion, Vol., No., 8, pp doi:.6/j.cnsns.6.. [] M. Syed, Improving the Mthemticl Solutions of Nonliner Differentil Equtions Using Different Control Methods, Ph.D. Thesis, Menofi University, Egypt, Novemer 6. [] M. Syed nd Y. S. Hmed, Stility nd Response of Nonliner Coupled Pitch-Roll Ship Model under Prmetric nd Hrmonic Excittions, Nonliner Dynmics, Vol. 6, No.,, pp. 7-. doi:.7/s [] M. Syed nd M. Kmel, Stility Study nd Control of Helicopter Blde Flpping Virtions, Applied Mthemticl Modelling, Vol. 5, No. 6,, pp doi:.6/j.pm... [5] M. Syed nd M. Kmel, : nd : Internl Resonnce Active Asorer for Non-Liner Virting System, Applied Mthemticl Modelling, Vol. 6, No.,, pp. -. doi:.6/j.pm [6] Y. A. Amer nd M. Syed, Stility t Principl Resonnce of Multi-Prmetriclly nd Externlly Excited Mechnicl System, Advnces in Theoreticl nd Applied Mechnics, Vol., No.,, pp. -. [7] M. Syed, Y. S. Hmed nd Y. A. Amer, Virtion Reduction nd Stility of Non-Liner System Sujected to Externl nd Prmetric Excittion Forces under Nonliner Asorer, Interntionl Journl of Contemporry Mthemticl Sciences, Vol. 6, No.,, pp [8] A. H. Nyfeh, Non-Liner Interctions, Wiley/Inter- Science, New York,.