Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

50, 2, pp. 639-652, Warsaw 2012

50th Anniversary of JTAM

ANALYTICAL SOLUTIONS FOR FREE OSCILLATIONS OF

BEAMS ON NONLINEAR ELASTIC FOUNDATIONS USING

THE VARIATIONAL ITERATION METHOD

Davood Younesian, Zia Saadatnia, Hassan Askari

Center of Excellence in Railway Transportation, School of Railway Engineering, Iran University of

Science and Technology, Narmak, Tehran, Iran; e-mail: younesian@iust.ac.ir

Closed form expressions are obtained for the dynamic response of an
elastic beam rested on a nonlinear foundation in this paper. The nonli-
near governing equation is solvedusing theVariational IterationMethod
(VIM). An iteration formulation is constructed based on the VIM and
the dynamic responses are then obtained. Frequency responses are pre-
sented in a closed form and their sensitivity with respect to the initial
amplitudes are investigated.Anumber of numerical simulations are then
carriedoutandperformanceandvalidity of the solutionprocedure is eva-
luated in the time domain. It is proved that the VIM is quite a reliable
and straightforward technique to solve the corresponding set of coupled
nonlinear differential equations.

Key words: beam, nonlinear oscillation, variational iteration method,
elastic foundation

1. Introduction

Vibration analysis of beam-type structures rested on nonlinear foundations
have recently received a remarkable amount of attention due to importan-
ce and various applications of the subject. Surveying the literature reveals
that the ballast dynamic behavior has been proved to be nonlinear in railway
tracks. Dynamic behavior of columns, piles and pipes supported along the-
ir length, typically by soils, are also some other applications of the subject.
It has been proved that the linearization in governing equations can some-
times lead to significant errors in the system modeling and identifications.
Consequently, free and forced vibration analysis of beams supported by non-
linear elastic and visco-elastic foundations have been widely focused in the



640 D. Younesian et al.

last decade. Applications of the Multiple Scales Method (MSM), Method of
Shaw and Pierre, Method of Normal Forms andMethod of King andVakakis
in free vibration analysis of a simply supported beam rested on a nonline-
ar elastic foundation have been summarized by Nayfeh (2000). Younesian et
al. (2005) and Kargarnovin et al. (2005) employed a perturbation technique
and obtained the main harmonic and also sub-harmonics of an infinite beam
supported by a nonlinear viso-elastic foundation traversed by a moving load.
An analytical solution was obtained by Bogacz and Czyczuła (2008) for the
response of a beam on a visco-elastic foundation subjected to moving distri-
buted loads. Vibrations of structures composed of beams and plates subjected
to various types of moving loads was analytically studied by Bogacz and Fri-
schmuth (2009). Santee andGoncalves (2006) used theMelnikovMethod and
obtained the frequency response and bifurcation diagrams for a harmonical-
ly excited beam rested on a nonlinear elastic foundation. Malekzadeh and
Vosoughi (2009) employed the Differential Quadrature Method (DQM) and
numerically studied large amplitude vibration of composite beams on nonli-
near elastic foundations. A weak form Quadrature Element Method (QEM)
was employed by Yihua et al. (2009) and themain natural frequency was nu-
merically obtained for vibrations of beams rested on nonlinear elastic founda-
tions. TheGalerkinmethodwas utilized bySenalp et al. (2010) and vibrations
of finite beams on linear and nonlinear elastic foundations were numerically
studied.

The orthogonality conditions for a transverse vibration mode equation of
a one-dimensional plate on an elasticWinkler foundation has been studied by
Zhang (2002). Baghani et al. (2011) andJafari-Talookolaei et al. (2011) respec-
tively employed Variational Iteration Method (VIM) and Homotopy Analysis
Method (HAM) and obtained the first natural frequency for unsymmetrical-
ly laminated composite beams rested on nonlinear elastic foundations. More
recently, external and internal-external resonances in finite beams rested on
nonlinear visco-elastic foundations subjected to moving loads were analyzed
using the MSM by Ansari et al. (2010, 2011). Surveying the literature de-
monstrates that the employed solution methods so far have some restrictions
and limitations. The numerical methods are not able to provide closed form
solutions, and consequently they are suitable for numerical case studies. The
utilized approximate methods are also limited to obtain only the first na-
tural frequency or they are sometimes practically inapplicable in providing
higher natural frequencies. The other constraint arises from the limitations
of approximate solutions in strong nonlinear systems. The present paper is
aimed at covering the limitations of earlier solution procedures in dealing



Analytical solutions for free oscillations of beams... 641

with the higher natural frequencies. Themain objectives of the present study
is to provide a straightforward solution procedure to generate a closed form
solution:

A) applicable for higher mode natural frequencies,

B) applicable for a strong nonlinear system.

There is no doubt that the closed form solutions are always of importance
because of their potential to provide adeeper and simpler engineering estimate
and judgment.Thevariational IterationMethod (VIM) (He, 2000) is employed
for this purpose. Application of the VIM has been proved to be quite efficient
in varieties of nonlinear systems governed by ordinary and partial differential
equations (Askari et al., 2010; He et al., 2010; Inan and Yildirim, 2010; Ozis
Yildirim, 2007; Wazwaz, 2008-2010; Yildirim, 2010; Yildirim and Ozis, 2009;
Younesian et al., 2011). Employing the VIM, analytical expressions are obta-
ined for the natural frequencies of free oscillation of an elastic beam rested
on a nonlinear foundation in this paper. The corresponding set of coupled
nonlinear differential equations is then derived from the governing partial dif-
ferential equation. The associated iteration formulation is then presented and
the closed form solution is consequently constructed. The frequency respon-
ses are obtained and illustratively discussed for a real mechanical system. A
number of numerical simulations are then carried out and the accuracy of the
solution procedure is evaluated.

2. Solution procedure

The case of a uniform finite beam resting on a non-linearWinkler foundation
is governed by Baghani et al. (2011) and Jafari-Talookolaei et al. (2011)

EIwxxxx+kw+αw
3+ρAwtt =0 (2.1)

in which the parameters E, I, A, and ρ are the modulus of elasticity, se-
cond moment of area, cross-sectional area, and the material density of the
beam, respectively. The other parameters k and α are the respective line-
ar and nonlinear parts of the foundation stiffness. The beam is assumed to
have homogeneous mass density and cross section along its length, and it is
modeled by the Euler-Bernoulli theory. The beam material is assumed to be
isotropic and mechanical properties of the foundation are uniform along the
beam length.



642 D. Younesian et al.

The approximate solution is constructed for a simply supported beam ba-
sed on theGalerkin approximation presented byNayfeh (2000) andYounesian
et al. (2008)

W(x,t)=
∞
∑

n=1

un(t)sin
nπx

L
(2.2)

The solution procedure is aimed at finding the main frequencies of the sym-
metric and un-symmetric mode shapes while the procedure can be straight-
forwardly expanded for even higher frequencies

2
∑

m=1

{[

ρAüm(t)+
(m4π4EI

L4
+k
)

um(t)
]

sin
mπx

L

}

+α

( 2
∑

m=1

um(t)sin
mπx

L

)3

=0

(2.3)

Using the orthogonality principle of the mode shapes, one can arrive at
(u1 =X,u2 =Y )

Ẍ+ω21X =−α1X
3−α3XY 2 Ÿ +ω22Y =−α2Y

3−α4X2Y (2.4)

in which

α1 =α2 =
3α

4ρA
α3 =α4 =

3α

2ρA
(2.5)

and

ω21 =
EIπ4

ρAL4
+
k

ρA
ω22 =

16EIπ4

ρAL4
+
k

ρA
(2.6)

Applying the variational iteration method (Ansari et al., 2011; He, 2000; He
et al., 2010; Inan and Yildirim, 2010; Ozis Yildirim, 2007; Wazwaz, 2008-
-2010;Yildirim, 2010;YildirimandOzis, 2009), onecan construct the following
iteration formulations

Xn+1 =Xn+
1

ω1

t
∫

0

sinω1(s− t)
(d2X

ds2
+ω21X+α1X

3+α3XY
2
)

ds

Yn+1 =Yn+
1

ω2

t
∫

0

sinω2(s− t)
(d2Y

ds2
+ω22Y +α2Y

3+α4YX
2
)

ds

(2.7)

Based on the iteration harmonic formulations recognized by the respective
amplitudes of A and B for X and Y , one can consequently arrive at



Analytical solutions for free oscillations of beams... 643

X1 =AcosΩ1t+
1

ω1

t
∫

0

sinΩ1(s− t)

·
[

A(ω21 −Ω
2
1)cosΩ1s+α1A

3cos3Ω1s+α3AB
2cosΩ1scos

2Ω1s
]

ds

(2.8)

Y1 =BcosΩ2t+
1

ω2

t
∫

0

sinΩ2(s− t)

·
[

B(ω22 −Ω
2
2)cosΩ2s+α2B

3cos3Ω2s+α4BA
2cosΩ2scos

2Ω1s
]

ds

inwhich Ω1 and Ω2 denote the nonlinear natural frequencies. Implementing a
sort of appropriatemathematical operations and simplifications, one can reach
to

X1 =
[

A+
α3AB

2

2(ω21 −Ω21)

+
α3AB

2

4

( 1

ω21 − (2Ω2+Ω1)2
+

1

ω21 − (2Ω2−Ω1)2
)

+
α1A

3

4

( 3

ω21 −Ω21
+

1

ω21 −9Ω21

)]

cosΩ1t−
α1A

3

4

(3cosΩ1t

ω21 −Ω21

)

+
cos3Ω1t

ω21 −9Ω21

−
α3AB

2

4

(2cosΩ1t

ω21 −Ω21
+
cos(2Ω2+Ω1)t

ω21 − (2Ω2+Ω1)2
+
2cos(2Ω2−Ω1)t
ω21 − (2Ω2−Ω1)2

)

(2.9)

Y1 =
[

B+
α4BA

2

2(ω22 −Ω22)

+
α4BA

2

4

( 1

ω22 − (2Ω1+Ω2)2
+

1

ω22 − (2Ω1−Ω2)2
)

+
α2B

3

4

( 3

ω22 −Ω22
+

1

ω22 −9Ω22

)]

cosΩ2t−
α2B

3

4

(3cosΩ2t

ω22 −Ω22
+
cos3Ω2t

ω22 −9Ω22

)

−
α4BA

2

4

(2cosΩ2t

ω22 −Ω22
+
cos(2Ω1+Ω2)t

ω22 − (2Ω1+Ω2)2
+
2cos(2Ω1−Ω2)t
ω22 − (2Ω1−Ω2)2

)

Eliminating the secular terms in X and Y directions, yields

A+
α3AB

2

2(ω21 −Ω21)
+
α3AB

2

4

( 1

ω21 − (2Ω2+Ω1)2
+

1

ω21 − (2Ω2−Ω1)2
)

+
α1A

3

4

( 3

ω21 −Ω21
+

1

ω21 −9Ω21

)

=0

(2.10)



644 D. Younesian et al.

B+
α4BA

2

2(ω22 −Ω22)
+
α4BA

2

4

( 1

ω22 − (2Ω1+Ω2)2
+

1

ω22 − (2Ω1−Ω2)2
)

+
α2B

3

4

( 3

ω22 −Ω22
+

1

ω22 −9Ω22

)

=0

Equations (2.10) implicitly generate the main frequencies of the symmetric
and un-symmetric modes of oscillations for the general case of A,B 6=0. For
a special case of

X(0)=A Y (0)= 0 (2.11)

one can arrive at

Ω1 =
1√
72

√

40ω21 +28α1A
2+
√

(40ω21 +28α1A
2)2−576ω21(α1A2+ω21)

(2.12)
For the other special case of

X(0)= 0 Y (0)=B (2.13)

one can reach to

Ω2 =
1√
72

√

40ω22 +28α2B
2+
√

(40ω22 +28α2B
2)2−576ω22(α2B2+ω22)

(2.14)

3. Numerical examples

The presented solution procedure is employed to obtain the frequency respon-
ses for a typical railway track system (Kargarnovin et al., 2005; Younesian et
al., 2005) with strong nonlinearity. The set of nonlinear frequency Equations
(2.10)2, (2.11) is solved and the first and second natural frequencies of the
beamare consequently obtained as two different functions of the initial condi-
tions. Mechanical and geometrical properties of the beam-foundation system
are listed in Table 1. Variations of the first and second natural frequencies
are illustrated in Figs.1 and 2. The relative error with respect to the linear
solutions is also demonstrated in these figures. As it is seen, the relative er-
ror can be evaluated up to 60% and 5%, respectively for the first and second
natural frequency. The corresponding latticed contour maps are illustrated in
Figs.3 and 4. The color gradient is correlated with nonlinear sensitivity of the
system with respect to the initial amplitudes. The elliptic-type contours are



Analytical solutions for free oscillations of beams... 645

Table 1. Mechanical and geometrical properties of the beam-foundation sys-
tem (Ansari et al., 2010, 2011)

Item Notation Value

Young’s modulus (steel) E 210GPa

Mass density ρ 7850kg/m3

Cross sectional area A 7.69 ·10−3m2
Secondmoment of area I 30.55 ·10−6m4
Beam length L 18m

First linear natural frequency ω1 98.86rad/s

Second llinear natural frequency ω2 157.8rad/s

Nonlinear stiffness coefficient α1 =α2 8.69 ·103
Nonlinear stiffness coefficient α3 =α4 1.74 ·104

Fig. 1. The first frequency ratio (nonliner/linear) with respect to the initial
amplitude

Fig. 2. The second frequency ratio (nonliner/linear) with respect to the initial
amplitude



646 D. Younesian et al.

seen to have an aspect ratio very close to one and they seem to be vertical in
Fig.3 and horizontal in Fig.4. This means that both natural frequencies have
almost similar sensitivity with respect to the initial amplitudes. In order to
evaluate the accuracy of the solution method in the time domain, a number
of numerical simulations are carried out in the following sections.

Fig. 3. First natural frequency contour map

Fig. 4. Second natural frequency contourmap

3.1. Case #1

A set of mutual infinitesimal initial conditions is assumed to be

X(0)= 0.01 Y (0)= 0.01 (3.1)

in this case. Solving the set of nonlinear equations of (2.9) and employing
the prescribed solution procedure for this case, the consequent closed form
solutions are obtained as



Analytical solutions for free oscillations of beams... 647

x(t)= 0.009989cos(10.019t)+2.702 ·10−6 cos(30.058t)
+5.563 ·10−7cos(88.971t)+9.3458 ·10−6 cos(68.932t)

Y (t)= 0.0103cos(39.4759t)+1.4738 ·10−7 cos(118.43t)
+2.19 ·10−6 cos(59.514t)−3.6877 ·10−6cos(19.427t)

(3.2)

The corresponding nonlinear differential equations are numerically solved in
parallel, and the numerical results are compared in the time domain in Fig.5.
The very close agreement can accordingly guarantee accuracy of the solution
procedure.

Fig. 5. Time history of the dynamic responses (A,B=0.01)

3.2. Case #2

For a set of larger initial amplitudes, i.e.

X(0)= 0.1 Y (0)= 0.1 (3.3)

Corresponding similar calculations have been carried out and the obtained
closed form solutions are listed as

x(t)= 0.0979cos(15.96t)+9.9115 ·10−4 cos(47.88t)
+4.5 ·10−4cos(98.76t)+9.9536 ·10−4cos(66.84t)

y(t)= 0.094cos(41.5t)+1.57 ·10−4 cos(124.5t)
+0.0011cos(73.32t)+0.003cos(9.48t)

(3.4)

The obtained analytical solutions are compared with the numerical ones in
Fig.6. It is seen that validity of the approximate closed form solutions is still
preserved in the time domain.



648 D. Younesian et al.

Fig. 6. Time history of the dynamic responses (A,B=0.1)

3.3. Case #3

In order to evaluate the accuracy of the solution procedure for a non-
conjugate initial condition, the following set of initial conditions is taken into
account

X(0)= 0.1 Y (0)= 0.05 (3.5)

The solution procedure ends up in the following closed form solutions

x(t)= 0.0985cos(13.68t)+0.00137cos(41.04t)

+1.2125 ·10−4cos(95.208t)+2.4134 ·10−4 cos(67.848t)
y(t)= 0.04826cos(40.762t)+2.0284 ·10−4cos(122.29t)
+7.0502 ·10−3cos(68.124t)+0.001578cos(13.404t)

(3.6)

A very good correlation is again detected between the numerical results and
analytical solutions (Fig.7).

Fig. 7. Time history of the dynamic responses (A=0.1,B=0.05)



Analytical solutions for free oscillations of beams... 649

4. Conclusions

Analytical solutions were obtained for free oscillation of an elastic beam sup-
ported by a nonlinear foundation in this paper. An iteration formulation was
presented based on the VIM and the dynamic responses were then obtained
in a closed form. The solution procedure was directed to obtain the main
symmetric and unsymmetric mode frequencies because of their importance in
dynamic characteristics of structures. The solution technique was planned to
be as simple as possible and also extendable to obtain higher (third andmore)
natural frequencies. Limitations of earlier solution procedures in dealing with
the higher natural frequencies were covered by a reliable and straightforward
technique. Characteristic equations were obtained as functions of initial am-
plitudes and the corresponding frequency responses were obtained in closed
forms.An error analysis was carried out, and it was proved that any lineariza-
tion could lead to a significant error especially for themain symmetricmode. It
was also found that both the symmetric and un-symmetric natural frequencies
had almost similar sensitivitywith respect to the initial amplitudes.Anumber
of numerical simulations were then directed to evaluate the performance and
validity of the solution procedure in the time domain. It was proved that the
VIM was a quite accurate and simple technique to find higher natural frequ-
encies of similar structures with strong nonlinearities and nonlinear couplings
between different modes of oscillations.

References

1. Ansari M., Esmailzadeh E., Younesian D., 2010, Internal-external reso-
nance of beams on non-linear viscoelastic foundation traversedbymoving load,
Nonlinear Dynamics, 61, 163-182

2. AnsariM., EsmailzadehE., YounesianD., 2011,Frequency analysis of fi-
nite beams onnonlinearKelvin-Voight foundation undermoving loads, Journal
of Sound and Vibration, 330, 1455-1471

3. Askari H., KalamiYazdi M., SaadatNia Z., 2010, Frequency analysis of
nonlinear oscillators with rational restoring force via He’s energy balance me-
thod and He’s variational approach,Nonlinear Science Letters A, 1, 425-430

4. Baghani M., Jafari-Talookolaei R.A., Salarieh H., 2011, Large ampli-
tudes free vibrations and post-buckling analysis of unsymmetrically laminated
composite beams on nonlinear elastic foundation,AppliedMathematical Model-
ling, 35, 130-138



650 D. Younesian et al.

5. Bogacz R., Czyczula W., 2008, Response of beam on visco-elastic founda-
tion tomoving distributed load, Journal of Theoretical andAppliedMechanics,
46, 763-775

6. Bogacz R., Frischmuth K., 2009, Vibration in sets of beams and plates
induced by traveling loads,Archive of Applied Mechanics, 79, 509-516

7. KargarnovinM.H.,YounesianD.,ThompsonD.J., JonesC.J.C., 2005,
Response of beams on nonlinear viscoelastic foundations to harmonic moving
loads,Computers and Structures, 83, 1865-1877

8. He J.H., 2000,Variational iterationmethod for autonomous ordinary differen-
tial systems,Applied Mathematics and Computation, 114, 115-123

9. He J.H.,WuG.C., Austin F., 2010,The variational iterationmethodwhich
should be followed,Nonlinear Science Letters A, 1, 1-30

10. InanA.,YildirimA., 2010,Comparisonbetweenvariational iterationmethod
and homotopy perturbationmethod for linear and nonlinear partial differential
equations with the nonhomogeneous initial conditions,Numerical Methods for
Partial Differential Equations, 26, 1581-1593

11. Jafari-Talookolaei R.A., SalariehH.,KargarnovinM.H., 2011,Ana-
lysis of large amplitude free vibrations of unsymmetrically laminated compo-
site beams on a nonlinear elastic foundation, Acta Mech., 219, 6575, DOI
10.1007/s00707-010-0439-x

12. Malekzadeh P., Vosoughi A.R., 2009, DQM large amplitude vibration of
composite beams on nonlinear elastic foundations with restrained edges,Com-
munications in Nonlinear Science and Numerical Simulation, 14, 906-915

13. Nayfeh A.H., 2000,Nonlinear Interactions, First ed. NewYork: JohnWiley

14. Ozis T., Yildirim A., 2007, Study of nonlinear oscillators with u1/3 force
by He’s variational iteration method, Journal of Sound and Vibration, 306,
372-376

15. SanteeD.M.,GoncalvesP.B., 2006,Oscillations of a beamon anon-linear
elastic foundation under periodic loads, Shock and Vibration, 13, 273-284

16. SenalpA.D.,ArikogluA.,Ozkol I.,DoganV.Z., 2010,Dynamic respon-
se of a finite length Euler-Bernoulli beam on linear and nonlinear viscoelastic
foundations to a concentratedmoving force, Journal ofMechanical Science and
Technology, 24, 1957-1961

17. WazwazA.M., 2008,Astudyon linear andnonlinearSchrodinger equationsby
the variational iterationmethod,Chaos, Solitons and Fractals, 37, 1136-1142

18. WazwazA.M., 2009a,Partial Differential Equations and SolitaryWaves The-
ory, First ed., NewYork: Springer



Analytical solutions for free oscillations of beams... 651

19. WazwazA.M., 2009b,Thevariational iterationmethod for analytic treatment
for linear and nonlinear ODEs, Applied Mathematics and Computation, 212,
120-134

20. Wazwaz A.M., 2010, The variational iteration method for solving linear and
nonlinearVolterra integral and integro-differential equations, International Jo-
urnal of Computer Mathematics, 87, 1131-1141

21. Yihua M., Li Q., Hongzhi Z., Vibration analysis of Timoshenko beams on
a nonlinear elastic foundation,Tsinghua Science and Technology, 14, 322-326

22. Yildirim A., 2010, Application of He’s variational iteration method to nonli-
near integro-differential equations, Zeitschrift für Naturforschung – Section A
Journal of Physical Sciences, 65, 418-422

23. YildirimA.,OzisT., 2009,SolutionsofSingular IVPsofLane-Emdentypeby
the variational iterationmethod,NonlinearAnalysis, Series A,TheoryMethods
and Applications, 70, 2480-2484

24. YounesianD., Askari H., SaadatNia Z., KalamiYazdiM., 2011, Frequ-
encyanalysis of stronglynonlinear generalizedDuffingoscillatorsusinghomoto-
pyperturbationmethodandvariational approach,Nonlinear Science LettersA,
2, 9-14

25. YounesianD.,KargarnovinM.H.,EsmailzadehE., 2008,Optimalpassi-
ve vibration control of Timoshenko beams with arbitrary boundary conditions
traversed by moving loads, Proceedings of the IMECHE, Part K: Journal of
Multi-Body Dynamics, 229, 179-188

26. YounesianD.,KargarnovinM.H.,ThompsonD.J., JonesC.J.C., 2005,
Parametrically excited vibration of aTimoshenko beamon randomviscoelastic
foundation subjected to a harmonic moving load, Nonlinear Dynamics, 45,
75-93

27. Zhang Y.S., 2002, Orthogonality conditions of the transverse vibration mo-
de function for a stepped one-dimensional plate on a Winkler foundation and
subjected to an in-plane force, Proceedings of the IMECHE, Part K: Journal
of Multi-body Dynamics, 216, 271-273

Analityczne rozwiązania problemu drgań swobodnych belek

umieszczonych na sprężystym podłożu wyznaczone za pomocą

iteracyjnej metody wariacyjnej

Streszczenie

Wpracy przedstawiono zamknięte formy rozwiązańopisujących odpowiedź dyna-
miczną elastycznej belki spoczywającej na nieliniowo sprężystym podłożu. Równanie



652 D. Younesian et al.

ruchu rozwiązanoprzypomocy iteracyjnejmetodywariacyjnej (VIM). Sformułowanie
iteracyjne skonstruowanewoparciuo tęmetodępozwoliłonawyznaczenie odpowiedzi
dynamicznej belki. Charakterystyki częstościowe przedstawiono w zamkniętej formie
i podkreślono ich wrażliwość na wartość amplitud początkowych. Następnie zapre-
zentowanowyniki kilku symulacji numerycznychw celu oszacowaniawydajności i do-
kładności zastosowanej procedury rozwiązywania równań ruchu w dziedzinie czasu.
Wykazano, że metoda VIM jest wystarczająco prosta i wiarygodnaw rozwiązywaniu
układu sprzężonych, nieliniowych równań różniczkowych.

Manuscript received Jane 22, 2011; accepted for print September 17, 2011