Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 4, pp. 959-968, Warsaw 2013

THE EFFECT OF QUINTIC NONLINEARITY ON THE INVESTIGATION OF

TRANSVERSELY VIBRATING BUCKLED EULER-BERNOULLI BEAMS

Hamid M. Sedighi, Arash Reza

Department of Mechanical Engineering, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

Jamal Zare

National Iranian South Oil Company (Nisoc), Ahvaz, Iran

e-mail: hmsedighi@gmail.com

A new formulation of vibrations of the axially loaded Euler-Bernoulli beam with quintic
nonlinearity is investigated in the present study. The beam nonlinear natural frequency as
a function of the initial amplitude is obtained. In this direction,modern powerful analytical
methods namely He’s Max-Min Approach (MMA) and Amplitude-Frequency Formulation
(AFF) are employed to approximate the frequency-amplitude relationshipof thebeamvibra-
tions.Afterwards, it is clearly shownthat thefirst term in the series expansions is sufficient to
produce a highly accurate approximation of the nonlinear system. Finally, preciseness of the
present analytical procedures is evaluated in contrast with numerical calculationmethods.

Key words: quintic nonlinearity, He’s max-min approach, amplitude-frequency formulation,
nonlinear vibration, buckled beam

1. Introduction

With evolution of technology, accurate comprehending of characteristics of beam vibrations is
extremely important to researchers and engineers. The dynamic response of simply supported
and clamped-clamped structures at large amplitudes of vibration can be encountered in many
engineering applications. In such cases, it is of interest to know how far the characteristics of
the dynamic response deviate from those defined via the linear theory. The problem of beam
vibrationswas recently investigated bymany researcherswith different boundary conditions and
hypotheses.These researches predict the nonlinear frequency of beamswhich are very important
for the design of many engineering structures. However, research on flexible beams has so far
been restricted to cubic nonlinearity. Literature which considered higher order of nonlinearities
is very limited (Sedighi et al., 2012d).
Nowadays, substantial progresses had been made in analytical solutions to nonlinear equ-

ationswithout small parameters. There have been several classical approaches employed to solve
governing nonlinear differential equations to study nonlinear vibrations including perturbation
methods, Hamiltonian approach (He, 2010; Sedighi and Shirazi, 2013), He’s max-min approach
(MMA) (He, 2008b; Sedighi et al., 2012a; Yazdi et al., 2010), HAM (Sedighi and Shirazi, 2011;
Sedighi et al., 2012d), parameter expansion method (He and Shou, 2007; Sedighi and Shira-
zi, 2012; Sedighi et al., 2011, 2012c,e), homotopy perturbation method (HPM) (Shadloo and
Kimiaeifar, 2011), multistage adomian decompositionmethod (Evirgen and Özdemir, 2011), va-
riational iterationmethod (Khosrozadeh et al., 2013; Sedighi et al., 2012a), modified variational
iteration method (Yang et al., 2012), Laplace transform method (Rafieipour et al., 2012), mo-
notone iteration schemes (Hasanov, 2011), multiple scales method (Hammad et al., 2011), and
Navier and Levy-type solution (Baferani et al., 2011; Naderi and Saidi, 2011). The application
of the new equivalent function to the deadzone, preload and saturation nonlinearities for finding
dynamical behavior of beam vibrations using PEM and HA was investigated by Sedighi and



960 H.M. Sedighi et al.

Shirazi (2012, 2013), Sedighi et al. (2011, 2012c,e). The MMA and AFF have been shown to
solve a large class of nonlinear problems efficiently, accurately and easily, with approximations
converging very rapidly to the solution. Usually, few iterations lead to high accuracy of the so-
lution. Themin-max approach and amplitude-frequency formulation proposed by He (2008a,b)
are proved to be a very effective and convenient way for handling the non-linear problems, One
iteration is sufficient to obtain a highly accurate solution.
Todevelop the comprehensive understanding onnonlinear frequency of beamvibrations, this

paper brings quintic nonlinearities into consideration. The analytical solutions for geometrically
nonlinear vibration of the Euler-Bernoulli beam including quintic nonlinearity usingMMA and
AFF is obtained. The nonlinear ordinary differential equation of the beamvibration is extracted
from the partial differential equation with first mode approximation, based on the Galerkin
theory. The results presented in this paper exhibit that the analytical methods are very effective
and convenient for nonlinear the beam vibration for which highly nonlinear governing equations
exist.

2. Equation of motion

Consider the Euler- Bernoulli beam of length l, moment of inertia I, mass per unit length m
and modulus of elasticity E, which is axially compressed by loading P as shown in Fig. 1.

Fig. 1. Configuration of a uniformEuler-Bernoulli beam (a) simply supported beam,
(b) clamped-clamped beam

Denoting by w the transverse deflection, the differential equation governing the equilibrium in
the deformed situation is derived as

d2

dx2

(

EIw′′(x,t)
√

[1+w′2(x,t)]3

)

+Pw′′(x,t)
(

1+
3

2
w′
2
)

+mẅ(x,t)= 0 (2.1)

where w′′(x,t)/
√

[1+w′2(x,t)]3 is the “exact” expression for the curvature, using the approxi-
mation

w′′(x,t)
√

[1+w′2(x,t)]3
∼=w

′′(x,t)
[

1−
3

2
w′
2
(x,t)+

15

8
w′
4
(x,t)

]

(2.2)

where the nonlinear term Pw′′(x,t)[1+3w′
2
/2] has been extracted from Sedighi et al. (2012d).

Governing quintic nonlinear equation (2.3) can be expressed as

EIw(4)
(

1−
3

2
w′
2
+
15

8
w′
4
)

−9EIw′′′w′w′′+
45

2
EIw′′′w′

3
w′′−3EIw′′

3

+
45

2
EIw′

2
w′′
3
+Pw′′

(

1+
3

2
w′
2
)

+mẅ=0

(2.3)



The effect of quintic nonlinearity on the investigation ... 961

which is subjected to the following boundary conditions:

— for simply supported (S-S) beam

w(0, t) =
∂2w

∂x2
(0, t)= 0 w(l,t)=

∂2w

∂x2
(l,t)= 0 (2.4)

— for clamped-clamped (C-C) beam

w(0, t) =w(l,t) = 0
∂w

∂x
(0, t) =

∂w

∂x
(l,t)= 0 (2.5)

Assuming w(x,t) = q(t)φ(x), where φ(x) is the first eigenmode of the beam vibration, it
can be expressed as:

— S-S beam

φ(x)= sin
πx

l
(2.6)

—C-C beam

φ(x)=
(x

l

)2(

1−
x

l

)2
(2.7)

applying the Bubnov-Galerkin method yields

l
∫

0

[

EIw(4)
(

1−
3

2
w′
2
+
15

8
w′
4
)

−9EIw′′′w′w′′+
45

2
EIw′′′w′

3
w′′−3EIw′′

3

+
45

2
EIw′

2
w′′
3
+Pw′′

(

1+
3

2
w′
2
)

+mẅ
]

φ(x) dx=0

(2.8)

By introducing the following non-dimensional variables

τ =

√

EI

ml4
q=
q

l
(2.9)

the non-dimensional nonlinear equation of motion about its first buckling mode can be written
as

d2q(τ)

dτ2
+γ1q(τ)+γ2[q(τ)]

3+γ3[q(τ)]
5 =0 (2.10)

where:

— S-S beam

γ1 =π
4
−
Pl2π2

EI
γ2 =−

3

8
π6−

3

8

Pl2π4

EI
γ3 =

15

64
π8 (2.11)

– C-C beam

γ1 =500.534−
12.142Pl2

EI
γ2 =−6.654−

0.1694Pl2

EI
γ3 =−0.3673 (2.12)



962 H.M. Sedighi et al.

3. He’s max-min approach

In many engineering problems, it is easy to findmaximum/minimum interval of the solution to
a nonlinear equation. From this maximum-minimum relationship called “He Chengtian inequ-
ality” which has millennia history, He (2008b) introduced approximated solutions for nonlinear
vibrating systems.
Consider a generalized nonlinear oscillator in the form

q̈+qf(q, q̇, q̈)= 0 q(0)=A q̇(0)= 0 (3.1)

According to the idea of the min-max approach, we choose a trial-function in the form

q=Acos(ωt) (3.2)

where ω is an unknown frequencywhich to be determined. Themethod implies that the square
of the frequency satisfies the following inequality

fmin ¬ω
2
¬ fmax (3.3)

where fmax and fmin are the maximum and minimum values of the function f, respectively.
According to the Chentian interpolation (He, 2008b), we obtain

ω2 =
fmin+kfmax
1+k

(3.4)

Thevalueof kcanbeapproximatelydeterminedbyvariousapproximatemethods.Sothe solution
to Eq. (3.1) can be expressed as

q=Acos

(

√

fmin+kfmax
1+k

t

)

(3.5)

To investigate the min-max procedure, Eq. (2.10) should be rewritten in the following form

d2q

dt2
+(γ1+γ2q

2+γ3q
4)q=0 (3.6)

Assuming the solution to the above equation in the form of Eq. (10), yields

ω2min = γ1 ω
2
max = γ1+γ2A

2+γ3A
4 (3.7)

Therefore, according to Eq. (3.3)

γ1 ¬ω
2
¬ γ1+γ2A

2+γ3A
4 (3.8)

and on assumption (3.4), we obtain

ω2 =
γ1+k(γ1+γ2A

2+γ3A
4)

1+k
(3.9)

Using the Bubnov-Galerkin procedure and substituting Eqs. (3.9) and (3.5) into Eq. (2.10)
results in the following value for the parameter k

k=
5γ3A

2+6γ2
2γ2+3γ3A

2
(3.10)

Substituting Eq. (3.10) into Eq. (3.9) yields the nonlinear frequency of the beam as a function
of the amplitude, as follows

ω(A)=

√

γ1+
3

4
γ2A2+

5

8
γ3A4 (3.11)



The effect of quintic nonlinearity on the investigation ... 963

4. Amplitude-frequency formulation

To solve nonlinear problems, an amplitude–frequency formulation for nonlinear oscillators was
proposed by He (2008a), which was deduced using an ancient Chinese mathematics method.
According to He’s amplitude-frequency formulation, u1 =Acosτ and u2 =Acos(ωτ) serve as
the trial functions. Substituting u1 and u2 into equation (2.10) results in the following residuals

R1 =−Acosτ+γ1Acosτ+γ2A
3cos3τ+γ3A

5cos5τ

R2 =−Aω
2cos(ωτ)+γ1Acos(ωτ)+γ2A

3cos3(ωτ)+γ3A
5cos5(ωτ)

(4.1)

According to the amplitude-frequency formulation, the above residuals can be rewritten in the
forms of weighted residuals (Khan and Akbarzade, 2012)

R11 =
4

T1

T1/4
∫

0

R1cosτ dτ T1 =2π

R22 =
4

T2

T2/4
∫

0

R2cos(ωτ) dτ T2 =
2π

ω

(4.2)

Applying He’s frequency-amplitude formulation

ω2 =
ω21R22−ω

2
2R11

R22−R11
(4.3)

where

ω1 =1 ω2 =ω (4.4)

then the approximate frequency can be obtained

ω(A)=

√

γ1+
3

4
γ2A2+

5

8
γ3A4 (4.5)

5. Results and discussion

To verify the soundness of the proposed solutions by asymptotic approaches, the authors plot
the analytical solutions for a simply supported and clamped-clamped beam at the side of cor-
responding numerical results in Fig. 2, where the first approximated amplitude-time curves of
a uniform beam subjected to axial compression is presented for different initial conditions. As
can be seen, the first order approximation of q(τ) from the analytical method is in excellent
agreement with numerical results from fourth-order Runge-Kuttamethod. The exact analytical
solutions reveal that the first term in the series expansions is sufficient to result in a highly accu-
rate solution of the problem. Furthermore, these equations provide excellent approximations to
the exact period regardless of the oscillation amplitude. Thematerial and geometric properties
adopted here have been prepared in the Appendix.
For a vibrating Euler-Bernoulli beam, the Euler-Lagrange equation is as follows

d2

dx2
[EIw′′(x,t)]+Pw′′(x,t)+mẅ(x,t)= 0 (5.1)

Applying the Bubnov-Galerkin method and using the first eigenmode of the simply supported
beam, yields

d2q

dt2
+γ1q(τ)= 0 (5.2)



964 H.M. Sedighi et al.

Fig. 2. Comparison between analytical and numerical results for different initial amplitudes, symbols:
numerical solution, solid line: analytical solutions, (a) S-S beam, (b) C-C beam

Figures 3 and 4 display the effect of the normalized amplitude on the nonlinear behavior of a
vibrating beam. FromEq. (4.5), the nonlinear natural frequency is a function of the amplitude,
which means when the oscillation amplitude becomes larger (for both S-S and C-C beam), the
accuracy of approximated frequencies in usual beam theory (γ2 = γ3 =0) and cubic nonlinear
beam (γ3 =0) decreases. It confirms that the normalized amplitude has a significant effect on
nonlinear behavior of the beams. From these figures it is observed that the results from the
usual beam theory are incompatible with the quintic nonlinear beamwhen the initial condition
becomes larger.

Fig. 3. The impact of nonlinear terms on dynamical behavior of the beam for A=0.3, – * – usual beam
theory, – • – quintic nonlinear beam, (a) S-S beam, (b) C-C beam

Fig. 4. The impact of nonlinear terms on dynamical behavior of the beam for A=0.4, – * – usual beam
theory, – • – quintic nonlinear beam, (a) S-S beam, (b) C-C beam



The effect of quintic nonlinearity on the investigation ... 965

In order to investigate the effect of parameter γ1 on the nonlinear behavior of the quintic
nonlinear beam, the natural frequency as a function of γ1 has been illustrated in Fig. 5 for
different amplitudes. It is observed that the difference between the nonlinear fundamental fre-
quency and the usual beam frequency increases with the vibration amplitude. Also, the percent
of error in approximating the natural frequency as a function of γ1 for different values of the
oscillation amplitude has been depicted in Fig. 6. The relative error of the approximated simple
beam theory frequency progressively increases at lower values of γ1 for all values of vibration
amplitudes.

Fig. 5. Comparison of fundamental frequencies of the usual beam and quintic nonlinear beam as a
function of γ1, (a) S-S beam, (b) C-C beam

Fig. 6. The percent of error in approximating the natural frequency of the usual beam as a function
of γ1, (a) S-S beam, (b) C-C beam

In order to demonstrate the necessity of quintic nonlinear terms, the percent of error in ap-
proximating the cubic natural frequency as a function of γ1 for different values of the oscillation
amplitude has been illustrated in Fig. 7.When the parameter γ1 decreases, the relative error of
the approximated cubic beam frequency increases, especially for the simply supported beam.

6. Conclusion

In the current study, twomodernpowerful analyticalmethods calledHe’smax-minapproachand
amplitude-frequency formulation were employed to solve the governing equation of vibration of
quintic nonlinear beams. It demonstrated that the fundamental frequency based upon the linear
theory and cubic nonlinear beam can be different from the natural frequency of the quintic



966 H.M. Sedighi et al.

Fig. 7. The percent of error in approximating the natural frequency of the cubic nonlinear beam as a
function of γ1, (a) S-S beam, (b) C-C beam

nonlinear beam at large vibration amplitudes. An excellent first-order analytical solution using
modernasymptotic approacheswas obtained.The soundness of the obtained analytical solutions
was verified by numerical methods.

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Manuscript received October 15, 2012; accepted for print April 9, 2013