research article an iteration scheme suitable for solving...
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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 582865 5 pageshttpdxdoiorg1011552013582865
Research ArticleAn Iteration Scheme Suitable for Solving Limit Cycles ofNonsmooth Dynamical Systems
Q X Liu Y M Chen and J K Liu
Department of Mechanics Sun Yat-sen University Guangzhou 510275 China
Correspondence should be addressed to Y M Chen chenyanmaohotmailcom
Received 19 July 2013 Accepted 26 September 2013
Academic Editor Mufid Abudiab
Copyright copy 2013 Q X Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The Mickens iteration method (MIM) is modified to solve self-excited systems containing nonsmooth nonlinearities andornonlinear damping terms If the MIM is implemented routinely the unknown frequency and amplitude of limit cycle (LC) wouldcouple to each other in complicated nonlinear algebraic equations at each iteration It is cumbersome to solve these algebraicequations especially for nonsmooth systems In the modified procedures the unknown frequency is substituted by the determinedvalue obtained at the previous iteration By this means the frequency is decoupled from the nonlinear terms Numerical examplesshow that the LCs obtained by the modified MIM agree well with numerical results The presented method is very suitable forsolving self-excited systems especially those with nonlinear damping and nonsmooth nonlinearities
1 Introduction
Recent years have witnessed the wide applications of iter-ation techniques such as the Mickens iteration method(MIM) [1ndash6] and the variational iteration method [7ndash10]In order to improve the efficiency Lim and Wu [11] Mar-inca and Herisanu [12] and Hu [13] modified the MIMrespectively
In principle the approximations can be obtained to anydesired accuracy by the MIM as long as the iteration pro-ceeds In theMIM algebraic equations are introduced at eachiteration to eliminate the so-called secular terms In the appli-cations in conservative oscillators the algebraic equationsare linear Nonlinear damping and nonsmooth terms appearwidely in many dynamical systems [14 15] As for the oscil-lators with nonlinear damping terms however very compli-cated nonlinear algebraic equations have to be solved at eachiteration [16] Moreover the algebraic equations cannot bededuced for systems with nonsmooth nonlinearities Itis necessary and worthwhile therefore to propose someapproaches to simplify the MIM This paper will present amodified iteration algorithm by decoupling the unknownfrequency from nonlinear terms
2 A Modified MIM
Consider a self-excited oscillator
119909 +119891 (119909 ) = 0 (1)
where the superscript denotes the differentiation with respectto time 119905 and119891(119909 ) is a nonlinear termwith damping termsAssume that system (1) has at least one limit cycle (LC) solu-tion Since the LC frequency and amplitude are independentof initial conditions they should be considered as unknownsto be determined at every iteration Denote the angularfrequency as 120596 and introduce the transformation as 120591 = 120596119905thus we rewrite (1) as
1205962
11990910158401015840
+ 1205962
119909 = 1205962
119909 minus 119891 (119909 1205961199091015840
) (2)
subject to the following initial conditions
119909 (0) = 120572 1199091015840
(0) = 0 (3)
where 120572 is the unknown LC amplitude and the superscriptdenotes the differentiation with respect to 120591 Note that 120572 willbe approximated as a series 120572
119896
by eliminating the secular
2 Mathematical Problems in Engineering
terms at each iteration stage In order to obtain the LC theMIM [16] can be given as
1205962
119896minus1
(11990910158401015840
119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
minus 119891 (119909119896minus1
120596119896minus1
1199091015840
119896minus1
)
119896 = 1 2
(4)
with the initial conditions being rewritten at each iteration as
119909119896
(0) = 120572119896
1199091015840
119896
(0) = 0 (5)
Note that the coefficient of the first harmonic in 119909119896minus1
remainsstill to be an unknown that is120572
119896minus1
This unknownwill couplewith the unknown frequency120596
119896minus1
which will result in a cou-pled nonlinear term (ie 120596
119896minus1
1199091015840
119896minus1
) in the right side of (4) Ifhigher powers of1199091015840 exist these termswill lead to very compli-cated functions in 120596
119896minus1
and 120572119896minus1
Different from conservativesystems 1205962
119896minus1
can no longer be considered as an independentunknown In order to simplify theMIM therefore amodifiedscheme is proposed as
1205962
119896minus1
(11990910158401015840
119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
minus 119891 (119909119896minus1
120596119896minus2
1199091015840
119896minus1
)
119896 = 1 2
(6)
As 119896 = 1 we choose 120596minus1
= 1205960
In the 119896th iteration 120596119896minus2
is agiven constant that is obtained at the (119896 minus 1)th iteration Thesquare of the unknown frequency that is1205962
119896minus1
can be treatedas an independent parameter because 120596
119896minus1
appears only in1205962
119896minus1
119909119896minus1
According to the initial conditions the startingiteration solution can be chosen as
1199090
(120591) = 1205720
cos 120591 (7)
It is obvious that as long as the series 120596119896
119896 = 1 2 and119909119896
119896 = 1 2 are convergent they must converge to theexact solutions The right-hand side of (6) can be expressedby Fourier series as
1205962
119896minus1
119909119896minus1
minus 119891 (119909119896minus1
120596119896minus2
1199091015840
119896minus1
)
=
120593(119896)
sum
119894=1
[119888119896minus1119894
(1205962
119896minus1
120572119896minus1
) cos (119894120591)
+ 119904119896minus1119894
(1205962
119896minus1
120572119896minus1
) sin (119894120591)]
(8)
where the harmonic coefficients 119888119896minus1119894
(1205962
119896minus1
120572119896minus1
) and119904119896minus1119894
(1205962
119896minus1
120572119896minus1
) are functions in 1205962119896minus1
and 120572119896minus1
Here 120593(119896) isa positive integer denoting the order of the highest harmonicApproximations120596
119896minus1
and 120572119896minus1
are determined by eliminatingthe secular terms that is letting
119888119896minus11
(1205962
119896minus1
120572119896minus1
) = 0 119904119896minus11
(1205962
119896minus1
120572119896minus1
) = 0
119896 = 1 2
(9)
These equations can be solved analytically if 1205962119896minus1
120572119896minus1
isconsidered as an independent unknown They can also benumerically solved by Newton-Raphson method The latteris employed in this study
Different from the existing procedures [16] as 119896 increases1205962
119896minus1
is always an independent unknown in the modifiedMIM Moreover unknown 120596
119896minus1
does not couple with non-linear terms It simplifies the MIM to a large extent as shownlater
3 Numerical Examples
Example 1 (system with nonlinear damping terms) Thevan der Pol equation is chosen to illustrate the previousprocedures more clearly
119909 +119909 + 120576 (1199092
minus 1) = 0 (10)
where 120576 is a given constant As known (10) has a stable LCsolution when 120576 gt 0 while an unstable one when 120576 lt 0
According to themodifiedMIM the corresponding itera-tion scheme is given as
119909119896
+ 1205962
119896minus1
119909119896
= 1205962
119896minus1
119909119896minus1
minus 119909119896minus1
minus 120576 (1199092
119896minus1
minus 1) 119896
119896 = 1 2
(11)
Introducing a new time variable 120591 = 120596119896minus1
119905 at each iterationstage we rewrite (11) as
1205962
119896minus1
(11990910158401015840
119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
minus 119909119896minus1
minus 120576120596119896minus1
(1199092
119896minus1
minus 1) 1199091015840
119896minus1
(12)
where the superscript denotes the derivative with respect to120591 The iteration algorithm begins with an initial solution
1199090
(120591) = 1205720
cos 120591 (13)
Then we obtain the governing equations in 1199091
(120591) as
1205962
0
(11990910158401015840
1
+ 1199091
) = (1205962
0
1205720
minus 1205720
) cos 120591 minus (1205761205720
1205960
minus 120576
1205723
0
1205960
4
) sin 120591
+ 120576
1205723
0
1205960
4
sin 3120591 1199091
(0) = 1205721
1199091015840
1
(0) = 0
(14)
Equating the coefficients of cos 120591 and sin 120591 to zeros results into
1205962
0
1205720
minus 1205720
= 0 1205761205720
1205960
minus 120576
1205723
0
1205960
4
= 0 (15)
which yields that 1205960
= 1 and 1205720
= 2 Substituting them into(16) we have
11990910158401015840
1
+ 1199091
= 2120576 sin 3120591 (16)
Considering initial conditions (14) we can obtain
1199091
= 1205721
cos 120591 + 3120576 sin 1205914
minus
120576 sin 31205914
(17)
where 1205721
is to be determined at the next iteration stage
Mathematical Problems in Engineering 3
Table 1 Comparison of the second-order frequency obtained by IS and LP method with the forth-order approximation obtained by LPmethod when 120576 = 1
120576 120596IS2
120596LP2
120596LP4
|120596IS2
minus 120596LP4
| |120596LP2
minus 120596LP4
|
1 0944799584 093750000 0943033854 176119864 minus 3 553119864 minus 3
05 0984820946 098437500 0984720866 100119864 minus 4 346119864 minus 4
025 0996121460 099609375 0996115367 609119864 minus 6 216119864 minus 5
According to iterative scheme (11) the equation in 1199092
(120591)
is deduced as
1205962
1
(11990910158401015840
2
+ 1199092
) =
9
sum
119894=1
[1198881119894
(1205962
1
1205721
) cos (1198941205961
119905)
+1199041119894
(1205962
1
1205721
) sin (1198941205961
119905)]
(18)
Equate the coefficients of cos 120591 and sin 120591 to zeros
31205962
1
4
minus
151205721
64
minus
1205723
1
4
minus
3
4
= 0
1205721
1205962
1
minus 1205721
minus
1205723
1
8
+
75
128
= 0
(19)
By solving (19) numerically we can determine 1205721
and 12059621
Here we obtain the second-order approximation and expandit as
120596IS2
= 1 minus
1205762
16
+
291205764
2048
+ 119900 (1205764
) (20)
According to [17] the Lindstedt-Poincare (LP) method pro-vides the second- and forth-order approximate frequency120596LP2
= 1minus1205762
16 and120596LP4
= 1minus1205762
16+171205764
3072 respectivelyThe attained approximation agrees well with the 4th-order LPsolution Table 1 indicates that 120596IS
2
is more accurate than 120596LP2
when compared with 120596LP4
Figure 1 shows the comparison of the phase planesbetween iteration solutions (119909
119896
) and numerical result Rapidconvergence of 119909
119896
to the numerical result can be observedNote that all numerical solutions are obtained by the fourth-order Runge-Kutta (RK) integration method When |120576| gt 1the iteration procedure presented by Chen and Liu [16] doesnot converge This is probably the difference between thestarting function (119909
0
= 1205720
cos 119905) and the exact solution istoo large The modified MIM is still effective for |120576| le 15 AsFigure 2 shows the LC solution with 120576 = 15 obtained by thepresented method is in excellent agreement with numericalone It is necessary to point out that the presented method isable to track unstable LCs whereas the RK method is not
Also plotted in Figure 2 are the results provided by the LPmethod [17]The iteration results aremuchmore precise thanthe 2nd and 4th-order LP approximations
It is necessary to point out that the presented method isable to track unstable LCs whereas the RK method is notFigure 3 shows an unstable LC of the van der Pol equationwith 120576 = minus1 obtained by the presentedmethod As shown theRK begins at the LC however the solution curve convergesto the equilibrium
IS2IS4
IS6RK
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus2 minus15 minus1 minus05 0 05 1 15 2
Figure 1The LC solutions of system (10) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896
In order to further demonstrate the merit of the modifiedMIM when applied to problems with nonlinear dampingterms we consider the following self-excited system [18]
119909 +119909 + 120576 [()2
minus 1] + 119891 () = 0 (21)
The nonlinear term contains high powers of that is 119891() =()3 If the originalMIM is employed the algebraic equations
governing 120596 will become very complicated Therefore itis necessary to employ the modified approach Figure 4indicates that the approximations obtained by the presentedmethod converge rapidly to the numerical solution as 119896increased
Example 2 (systemwith nonsmooth nonlinearity) Themod-ified MVIM is further applied to nonsmooth dynamicalsystem expressed as
119909 +119891 (119909 ) + 120578119892 (119909) = 0 (22)
Here 119891(119909 ) is a nonlinear damping term and 119892(119909) is anonsmooth function If substituting the 119909
119896minus1
into 119892(119909) onaccount of119909
119896minus1
contained unknown quantities (120572119896minus1
) so119892(119909)can not be expanded as Fourier progression by numericalintegration To this end (9) cannot be deduced by eliminating
4 Mathematical Problems in EngineeringVe
loci
ty
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
RKIS
LP2LP4
Figure 2 The LC solutions of system (10) with 120576 = 15 obtained bythe modified MIM RK method and LP method respectively Theiteration solution is denoted as IS and the 119896th-order LP approxi-mation as LP119896
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
ISRK
Figure 3 Comparison of the LC solution of system (10) with 120576 = minus1provided by the modified MIM and by RK method respectively
the secular terms Likewise we present the following iterationscheme
1205962
119896minus1
(119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
+ 119891 (119909119896minus1
120596119896minus2
119896minus1
) + 120578119892 (119909119896minus2
)
(23)
In this scheme 119892(119909119896minus2
) can be expanded as a Fourier seriessince 120572
119896minus2
has been determined at the previous iterationLet us consider a van der Pol type oscillator with a non-
smooth function as
+ 119909 + 120576 (1 minus 1199092
) + 120578119892 (119909) = 0 (24)
Velo
city
minus15
minus1
minus05
0
05
1
15
Displacementminus15 minus1 minus05 0 05 1 15
RKISk = 3
ISk = 5
Figure 4The LC solutions of system (21) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896D
ispla
cem
ent
2
15
1
05
0
minus05
minus1
minus15
minus2
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 5 LC solutions of system (24) with (25) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
with
119892 (119909) =
119909 minus 1 119909 ge 1
0 minus1 lt 119909 lt 1
119909 + 1 119909 le minus1
(25)
Figure 5 shows the LC of system (14) with 120576 = 1 and 120578 =05 The 5th-order approximations obtained by the presented
Mathematical Problems in Engineering 5D
ispla
cem
ent
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus25
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 6 LC solutions of system (24) with (26) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
method agree well with the numerical solution when thenonsmooth term is given as
119892 (119909) = sgn (119909) =
1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(26)
The LC can also be obtained very accurate as Figure 6 shows
4 Conclusions
The Mickens iteration method (MIM) has been modifiedso that it is suitable for solving LC solutions of self-excitedsystemswith nonsmooth andor damping nonlinearities Dif-ferent from the routinely-used MIM the modified methoddecouples the unknown frequency from nonlinear termsThis modification simplifies the MIM significantly Numeri-cal examples show the feasibility and validity of the presentedmethod which implies that it could be applicable to morenonlinear dynamical systems especially those with nonlineardamping terms and nonsmooth nonlinearities
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (11002088 11272361 11172333) Doc-toral Program Foundation of Ministry of Education ofChina (20130171110039) Guangdong Province Natural Sci-ence Foundation (S2012040007920 S2013010013802) Fun-damental Research Funds for the Central Universities(13lgzd06) and the Guangdong Province Science and Tech-nology Program (2012A030200011)
References
[1] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[2] R E Mickens ldquoHarmonic balance and iteration calculationsof periodic solutions to 119910
10158401015840
+ 119910minus1
= 0rdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 968ndash972 2007
[3] R EMickens ldquoA generalized iteration procedure for calculatingapproximations to periodic solutions of ldquotruly nonlinear oscilla-torsrdquordquo Journal of Sound andVibration vol 287 no 4-5 pp 1045ndash1051 2005
[4] S Bhattacharjee and J K Bhattacharjee ldquoLindstedt Poincaretechnique applied to molecular potentialsrdquo Journal of Mathe-matical Chemistry vol 50 no 6 pp 1398ndash1410 2012
[5] H Hu and J H Tang ldquoA classical iteration procedure validfor certain strongly nonlinear oscillatorsrdquo Journal of Sound andVibration vol 299 no 1-2 pp 397ndash402 2007
[6] J I Ramos ldquoOn Linstedt-Poincare technique for the quinticDuffing equationrdquo Applied Mathematics and Computation vol193 no 2 pp 303ndash310 2007
[7] F K Yin J Q Song and X Q Cao ldquoCouple of the variationaliteration method and Legendre wavelets for nonlinear partialdifferential equationsrdquo Journal of Applied Mathematics vol2013 Article ID 157956 11 pages 2013
[8] M T Atay and O Kilic ldquoThe semianalytical solutions for stiffsystems of ordinary differential equations by using variationaliteration method and modified variational iteration methodwith comparison to exact solutionsrdquoMathematical Problems inEngineering vol 2013 Article ID 143915 11 pages 2013
[9] A-J Chen ldquoResonance analysis for tilted support spring cou-pled nonlinear packaging system applying variational iterationmethodrdquo Mathematical Problems in Engineering vol 2013Article ID 384251 4 pages 2013
[10] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008
[11] C W Lim and B S Wu ldquoA modified Mickens procedure forcertain non-linear oscillatorsrdquo Journal of Sound and Vibrationvol 257 no 1 pp 202ndash206 2002
[12] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006
[13] H Hu ldquoSolutions of the Duffing-harmonic oscillator by aniteration procedurerdquo Journal of Sound and Vibration vol 298no 1-2 pp 446ndash452 2006
[14] Y M Chen G Meng and J K Liu ldquoA new method for Fourierseries expansions applications in rotor-seal systemsrdquoMechanicsResearch Communications vol 38 no 5 pp 399ndash403 2011
[15] T Pirbodaghi M T Ahmadian and M Fesanghary ldquoOn thehomotopy analysis method for non-linear vibration of beamsrdquoMechanics Research Communications vol 36 no 2 pp 143ndash1482009
[16] Y M Chen and J K Liu ldquoA modified Mickens iterationprocedure for nonlinear oscillatorsrdquo Journal of Sound andVibration vol 314 no 3ndash5 pp 465ndash473 2008
[17] A H Nayfeh Introduction to Perturbation Techniques Wiley-Interscience New York NY USA 1993
[18] W J Ding Self-Excited Vibration Tsinghua University PressBeijing China 2009
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2 Mathematical Problems in Engineering
terms at each iteration stage In order to obtain the LC theMIM [16] can be given as
1205962
119896minus1
(11990910158401015840
119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
minus 119891 (119909119896minus1
120596119896minus1
1199091015840
119896minus1
)
119896 = 1 2
(4)
with the initial conditions being rewritten at each iteration as
119909119896
(0) = 120572119896
1199091015840
119896
(0) = 0 (5)
Note that the coefficient of the first harmonic in 119909119896minus1
remainsstill to be an unknown that is120572
119896minus1
This unknownwill couplewith the unknown frequency120596
119896minus1
which will result in a cou-pled nonlinear term (ie 120596
119896minus1
1199091015840
119896minus1
) in the right side of (4) Ifhigher powers of1199091015840 exist these termswill lead to very compli-cated functions in 120596
119896minus1
and 120572119896minus1
Different from conservativesystems 1205962
119896minus1
can no longer be considered as an independentunknown In order to simplify theMIM therefore amodifiedscheme is proposed as
1205962
119896minus1
(11990910158401015840
119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
minus 119891 (119909119896minus1
120596119896minus2
1199091015840
119896minus1
)
119896 = 1 2
(6)
As 119896 = 1 we choose 120596minus1
= 1205960
In the 119896th iteration 120596119896minus2
is agiven constant that is obtained at the (119896 minus 1)th iteration Thesquare of the unknown frequency that is1205962
119896minus1
can be treatedas an independent parameter because 120596
119896minus1
appears only in1205962
119896minus1
119909119896minus1
According to the initial conditions the startingiteration solution can be chosen as
1199090
(120591) = 1205720
cos 120591 (7)
It is obvious that as long as the series 120596119896
119896 = 1 2 and119909119896
119896 = 1 2 are convergent they must converge to theexact solutions The right-hand side of (6) can be expressedby Fourier series as
1205962
119896minus1
119909119896minus1
minus 119891 (119909119896minus1
120596119896minus2
1199091015840
119896minus1
)
=
120593(119896)
sum
119894=1
[119888119896minus1119894
(1205962
119896minus1
120572119896minus1
) cos (119894120591)
+ 119904119896minus1119894
(1205962
119896minus1
120572119896minus1
) sin (119894120591)]
(8)
where the harmonic coefficients 119888119896minus1119894
(1205962
119896minus1
120572119896minus1
) and119904119896minus1119894
(1205962
119896minus1
120572119896minus1
) are functions in 1205962119896minus1
and 120572119896minus1
Here 120593(119896) isa positive integer denoting the order of the highest harmonicApproximations120596
119896minus1
and 120572119896minus1
are determined by eliminatingthe secular terms that is letting
119888119896minus11
(1205962
119896minus1
120572119896minus1
) = 0 119904119896minus11
(1205962
119896minus1
120572119896minus1
) = 0
119896 = 1 2
(9)
These equations can be solved analytically if 1205962119896minus1
120572119896minus1
isconsidered as an independent unknown They can also benumerically solved by Newton-Raphson method The latteris employed in this study
Different from the existing procedures [16] as 119896 increases1205962
119896minus1
is always an independent unknown in the modifiedMIM Moreover unknown 120596
119896minus1
does not couple with non-linear terms It simplifies the MIM to a large extent as shownlater
3 Numerical Examples
Example 1 (system with nonlinear damping terms) Thevan der Pol equation is chosen to illustrate the previousprocedures more clearly
119909 +119909 + 120576 (1199092
minus 1) = 0 (10)
where 120576 is a given constant As known (10) has a stable LCsolution when 120576 gt 0 while an unstable one when 120576 lt 0
According to themodifiedMIM the corresponding itera-tion scheme is given as
119909119896
+ 1205962
119896minus1
119909119896
= 1205962
119896minus1
119909119896minus1
minus 119909119896minus1
minus 120576 (1199092
119896minus1
minus 1) 119896
119896 = 1 2
(11)
Introducing a new time variable 120591 = 120596119896minus1
119905 at each iterationstage we rewrite (11) as
1205962
119896minus1
(11990910158401015840
119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
minus 119909119896minus1
minus 120576120596119896minus1
(1199092
119896minus1
minus 1) 1199091015840
119896minus1
(12)
where the superscript denotes the derivative with respect to120591 The iteration algorithm begins with an initial solution
1199090
(120591) = 1205720
cos 120591 (13)
Then we obtain the governing equations in 1199091
(120591) as
1205962
0
(11990910158401015840
1
+ 1199091
) = (1205962
0
1205720
minus 1205720
) cos 120591 minus (1205761205720
1205960
minus 120576
1205723
0
1205960
4
) sin 120591
+ 120576
1205723
0
1205960
4
sin 3120591 1199091
(0) = 1205721
1199091015840
1
(0) = 0
(14)
Equating the coefficients of cos 120591 and sin 120591 to zeros results into
1205962
0
1205720
minus 1205720
= 0 1205761205720
1205960
minus 120576
1205723
0
1205960
4
= 0 (15)
which yields that 1205960
= 1 and 1205720
= 2 Substituting them into(16) we have
11990910158401015840
1
+ 1199091
= 2120576 sin 3120591 (16)
Considering initial conditions (14) we can obtain
1199091
= 1205721
cos 120591 + 3120576 sin 1205914
minus
120576 sin 31205914
(17)
where 1205721
is to be determined at the next iteration stage
Mathematical Problems in Engineering 3
Table 1 Comparison of the second-order frequency obtained by IS and LP method with the forth-order approximation obtained by LPmethod when 120576 = 1
120576 120596IS2
120596LP2
120596LP4
|120596IS2
minus 120596LP4
| |120596LP2
minus 120596LP4
|
1 0944799584 093750000 0943033854 176119864 minus 3 553119864 minus 3
05 0984820946 098437500 0984720866 100119864 minus 4 346119864 minus 4
025 0996121460 099609375 0996115367 609119864 minus 6 216119864 minus 5
According to iterative scheme (11) the equation in 1199092
(120591)
is deduced as
1205962
1
(11990910158401015840
2
+ 1199092
) =
9
sum
119894=1
[1198881119894
(1205962
1
1205721
) cos (1198941205961
119905)
+1199041119894
(1205962
1
1205721
) sin (1198941205961
119905)]
(18)
Equate the coefficients of cos 120591 and sin 120591 to zeros
31205962
1
4
minus
151205721
64
minus
1205723
1
4
minus
3
4
= 0
1205721
1205962
1
minus 1205721
minus
1205723
1
8
+
75
128
= 0
(19)
By solving (19) numerically we can determine 1205721
and 12059621
Here we obtain the second-order approximation and expandit as
120596IS2
= 1 minus
1205762
16
+
291205764
2048
+ 119900 (1205764
) (20)
According to [17] the Lindstedt-Poincare (LP) method pro-vides the second- and forth-order approximate frequency120596LP2
= 1minus1205762
16 and120596LP4
= 1minus1205762
16+171205764
3072 respectivelyThe attained approximation agrees well with the 4th-order LPsolution Table 1 indicates that 120596IS
2
is more accurate than 120596LP2
when compared with 120596LP4
Figure 1 shows the comparison of the phase planesbetween iteration solutions (119909
119896
) and numerical result Rapidconvergence of 119909
119896
to the numerical result can be observedNote that all numerical solutions are obtained by the fourth-order Runge-Kutta (RK) integration method When |120576| gt 1the iteration procedure presented by Chen and Liu [16] doesnot converge This is probably the difference between thestarting function (119909
0
= 1205720
cos 119905) and the exact solution istoo large The modified MIM is still effective for |120576| le 15 AsFigure 2 shows the LC solution with 120576 = 15 obtained by thepresented method is in excellent agreement with numericalone It is necessary to point out that the presented method isable to track unstable LCs whereas the RK method is not
Also plotted in Figure 2 are the results provided by the LPmethod [17]The iteration results aremuchmore precise thanthe 2nd and 4th-order LP approximations
It is necessary to point out that the presented method isable to track unstable LCs whereas the RK method is notFigure 3 shows an unstable LC of the van der Pol equationwith 120576 = minus1 obtained by the presentedmethod As shown theRK begins at the LC however the solution curve convergesto the equilibrium
IS2IS4
IS6RK
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus2 minus15 minus1 minus05 0 05 1 15 2
Figure 1The LC solutions of system (10) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896
In order to further demonstrate the merit of the modifiedMIM when applied to problems with nonlinear dampingterms we consider the following self-excited system [18]
119909 +119909 + 120576 [()2
minus 1] + 119891 () = 0 (21)
The nonlinear term contains high powers of that is 119891() =()3 If the originalMIM is employed the algebraic equations
governing 120596 will become very complicated Therefore itis necessary to employ the modified approach Figure 4indicates that the approximations obtained by the presentedmethod converge rapidly to the numerical solution as 119896increased
Example 2 (systemwith nonsmooth nonlinearity) Themod-ified MVIM is further applied to nonsmooth dynamicalsystem expressed as
119909 +119891 (119909 ) + 120578119892 (119909) = 0 (22)
Here 119891(119909 ) is a nonlinear damping term and 119892(119909) is anonsmooth function If substituting the 119909
119896minus1
into 119892(119909) onaccount of119909
119896minus1
contained unknown quantities (120572119896minus1
) so119892(119909)can not be expanded as Fourier progression by numericalintegration To this end (9) cannot be deduced by eliminating
4 Mathematical Problems in EngineeringVe
loci
ty
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
RKIS
LP2LP4
Figure 2 The LC solutions of system (10) with 120576 = 15 obtained bythe modified MIM RK method and LP method respectively Theiteration solution is denoted as IS and the 119896th-order LP approxi-mation as LP119896
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
ISRK
Figure 3 Comparison of the LC solution of system (10) with 120576 = minus1provided by the modified MIM and by RK method respectively
the secular terms Likewise we present the following iterationscheme
1205962
119896minus1
(119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
+ 119891 (119909119896minus1
120596119896minus2
119896minus1
) + 120578119892 (119909119896minus2
)
(23)
In this scheme 119892(119909119896minus2
) can be expanded as a Fourier seriessince 120572
119896minus2
has been determined at the previous iterationLet us consider a van der Pol type oscillator with a non-
smooth function as
+ 119909 + 120576 (1 minus 1199092
) + 120578119892 (119909) = 0 (24)
Velo
city
minus15
minus1
minus05
0
05
1
15
Displacementminus15 minus1 minus05 0 05 1 15
RKISk = 3
ISk = 5
Figure 4The LC solutions of system (21) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896D
ispla
cem
ent
2
15
1
05
0
minus05
minus1
minus15
minus2
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 5 LC solutions of system (24) with (25) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
with
119892 (119909) =
119909 minus 1 119909 ge 1
0 minus1 lt 119909 lt 1
119909 + 1 119909 le minus1
(25)
Figure 5 shows the LC of system (14) with 120576 = 1 and 120578 =05 The 5th-order approximations obtained by the presented
Mathematical Problems in Engineering 5D
ispla
cem
ent
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus25
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 6 LC solutions of system (24) with (26) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
method agree well with the numerical solution when thenonsmooth term is given as
119892 (119909) = sgn (119909) =
1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(26)
The LC can also be obtained very accurate as Figure 6 shows
4 Conclusions
The Mickens iteration method (MIM) has been modifiedso that it is suitable for solving LC solutions of self-excitedsystemswith nonsmooth andor damping nonlinearities Dif-ferent from the routinely-used MIM the modified methoddecouples the unknown frequency from nonlinear termsThis modification simplifies the MIM significantly Numeri-cal examples show the feasibility and validity of the presentedmethod which implies that it could be applicable to morenonlinear dynamical systems especially those with nonlineardamping terms and nonsmooth nonlinearities
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (11002088 11272361 11172333) Doc-toral Program Foundation of Ministry of Education ofChina (20130171110039) Guangdong Province Natural Sci-ence Foundation (S2012040007920 S2013010013802) Fun-damental Research Funds for the Central Universities(13lgzd06) and the Guangdong Province Science and Tech-nology Program (2012A030200011)
References
[1] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[2] R E Mickens ldquoHarmonic balance and iteration calculationsof periodic solutions to 119910
10158401015840
+ 119910minus1
= 0rdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 968ndash972 2007
[3] R EMickens ldquoA generalized iteration procedure for calculatingapproximations to periodic solutions of ldquotruly nonlinear oscilla-torsrdquordquo Journal of Sound andVibration vol 287 no 4-5 pp 1045ndash1051 2005
[4] S Bhattacharjee and J K Bhattacharjee ldquoLindstedt Poincaretechnique applied to molecular potentialsrdquo Journal of Mathe-matical Chemistry vol 50 no 6 pp 1398ndash1410 2012
[5] H Hu and J H Tang ldquoA classical iteration procedure validfor certain strongly nonlinear oscillatorsrdquo Journal of Sound andVibration vol 299 no 1-2 pp 397ndash402 2007
[6] J I Ramos ldquoOn Linstedt-Poincare technique for the quinticDuffing equationrdquo Applied Mathematics and Computation vol193 no 2 pp 303ndash310 2007
[7] F K Yin J Q Song and X Q Cao ldquoCouple of the variationaliteration method and Legendre wavelets for nonlinear partialdifferential equationsrdquo Journal of Applied Mathematics vol2013 Article ID 157956 11 pages 2013
[8] M T Atay and O Kilic ldquoThe semianalytical solutions for stiffsystems of ordinary differential equations by using variationaliteration method and modified variational iteration methodwith comparison to exact solutionsrdquoMathematical Problems inEngineering vol 2013 Article ID 143915 11 pages 2013
[9] A-J Chen ldquoResonance analysis for tilted support spring cou-pled nonlinear packaging system applying variational iterationmethodrdquo Mathematical Problems in Engineering vol 2013Article ID 384251 4 pages 2013
[10] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008
[11] C W Lim and B S Wu ldquoA modified Mickens procedure forcertain non-linear oscillatorsrdquo Journal of Sound and Vibrationvol 257 no 1 pp 202ndash206 2002
[12] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006
[13] H Hu ldquoSolutions of the Duffing-harmonic oscillator by aniteration procedurerdquo Journal of Sound and Vibration vol 298no 1-2 pp 446ndash452 2006
[14] Y M Chen G Meng and J K Liu ldquoA new method for Fourierseries expansions applications in rotor-seal systemsrdquoMechanicsResearch Communications vol 38 no 5 pp 399ndash403 2011
[15] T Pirbodaghi M T Ahmadian and M Fesanghary ldquoOn thehomotopy analysis method for non-linear vibration of beamsrdquoMechanics Research Communications vol 36 no 2 pp 143ndash1482009
[16] Y M Chen and J K Liu ldquoA modified Mickens iterationprocedure for nonlinear oscillatorsrdquo Journal of Sound andVibration vol 314 no 3ndash5 pp 465ndash473 2008
[17] A H Nayfeh Introduction to Perturbation Techniques Wiley-Interscience New York NY USA 1993
[18] W J Ding Self-Excited Vibration Tsinghua University PressBeijing China 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Table 1 Comparison of the second-order frequency obtained by IS and LP method with the forth-order approximation obtained by LPmethod when 120576 = 1
120576 120596IS2
120596LP2
120596LP4
|120596IS2
minus 120596LP4
| |120596LP2
minus 120596LP4
|
1 0944799584 093750000 0943033854 176119864 minus 3 553119864 minus 3
05 0984820946 098437500 0984720866 100119864 minus 4 346119864 minus 4
025 0996121460 099609375 0996115367 609119864 minus 6 216119864 minus 5
According to iterative scheme (11) the equation in 1199092
(120591)
is deduced as
1205962
1
(11990910158401015840
2
+ 1199092
) =
9
sum
119894=1
[1198881119894
(1205962
1
1205721
) cos (1198941205961
119905)
+1199041119894
(1205962
1
1205721
) sin (1198941205961
119905)]
(18)
Equate the coefficients of cos 120591 and sin 120591 to zeros
31205962
1
4
minus
151205721
64
minus
1205723
1
4
minus
3
4
= 0
1205721
1205962
1
minus 1205721
minus
1205723
1
8
+
75
128
= 0
(19)
By solving (19) numerically we can determine 1205721
and 12059621
Here we obtain the second-order approximation and expandit as
120596IS2
= 1 minus
1205762
16
+
291205764
2048
+ 119900 (1205764
) (20)
According to [17] the Lindstedt-Poincare (LP) method pro-vides the second- and forth-order approximate frequency120596LP2
= 1minus1205762
16 and120596LP4
= 1minus1205762
16+171205764
3072 respectivelyThe attained approximation agrees well with the 4th-order LPsolution Table 1 indicates that 120596IS
2
is more accurate than 120596LP2
when compared with 120596LP4
Figure 1 shows the comparison of the phase planesbetween iteration solutions (119909
119896
) and numerical result Rapidconvergence of 119909
119896
to the numerical result can be observedNote that all numerical solutions are obtained by the fourth-order Runge-Kutta (RK) integration method When |120576| gt 1the iteration procedure presented by Chen and Liu [16] doesnot converge This is probably the difference between thestarting function (119909
0
= 1205720
cos 119905) and the exact solution istoo large The modified MIM is still effective for |120576| le 15 AsFigure 2 shows the LC solution with 120576 = 15 obtained by thepresented method is in excellent agreement with numericalone It is necessary to point out that the presented method isable to track unstable LCs whereas the RK method is not
Also plotted in Figure 2 are the results provided by the LPmethod [17]The iteration results aremuchmore precise thanthe 2nd and 4th-order LP approximations
It is necessary to point out that the presented method isable to track unstable LCs whereas the RK method is notFigure 3 shows an unstable LC of the van der Pol equationwith 120576 = minus1 obtained by the presentedmethod As shown theRK begins at the LC however the solution curve convergesto the equilibrium
IS2IS4
IS6RK
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus2 minus15 minus1 minus05 0 05 1 15 2
Figure 1The LC solutions of system (10) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896
In order to further demonstrate the merit of the modifiedMIM when applied to problems with nonlinear dampingterms we consider the following self-excited system [18]
119909 +119909 + 120576 [()2
minus 1] + 119891 () = 0 (21)
The nonlinear term contains high powers of that is 119891() =()3 If the originalMIM is employed the algebraic equations
governing 120596 will become very complicated Therefore itis necessary to employ the modified approach Figure 4indicates that the approximations obtained by the presentedmethod converge rapidly to the numerical solution as 119896increased
Example 2 (systemwith nonsmooth nonlinearity) Themod-ified MVIM is further applied to nonsmooth dynamicalsystem expressed as
119909 +119891 (119909 ) + 120578119892 (119909) = 0 (22)
Here 119891(119909 ) is a nonlinear damping term and 119892(119909) is anonsmooth function If substituting the 119909
119896minus1
into 119892(119909) onaccount of119909
119896minus1
contained unknown quantities (120572119896minus1
) so119892(119909)can not be expanded as Fourier progression by numericalintegration To this end (9) cannot be deduced by eliminating
4 Mathematical Problems in EngineeringVe
loci
ty
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
RKIS
LP2LP4
Figure 2 The LC solutions of system (10) with 120576 = 15 obtained bythe modified MIM RK method and LP method respectively Theiteration solution is denoted as IS and the 119896th-order LP approxi-mation as LP119896
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
ISRK
Figure 3 Comparison of the LC solution of system (10) with 120576 = minus1provided by the modified MIM and by RK method respectively
the secular terms Likewise we present the following iterationscheme
1205962
119896minus1
(119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
+ 119891 (119909119896minus1
120596119896minus2
119896minus1
) + 120578119892 (119909119896minus2
)
(23)
In this scheme 119892(119909119896minus2
) can be expanded as a Fourier seriessince 120572
119896minus2
has been determined at the previous iterationLet us consider a van der Pol type oscillator with a non-
smooth function as
+ 119909 + 120576 (1 minus 1199092
) + 120578119892 (119909) = 0 (24)
Velo
city
minus15
minus1
minus05
0
05
1
15
Displacementminus15 minus1 minus05 0 05 1 15
RKISk = 3
ISk = 5
Figure 4The LC solutions of system (21) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896D
ispla
cem
ent
2
15
1
05
0
minus05
minus1
minus15
minus2
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 5 LC solutions of system (24) with (25) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
with
119892 (119909) =
119909 minus 1 119909 ge 1
0 minus1 lt 119909 lt 1
119909 + 1 119909 le minus1
(25)
Figure 5 shows the LC of system (14) with 120576 = 1 and 120578 =05 The 5th-order approximations obtained by the presented
Mathematical Problems in Engineering 5D
ispla
cem
ent
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus25
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 6 LC solutions of system (24) with (26) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
method agree well with the numerical solution when thenonsmooth term is given as
119892 (119909) = sgn (119909) =
1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(26)
The LC can also be obtained very accurate as Figure 6 shows
4 Conclusions
The Mickens iteration method (MIM) has been modifiedso that it is suitable for solving LC solutions of self-excitedsystemswith nonsmooth andor damping nonlinearities Dif-ferent from the routinely-used MIM the modified methoddecouples the unknown frequency from nonlinear termsThis modification simplifies the MIM significantly Numeri-cal examples show the feasibility and validity of the presentedmethod which implies that it could be applicable to morenonlinear dynamical systems especially those with nonlineardamping terms and nonsmooth nonlinearities
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (11002088 11272361 11172333) Doc-toral Program Foundation of Ministry of Education ofChina (20130171110039) Guangdong Province Natural Sci-ence Foundation (S2012040007920 S2013010013802) Fun-damental Research Funds for the Central Universities(13lgzd06) and the Guangdong Province Science and Tech-nology Program (2012A030200011)
References
[1] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[2] R E Mickens ldquoHarmonic balance and iteration calculationsof periodic solutions to 119910
10158401015840
+ 119910minus1
= 0rdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 968ndash972 2007
[3] R EMickens ldquoA generalized iteration procedure for calculatingapproximations to periodic solutions of ldquotruly nonlinear oscilla-torsrdquordquo Journal of Sound andVibration vol 287 no 4-5 pp 1045ndash1051 2005
[4] S Bhattacharjee and J K Bhattacharjee ldquoLindstedt Poincaretechnique applied to molecular potentialsrdquo Journal of Mathe-matical Chemistry vol 50 no 6 pp 1398ndash1410 2012
[5] H Hu and J H Tang ldquoA classical iteration procedure validfor certain strongly nonlinear oscillatorsrdquo Journal of Sound andVibration vol 299 no 1-2 pp 397ndash402 2007
[6] J I Ramos ldquoOn Linstedt-Poincare technique for the quinticDuffing equationrdquo Applied Mathematics and Computation vol193 no 2 pp 303ndash310 2007
[7] F K Yin J Q Song and X Q Cao ldquoCouple of the variationaliteration method and Legendre wavelets for nonlinear partialdifferential equationsrdquo Journal of Applied Mathematics vol2013 Article ID 157956 11 pages 2013
[8] M T Atay and O Kilic ldquoThe semianalytical solutions for stiffsystems of ordinary differential equations by using variationaliteration method and modified variational iteration methodwith comparison to exact solutionsrdquoMathematical Problems inEngineering vol 2013 Article ID 143915 11 pages 2013
[9] A-J Chen ldquoResonance analysis for tilted support spring cou-pled nonlinear packaging system applying variational iterationmethodrdquo Mathematical Problems in Engineering vol 2013Article ID 384251 4 pages 2013
[10] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008
[11] C W Lim and B S Wu ldquoA modified Mickens procedure forcertain non-linear oscillatorsrdquo Journal of Sound and Vibrationvol 257 no 1 pp 202ndash206 2002
[12] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006
[13] H Hu ldquoSolutions of the Duffing-harmonic oscillator by aniteration procedurerdquo Journal of Sound and Vibration vol 298no 1-2 pp 446ndash452 2006
[14] Y M Chen G Meng and J K Liu ldquoA new method for Fourierseries expansions applications in rotor-seal systemsrdquoMechanicsResearch Communications vol 38 no 5 pp 399ndash403 2011
[15] T Pirbodaghi M T Ahmadian and M Fesanghary ldquoOn thehomotopy analysis method for non-linear vibration of beamsrdquoMechanics Research Communications vol 36 no 2 pp 143ndash1482009
[16] Y M Chen and J K Liu ldquoA modified Mickens iterationprocedure for nonlinear oscillatorsrdquo Journal of Sound andVibration vol 314 no 3ndash5 pp 465ndash473 2008
[17] A H Nayfeh Introduction to Perturbation Techniques Wiley-Interscience New York NY USA 1993
[18] W J Ding Self-Excited Vibration Tsinghua University PressBeijing China 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in EngineeringVe
loci
ty
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
RKIS
LP2LP4
Figure 2 The LC solutions of system (10) with 120576 = 15 obtained bythe modified MIM RK method and LP method respectively Theiteration solution is denoted as IS and the 119896th-order LP approxi-mation as LP119896
Velo
city
3
2
1
0
minus1
minus2
minus3
Displacementminus25 minus2 minus15 minus1 minus05 0 05 1 15 2 25
ISRK
Figure 3 Comparison of the LC solution of system (10) with 120576 = minus1provided by the modified MIM and by RK method respectively
the secular terms Likewise we present the following iterationscheme
1205962
119896minus1
(119896
+ 119909119896
) = 1205962
119896minus1
119909119896minus1
+ 119891 (119909119896minus1
120596119896minus2
119896minus1
) + 120578119892 (119909119896minus2
)
(23)
In this scheme 119892(119909119896minus2
) can be expanded as a Fourier seriessince 120572
119896minus2
has been determined at the previous iterationLet us consider a van der Pol type oscillator with a non-
smooth function as
+ 119909 + 120576 (1 minus 1199092
) + 120578119892 (119909) = 0 (24)
Velo
city
minus15
minus1
minus05
0
05
1
15
Displacementminus15 minus1 minus05 0 05 1 15
RKISk = 3
ISk = 5
Figure 4The LC solutions of system (21) with 120576 = 1 obtained by themodifiedMIM and RKmethod respectivelyThe 119896th-order solution(119909119896
) is represented by IS119896D
ispla
cem
ent
2
15
1
05
0
minus05
minus1
minus15
minus2
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 5 LC solutions of system (24) with (25) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
with
119892 (119909) =
119909 minus 1 119909 ge 1
0 minus1 lt 119909 lt 1
119909 + 1 119909 le minus1
(25)
Figure 5 shows the LC of system (14) with 120576 = 1 and 120578 =05 The 5th-order approximations obtained by the presented
Mathematical Problems in Engineering 5D
ispla
cem
ent
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus25
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 6 LC solutions of system (24) with (26) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
method agree well with the numerical solution when thenonsmooth term is given as
119892 (119909) = sgn (119909) =
1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(26)
The LC can also be obtained very accurate as Figure 6 shows
4 Conclusions
The Mickens iteration method (MIM) has been modifiedso that it is suitable for solving LC solutions of self-excitedsystemswith nonsmooth andor damping nonlinearities Dif-ferent from the routinely-used MIM the modified methoddecouples the unknown frequency from nonlinear termsThis modification simplifies the MIM significantly Numeri-cal examples show the feasibility and validity of the presentedmethod which implies that it could be applicable to morenonlinear dynamical systems especially those with nonlineardamping terms and nonsmooth nonlinearities
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (11002088 11272361 11172333) Doc-toral Program Foundation of Ministry of Education ofChina (20130171110039) Guangdong Province Natural Sci-ence Foundation (S2012040007920 S2013010013802) Fun-damental Research Funds for the Central Universities(13lgzd06) and the Guangdong Province Science and Tech-nology Program (2012A030200011)
References
[1] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[2] R E Mickens ldquoHarmonic balance and iteration calculationsof periodic solutions to 119910
10158401015840
+ 119910minus1
= 0rdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 968ndash972 2007
[3] R EMickens ldquoA generalized iteration procedure for calculatingapproximations to periodic solutions of ldquotruly nonlinear oscilla-torsrdquordquo Journal of Sound andVibration vol 287 no 4-5 pp 1045ndash1051 2005
[4] S Bhattacharjee and J K Bhattacharjee ldquoLindstedt Poincaretechnique applied to molecular potentialsrdquo Journal of Mathe-matical Chemistry vol 50 no 6 pp 1398ndash1410 2012
[5] H Hu and J H Tang ldquoA classical iteration procedure validfor certain strongly nonlinear oscillatorsrdquo Journal of Sound andVibration vol 299 no 1-2 pp 397ndash402 2007
[6] J I Ramos ldquoOn Linstedt-Poincare technique for the quinticDuffing equationrdquo Applied Mathematics and Computation vol193 no 2 pp 303ndash310 2007
[7] F K Yin J Q Song and X Q Cao ldquoCouple of the variationaliteration method and Legendre wavelets for nonlinear partialdifferential equationsrdquo Journal of Applied Mathematics vol2013 Article ID 157956 11 pages 2013
[8] M T Atay and O Kilic ldquoThe semianalytical solutions for stiffsystems of ordinary differential equations by using variationaliteration method and modified variational iteration methodwith comparison to exact solutionsrdquoMathematical Problems inEngineering vol 2013 Article ID 143915 11 pages 2013
[9] A-J Chen ldquoResonance analysis for tilted support spring cou-pled nonlinear packaging system applying variational iterationmethodrdquo Mathematical Problems in Engineering vol 2013Article ID 384251 4 pages 2013
[10] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008
[11] C W Lim and B S Wu ldquoA modified Mickens procedure forcertain non-linear oscillatorsrdquo Journal of Sound and Vibrationvol 257 no 1 pp 202ndash206 2002
[12] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006
[13] H Hu ldquoSolutions of the Duffing-harmonic oscillator by aniteration procedurerdquo Journal of Sound and Vibration vol 298no 1-2 pp 446ndash452 2006
[14] Y M Chen G Meng and J K Liu ldquoA new method for Fourierseries expansions applications in rotor-seal systemsrdquoMechanicsResearch Communications vol 38 no 5 pp 399ndash403 2011
[15] T Pirbodaghi M T Ahmadian and M Fesanghary ldquoOn thehomotopy analysis method for non-linear vibration of beamsrdquoMechanics Research Communications vol 36 no 2 pp 143ndash1482009
[16] Y M Chen and J K Liu ldquoA modified Mickens iterationprocedure for nonlinear oscillatorsrdquo Journal of Sound andVibration vol 314 no 3ndash5 pp 465ndash473 2008
[17] A H Nayfeh Introduction to Perturbation Techniques Wiley-Interscience New York NY USA 1993
[18] W J Ding Self-Excited Vibration Tsinghua University PressBeijing China 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5D
ispla
cem
ent
25
2
15
1
05
0
minus05
minus1
minus15
minus2
minus25
Velocityminus3 minus2 minus1 0 1 2 3
R-KPresent method
Figure 6 LC solutions of system (24) with (26) (120576 = 1 120578 = 05)provided by the modified MIM and by RK method respectively
method agree well with the numerical solution when thenonsmooth term is given as
119892 (119909) = sgn (119909) =
1 119909 gt 0
0 119909 = 0
minus1 119909 lt 0
(26)
The LC can also be obtained very accurate as Figure 6 shows
4 Conclusions
The Mickens iteration method (MIM) has been modifiedso that it is suitable for solving LC solutions of self-excitedsystemswith nonsmooth andor damping nonlinearities Dif-ferent from the routinely-used MIM the modified methoddecouples the unknown frequency from nonlinear termsThis modification simplifies the MIM significantly Numeri-cal examples show the feasibility and validity of the presentedmethod which implies that it could be applicable to morenonlinear dynamical systems especially those with nonlineardamping terms and nonsmooth nonlinearities
Acknowledgment
This work is supported by the National Natural ScienceFoundation of China (11002088 11272361 11172333) Doc-toral Program Foundation of Ministry of Education ofChina (20130171110039) Guangdong Province Natural Sci-ence Foundation (S2012040007920 S2013010013802) Fun-damental Research Funds for the Central Universities(13lgzd06) and the Guangdong Province Science and Tech-nology Program (2012A030200011)
References
[1] R E Mickens ldquoIteration procedure for determining approx-imate solutions to nonlinear oscillator equationsrdquo Journal ofSound and Vibration vol 116 no 1 pp 185ndash187 1987
[2] R E Mickens ldquoHarmonic balance and iteration calculationsof periodic solutions to 119910
10158401015840
+ 119910minus1
= 0rdquo Journal of Sound andVibration vol 306 no 3ndash5 pp 968ndash972 2007
[3] R EMickens ldquoA generalized iteration procedure for calculatingapproximations to periodic solutions of ldquotruly nonlinear oscilla-torsrdquordquo Journal of Sound andVibration vol 287 no 4-5 pp 1045ndash1051 2005
[4] S Bhattacharjee and J K Bhattacharjee ldquoLindstedt Poincaretechnique applied to molecular potentialsrdquo Journal of Mathe-matical Chemistry vol 50 no 6 pp 1398ndash1410 2012
[5] H Hu and J H Tang ldquoA classical iteration procedure validfor certain strongly nonlinear oscillatorsrdquo Journal of Sound andVibration vol 299 no 1-2 pp 397ndash402 2007
[6] J I Ramos ldquoOn Linstedt-Poincare technique for the quinticDuffing equationrdquo Applied Mathematics and Computation vol193 no 2 pp 303ndash310 2007
[7] F K Yin J Q Song and X Q Cao ldquoCouple of the variationaliteration method and Legendre wavelets for nonlinear partialdifferential equationsrdquo Journal of Applied Mathematics vol2013 Article ID 157956 11 pages 2013
[8] M T Atay and O Kilic ldquoThe semianalytical solutions for stiffsystems of ordinary differential equations by using variationaliteration method and modified variational iteration methodwith comparison to exact solutionsrdquoMathematical Problems inEngineering vol 2013 Article ID 143915 11 pages 2013
[9] A-J Chen ldquoResonance analysis for tilted support spring cou-pled nonlinear packaging system applying variational iterationmethodrdquo Mathematical Problems in Engineering vol 2013Article ID 384251 4 pages 2013
[10] V Marinca N Herisanu and C Bota ldquoApplication of the vari-ational iteration method to some nonlinear one-dimensionaloscillationsrdquoMeccanica vol 43 no 1 pp 75ndash79 2008
[11] C W Lim and B S Wu ldquoA modified Mickens procedure forcertain non-linear oscillatorsrdquo Journal of Sound and Vibrationvol 257 no 1 pp 202ndash206 2002
[12] VMarinca andN Herisanu ldquoAmodified iteration perturbationmethod for somenonlinear oscillation problemsrdquoActaMechan-ica vol 184 no 1ndash4 pp 231ndash242 2006
[13] H Hu ldquoSolutions of the Duffing-harmonic oscillator by aniteration procedurerdquo Journal of Sound and Vibration vol 298no 1-2 pp 446ndash452 2006
[14] Y M Chen G Meng and J K Liu ldquoA new method for Fourierseries expansions applications in rotor-seal systemsrdquoMechanicsResearch Communications vol 38 no 5 pp 399ndash403 2011
[15] T Pirbodaghi M T Ahmadian and M Fesanghary ldquoOn thehomotopy analysis method for non-linear vibration of beamsrdquoMechanics Research Communications vol 36 no 2 pp 143ndash1482009
[16] Y M Chen and J K Liu ldquoA modified Mickens iterationprocedure for nonlinear oscillatorsrdquo Journal of Sound andVibration vol 314 no 3ndash5 pp 465ndash473 2008
[17] A H Nayfeh Introduction to Perturbation Techniques Wiley-Interscience New York NY USA 1993
[18] W J Ding Self-Excited Vibration Tsinghua University PressBeijing China 2009
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of