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To cite this version:
Tien Hoang, Denis Duhamel, Gilles Forêt, Honoré P. Yin, Pierre Argoul. Frequency dependent it-
eration method for forced nonlinear oscillators. Applied Mathematical Modelling, Elsevier, 2017, 42,
pp.441 - 448. �10.1016/j.apm.2016.10.012�. �hal-01695465�
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Frequency dependent iteration method for forced nonlinear oscillators
T. Hoanga,∗, D. Duhamela, G. Foreta, H.P. Yina, P. Argoula
aUniversit´e Paris-Est, Navier (UMR 8205 ENPC-IFSTTAR-CNRS), Ecole Nationale des Ponts et Chauss´ees, 6 et 8 Avenue Blaise Pascal, Cit´e Descartes, Champs-sur-Marne,77455
Marne-la-Vall´ee Cedex 2, France
Abstract
A new iteration method for nonlinear vibrations has been developed by de- composing the periodic solution in two parts corresponding to low and high harmonics. For a nonlinear forced oscillator, the iteration schema is proposed with different formulations for these two parts. Then, the schema is deduced by using the harmonic balance technique. This method has proven to converge to the periodic solutions provided that a convergence condition is satisfied. The convergence is also demonstrated analytically for linear oscillators. Moreover, the new method has been applied to Duffing oscillators as an example. The numerical results show that each iteration schema converges in a domain of the excitation frequency and it can converge to different solutions of the nonlinear oscillator.
Keywords: Forced nonlinear oscillator, harmonic balance method, iteration procedure, Duffing oscillator.
1. Introduction
The harmonic balance method (HBM) has many applications in nonlinear dynamics. The original principle of this method is to express the periodic so- lution in terms of Fourier series with limited numbers of harmonics and to substitute this expression to the dynamic equation in order to find out balance
5
∗Corresponding author
Email address: tien.hoang@enpc.fr (T. Hoang)
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of all harmonics. A typical difficulty of this method is linked to the dependence of the quality of the approach on the way to carry enough terms in the solution and check the order of the coefficients for all the neglected harmonics [1, 2].
Some alternative techniques based on HBM have been developed in order to solve this difficulty. The Galerkin procedure [3–6] was used to calculate
10
incrementally the Fourier coefficients of the classical HBM. Another technique called the iteration procedures were presented by Mickens [7, 8] and applied to different kinds of non-linearities [9, 10]. The high dimensional harmonic balance (HDHB) was developed by Hall et al. [11]. This method is based on a constant Fourier transformation matrix which is an approximation of the matrix related
15
to the non-linearity deduced from the classical HBM. This method has some advantages in calculation in case of high dimension, but it can lead to non- physical solutions [12–14]. Recently, other new methods based on HBM were developed by Cochelin et al. [15–17] and Ju et al. [18–20].
In this paper, we propose a new method for forced nonlinear oscillators based
20
on the iteration procedures and HBM. The existing methods propose to use the same iteration schema for all harmonics. This new method provides different schemas which adapt to the frequency of the force. By decomposing the pe- riodical solution in low and high harmonic components, an iteration schema is presented and then developed by using HBM. The schemas are proved to
25
converge to the periodic solution, provided that a sufficient condition is satis- fied. Thus, the harmonic decomposition is a new way to build iteration schemas which could be applied to other problems using HMB (e.g. the high dimensional problems). Moreover, this method is demonstrated to converge to the analytic solution of a linear oscillator and it is applied to a forced Duffing oscillator. The
30
numerical results show that each iteration schema converges in each range of the excitation frequency and it can converge to different solutions of the oscillator.
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2. Iteration schemas
Let’s consider a forced nonlinear oscillator given by d2u
dt2 +βu˙+ω20u+f(u,u) =˙ Acos Ωt (1) wheref(u,u) is a nonlinear function. This equation is rewritten to read˙
d2u
dt2 + Ω2u= (Ω2−ω20)u−βu˙ −f(u,u) +˙ Acos Ωt≡G(u,u)˙ (2) A classical iteration schema is often proposed as follows
d2uk+1
dt2 + Ω2uk+1=L(G(ui,u˙j)i,j≤k) (3) whereLis a linear form which is chosen for each functionG(u,u) in order to meet˙ the convergence. Now we will build a different iteration schema by considering the periodic solution defined by series{qnk},{pnk}(with 1≤n≤N) as follows
uN k(t) =q0k
2 +
N
X
n=1
qnkcosnΩt+pnksinnΩt (4) For eachN (0 ≤ N ≤N), we decomposeuN k into two terms which corre- spond to low and high harmonics as follows
35
Xk(t) = q0k
2 +
N
X
n=1
qnkcosnΩt+pnksinnΩt (5)
Yk(t) =
N
X
n=N+1
qnkcosnΩt+pnksinnΩt (6)
uN k(t) = Xk(t) +Yk(t) (7)
Then, the proposed iteration schema is d2Yk+1
dt2 +βX˙k+1+ω02Xk+1=Acos Ωt−f(uN k,u˙N k)
−d2Xk
dt2 −βY˙k−ω20Yk
(8)
Here, we take the initial values (i.e. k = 1) by qn1 = pn1 = 0. Now we will develop this schema by using the harmonic balance technique. By performing
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the Fourier series development of equation (8), we obtain the following results 1
T Z T
0
d2Yk+1
dt2 +βX˙k+1+ω20Xk+1
cosnΩt dt=Aδ1n− Cnk
−1 T
Z T 0
d2Xk
dt2 +βY˙k+ω02Yk
cosnΩt dt
(9)
1 T
Z T 0
d2Yk+1
dt2 +βX˙k+1+ω20Xk+1
sinnΩtdt=−Snk
−1 T
Z T 0
d2Xk
dt2 +βY˙k+ω20Yk
sinnΩt dt
(10)
whereT = 2πΩ; δ1n= 1 if n= 1 andδ1n = 0 if other. Cnk,Snk are the Fourier coefficients off(uN k,u˙N k)
Cnk= 2 T
Z T 0
f(uN k,u˙N k) cosnΩtdt Snk= 2
T Z T
0
f(uN k,u˙N k) sinnΩtdt
(11)
Thereafter, by substituting equations (5) and (6) into equations (9) and (10), we obtain
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For n= 0
ω20q0(k+1)=−C0k
For 0< n≤ N
βnΩpn(k+1)+ω20qn(k+1)=Aδn1− Cnk+n2Ω2qnk
−βnΩqn(k+1)+ω20pn(k+1)=−Snk+n2Ω2pnk
For N < n≤N
−n2Ω2qn(k+1)=Aδ1n− Cnk−ω02qnk−βnΩpnk
−n2Ω2pn(k+1)=−Snk−ω20pnk+βnΩqnk
Finally, we have deduced the iteration schema (8) to the results in Table 1 where{pnk},{qnk}are represented as recurrent sequences. By taking the initial values (pn1 =qn1 = 0), we can compute these series for a number of iteration K to get the solution of equation (1). Here, we have one iteration schema
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corresponding to each value ofN. In the next section, we will prove that these
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schemas converge to the periodic solution of equation (1) if a sufficient condition is satisfied.
Table 1: Frequency dependent iteration schema
n= 0 qn(k+1)=−C0k
ω02 ; pn(k+1)= 0
0< n≤ N
pn(k+1)= n2Ω2(βnΩqnk+ω02pnk)
(βnΩ)2+ω04 −βnΩCnk+ω02Snk+βnΩAδn1
(βnΩ)2+ω40 qn(k+1)= n2Ω2(−βnΩpnk+ω02qnk)
(βnΩ)2+ω40 +βnΩSnk−ω02Cnk+ω02Aδn1
(βnΩ)2+ω04
N < n≤N
qn(k+1)= ω20
n2Ω2qnk+ β
nΩpnk+ Cnk
n2Ω2 −Aδn1
n2Ω2 pn(k+1)= ω20
n2Ω2pnk− β
nΩqnk+ Snk n2Ω2
3. Sufficient condition of convergence
Lemma: Iff ∈ C1 and if the series of functions{uN k} defined by equation (4) and the schema in Table 1 is uniformly convergent when N, k tends to
50
infinity, their limit is a periodic solution of equation (1).
Demonstration: Suppose that uN k(t) converges uniformly to u(t). We have to prove that
(i)u(t) is periodic of periodT (ii)u(t) is a solution of equation (1).
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We prove (i) by contrary. Suppose that it exists τ such that|u(τ)−u(τ+ T)|=a >0. BecauseuN k(t) is uniformly convergent, for 0<2ǫ < ait exists M, Ksuch that∀N > M, k > K we have
|uN k(τ)−u(τ)|< ǫ
|uN k(τ+T)−u(τ+T)|< ǫ In addition, we haveuN k(τ) =uN k(τ+T). Thus
a=|u(τ)−u(τ+T)| ≤ |uN k(τ)−u(τ)|+|uN k(τ)−u(τ+T)|<2ǫ
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The last inequality cannot be true because 2ǫ < a. By consequence, u(t) is periodic.
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To prove (ii), it is sufficient to show that∀ǫ >0,∃M′, K′ such that
|¨uN k+ω20uN k+f(uN k,u˙N k)−Kcos Ωt|< Cǫ∀N > M′, k > K′ (12) whereC is a constant.
By using the triangular inequality, we have
¨uN(k+1)+ω02uN(k+1)+f(uN(k+1),u˙N(k+1))−Γ cos Ωt
≤ω20|Yk+1−Yk| +β
Y˙k+1−Y˙k
+
d2
dt2(Xk+1−Xk)
+
f uN(k+1),u˙N(k+1)
−f(uN k,u˙N k)
+
d2
dt2(Yk+1+Xk) +β( ˙X(k+1)+ ˙Yk) +ω02(Xk+1+Yk) +f(uN k,u˙N k)−Acos Ωt We notefa, fb, fc, fdandfethe five terms of the last inequality, that means
fa=ω02|Yk+1−Yk| fb=β
Y˙k+1−Y˙k
fc=
d2
dt2(Xk+1−Xk) fd=
f uN(k+1),u˙N(k+1)
−f(uN k,u˙N k) fe=
d2
dt2(Yk+1+Xk) +β( ˙X(k+1)+ ˙Yk)
+ω02(Xk+1+Yk) +f(uN k,u˙N k)−Acos Ωt
In order to prove (12), we have to prove thatfa, fb, ..., feare measured byǫ.
Because uN k is uniformly convergent, it is clear that fa is measured by ǫ.
To prove thatfb andfc are measured byǫ, it is sufficient, for example, to take the definition ofXmk from equation (5) with remark that
65
fc =
d2
dt2(X(k+1)−Xk)
=
m
X
n=1
n2Ω2
(pn(k+1)−pnk) sinnΩt+ (qn(k+1)−qnk) cosnΩt
≤
m
X
n=1
n2Ω2
pn(k+1)−pnk
+
qn(k+1)−qnk
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As the series {uN k} converge uniformly, the seriespnk, qnk are uniformly con- vergent whenk tends to infinity. Thus,fb andfc are measured byǫ.
In order to measure fd, we use a following property: for f ∈ C1, it exists C such that|f(u)−f(˜u)| ≤Cǫwhen|u−u|˜ < ǫ. As the seriesuN k converges uniformly, we can choseklarge enough for|uN(k+1)−uN k|< ǫ. By consequence,
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fd is measured byǫ.
Forfe, by using the iteration schema in Table 1, we obtain
d2
dt2(Y(k+1)+Xk) + Ω2(X(k+1)−Xk) +ω02uN k+f(uN k,u˙N k)−Acos Ωt
=
f(umk,u˙mk)− C0k
2 +
m
X
n=1
CnkcosnΩt+SnksinnΩt
!
Thus, fe is exactly the difference between the function f(uN k,u˙N k) and its Fourier series development. BecauseuN kis periodic and bounded,f(uN k,u˙N k) is also periodic and bounded. Thus, this difference is measured by ǫ when N is large enough. Therefore, the five terms are measured byǫ, and the lemma is
75
proved.
4. Examples 4.1. Linear oscillator
We consider a forced linear oscillator, i.e. f(u,u) = 0. The iteration schema˙ in Table 1 becomes
n= 0 q0(k+1)= 0
1≤n≤ N pn(k+1)=n2Ω2(βnΩqnk+ω02pnk)
(βnΩ)2+ω04 + βnΩAδn1
(βnΩ)2+ω04 qn(k+1)=n2Ω2(−βnΩpnk+ω20qnk)
(βnΩ)2+ω04 + ω02Aδn1
(βnΩ)2+ω04 N ≤n≤N qn(k+1)= ω02
n2Ω2qnk+ β
nΩpnk−Aδn1
n2Ω2 pn(k+1)= ω20
n2Ω2pnk− β nΩqnk
We will prove that when the schema converges, its limit is the analytic solution.
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CaseN = 0
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Forn≥2, we have
qn(k+1)+ ipn(k+1)= ω02
n2Ω2 − iβ nΩ
(qnk+ ipnk) qn(k+1)−ipn(k+1)=
ω02 n2Ω2 + iβ
nΩ
(qnk−ipnk)
The last equations define two geometrical sequences and they converge if and only if
ω20
n2Ω2 +nΩiβ
<1; their limits are zeros (i.e.,qnk, pnk→0 whenk→ ∞).
Forn= 1, we have
q1(k+1)+ ip1(k+1)−A1= ω20
Ω2 −iβ Ω
(q1k+ ip1k−A1) q1(k+1)−ip1(k+1)−A2=
ω20 Ω2 +iβ
Ω
(q1k−ip1k−A2)
(13)
where A1 = −A/ Ω2−ω02+ iβΩ
and A2 = −A/ Ω2−ω02−iβΩ
. The last equations define also two geometrical sequences which converge to zeros with the same condition for n≥ 2. By consequence, the limits of {q1k} and{p1k}
are
q1= (ω02−Ω2)A (ω02−Ω2)2+ (βΩ)2 p1= βΩA
(ω20−Ω2)2+ (βΩ)2
(14)
Equation (14) is exactly the analytic solution of the forced linear oscillator.
Thus, the sufficient condition is thus justified in this case. In addition, equa- tion (13) shows that the schema converges linearly and the convergence rate is calculated by
µ=
ω02 Ω2 +iβ
Ω
(15) The condition of the convergenceµ <1 can be rewritten as Ω4−ω40>(βΩ)2. It means that Ω is larger than the resonance frequency of the oscillator Ωr. CaseN = 1
85
Forn≥2, we have the same results with the previous case, i.e.,qnk, pnk→0 whenk→ ∞if and only if
ω20
n2Ω2 +nΩiβ
<1 forn≥2 .
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Forn= 1, we have
q1(k+1)+ ip1(k+1)−A1= Ω2
−iβΩ +ω02(q1k+ ip1k−A1) q1(k+1)−ip1(k+1)−A2= Ω2
iβΩ +ω20(q1k−ip1k−A2)
We see that the last two equations define also two geometrical sequences which converge if and only if
Ω2 iβΩ+ω20
<1. The condition of convergence for n = 1 andn≥2 of the schema can be rewritten as follows
Ω4−ω04<(βΩ)2 16Ω4−ω04>4(βΩ)2
(16)
We deduce that the iteration schemaN = 1 converges linearly when Ω is smaller than the resonance frequency and the convergence rate is
µ=
Ω2 iβΩ +ω02
(17) CaseN ≥2
Similarly, the schema converges linearly to the analytic solution if and only
if
N4Ω4−ω40<(βNΩ)2
(N + 1)4Ω4−ω40>(β(N + 1)Ω)2
(18)
Remark: From equations (13) and (17), when Ω≃Ωr, the convergence rate µcan be approached by
µ≃
1−∆Ω Ωr
2
(19) where ∆Ω =|Ω−Ωr|. It means that µK ≃1−2K∆ΩΩr. Thus, the number of iteration can be estimated byK∼12∆ΩΩr.
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Numerical example
The iteration method is used for a forced linear oscillator with ω0= 1, A= 1. Figure 1 shows the amplitude A = p
q12+p21 in function of the excitation frequency Ω with different values of the damping coefficientβ. Here, we take
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the number of harmonicsN = 3 and the number of iteration K = 300 for all
95
frequencies Ω. The figure shows that the iteration method converges to the analytic solution. Particularly, the schema withN = 0 is convergent for low frequency Ω and the schema with N = 1 is convergent for high frequency Ω.
This numerical result agrees well with the previous demonstration.
Frequency Ω
0.5 1 1.5 2 2.5
AmplitudeA
0 1 2 3 4
N = 0 N = 1 Analytic β= 0.3
β= 0.7
β= 1
Figure 1: Forced linear oscillator by analytic and iteration methods
4.2. Duffing oscillator
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Let’s consider another example, a forced Duffing oscillator. This oscillator has many applications in physics and it can be described by
¨
u+βu˙+u+εu3=Acos Ωt (20) Whenεis small, the perturbation techniques [1, 21] can be applied for this oscillator. This technique demonstrates that the periodical solution depends on the frequency Ω by
A2 (
β2Ω2+
1−Ω2+3 4εA2
2)
=A2, tanϑ= βΩ
1−ω2+34εA2 (21) where A = p
q12+p21 is the amplitude and tanϑ = p1/q1 is the phase of the first harmonic.
Figure 2 shows the numerical results with differentN for a forced Duffing oscillator with parametersω= 1, β= 0.1, ε= 0.01 andA= 1. In this example,
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we take the number of harmonics N = 7 et the number of iterationK = 200
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for all frequencies Ω. Here, the schema is considered to converge when the amplitude of the last iteration changes less than 1%. We see that each iteration schema converges in each frequency domain of Ω. Particularly, in the domain I, two schemas converges to two different solutions, and in domain II, different schemas converge to the same solution.
Frequency Ω
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
AmplitudeA
0 2 4 6 8 10
N = 0 N = 1 N = 2 N = 3 PT
II I
Figure 2: Forced Duffing oscillator by the perturbation technique (PT) and the iteration method. Parameter: ω= 1, β= 0.1, ε= 0.01 andA= 1
110
Figure 3 shows the results for the Duffing oscillator with different nonlinear parameterε. We see that the schema N = 0 converges for high frequency Ω and the schema N = 1 converges for low frequency Ω. There is no schema of Table 1 converge to the unstable solution of the Duffing oscillator.
In order to estimate the convergence rate of the schemas, the numerical error of equation (20) is calculated by
O(ε) = max|¨u+βu˙ +u+εu3−Acos Ωt|
where u(t) is the numerical result in each iteration. Figure 4 shows the error
115
versus the number of iteration for the Duffing oscillator withε= 0.01,N= 10 andK= 1000. We can see that the schemas converge in logarithmic scale of the number of iteration. Moreover, when the frequency Ω approaches the resonance frequency (green cycle markers), the convergence is relatively slow.
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Frequency Ω
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5
AmplitudeA
0 2 4 6 8 10 12
N = 0 N = 1 PT
ε= 0
ε= 0.005 ε= 0.01
Figure 3: Forced Duffing oscillator by the perturbation technique (PT) and the iteration method
0 0.2 0.4 0.6 0.8 1
Number of iteration Ω = 1.2 Ω = 1.2
Ω = 1.0 Ω = 1.4
103 102
101
Numericalerror(1E-3)
Figure 4: Numerical error in each iteration with the schemaN = 0 (triangle) andN = 1 (cycle) for the Duffing oscillator
. 5. Conclusion
120
When the periodic solution of a dynamical system is decomposed into two parts corresponding low and high harmonics, we can propose an iteration schema with different formulations for the two parts. This decomposition can be built to adapt the system in term of convergence. The application for forced Duff- ing oscillators show that each schema converges in each range of the excitation
125
frequency and we can reach different solutions by using different schema.
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