Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 50, 1, pp. 215-229, Warsaw 2012 50th anniversary of JTAM MOTION OF A RIGID ROD ROCKING BACK AND FORTH AND CUBIC-QUINTIC DUFFING OSCILLATORS Seyed S. Ganji Young Researchers Club, Science and Research Branch, Islamic Azad University, Tehran, Iran e-mail: r.alizadehganji@gmail.com Amin Barari Aalborg University, Department of Civil Engineering, Aalborg, Denmark S. Karimpour Semnan University, Department of Civil and Structural Engineering, Semnan, Iran G. Domairry Babol University of Technology, Department of Mechanical Engineering, Babol, Iran In this work, we implemented the first-order approximation of the Ite- ration PerturbationMethod (IPM) for approximating the behavior of a rigid rod rocking back and forth on a circular surface without slipping aswell asCubic-QuinticDuffingOscillators.Comparing the results with the exact solution, has led us to significant consequences.The results re- veal that the IPM is very effective, simple and convenient to systems of nonlinear equations. It is predicted that IPMcan be utilized as a widely applicable approach in engineering. Key words: nonlinear oscillation, iteration perturbation method (IPM), rocking rigid rod, cubic-quintic Duffing oscillator 1. Introduction With the rapid development of nonlinear science, it appears an ever-increasing interest of scientists and engineers in the analytical asymptotic techniques for nonlinear problems. Though it is easy for us now to find solutions to linear systems by means of numerical simulations, it is still very difficult to solve nonlinear problems analytically. Duffing oscillators comprise one of the cano- nical examples of Hamilton systems. However, simple generalizations of such 216 S.S. Ganji et al. oscillators, such as cubic-quintic Duffinng oscillators, have not been studied extensively (Hamdan and Shabaneh, 1997; Lin, 1999; Wu et al., 2006). Be- lendez et al. (2011) presented a closed-form solution for the quintic Duffing equation using a cubication method. The restoring force is expanded in Che- byshev polynomials through their work and the governing nonlinear equation is approximated by a cubic Duffing equation in which the coefficients for the linear and cubic terms depend on the initial amplitude. The coupled New- ton method with harmonic balancing was also utilized by Lai et al. (2009) for approximating higher-order solutions for strongly nonlinear Dufing oscil- lators with the cubic-quintic nonlinear restoring force. In addition, Ganji et al. (2009a) applied a new approximatemethod, so-called Energy BalanceMe- thod, to analyze these types of nonlinear oscillators with different engineering parameters of α, β and γ. Principally, analytical methods to solve a nonlinear oscillator are limited to the perturbation approach (Nayfeh, 1981). However, as with other analyti- cal techniques, certain limitations restrict thewide application of perturbation methods, themost important of which is the dependence of thesemethods on the existence of a small parameter in the equation. Disappointingly, themajo- rity of nonlinear problems have no small parameter at all. Even in cases where a small parameter does exist, the determination of such a parameter does not seem to follow any strict rule, and is rather problem-specific. Furthermore, the approximate solutions solved by the perturbation methods are valid, in most cases, only for small values of the parameters. It is obvious that all the- se limitations come from the assumption of the small parameter. Therefore, new analytical techniques should be developed to overcome these analytical deficiencies (Barari et al., 2008; Sfahani et al., 2010). Bayat et al. (2010) employed the Energy Balance Method to obtain ana- lytical expressions for the non-linear fundamental frequency and deflection of Euler-Bernoulli beams. Their approximations were valid for a wide range of vibration amplitudes, unlike the solutions obtained by other analytical tech- niques, such as perturbationmethods. The periodic solution for nonlinear free vibration of conservative, coupled mass-spring systems with linear and nonli- near stiffnesses as well as two mass-spring systems and buckling of a column were investigated within theworks presented byBayat et al. (2011) andGanji et al. (2011). In the first work, the energy balance methodology was utilized for the approximations while, in the latter, after finding themaximal andmi- nimal solution thresholds of the nonlinear problem, an approximate solution to the nonlinear equation was easily achieved usingHe Chengtian’s interpola- tion. The other techniques recently proposed to eliminate the small parameter Motion of a rigid rod rocking back... 217 are listed as: homotopy perturbation (Barari et al., 2008; Belendez et al., 2007; He, 2005; Sfahani et al., 2010; Yıldırım and Özis, 2007; Miansari et al., 2010), differential transformation (Ganji et al., 2010; Omidvar et al., 2010), max-min (Ibsen et al., 2010; Ganji et al., 2011), parameterized perturbation (Barari et al., 2011), frequency-amplitude formulation (Fereidon et al., 2011; Ganji et al., 2009b), harmonic balance (Gottlieb, 2006; Lim et al., 2006), energy balance (Bayat et al., 2010, 2011; Ganji et al., 2009d; Momeni et al., 2011; Sfahani et al., 2011), variational iteration (Barari et al., 2008; Fouladi et al., 2010; Hosseinzadeh et al., 2010) and variational approach (He, 2006; Ganji et al., 2009c). In this letter, we present the periodic solution based on the iteration perturbation method (IPM) (He, 2001) for nonlinear oscillators. Themethod is applied to two cases, and the results are compared with those obtained by the exact solutions. In Sections 4 and 5, the cubic-quintic Duffing oscillator (Hamdan andShabaneh, 1997) andmotion of a rigid rod rockingback (Nayfeh andMook, 1979; Wu et al., 2003) are analyzed as well. Thementioned problems can be written in the following forms x′′+f(x)= 0 f(x)=αx+βx3+γx5 x(0)=A x′(0)= 0 (1.1) and ( 1 12 + 1 16 u2 ) u′′ 2 + 1 16 uu′ 2 + g 4l ucosu=0 u(0)=β du dt (0)= 0 (1.2) where g > 0 and l > 0 are known positive constants. 2. Basic idea of the iteration perturbation method In this paper, we consider the following differential equation u′′+f(u,u′,u′′, t)= 0 (2.1) We introduce the variable y = du/dt, and then Eq. (2.1) can be replaced by an equivalent system u′(t)= y(t) y′(t)=−f(u,y,y′, t) (2.2) 218 S.S. Ganji et al. Assume that its initial approximate guess can be expressed as u(t)=Acos(ωt) (2.3) where ω is the angular frequency of oscillation. Then we have u′(t)=−Aω sin(ωt)= y(t) u′′(t)=−Aω2cos(ωt)= y′(t) (2.4) Substituting Eqs. (2.3) and (2.4) into Eq. (2.2)2, we obtain y′(t)=−f(u,y,y′, t)=− ∞ ∑ n=0 α2n+1cos[(2n+1)ωt] (2.5) Substituting Eq. (2.5) into Eq. (2.2)2, yields y′(t)=−[α1cos(ωt)+α3cos(3ωt)+ . . .] (2.6) Integrating Eq. (2.6), gives y(t)=− α1 ω sin(ωt)− α3 3ω sin(3ωt)− . . . (2.7) Comparing Eqs. (2.4)1 and (2.7), we obtain (2.8)−Aω=−α1 ω ω= √ α1 A T =2π √ A α1 (2.8) 3. Illustration of the problems In this Section, IPMwhichwas presented in Section 2 is applied to two smooth oscillators with odd nonlinearities in the displacement, and the results are compared with the exact solution. Case 1. In this example, we consider the following nonlinear oscillator (Lim andWu, 2003; Ramos, 2009) u′′+ u3 1+u2 =0 u(0)=A u′(0)= 0 (3.1) From Eq. (3.1), we have u′′ =−u′′u2−u3 u(0)=A u′(0)= 0 (3.2) Motion of a rigid rod rocking back... 219 Equation (3.2) is equivalent to the two-dimensional system u′ = y y′ =−y′u2−u3 (3.3) Substituting u=Acos(ωt) into the right-hand side of Eqs. (3.3), gives u′ =−Aω sin(ωt)= y y′ =A3cos3(ωt)(ω2−1) (3.4) It is possible to perform the following Fourier series expansion A3cos3(ωt)(ω2−1)=α1cos(ωt)+α3cos(3ωt)+ . . . α1 = 4 π π/2 ∫ 0 A3cos4(θ)(ω2−1) dθ= 3A3(ω2−1) 4 α3 = 4 π π/2 ∫ 0 A3cos3θcos(3θ)(ω2−1) dθ= A 3(ω2−1) 4 (3.5) Substituting Eqs. (3.5) into Eq. (3.4)2, yields y′ = A3(ω2−1) 4 [3cos(ωt)+cos(3ωt)] (3.6) By integrating Eq. (3.6), we obtain y= A3(ω2−1) 4 ∫ [3cos(ωt)+cos(3ωt)] dt = A3(ω2−1) ω [3 4 sin(ωt)+ 1 12 sin(3ωt) ] (3.7) Comparing Eqs. (3.4)1 and (3.7), gives ω= 3A√ 9A2+12 T = 2π √ 9A2+12 3A (3.8) The exact frequency ωex of Eqs. (3.15) is (Lim andWu, 2003) ωex =π [ 2 π/2 ∫ 0 A2cos2θ √ A2cos2θ+ln ( 1− A2cos2 θ 1+A2 ) dθ ]−1 (3.9) In case 1, we assume A = 0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, and 100. The obtained exact results are expressed in Eq. (3.8). The results for the 220 S.S. Ganji et al. Table 1.Comparison between IPM and exact solution for Example 1 A ω ωex |(ωex−ω)/ωex| 0.01 0.00866 0.00847 2.242 0.05 0.04326 0.04232 2.22 0.1 0.08627 0.08439 2.22 0.5 0.39736 0.38737 2.58 1.0 0.65465 0.63678 2.81 5.0 0.97435 0.96698 0.763 10.0 0.99340 0.99092 0.250 50.0 0.99973 0.99961 0.012 100.0 0.99993 0.99990 0.003 approximate frequency ωwith the exact frequency ωex are also compared and tabulated in Table 1. From the illustrated results, the maximum error 2.22% can be obtained. Hence, it is concluded that there is an excellent agreement with the exact solutions for the nonlinear systems. Case 2. This example corresponds to u′′+ u 1+εu2 =0 u(0)=A u′(0)= 0 (3.10) From Eq. (3.10), we have u′′ =−u′′εu2−u u(0)=A u′(0)= 0 (3.11) Equation (3.11) is equivalent to the two-dimensional system u′ = y y′ =−y′εu2−u (3.12) Substituting u=Acos(ωt) into the right-hand side of Eqs. (3.12), gives u′ =−Aωsin(ωt)= y y′ =Acos(ωt)[A2εω2cos2(ωt)−1] (3.13) It is possible to carry out the following Fourier series expansion Acos(ωt)[A2εω2cos2(ωt)−1]=α1cos(ωt)+ . . . α1 = 4 π π/2 ∫ 0 Acos2θ[A2εω2cos2(θ)−1] dθ= A(3A 2ω2ε−4) 4 (3.14) Substituting Eqs. (3.14) into Eq. (3.13)2, yields y′ = A(3A2ω2ε−4) 4 cos(ωt)+ . . . (3.15) Motion of a rigid rod rocking back... 221 Integration of Eq. (3.15) leads to y= ∫ (A(3A2ω2ε−4) 4 cos(ωt)+. . . ) dt= A(3A2ω2ε−4) 4ω sin(ωt)+. . . (3.16) Comparing Eqs. (3.13)1 and (3.16), gives ω= 2√ 3εA2+4 T =π √ 3εA2+4 (3.17) Equation (3.17)1 gives the same frequency as the one resulting from the ap- plication of the harmonic balance method to Eq. (3.10). It is also exactly the same as that obtained by the artificial parameter Linstedt-Poincare method (Ramos, 2009). 4. Cubic-quintic Duffing equations Now, we consider the nonlinear cubic-quintic Duffing equations. From Eq. (1.1), we have x′′ =−αx−βx3−γx5 (4.1) Equation (4.1) is equivalent to the two-dimensional system x′ = y y′ =−αx−βx3−γx5 (4.2) Substituting u=Acos(ωt) into the right-hand side of Eqs. (4.2), gives x′ =−Aω sin(ωt)= y y′ =−Acos(ωt)[α+βA2cos2(ωt)+γA4cos4(ωt) (4.3) Expanding the above in the Fourier series, we have −Acos(ωt)[α+βA2cos2(ωt)+γA4cos4(ωt)] =α1cos(ωt)+ . . . α1 = 4 π π/2 ∫ 0 Acos2θ[α+βA2cos2θ+γA4cos4θ] dθ =4A (α 4 + 3βA2 16 + 5γA4 32 ) (4.4) 222 S.S. Ganji et al. Substituting Eqs. (4.4) into Eq. (4.3)2, yields y′ =4A (α 4 + 3βA2 16 + 5γA4 32 ) cos(ωt)+ . . . (4.5) Integrating Eq. (4.4)2, yields y= ∫ [ 4A (α 4 + 3βA2 16 + 5γA4 32 ) cos(ωt)+ . . . ] dt = 4A ω (α 4 + 3βA2 16 + 5γA4 32 ) sin(ωt)+ . . . (4.6) Comparing Eqs. (4.3)1 and (4.6), gives ω= √ 16α+12A2β+10γA4 4 T = 8π √ 16α+12A2β+10γA4 (4.7) The exact frequency ωex for the cubic-quintic Duffing oscillator is (Wu et al., 2003) ωe(A)=πk1 ( 2 π/2 ∫ 0 1 √ 1+k2 sin 2 t+k3 sin 4 t dt )−1 (4.8) where k1 = √ α+ βA2 2 + γA4 3 k2 = 3βA2+2γA4 6α+3βA2+2γA4 k3 = 2γA4 6α+3βA2+2γA4 The above result from Eq. (4.7)1 is in good agreement with the result obtained by the exact solution as given in Eq. (4.8). Comparisons between the IPM and exact solutions for the cubic-quintic Duffing system are illustrated in Fig.1 and Table 2. 5. Motion of a rocking rigid rod In this Section, we present an example of motion of a rigid rod rocking back and forth on a circular surface without slipping as presented in Eq. (1.2) u′′ =− 3 4 u2u′′− 3 4 uu′ 2− 3gucosu l (5.1) Motion of a rigid rod rocking back... 223 Fig. 1. Comparison between IPM and the exact solution for cubic-quintic Duffing oscillator (Eq. (1.1)); (a) α=1, β=10, γ=100,A=0.1, (b) α=1, β=1, γ=1, A=1.0 Table 2.Comparison between IPM and exact solution for cubic-quintic Duf- fing oscillator A α=β= γ=1 α=1, β=10, γ=100 ω ωex B ω ωex B 0.1 1.00377 1.00377 0.0 1.03983 1.03970 0.01250 0.5 1.10750 1.10654 0.06757 2.60408 2.52469 3.14468 1 1.54110 1.52359 1.14926 8.42615 8.01005 5.19472 5 20.2577 19.1815 5.61061 198.119 187.199 5.83318 10 79.5361 75.1774 5.79795 791.044 747.323 5.85038 50 1976.90 1867.57 5.85413 19764.71 18671.34 5.85587 100 7906.17 7468.83 5.85553 79057.42 74683.91 5.85602 500 197642.83 186709.04 5.85606 1976424.01 1867085.99 5.85608 1000 790569.89 746834.69 5.85608 7905694.62 7468342.49 5.85608 ωex – Ramos (2009); B= |(ωex−ω)/ωex| Equation (5.1) is equivalent to the two-dimensional system u′ = y y′ =− 3 4 u2y′− 3 4 uy2− 3gucosu l (5.2) Substituting u=Acos(ωt) into the right-hand side of Eqs. (5.2), gives x′ =−Aω sin(ωt)= y y′ =− 3 4l Acos(ωt)[−2A2ω2lcos2(ωt)+A2ω2l+4gcos(Acos(ωt))] (5.3) 224 S.S. Ganji et al. with the application of Fourier series expansion, we have − 3 4l Acos(ωt)[−2A2ω2lcos2(ωt)+A2ω2l+4gcos(Acos(ωt))] =α1cos(ωt)+ . . . α1 = 4 π π/2 ∫ 0 − 3 4l Acos2θ[−2A2ω2lcos2θ+A2ω2l+4gcos(Acos(θ))] dθ = 3 8l [A3ω2l−16gAJ(0,A)+16gJ(1,A)] (5.4) where J – Bessel function. Substituting Eqs. (5.4) into Eq. (5.3)2, yields y′ = 3 8l [A3ω2l−16gAJ(0,A)+16gJ(1,A)]cos(ωt)+ . . . (5.5) Integrating Eq. (5.5), yields y= ∫ ( 3 8l [A3ω2l−16gAJ(0,A)+16gJ(1,A)]cos(ωt)+ . . . ) dt = 3 8ωl [A3ω2l−16gAJ(0,A)+16glJ(1,A)]sin(ωt)+ . . . (5.6) Comparing Eqs. (5.2)2 and (5.5), gives ω= √ 48[lA(3A2 +8)g(AJ(0,A)−J(1,A))] lA(3A2+8) T = 2πlA(3A2+8) √ 48[lA(3A2 +8)g(AJ(0,A)−J(1,A))] (5.7) The exact period Tex for Eq. (1.2) is (Wu et al., 2003) Tex =4 √ l 3g π/2 ∫ 0 √ (4+3β2 sin2ϕ)β2 cos2ϕ 8[β sinβ+cosβ−β sinϕsin(β sinϕ)− cos(β sinϕ)] dϕ (5.8) For comparison, the approximate period computed by Eq. (5.7)2, and the exact period Tex obtained by Eq. (5.8) are given in Fig.2 and Table 3. Motion of a rigid rod rocking back... 225 Fig. 2. Comparison between IPM and the exact solution for motion of the rocking rigid rod (Eq. (1.2)); (a) g= l=1,A=0.10π, (b) g= l=1,A=0.20π, (c) g= l=1,A=0.30π Table 2.Comparison between IPM and exact solution for motion of the roc- king rigid rod, when g= l=1 β T Tex |(Tex−T)/Tex| 0.05π 3.66129 3.66109 0.0054 0.10π 3.76394 3.76397 0.0008 0.15π 3.94064 3.94086 0.0056 0.20π 4.20116 4.20292 0.04187 0.25π 4.56246 4.56948 0.15363 0.30π 5.05355 5.07728 0.46738 0.35π 5.72584 5.79770 1.23946 0.40π 6.67785 6.89564 3.1584 226 S.S. Ganji et al. 6. Conclusions In this paper, the IPMhas been implemented in order to analyze the equation of motion associated with a rocking rigid rod as well as cubic-quintic Duffing oscillators. We conclude from the obtained results that the IPM is an effi- cient method for finding periodic solutions for non-linear oscillatory systems. 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Yıldırım A., Özis T., 2007, Solutions of singular IVPs of Lane-Emden type by homotopy perturbationmethod,Physics Letters A, 369, 1/2, 70 Ruch pręta toczącego się wahadłowo po płaszczyźnie oraz oscylatora Duffinga piątego stopnia Streszczenie Wpracy omówionopierwszorzędowąaproksymację zachowania się sztywnegoprę- ta toczącego się bez poślizgu ruchemwahadłowympo kołowej powierzchni za pomocą iteracyjnejmetodyperturbacyjnej (IPM).Tę samąmetodę zastosowanotakże do ana- lizy dynamiki oscylatoraDuffinga piątego stopnia. Porównanie otrzymanychwyników z rozwiązaniem dokładnym doprowadziło do istotnych wniosków. Wykazano przede wszystkim wysoką efektywność metody IPM przy jej jednoczesnej prostocie i wygo- dzie w stosowaniu do nieliniowych równań ruchu. Autorzy podkreślają duże walory aplikacyjne metody IPMw praktyce inżynierskiej. Manuscript received February 28, 2011; accepted for print May 13, 2011