Duffing Oscillator with Heptic Nonlinearity under Single Periodic Forcing

Transcription

1 International Journal of Mechanics and Applications 3, 3(: DOI:.593/j.mechanics.33. Duffing Oscillator with Heptic Nonlinearity under Single Periodic Forcing M. O. Oyesanya *, J. I. Nwamba Department of Mathematics, University of Nigeria, Nsukka 4, Nigeria Abstract In this paper, the Homotopy Analysis Method (HAM is used to obtain an accurate analytical two-term approximate solution to the positively damped cubic-quintic-heptic Duffing equation with algebraically decaying amplitude as well as a single periodic forcing. This paper also presents the interesting behavior of the non-zero au xiliary parameter which provides a convenient way to adjust and control the convergence of the approximations. Our analysis shows that neither the strength of the damping nor that of the forcing exerts any influence on the auxiliary parameter of the nonlinearity. We observe also that the degree of damping is elicited by the degree of nonlinearity and the initial guesses of the time constants Keywords Homotopy Analysis, Duffing Oscillator, Heptic Nonlinearity, Periodic Forcing, Damping. Introduction The Duffing equations have been presented in the literature considering various types of nonlinearity. Pirbodaghi et al[] have worked on the Duffing equations with cubic and quintic nonlinearities. Some authors[], [3],[4],[5][6 ],[7] and[8] have investigated many kinds of nonlinear oscillatory systems in physics, mechanics and engineering. Duffing equations with cubic and quintic nonlinearities have not been extensively studied as the one with cubic nonlinearity because of its complexity[9,,, ]. The cubic nonlinear Duffing oscillator had been studied extensively and has been used as model for seismic analysis[3] and for earthquake prediction[5]. Recently the cubic-quintic nonlinearity in Duffing oscillator has been engaging attention[4, 6, 5] and very interesting features have been detected which were non-existent in the cubic nonlinear model. Interesting results have been obtained from the geometric and analytic studies of the un-damped and unforced cubic-quintic Duffing equation using different methods[, 6, 7]. It has been observed that the cubic-quintic nonlinearity gives vent to damping in a morepronounced way than the cubic nonlinearity. We therefore in this paper investigate the influence of higher nonlinearity particularly heptic nonlinearity on the damping of Duffing oscillator. The nth-order Duffing equation with viscous damping and periodic forcing can be generally expressed as * Corresponding author: moyesanya@yahoo.com (M. O. Oyesanya Published online at Copyright 3 Scientific & Academic Publishing. All Rights Reserved N M n+ n m m n= m= ( cos sin u + δu + au = F ωt+ X ωt (. Where δ is the damping coefficient, and an, Fm, X m are arbitrary constants, NM<, In this paper the homotopy analysis method[8, 9,]which has been effectively applied to a wide variety of problems in applied mathematics, physics and engineering is applied to the positively damped cubic-quintic-heptic Duffing equation with a single sinusoidal forcing governed by ut ( = f[ ut (, ut (, F], t (. and subject to the initial conditions u( = a, u ( = b (. The cubic-quintic-heptic Duffing equation can be used to model a classical particle in a double-well potential. It was used in[] to model the dynamic behavior of a cargo system under crossover indirect tie-down. As was noted in[], the homotopy analysis method which was proposed by introducing an auxiliary parameter gg to construct a new kind of homotopy in a more general form has the following advantages:. It is valid even if a given nonlinear problem does not contain any small/large parameters at all.. It equips us with a convenient way to adjust and control convergence regions of series of analytic approximations. 3. It can be efficiently employed in approximating a nonlinear problem by choosing different sets of base functions. We investigate here factors that influence the non-zero auxiliary parameter g.

2 36 M. O. Oyesanya et al.: Duffing Oscillator with Heptic Nonlinearity under Single Periodic Forcing Free oscillations of a positively damped system have two different time scales. One is related to the frequency of oscillationandthe other to the decaying amplitude of oscillation. It is clear that the free oscillat ion of positively damped systems can be expressed by the set of base functions m m ζt + sin nωt, + ζt cos nωt m, n (.3 {( ( ( ( } where the two unknown time-constants ωωωωωωωωωω relate to the two time scales τ = ωt, τ = ζt (.4 respectively. Having noted this, it will be reasonable to say that the set of base functions given by (.3 can as well be used to express the forced oscillation of a positively damped system as long as the forcing is a function of one of the two time scales. In this regard, we suggest a periodicforcing with amp litude ωω and given by Fcosω t (.5 Under the transformation (.4 one has uu(tt = uu(ττ,ττ, and the original governing equation (. becomes u u u ω + ωζ + ζ = f ωu u, F τ + ζ ( τ τ (.6 τ τ τ τ subject to the initial conditions when ( ( τ, τ u( τ, τ u u τ τ a ω ζ b, =, + =, τ τ τ = = (.7 τ. Basic Ideas A real function can be efficiently represented by a better set of base functions. Hence, the need to approximate a given nonlinear problem by a proper set of base functions equips us with the initiative to apply the homotopy analysis method. Clearly, according to (.3 and the definition (.4, the considered problem can be represented by (.3 such that m u( τ, τ = ( + ζt ηmn, sin nωt+ ϑmn, cos nωt (. m= n= Where ηmn,, ϑmn, arecoefficients. This provides us with a rule, called the Rule of Solution Expression[5]. As was noted in[], HAM is based on continuous variations from the initial guesses to the exact solution of a considered problem. For the problem under consideration, one constructs the continuous mapping ( τ, τ ( τ, τ,, ω Ω(, ζ Λ( in such a way that U(,, q, ( q, ( q u U q q q τ τ Ω Λ vary fro m their initial guesses uu (ττ,ττ, ωω, ζζ to their exact solutions uu(ττ, ττ, ωω, ζζ respectively as qq ( the embedding parameter increases from to. In line with the above reasons one constructs a family in q of nonlinear differential equations with initial conditions ( ( ( { τ τ τ τ } ( τ τ q L U,, q L u, = gpn U,, q (. ( q U τ, τ, = ϕτ, = τ = (. u u ω + ζ = ψ, τ = τ = τ τ (.3 And where uu (ττ, ττ is an in itial guess of uu(ττ,ττ, NN is a nonlinear operator defined by NN[ (ττ, ττ, qq] = Ω (qq uu + Ω(qqΛ(qq uu + Λ (qq uu ff Ω(qq + Λ(qq, uu, FF(ττ ττ ττ ττ ττ ττ ττ (.4 LLLLLLLL LL are two auxiliary linear operators defined by LL[ (ττ,ττ, qq] = Ω + ΩΛ( + ττ ττ + Λ ( + ττ ττ ττ + λλ ττ Ω + Λ( + ττ ττ + λλ ττ (.5 and LL [uu (ττ, ττ ] = ω uu + ω ττ ζ ( + ττ uu + ζ ττ ττ ( + ττ uu uu + λλ ττ ω + ζ ττ ( + ττ uu + λλ ττ uu (.6 in wh ich ω and ζ are initial guesses of the time-constants ωωωωωωωωωω respectively, and λλ,λλ are two constants to be determined later in terms of ζ and ω respectively. The auxiliary non-zero parameter gg equips us with a convenient way to adjust and control the convergence of approximations such that a properly chosen gg guarantees the convergence of the resulting series to be given later at qq =. Under the Rule of Solution Expression[], one chooses uu (ττ, ττ = aa cos ττ + (aa cos ττ + bb sinττ ( + ττ (.7 as the initial guess of uu(ττ, ττ, where aa, aa aaaaaabb are unknown constants to be determined later. When qq =, Eq. (. has the solution (.7 with

3 International Journal of Mechanics and Applications 3, 3(: Ω( = ωω, Λ( = ζζ (.8 and when qq =, Eqs. (.-(.3 becomes exactly (.6 and (.7 as long as Ω( = ωωωωωωωωλ( = ζζ. Thus, (ττ,ττ, = uu(ττ, ττ (.9 Therefore as qq increases from to, truly varies fro m the initial t rial uu to the exact solution uu of the original equations (.6 and (.7; so do Ω and Λ vary from the initial guesses to the time-constants ωωωωωωωωωω respectively. Equations (.-(.3 are the zeroth-order deformation equations. With the nature of (.8, Ω(qq, (ττ, ττ, qqaaaaaaλ(qq can be expanded in power series of qq by Taylor s theorem as: where, ωω [kk] Ω(qq = ωω + + kk = qq kk (. [kk] + ζζ uu [kk] + Λ(qq = ζζ + kk = qq kk (. (ττ,ττ, = uu (ττ,ττ + kk = qq kk (. [kk ωω ] = ddkk Ω(qq [kk, qq = ζζ ] ddqq kk = dd kk Λ (qq [kk, qq = uu ] ddqq kk = kk qqkk, qq = (.3 are the kth-order deformation derivatives. The auxiliary non-parameter gg influences the convergence of the series (. (.. A convergent series given by the HAM (atqq = must be an exact solution of the considered problem as was proved in[8]. Hence, one obtains ω = ωω + + kk = ωω kk (.4 ζζ = ζζ + + kk = ζζ kk (.5 uu(ττ,ττ = uu (ττ, ττ + + uu kk (ττ, ττ kk = (.6 where ωω kk = ωω [kk], ζζ kk = ζζ [kk], uu kk (ττ,ττ = uu [kk] (ττ,ττ (.7 If we d ifferentiate Equations. (. (.3 kk times with respect to qq and then set qq = and finally divide them by kk!. We obtain the so-called kkkkh-order deformation equation[] LL [uu kk (ττ, ττ χχ kk uu kk (ττ, ττ ] = ggrr kk (ττ, ττ WW kk (ττ,ττ + χχ kk WW kk (ττ, ττ (.8 subject to the corresponding initial conditions at (ττ =, ττ = uu kk (ττ,ττ, qq = φφ (.9 where and And ωω uu kk ττ + ζζ uu kk ττ = (. WW kk (ττ, ττ = kk nn jj = ωω jj ωω nn jj uu kk nn nn ττ + jj = ωω jj ζζ nn jj ( + ττ uu kk nn + nn ζζ ττ ττ jj = jj ζζ nn jj ( + ττ uu kk nn nn= ττ + λλωωnn uukk nn ττ+ζζnn+ττ uukk nn ττ (. in wh ich NN[ ] is given by (.4 and from (., the initial condition becomes RR kk (ττ, ττ = dd kk NN [ ], aaaa qq = (. (kk! ddqq kk mm uu kk = ωω kk kk, kk χχ kk =, kk > mm uu kk = ζζ kk kk + kk uu ττ ττ jj (ττ, ττ jj = = ψψ (.4 Now to determine λλ aaaaaaλλ, we demand that for any constants PP aaaaaapp the equation LL [( + ττ (PP sinττ + PP cos ττ ] = (.5 holds. Then λλ = ζζ, λλ = ωω (.6 Hence, the initial guesses ωω, ζζ play a crucial role in determin ing the in itial guess uu (ττ, ττ and the auxiliary linear operatorsllllnnnnll. However we must take note of the existence of the terms ( + ττ sin(ττ and ( + ττ cos(ττ on the right hand side of (.8. Their existence goes contrary to the Rule of Solution Expression, which is clearly described by (.. Therefore, for a uniformly valid solution, one has to set the coefficients of these two terms to zero. We must as well note that for un-damped systems, WW kk (ττ, ττ = WW kk (ττ,ττ = δδ = ζζ = (.7 (.3

4 38 M. O. Oyesanya et al.: Duffing Oscillator with Heptic Nonlinearity under Single Periodic Forcing It is very important not to forget that for the particular problem (positively-damped with algebraically decaying amplitude, we must select suitable aa, ωω aaaaaaaa init ially. 3. Application of HAM Writing (. explicitly under the transformations given in (.4, we obtain ωω uu + ωωωω uu + ζζ uu + δδ ωω + ζζ + αααα + ββuu 3 + μμuu 5 + ρρuu 7 = FF cos ττ ττ ττ ττ ττ ττ ττ (3. subject to the corresponding initial conditions uu(ττ,ττ = aa, ωω (ττ,ττ + ζζ (ττ,ττ =, wwheeeeττ ττ ττ = ττ = (3. Equation (3. can also be obtained from (. by setting MM =, XX mm =, NN = 3, aa = αα, aa = ββ, aa = μμ, aaaaaaaa 3 = ρρ, after which the transformation depicted by (.4 is used. We construct such a family of equations as described by (. where the auxiliary linear operator is given by LL [uu (ττ, ττ ] = ω uu + ω ττ ζ ( + ττ uu + ζ ττ ττ ( + ττ uu uu + ζζ ττ ω + ζ ττ ( + ττ uu + ωω ττ uu (3. and the nonlinear operator with the help of (3. is given by NN[ (ττ, ττ, qq] = ωω + ωωωω + ζζ + δδ ωω + ζζ + αα +ββ 3 + μμ 5 + ρρ 7 FF cos ττ ττ ττ ττ ττ ττ ττ (3.3 Employing (. and (3., we choose our initial guess of uu(ττ, ττ as uu (ττ, ττ = aa( + ττ cos ττ + ζζ sinττ ωω (3.4 We set kk = in (.8 to obtain LL [uu (ττ,ττ uu (ττ, ττ ] = ggrr (ττ, ττ WW (ττ, ττ (3.5 For uniformly valid solution, two linear algebraic equations were obtained and solved after setting to zero the coefficients of ( + ττ sin(ττ and ( + ττ cos(ττ iiii (3.5. The results obtained respectively are as follows: gg ζζ = ωω ζζ ωω +ζζ + BBωω ζζ δδ ωω ζζ BB ωω δδ ωω + ζζ ωω + FFζζ ( +ττ + 5BBωω μμ aa 4 + 7BBBB aa 6 9 5BB +3BB4 +BB6 (3.6 +ττ ζζ +ττ aa 4(+ττ 4 8 (+ττ 6 ωω = gg ωω BBζζ ( + ττ + BBζζ ( + ττ + δδδδ δδδδ (+ττ + αα FF (+ττ ωω aaωω 3ββ aa + (+ττ 8ωω [ + BB ] + 5μμaa 4 (+ττ 4 + 8ωω BB+BB4+ρρaa6+ττ 68ωω+35BB+4BB4+35BB6 (3.7 where BB = ζζ. ωω After eliminating terms that brings non-uniformity equation (3.5 becomes LL [uu (ττ,ττ ] = gg TTTTTTTT (tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt (3.8 Equation (3.8 has the solution uu (ττ, ττ = uu PP + ( + ττ [aa cos ττ + bb sinττ ] (3.9 where uu PP (ττ, ττ = gg[(pp KK + PP KK + PP 3 KK 3 cos 3ττ + (PP 4 KK 4 + PP 5 KK 5 + PP 6 KK 6 sin3ττ + (PP 7 KK 7 + PP 8 KK 8 cos 5ττ + (PP 9 KK 9 + PPKKsin5ττ+PPKKcos7ττ+PPKKsin7ττ (3. PP = ββ aa 3 3BB ( +ττ 3, PP = 5μμaa5 BB 3BB 4 (+ττ 5, PP 3 = ρρ aa 7 9BB 5BB 4 3BB 6 ( +ττ 7 (3. 6ζζ 4 8ωω +ωω ζζ 6 8ωω +4ωω ζζ 8ωω +36ωω ζζ KK = 6ζζ 8ωω, KK +44ωω = ζζ ζζ ζζ 8ωω, KK +576ωω 3 = 4ζζ ζζ 4ζζ 8ωω (3. +96ωω ζζ PP 4 = ββaa 3 BB 3 3BB (+ττ 3, PP 5 = 5BBBB aa 5 3 +BB BB 4 ( +ττ 5, PP 6 = BBBB aa 7 +3BB 3BB 4 3BB 6 (+ττ 7 (3.3 6ζζ 4 8ωω ωω ζζ 6 8ωω 4ωω ζζ 8ωω 36ωω ζζ KK 4 = 6ζζ 8ωω, KK +44ωω 5 = ζζ ζζ ζζ 8ωω,KK +576ωω 6 = 4ζζ ζζ 4ζζ 8ωω ( ωω ζζ PP 7 = μμaa5 BB +5BB 4 (+ττ 5, PP 6 8 = 7ρρ aa 7 BB 5BB 4 +5BB 6 (+ττ 7 4ζζ 4ωω +6ωω ζζ KK 8 = 4ζζ 4ωω, PP +36 ωω 9 = μμμμ aa 5 5 BB +5BB4 ( +ττ ζζ 6 ζζ 4ωω 4ωω ζζ KK 9 = ζζ 4ωω, KK +6ωω = 4ζζ ζζ 4ζζ 4ωω, PP +36ωω ζζ 4ζζ 48ωω +84ωω ζζ KK = 4ζζ 48ωω, PP +756 ωω ζζ Fro m (3., we obtain 5 4ωω 6ωω ζζ = 7BBBB aa 7 5BB +3BB4 BB 6 7 ( +ττ 7, KK 7 = ζζ 4ωω +4ωω ζζ ζζ 4ωω +6ωω ζζ (3.5, PP = 7BBBB aa 7 5 5BB 9BB 4 +BB 6 (+ττ 7 = 7ρρ aa 7 7 3BB +5BB 4 BB 6 (+ττ 7,KK = (3.6 (3.7 4ζζ 48ωω 84ωω ζζ 4ζζ 48ωω +756ωω ζζ (3.8

5 International Journal of Mechanics and Applications 3, 3(: aa = gg[pp 7 KK 7 + PP 8 KK 8 + PP KK + PP KK + PP KK + PP 3 KK 3 ], wwheeeeeeττ = ττ = (3.9 bb = ζζ ωω [aa + gg(pp 7 KK 7 + PP 8 KK 8 + PP KK + PP KK + PP KK + PP 3 KK 3 ] Following the same procedure, one can obtain ωω, ζζ, uu (ττ, ττ and so on. We must note that for the application done above, we have chosenaa =. This choice is not mandatory, but aa must be chosen to be so close to zero or zero. The general rule for choosing aa was also given in[]. 4. Results aa ζζ + 5gg(PP ωω 9 KK 9 + PP KK + 7gggg KK + 3gg(PP 4 KK 4 + PP 5 KK 5 + PP 6 KK 6, wwheeeeeeττ = ττ = (3. approximation to its frequency and the first-order approximate solution to the un-damped cubic-quintic-heptic Duffing equation with a single periodic forcing which can be obtained from (. by taking MM =, XX mm =, NN = 3, aa = αα, aa = ββ, aa = μμ, aaaaaa aa 3 = ρρ and given by ωω dd uu + αααα + ddττ ββuu3 + μμuu 5 + ρρuu 7 = FF cos ττ (4. having applied the transformation in (.4 and subject to the corresponding initial conditions In this section we use the results given above to obtain the solutions to. Positively damped cubic-quintic Duffing equation with a single periodic forcing.. Positively damped and unforced cubic-quintic-heptic Duffing equation. 3. Un-damped cubic-quintic-heptic Duffing equation with a single periodic forcing. 4. Un-damped and unforced cubic-quintic-heptic Duffing equation. 5. Un-damped cubic-quintic Duffing equation with a single periodic forcing. 6. Un-damped and unforced cubic-quintic Duffing equation. After applying the transformation in (.4 and subject to the initial conditions prescribed in (3., the positively damped cubic-quintic Duffing equation with a single periodic forcing is obtained from (. by setting MM =, XX mm =, NN =, aa = αα, aa = ββ, aa = μμ giving ωω uu uu + ωωωω + ζζ uu ττ ττ ττ ττ +δδ ωω + ζζ + αααα + ββuu 3 + μμuu 5 = FF cos ττ ττ ττ (4. Setting ρρ = in (3.9, (3.7 and (3.6 one obtains the first-order approximate solution to (4. with the same initial guess given in (3.4. Consequently, other higher-order approximate solutions to (4. can as well be obtained provided ρρ =. After applying the transformation in (.4 and subject to the initial conditions prescribed in (3. the positively damped and unforced cubic-quintic-heptic Duffing equation is obtained by setting MM =, XX mm =, FF mm =, NN = 3, aa = αα, aa = ββ, aa = μμ, aaaaaa aa 3 = ρρin (. as ωω uu + ωωωω uu + ζζ uu ττ ττ ττ + δδ ωω + ζζ ττ ττ ττ +αααα + ββuu 3 + μμuu 5 + ρρuu 7 = (4. Setting FF = in (3.7 and (3.6, we obtain the first-order approximate solution to (4. as given in (3.9 with the same initial guess function given in (3.4. The other higher-order approximate solutions to (4. can also be obtained as long as FF remains zero. Setting δδ = ζζ = in (3.9, (3.7, (3.6 and (3.4 one obviously obtains the initial guess function,the first uu(ττ = aa, ωω dddd (ττ dddd =, wwheeeeee = (4.3 Similarly one can obtain the higher-order approximate solutions to (4.. From (. as well as (4., one can obtain the un-damped and unforced cubic-quintic-heptic Duffing equation subject to the initial conditions given by (4.3 and its initial guess function,first frequency approximation and the first-order approximate solution employing (3.9, (3.7, (3.6 and (3.4, provided δδ = ζζ = by taking MM =, XX mm =, FF mm =, NN = 3, aa = αα, aa = ββ, aa = μμ, aaaaaa aa 3 = ρρ. Higher-order approximate solutions can as well be obtained. Setting δδ = ζζ = in (3.9, (3.7, (3.6 and (3.4 one also obtains the initial guess function,the first frequency approximation and the first-order approximate solution to the un-damped cubic-quintic Duffing equation with a single periodic forcing which can be obtained from (. by taking MM =, XX mm =, NN =, aa = αα, aa = ββ, aaaaaa aa = μμ and given by ωω dd uu ddττ + αααα + ββuu3 + μμuu 5 = FF cos ττ (4.4 having applied the transformation in (.4 and subject to the corresponding initial conditions uu(ττ = aa, ωω dddd (ττ dddd =, wwheeeeee = (4.5 Similarly one can obtain the higher-order approximate solutions to (4.4. Finally the un-damped and unforced cubic-quintic Duffing equation can also be obtained from (. by setting MM =, XX mm =, FF mm =, NN =, aa = αα, aa = ββ, aa = μμ, aaaaaa δδ = and is given by ωω dd uu ddττ + αααα + ββuu3 + μμuu 5 = (4.6 where ζζ =, after applying the transformation in (.4 and subject to the initial conditions prescribed in (4.3. By enforcingζζ = δδ = FF = in (3.9,(3.7,(3.6 and (3.4, one as well obtains the first-order approximate solution and the first frequency approximation to (4.6 as given in (3.9. The other higher-order approximate solutions to (4. can also be obtained as long as FF remains zero. We note that the first frequency approximation and the first-order approximate solution to (4.6 obtained by setting some parameters above equal to zero is exactly the same results obtained in[]. Below are presented some of the simulations we did, in Figure One and Figure Two, we observe the behavior of the damped and forced cubic-quintic-heptic Duffing

6 4 M. O. Oyesanya et al.: Duffing Oscillator with Heptic Nonlinearity under Single Periodic Forcing oscillator.figurethree and Figure Four depict the behavior of the damped and forced cubic-quintic Duffing oscillator. We also did a lot of simulations using different values for our parameters to observe the behavior of the auxiliary non-zero parameter which controls the convergence of our approximations.interestingly neitherthe strength of the dampingnor that of the forcing exert any influenceon gg for all the different cases we treated. It was rather the initial guesses of the time constantsωω aaaaaa ζζ, and of course as was Ta ble. Behaviorof the auxiliary parameter g αα ββ μμ ρρ δδ aa ζζ FF ωω gg noted in[] the strength of the nonlinearity that determines and influences gg. Furthermore, just as was noted in[], problems with stronger nonlinearity demands that we choose gg such that its absolute value lies very close to zero but must not be zero. The table below depictssome of these interesting behaviorsof gg. (Good approx. < ± (not good 4 4 (Good approx. < ± (not good (Good approx. < ± (not good (Good approx. < ± 8 (not good 4 4 (Good approx. < ± (not good (Good approx. < ± (not good (Good approx. < ± (not good (Good approx. < ± (not good (Good approx. < ± (not good 5 3 (Good approx. < ± 9 (not good (Good approx. < ± (not good 6

7 International Journal of Mechanics and Applications 3, 3(: Fi gure. αα = ββ = μμ = ρρ = aa = δδ = ωω = ζζ =. Behavior of the displacement of the damped and forced cubic-quintic-heptic Duffing oscillator as time increases for ω = ζ = Fi gure. αα = ββ = μμ = ρρ = aa = δδ = ωω =, ζζ =.Behavior of the displacement of the damped and forced cubic-quintic-heptic Duffing oscillator as time increases for ω = aaaaaa ζ = /

8 4 M. O. Oyesanya et al.: Duffing Oscillator with Heptic Nonlinearity under Single Periodic Forcing Figure 3. αα = ββ = μμ = aa = δδ = ωω = ζζ =.Behavior of the displacement of the damped and forced cubic-quintic Duffing oscillator as time increases for ω = ζ = Fi gure 4. αα = ββ = μμ = aa = δδ = ωω =, ζζ =.Behavior of the displacement of the damped and forced cubic-quintic Duffing oscillator as time increases for ω = aaaaaa ζ = / 5. Conclusions In this paper, the homotopy analysis method (HAM was used to obtain analytic and uniformly-valid approximate solutions to the damped and driven, free oscillating, as well as the un-damped and un-driven Duffing oscillator equations with different nonlinearities.it was observed as noted in[6] that the degree of damping is elicited by the degree of nonlinearity as can be seen in Figures -4. Hence one can use increased nonlinearity to reduce the effect of external forcing as well as negative damping. We also observed that apart from the strength of the nonlinearity of a given problem, the initial guesses ωω aaaaaa ζζ, chosen for any given problem aids in determining a suitable auxiliary non-zero parameter such that, givengg cc < gg gg dd, where gg cc > and gg dd. It is also observed that every chosen g satisfying

9 International Journal of Mechanics and Applications 3, 3(: gg dd > gg gg cc gives a good approximation while every chosen g satisfying gg dd gg > gg cc does not give a good approximation. There is need to consider and investigate the degree of nonlinearity that can reduce the effect of external forcing for Duffing oscillator with multiple forcing functions and investigate the influence and the range of validity of the auxiliary parameter. Stability analysis also should engage attention in subsequent work. Highlights. Neither the strength of the damping nor that of the forcing exerts any influence on the auxiliary parameter of the nonlinearity.. Factors that affect the nonzero auxiliary parameter (which controls the convergence of the approximate solutions obtained in HAM are shown and discussed. 3. The degree of damping is elicited by the degree of nonlinearity and the initial guesses of the time constants REFERENCES [] Pirbodaghi, T., Hoseni, S. H., Ahmadian, M. T., Farrahi, G. H. (9 Duffing equations with cubic and quintic nonlinearities. J. of Computers and Mathematics with Applications 57, [] Nayfeh, A. H., Mook, D. T. (979 Nonlinear oscillations, John Wiley &Sons Inc., New York. [3] Sedighi, H. M., Shirazi, K. H., Zare, J. ( An analytic solution of transversal oscillation of quintic nonlinear beam with homotopy analysis method. Int. J. of Nonlinear Mechanics 47, [4] Guckenheimer, J. and Philip Holmes (983Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Number 4 in Applied Mathematical Sciences, Springer-Verlag, New York, NY. [5] Oyesanya, M. O. (8 Duffing Oscillator as a model for predicting earthquake occurrence, J. of NAMP, [6] Oyesanya, M. O., Nwamba, J. I. ( Stability Analysis of damped cubic-quintic Duffing Oscillator, World Journal of Mechanics (In Press. [7] Rand, R. H. (3 Lecture notes on nonlinear vibrations, a free online book available at randdocs/nlvibe45.pdf. [8] Mickens, R. E. (996 Oscillations in Planar Dynamic Systems, World Scientific, Singapore. [9] Wu, B. S., Sun, W. P., Lim, W. C. (6 An analytic approximate technique for a class of strongly nonlinear oscillators, Int. Journal of Nonlinear Mechanics, 4, [] Lin, J. (999 A new approach to Duffing equation with strong and high nonlinearity, Communications in Nonlinear Science and Numerical Simulation, 4, [] Hamidan, M. N., Shabaneh, N. H. (997 On the large amplitude free vibration of a restrained uniform beam carrying an intermediate lumped mass, Journal of Sound and Vibration, 99, [] Lai, S. K., Lim, C. W., Wu, B. S., Wang, S., Zeng, Q. C., He, X. F. (9 Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators, Journal of Applied Mathematical Modelling, 33, [3] Correig, A. M. and Urquizu ( Some dynamical aspects of microseismtime series, Geophys. J. Int. 49, [4] Belandez, A., Bernabeu, G., Frances, J., Mendez, D. I., Marini, S. ( An accurate approximate solution for the quintic Duffing oscillator equation. J. of Mathematical and Computer Modelling 5, [5] Chua, V. (3 Cubic-Quintic Duffing Oscillators, [6] Kargar, A., Akbarzade, M. ( An analytic solution of nonlinear cubic-quintic Duffing equation using Global Error Minimization Method, J. Adv. Studies Theor. Phys., 6 (, [7] Farzaneh, Y., Tootoonchi, A. A. ( Global error minimization method for solving strongly nonlinear oscillator differential equations. J. of Computers and Mathematics with Applications 59, 8, [8] Liao, S. J. (99 The proposed homotopy analysis techniques for the solutions of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University. [9] Liao, S. J. (999 (a A simple way to enlarge the convergence region of perturbation approximations, Int. J. Nonlinear Dynamics, 9 (, 9-. [] Liao, S. J. (3 An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude, Int. J. Nonlinear Mech. 38, [] Lesage, J. C., Liu, M. C. (8 On the investigation of a restrained cargo system modeled as a Duffing oscillator of various orders, Proceedings of ECTC (Early Career Technical Conference ASME, Maimi, Florida, USA. [] Ganji, S. S., Ganji, D. D., Babazadeh, H., Karimpour, S. (8 Variational approach Method for nonlinear oscillations of the motion of a Rigid Rod rocking back and cubic-quintic Duffing oscillators, Progress inelectromagnetic Research M, 4, 3-3.