Accepted Manuscript

Global residue harmonic balance method for Helmholtz-Duffing oscillator

Peijun Ju

PII: S0307-904X(14)00513-7DOI: http://dx.doi.org/10.1016/j.apm.2014.10.029Reference: APM 10182

To appear in: Appl. Math. Modelling

Received Date: 26 January 2013Revised Date: 2 April 2014Accepted Date: 13 October 2014

Please cite this article as: P. Ju, Global residue harmonic balance method for Helmholtz-Duffing oscillator, Appl.Math. Modelling (2014), doi: http://dx.doi.org/10.1016/j.apm.2014.10.029

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Global residue harmonic balance method for Helmholtz-Duffing

oscillator

Peijun Ju *

School of Control Science and Engineering, Shandong University, Jinan 250061, China

Abstract

In this paper, an improved harmonic balance method was proposed to solve the

Duffing-harmonic equation. This method is called the global residue harmonic

balance method (GRHBM). Unlike other harmonic balance methods, all the former

global residual errors are introduced in the present approximation to improve the

accuracy. A maximum absolute value of the relative errors for the second-order

approximation is less than 0.1165%. Comparison of the result obtained using this

approach with the exact one and existing results reveals that the high accurate,

simplicity and efficiency of the proposed procedure. The method can be easily

extended to other strongly nonlinear oscillators.

Key words: Global residue harmonic balance method, Duffing-harmonic equation,

Approximate solutions, Nonlinear oscillators

* Tel: +08613325277995 E-mail address: jpj615@163.com

1. Introduction

The study of nonlinear equations has drawn much attention in recent years.

Various powerful mathematical techniques were used to obtain approximate analytical

solutions, such as harmonic balance method [1-7], energy balance method [8-11],

Hamiltonian approach [12-13], the homotopy perturbation method [15-19] and other

new methods[20-24].

As a class of dynamical systems with strong nonlinearities and larger oscillation

amplitudes, Duffing-harmonic oscillator system has been recently the focus of a great

deal of research. Many techniques have been developed to determine analytical

approximations to the periodic solution of Duffing-harmonic oscillator system.

Mickens [1] and Hu and Tang [2] used the harmonic balance method to investigate the

analytical approximation. Radhakrishnan et al. solve Duffing-harmonic oscillator

system by numerical integration of its energy relation [3]. Lim et al. presented a

Newton’s harmonic balance method to solve the Duffing-harmonic oscillator system,

and their approximations give accurate results for a large range of the initial

amplitudes [4,5]. By using He’s energy balance method, Õzis and Yildirim obtained

an accurate result [8]. Beléndez et al. obtain small relative error between the

approximate frequency and the exact one, by introducing a cubication method and the

modified homotopy perturbation method [16-18]. D.D. Ganji et al. obtained the

approximate frequency by He’s energy balance method (HEBM) [10], while S.S.

Ganji et al. by max-min approach (MMA) [21]. Guo et al. presented an Iterative

homotopy harmonic balancing approach to solve the system, and their second-order

approximation give accurate results [6].

In this paper, based on the ideas of homotopy perturbation and the residue

harmonic balance method [7], we put forward a novel approximate method, namely

the global residue harmonic balance method, to determine the periodic solutions of

Duffing-harmonic oscillator systems. To obtain higher-order analytical

approximations, all the former residual errors are considered in the solving process of

present order approximation.

2．．．．Basic ideas of the global iterative harmonic balance approach

For the sake of simplicity, we have been considered systems governed by

equations having the form

( )u f u=&& , (0)u A= , (0) 0u =& , (1)

where du

udt

=& and f is a nonlinear function. For convenience, we assume

( ) ( )f u f u− = − .

With a new independent variable tτ ω= , where ω is the frequency of system,

Eq. (1) becomes

2 ( )u f uω ′′ = , (0)u A= , (0) 0u′ = , (2)

where du

udτ

′ = .

Considering the periodic solution does exist, it may be better to approximate the

solution by finite sums of trigonometric functions, i.e.

1

( ) cos( ),M

k

k

u A kτ τ=

=∑ (3)

where 2

0

1( )cos( )kA f t kt dt

π

π= ∫ .

First, we consider only one-term expansion into Eq. (3), and take the initial

approximation of ( )u τ and ω satisfying initial conditions in Eq. (2) as

0 ( ) cos( ),u Aτ τ= 2 20ω ω= , (4)

Substituting Eq. (4) into Eq. (2) and equating the coefficient of cos( )τ to zero, we

can determine the parameter 0ω . Then, the zero-order approximation is in the form of

Eq. (4), where 0tτ ω= .

Using Eq. (3) and the harmonic balance method, in order to obtain the k th-order

( 1,2, ,k = L ) approximation, we may assume

1

( )0

( ) ( ) ( )k

i k

i

u u puτ τ τ−

=

= +∑ ,1

2 20 ( )

1

,k

i k

i

pω ω ω ω−

=

= + +∑ (5)

where p is the order parameter, and

(0) 0( ) ( )u uτ τ= , 1

( ) ( )0

( ) ( ) ( )i

i j i

j

u u uτ τ τ−

=

= +∑ , 1,..., 1i k= − ,

3, 5, 2 1,( ) (cos( ) cos(3 )) (cos( ) cos(5 )) (cos( ) cos((2 1) ))i i i i iu a a a iτ τ τ τ τ τ τ+= − + − + + − +L ,

1,...,i k= , (6)

(0) 0ω ω= , 1

2 2( ) 0 ( )

1

i

i j i

j

ω ω ω ω−

=

= + +∑ , 1,..., 1i k= − , (7)

where 3,ia , 5,ia , … , 2 1,i ia + and i

ω ( 1,...,i k= ) are unknown parameters to be

determined. We first substitute Eq. (5) into Eq. (3) to obtain the coefficient of p as a

function 3 5 2 1,( , , , , , )k k k k k k

F a a aτ ω+

L . Then, calculating the residual errors of

( 1)k − th-order approximation, i.e.

21 ( 1) ( 1) ( 1)( ) ( ).k k k kR u f uτ ω− − − −

′′= − (8)

If 1( ) 0k

R τ−

= then ( 1) ( )ku τ− happens to be the exact solution. Generally such

case will not arise for nonlinear problems. At this moment, we will use this residual

error to obtain a more accurate higher-order approximation. Our option is taking a

new equation:

3 5 2 1, 1( , , , , , ) ( ) 0.k k k k k k kF a a a Rτ ω τ+ −+ =L (9)

Equating the coefficients of cos( )τ , cos(3 )τ , … , cos((2 1) )k τ+ to zero in Eq.

(9), we can easily obtain the unknown parameters 3,ka , 5,ka , … , 2 1,k ka + and k

ω for

the number of equations equal to the number of the parameters.

Then, the k th-order approximate periodic solution and frequency can be

obtained in the form

( ) (0) (1) ( 1)( ) ( ) ( ) ( ) ( )k k ku u u u uτ τ τ τ τ−= + + + +L , 2 2( ) 0 (1) ( 1)k k kω ω ω ω ω−= + + +L . (10)

3. Duffing-harmonic oscillator

Conservative nonlinear oscillatory systems can often be modeled by potentials

having a rational form for the potential energy, which lead to the equation of motion

3

20.

1

uu

u+ =

+&& (11)

For small u , the equation is that of a Duffing-type nonlinear oscillator, while for

large u ,the equation approximates that of a linear harmonic oscillator, hence Eq. (11)

is called the Duffing-harmonic oscillator. The above system has no possible small

parameters. Therefore, the classical perturbation methods do not apply to such a

problem.

If ω is the frequency of Duffing-harmonic oscillator, Eq. (11) can be rewritten

as

2 2 3( ) 0.u u u uω ′′ ′′+ + = (12)

where

,tτ ω= du

udτ

′ = , (13)

with the initial conditions

(0) ,u A= (0) 0u′ = . (14)

3.1 Zero-order approximation

First, we consider

0 ( ) cos( ),u Aτ τ= 2 20ω ω= , (15)

Substituting Eq. (15) into Eq. (12), yields

2 2 3 3 2 3 30 0 0

3 3 1 1cos( ) cos(3 ) 0

4 4 4 4A A A A Aω ω τ ω τ

− − + + − + =

. (16)

Equating the coefficient of cos( )τ to zero, we obtain

220 2

3

4 3

A

Aω =

+. (17)

At this time, we know the zero-order approximate periodic solution in the form

Eq. (15).

3.2 First-order approximation

To obtain the rest of the nonlinear correction, we consider

0 1( ) ( ) ( )u u puτ τ τ= + , 2 20 1,pω ω ω= + (18)

where p is a bookkeeping parameter. We assume

1 31( ) (cos( ) cos(3 ))u aτ τ τ= − , (19)

where 31a and the former parameter 1ω are two unknown constants that will be

determined later.

Substituting Eq. (18) into Eq. (12) and taking the coefficients of the p , we obtain

a function 1 31 1( , , )F aτ ω .

( )

( ) ( )( )

2 4 4 2 21 31 1 31 31 12

22 2 2 2 2

31 31 1 312

1( , , ) 12 24 (9 16 24 ) cos( )

16 12

48 96 (4 3 ) cos(3 ) 24 12 cos(5 )16 12

F a A a A a A A A tA

AA a a A A t A A a t

A

τ ω ω

ω

= + − − −+

+ + − − + −+

(20)

Now, substituting Eq. (15) and Eq. (17) into Eq. (12) it results the following

residual

2 3 30 0

1 1( ) cos(3 ).

4 4R A Aτ ω τ

= − +

(21)

Following Eq. (9), we have

1 31 1 0( , , ) ( ) 0.F a Rτ ω τ+ = (22)

Equating the coefficients of cos( )τ and cos(3 )τ to zero in Eq. (22), we obtain

2 2

1 2 4 2

4 (1 2 ),

(39 10 32)(4 3 )

A A

A A Aω

+= −

+ + +

2

31 2 4

1 (4 3 ),

3 39 10 32

A Aa

A A

+= −

+ + (23)

and the first-order approximate periodic solution in the form

(1) 0 1 31 31( ) ( ) ( ) ( ) cos( ) cos(3 ),u u u A a aτ τ τ τ τ= + = + − (24)

2 2(1) 0 1,ω ω ω= + (25)

where (1)tτ ω= .

3.3 Second-order approximation

Based on the first-order approximations, the periodic solution and the frequency

of Eq. (12) can be further expressed as

0 1 2( ) ( ) ( ) ( )u u u puτ τ τ τ= + + , 2 20 1 2.pω ω ω ω= + + (26)

and

2 32 52( ) (cos cos3 ) (cos cos5 ),u a aτ τ τ τ τ= − + − (27)

where 32a , 52a and the former parameter 2ω are unknown constants that will be

determined later.

Substituting Eq. (26) into Eq. (12) and taking the coefficients of the p , we obtain

the function 2 32 52 2( , , , )F a aτ ω . Substituting Eq. (24) and Eq. (25) into Eq. (12) it

results the residual error function 1( )R τ . Taking the two functions into following

equation

2 32 52 2 1( , , , ) ( ) 0.F a a Rτ ω τ+ = (28)

Equating the coefficients of cos( )τ , cos(3 )τ and cos(5 )τ to zero in Eq. (28),

we obtain

21

2 2 4

4,

(39 10 32)

A X

A Aω = −

+ + ∆

232 2 4

4,

9(39 10 32)

AXa

A A= −

+ + ∆

352

2,

3

AXa =

∆ (29)

where the values ( 1,2,3)i

X i = and ∆ are presented in Appendix.

The second-order approximate periodic solution is

(2) 0 1 2 31 32 52 31 32 52( ) ( ) ( ) ( ) ( )cos( ) ( )cos(3 ) cos(5 ),u u u u A a a a a a aτ τ τ τ τ τ τ= + + = + + + − + −

2 2(2) 0 1 2 ,ω ω ω ω= + + (30)

where (2)tτ ω= .

It should be clear how the procedure works for constructing further higher-order

approximations.

4 Result and discussion

In order to illustrate the applicability, accuracy and effectiveness of the proposed

approach, we compare the analytical approximate frequency and periodic solution

with the exact ones.

The exact frequency can be obtained from the following complicated relation

that is given in [5].

12

2 202 2

2

cos( )

2 coscos ln[1 ]

1

e

A tdt

A tA t

A

ππ

ω −=

+ −+

∫ (31)

We denote 100( )i i e e

E ω ω ω= − as the relative error of the i th-order

approximation for Duffing-harmonic oscillator.

Table 1 shows the comparison between the present and existing solutions,

namely the second-order approximations obtained with GRHBM, the first-order

approximation to the frequency of oscillation obtained by Guo et al., by means of an

iterative homotopy harmonic balance method (IHHBM) [6] , frequency obtained by

He’s energy balance method (HEBM) [10] and max-min approach (MMA) [21].

Table 1 Comparison of approximations and exact frequencies for Duffing-harmonic oscillator.

The results presented in Table 1 clearly show that the methods GRHBM and

IHHBM achieved more excellent agreement than HEBM and MMA. The GRHBM

result is more (respectively, slightly less) accurate than that of IHHBM for 1A ≤

(respectively, 5A ≥ ) . For example, when 0.5A = the relative error of the

second-order frequency is -0.0012%, the former result obtained by Guo et al., is

-0.4967% [6].

A e

ω GRHBM IHHBM HEBM MMA

0.01

0.05

0.1

0.5

1.0

5.0

10

50

100

0.00847

0.04232

0.08439

0.38737

0.63678

0.96698

0.99092

0.99961

0.99990

0.008472(0.0283%)

0.042322(0.0043%)

0.084394(0.0049%)

0.38737(-0.0012%)

0.636795(0.0023%)

0.968107(0.1165%)

0.991591(0.0677%)

0.999657(0.0047%)

0.999914(0.0014%)

0.008478(0.0945%)

0.042344(0.0567%)

0.084418(0.0332%)

0.38545(-0.4967%)

0.63136(-0.851%)

0.96667(-0.0319%)

0.99090(-0.0024%)

0.99961(-0.0002%)

0.999901(0.0001%)

0.00866(2.2432%)

0.04326(2.2212%)

0.08627(2.2278%)

0.39638(2.3259%)

0.65164(2.3336%)

0.97343(0.6670%)

0.99314(0. 2240%)

0.99973(0.0120%)

0.99999(0.0090%)

0.00866(2.2432%)

0.04326(2.2212%)

0.08627(2.2278%)

0.39736(2.5789%)

0.65465(2.8063%)

0.97435(0.6691%)

0.99340(0.2503%)

0.99973(0.0120%)

0.99993(0.0030%)

It is worth noting that, for the maximal absolute value of the relative errors, the

approximate frequency obtained in this paper, combining IHHBM, is slightly more

accurate (0.2895%( 1E ) and 0.1165% ( 2E ) versus 0.851%[6]) than the first-order

approximate frequency obtained by Guo et al. [6].

The periodic solution achieved by numerical integration of Eq. (12) using

Runge-Kutta scheme and the second-order approximate periodic solution given by Eq.

(30) are plotted in Figs1-4. These figures represent, respectively, four different

amplitudes 0.01A = , 0.5, 5 and 50. They show that the approximate periodic solution

provides excellent approximation comparing to the exact periodic solution for small

as well as for large amplitude of oscillation.

5. Conclusion

A global residue harmonic balance method (GRHBM) has been used to obtain

three approximate frequencies and periodic solutions for the Duffing-harmonic

oscillator. We obtain an excellent agreement for all the residual error is used to

proceeding with harmonic balancing. The results will be helpful in study other

strongly nonlinear oscillators. Recently, some new methods were proposed by Y.

Khan et al. in [22-24] which are easy to use and friendly methods. A future work is to

compare GRHBM with these new methods.

Fig. 1 Comparison between analytical approximate solutions and exact solution for 0.01A =

Fig. 2 Comparison between analytical approximate solutions and exact solution for 0.5A =

Fig. 3 Comparison between analytical approximate solutions and exact solution for 5A =

Fig 4 Comparison between analytical approximate solutions and exact solution for 50A =

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Appendix

The values ( 1, 2,3)i

X i = and ∆ in Eq. (29) are presented as below

1X =(1749600000000 6A +51379920000000 4A +698057352000000 2A +5827905547200000) 28A

+(33501446346240000 4A +140840415446832000 2A +449008725898404000) 22A

+(1110938790236693040 4A +2164791346739164872 2A +3349676218711307796) 16A

+(4124299437382437900 4A +4020561823848634167 2A +3059478252685316576) 10A

+(1768736508843204096 4A +741310535886624768 2A +207616456123678720) 4A

+33188673240170496 2A +2028721749884928,

2X =(5904900000000 6A +189022410000000 4A +2820893202000000 2A +26080808976000000)

30A +(167438166962940000 4A +792861943102488000 2A +2870799782408657100) 24A

+(8130057620217126030 4A +18262949906675678562 2A +32796531300124106163) 18A

+(47207173907726830701 4A +54339065166680506062 2A +49637428874998281144) 12A

+(35472746687173366800 4A +19372704358011629184 2A +7786374240783392768) 6A

+2162332794660438016 4A +368560237035651072 2A +28812435901120512,

3X =(218700000000 6A +6215454000000 4A +81545648400000 2A +654185388420000) 26A

+(3581997701040000 6A +14144492109310200 4A +41465266496357400 2A +91407280917881286

) 18A +(151432124231541141 4A +185469101630190189 2A +160640096015382210) 12A

+(86882476987596360 4A +14380666515959808 2A -18021998043072512) 6A

-15882568932057088 4A -5645816199905280 2A -803408235724800,

∆ =(6123600000000 6A +211205880000000 4A +3404341548000000 2A +34028121546000000) 28A

+(236059546632120000 6A +1205418822799968000 4A +4689648931579491600 2A

+14193355896848141280) 20A +(33822874705790864268 6A +63813346845672803868 4A

+95309456481644416311 2A +112041973038923497908) 12A +(102396952803640001008 4A

+71229035343489706624 2A +36422203401630900224) 6A +12901760886549168128 4A

+2827110614223028224 2A +288586791747846144.