An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities
An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but unlike them the obtained from the...
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irk-123456789-1742012021-01-08T01:26:23Z An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities Mohammadian, M. Pourmehran, O. Ju, P. An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but unlike them the obtained from the previous step errors are considered in the present step to increase the accuracy of the solution. A comparison of results with those obtained by exact solution and other approximate analytical techniques confirms an accuracy of the method. It is shown that the achieved approximate solutions are valid for both small and large amplitudes of oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and quintic terms on the frequency is discussed. Запропонована ітераційна процедура для отримання наближених розв’язків високого порядку консервативної системи, яка містить осцилятор з відновлювальною силою, що описується третім і п’ятим порядками нелінійності. Запропонований метод аналогічний до класичних методів гармонічного балансу, однак на відміну від них тут похибки, отримані на попередньому кроці, розглядаються на наступному кроці з метою підвищення точності розв’язку. Порівняння результатів з результатами, отриманими як точний розв’язок і іншими наближеними аналітичними методиками підтверджує точність методу. Показано, що отримані наближені розв’язки вірні як для малих, так і для великих амплітуд коливань і можуть узгоджуватись з точним розв’язком з високим рівнем точності при низьких порядках наближень. Далі обговорюється вплив членів третього і п’ятого порядків на основі отриманих аналітичних розв’язків. 2018 Article An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities / M. Mohammadian, O. Pourmehran, P. Ju // Прикладная механика. — 2018. — Т. 54, № 4. — С. 113-124. — Бібліогр.: 17 назв. — англ. 0032-8243 http://dspace.nbuv.gov.ua/handle/123456789/174201 en Прикладная механика Інститут механіки ім. С.П. Тимошенка НАН України |
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An iterative procedure is suggested for obtaining the higher-order approximate solutions of a conservative system comprising an oscillator with cubic and quintic restoring force function. The proposed method is similar to the traditional harmonic balance methods but unlike them the obtained from the previous step errors are considered in the present step to increase the accuracy of the solution. A comparison of results with those obtained by exact solution and other approximate analytical techniques confirms an accuracy of the method. It is shown that the achieved approximate solutions are valid for both small and large amplitudes of oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and quintic terms on the frequency is discussed. |
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Mohammadian, M. Pourmehran, O. Ju, P. |
| spellingShingle |
Mohammadian, M. Pourmehran, O. Ju, P. An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities Прикладная механика |
| author_facet |
Mohammadian, M. Pourmehran, O. Ju, P. |
| author_sort |
Mohammadian, M. |
| title |
An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities |
| title_short |
An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities |
| title_full |
An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities |
| title_fullStr |
An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities |
| title_full_unstemmed |
An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities |
| title_sort |
iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities |
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Інститут механіки ім. С.П. Тимошенка НАН України |
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2018 |
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http://dspace.nbuv.gov.ua/handle/123456789/174201 |
| citation_txt |
An iterative approach for obtaining nonlinear frequency of a conservative oscillator with strong nonlinearities / M. Mohammadian, O. Pourmehran, P. Ju // Прикладная механика. — 2018. — Т. 54, № 4. — С. 113-124. — Бібліогр.: 17 назв. — англ. |
| series |
Прикладная механика |
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2025-07-15T11:07:27Z |
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| fulltext |
2018 П Р И К Л А Д Н А Я М Е Х А Н И К А Том 54, № 4
ISSN0032–8243. Прикл. механика, 2018, 54, № 4 113
M . M o h a m m a d i a n a * , O . P o u r m e h r a n b , P . J u c
AN ITERATIVE APPROACH FOR OBTAINING NONLINEAR FREQUENCY OF
A CONSERVATIVE OSCILLATOR WITH STRONG NONLINEARITIES
a Department of Mechanical Engineering, Kordkuy center,
Gorgan branch, Islamic Azad University, Kordkuy, Iran.
b Young Researchers and Elite Club, Gorgan Branch, Islamic Azad University, Gorgan, Iran.
c School of Mathematics and Statistics, Taishan University, Taian, PR China.
* Corresponding author: P.O.B. 48811644479 Kordkuy, mo.mohammadyan@gmail.com
Abstract. An iterative procedure is suggested for obtaining the higher-order approximate
solutions of a conservative system comprising an oscillator with cubic and quintic restoring
force function. The proposed method is similar to the traditional harmonic balance methods
but unlike them the obtained from the previous step errors are considered in the present step to
increase the accuracy of the solution. A comparison of results with those obtained by exact
solution and other approximate analytical techniques confirms an accuracy of the method. It is
shown that the achieved approximate solutions are valid for both small and large amplitudes of
oscillation and can meet the exact solutions with a high level of accuracy in the lower-order of
approximations. Furthermore, using the obtained analytical solutions, the effect of cubic and
quintic terms on the frequency is discussed.
Key words: harmonic balance method, Duffing oscillator, аnalytical solution, frequency.
1. Introduction.
A large bunch of the mechanical systems are involved with nonlinear governing equa-
tion of motion which solving these equations is an important issue for the sake of their non-
linear behavior [1-3]. Transverse vibrations of a beam with large amplitude and implementa-
tion of nonlinear springs in a mechanical system are two examples which can be derived as
a nonlinear ordinary differential equation. Recently, many different methods have been de-
veloped to obtain the approximate solutions of such problems. Some of them are: harmonic
balance method [4-6], energy balance method [7, 8], Hamiltonian approach [9], He’s ampli-
tude–frequency formulation [10] and variational iteration method [11]. In these methods,
unlike the classical perturbation method, the presence of small parameter is not required.
Duffing equation is a well-known nonlinear differential equation which is composed of
third and fifth orders of nonlinearities. Many applications of this equation can be found in
some engineering systems such as free vibrations of a restrained uniform beam including a
lumped mass in its middle region [12], the nonlinear dynamics of slender elastic, and the
Pochhammer– Chree (PC) equation [13, 14].
A combination of homotopy analysis method and Pade technique was employed by
Pirbodaghi et al. [15] to obtain the analytical approximate solution of the cubic-quintic
Duffing equation. Ganji et al. [8] considered different parameters and applied energy bal-
ance technique on the approximate frequencies of the mentioned equation. In another re-
search, Newton’s method and harmonic balance technique were coupled by Lai et al. [16]
for solving higher-order approximations of Duffing oscillators with cubic-quintic nonlinear
restoring force. They showed that their results are valid for both small and large amplitudes.
114
Also, Khan et al. [14] used coupled homotopy and variational formulation to analyze the
aforementioned oscillator. They obtained the first four approximate formulas and concluded
that the proposed method observe a good agreement with the exact solution. Moreover, Zun-
iga [17] employed the Jacobi elliptic functions for the Duffing oscillator and showed that the
exact frequency of the system is including the complete elliptic integral of the first kind.
In the current paper, an approach is employed to solve and obtain the analytical ap-
proximate solutions of the cubic-quintic Duffing oscillator. This method has a main differ-
ence with other traditional harmonic balance methods such that all the errors obtained in the
previous approximation are used in the present one. Moreover, to obtain the unknown con-
stants we are dealing with only a set of linear equations which can be easily solved. The
comparison that will be done between the presented method, exact solution, coupled ho-
motopy-variational formulation as well as He’s energy balance method, will confirm the
ability and high precision of this method.
2. Description of the problem.
Duffing oscillator with the cubic and quintic terms of nonlinearity is a conservative sys-
tem and can be defined with the following second-order differential equation [8, 12, 17]
3 5
1 2 3 0,x x xx (1)
where, the double dots superscript represents the second differential respect to time (t). The
coefficients 1 , 2 and 3 are supposed to get different values. For the case with 1 0
and 2 3 0 , Eq. (1) represents the simple harmonic vibration of an oscillator with the
frequency of 1 . If 2 0 and 3 0 , Eq. (1) can be related to Duffing equation with
only cubic term of nonlinearity. On the other hand, the quintic Duffing equation is con-
structed by considering 2 0 and 3 0 . For the other cases, Eq. (1) is named as cubic-
quintic Duffing equation where 2 and 3 are not equal to zero.
The initial conditions for Eq. (1) are considered as:
˙
0 ; 0 0.
dx
x A x
dt
(2)
3. Basic idea of the proposed iterative approach.
To illustrate how the method works, it is considered that the governing differential
equation of system is as follows:
; 0 ; 0 0,u f u u A u (3)
where u is the second differentiation respects to t. It is assumed that ( )f u is an odd func-
tion. Introducing an independent variable t , Eq. (3) can be rewritten as below:
2 ; 0 ; 0 0,u f u u A u (4)
where ''u is the second differentiation respect to . Also, is the frequency of system in
which should be determined. The solution of Eq. (4) is assumed to be periodic and the sim-
plest form of it that would satisfy the initial conditions is given by the following:
2 2
0 0( ) cos( ) .;u A (5)
Substituting Eq. (5) into Eq. (4) and then setting up the coefficient of cos( ) as zero,
the parameter 0 can be determined. Hence, the zero-order approximation of Eq. (3) is as
follows:
0 0( ) cosu A t (6)
115
Eq. (1) is a nonlinear problem and therefore the above solution is not the exact one.
Hence, substituting Eq. (6) into Eq. (4), the error for the zero-order approximation derives
as:
2
0 0 0 0( ) ( )R u f u . (7)
The above error is kept to be used in the next order of approximation. To obtain the
first-order approximation, the following assumption is supposed:
2 2
0 1 0 1( ) ( ) ); .(u u pu p (8)
1( ) (cos ( ) cos(3 )).u B (9)
The coefficient p is the order parameter and takes the values in interval [0,1]. In addi-
tion, 1 and B are two unknown constants that will be determined later. In this time, Eq.
(8) is substituted into Eq. (4) and then from the governing equation, the coefficients of p are
considered as 1 1( , , )F B . Now, the following relation is considered:
1 1 0, , )( ( ) 0.F B R (10)
Eq. (10) means that in the current method, unlike other traditional harmonic balance
methods, the error obtained in the previous step is considered in the present step to increase
the accuracy of solution. Equating the coefficients of cos( ) and cos(3 ) to zero in Eq.
(10), two linear equations composed of the unknown constants 1 and B can be achieved.
Once the aforementioned constant parameters are obtained, the first-order approximation is
written as:
2 2
(1) 0 11 1( ) ( ) cos ( ) cos ( ); ; .u A B B (11)
Substituting Eq. (11) into Eq. (4), the error for the first-order approximation becomes
as:
2
1 (1) (1) 1( ) ( ).R u f u (12)
Similarly, the above error is kept to be used in the next step. In the following, to obtain
the second-order approximation, it is assumed as:
2 2
(1) 2 (1) 2( ) ( ) ( ;);u u pu p (13)
2 ( ) (cos( ) cos(3 )) ( ) cos(5 ));u C D (14)
where 2 , C and D are three unknown constants which should be determined. Similar to the
previous step, Eq. (13) is substituted into Eq. (4) and then the coefficients of p are consid-
ered as 2 2( , , , ).F C D Here, in order to increase the accuracy of the second-order approx-
imation, the error 1( )R obtained in Eq. (12) is added to 2F and consequently a new equa-
tion is obtained as follows:
2 2 1, , , .( ) ( ) 0F C D R (15)
Equating the coefficients of cos ( ), cos (3 ) and cos (5 ) to zero in Eq. (15), three
linear equations including the unknown constant parameters 2 , C and D can be achieved.
As the aforementioned constants are obtained, the second-order approximation is written as
116
2 ( ) ( ) cos( ) ( ) cos(3 ) cos(5 ),u A B C D B C D (16)
where,
2 2
(2) (2) (1) 2; .t (17)
Similar to the above steps, the higher-order approximations can also be derived. It
should be noted that using Eqs. (10) and (15) in the process of solving nonlinear oscillator is
the main difference between the current method and other classical harmonic balance tech-
niques. Moreover, these equations (i.e. Eqs. (10) and (15)) lead to a system of linear equa-
tions which can be easily solved. The convergence of the proposed method is explained in
the Appendix.
4. Implementation of the method to Duffing oscillator.
In this section, the approach is used to obtain the approximate solution of Eq. (1). Pri-
marily, introducing an independent variable, t , contributes to transform the mentioned
equation to the following:
2 3 5
1 2 3 0,x x x x (18)
where x and are the second differentiation respect to , and the frequency of system,
respectively. Also, the initial conditions change towards (0)x A and 0.(0)x
4.1. Zero-order approximation.
The simplest form of equation in which can satisfy the initial conditions of Eq. (18) is as
follows:
2 2
0 0( () cos ); .x A (19)
Substituting above equation into Eq. (18), yields
2 3 5 3 5 5
0 1 2 3 2 3 3
3 5 1 5 1
cos cos 3 cos 5 0.
4 8 4 16 16
A A A A A A A
(20)
To avoid secular terms in the next step, the coefficient of cos ( ) should be identical to
zero. Applying this approach, the zero-order approximation (i.e., 0 ) is obtained as follows
2 4
0 1 2 3
3 5
4 8
A A (21)
So, the zero-order analytical approximate solution of Eq. (1) is written as
0 0cos( ).x A t (22)
The error for the zero-order approximation obtained as Eq. (23) which will be kept to
be used in the next order of approximation:
3 5 5
0 2 3 3
1 5 1
cos 3 cos 5 .
4 16 16
R A A A
(23)
4.2. First-order approximation.
To obtain the first-order approximation, the following assumption is considered
2 2
(0) 1 (0) 1( ) ( ) ( ;);x x px p (24)
1( ) cos( ) cos(3 ) .x B (25)
Eq. (24) is substituted into Eq. (18) and then considering the coefficients of the p, we
obtain a function as 1 1( , , )F B . According to Eq. (10), we have
1 1 0, , )( ( ) 0.F B R (26)
117
In above equation the coefficients of cos ( ) , cos(3 ) should be vanished. Therefore,
we have two linear equations as follows
5 2
3 1 2
15 3
0;
16 4
A B A A B
5 3 4 2
3 2 1 3 2
5 1 85
8 6 0.
16 4 16
A A B A B A B (27)
The solutions of above relations results in:
3 2 4 2 2
3 3 2 2
1 4 2
1 3 2
(25 40 16 )3
;
16 128 85 96
A A A
A A
(28)
3 2
3 2
4 2
1 3 2
(5 4 )
.
128 85 96
A A
B
A A
(29)
According to Eq. (11), the first-order analytical approximate solution of Eq. (1) is writ-
ten as
3 2 3 2
3 2 3 2
1 4 2 4 2
1 3 2 1 3 2
(5 4 ) (5 4 )
cos cos 3 ,
128 85 96 128 85 96
A A A A
x A t t
A A A A
(30)
where is as follows
2
0 1
4 2 4 2 4 2 2 8
3 2 2 2 1 3 1 1 3
4 2
3 2 1
465 276 768 660 512 193,75
.
340 384 512
A A A A A
A A
(31)
Substituting Eq. (30) into Eq. (18), the error is obtained as function 1 )(R .
4.3. Second-order approximation.
As in previous steps, for the second-order approximation, it is assumed
2 2
(1) 2 (1) 2( ) ( ) ( ;);x x px p (32)
2 ( ) (cos( ) cos (3 )) (cos( ) cos(5 )).x C D (33)
Eq. (32) is substituted into Eq. (18) and then from the consequent equation, the coeffi-
cients of p are considered as 2 2( , , , ).F C D Now, the error 1 )(R is added to 2F and con-
sequently a new equation is obtained as follow:
2 2 1, , , .( () ) 0F C D R (34)
Equating the coefficients of cos ( ), cos (3 ) and cos(5 ) to zero in Eq. (34), one can
obtain three linear equations, consequently. By solving these equations, three unknown con-
stants 2 , C and D can be determined. Given the large amount of space of correlations, we
evaded to cite them here. But, their numerical values are reported in the results section.
Eventually, the second-order analytical approximation of Eq. (1) is
2 2 2 2( ) cos( ) ( ) cos(3 ) cos (5 );x A B C D t B C t D t (35)
2 2
(2) 0 1 2. (36)
118
5. Results and discussion.
In this section, to assess the analytical approximate solutions achieved by the presented
method, the obtained results are compared with the exact solution as well as those of litera-
ture. The exact frequency is given by [13, 16]:
1
1
2 42 2
2 30
,
2 (1 sin sin )
Exact
k
k k d
(37)
where:
2 4
1 1 2 3 ;
2 3
A A
k (38)
2 4
2 3
2 2 4
1 2 3
3 2
;
6 3 2
A A
k
A A
(39)
4
3
3 2 4
1 2 3
2
;
6 3 2
A
k
A A
(40)
Considering various parameters of system in rank order include as 1 2 3 1;
1 2 35, 3, 1; ; and 1 2 31, 10, 100, the approximate frequencies are ob-
tained and illustrated in Tables 1 – 3.
Table 1
A Exact Zero-order First-order Second-order Khan et al, [14] Ganji et al, [8]
0,1 1,0037770 1,003774128 1,003772940 1,003772938 1,0031009 1,00377306
0,5 1,1065487 1,107502822 1,106575472 1,106545257 1,0877056 1,10635650
1,0 1,5235914 1,541103501 1,525073610 1,523748195 1,4456576 1,10635650
5 19,1815720 20,25771458 19,37354774 19,22145533 17,8276787 19,608880
10 75,1776276 79,53615530 75,97375093 75,34539761 69,8760834 76,889585
50 1867,5796 1976,898075 1887,694912 1871,836939 1735,9103 1910,33222
100 7468,8525 7906,168540 7549,340082 7485,889922 6942,2827 7639,85509
500 186709,59 197642,83 188721,99 187135,59 173546,2 190984,592
1000 746836,94 790569,89 754886,52 748540,91 694183,44 763936,894
Comparison of the frequencies obtained by current approach with the exact solution and
the other methods for 1 2 3 1.
Table 2
A Exact Zero-order First-order Second-order Khan et al, [14] Ganji et al, [8]
0,1 2,2411156 2,241107427 2,241106482 2,241106482 2,2402105 2,241102478
0,5 2,3661575 2,366762028 2,366156024 2,366148280 2,3434565 2,366246867
1,0 2,7962794 2,806243040 2,796695889 2,796295935 2,7566507 2,798963393
5 20,2164536 21,25735167 20,39110224 20,25141913 18,7895069 20,64011142
10 76,1700134 80,49844720 76,94867693 76,33263727 70,7962723 77,88483819
50 1868,5568 1977,847315 1888,654708 1872,808907 1736,8159 1911,314776
100 7469,8296 7907,117362 7550,299393 7486,861400 6943,1880 7640,837246
500 186710,58 197643,7768 188722,9500 187136,5622 173547,11 190985,5701
1000 746837,94 790570,8380 754887,4839 748541,8778 694184,38 763937,8765
119
Comparison of the frequencies obtained by current approach with the exact solution and
the other methods for 1 2 35, 3, 1.
Table 3
A Exact Zero-order First-order Second-order Khan et al, [14] Ganji et al, [8]
0,1 1,0397019 1,039831717 1,039699840 1,039697865 1,0325994 1,039642196
0,5 2,5247023 2,604083332 2,535046776 2,526416372 2,3542032 2,554014562
1,0 8,0100698 8,426149772 8,080690531 8,024286085 7,4440041 8,176911017
5 187,19966 198,1186513 189,2033240 187,6230518 174,00040 191,4770915
10 747,32526 791,0442465 755,3661805 749,0266516 694,63605 764,4279087
50 18671,400 19764,70974 18872,63073 18713,99620 17355,027 19098,90111
100 74684,133 79057,41585 75489,08410 74854,52775 69418,750 76394,13136
500 1867091,6 1976424,012 1887215,592 1871351,538 1735457,9 1909841,499
1000 7468365,0 7905694,625 7548860,931 7485404,694 6941830,5 7639364,525
Comparison of the frequencies obtained by current approach with the exact solution and
the other methods for 1 2 31, 10, 100.
In these tables, the third-order approximate values achieved by coupled homotopy-
variational formulation (CHVF-3) [14] as well as the results of He’s energy balance method
(HEBM) [8] have been also reported. As can be seen, the values of the second-order meets a
good agreement with the exact solution. In order to have a better comparison, for example,
the errors of the current approach as well as two mentioned methods for the system parame-
ters 1 2 35, 3, 1 are depicted in Fig. 1. The correlation of the error is considered as
follows:
Error % 100,Exact
Exact
(41)
where Exact is obtained from Eq. (37).
Figure 1. Relative errors for the approximate frequencies
respect to amplitude, 1 2 3( 5, 3, 1) .
120
Fig. 1 shows that the relative error of the current method decreases with increasing the
approximation order. Moreover, with increasing the amplitude (i.e., A), the error approaches
to approximately 7% and 2,3% for the CHVF-3 and HEBM, respectively; While it goes to-
ward almost 5,9%, 1.1% and 0,2% for the zero, first and second order approximations, re-
spectively. It is astounding by this figure that merely the first-order approximate solution of
the current method leads to accurate results with a relative error less than the third-order
coupled homotopy-variational formulation (CHVF-3) as well as He’s energy balance meth-
od (HEBM). Similar results have been obtained using other system parameters. Hence, it
can be concluded that the frequencies achieved by the proposed method can meet the exact
ones in the lower-order approximations.
The displacement of the oscillator (i.e., x(t)) considering various system parameters for
large amplitude (here A=10) is depicted in Figs. 2 and 3.
Figure 2. The Displacement of oscillator respect to time
for 1 2 310, 5, 3, 1A
Figure 3. The Displacement of oscillator respect to time
for 1 2 310, 1, 10, 100A
121
These figures reveal that the GRHBM is an accurate method especially in the first and
second orders of approximation and has sufficient ability in solving strong nonlinear prob-
lems such as current cubic-quintic Duffing oscillator.
The analytical relation obtained in section 4.3, is employed to investigate the effect of
nonlinear terms of Eq. (1) on the oscillator frequency. To this end, the coefficients of the
cubic and quintic terms (i.e., 2 and 3 ) are supposed to change from 10 to 50 and 0 to
100, respectively. The variation of the frequency for various values of amplitude is illustrat-
ed in Figs. 4 – 6.
Figure 4. Variation of frequency respect to 3 for various values of 2
for 10,1; 1.A
Figure 5. Variation of frequency respect to 3 for various values of 2
for 1 1.1,A
122
Figure 6. Variation of frequency respect to 3 for various values of 2
for 1 1.5,A
Fig. 4 shows that for small amplitude ( 0,1)A and a specific value of 2 , the fre-
quency is approximately constant and is not influenced by the variation of 3. Also, this
figure shows that for a specific value of 3, increasing the 2 contributes to increment of
frequency. Figs. 5 and 6 reveal that for large amplitudes and any value of 2 , the frequency
increases with increasing 3. Moreover, as can be seen in Fig. 6, for large values of ampli-
tude the frequency is only influenced by the variation of 3.
6. Conclusion.
In this paper, an iterative approach was employed for obtaining the analytical approxi-
mate frequencies and displacement of a conservative system comprising an oscillator with
cubic and quintic nonlinearities. Using the achieved analytical expressions, the effect of
nonlinear terms on the nonlinear frequency was investigated. The results showed that the
cubic term is more impressive in the lower amplitudes and its effect gradually vanishes with
increment of oscillator amplitude. Also, the quintic term plays a significant role in nonlinear
frequency regarding the higher values of amplitude. Moreover, comparing the results in
terms of the other methods showed the applicability and accuracy of the proposed method
such that it can approach to the exact solution in the lower-order approximations. The men-
tioned method is very simple to implement and is not restricted to the presence of small pa-
rameter in system and its results are valid for a wide range of system parameters along with
both small and large amplitudes. Therefore, it can be easily extended and employed for the
other strong nonlinear oscillators due to its great potential.
Appendix.
The proposed method provides a series solution for the problem as follows
0 1 2
0
( ) ( ) ( ) ( ) ... ( ).i
i
u t u t u t u t u t
(A1)
The above solution converges if there is 0 1 such that
123
1 0 0( ) ( ) ; ; .i iu t u t i i i N (A2)
In order to proof, the sequence nq is first defined as follows
0
( ) ( 0,1, 2, ...).
n
n i
i
q u t n
(A3)
Then
0
0
1
1 1 ... .n i
n n n n iq q u u u
(A4)
For every 0, ,j k N j k i we have
0 0 0 0
0 0 0 0
1 1 2 1
1 1 2 1 1 1
1 1 1
( ) ( ) ... ( )
... ...
1
... .
1
j k j j j j k k
j j j j k k j j k
j k
j i j i k i k i
i i i i
q q q q q q q q
q q q q q q u u u
u u u u
(A5)
Consequently
0
0
1
, ,
1
0 lim lim 0.
1
j k
k i
j k ij k j k
q q u
(A6)
Also,
,
lim 0j kj k
q q
due to have 0 1. Sequence nq is a Cauchy sequence
and according to its definition in the Hilbert space, shows that the solution
0
( )i
i
u t
is con-
vergent.
РЕЗЮМЕ. Запропонована ітераційна процедура для отримання наближених розв’язків високо-
го порядку консервативної системи, яка містить осцилятор з відновлювальною силою, що описується
третім і п’ятим порядками нелінійності. Запропонований метод аналогічний до класичних методів
гармонічного балансу, однак на відміну від них тут похибки, отримані на попередньому кроці,
розглядаються на наступному кроці з метою підвищення точності розв’язку. Порівняння результатів
з результатами, отриманими як точний розв’язок і іншими наближеними аналітичними методиками
підтверджує точність методу. Показано, що отримані наближені розв’язки вірні як для малих, так і
для великих амплітуд коливань і можуть узгоджуватись з точним розв’язком з високим рівнем точ-
ності при низьких порядках наближень. Далі обговорюється вплив членів третього і п’ятого порядків
на основі отриманих аналітичних розв’язків.
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From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 24.10.2017 Утверждена в печать 30.01.2018
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