Springerplus. 2016; 5: 456.
Published online 2016 April 14.
PMCID: PMC4831959

# An analytical coupled technique for solving nonlinear large-amplitude oscillation of a conservative system with inertia and static non-linearity

## Abstract

Based on a new trial function, an analytical coupled technique (a combination of homotopy perturbation method and variational method) is presented to obtain the approximate frequencies and the corresponding periodic solutions of the free vibration of a conservative oscillator having inertia and static non-linearities. In some of the previous articles, the first and second-order approximations have been determined by the same method of such nonlinear oscillator, but the trial functions have not been satisfied the initial conditions. It seemed to be a big shortcoming of those articles. The new trial function of this paper overcomes aforementioned limitation. The first-order approximation is mainly considered in this paper. The main advantage of this present paper is, the first-order approximation gives better result than other existing second-order harmonic balance methods. The present method is valid for large amplitudes of oscillation. The absolute relative error measures (first-order approximate frequency) in this paper is 0.00 % for large amplitude A = 1000, while the relative error gives two different second-order harmonic balance methods: 10.33 and 3.72 %. Thus the present method is suitable for solving the above-mentioned nonlinear oscillator.

Keywords: Homotopy perturbation method (HPM) and variational approach, Non-linear oscillation, Large amplitude, Cantilever beam

## Background

Nonlinear oscillation problems are important in physical sciences, mechanical structures and engineering structures. Nonlinear vibration of oscillation systems are modeled by nonlinear differential equations. It is almost difficult to get exact solution for such nonlinear differential equations. Several methods have been used to solve weakly (small parameters, so-called perturbation parameters) nonlinear differential equations. Among all, most widely used technique is perturbation method (Marion 1970; Krylov and Bogoliubov 1947; Bogoliubov and Mitropolskii 1961; Nayfeh 1973; Nayfeh and Mook 1979; Nayfeh 1981). The perturbation method is not applied when a small parameter is absent in a nonlinear problem. Nonlinear of planar, large-amplitude free vibrations of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span is one of the problems that does not contain small parameter. In general, such problem is not always possible to get exact solution because of their complexity and thus the analytical approximate techniques must be needed to solve such problem. Moreover, there have been many strongly nonlinear problems arising in both science and engineering. To eliminate the limitations of classical perturbation technique, many analytical techniques such as variational iterative method (He et al. 2010; Herisanu and Marinca 2010a, b), variational method (He 2007; Kaya et al. 2010; Khan et al. 2011), energy balance method (EBM) (He 2002, 2006), homotopy analysis method (Liao 2003) used to solve strongly nonlinear problems. Recently, Khan et al. (2013) generalized the standard homotopy analysis method to solve nonlinear oscillators with rational terms. Moreover, Khan and Mirzabeigy (2014) has been improved He’s energy balance method, especially the second-order approximation is considered here.

In this paper, an analytical coupled method [a combination of homotopy perturbation method (He 2004) and variational method (He 2007)], along with a new trial function, has been presented to obtain the approximate frequency and the corresponding periodic solution of the strongly nonlinear oscillation of a conservative oscillator having inertia and static non-linearities (Akbarzade and Khan 2012; Hamdan and Dado 1997; Wu et al. 2003; Herisanu and Marinca 2010a, b). The new trial function of the present paper has satisfied the initial conditions. The results obtained in this paper (first-order approximate frequencies) are much better result for large values of amplitude than other existing results (Hamdan and Dado 1997; Wu et al. 2003). The method is very easy and straightforward.

## Formulation and solution method

Consider the nonlinear oscillator (Hamdan and Dado 1997; Wu et al. 2003; Herisanu and Marinca 2010a, b)

$d2udt2+u+αu2d2udt2+αududt2+βu3=0,$
1

subject to the initial conditions

$u(0)=A,dudt(0)=0.$
2

By considering the nonlinear oscillator, Eq. (1), the following homotopy can be constructed:

uω2up[α u2uα uu′2β u3 + (1 - ω2)u] = 0,
3

where p ∈  [0, 1] and ω is an unknown angular frequency of the nonlinear oscillator which is further to be determined. When p = 0, Eq. (3) becomes the linearized equation, uω2u = 0. When p = 1, it turns out to be the original one.

Let us consider that the periodic solution to Eq. (1) may be written as a power series in p:

uu0pu1p2u2 +  ⋯ .
4

Substituting Eq. (4) into Eq. (3) and equating the coefficients of p0 and p1, we obtain

$u0″+ω2u0=0,u0(0)=A,u0′(0)=0,$
5

and

$u1″+ω2u1+αu02u0″+αu0u0′2+βu03+(1-ω2)u0=0,u1(0)=0,u1′(0)=0.$
6

The solution of Eq. (5) is u0Acosω t, where ω will be determined from the variational formulation for u1, which reads:

$J(u1)=∫0T-12u1′2u1″+12ω2u12+αu02u0″u1+αu0u0′2u1+βu03u1+(1-ω2)u0u1dt,T=2πω.$
7

In previous article (Akbarzade and Khan 2012), a trial function was chosen in the following form:

u1(t) = B(cosω t - ⅓cos5ω t)
8

The accuracy of the first-order approximate solution, Akbarzade and Khan (2012) was chosen the trial function in the following form:

$u1(t)=B1cosωt-13cos3ωt+B313cos3ωt-35cos5ωt+57cos7ωt.$
9

Here, we observe that the trial functions Eqs. (8)–(9) are not satisfied the initial conditions $u1(0)=0,u1′(0)=0$ when substitutes t = 0 in the Eqs. (8)–(9). It is the main shortcomings of the article Akbarzade and Khan (2012).

In this paper, the limitation of the article Akbarzade and Khan (2012) has been removed by choosing a simple new trial function in the following form:

$u1(t)=Bcosωt-13cos3ωt-23cos5ωt.$
10

The new trial function given in Eq. (10) is satisfied the initial conditions $u1(0)=0,u1′(0)=0.$ The trial function given in Eq. (10) makes the solution rapidly converges; furthermore, the determination of first-order approximation is very easy.

Substituting u1 into functional Eq. (7), we obtain the following result:

$J(A,B,ω)=Bπ9A+6A3β-(9A+52B+3αA3)ω29ω.$
11

Setting:

$∂J∂B=0,∂J∂ω=0.$
12

Solving Eq. (12), we obtain the first approximate frequency as a function of amplitude as

$ω=ω0=3+2A2β3+αA2,$
13

where ω0 is the first-order analytical approximate frequency.

Therefore, the first-order approximate solution of Eq. (1) becomes

u(t) = Acosω t
14

where ω is given in Eq. (13).

Thus, the determination of first-order approximation is very easy and straightforward. On the other hand, the determination of second-order approximation of the article (Herisanu and Marinca 2010a, b) is very laborious process; thus, seems to be complex.

## Results and discussion

An analytical coupled technique [combining of the homotopy perturbation method (He 2004) and variational method (He 2007)], along with a simple new trial function, has been presented to determine the approximate frequency and the corresponding periodic solution of the above-mentioned nonlinear oscillator given by Eq. (1). Recently, some authors (Akbarzade and Khan 2012; Hamdan and Dado 1997; Wu et al. 2003; Herisanu and Marinca 2010a, b) have determined the approximate frequencies and the periodic solutions of such nonlinear oscillator. They (Akbarzade and Khan 2012; Hamdan and Dado 1997; Wu et al. 2003; Herisanu and Marinca 2010a, b) were obtained second-order approximation because their first-order approximation did not provide better result. On the other hand, the solution procedures of the article (Herisanu and Marinca 2010a, b) are not easy and it is very laborious process also. In this situation, the first-order approximation of the present paper gives significantly better result than other existing second-order approximations (Hamdan and Dado 1997; Wu et al. 2003).

To verify the efficiency and accuracy of the present method, the approximate frequencies have been obtained for several amplitudes when αβ = 1 and αβ = 2 and have been compared those results with other existing harmonic balance methods (Hamdan and Dado 1997; Wu et al. 2003). All results are shown respectively in Tables 1 and and2.2. The absolute relative errors of the present paper (first-order frequencies) have been compared with the numerical frequency and give less than 0.00 % in the limit as A →  whereas the absolute relative errors of the first and second-order analytical approximations (obtained by Hamdan and Dado 1997) give less than 13.40 and 10.33 %, respectively. On the other hand, the absolute relative errors of the first and second-order analytical approximations (obtained by Wu et al. 2003) give less than 13.40 and 3.72 %, respectively. Thus, the convergent rate of the present method is very faster than Hamdan and Dado (1997); Wu et al. (2003).

Comparison between the numerical frequency ω, the approximate frequency obtained by present method (given in Eq. 13) and other existing frequencies (Hamdan and Dado 1997; Wu et al. 2003) for αβ = 1 as well as several large amplitudes
Comparison between the numerical frequency ω, the approximate frequency obtained by present method (given in Eq. 13) and other existing frequency (Hamdan and Dado 1997; Wu et al. 2003) for αβ = 2 as well as several large amplitudes

Next, the approximate solution of Eq. (1) has been determined by using present method and harmonic balance method (Wu et al. 2003) for αβ = 1,  A = 10 and shown in Fig. 1. Finally, the approximate solution of Eq. (1) has been determined by using present method and harmonic balance method (Wu et al. 2003) for αβ = 2,  A = 10 and shown in Fig. 2. All figures include numerical solution obtained by fourth order Runge–Kutta method.

Comparison of the analytical approximate periodic solution obtained by present method (denoting by circles line) with numerical solution obtained by fourth order Runge–Kutta method (denoted by solid line) and also with the first-order (denoted ...
Comparison of the analytical approximate periodic solution obtained by present method (denoting by circles line) with numerical solution obtained by fourth order Runge–Kutta method (denoted by solid line) and also with the first-order (denoted ...

From all the figures, we see that the first-order approximate solution obtained by harmonic balance method deviates from numerical solution. Moreover, the second-order approximate solution obtained by harmonic balance method does not better agreement with the corresponding numerical solution. On the other hand, the first-order approximate solution obtained by present method gives excellent agreement with the corresponding numerical solution. Therefore, the present method is suitable for solving Eq. (1) than Akbarzade and Khan (2012); Hamdan and Dado (1997); Wu et al. (2003); Herisanu and Marinca (2010a, b) for strong nonlinearity as well as large amplitudes of oscillation.

## Conclusion

In this paper, a simple analytical technique has been presented to solve of nonlinear oscillations of planar, flexural large amplitudes free vibration of a slender, inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. Generally, the first-order approximation is considered in this paper. The first-order approximation gives rapidly converges to the corresponding numerical solution. The present method gives better result than other existing results for large amplitudes of oscillation. It has been proved that the present method is very effective and convenient and provides more accurate result for solving strongly nonlinear oscillators.

## Authors’ contributions

MAR and MSA prepared the manuscript. Both authors read and approved the final manuscript.

### Acknowledgements

The authors are grateful to the reviewers for their helpful comments/suggestions in improving the manuscript. The authors are also grateful to Mr. Harun Or Rashid, Lecturer (English), for his help in language.

### Competing interests

The author declares that they have no competing interests.

## Contributor Information

Md. Abdur Razzak, db.ca.teur@m_ar.

Md. Shamsul Alam, ten.dbarbil@daehhtam.

## References

• Akbarzade M. Coupled method of homotopy perturbation method and variational approach for solution to nonlinear cubic-quintic duffing oscillator. Adv Theor Appl Mech. 2010;3(7):329–337.
• Akbarzade M, Khan Y. Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions. Math Comput Model. 2012;55:480–489. doi: 10.1016/j.mcm.2011.07.043.
• Akindeinde SO. Homotopy perturbation method for the strongly nonlinear Darcy-Forscheimer model. Math Theor Model. 2015;5(9):78–84.
• Aminikhaha H, Hemmatnezhad M. An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations. Appl Math Lett. 2011;24:1502–1508. doi: 10.1016/j.aml.2011.03.032.
• Bogoliubov NN, Mitropolskii YA. Asymptotic methods in the theory of nonlinear oscillations. New York: Gordan and Breach; 1961.
• Ganji DD, Sadighi A. Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations. Int J Non Sci Num Simul. 2006;7:411–418.
• Ghorbani A, Nadjafi JS. He’s homotopy perturbation method for calculating adomian polynomials. Int J Non Sci Num Simul. 2007;8:229–232.
• Hamdan MN, Dado MHF. Large amplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. J Sound and Vib. 1997;206:151–168. doi: 10.1006/jsvi.1997.1081.
• He JH. Homotopy perturbation technique. Comput Methods Appl Mech Eng. 1999;3:257–262. doi: 10.1016/S0045-7825(99)00018-3.
• He JH. New perturbation technique which is also valid for large parameters. J Sound Vib. 2000;229:1257–1263. doi: 10.1006/jsvi.1999.2509.
• He JH. Preliminary report on the energy balance for nonlinear oscillations. Mech Res Commun. 2002;29:107–111. doi: 10.1016/S0093-6413(02)00237-9.
• He JH. The homotopy perturbation method for nonlinear oscillators with discontinuous. Appl Math Comput. 2004;151:287–292. doi: 10.1016/S0096-3003(03)00341-2.
• He JH. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B. 2006;20:1141–1199. doi: 10.1142/S0217979206033796.
• He JH. Variational approach for nonlinear oscillators. Chaos Solitions Fractals. 2007;34(5):1430–1439. doi: 10.1016/j.chaos.2006.10.026.
• He JH, Wu GC, Austin F. The variational iterative method which should be followed. Nonlinear Sci Lett A. 2010;1(1):1–30.
• Herisanu N, Marinca V. A modified variational iterative method for strongly nonlinear oscillators. Nonlinear Sci Lett A. 2010;1(2):183–192.
• Herisanu N, Marinca V. Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica. 2010;45:847–855. doi: 10.1007/s11012-010-9293-0.
• Herisanu N, Marinca V. Optimal homotopy perturbation method for non-conservative dynamical system of a rotating electrical machine. Zeitschrift fur Naturforschung A. 2012;67:509–516.
• Kaya MO, Durmaz S, Demirbag SA. He’s variational approach to multiple coupled nonlinear oscillators. Int J Non Sci Num Simul. 2010;11(10):859–865.
• Khan Y, Mirzabeigy A. Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator. Neural Comput Appl. 2014;25:889–895. doi: 10.1007/s00521-014-1576-2.
• Khan Naj A, Ara A, Khan Nad A. On solutions of the nonlinear oscillators by modified homotopy perturbation method. Math Sci Lett. 2014;3(3):229–236. doi: 10.12785/msl/030315.
• Khan Y, Faraz N, Yildirim A. New soliton solutions of the generalized Zakharov equations using He’s variational approach. Appl Math Lett. 2011;24:965–968. doi: 10.1016/j.aml.2011.01.006.
• Khan Y, Akbarzadeb M, Kargar A. Coupling of homotopy and the variational approach for a conservative oscillator with strong odd-nonlinearity. Sci Iran A. 2012;19(3):417–422. doi: 10.1016/j.scient.2012.04.004.
• Khan Y, Fardib M, Boroujenib FH. Nonlinear oscillators with rational terms: a new semi-analytical technique. Sci Iran A. 2013;20(4):1153–1160.
• Krylov NN, Bogoliubov NN. Introduction to nonlinear mechanics. New Jersey: Princeton University Press; 1947.
• Liao SJ. Beyond perturbation: introduction to homotopy analysis method. Boca Raton: CRC Press; 2003.
• Marinca V, Herisanu N. Optimal homotopy perturbation method for strongly nonlinear differential equations. Non Sci Lett A. 2010;1:273–280.
• Marinca V, Herisanu N. Nonlinear dynamic analysis of an electrical machine rotor-bearing system by the optimal homotopy perturbation method. Comput Math Appl. 2011;61:2019–2024. doi: 10.1016/j.camwa.2010.08.056.
• Marion JB. Classical dynamics of particles and system. San Diego: Harcourt Brace Jovanovich; 1970.
• Nayfeh AH. Perturbation methods. New York: Wiley; 1973.
• Nayfeh AH. Introduction to perturbation techniques. New York: Wiley; 1981.
• Nayfeh AH, Mook DT. Nonlinear oscillations. New York: Wiley; 1979.
• Rafei M, Ganji DD, Daniali H. Solution of the epidemic model by homotopy perturbation method. Appl Math Comput. 2007;187:1056–1062. doi: 10.1016/j.amc.2006.09.019.
• Suleman M, Wu Q. Comparative solution of nonlinear quintic cubic oscillator using modified homotopy perturbation method. Adv Math Phys. 2015
• Wang F, Li W, Zhang H. A new extended homotopy perturbation method for nonlinear differential equations. Math Comput Model. 2012;55:1471–1477. doi: 10.1016/j.mcm.2011.10.029.
• Wu BS, Lim CW, Ma YF. Analytical approximation to large-amplitude oscillation of a non-linear conservative system. Int J Nonlinear Mech. 2003;38:1037–1043. doi: 10.1016/S0020-7462(02)00050-1.

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