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Adv. Studies Theor. Phys., Vol. 5, 2011, no. 8, 349 - 356

Solution of Nonlinear Oscillators Using Global Error

Minimization Method

Mehdi Akbarzade1

Department of Mechanical Engineering, Quchan Branch

Islamic Azad University- Quchan- Iran

Jafar Langari

Department of Mechanical Engineering, Quchan BranchIslamic Azad University- Quchan- Iran

Abstract

A modified variational approach called Global Error Minimization (GEM) method isdeveloped for obtaining an approximate closed-form analytical solution for nonlinear

oscillator differential equations. The proposed method converts the nonlineardifferential equation to an equivalent minimization problem. A trial solution is

selected with unknown parameters. Next, the GEM method is used to solve the

minimization problem and to obtain the unknown parameters. This will yield theapproximate analytical solution of the nonlinear ordinary differential equations

(ODEs). This approach is simple, accurate and straightforward to use in identifying

the solution.

Keywords: Nonlinear Oscillators; Analytical approximate solutions; Global Error

Minimization method.

1- Introduction

Most phenomena in our world are essentially nonlinear and are described by

nonlinear ordinary differential equations. Solving nonlinear ODEs is thus of greatimportance for gaining insight into real-world or engineering problems. However,

1Corresponding author: mehdiakbarzade@yahoo.com

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Solution of nonlinear oscillators 351

( )01

( ) cos( ) sin( )n n

n

u t a a n t b n t

=

= + + (4)Where 0a , na , nb are constants. These unknown constants could not be determined for

the case of infinite Fourier series. However, we can approximate Eq. (4) by a finite

series [10]:

( )01

( ) cos( ) sin( )m

n n

n

u t a a n t b n t

=

= + + (5)

Various methods have been developed for determining the unknown constants used

in Eq. (5) [8, 10]. In this paper, a natural and efficient method will be developed for

determining these unknowns. The nonlinear problem (1) is first converted to theminimization problem (3). We directly substitute the trial solution (5) in the

minimization problem. The solutions of the minimization problem are the unknownconstants of Eq. (5).

3- Application

3-1 Example 1

First we consider the motion of a ball bearing oscillation in a glass tube that is bent

into a curve such that the restoring force depends upon the cube of the displacement

u. the governing equation, ignoring frictional losses, is [7]:3 0 , (0) , (0) 0u u u A u + = = = (6)

We begin the procedure with the simplest trial solution:

1 1 1( ) cos( ), (0) , (0) 0u t A t u A u = = = (7)

Next, we convert Eq. (9) to the minimization problem (3):

( )2

3

1 10

2( , , ) ,

T

E u u u u u dt T

= + = (8)

By replacing 1( ) cos( )u t A t = in Eq. (8) and performing the integration we get:

( )2 4 2 2 4 2

24 36 151

24

A A AE

+ =

(9)

The solution could be found through the condition 0 :E

=

19 3 39

6A = +

(10)

Its period can be written in the form:1 17.1587T A

= (11)

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352 M. Akbarzade and J. Langari

The exact period [7] is -1 17.4163T A = . Therefore, it can be easily proved that the

maximal relative error is less than 3.61%.

If there is no small parameter in the equation, the traditional perturbation methodscannot be applied directly.

Fig. 1. Comparison of the GEM Method with the exact solution( 1, =1)A = .

3.2- Example 2

We consider the quadratic nonlinear oscillator [7]:5 0u u u + + = (12)

With initial condition of:

(0) , (0) 0u A u = = (13)

We convert Eq. (9) to the minimization problem (3):

( )2

5

1 1 10

2( , , ) ,

T

E u u u u u u dt T

= + + = (14)

By replacing 1( ) cos( )u t A t = in Eq. (14) and performing the integration we get:

( )( )2 2 2 8 4 4 21 1 2400 945 1920 2400 1920 38401920

E A A A A

= + + + + (15)

The solution could be found through the condition 0 :E

=

4 4 81 48 30 3 1024 1280 47812

A A A= + + + + (16)

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Solution of nonlinear oscillators 353

To show the remarkable accuracy of the obtained result, the approximate period with

the Hes Energy Balance Method [3] and HPM [7] are compared in table 1.

Table 1 Comparison of the GEM Method with the Energy Balance Method, HPM

and Numerical.

A Energy

Balance

frequency

[11]

Homotopy

Perturbation

frequency [24]

GEM

frequency

Numerical

frequency

0.1 1.00003 1.00003 1.00003 1.00003

0.2 1.00047 1.00050 1.00050 1.00050

0.3 1.00236 1.00253 1.00254 1.00253

0.5 1.01807 1.01934 1.01941 1.01940

1.0 1.25831 1.27475 1.28082 1.28079

5.0 19.1202 19.7895 20.3919 19.9719

10.0 76.3828 79.0633 81.4807 80.0801

3.3- Example 3

Consider a relative harmonic oscillator [7]:3

2 2(1 ) 0x x x + + = (17)

We make a transformation:

21

ux

u

=+

(18)

It follows:

2 3/ 2(1 )

ux

u

=

+

(19)

Substituting the expression of Eq. (18) for x into Eq. (17):3/ 2

22 3/ 2

2 2 3/ 2(1 ) 1

1 (1 )u xx x x x

u u = = =

+ + (20)

Comparing Eq. (19) and Eq. (20), we have:u x= (21)So Eq. (17) becomes:

20

1

uu

u

+ =+

(22)

We rewrite:

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354 M. Akbarzade and J. Langari

2 2 2(1 ) 0u u u + = (23)

With initial condition of:(0) , (0) 0u A u = = (24)

We convert Eq. (23) to the minimization problem (3):

( )2

2 2 2

1 1 10

2( , , ) (1 ) ,

T

E u u u u u u dt T

= + = (25)

By replacing 1( ) cos( )u t A t = in Eq. (25) and performing the integration we get:

( )4 2 8 4 8 8 4 4 21 1

144 240 105 144 288 240192

E A A A A

= + + +

(26)

The solution could be found through the condition 0 :E

=

( )( )( )

2 4

11

3 42 4 2 2 44

1 1

7 80 35 48

1372 30 36 1635 3840 2304 80 35 48

A A

A A A A A

=+ +

+ + + + + +

(27)

Table 2 Comparison of the GEM Method with the Hes frequency-amplitudeformulation and HPM and Numerical.

A Hes frequency-

amplitude

formulation

Homotopy

Perturbation

Method [24]

GEM

frequency

Numerical

frequency

0.1 0.9980 0.9980 0.9979 0.9979

0.2 0.9922 0.9922 0.9918 0.9917

0.3 0.9828 0.9828 0.9820 0.9818

0.5 0.9554 0.9554 0.9535 0.9552

1.0 0.8633 0.8633 0.8575 0.85015.0 0.4671 0.4671 0.4581 0.4510

10.0 0.3333 0.3333 0.3266 0.3218

4- Conclusions

The GEM method was successfully applied to some strong nonlinear equations. Thesolution procedure of GEM method is of deceptive simplicity and the insightful

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Solution of nonlinear oscillators 355

solutions obtained are of high accuracy even for the one-order approximation. The

method is useful to obtain analytical solution for all oscillators and vibrationproblems, such as in the fields of civil structures, fluid mechanics, electromagnetics

and waves, etc. Oscillator problems are very frequently encountered in all of the

mentioned major fields of science and engineering. The GEM method provides aneasy and direct procedure for determining approximations to the periodic solutions.

The GEM method is a well-established method for analysing nonlinear systems

which can be easily extended to any nonlinear equation. The accuracy and efficiencyof the method was demonstrated by presenting some examples.

References

[1] A. Fereidoon, Y. Rostamiyan, M. Akbarzade, Davood Domiri Ganji, Applicationof Hes homotopy perturbation method to nonlinear shock damper dynamics,

Archive of Applied Mechanics, 80 (6), (2010) 641-649.

[2] Demirbag SA, Kaya MO, Zengin FO, Application of Modified He's Variational

Method to Nonlinear Oscillators with Discontinuities, Int. J. Nonlin. Sci. Num.,

10(1) (2009) 27-31.

[3] D. D. Ganji, N. Ranjbar Malidarreh, M. Akbarzade, Comparison of EnergyBalance Period for Arising Nonlinear Oscillator Equations (Hes energy balanceperiod for nonlinear oscillators with and without discontinuities), Acta Applicandae

Mathematicae, 108 (2009), 353-362.

[4] J.H He, Variational approach for nonlinear oscillators, Chaos Solitons andFractals, 34 (2007) 1430-1439.

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863 - 870

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OSCILLATORS WITHn

u FORCE BY HES ENERGY BALANCE METHOD ,Progress In Electromagnetics Research C, Vol. 3, 5766 (2008).

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356 M. Akbarzade and J. Langari

[8] J.H. He, Some asymptotic methods for strongly nonlinear equations,International Journal of Modern Physics B, 20 (10) (2006) 1141-1199.

[9] K.P. Badakhshan, A.V. Kamyad, A. Azemi, Using AVK method to solvenonlinear problems with uncertain parameters, Applied Mathematics and

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[10] Yadollah Farzaneh, Ali Akbarzadeh Tootoonchi, Global Error Minimization

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[11] Zengin FO, Kaya MO, Demirbag SA Modified Application of parameter-

expansion method to nonlinear oscillators with discontinuities, Int. J. Nonlin. Sci.Num, 9 (2008) 267-270.

Received: February 2011