JOURNAL OF SOUND AND
VIBRATION
Journal of Sound and Vibration 299(2007)331–338
Short Communication
Solution of a mixed parity nonlinear oscillator:
Harmonic balance
H.Hu ?
School of Civil Engineering,Hunan University of Science and Technology,Xiangtan,Hunan 411201,PR China
Received 2March 2006;received in revised form 4June 2006;accepted 8June 2006
Available online 22August 2006
Abstract
The method of harmonic balance is used to calculate ?rst-order approximations to the periodic solutions of a mixed parity nonlinear oscillator.First,the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction.Then the two auxiliary equations,where the restoring force functions are odd,are solved by using a ?rst harmonic term (without a constant).Therefore,we obtain the two approximate solutions to the mixed parity nonlinear oscillator.One is expressed in terms of the exact amplitude in the negative direction,the other in terms of the approximate
amplitude.These solutions are more accurate than the second approximate solution obtained by the Lindstedt–Poincare
method for large amplitudes.
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1.Introduction
A differential equation modeling many of the important features of nonlinear oscillations is [1]
d 2x
d ˉt
ta 1x ta 2x 2ta 3x 3?0,(1)
where ea 1;a 2;a 3Tare non-negative parameters.The method of harmonic balance is a useful technique [2,3]for
solving nonlinear oscillatory problems.It has been found to work well when all terms in the dependent variable have odd parity [4].Nayfeh and Mook [1,pp.59–61]have cautioned against use of the method when terms of mixed parity are involved,pointing out that for full consistency,a second harmonic term (as well as a constant)must be taken into account in the solution expression.The main purpose of this communication is to use the ?rst-order harmonic balance method to determine analytical approximations to the periodic solutions of mixed parity nonlinear oscillators by resorting to two auxiliary equations.This work represents a companion to previous work on the quadratic nonlinear oscillator [5]:
€x
tx t x 2?0.(2)
For convenience,de?ning x ?????????????a 1=a 3p y and ˉt ?t =?????
a 1p ,Eq.(1)is reduced to the following equation:
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0022-460X/$-see front matter r 2006Elsevier Ltd.All rights reserved.doi:10.1016/j.jsv.2006.06.046
?Tel.:+8607328290701.
E-mail address:huihuxt@https://www.doczj.com/doc/662968577.html,.
€y
ty t y 2ty 3?0,(3)
where ?a 2=?????????
a 1a 3p and overdots denote differentiation with respect to time,t .Let the initial conditions be
y e0T?A 40;
_y
e0T?0.(4)
By the use of scaling [6,pp.398–400]and the Lindstedt–Poincare
perturbation method,Mickens [2,pp.68–71]obtained the second approximation for Eqs.(3)and (4):
y M
?y M ey ;A T?A cos y t A 2
6
eà3t2cos y tcos 2y TtA 33à 2t174 2à27288 cos y t 23
cos 2y t2 2t3
32 cos 3y ,
e5T
where
y ?o M t ;
o M ?1te9à10 2TA 2=24;
0o A 51.
(6a,b,c)
(Eqs.(5)and (6)can be obtained by letting a ? and b ?1in Eqs.(2.105)and (2.106)of Ref.[2].)The
corresponding approximate period of the oscillation is
T M ?2p =o M ?2p ?1te9à10 2TA 2=24 à1.
(7)
We will see that the ?rst approximations obtained in this paper are more accurate than Mickens’s results for large amplitudes.
2.The amplitude B in the negative direction Eq.(3)can be rewritten as
_y
d _y tey t y 2ty 3Td y ?0.(8)
Integrating of this equation gives the ?rst integral
_y
22ty 22t y 33ty 44
?h ,(9)where h is a constant of integration.The behavior of mixed parity nonlinear oscillators is different for positive and negative directions.Assume that the system oscillates between asymmetric limits [àB ,A ](B 40).Noting
that when y ?A and y ?àB ,the corresponding _y
?0,we have from Eq.(9)B 22à B 33tB 44?A 22t A 33tA 4
4
.(10)Solving for B with MATLAB gives the following exact solution:
B e ?A 3t4 9t
C 1=39à1à8 227t2 A 9tA 23
6
C ,
(11)
where
C ?270A 2e tA Tt18A e27à4 2Tt4 e16 2à81Tt54?54e1t A 3t A 5Tt3 2e3A 4à4T
t108A 2eA 2à 2Tt9A 2e15t3A 4Tà8 A e9t 2A 2Tt16 3A e1t A T 1=2.
e12T
We now seek an approximate expression for B .Actually,the smaller is,the more ‘‘cubic’’Eq.(3)is.In this
case,B is close to A even though A is not small.Therefore,we have
B ?A tD B ,
(13)where D B !0when !0.Let
h eB ; T?B 22à B 33tB 4
4
;h B eB ; T?
q h
q B
.(14a,b)
H.Hu /Journal of Sound and Vibration 299(2007)331–338
332
Substituting Eq.(13)into Eq.(14a)results in
h eB ; T%h eA ; Tth B eA ; TD B ?A 22à A 33tA 4
4
teA à A 2tA 3TD B .(15)
From Eqs.(10)and (15),we obtain
D B ?2 A 2
3e1à A tA T
.
(16)
Thus,the approximate expression for B is
B a ?B ?A tD B ?A t
2 A 2
3e1à A tA 2T
.(17)We also have an approximate expression from Eq.(5)
B M ?y M ep ;A T ?A 1t2 A 31t2 A
3
.(18)
For comparison,the exact amplitude B e and the approximate amplitudes computed by Eqs.(17)and (18),
respectively,are listed for ?1in Table 1.Table 1shows that B M is somewhat more accurate than B a if A p 0:4,but when A X 0:6B a is more accurate than B M .3.Solutions of the two auxiliary equations
Based on the discussion in Ref.[5],we ?rst consider the following auxiliary equation:
€y
ty t y 2sgn ey Tty 3?0;y e0T?A ;
_y
e0T?0,(19)
where sgn ey Tis the sign function,equal to +1if y 40,0if y ?0and à1if y o 0.Let o A be the angular
frequency of Eq.(19).Then substituting
y ?A cos o A t ?A cos y
(20)
into Eq.(19)and taking into account that [5]
eA cos y T2
sgn eA cos y T?A cos y j j A cos y ?8A 2
3p
cos y thigher order harmonics,(21)
we obtain
ào 2A t1t8 A 3p t3A 24
A cos y thigher order harmonics ?0.(22)
Table 1
Comparison of the approximate amplitudes with the exact amplitude for ?1A B e
B M
B a
0.10.10710.10710.10730.20.22990.23020.23170.40.52290.53510.54040.50.68460.72220.72220.60.84640.93600.91580.8 1.1511 1.4542 1.30791.0 1.4257 2.1111 1.66672.0 2.57898.2222 2.88895.0 5.648877.2222 5.793710.010.6618521.1111
10.7326100.0
100.6666
4.5121?105
100.6733
H.Hu /Journal of Sound and Vibration 299(2007)331–338
333
Setting the coef?cient of cos o A t equal to zero and solving for o A yields
o A ???????????????????????????????
1t8 A 3p t3A 2
4s .(23)
Therefore,a ?rst approximation to the periodic solution of Eq.(19)is y ?A cos ??????????????????????????????
1t8 A 3p t3A 2
4s t .
(24)
The corresponding approximate period of the oscillation is
T A ?2p A ?2p 1t8 A t3A 2
à1=2
.
(25)
Now we consider the second auxiliary equation:
€y
ty à y 2sgn ey Tty 3?0;y e0T?B ;
_y
e0T?0.(26)The ?rst approximation to Eq.(26)is assumed to be
y ?B cos o B t ?B cos y ,
(27)
where o B is the angular frequency of Eq.(26).Substituting Eq.(27)into Eq.(26)and noting the relation given
in Eq.(21),we have
ào 2
B t1à8 B 3p t3B 24
B cos y thigher order harmonics ?0.(28)Appling the harmonic balance technique gives
o B ???????????????????????????????
1à8 B 3p t3B 2
4
s .(29)
Then from Eq.(27)we obtain
y ?B cos ??????????????????????????????1à8 B 3p t3B 2
4
s t .
(30)
The corresponding approximate period of the oscillation to Eq.(26)is
T B ?2p B ?2p 1à8 B t3B 2
à1=2
.
(31)
4.Results and discussion
The ?rst approximate period T 1and the corresponding periodic solution y 1et Tto Eqs.(3)and (4)are,
respectively,
T 1?T A tT B 2?p 1t8 A 3p t3A 24 à1=2t1à8 B 3p t3B
24
à1=2
"#,(32)y 1et T?A cos o A t ;
0p t p
T A
4
,(33a)y 1et T?B cos o B
t àT A 4tT B
4
;T A 4p t p T A 4tT B
2
,(33b)
H.Hu /Journal of Sound and Vibration 299(2007)331–338
334
y 1et T?A cos o A
t tT A 2àT B
2
;T A 4tT B
2
p t p T 1.(33c)
If B ?B e (B a )in Eqs.(32)and (33),then we will use T 1Be (T 1Ba )and y 1Be (y 1Ba )to denote T 1and y 1,
respectively.
The exact period T e to Eqs.(3)and (4)is
T e ?Z A 02d x ???????????????????????????????????????????????????????????????????????A 2àx 2t2 eA 3àx 3Tt1eA 4àx 4Tq tZ B
02d x
???????????????????????????????????????????????????????????????????????B 2àx 2à2 eB 3àx 3Tt1eB 4àx 4Tq ,(34)
where B is given,in terms of A ,in Eq.(11).
For comparison,the exact period T e obtained by integrating Eq.(34)and the approximate periods T M (Eq.(7)),T 1Be (B ?B e in Eq.(32))and T 1Ba (B ?B a in Eq.(32))are listed in Table 2for ?1.The percentage errors are de?ned as 100?T M eT 1Be ;T 1Ba TàT e =T e .Table 2indicates that there is not much difference between the exact period and the approximate periods if A p 0:2.But T 1Be and T 1Ba are more accurate than T M when A X 0:4.T M is not valid for large amplitudes.
Comparisons between the numerical solution y Num of Eqs.(3)and (4)with the approximate solutions y M (Eq.(5)),y 1Be and y 1Ba are shown in Figs.1and 2for the time in one exact period ( ?1).Fig.1shows that y 1Be and y 1Ba are more accurate than y M .When A X 1y M is not valid.Therefore,y M does not appear in Fig.2.5.Conclusions
A mixed parity nonlinear oscillator modeled by Eqs.(3)and (4)has been attacked by the ?rst-order harmonic balance method.First,the amplitude
B in the negative direction is expressed in terms of the amplitude A .Then the method is applied to the two auxiliary equations (19)and (26),where the restoring force functions are odd.The ?rst approximate periods T 1Be and T 1Ba are more accurate than the second
approximate period T M obtained by the Lindstedt–Poincare
method when A X 0:2,and the approximate solutions y 1Be and y 1Ba are more accurate than the second approximate solution y M for large amplitudes.Furthermore,if we only use the approximate amplitude B a ,the ?rst-order harmonic balance is much simpler than the second-order perturbation.In fact,if we obtain the approximate solutions to Eq.(19),then we do not need to actually solve Eq.(26).Letting ?à and A ?B in Eqs.(23),(24)and (25)gives Eqs.(29),(30)and (31)immediately.Obviously,the approach in this paper can be applied to other types of mixed parity nonlinear oscillators.
Although our result complements the analysis of Mickens [2,pp.68–71]for small amplitudes,there is clearly room for improvement of the accuracy of the approximate solutions y 1Be and y 1Ba ,especially when A X 1.It is very dif?cult to use the method of harmonic balance to construct higher-order analytical approximations because it requires analytical solutions of sets of complicated nonlinear algebraic equations.
Table 2
Comparison of approximate periods with the corresponding exact period to Eqs.(3)and (4)for ?1A T e
T M (%error)T 1Be (%error)T 1Ba (%error)0.1 6.28559 6.28580(0.0035) 6.28526(à0.0052) 6.28554(à0.00007)0.2 6.28794 6.29367(0.0911) 6.28673(à0.0193) 6.28865(0.0113)0.4 6.20413 6.32535(1.9539) 6.20119(à0.0475) 6.20331(à0.0132)0.5 6.05759 6.34932(4.8159) 6.05370(à0.0642) 6.03580(à0.3597)0.6 5.83368 6.37887(9.3455) 5.82692(à0.1160) 5.75941(à1.2732)0.8 5.27286 6.45533(22.4256) 5.25371(à0.3631) 5.03897(à4.4357)1.0 4.72140 6.55637(38.8649) 4.68905(à0.6853) 4.38081(à7.2137)2.0 2.975777.53982(153.3738) 2.92793(à1.6076) 2.74902(à7.6201)5.0 1.36965à150.79645(—) 1.34144(à2.0595) 1.32275(à3.4245)10.00.71463à1.98416(—)0.69932(à2.1420)0.69686(à2.4868)100.0
0.073913
à0.015116
(—)
0.072308
(à2.1720)
0.072305
(à2.1752)
H.Hu /Journal of Sound and Vibration 299(2007)331–338
335
Thus,in this paper we restrict our investigation by the ?rst harmonic only.We will apply a modi?ed iteration procedure [7]to Eqs.(3)and (4)for second-order approximations in another paper.
The mixed parity nonlinear differential equation (1)occurs quite widely in a variety of engineering applications such as the nonlinear free vibrations of laminated plates [2,8–10].If a 2o 0and a 3o 0,then we may let a 2?àb 2and a 3?àb 3.Thus,de?ning x ?????????????a 1=b 3p y and ˉt ?t =?????a 1p ,Eq.(1)is reduced to
€y
ty à y 2ày 3?0,(35)
where ?b 2=?????????
a 1
b 3p .Obviously,the present method can also be used to deal with this equation.
The main difference between this paper and Ref.[5]is that here we present a method for approximate expressions for B .When the restoring force function is a combination of quadratic,cubic and quintic terms,there is no exact expression for B .But we can still use the method in this paper to obtain approximate solutions for B .
Because the behavior of quadratic or mixed parity oscillators is different for positive and negative directions,quadratic and mixed parity oscillators are more complicated than cubic oscillators.One limitation of the results developed in this paper is the requirement that the initial conditions should be y e0T?A 40and _y
e0T?0.It needs further research to solve Eq.(3)with arbitrary initial conditions by using the present method.
-1.5
-1
-0.5
0.5
1
Time t
D i s p l a c e m e n t y
1
2
3456
-0.5
-0.4-0.3-0.2-0.100.10.20.3
0.4Time t
D i s p l a c e m e n t y
1
2
34
5
6
-0.2
-0.15-0.1-0.0500.050.10.150.2Time t D i s p l a c e m e n t y
(a)(b)
(c)
https://www.doczj.com/doc/662968577.html,parison of the approximate solutions y M (dash-dot curve),y 1Be (dotted curve)and y 1Ba (dashed curve)with the numerical solution y Num (solid curve)for ?1:(a)A ?0:2;(b)A ?0:4;(c)A ?1:0.
H.Hu /Journal of Sound and Vibration 299(2007)331–338
336
In this paper,the approximate solution to Eqs.(3)and (4)is obtained with the aid of the two auxiliary equations (19)and (26).In general,consider the nonlinear oscillator modeled by
€y
tf odd ey Ttf even ey T?0;y e0T?A 40;
_y
e0T?0,(36)
where
f odd eày T?àf odd ey T;
f even eày T?f even ey T.
(37a,b)
When y X 0,Eq.(36)is equivalent to
€y
tf odd ey Ttsgn ey Tf even ey T?0;y e0T?A 40;
_y
e0T?0,(38)
in which the restoring force function f ey T?f odd ey Ttsgn ey Tf even ey Tis odd.If y o 0,then substituting y ?àˉ
y (ˉy 40)into Eq.(36)gives
€ˉy tf odd eˉy Tàf even eˉy T?0;
ˉy e0T?B 40;
_ˉy e0T?0.
(39)
For ˉy 40,Eq.(39)is also equivalent to
€ˉy tf odd eˉy Tàsgn eˉy Tf even eˉy T?0;
ˉy e0T?B 40;
_ˉy e0T?0,
(40)
-5-4-3-2-1012345Time t
D i s p l a c e
m e n t y
0.1
0.2
0.3
0.40.50.60.7
-10
-8-6-4-202468
10Time t
D i s p l a c e m e n t y
-100
-80-60-40-20020406080100Time t
D i s p l a c e m e n t y
(a)(b)
(c)
https://www.doczj.com/doc/662968577.html,parison of the approximate solutions y 1Be (dotted curve)and y 1Ba (dashed curve)with the numerical solution y Num (solid curve)for ?1:(a)A ?5:0;(b)A ?10:0;(c)A ?100:0.
H.Hu /Journal of Sound and Vibration 299(2007)331–338
337
where the restoring force function f eˉy T?f odd eˉy Tàsgn eˉy Tf even eˉy Tis also odd.Like Eq.(10),the relation between A and B is described by the following equation:Z A 0
f odd ey Td y tZ A 0
f even ey Td y ?Z àB 0
f odd ey Td y tZ àB 0
f even ey Td y ?Z B 0
f odd ey Td y àZ B
f even ey Td y :(41)
Obviously,the two auxiliary equations of Eq.(36)are Eqs.(38)and (40).Acknowledgements
This work was supported by Scienti?c Research Fund of Hunan Provincial Education Department (Project no.04C245).References
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