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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 518684, 7 pages
Research Article

High-Order Energy Balance Method to Nonlinear Oscillators

Faculty of Aeronautics and Astronautics, Istanbul Technical University, Maslak, 34469 Istanbul, Turkey

Received 1 April 2012; Revised 25 June 2012; Accepted 9 July 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 Seher Durmaz and Metin Orhan Kaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Energy balance method (EBM) is extended for high-order nonlinear oscillators. To illustrate the effectiveness of the method, a cubic-quintic Duffing oscillator was chosen. The maximum relative errors of the frequencies of the oscillator read 1.25% and 0.6% for the first- and second-order approximation, respectively. The third-order approximation has an accuracy as high as 0.008%. Excellent agreement of the approximated frequencies and periodic solutions with the exact ones is demonstrated for several values of parameters of the oscillator.

1. Introduction

A great deal of work has been devoted to the nonlinear problems encountered in the fields of applied mathematics, physics, and engineering sciences. In general, the analytical approximation to the solution of a given nonlinear problem is difficult, sometimes impossible; plenty of techniques based on numerical methods have been implemented. Among them are the variational iteration method [1, 2], the harmonic balance method [3, 4], and energy balance method [58] developed to solve nonlinear differential equations. In this study, we have investigated the application of high-order energy balance method to cubic-quintic Duffing oscillator. The nonlinear frequencies are calculated for the first-, second-, and third-order EBM and compared with the results of different techniques.

2. The Basic Idea of Energy Balance Method

This section briefly introduces energy balance method proposed by He [5]. In this method, a variational principle for the oscillation is established, then the corresponding Hamiltonian is considered from which the angular frequency can be easily obtained by Galerkin method.

Let us consider the motion of a general oscillator with the initial conditions in the form 𝑢+𝑓(𝑢)=0,𝑢(0)=𝐴,𝑢(0)=0,(2.1) where 𝐴 is the initial amplitude.

Its variational can be written as 𝐽(𝑢)=0𝑇/412𝑢2+𝐹(𝑢)𝑑𝑡.(2.2) Here 𝑇=2𝜋/𝜔 is the period of the nonlinear oscillation and 𝐹(𝑢)=𝑓(𝑢)𝑑𝑢.

The Hamiltonian of (2.1) can be written in the form: 1𝐻(𝑢)=2𝑢2+𝐹(𝑢).(2.3) In (2.2) the kinetic energy (𝐸) and potential energy (𝑃) can be, respectively, expressed as 𝐸=(1/2)𝑢2, 𝑃=𝐹(𝑢).

Throughout the oscillation since the system is conservative, the total energy remains unchanged during the motion; the Hamiltonian of the oscillator becomes a constant value, 𝐻=𝐸+𝑇=𝐹(𝐴).(2.4) For the first-order approximation, the following trial function can be assumed: 𝑢(𝑡)=𝐴cos𝜔𝑡.(2.5) Substituting (2.5) into (2.3) yields the following residual: 𝑅𝐴(𝑡)=22𝜔2sin2𝜔𝑡+𝐹(𝐴cos𝜔𝑡)𝐹(𝐴).(2.6) The residual is forced to zero, in an average sense, by setting weighted integrals of the residual to zero 0𝑇/4𝑅(𝑡)𝑤𝑛𝑑𝑡=0,𝑛=1,2,,(2.7) where 𝑤𝑛 is a set of weighting function (or test).

There are lots of weighting functions, that is, Galerkin, least squares, collocation and so forth. In this study, we used Galerkin method as a weighting function.

3. High-Order Energy Balance Method

In order to extend He’s energy balance method, let us assume that the solution of (2.1) can be expressed as 𝑢=𝐴1cos𝜔𝑡+𝐴2cos3𝜔𝑡++𝐴𝑛cos(2𝑛1)𝜔𝑡.(3.1) From the initial conditions, the coefficients should satisfy the following constrain:𝐴=𝐴1+𝐴2++𝐴𝑛.(3.2a) One of these parameters can be chosen as a dependent parameter. Hence, 𝐴𝑛=𝐴𝐴1𝐴2𝐴𝑛1.(3.2b)By inserting (3.1) into (2.7), the following systems can be obtained: 0𝑇/4𝑅(𝑡)cos𝜔𝑡𝑑𝑡=0,0𝑇/4𝑅(𝑡)cos3𝜔𝑡𝑑𝑡=0,0𝑇/4𝑅(𝑡)cos(2𝑛1)𝜔𝑡𝑑𝑡=0.(3.3)

4. Example

A cubic-quintic Duffing oscillator is considered. In the following sections, the nonlinear frequencies will be compared with the results of different techniques to illustrate the efficiency and accuracy of energy balance method.

The governing differential equation of this oscillator is in the form of𝑢+𝛼𝑢+𝜀𝑢3+𝜆𝑢5=0,where𝛼0(4.1a) with the initial conditions 𝑢(0)=𝐴,𝑢(0)=0.(4.1b)The Hamiltonian of (4.1a) is given as follows: 1𝐻(𝑢)=2𝑢2+12𝑢2+14𝜀𝑢4+16𝜆𝑢6=12𝛼𝐴2+14𝜀𝐴4+16𝜆𝐴6.(4.2) For the first-order approximation, assume that 𝑢(𝑡) is in the following form: 𝑢(𝑡)=𝐴cos𝜔𝑡.(4.3) Substituting the first approximation into (4.2) yields 𝑅11(𝑡)=2𝐴2𝜔2sin21(𝜔𝑡)+4𝜀𝐴4cos41(𝜔𝑡)+6𝜆𝐴6cos61(𝜔𝑡)+2𝛼𝐴2cos2(𝜔𝑡)𝜀𝐴44𝛼𝐴22𝜆𝐴64.(4.4) First-order approximation can be obtained by setting 0𝑇/4𝑅1(𝑡)cos(𝜔𝑡)𝑑𝑡=0,𝑇=2𝜋𝜔.(4.5) The amplitude-frequency relationship for the first approximation is obtained as 𝜔(𝐴)=70𝛼+49𝜀𝐴2+38𝜆𝐴4.70(4.6) To obtain a more accurate result, let us define 𝑢 as follows: 𝑢(𝑡)=𝐴1cos𝜔𝑡+𝐴𝐴1cos3𝜔𝑡(4.7) Substituting (4.7) into (4.2) results in the following residual: 𝑅2𝜔(𝑡)=22𝐴1sin(𝜔𝑡)+3𝐴𝐴1sin(3𝜔𝑡)2+𝛼2𝐴1cos(𝜔𝑡)+𝐴𝐴1cos(3𝜔𝑡)2+𝜀4𝐴1cos(𝜔𝑡)+𝐴𝐴1cos(3𝜔𝑡)4+𝜆6𝐴1cos(𝜔𝑡)+𝐴𝐴1cos(3𝜔𝑡)6𝛼𝐴22𝜀𝐴44𝜆𝐴66.(4.8) We set0𝑇/4𝑅2(𝑡)cos(𝜔𝑡)𝑑𝑡=6550410𝜆𝐴614096128𝜆𝐴5𝐴1+𝐴48898327𝜀+59159040𝜆𝐴21128𝐴3131461𝜀𝐴1+979520𝜆𝐴31+32𝐴21831402𝛼+723520𝐴21𝜀+946176𝜆𝐴41+3187041𝜔248𝐴𝐴1461890𝛼+1033600𝐴21𝜀+2150400𝜆𝐴41+4572711𝜔2+2𝐴27066917𝛼+22697856𝐴21𝜀+77271040𝜆𝐴41+63602253𝜔2=0,(4.9a)0𝑇/4𝑅2(𝑡)cos(3𝜔𝑡)𝑑𝑡=5265546𝜆𝐴6+83291904𝜆𝐴5𝐴1+𝐴46789783𝜀313950720𝜆𝐴21+2176𝐴345429𝜀𝐴1+301120𝜆𝐴31+6𝐴21616615𝛼37333632𝜀𝐴21128921600𝜆𝐴4114549535𝜔296𝐴211016158𝛼+1012928𝜀𝐴21+1396736𝜆𝐴41+1801371𝜔2+48𝐴𝐴12678962𝛼+5002624𝜀𝐴21+10309632𝜆𝐴41+4572711𝜔2=0.(4.9b)

By solving (4.9a)-(4.9b) simultaneously, one can obtain the second-order approximate amplitude-frequency relation. For different  𝜀𝐴2  values, the approximate frequencies are given in Table 1.

Table 1: Comparison of the first-, second-, and third-order frequencies for 𝛼=1 and 𝜆=0.

Moreover, the accuracy of results will be further improved by defining 𝑢 in the following form: 𝑢(𝑡)=𝐴1cos𝜔𝑡+𝐴2cos3𝜔𝑡+𝐴𝐴1𝐴2cos5𝜔𝑡.(4.10) Substituting (4.10) into (4.2), we get the following residual for the third-order approximation: 𝑅31(𝑡)=2𝜔2𝐴1sin(𝜔t)+3𝐴2sin(3𝜔𝑡)+5𝐴𝐴1𝐴2sin(5𝜔𝑡)2+12𝛼𝐴1cos(𝜔𝑡)+𝐴2cos(3𝜔𝑡)+𝐴𝐴1𝐴2cos(5𝜔𝑡)2+14𝜀𝐴1cos(𝜔𝑡)+𝐴2cos(3𝜔𝑡)+𝐴𝐴1𝐴2cos(5𝜔𝑡)4+16𝜆𝐴1cos(𝜔𝑡)+𝐴2cos(3𝜔𝑡)+𝐴𝐴1𝐴2cos(5𝜔𝑡)6𝛼𝐴22𝜀𝐴44𝜆𝐴66.(4.11) Inserting (4.11) into (3.3) for 𝑛=3 and using the same procedure explained above, we get three weighted integrals. Solving these three equations simultaneously, the amplitude-frequency relation for the third-order approximation is obtained. For higher-order approximations, the similar procedures can be applied, however, the accuracy of the third-order approximation is appropriate for several values of parameters 𝛼, 𝜀, and 𝜆.

In the following, the nonlinear frequencies for the cubic-quintic oscillator are calculated for two different cases: (i) 𝜆=0 and (ii) 𝜆0.

The first case considered here corresponds to the cubic Duffing oscillator. For 𝜆=0, the results of the nonlinear frequencies are obtained by the first-, second-, and third-order energy balance method and compared with the results of [4, 9]. In the second case, for 𝜆0, the nonlinear frequencies are given in Table 2. Additionally, the numerical solution for all cases is acquired by standard Runge-Kutta method (R-K).

Table 2: Comparison of the first-, second-, and third-order frequencies for 𝛼=1, 𝜀=5.

In Table 1, the relative errors for the first-order approximation read 1.25%, while this error reduces to 0.59% in the second approximation. We observe that the differences between the third-order and the exact frequencies are sufficiently small.

As seen in Table 2, the results of third order approximation are in very good agreement between the numerical results. The comparison of approximate and numerical solutions can also be found in Figure 1. It can be seen that the first-, and the second-order results have slight differences compared to the numerical solution. However, the third-order approximation is overlapping with the numerical solution.

Figure 1: Approximate and numerical solutions for (a)  𝐴=𝛼=1,  𝜀=100,  𝜆=0; (b)  𝐴=𝛼=1,  𝜀=𝜆=100.

5. Conclusion

In this paper, energy balance method is extended for high-order solutions. The first-order approximate frequency for Duffing oscillator gives 1.25% relative error, while the second-, and the third-order approximated frequencies reach 0.59% and 0.008% relative errors, respectively. Moreover, relative errors in high-order energy balance reduce to smaller values than global error minimization and harmonic balance methods. Consequently, we can state that extended method is very effective and convenient for the cubic-quintic Duffing oscillator.


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