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Engineers, Part C: Journal of MechanicalProceedings of the Institution of Mechanical
http://pic.sagepub.com/content/early/2013/01/24/0954406213475561The online version of this article can be found at:
DOI: 10.1177/0954406213475561
online 24 January 2013publishedProceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
Hamid M. Sedighi and Kourosh H. ShiraziAsymptotic approach for nonlinear vibrating beams with saturation type boundary condition
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Original Article
Asymptotic approach for nonlinear vibrating beams with saturation typeboundary condition
Hamid M Sedighi and Kourosh H Shirazi
Date received: 1 November 2012; accepted: 20 December 2012
Abstract
This article attempts to analyze the complicated vibrational behavior of the Euler–Bernoulli beam exposed to saturated
nonlinear boundary condition through proposing an innovative precise equivalent function. In this direction, the beamvibrational response is attained by way of a new effective analytical method namely Hamiltonian approach. Despite all theprocedures based on perturbation methods that deadzone or saturation dead-band parameter is omitted during inte-gration, this study indicates that how using Hamiltonian approach, the impact of dead-band parameter is taken intoaccount leading to higher accuracy of the approximated solution. Finally, the precision of the proposed equivalentfunction is evaluated in comparison with the numerical solutions, giving excellent results.
Keywords
Accurate equivalent function, saturation nonlinearity, nonlinear vibration of beam, Hamiltonian approach
Introduction
Most of the beam structures may operate in the non-
linear range during lifetime due to sources of nonli-
nearity such as large deformation effects or boundary
conditions. In the case of non-differentiable nonli-
nearity like saturation type, the analytical solutions
related to the nonlinear problems become very com-
plex. Engineering mechanisms, as electrical power sys-
tems, usually involve saturation type nonlinearities.1
This nonlinearity, due to its inherent difficulty, has
not been modeled exactly by researchers, till present.The approximation of this nonlinear condition in
order to obtain the analytical solution of dynamical
systems behavior has been always the major difficulty
of engineer’s computations. Xin et al.2 analyzed the
impact of saturation nonlinearities and disturbance
rejection on power system small-signal stability
based on the estimated stability region and maximum
endurable disturbance rejection. In the other research,
Xin et al.3 studied the class of linear dynamical sys-
tems subject to saturation nonlinearities and approxi-
mated the considered mechanisms by singular
perturbation dynamical systems based on the notion
of Pade approximation. A frequency-domain criterionfor the elimination of limit cycles in a class of digital
filters using saturation nonlinearity was presented by
Singh.4 The objective of this study is to introduce the
innovative exact equivalent fraction (EF) for satur-
ation nonlinearity as a boundary condition and to
implement the Hamiltonian approach (HA)5 in the
nonlinear beam vibrations.
The introduced function is suitable for analytical
studies of nonlinear dynamical systems using approxi-
mated approaches such as energy balance method,6
variational iteration method,7 modified variational
iteration method,8 Lindstedt–Poincare ´ method,9
Pade ´ technique,10 max-min approach,11 HAM,12,13
parameter expansion method,14–17
amplitude–fre-quency formulation,18 homotopy perturbation trans-
form method,19 Laplace transform method20 and
homotopy perturbation method21–23 as well as numer-
ical studies in direct simulations. A survey of some
recent developments in asymptotic techniques for
strongly nonlinear equations has been investigated
by He.24
Department of Mechanical Engineering, Shahid Chamran University,
Ahvaz, Iran
Corresponding author:Hamid M Sedighi, Department of Mechanical Engineering, Shahid
Chamran University, Ahvaz, Iran.
Email: [email protected]
Proc IMechE Part C:
J Mechanical Engineering Science
0(0) 1–8
! IMechE 2013
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Approximate methods for studying nonlinear
vibrations of distributed parameter systems are
important for investigative and/or designing pur-
poses. Usually, in the analytical procedures in order
to approximate the nonlinear responses, deadzone
and saturation dead-band parameter is deleted
during integration.25–27 This study indicates thatusing HA, the impact of this parameter is included
and so the accuracy of the approximated solution
increases.
The current work attempts to compute nonlinear
frequency–amplitude relation in a cantilever beam in
the presence of saturation type nonlinear boundary
condition. Recently a new powerful method namely
Hamiltonian approach (HA) proposed by He5 has
proven to be a very effective and convenient way of
handling nonlinear problems and has been success-
fully developed and applied to various engineering
problems.
28–32
Previously, He
6
introduced the energybalance method based on collocation and the
Hamiltonian. This approach is very simple but
strongly depends on the chosen location point.
Recently, He5 proposed the Hamiltonian approach.
This approach is a kind of energy method with a
vast application in conservative oscillatory systems.
He et al.28 used HA to establish frequency–amplitude
relationship of Duffing-harmonic oscillator. Some
new asymptotic methods for the solitary solutions of
nonlinear differential equations, nonlinear differen-
tial-difference equations, and nonlinear fractional dif-
ferential equations were studied by He.29 Sedighi
et al.30 have presented the advantages of some effect-ive analytical approaches such as min-max approach,
parameter expansion method, Hamiltonian approach,
variational iteration method and energy balance
method on the asymptotic solutions of governing
equation of transversely vibrating cantilever beams.
The objective of this article is to substantiate the
acceptable ability of HA in predicting the analytical
response of nonlinear systems dealing with nondiffer-
entiable nonlinearities. The innovative EF for nondif-
ferentiable saturated nonlinearity has been employed
in the analytical procedures, here. The effects of vibra-
tion amplitude besides saturation dead-band param-eter on natural frequency are taken into
consideration. The results presented in this article
exhibit that the analytical method is very effective
and high-accuracy for nonlinear vibration for which
the highly nonlinear governing equations exist.
Mathematical formulation
The cantilever beam studied in this work has length L,
mass per unit length of the beam m, moment of inertia
I and modulus of elasticity E as shown in Figure 1.
Assume that the Euler–Bernoulli theorem can be
adopted. Crespo da Silva and Glenn33 derived theequations of motion governing the nonlinear nonpla-
nar vibrations of Euler–Bernoulli beams. The integral
partial-differential equations are simplified to the case
of planar motion of cantilever beam under transverse
excitation. The governing partial differential equation
for the nonlinear flexural vibration of the beam is, asfollows
m €v þ EIviv þ EI v0 v0v00ð Þ0Â Ã0þ 1
2m v0
Z xL
@2
@t2
Z x0
v02dx
24
35dx
8<:
9=;
0
¼ 0 ð1Þ
Here x is the axial coordinate which is measured from
the origin, v denotes the lateral vibration in y direc-
tion. The boundary conditions for the beam including
saturation nonlinear type can be expressed as
v 0, tð Þ ¼ @v
@x0, tð Þ ¼ 0,
@2v
@x2L, tð Þ ¼ 0,
EI @3v
@x3L, tð Þ ¼ F sat L, tð Þ ð2Þ
where F sat L, tð Þ is nonlinear boundary condition at its
end as shown in Figure 2 and is described by the fol-
lowing nonlinear saturation formula
F sat vð Þ ¼ f 1 v
ð Þv5
À
k pv À 4v4
f 2 vð Þ v4
8<: ð3Þ
Figure 1. Configuration of cantilever beam with saturation
type boundary condition.
2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)
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where k p is the constant of primary spring. Assuming
v x, tð Þ ¼ q tð Þ’ xð Þ, where ’ xð Þ is the first eigenmode of
the clamped–free beam and can be expressed as
’ xð Þ ¼ cosh xð Þ À cos xð Þ
Àcosh Lð Þ þ cos Lð Þsinh Lð Þ þ sin Lð Þ
sinh xð Þ À
sin xð Þð Þ
ð4Þ
where ¼ 1:875 is the root of characteristic equation
for first eigenmode. Applying the weighted residual
Bubnov–Galerkin method yields
Z L0
m €v þ EIviv þ EI v0 v0v00ð Þ0Â Ã0
þ 1
2m v0
Z x
L
@2
@t2 Z x
0
v02dx
24
35dx
8<:
9=;
01A’ xð Þdx ¼ 0 ð5Þ
to implement the end nonlinear boundary condition,
applying integration by parts on equation (5), it is
converted to the following
Z L0
m €vþEI v0 v0v00ð Þ0Â Ã0
þ1
2m v0
Z xL
@2
@t2
Z x0
v02dx
2
4
3
5dx
8<:
9=;
1
A’ xð Þdx
þZ L0
EIviv’ xð Þdx¼0 ð6Þ
Z L0
m €v þ EI v0 v0v00ð Þ0Â Ã0
þ 1
2m v0
Z xL
@2
@t2
Z x0
v02dx
24
35dx
8<:
9=;1A’ xð Þdx
þ EIv000’ xð Þ
L
0ÀZ
L
0
EIv000d ’ xð Þð Þ ¼ 0 ð7Þ
In the above equation the boundary condition term
EIv000 L, tð Þ is replaced by F sat L, tð Þ. By introducing the
following nondimensional variables
¼ ffiffiffiffiffiffiffiffiffi
EI
mL4
r t, q ¼ q
Lð8Þ
and applying the weighted residual Bubnov–Galerkin
method, the nondimensional nonlinear equation of
motion can be expressed as
€q þ 1q þ 2q3 þ 3q _q2 þ 4q2€q þ F sat L, tð Þ ¼ 0
ð9Þ
where
1 ¼ 12:3624,2 ¼ 40:44, 3 ¼ 4 ¼ 4:6 ð10Þ
To solve nonlinear ordinary equation (9) analytically,
the saturation condition F sat, must be formulated,
properly. The EF for the shifted Heaviside function
is expressed in the following relation
H v À að Þ ¼ 1
2þ 1
2
v À aj jv À a
ð11Þ
In this article, we introduce novel exact equivalentfunction for this nonlinearity as
F sat vð Þ ¼ k pv þ Àk pv þ f 2 vð ÞÀ ÁH v À að Þ
þ Àk pv þ f 1 vð ÞÀ ÁH Àv À að Þ ð12Þ
Using this new definition of F sat, and setting
f 1 vð Þ ¼ f 2 vð Þ ¼ ksv equation (9) can be rewritten as
follows
€q þ 01q þ 2q3 þ 3q _q2 þ 4q2
€q
þ5 q
À0 À q
þ0 À Á ¼
0
ð13-a
Þwhere
01 ¼ 1 þ p þ s, 5 ¼ s
2, 0 ¼
2L,
p ¼ 4k pL3
EI , s ¼ 4ksL3
EI ð13-bÞ
Overview of the Hamiltonian approach
Consider a general form of nonlinear differential
equation
€q þ f q, _q, €qð Þ ¼ 0, q 0ð Þ ¼ A, _q 0ð Þ ¼ 0, ð14Þ
Figure 2. Plot of generalized saturation nonlinearity.
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Establishing variational principle for equation (14)
proposed by He5 yields
J qð Þ ¼Z T =
4
0
À _q2
2þ F q, _q, €qð Þ
dt, ð15Þ
where T is the period of the nonlinear oscillator.
In the functional (15), _q2=2 is the kinetic energy and
F q, _q, €qð Þ satisfies @F =@q ¼ f . Throughout the oscilla-
tion since the system is conservative, the total energy
remains unchanged during the motion. Hamiltonian
of the oscillator becomes a constant value H ¼ K :E :þP:E : ¼ H 0, where the terms K :E : and P:E : are kinetic
and potential energies of the system, respectively. The
new function H qð Þ can be written as5
^H qð Þ ¼ Z
T =4
0
_q2
2 þ F q, _q, €qð Þ dt ¼T
4 H 0 ð16Þ
It is obvious that
@H
@T ¼ H 0
4ð17Þ
and then the frequency–amplitude relation is obtained
by setting
@
@Ai
@H
@T ¼0 or
@
@Ai
@H
@ 1=!ð Þ ¼
0
ð18
Þ
Approximation by Hamiltonian
technique
Assuming q ¼ Pni ¼1 Ai cos i ! ð Þ as an approximate
solution. Using first term approximation, the solution
can be assumed as
q ¼ A cos ! ð Þ ð19ÞLet us consider the nonlinear equation (13-a) the
function F q, _q, €qð Þ can be expressed as
F q, _q, €qð Þ ¼ _q2
21 þ 4q2À Áþ 0
1
2q2 þ 2
4q4
þ Z 5 q À 0 À q þ 0 À Á dq |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}I :P:
ð20Þ
The last part of equation (20) yields
I :P: ¼ 5
Àq2=2 þ 0q q40
q2=2 À 0q þ 02 q4 0
&
À
Àq2=2 À 0q q4À 0
q2
=2 þ 0q þ 02
q4À 0& ! ð21
ÞAssuming q ¼ A cos ! ð Þ and A4 0, in the one-
fourth of oscillation period, equation (21) can be sim-
plifies as
I :P:¼À5
q2
2þ0qþ02
þ Àq2=2þ0q q40
q2=2À0qþ02 q40
(
ð22Þ
Therefore, the Hamiltonian of the system is easily
established as
H qð Þ ¼Z T =4
0
_q2
2þ 0
1
2q2 þ 2
4q4 þ 4
2q2
_q2
À5
1
2q2 þ 0q þ 02
dt
þZ 1=!ÂcosÀ1 0=Að Þ
0
51
2q2 À 0q þ 02
dt
þ Z T =4
1=!ÂcosÀ1 0=Að Þ
5 À 12
q2 þ 0q
dt
ð23Þ
Substituting equation (19) into (23) leads to
setting
@
@A
@H
@ 1=!ð Þ
¼ 0:39233!
2A3 À 0:5892A3
þ 0:7854 !2 À 01 þ 25À ÁA À 5A cos À1 0
A À 5
03
A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2 À 02p þ 5
0A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 À 02
p ¼ 0 ð25Þ
H ¼ 1
!0:14732A4 þ 0:39270
1A2 þ 0:3927A2!2 À 0:39275A2 þ 0:09823A4!2 À 1:5708502
ÀÀ1:55A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 À 0
A
2s þ 0:55A2 cos À1 0
A
þ 5
2 cos À1 0A
À 0:1255A2
1A ð24Þ
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From equation (25) we can easily find that the solution ! is
! Að Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:52A2 þ 2 0
1 À 25
À Áþ 2:545 cos À1 0A
À Áþ 03
A2 ffiffiffiffiffiffiffiffiffiffi
A2À02p À 0 ffiffiffiffiffiffiffiffiffiffi
A2À02p
3A2 þ 2
vuut ð26Þ
Replacing ! from equation (26) into equation (19) yields
q ð Þ ¼ A cos
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1:52A2 þ 2 0
1 À 25
À Áþ 2:545 cosÀ1 0A
À Áþ 03
A2 ffiffiffiffiffiffiffiffiffiffi
A2À02p À 0 ffiffiffiffiffiffiffiffiffiffi
A2À02p
3A2 þ 2
vuut
0B@
1CA ð27Þ
Figure 3. Comparison of the results of analytical solutions with the numerical simulations for 0¼ 0:5 A , p¼s¼ 0:1. (a) Time
history; (b) Phase portrait. Symbols: numerical solution; Solid line: analytical solutions.
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Results and discussion
In order to demonstrate the integrity of proposed EF
and solutions by HA, the analytical results together
with the corresponding numerical results have been
presented, graphically. First-order approximation of
q
ð Þusing HA analytic method and introduced EF,
exhibits a good agreement with numerical resultsfrom fourth-order Runge–Kutta method, as depicted
in Figure 3. In addition, the phase portrait curves for
different amplitudes A ¼ 0.01, 0.05, 0.08, 0.1 and 1.2 is
plotted to reveal the high accuracy of asymptotic solu-
tions. Consequently, the phase-space curves generated
from HA are thoroughly acceptable as verified by
numerical results.
To explain the effect of saturation dead-band par-
ameter 0 on the nonlinear response of beam vibration,
the normalized frequency as a function of amplitude A
is shown in Figure 4 for different values of 0. For the
same normalized amplitudes, the frequency of beam
vibration shifts downward, when the saturation par-
ameter 0 moves upward. It is evident that the analyticsolution converges rapidly and is valid for a wide
range of saturation parameter and initial conditions.
The influence of primary and secondary spring par-
ameters p, s on the natural frequency has been illu-
strated in Figure 5 as a function of normalized
amplitude of cantilever beam vibration. As can be
observed, the greater spring parameters p, s
Figure 4. Fundamental normalized frequency vs. normalized amplitude: effect of parameter 0.
Figure 5. Fundamental normalized frequency vs. normalized amplitude: effect of parameters p, s.
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produce larger limit cycle frequency. It is inferred
from Figure 4 and 5 that the fundamental frequency
increases as the vibration amplitude gets larger,
Regardless of dead-band and spring parameters.
Furthermore, the effect of different boundary con-
ditions (free, saturated and dead-zone) on the asymp-
totic solution of cantilever beam vibration has been
studied through Figure 6. Free end cantilever beam
has the less nonlinear frequency. However, the fre-
quency of beam under saturated boundary condition
is larger than similar beam under dead-zone boundary
condition. It should be pointed out that dead-zoneboundary condition appears via substituting p ¼ 0
in the equation of motion.
The impact of saturation parameter is also investi-
gated in the following. As illustrated in Figure 7,
increasing saturation parameter causes the vibrating
frequency to increase and consequently makes the
time period of cantilever beam oscillation decrease.
Conclusion
An advanced effective asymptotic method namely HA
was employed to establish the frequency–amplitude
relationship of vibrating cantilever beam under satur-
ation type boundary condition. In this direction, theinnovative EF for nondifferentiable saturated nonli-
nearity has been engaged to predict the analytic
Figure 6. Comparison between solutions of vibrating cantilever beam with free, saturated and dead-zone boundary conditions.
Figure 7. The impact of saturated boundary condition on the asymptotic solution.
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response of nonlinear beam vibrations. The saturated
nonlinearity was rewritten precisely using continuous
functions. The accuracy of the obtained results using
introduced EF, confirms the strength of the presented
modeling.
FundingThis research received no specific grant from any funding
agency in the public, commercial, or not-for-profit sectors.
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