Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 287-296, Warsaw 2013

APPLICATION OF HIGHER ORDER HAMILTONIAN APPROACH TO

NONLINEAR VIBRATING SYSTEMS

Hassan Askari, Zia Saadat Nia

IranUniversity of Science andTechnology, Center of Excellence inRailwayTransportation, School of Railway Engineering,

Tehran, Iran

Ahmet Yildirim

Ege University, Department of Mathematics, Bornova, Turkey; e-mail: ahmet.yildirim@ege.edu.tr

Mohammed K. Yazdi

Iran University of Science and Technology, School of Mechanical Engineering, Tehran, Iran

Yasir Khan

Zheijiang University, Department of Mathematics, Hangzhou, China

The higher order Hamiltonian approach is utilized to elicit approximate solutions for two
nonlinear oscillation systems. Frequency-amplitude relationships and themodel of buckling
of a column and mass-spring system are scrutinized in this paper. First, second and third
approximate solutions of examples are achieved, and the frequency responses of the systems
areverifiedbyexactnumerical solutions.According to thenumerical results,wecanconclude
that the Hamiltonian approach is an applicablemethod for solving the nonlinear equations,
and the accuracy of this method in the second and third approximates is very high and
reliable. The achieved results of this paper demonstrate that this method is powerful and
uncomplicated for solving of sophisticated nonlinear problems.

Key words: higher order Hamiltonian approach, Duffing equation, analytical solutions

1. Introduction

Since the nonlinear science has been emerged in real world uses, there is a cause for increasing
attention of scientists and engineers in analytical approaches for nonlinear problems (He, 2006).
Recently,many scientists have proposed andmodified a lot ofmethods for solving nonlinear equ-
ations (Nayfeh andMook, 1979). He (2002) have invented several non-perturbative approaches
such as energy balancemethod (EBM), variational approach (He, 2007),max-min approach (He,
2008b), Hamiltonian approach (He, 2010) and frequency amplitude formulation (He, 2008a). Ba-
sed on He’s methods, many researchers have evaluated diverse kinds of nonlinear problems. For
instance, D.D. Ganji et al. (2010) and S.S. Ganji et al. (2009) used energy balance method for
solving Van der Pol damped equations and relativistic oscillator. Momeni et al. (2011) andOzis
andYildirim (2007) employed EBM for solving theDuffing harmonic equation. Simiraly, a non-
linear oscillator with discontinuity was analyzed by D.D. Ganji et al. (2009) by means of this
approach. Also, Younesian et al. (2010a) analyzed the generalized Duffing equation by it. The
variational approach was applied for solving the relativistic oscillator (He, 2007), generalized
Duffing equation (Younesian et al., 2010a), oscillator with a fractional power (Younesian et al.,
2010b), Duffing harmonic oscillator (Askari et al., 2010). The frequency amplitude formulation
was incorporated by Cai andWu (2009), Younesian et al. (2010a), Kalami et al. (2010), Ren et
al. (2009), Zhang et al. (2009) and Zhao (2009) for solving the relativistic harmonic oscillator,
generalized Duffing equation, autonomous conservative nonlinear oscillator, nonlinear oscillator



288 H. Askari et al.

withdiscontinuity, Schrödinger equation andnonlinear oscillator with an irrational force, respec-
tively. Moreover, the max-min approach was used for analyzing the relativistic oscillator (Shen
andMo, 2009), buckling of a column (Ganji et al., 2011), twomass spring system (Ganji et al.,
2011), nonlinear oscillator with discontinuity (Zeng, 2009) and a nonlinear oscillation system of
motion of a rigid rod rocking back[(Ganji et al., 2010). In this paper, two kinds of systems with
the same form of nonlinear equation are analyzed. Figure 1 describes a model of buckling of a
column (Nayfeh andMook, 1979). The vibration of this systemwas investigated byGanji et al.
(2011) and Nayfeh andMook (1979).

Fig. 1. Model of buckling of a column (Nayfeh andMook, 1979)

Figure 2 shows the physical model of Duffing equation with a constant coefficient. This
system was examined byMehdipour et al. (2010) bymeans of of the energy balance method.

Fig. 2. Physical model of Duffing equation (Rao, 2006)

In the presentwork, theHamiltonian approach is used for solving the governing equations of
the above problems.Thismethodwas invented by J.H.He, and it has been used for evaluating a
largenumberofnonlinearproblems.Thenonlinear oscillatorwith a fractional power (Cveticanin,
2010), nonlinear oscillator with discontinuity (Yildrim et al., 2011c), nonlinear oscillator with
rational and irrational elastic forces (Yildrim et al., 2011a), nonlinear oscillations of a punctual
charge in the electric field of a charged ring (Yildrim et al., 2011b), nonlinear vibration of a rigid
rod rockingbackKhan et al., 2010) have been solvedbymeans of this potent and straightforward
method. Furthermore, Yilidrim et al. (2012) have demonstrated the relationship of this method
with the variational approach. The frequency-amplitude relationship is then obtained in an
analytical form. Also, the obtained frequency responses of the systems are compared with the
exact numerical solutions. In addition, the achieved results are compared with the results of the
max-min approach that were obtained by Ganji et al. (2011). Moreover, according to Yildirim
et al. (2012), it is stated that the variational approach leads to the same results for this systems
even for higher order approximations. Furthermore, results of several papers are developed to
obtain the frequency amplitude relationship of this system.



Application of higher order Hamiltonian approach... 289

2. Mathematical modeling

In this section, we consider a column as shown in Fig. 1. The mass m moves in the horizontal
direction only. Using thismodel that represents the column, we demonstrate how one can study
its static stability by determining the nature of the singular point at x = 0 of the dynamic
equations (Nayfeh and Mook, 1979; Ganji et al., 2011). Avoiding the weight of springs and
columns, the governing equation for motion of m is (Nayfeh andMook, 1979)

mü+
(
k1−

2P

l

)
u+
(
k3−

P

l3

)
u3+ . . .=0 (2.1)

where the spring force is given by

FSpring = k1u+k3u
3+ . . . (2.2)

This equation can be put in the general form

ü+α1u+α3u
3+ . . .=0 (2.3)

Also, the g overning equation for the model shown in Fig. 2 is obtained as

ü+
K1
m
u+

K2
2mh2

u3 =0 (2.4)

3. Solution procedure

Consider the following equation which describes the well knownDuffing equation

ü+α1u+α3u
3 =0 u(0)=A u̇(0)= 0 (3.1)

where for the first system

α1 =
1

m

(
k1−

2P

l

)
α3 =

1

m

(
k3−

p

l3

)
(3.2)

and for the second one

α1 =
K1
m

α2 =
K2
2mh2

(3.3)

Based on the first order of the Hamiltonian approach introduced by He (2010), a solution for
Eq. (3.1) is assumed as

u=Acosωt (3.4)

with satisfying the initial conditions. Its Hamiltonian can be easily obtained, which reads

H =
1

2
u̇2+

α1
2
u2+

α3
2
u4 (3.5)

Integrating Eq. (3.6) with respect to time from 0 to T/4, we have

H̃(u)=

T/4∫

0

(1
2
u̇2+

α1
2
u2+

α3
2
u4
)
dt (3.6)



290 H. Askari et al.

Substituting Eq. (3.4) into Eq. (3.6), leads to

H̃(u)=

T/4∫

0

(1
2
A2ω2 sin2ωt+

α1
2
A2cos2ωt+

α3
4
A4cos4ωt

)
dt

=

π/2∫

0

(1
2
A2ω sin2ωt+

α1
2ω
A2cos2ωt+

α3
4ω
A4cos4ωt

)
dt

=
π

8
A2ω+

α1π

8ω
A2+

α3π

64ω
A4

(3.7)

Setting

∂

∂A

(
∂H̃

∂ 1
ω

)
=
π

4
Aω2+

α1π

4
A+
4α3π

16
A3 =0 (3.8)

and consequently, the obtained frequency equals to

ω=

√
α1+

3

4
α3A2 (3.9)

The energy balance method (Mehdipour et al., 2010), varioational approach (Yildirim et al.,
2012), harmonic balance method (Yildirim et al., 2012) and the max-min (Ganji et al., 2011)
approach the same to result for this problem.

3.1. Second order Hamiltonian approach

Inorder to improve theaccuracyof this approach, the followingperiodic solution is considered
(Yildirim et al., 2011c; Durmaz et al., 2010)

u= acosωt+bcos3ωt (3.10)

where the initial condition is

A= a+ b (3.11)

Substituting Eq. (3.11) into Eq. (3.6), we obtain

H̃(u)=

T/4∫

0

[1
2
(aω sinωt+3bω sin3ωt)2+

1

2
α1(acosωt+ bcos3ωt)

2

+
1

4
α3(acosωt+ bcos3ωt)

4
]
dt=

π/2∫

0

[1
2
ω(asint+3bsin3t)2

+
1

2ω
α1(acost+ bcos3t)

2+
1

4ω
α3(acos t+ bcos3t)

4
]
dt

=
π

8
ω(a2+9b2)+

π

8ω
α1(a

2+ b2)+
π

64ω
α3(3a

4+4a3b+12a2b2+3b4)

(3.12)

Setting

∂

∂a

(
∂H̃

∂ 1
ω

)
=−
π

4
aω2+

π

4
α1a+

π

64
α3(12a

3+12a2b+24ab2)= 0

∂

∂b

(
∂H̃

∂ 1
ω

)
=−
18πb

8
ω2+

π

4
α1b+

π

64
α3(4a

3+24a2b+24ab2)=0

(3.13)



Application of higher order Hamiltonian approach... 291

After somemathematical simplifications, it is achieved that

a=0.95714A b=0.04289A (3.14)

and the frequency-amplitude relationship can be written as

ωSHA =
√
α1+0.7205α3A2 (3.15)

3.2. Third order Hamiltonian approach

Consider the following periodic equation as the response to Eq. (3.1)

u= acosωt+bcos3ωt+ ccos5ωt (3.16)

where

A= a+ b+ c (3.17)

Substituting Eq. (22) into Eq. (3.6), we obtain

H̃(u)=

T/4∫

0

[1
2
(aω sinωt+3bω sin3ωt+5bω sin5ωt)2

+
1

2
α1(acosωt+ bcos3ωt+ ccos5ωt)

2+
1

4
α3(acosωt+ bcos3ωt+ccos5ωt)

4
]
dt

=

π/2∫

0

[1
2
ω(asint+3bsin3t+5ccosωt)2+

1

2ω
α1(acos t+cos3t+5ccos t)

2

+
1

4ω
α3(acost+ bcos3t+5ccosωt)

4
]
dt=

π

8
ω(a2+9b2+25c2)+

π

8ω
α1(a

2+ b2+ c2)

+
π

64ω
α3(3a

4+3b4+3c4+12b2c2+12ab2c+12a2bc+4a3b+12a2c2+12a2b2)

(3.18)

Setting

∂

∂a

(
∂H̃

∂ 1
ω

)
=−
π

4
aω2+

π

4
α1a+

π

64
α3(12a

3+12a2b+12b2c+24ab2+24abc+24ac2)= 0

∂

∂b

(
∂H̃

∂ 1
ω

)
=−
18πb

8
ω2+

π

4
α1b+

π

64
α3(4b

3+24bc2+24abc+12a2b+4a3+24a2b)= 0

(3.19)
∂

∂c

(
∂H̃

∂ 1
ω

)
=−
50πc

8
ω2+

π

4
α1c+

π

64
α3(12c

3+24b2c+12ab2+12a2b+24a2c)= 0

Then, after some simplifications, we obtain

a=0.955091A b=0.0430519A c=0.0018569A (3.20)

Finally, the natural frequency of the system equals to

ωTHA =
√
α1+0.7178α3A2 (3.21)



292 H. Askari et al.

4. Discussion and numerical results

The presented solution procedures are used to obtain frequency responses. Variations of the
natural frequencies are illustrated in Figs. 3 and 4 for Example 1. The frequency responses are
tabulated for some special cases. According toTable 1, it was demonstrated thatwhen the order
of the proposed method increases, higher agreement and more accurate results are obtained.
The time history obtained for the initial condition is illustrated in Fig. 5. It is seen that in the
time domain, a very excellent correlation is still preserved.

Fig. 3. The frequency ratio (nonlinear/linear) with respect to initial amplitudes for Example 1;

K1 =500,K3 =500,m=50, p=150, l=10

Fig. 4. The frequency ratio (nonlinear/linear) with respect to initial amplitudes for Example 1;
K1 =10,K3 =5,m=1, p=1, l=1

Table 1.Comparison of approximate and exact frequencies for Example 1.

(m,l,p) (k1,k3) A ωFHA ωSHA ωTHA ωExact

(1,1,1) (10,5) 1 3.2015 3.1877 3.1861 3.1861

(10,10,10) (10,50) 10 19.3816 18.9985 18.9539 18.9528

(50,25,40) (30,100) 20 24.5052 24.0202 23.9636 23.9623

(100,50,150) (70,20) 100 38.7357 37.9687 37.8793 37.8772

(1000,500,1000) (500,500) 1 0.9332 0.9253 0.92444 0.92442

For Example 2, the numerical results are obtained, and in Table 2 frequency responses of
the system are given and analyzed for some special cases. To show and prove the accuracy of
these analytical methods, comparisons of analytical and exact results for the practical cases are
presented in Fig. 6.



Application of higher order Hamiltonian approach... 293

Fig. 5. Time history of dynamic responses (A=1,M =1, P =1,L=1,K1 =10,K3 =5)

Table 2.Comparison of approximate and exact frequencies for Example 2

(h,m) (k1,k2) A ωFHA ωSHA ωTHA ωExact

(1,1) (10,5) 1 3.44601 3.4353 3.4340 3.4340

(1,10) (10,50) 1 1.6955 1.6737 1.6712 1.6711

(10,10) (30,100) 2 1.7748 1.7731 1.77297 1.77296

(1,100) (70,20) 5 1.6046 1.5815 1.5789 1.5788

Fig. 6. Time history of dynamic responses (A=1,m=1, h=1,K1 =10,K2 =5)

Table 3 reveals the achieved frequency-amplitude relationship for the objective problem
of this paper. The results of diverse kinds of approaches are illustrated in this Table using
corresponding references.

5. Conclusion

In this paper, two dynamic systemswere considered, where in both cases the governing equation
was expressed as the Duffing equation. The Hamiltonian approach was then applied in three
orders to find the approximate periodic solution of this equation. The accuracy of solution pro-
cedures was evaluated by comparing the obtained results with the exact ones in time histories
and tables. The effects of nonlinear parameters and initial amplitudes on the natural frequency
were also illustrated in two figures. It was proved that as the order of the proposed approach
increases, higher agreement and more accurate results are obtained. Indeed, it can be conclu-
ded that the higher order Hamiltonian approach is a valid and strong method in evaluating
conservative nonlinear oscillatory systems even for large amplitudes and strong nonlinearity.
Furthermore, according to Ganji et al. (2011), the max-min approach and the Hamiltonian ap-



294 H. Askari et al.

Table 3.Obtained frequency-amplitude relationship from related references

Approach Frequency-amplitude relationship

Energy balance method (Younesian et
al., 2010a), max-min (Ganji et al.,
2011); approach, frequency-amplitude
formulation (Younesian et al., 2010a);
homotopy perturbation (Younesian et
al., 2011); harmonic balance method
(Brléndez et al., 2011)

Example 1: ω=

√(
k1−

2p

l

)
+
3

4

(
k3−

p

l3

)
A2

Example 2: ω=

√
k1
m
+
3

8

k2
mh2
A2

Modified energy balance method Example 1: ω=

√(
k1−

2p

l

)
+
7

10

(
k3−

p

l3

)
A2

(Younesian et al., 2011)

Example 2: ω=

√
k1
m
+
7

20

k2
mh2
A2

Simple approach (Ren and He, 2009) Example 1: ω=

√(
k1−

2p

l

)
+
7

9

(
k3−

p

l3

)
A2

Example 2: ω=

√
k1
m
+
7

18

k2
mh2
A2

proach have the same results for this problem in the first approximation. In addition, basing
on Yildirim et al. (2012), we can state that the variational approach can lead to similar results
for this problem even for a higher order of the approximation. Besides, the harmonic balance
method gives the same result for the objective systems (Beléndez et al., 2011).Moreover, results
of lots of other papers were developed for the systems considered in this paper, and they are
listed in Table 3.

References

1. Askari H., Kalami Yazdi M., Saadatnia Z., 2010, Frequency analysis of nonlinear oscilla-
tors with rational restoring force via He’s energy balance method and He’s variational approach,
Nonlinear Science Letters A, 1, 425-430

2. Beléndez A., Arribas E., Francés J., Pascual I., 2011, Notes on application of the Hamil-
tonian approach to nonlinear oscillators with rational and irrational elastic terms, Mathematical
and Computer Modelling, http://dx.doi.org/10.1016/j.mcm.2011.06.024

3. Cai X.C., Wu W.Y., 2009, He’s frequency formulation for the relativistic harmonic oscillator,
Computer and Mathematic with Application, 58, 2358-2359

4. CveticaninL.,KalamiYazdiM., SaadatniaZ.,AskariH., 2010,ApplicationofHamiltonian
approach to the generalized nonlinear oscillator with fractional power, International Journal of
Nonlinear Sciences and Numerical Simulation, 11, 997-1002

5. Durmaz S., Altay D., Kaya M.O., 2010, High order Hamiltonian approach to nonlinear oscil-
lators, International Journal of Nonlinear Sciences and Numerical Simulation, 11, 565-570

6. Ganji D.D., Esmaeilpour M., Soleimani S., 2010, Approximate solutions to Van der Pol
damped nonlinear oscillators by means of He’s energy balance method, International Journal of
Computer Mathematics, 87 2014- 2023

7. Ganji D.D., Ranjbar Malidarreh N., Akbarzade M., 2009, Comparison of energy balance
period with exact period for arising nonlinear oscillator equations (He’s energy balance period



Application of higher order Hamiltonian approach... 295

for nonlinear oscillators with and without discontinuities), Acta Applicandae Mathematicae, 108,
353-362

8. Ganji S.S., Barari A., Ganji D.D., 2011, Approximate analysis of two mass-spring systems
and buckling of a column,Computers and Mathematics with Applications, 61, 1088-1095

9. Ganji S.S., Ganji D.D., Davodi A.G., Karimpour S., 2010, Analytical solution to nonlinear
oscillation system of the motion of a rigid rod rocking back using max-min approach, Applied
Mathematical Modelling, 34, 2676-2684

10. Ganji S.S., Ganji D.D., Ganji Z.Z., Karimpour S., 2009, Periodic solution for strongly non-
linear vibration systems by He’s energy balance method, Acta Applicandae Mathematicae, 106,
79-92

11. He J.H., 2002, Preliminary report on the energy balance for nonlinear oscillations, Mechanics
Research Communications, 29, 107-111

12. He J.H., 2006, Some asymptotic methods for strongly nonlinear equations, International Journal
of Modern Physics B, 20, 10, 1141-1199

13. He J.H., 2007, Variational approach for nonlinear oscillators, Chaos Solitons and Fractals, 34,
1430-1439

14. He J.H., 2008a, An improved amplitude-frequency formulation for nonlinear oscillators, Interna-
tional Journal of Nonlinear Sciences and Numerical Simulation, 9, 2, 211-212

15. He J.H., 2008b, Max-min approach to nonlinear oscillators, International Journal of Nonlinear
Sciences and Numerical Simulation, 9, 2, 207-210

16. He J.H., 2010, Hamiltonian approach to nonlinear oscillators,Physics Letters A, 374, 2312- 2314

17. Khan Y., Wu Q., Askari H., Saadatnia Z., Kalami Yazdi M., 2010, Nonlinear vibration
analysis of a rigid rod on a circular surface via Hamiltonian approach,Mathematical and Compu-
tational Applications, 15, 974-977

18. Mehdipour I., Ganji D.D.,MozaffariM., 2010,Application of the energy balancemethod to
nonlinear vibrating equations,Current Applied Physics, 10, 104-112

19. Momeni M., Jamshidi N., Barari A., Ganji D.D., 2011, Application of He’s energy balance
method to Duffing-harmonic oscillators, International Journal of Computer Mathematics, 88, 1,
135-144

20. Nayfeh A.H., Mook D.T., 1979,Nonlinear Oscillations, JohnWiley & Sons, NewYork

21. Ozis T., YildirimA., Determination of the frequency-amplitude relation for a Duffing-harmonic
oscillator by the energy balancemethod,Computers andMathematics with Applications, 54, 1184-
1187

22. Ren Z.F., He J.H., 2009, A simple approach to nonlinear oscillators, Physics Letters A, 373,
3749-3752

23. Ren Z.-F., Liu G.-Q., Kang Y.-X., Fan H.-Y., Li H.-M., Ren X.-D., Gui W.-K., 2009,
Application of He’s amplitude-frequency formulation to nonlinear oscillators with discontinuities,
Physica Scripta, 80, 4, 045003

24. Rao S.S., 2006,Mechanical Vibrations Book, Fourth ed., ISBN: 978-964-9585-5-0

25. Shen Y.Y., Mo L.F., 2009, The max-min approach to a relativistic equation, Computers and
Mathematics with Applications, 58, 2131-2133

26. YazdiM.K., KhanY.,MadaniM., Askari H., Saadatnia Z., YildirimA., Analytical solu-
tions for autonomous conservative nonlinear oscillator, International Journal of Nonlinear Sciences
and Numerical Simulation, 11, 11, 979-984

27. YildirimA., Saadatnia Z.,AskariH., 2011a,Application of theHamiltonian approach to non-
linear oscillatorswith rational and irrational elastic terms,Mathematical and ComputerModelling,
54, 697-703



296 H. Askari et al.

28. Yildirim A., Askari H., Saadatnia Z., Kalami Yazdi M., Khan Y., 2011b, Analysis of
nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian
approach and the energy balance method, Computers and Mathematics with Applications, 62,
486-490

29. Yildirim A., Saadatnia Z., Askari H., Khan Y., Kalami Yazdi M., 2011c, Higher order
approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach, Applied
Mathematics Letters, 24, 2042-2051

30. YildirimA.,AskariH.,KalamiYazdiM.,KhanY., 2012,Arelationshipbetween threeanaly-
tical approaches tononlinearproblems,AppliedMathematics Letters, doi:10.1016/j.aml.2012.02.001

31. Younesian D., Askari H., Saadatnia Z., KalamiYazdi M., 2010a, Frequency analysis of
strongly nonlinear generalized Duffing oscillators using He’s frequency-amplitude formulation and
He’s energy balance method,Computer and Mathematic with Application, 59, 3222-3228

32. Younesian D., Askari H., Saadatnia Z., Kalami Yazdi M., 2011, Free vibration analysis
of strongly nonlinear generalized duffing oscillators using He’s variational approach and homotopy
perturbationmethod, Nonlinear Science Letters A, 2, 11-16

33. Younesian D., Askari H., Saadatnia Z., Yildirim A., 2010b, Periodic solutions for the
generalized nonlinear oscillators containing fraction order elastic force, International Journal of
Nonlinear Sciences and Numerical Simulation, 11, 1027-1032

34. Zeng D.Q., 2009, Nonlinear oscillator with discontinuity by the max-min approach,Chaos, Soli-
tons and Fractals, 42, 2885-2889

35. ZhangY.N.,XuF.,DengL.L., 2009,Exact solution for nonlinear Schrödinger equationbyHe’s
frequency formulation,Computers and Mathematics with Applications, 58, 2449-2451

36. Zhao L., 2009, He’s frequency-amplitude formulation for nonlinear oscillators with an irrational
force,Computers and Mathematics with Applications, 58, 2477-2479

Zastosowanie metody Hamiltona wyższego rzędu w zagadnieniu drgań układów

nieliniowych

Streszczenie

W pracy przedstawiono zastosowanie metody Hamiltona wyższego rzędu do wyznaczania przybliżo-
nych rozwiązańanalitycznychdladwóchnieliniowychukładówdrgających.Szczegółowejanalizie poddano
charakterystyki amplitudowo-częstościowe modelu ściskanej belki oraz dyskretnego układu sprężysto-
inercyjnego. Otrzymano przybliżone rozwiązania pierwszego, drugiego i trzeciego rzędu, a odpowiedzi
częstościowe układów porównano z dokładnymi rezultatami symulacji numerycznych. Na ich podstawie
oceniono, żemetodaHamiltona jest stosowalnadlaukładównieliniowych,aprzybliżeniadrugiego i trzecie-
go rzędu stanowią rozwiązania analityczne owysokiej dokładności. Uzyskanew pracywyniki przekonują,
że zaproponowana metoda jest prostym i jednocześnie bardzo skutecznym narzędziem rozwiązywania
nieliniowych problemów układówmechanicznych o dużym stopniu złożoności.

Manuscript received March 24, 2012; accepted for print June 14, 2012