 Applied Mathematics, 2011, 2, 1469-1478 doi:10.4236/am.2011.212209 Published Online December 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM The Analytical and Numerical Solutions of Differential Equations Describing of an Inclined Cable Subjected to External and Parametric Excitation Forces Mohamed S. Abd Elkader1,2 1Department of Mathematics and Statistics, Faculty of Science, Taif University, El-Taif, Kingdom of Saudi Arabia 2Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt E-mail: moh_6_11@yahoo.com Received October 21, 2011; revised November 22, 2011; accepted November 30, 2011 Abstract The analytical and numerical solutions of the response of an inclined cable subjected to external and para-metric excitation forces is studied. The method of perturbation technique are applied to obtained the periodic response equation near the simultaneous principal parametric resonance in the presence of 2:1 internal reso-nance of the system. All different resonance cases are extracted. The effects of different parameters and worst resonance case on the vibrating system are investigated. The stability of the system are studied by us-ing frequency response equations and phase-plane method. Variation of the parameters α2, α3, β2, γ2, η2, γ3, η3, f2 leads to multi-valued amplitudes and hence to jump phenomena. The simulation results are achieved using MATLAB 7.6 programs. Keywords: Perturbation Method, Resonance, Chaotic Response, Stability 1. Introduction Cable structures play an important role in many engi-neering fields, such as civil, ocean and electric engineer-ing. Arafat and Nayfeh  studied the motion of shallow suspended cables with primary resonance excitation. The method of multiple scales is applied to study nonlinear response of this suspended cables and its stability and the dynamic solutions. Some interesting work on the nonlin- ear dynamics of cables to the harmonic excitations can be found in the review articles by Rega [2,3]. Nielsen and Kierkegaard  investigated simplified models of inclined cables under super and combinatorial harmonic excitation and gave analytical and purely numerical re- sults. Zheng, Ko and Ni  considered the super-har- monics and internal resonance of a suspended cable with almost commensurable natural frequencies. Zhang and Tang  investigated the chaotic dynamics and global bifurcations of the suspended inclined cable under com- bined parametric and external excitations. Nayfeh et al.  investigated the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in 2:1 and 1:1 in- ternal resonances. Chen and Xu  investigated the glo- bal bifurcations of the inclined cable subjected to a har- monic excitation leading to primary resonances with the external damping by using averaging method. Kamel and Hamed , studied the nonlinear behavior of an inclined cable subjected to harmonic excitation near the simulta-neous primary and 1:1 internal resonance using multiple scale method. Abe  investigated the accuracy of non- linear vibration analyses of a suspended cable, which possesses quad ratic and cub ic nonlinearities, with 1:1 in - ternal resonance. The nonlinear dynamics of suspend cable structures have been studied with 2:1 internal reso-nances by the authors [11,12]. Experimental studies of this problem have been conducted by Alaggio and Rega  and Rega and Allagio , however explicit stabil-ity regions for t he semi-triv i al sol ut i on h ave not be en cal - culated analytically. Here, we use a modal model to com- pute the insta b i li t y b o un d a ry f or a range of excitati on fre- quencies close to the 2:1 resonance for an inclined cable, including nonlinear modal interaction. The out-of-plane M. S. ABD ELKADER 1470 dynamic stability of in clined cables subj ected to in-plane vertical support excitation is investigated by Gonzalez-Buelga et al. . Perkins  examined the effect of one support motion on the three-dimensional nonlinear response. Using the Galerkin method, he constructed a two-degree-of-freedom model to analyze the 2:1 internal resonance. Lee and Perkins  extended the work to in- clude second-order perturbations and multiple internal resonances. Still, the focus was on the 2:1 internal reso- nance, whereas the excitation was changed to a harmoni- cally varying load per unit length acting in the static equilibrium plane. Lee and Perkins  also used a three-degree-of-freedom model to simulate non-linear response of suspended, inclined cables driven by planar excitation and determined the existence and stability of four classes of periodic solutions. Eissa and Sayed [19-21] and Sayed , studied the effects of different active controllers on simple and spring pendulum at the primary resonance via negative velocity feedback or its square or cubic. Sayed and Hamed  studied the response of a two-degree-of-freedom system with quadratic coupling under parametric and harmonic excitations. The method of multiple scale perturbation technique is applied to solve the non-linear differential equations and obtain approximate solutions up to and including the second-order approximations. Sayed and Kamel [24,25] investigated the effects of different con-trollers on the vibrating system and the saturation control to reduce vibrations due to rotor blade flapping motion. The stability of the ob tained numerical solution is inves-tigated using both phase plane methods and frequency response equations. Amer and Sayed , studied the response of one-degree-of freedom, non-linear system under multi-parametric and external excitation forces simulating the vibration o f the cantilever beam. Variation of some parameters leads to multi-valued amplitudes and hence to jump phenomena. Sayed et al. , investigated the non-linear dynamics of a two-degree-of freedom vi- bration system including quadratic and cubic non-lin- earities subjected to external and parametric excitation forces. The stability of the system is investigated using both frequency response curves and phase-plane trajecto- ries. The effects of different parameters of the system are studied numerically. This work deals with model having two-degree-of-freedom nonlinear system subjected to external and pa-rametric excitation forces describes the vibrations of an inclined cable. The method of multiple scales perturba-tion is applied to obtain modulation response equations near the simultaneous principal parametric resonance in the presence of 2:1 internal resonance (222 and 122). The stability of the proposed analytic nonlin-ear solution near the above case is studied and the stabil-ity condition is determined. The effect of different pa-rameters on the steady state response of the vibrating system is studied and discussed from the frequency re-sponse curves. The numerical solution and chaotic re-sponses of the nonlinear system of an inclined cable for some different parameters are also studied. A compari-son with previously published work is included. 2. Mathematical Analysis Our attention is focused on an elastic-sag hanging at fixed supports and excited by harmonic and parametric distributed vertical forcing in plane. The two-degree-of-freedom describing the nonlinear dynamics of cable shown in Figure 1, can be written as: 2223211 222 220xcxx xyxxy    (1) 232233 3112 22cos cosycyy xyyxftyf t  2y (2) where x and denote in-plane and out-of-plane dis-placements, respectively, and dots denote derivatives with respect to the time t. The parameters 1 and 2 are the viscous damping coefficients, 1yc c and 2 are the natural frequencies associated with in-plane and out-of-plane modes 1 and 2 are the excitation frequencies, f1 and f2 are the excitation forces amplitude, 222,,, 233,, and 3 are the coefficients of nonlinear pa-rameters. The linear viscous damping forces, the exciting forces and nonlinear parameters are assumed to be 22 222112 222ˆˆˆ ˆˆˆ,, , ,ˆˆˆ,1, 22, 3nnsssssccc cffns,s  where  is a small perturbation parameter and 01. For the convenience of the analysis of Equa-tions (1)-(2), the non-dimensional parameter  is in-troduced. We ca n obt ai n 22 2112 223 222ˆˆˆ2(ˆˆ()02)xcx xxyxxy    (3) 22 2322233 32112 2ˆˆˆˆ2(ˆˆ(coscos ))ycy yxyyxyftyf t   (4) The parameters 23 2ˆˆˆ,,ˆˆˆˆˆ,,,,cc are of the order of 1 and the parameters 12232312 are of the order of 2. The approximate solution of Equations (3)-(4) can be obtained using the method of multiple scales . Let ˆˆˆ,,,f f0012 101222012(;) (,,)(,,)(,,)xtxTTT xTTTxTTT (5) Copyright © 2011 SciRes. AM M. S. ABD ELKADER1471 Figure 1. A schematic of inclined cable under combined excitations. 0012 101222012(;)(,,)(,,)(,,)yty TTTyTTTyTTT (6) where, nnTt (n = 0, 1, 2) are the fast and slow time scales respectively. In terms of 0 and , the time derivatives transform according to 1,TT 2T20122222001102ddd2(2dDD DtDDDDDDt  2) (7) where nn. Substituting Equations (5)-(6) and (7) into Equations (3)-(4) and equating the coefficients of similar powers of  in both sides, we obtain the differen-tial equations as follo ws: DT Order 0(): 22010()Dx00 (8) 22020()Dy (9) Order 1(): 22 2011 010202ˆˆ()2DxDDxx 20y0y (10) 2202101030ˆ()2DyDDyx 2 (11) Order (): 22 201210011 02000201 2013220 2001()2 2ˆˆˆ222ˆˆDxDxDDxDDxcDxxxyyxxy  (12) 22 202210 011 02032003 0101302300 110 0220()2 2ˆˆˆ2()ˆˆˆcos cosDyDyDDyDDycDyxyyxyyxfTyfT (13) The solution of Equations (8)-(9) can be expressed in the complex form: 01210( ,)exp()xAT Ti Tcc (14) 012 20( ,)exp()yBTTiTcwhere cc denotes the complex conjugate of the preceding terms and A, are complex functions in 1 and 2T which determined through the elimination of secular and small-divisor terms from the first and second-order of approximations. BTIn this case, we analyze the case where 222 and 122. To describe quantitatively the nearness of the resonances, we introduce the detuning parameters 1 and 2 according to 221ˆ2, 212ˆ2. Substituting Equations ( 14)-( 15) into Equations (10)-(11) and eliminating the secular terms leads to the solvability conditions for the first-order expansion as: 212 211ˆˆ2exp(iDA BiT )0 (16) 21 321ˆˆ2exp(iDB AB iT )0 (17) After eliminating the secular terms, the particular solu-tions of Equations (10)-(11) a re given by : 2222110222111ˆˆˆexp(2 )3xAiT AABBcc (18) 3102221212ˆexpyABiTcc(19) Now substituting Equations (14)-(15) and Equations (18)-(19) into Equations (12)-(13), the following are ob-tained 220122211121211()ˆ(22)exp( )DxDA icA iDAABBAAiTNSTcc  0 (20) 2202222122223 422220()ˆ22 exp(1ˆexp( ())2DyDB icB iDBAABBBiTfBiTNSTcc  0) (21) where 23 2212222212 1ˆˆˆˆ24ˆ2,()    222221ˆ10 ˆ33 , 2323332222121ˆˆˆ2ˆ2,()  234321ˆˆ2ˆ3  c (15) and NST stands for non-secular terms. Eliminating the Copyright © 2011 SciRes. AM M. S. ABD ELKADER 1472 secular terms leads to the solvability conditions for the second-order expansion 222111 211ˆ22iDADA icAABBAA  (22) 222 1324222ˆ221ˆˆexp( )2iDBDBicBAABBBf BiT  11 (23) Stability Analysis of Nonlinear Solutions From Equation (7), multiplying both sides be 12,i 22i we get 21111d22 2dAiiDAit2DA (24) 2221d22 2dBiiDBit22DB (25) To analyze the solutions of Equations (16)-(17) and Equations (22)-(23), we express A and in the polar form B1212 12(, )(2),(, )(2)iiATTae BTTbe (26) where a , b and (1,2ss) are the steady state ampli- tudes and phases o f the motion respectively. Substituting Equations (26), (16)-(17) and Equations (22)-(23) into Equations (24)-(25) and equating the real and imaginary parts we obtain the following equations describing the modulation of the amplitudes and phases: 22221211sin48acab2  (27) 22221221123523 621112cos488816abab a  (28) 23 322122222sin sin448fbcbab b   (29) 232322223210 122cos48cos4babfbb  9ab (30) where 237 2389103222222 225678123411122 21121,8832 32,,,, , ,ˆˆand2 ,2TTForm the system of Equations (27)-(30) to have sta-tionary solutions, the following conditions must be satis-fied: 120ab (32) It follows from Equation (31) that 21111,22  (33) Hence, the steady state solutions of Equations (27)-(30) are given by 222212211sin 048ca b  (34) 23622212 2211252321211() cos4880816abab   a (35) 23 322222221sinsin 0448fcbab b   (36) 223 31223210 12221cos248cos 04babfbb   9ab (37) Solving the resulting algebraic equations for the fixed points of the practical case where , that is non-planar motions, we obtain the following frequency response equations 0, 0ab222222426412 112322 6121 1224212()2( )2( )20acaababab aab  (38) 2222 242 262212910 91242422210 19 10422242 12214216cos()02bcbabbbafbababfab 2b(39) where 2    5236 22212322111112,,88416 8       (31) and 23 3422248 . The stability of the obtained fixed points for the simul-Copyright © 2011 SciRes. AM M. S. ABD ELKADER1473 taneous primary, principal parametric and 2:1 internal resonance case is determined and studied as follows: one lets 101110 1101,and ss saaabbb   (40) where a10, b10 and 0s are the solutions of Equations (34)-(37) and a11, b11, 1s are perturbations which are assumed to be small compared to a10, b10 and 0s. Sub-stituting Equation (40) into Equations (27)-(30), using Equations (34)-(37) and keeping only the linear terms in a11, b11, 1s we obtain: 2113 10202131020 11111cos2sinacaKbKb b  (41) 2121106 1021 4209 101110 10231041020202110310 4101201 102010 10102610210 10101110 22213() cos 28sin sin2cos 2cos23cos4si2ba aaaabaababba bafbbbf      10 11n (42) 114102011224102011224101020 211010 11102sinsin sin4cos cos4bb afca bfab b   (43) 4 2091011 4102021241091012010 101110 101022111010 112cos 2sincos3 cos2sin2aa aaafbbb bf  24b (44) The system of Equations (41)-(44) are first order autonomous ordinary differential equations and the sta-bility of a particular fixed point with respect to an infini-tesimal disturbance proportional to exp( )t is deter-mined by eigenvalues of the Jacobian matrix of the right hand sides of Equations (41)-(44). The zeros of the char-acteristic equation are given by 43212340LL LL (45) where, and are functions of the parameters (1 23223232112212123,,LLL,, ,cc4L,,,,,,,,, ,, ,,ab f). According to the Routh-Hurwitz criterion the necessary and sufficient conditions for all the roots of Equation (45) to possess negative real parts are: 2112331231440,0,0, 0LLLLLLLL LLL  (46) The system is stable if the eigenvalues have negative real parts, otherwise is unstable. In the frequency re-sponse curves, solid/dotted lines denote stable/ unstable periodic responses, respectively. 3. Results and Discussion The response of the two-degree-of-freedom nonlinear system under both parametric and external excitations is studied. The solution of this system is determined up to and including the second order approximation by apply-ing the multiple time scale perturbation. The steady state solution and its stability are determined and representa-tive numerical results are included. The stability zone and effects of the different parameters are discussed us-ing frequency response curve. The stability of the nu-merical solution is studied also using the phase-plane method. Some of the resulting resonance cases are con-firmed applying well-known numerical techniques. The effects of the some different parameters on the vibrating system behavior are investigated and discussed. 3.1. Numerical Solution Figure 2 shows that the response of the inclined cable for the non-resonant at the practical values of the pa-rameters c1 = 0.0002, c2 = 0.03, α2 = 0.2, β2 = 0.5, γ2 = 0.3, η2 = 0.5, α3 = 0.03, η3 = 0.05, γ3 = 0.04, 1 = 2, 2 = 0.01, 1 = 2.75, 2 = 3.2, 1 = 1.2, 2 = 1.5. It can be seen from this figure that the steady state amplitude is about 0.005 with dynamic chaotic behavior for the in- plane mode and about 0.18 with multi-limit cycle for the out-of-plane mode. The amplitudes decreasing with in-creasing time and tend to steady state motion and have stable solution. The worst resonance case is also con-firmed numerically as shown in Figure 3. From this fig-ure, it can be notice that the maximum steady state am-plitude of the in-plane mode is about 130 times that of basic case with multi-limit cycle, while the maximum amplitude of out-of-plane mode is about 4 times of the basic case with chaotic motion. Effects of external and parametric excitation forces f1 and f2. Copyright © 2011 SciRes. AM M. S. ABD ELKADER 1474 0100 200 300-0.0500.05TimeA mpli tude(x) 0100 200 300-0.500.5TimeAm plitude(y) Figure 2. Non-resonance system behavior (basic case) Ω1 ≠ ω1 ≠ ω2. 0100 200 300-101TimeAmplitude(x) 0100 200 300-101TimeAm plitude(y ) Figure 3. Simultaneous principal parametric resonance in the presence of 2:1 internal resonance (2212and 2 2). For increasing the amplitude of the external or parame- tric excitation forces f1 or f2, we observe that the modes of vibration have increasing magnitudes and there exist chaotic dynamic motion as shown in Figures 4 and 5. 3.2. Frequency Response Curves The frequency response Equations (38)-(39) are nonlin-ear algebraic equations in the amplitudes of the system (in-plane mode) and b(out-of-plane mode). The stabil-ity of a fixed point solution is studied by examination of the eigenvalues of Equation (45). The numerical results of Equations (38) and (39) are plotted in Figures 6-8. aFigure 6, show the frequency response curves of the two modes of inclined cable against detuning parameter 1. From the geometry of the figures we observe that the amplitudes have two branches and these branches are bent to the right, the bending leads to multi-valued solu-tions and hence the effective nonlinearity is hardening type. In Figure 6(a), there are two branches of nontrivial solution such that the left branch stable and the right branch lose stability as 10.4. Figure 6(b), show that the steady state amplitudes are increasing for increasing 0100 200 300-10-50510TimeAmpli tude(x) 0100 200 300-10-50510TimeAmplitude(y) Figure 4. Effects of increasing value of external excitation force f1 = 5. 0100 200300-4-2024TimeAmplitude(x) 0100 200 300-4-2024TimeAmplitude(y) Figure 5. Effects of increasing value of parametric excita-tion force f2 = 3. parametric excitation force 2. The region of instability for two modes is increasing for increasing 2. For in creasing nonlinear parameter 2ff (i.e. 21) as shown in Figure 6(c), we show that the regions of definition are decreasing and the two branches of the steady state am- plitude curve are contracted and give one continuous curve which is stable and response amplitude of the in- plane mode is increased. Figure 6(d) show that the re- sponse amplitudes of the inclined cable are increasing for Copyright © 2011 SciRes. AM M. S. ABD ELKADER Copyright © 2011 SciRes. AM 1475 0 1 2300.10.20.30.40.5a0 1 2 300. 511. 522. 53b 0246800.511.522.5a024680246b (a) (b) 00.5 11.5 22.50.20.40.60.81a00.5 11.5 22.50123b 00.5 11.5 22.5 3012345a00.5 11.5 22.5 300.511.522.53b (c) (d) 0 1 2 300.511.522.5a0 12300.511.522.53b 0123123456a0 12301234b (e) (f) Figure 6. (a): Frequency response curves for amplitudes against σ1; (b): Frequency response curve for increasing parametric excitation force f2 = 3.0; (c): Frequency response curve for increasing nonlinear parameter β2 = 1.0; (d): Frequency response curve for decreasing nonlinear parameter η3 = 0.1; (e): Frequency response curve for increasing nonlinear parameter γ2 = 1.8; (f): Frequency response curve for negative value of nonlinear parameter γ3 = –0.4. decreasing nonlinear parameter 3 and the regions of multi-valued and instability of two modes are increasing. The regions of instability solutions are increasing for increasing nonlinear parameter 2 as shown in Figure 6(e). Figure 6(f) shows that for negative value of non- linear parameter 3 the response amplitudes are in-creasing and the stability solution are decreasing with increasing region of multi-valued. Figure 7, represent the variation of the amplitudes of the inclined cable against the detuning parameter 2. In Figure 7(a), we see that each mode of the inclined cable has one continuous curve and single valued solution and it is symmetric about the origin and it is noticed that the in-plane mode reaches maximum value at 20 and the out-of-plane mode reaches minimum value at the same value of 2. Also, it intersects in two points and these modes have stable and unstable solutions. From Figure 7(b), we observe that for increasing parametric excitation force f2 the symmetric branch moves up with increased magnitudes and the region of stability is in-creased. For increasing nonlinear parameter 3, we note that the amplitudes of the two modes of the inclined ca-ble have decreasing magnitudes and increasing stable solutions, as shown in Figure 7(c). The steady state am-plitudes of the two modes are increasing for decreasing nonlinear parameter 3 as shown in Figure 7(d). Also, the region of stability solutions is increased. From Fig-ure 7(e) we observe that the steady state amplitudes a and b of the two modes are increasing for decreasing value of nonlinear parameters 3 respectively with increasing stable solutions. The stability solution is de-creasing as the nonlinear parameter 2 is increase and the curves are shifted to the right and has hardening phe-nomena and there exists jump phenomena, as shown in Figure 7(f). Figure 8 represent force-response curves for the non- linear solution of the case of simultaneous principal pa-rametric resonance in the presence of 2:1 internal reso-nances. In this figure the amplitudes of the inclined cable are plotted as a function of the parametric excitation force f2. Figure 8 shows that the response amplitudes of the inclined cable have a continuous curve and the curve has stable and unstable solutions. 4. Comparison with Published Work In comparison with the previous work , we have the global bifurcation of this inclined cable leading to pri-mary resonances and 1:1 internal resonance is investi-gated. A new global perturbation technique is employed to analyze Shilnikov type homoclinic orbits and chaotic dynamics in the inclined cable. Kamel and Hamed , M. S. ABD ELKADER Copyright © 2011 SciRes. AM 1476         Amplitudesab -2 -1 0 1 200.20.40.60.8a-1 -0.500.5 10.20.40.60.811.2bf2=0.3f2=0.3f2=0.6f2=0.6f2=0.15f2=0.15 (a) (b) -1.5 -1 -0.5 00.5 11.500.10.20.30.40.5a-1.5 -1 -0.5 00.5 11.50. 20. 30. 40. 50. 60. 70. 8b  -0.4 -0.200.2 0.4 0.60.10.20.30.40.50.6a-0.4 -0.200.2 0.40.20.40.60.8beta3=1.6eta3=1.6eta3=0.6eta3=0 .6eta3=0.2 eta3=0.2 (c) (d) -0.2 -0.100.1 0.2 0.300.20.40.60.8a-0.2 -0.1 00.1 0.2 0.30. 20. 40. 60. 8b -2 -1 0 12300.511.5a-2 -1 0 1 230.20.30.40.50.60.70.8b  2 (e) (f) Figure 7. (a): Frequency response curves for simultaneous principal parametric resonance in the presence of 2:1 internal resonance 22 12and 2 ; (b): Frequency response curve for parametric excitation force f2; (c): Frequency response curve for nonlinear parameter γ3; (d): Frequency response curve for nonlinear parameter η3; (e): Frequency response curve for nonlinear parameter α3; (f): Frequency response curve for nonlinear parameter α2. 0246810 1200. 511. 5f2a 0246810012345f2b Figure 8. Force response curves for (2212,22). studied the nonlinear behavior of an inclined cable sub-jected to harmonic excitation near the simultaneous pri-mary and 1:1 internal resonance by using multiple scale method. In this paper, periodic and chaotic response of a dis-cretization two-degree-of-freedom model of a suspended inclined cable, containing a 2:1 internal resonance, sub-ject to harmonic external and parametric excitation are obtained. The stable/unstable periodic solutions are de-termined using the method of multiple scale and are pre-sented through frequency response plots. Chaotic re-sponses are determined by numerical integration of the governing ordinary differential equations of motion. Var-iation of the parameters 23222,, ,,,332,,fleads M. S. ABD ELKADER1477 to multi-valued amplitudes and hence to jump phenom-ena. 5. Conclusions Cables are very efficient structural members and hence have been widely used in many long-span structures, including suspension, roofs and guyed towers. The nonlinear dynamic response of the nonlinear system sub-jected to external and parametric excitations is investi-gated. The method of multiple scales is applied to obtain the solution of the considered system up to second order approximation. The numerical solutions and chaotic re-sponse of this nonlinear system are investigated. The stability of the proposed analytic nonlinear solution is studied at worst resonance case which is the simultane-ous principal parametric resonance in the presence of 2:1 internal resonances. The modulation equations of the amplitudes and phases are obtained and steady state solu-tions are determined. The effects of some nonlinear pa-rameters on the steady state response of the vibrating cable leading to multi-valued solutions. From the analy-sis the following may be concluded. 1) For the resonance case 22122, 2 we note that the steady state amplitude is increased to about 130% compared to basic case with multi-limit cycle, and it is better to avoid this resonance case as working condi-tions for the system. 2) The steady state amplitude of the system are in-creasing for increasing external or parametric excitation force, and for large values of the system become unsta-ble. 3) Variation of α2, α3, β2, γ2, η2, γ3, η3, f2 leads to multi-valued amplitudes and hence jump phenomena. 4) For increasing parametric excitation force f2 or negative value of the nonlinear parameter γ3 we observe that the steady state amplitudes of the two modes are increasing with increasing instability solutions. 5) Increasing of the nonlinear parameters η3 or γ3 can reduce the amplitude of the system and obtain the effect of reduction of the amplitude. 6) Variation of the parameter α2 leads to multi-valued amplitudes and hence to jump phenomena. 7) For increasing parametric excitation force f2 or de-creasing nonlinear parameter α3 we show that the steady state amplitudes of the two modes are increasing. 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