Applied Mathematics, 2011, 2, 1469-1478
doi:10.4236/am.2011.212209 Published Online December 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
The Analytical and Numerical Solutions of Differential
Equations Describing of an Inclined Cable Subjected to
External and Parametric Excitation Forces
Mohamed S. Abd Elkader1,2
1Department of Mathematics and Statistics, Faculty of Science, Taif University, El-Taif,
Kingdom of Saudi Arabia
2Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University,
Menouf, Egypt
E-mail: moh_6_11@yahoo.com
Received October 21, 2011; revised November 22, 2011; accepted November 30, 2011
Abstract
The analytical and numerical solutions of the response of an inclined cable subjected to external and para-
metric excitation forces is studied. The method of perturbation technique are applied to obtained the periodic
response equation near the simultaneous principal parametric resonance in the presence of 2:1 internal reso-
nance of the system. All different resonance cases are extracted. The effects of different parameters and
worst resonance case on the vibrating system are investigated. The stability of the system are studied by us-
ing frequency response equations and phase-plane method. Variation of the parameters α2, α3, β2, γ2, η2, γ3, η3,
f2 leads to multi-valued amplitudes and hence to jump phenomena. The simulation results are achieved using
MATLAB 7.6 programs.
Keywords: Perturbation Method, Resonance, Chaotic Response, Stability
1. Introduction
Cable structures play an important role in many engi-
neering fields, such as civil, ocean and electric engineer-
ing. Arafat and Nayfeh [1] studied the motion of shallow
suspended cables with primary resonance excitation. The
method of multiple scales is applied to study nonlinear
response of this suspended cables and its stability and the
dynamic solutions. Some interesting work on the nonlin-
ear dynamics of cables to the harmonic excitations can
be found in the review articles by Rega [2,3]. Nielsen
and Kierkegaard [4] investigated simplified models of
inclined cables under super and combinatorial harmonic
excitation and gave analytical and purely numerical re-
sults. Zheng, Ko and Ni [5] considered the super-har-
monics and internal resonance of a suspended cable with
almost commensurable natural frequencies. Zhang and
Tang [6] investigated the chaotic dynamics and global
bifurcations of the suspended inclined cable under com-
bined parametric and external excitations. Nayfeh et al.
[7] investigated the nonlinear nonplanar responses of
suspended cables to external excitations. The equations
of motion governing such systems contain quadratic and
cubic nonlinearities, which may result in 2:1 and 1:1 in-
ternal resonances. Chen and Xu [8] investigated the glo-
bal bifurcations of the inclined cable subjected to a har-
monic excitation leading to primary resonances with the
external damping by using averaging method. Kamel and
Hamed [9], studied the nonlinear behavior of an inclined
cable subjected to harmonic excitation near the simulta-
neous primary and 1:1 internal resonance using multiple
scale method. Abe [10] investigated the accuracy of non-
linear vibration analyses of a suspended cable, which
possesses quad ratic and cub ic nonlinearities, with 1:1 in -
ternal resonance. The nonlinear dynamics of suspend
cable structures have been studied with 2:1 internal reso-
nances by the authors [11,12]. Experimental studies of
this problem have been conducted by Alaggio and Rega
[13] and Rega and Allagio [14], however explicit stabil-
ity regions for t he semi-triv i al sol ut i on h ave not be en cal -
culated analytically. Here, we use a modal model to com-
pute the insta b i li t y b o un d a ry f or a range of excitati on fre-
quencies close to the 2:1 resonance for an inclined cable,
including nonlinear modal interaction. The out-of-plane
M. S. ABD ELKADER
1470
dynamic stability of in clined cables subj ected to in-plane
vertical support excitation is investigated by Gonzalez-
Buelga et al. [15]. Perkins [16] examined the effect of
one support motion on the three-dimensional nonlinear
response. Using the Galerkin method, he constructed a
two-degree-of-freedom model to analyze the 2:1 internal
resonance. Lee and Perkins [17] extended the work to in-
clude second-order perturbations and multiple internal
resonances. Still, the focus was on the 2:1 internal reso-
nance, whereas the excitation was changed to a harmoni-
cally varying load per unit length acting in the static
equilibrium plane. Lee and Perkins [18] also used a
three-degree-of-freedom model to simulate non-linear
response of suspended, inclined cables driven by planar
excitation and determined the existence and stability of
four classes of periodic solutions.
Eissa and Sayed [19-21] and Sayed [22], studied the
effects of different active controllers on simple and spring
pendulum at the primary resonance via negative velocity
feedback or its square or cubic. Sayed and Hamed [23]
studied the response of a two-degree-of-freedom system
with quadratic coupling under parametric and harmonic
excitations. The method of multiple scale perturbation
technique is applied to solve the non-linear differential
equations and obtain approximate solutions up to and
including the second-order approximations. Sayed and
Kamel [24,25] investigated the effects of different con-
trollers on the vibrating system and the saturation control
to reduce vibrations due to rotor blade flapping motion.
The stability of the ob tained numerical solution is inves-
tigated using both phase plane methods and frequency
response equations. Amer and Sayed [26], studied the
response of one-degree-of freedom, non-linear system
under multi-parametric and external excitation forces
simulating the vibration o f the cantilever beam. Variation
of some parameters leads to multi-valued amplitudes and
hence to jump phenomena. Sayed et al. [27], investigated
the non-linear dynamics of a two-degree-of freedom vi-
bration system including quadratic and cubic non-lin-
earities subjected to external and parametric excitation
forces. The stability of the system is investigated using
both frequency response curves and phase-plane trajecto-
ries. The effects of different parameters of the system are
studied numerically.
This work deals with model having two-degree-of-
freedom nonlinear system subjected to external and pa-
rametric excitation forces describes the vibrations of an
inclined cable. The method of multiple scales perturba-
tion is applied to obtain modulation response equations
near the simultaneous principal parametric resonance in
the presence of 2:1 internal resonance (22
2
 and
12
2
). The stability of the proposed analytic nonlin-
ear solution near the above case is studied and the stabil-
ity condition is determined. The effect of different pa-
rameters on the steady state response of the vibrating
system is studied and discussed from the frequency re-
sponse curves. The numerical solution and chaotic re-
sponses of the nonlinear system of an inclined cable for
some different parameters are also studied. A compari-
son with previously published work is included.
2. Mathematical Analysis
Our attention is focused on an elastic-sag hanging at
fixed supports and excited by harmonic and parametric
distributed vertical forcing in plane. The two-degree-of-
freedom describing the nonlinear dynamics of cable
shown in Figure 1, can be written as:
22232
11 222 2
20xcxx xyxxy
 
 
  (1)
23
2233 3
112 2
2
cos cos
ycyy xyyx
ftyf t


 
 2
y
(2)
where
x
and denote in-plane and out-of-plane dis-
placements, respectively, and dots denote derivatives
with respect to the time t. The parameters 1 and 2 are
the viscous damping coefficients, 1
y
c c
and 2
are the
natural frequencies associated with in-plane and out-of-
plane modes 1 and 2 are the excitation frequencies, f1
and f2 are the excitation forces amplitude, 222
,,,

233
,,
and 3
are the coefficients of nonlinear pa-
rameters. The linear viscous damping forces, the exciting
forces and nonlinear parameters are assumed to be
22 222
112 2
22
ˆˆˆ ˆ
ˆˆ
,, , ,
ˆ
ˆˆ
,
1, 22, 3
nnsss
ss
ccc cff
ns
,
s


 


where
is a small perturbation parameter and
0
1. For the convenience of the analysis of Equa-
tions (1)-(2), the non-dimensional parameter
is in-
troduced. We ca n obt ai n
22 2
112 2
23 2
22
ˆ
ˆ
ˆ
2(
ˆˆ
()0
2
)
x
cx xxy
xxy

 
 

  (3)
22 232
2233 3
2112 2
ˆˆˆ
ˆ
2(
ˆˆ
(coscos )
)
y
cy yxyyxy
ftyf t

 

  (4)
The parameters 23 2
ˆ
ˆˆ
,,

ˆˆˆ
ˆˆ
,,,,cc

are of the order of 1 and
the parameters 12232312
are of the order
of 2. The approximate solution of Equations (3)-(4) can
be obtained using the method of multiple scales [28]. Let
ˆˆ
ˆ
,,,f f
0012 1012
22012
(;) (,,)(,,)
(,,)
x
txTTT xTTT
xTTT


(5)
Copyright © 2011 SciRes. AM
M. S. ABD ELKADER1471
Figure 1. A schematic of inclined cable under combined
excitations.
0012 1012
22012
(;)(,,)(,,)
(,,)
yty TTTyTTT
yTTT


(6)
where, n
n
Tt
(n = 0, 1, 2) are the fast and slow time
scales respectively. In terms of 0 and , the time
derivatives transform according to 1
,TT 2
T
2
012
2222
00110
2
d
d
d2(2
d
DD D
t
DDDDDD
t


 
 2
)
(7)
where nn
. Substituting Equations (5)-(6) and
(7) into Equations (3)-(4) and equating the coefficients of
similar powers of in both sides, we obtain the differen-
tial equations as follo ws:
DT 
Order 0
()
:
22
010
()Dx
0
0
(8)
22
020
()Dy (9)
Order 1
()
:
22 2
011 010202
ˆ
ˆ
()2DxDDxx

 
2
0
y
0
y
(10)
22
02101030
ˆ
()2DyDDyx
 
2
(11)
Order ()
:
22 2
01210011 020
00201 201
32
20 200
1
()2 2
ˆ
ˆ
ˆ
222
ˆˆ
DxDxDDxDDx
cDxxxyy
xxy


 


(12)
22 2
02210 011 020
3
2003 010130
2
300 110 0220
()2 2
ˆˆ
ˆ
2()
ˆˆ
ˆcos cos
DyDyDDyDDy
cDyxyyxy
yxfTyfT



 
(13)
The solution of Equations (8)-(9) can be expressed in
the complex form:
01210
( ,)exp()
x
AT Ti Tcc
 (14)
012 20
( ,)exp()yBTTiTc
where cc denotes the complex conjugate of the preceding
terms and
A
, are complex functions in 1 and 2
T
which determined through the elimination of secular and
small-divisor terms from the first and second-order of
approximations.
BT
In this case, we analyze the case where 22
2
 and
12
2
. To describe quantitatively the nearness of the
resonances, we introduce the detuning parameters 1
and 2
according to 221
ˆ
2

, 2
12
ˆ
2

.
Substituting Equations ( 14)-( 15) into Equations (10)-(11)
and eliminating the secular terms leads to the solvability
conditions for the first-order expansion as:
2
12 21
1ˆˆ
2exp(
iDA BiT
 
)0

(16)
21 321
ˆˆ
2exp(iDB AB iT
 
)0
(17)
After eliminating the secular terms, the particular solu-
tions of Equations (10)-(11) a re given by :
2
222
110
222
111
ˆ
ˆˆ
exp(2 )
3
x
AiT AABB


cc

(18)



3
10
2
2
212
12
ˆexpyABi


Tcc




(19)
Now substituting Equations (14)-(15) and Equations
(18)-(19) into Equations (12)-(13), the following are ob-
tained
22
012
22
1112121
1
()
ˆ
(22)exp( )
Dx
DA icA iDAABBAAiT
NSTcc
 

 

0
(20)

22
022
22
122223 42
2220
()
ˆ
22 exp(
1ˆexp( ())
2
Dy
DB icB iDBAABBBiT
fBiTNSTcc
 

 

0
)
(21)
where
23 22
12
222
212 1
ˆˆ
ˆˆ
24ˆ
2,
()
 
 
 


2
2
22
2
1
ˆ
10 ˆ
3
3
 

,
2
323
33
2
22
21
21
ˆˆˆ
2ˆ
2,
()


 




23
43
2
1
ˆˆ
2ˆ
3

 

c
 (15) and NST stands for non-secular terms. Eliminating the
Copyright © 2011 SciRes. AM
M. S. ABD ELKADER
1472
secular terms leads to the solvability conditions for the
second-order expansion
22
2111 2
11
ˆ
22
iDADA icAABBAA

  (22)
2
22 13
2
42
22
ˆ
22
1ˆˆ
exp( )
2
iDBDBicBAAB
BBf BiT

 
 11
(23)
Stability Analysis of Nonlinear Solutions
From Equation (7), multiplying both sides be 1
2,i
2
2i
we get
2
1111
d
22 2
d
A
iiDAi
t


2
DA
(24)
2
221
d
22 2
d
B
iiDBi
t


22
DB
(25)
To analyze the solutions of Equations (16)-(17) and
Equations (22)-(23), we express
A
and in the polar
form B
12
12 12
(, )(2),(, )(2)
ii
A
TTae BTTbe
 (26)
where a , b and (1,2
ss)
are the steady state ampli-
tudes and phases o f the motion respectively. Substituting
Equations (26), (16)-(17) and Equations (22)-(23) into
Equations (24)-(25) and equating the real and imaginary
parts we obtain the following equations describing the
modulation of the amplitudes and phases:
2
222
1
2
1
1
sin
48
acab

2

 


(27)
2
222
12
2
11
23
523 6
2
11
12
cos
48
88
16
ab
ab a











 


(28)
23 32
21
2
222
2
sin sin
44
8
f
bcbab b
 



 



(29)
2
323
22
2
2
32
10 1
2
2
cos
48
cos
4
bab
f
bb




 



 
9
ab
(
30)
where

2
37 238
910
32
22
2
22 22
56781234
11122 2112
1
,
88
32 32
,,,, , ,
ˆˆ
and2 ,2
TT

Form the system of Equations (27)-(30) to have sta-
tionary solutions, the following conditions must be satis-
fied:
12
0ab



(32)
It follows from Equation (31) that
2111
1,
22
 

 (33)
Hence, the steady state solutions of Equations (27)-(30)
are given by
2
222
12
2
11
sin 0
48
ca b




(34)
23
6
222
12 2
2
11
2
523
2
12
1
1
() cos
48
8
0
816
ab
ab

 





 





 



a
(35)
23 32
2
222
2
21
sinsin 0
44
8
f
cbab b
 




 (36)
2
23 3
12
2
32
10 1
2
2
2
1cos
24
8
cos 0
4
bab
f
bb
 



 



 
9
ab
(37)
Solving the resulting algebraic equations for the fixed
points of the practical case where , that is
non-planar motions, we obtain the following frequency
response equations
0, 0ab
2222224264
12 1123
22 6
121 122
42
12
()
2( )2( )
20
acaabab
ab a
ab
 
 




(38)
2222 242 2622
12910 91
2
42422
2
10 19 1042
2
2
42 12
2
1
4
216
cos()0
2
bcbabbba
f
babab
fab




 

2
b
(39)
where
2


 




 






 

 
5236 222
123
22
11
1
1
12
,,
884
16 8



 
 


 

 

(31) and 23 3
422
24
8
 


.
The stability of the obtained fixed points for the simul-
Copyright © 2011 SciRes. AM
M. S. ABD ELKADER1473
taneous primary, principal parametric and 2:1 internal
resonance case is determined and studied as follows:
one lets
101110 11
01
,
and ss s
aaabbb
 
 
 (40)
where a
10, b
10 and 0
s
are the solutions of Equations
(34)-(37) and a
11, b11, 1
s
are perturbations which are
assumed to be small compared to a10, b10 and 0
s
. Sub-
stituting Equation (40) into Equations (27)-(30), using
Equations (34)-(37) and keeping only the linear terms in
a11, b11, 1
s
we obtain:


2
113 102021
31020 11
111cos
2sin
acaKb
Kb b
 
(41)
21
2
1106 10
21 4209 1011
10 10
2
310
410202021
10
310 410
1201 1020
10 1010
2
6102
10 101011
10 2
2
2
1
3
() cos 2
8
sin sin
2cos 2cos
2
3cos
4
si
2
ba aa
aa
b
aa
ba
b
ba b
af
bb
b
f
 
 

 





 



 

10 11
n




(42)

114102011
2
24102011
2
2
4101020 211010 11
10
2
sin
sin sin
4
cos cos
4
bb a
f
ca b
f
ab b


 

 

(43)

4 2091011 4102021
2
410910
1
2010 1011
10 10102
2
11
10
10 11
2
cos 2sin
cos3 cos
2
sin
2
aa a
aaf
b
bb b
f
 


 



2
4
b
(44)
The system of Equations (41)-(44) are first order
autonomous ordinary differential equations and the sta-
bility of a particular fixed point with respect to an infini-
tesimal disturbance proportional to exp( )t
is deter-
mined by eigenvalues of the Jacobian matrix of the right
hand sides of Equations (41)-(44). The zeros of the char-
acteristic equation are given by
432
1234
0LL LL


(45)
where, and are functions of the parameters
(1 23223232112
212
123
,,LLL
,, ,cc
4
L
,,,,,,,,, ,, ,,ab f

).
According to the Routh-Hurwitz criterion the necessary
and sufficient conditions for all the roots of Equation
(45) to possess negative real parts are:
2
11233123144
0,0,0, 0LLLLLLLL LLL 
(46)
The system is stable if the eigenvalues have negative
real parts, otherwise is unstable. In the frequency re-
sponse curves, solid/dotted lines denote stable/ unstable
periodic responses, respectively.
3. Results and Discussion
The response of the two-degree-of-freedom nonlinear
system under both parametric and external excitations is
studied. The solution of this system is determined up to
and including the second order approximation by apply-
ing the multiple time scale perturbation. The steady state
solution and its stability are determined and representa-
tive numerical results are included. The stability zone
and effects of the different parameters are discussed us-
ing frequency response curve. The stability of the nu-
merical solution is studied also using the phase-plane
method. Some of the resulting resonance cases are con-
firmed applying well-known numerical techniques. The
effects of the some different parameters on the vibrating
system behavior are investigated and discussed.
3.1. Numerical Solution
Figure 2 shows that the response of the inclined cable
for the non-resonant at the practical values of the pa-
rameters c1 = 0.0002, c2 = 0.03, α2 = 0.2, β2 = 0.5, γ2 =
0.3, η2 = 0.5, α3 = 0.03, η3 = 0.05, γ3 = 0.04, 1 = 2, 2 =
0.01, 1 = 2.75, 2 = 3.2,
1 = 1.2,
2 = 1.5. It can be
seen from this figure that the steady state amplitude is
about 0.005 with dynamic chaotic behavior for the in-
plane mode and about 0.18 with multi-limit cycle for the
out-of-plane mode. The amplitudes decreasing with in-
creasing time and tend to steady state motion and have
stable solution. The worst resonance case is also con-
firmed numerically as shown in Figure 3. From this fig-
ure, it can be notice that the maximum steady state am-
plitude of the in-plane mode is about 130 times that of
basic case with multi-limit cycle, while the maximum
amplitude of out-of-plane mode is about 4 times of the
basic case with chaotic motion.
Effects of external and parametric excitation forces f1
and f2.
Copyright © 2011 SciRes. AM
M. S. ABD ELKADER
1474
0100 200 300
-0.05
0
0.05
Time
A mpli tude(x)
0100 200 300
-0.5
0
0.5
Time
Am plitude(y)
Figure 2. Non-resonance system behavior (basic case) 1
ω1 ω2.
0100 200 300
-1
0
1
Time
Amplitude(x)
0100 200 300
-1
0
1
Time
Am plitude(y )
Figure 3. Simultaneous principal parametric resonance in the
presence of 2:1 internal resonance (221
2and 2 2

).
For increasing the amplitude of the external or parame-
tric excitation forces f1 or f2, we observe that the modes
of vibration have increasing magnitudes and there exist
chaotic dynamic motion as shown in Figures 4 and 5.
3.2. Frequency Response Curves
The frequency response Equations (38)-(39) are nonlin-
ear algebraic equations in the amplitudes of the system
(in-plane mode) and b(out-of-plane mode). The stabil-
ity of a fixed point solution is studied by examination of
the eigenvalues of Equation (45). The numerical results
of Equations (38) and (39) are plotted in Figures 6-8.
a
Figure 6, show the frequency response curves of the
two modes of inclined cable against detuning parameter
1
. From the geometry of the figures we observe that the
amplitudes have two branches and these branches are
bent to the right, the bending leads to multi-valued solu-
tions and hence the effective nonlinearity is hardening
type. In Figure 6(a), there are two branches of nontrivial
solution such that the left branch stable and the right
branch lose stability as 10.4
. Figure 6(b), show that
the steady state amplitudes are increasing for increasing
0100 200 300
-10
-5
0
5
10
Time
Ampli tude(x)
0100 200 300
-10
-5
0
5
10
Time
Amplitude(y)
Figure 4. Effects of increasing value of external excitation
force f1 = 5.
0100 200300
-4
-2
0
2
4
Time
Amplitude(x)
0100 200 300
-4
-2
0
2
4
Time
Amplitude(y)
Figure 5. Effects of increasing value of parametric excita-
tion force f2 = 3.
parametric excitation force 2. The region of instability
for two modes is increasing for increasing 2. For in
creasing nonlinear parameter 2
f
f
(i.e. 21
) as shown
in Figure 6(c), we show that the regions of definition are
decreasing and the two branches of the steady state am-
plitude curve are contracted and give one continuous
curve which is stable and response amplitude of the in-
plane mode is increased. Figure 6(d) show that the re-
sponse amplitudes of the inclined cable are increasing for
Copyright © 2011 SciRes. AM
M. S. ABD ELKADER
Copyright © 2011 SciRes. AM
1475
0 1 23
0
0.1
0.2
0.3
0.4
0.5
a
0 1 2 3
0
0. 5
1
1. 5
2
2. 5
3
b
02468
0
0.5
1
1.5
2
2.5
a
02468
0
2
4
6
b
(a) (b)
00.5 11.5 22.5
0.2
0.4
0.6
0.8
1
a
00.5 11.5 22.5
0
1
2
3
b
00.5 11.5 22.5 3
0
1
2
3
4
5
a
00.5 11.5 22.5 3
0
0.5
1
1.5
2
2.5
3
b
(c) (d)
0 1 2 3
0
0.5
1
1.5
2
2.5
a
0 123
0
0.5
1
1.5
2
2.5
3
b
0123
1
2
3
4
5
6
a
0 123
0
1
2
3
4
b
(e) (f)
Figure 6. (a): Frequency response curves for amplitudes against σ1; (b): Frequency response curve for increasing parametric
excitation force f2 = 3.0; (c): Frequency response curve for increasing nonlinear parameter β2 = 1.0; (d): Frequency response
curve for decreasing nonlinear parameter η3 = 0.1; (e): Frequency response curve for increasing nonlinear parameter γ2 = 1.8;
(f): Frequency response curve for negative value of nonlinear parameter γ3 = –0.4.
decreasing nonlinear parameter 3
and the regions of
multi-valued and instability of two modes are increasing.
The regions of instability solutions are increasing for
increasing nonlinear parameter 2
as shown in Figure
6(e). Figure 6(f) shows that for negative value of non-
linear parameter 3
the response amplitudes are in-
creasing and the stability solution are decreasing with
increasing region of multi-valued.
Figure 7, represent the variation of the amplitudes of
the inclined cable against the detuning parameter 2
. In
Figure 7(a), we see that each mode of the inclined cable
has one continuous curve and single valued solution and
it is symmetric about the origin and it is noticed that the
in-plane mode reaches maximum value at 20
and
the out-of-plane mode reaches minimum value at the
same value of 2
. Also, it intersects in two points and
these modes have stable and unstable solutions. From
Figure 7(b), we observe that for increasing parametric
excitation force f2 the symmetric branch moves up with
increased magnitudes and the region of stability is in-
creased. For increasing nonlinear parameter 3
, we note
that the amplitudes of the two modes of the inclined ca-
ble have decreasing magnitudes and increasing stable
solutions, as shown in Figure 7(c). The steady state am-
plitudes of the two modes are increasing for decreasing
nonlinear parameter 3
as shown in Figure 7(d). Also,
the region of stability solutions is increased. From Fig-
ure 7(e) we observe that the steady state amplitudes a
and b of the two modes are increasing for decreasing
value of nonlinear parameters 3
respectively with
increasing stable solutions. The stability solution is de-
creasing as the nonlinear parameter 2
is increase and
the curves are shifted to the right and has hardening phe-
nomena and there exists jump phenomena, as shown in
Figure 7(f).
Figure 8 represent force-response curves for the non-
linear solution of the case of simultaneous principal pa-
rametric resonance in the presence of 2:1 internal reso-
nances. In this figure the amplitudes of the inclined cable
are plotted as a function of the parametric excitation
force f2. Figure 8 shows that the response amplitudes of
the inclined cable have a continuous curve and the curve
has stable and unstable solutions.
4. Comparison with Published Work
In comparison with the previous work [8], we have the
global bifurcation of this inclined cable leading to pri-
mary resonances and 1:1 internal resonance is investi-
gated. A new global perturbation technique is employed
to analyze Shilnikov type homoclinic orbits and chaotic
dynamics in the inclined cable. Kamel and Hamed [9],
M. S. ABD ELKADER
Copyright © 2011 SciRes. AM
1476
    
 
 
 
 
Amplitudes
a
b
-2 -1 0 1 2
0
0.2
0.4
0.6
0.8
a
-1 -0.500.5 1
0.2
0.4
0.6
0.8
1
1.2
b
f
2
=0.3
f
2
=0.3
f
2
=0.6
f
2
=0.6
f
2
=0.15
f
2
=0.15
(a) (b)
-1.5 -1 -0.5 00.5 11.5
0
0.1
0.2
0.3
0.4
0.5
a
-1.5 -1 -0.5 00.5 11.5
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
0. 8
b




-0.4 -0.200.2 0.4 0.6
0.1
0.2
0.3
0.4
0.5
0.6
a
-0.4 -0.200.2 0.4
0.2
0.4
0.6
0.8
b
eta3=1.6
eta3=1.6
eta3=0.6
eta3=0 .6
eta3=0.2 eta3=0.2
(c) (d)
-0.2 -0.100.1 0.2 0.3
0
0.2
0.4
0.6
0.8
a
-0.2 -0.1 00.1 0.2 0.3
0. 2
0. 4
0. 6
0. 8
b




-2 -1 0 123
0
0.5
1
1.5
a
-2 -1 0 1 23
0.2
0.3
0.4
0.5
0.6
0.7
0.8
b




2
(e) (f)
Figure 7. (a): Frequency response curves for simultaneous principal parametric resonance in the presence of 2:1 internal
resonance 22 1
2and 2 

; (b): Frequency response curve for parametric excitation force f2; (c): Frequency response
curve for nonlinear parameter γ3; (d): Frequency response curve for nonlinear parameter η3; (e): Frequency response curve
for nonlinear parameter α3; (f): Frequency response curve for nonlinear parameter α2.
0246810 12
0
0. 5
1
1. 5
f
2
a
0246810
0
1
2
3
4
5
f
2
b
Figure 8. Force response curves for (221
2,2
2


).
studied the nonlinear behavior of an inclined cable sub-
jected to harmonic excitation near the simultaneous pri-
mary and 1:1 internal resonance by using multiple scale
method.
In this paper, periodic and chaotic response of a dis-
cretization two-degree-of-freedom model of a suspended
inclined cable, containing a 2:1 internal resonance, sub-
ject to harmonic external and parametric excitation are
obtained. The stable/unstable periodic solutions are de-
termined using the method of multiple scale and are pre-
sented through frequency response plots. Chaotic re-
sponses are determined by numerical integration of the
governing ordinary differential equations of motion. Var-
iation of the parameters 23222
,, ,,,

332
,,f

leads
M. S. ABD ELKADER1477
to multi-valued amplitudes and hence to jump phenom-
ena.
5. Conclusions
Cables are very efficient structural members and hence
have been widely used in many long-span structures,
including suspension, roofs and guyed towers. The
nonlinear dynamic response of the nonlinear system sub-
jected to external and parametric excitations is investi-
gated. The method of multiple scales is applied to obtain
the solution of the considered system up to second order
approximation. The numerical solutions and chaotic re-
sponse of this nonlinear system are investigated. The
stability of the proposed analytic nonlinear solution is
studied at worst resonance case which is the simultane-
ous principal parametric resonance in the presence of 2:1
internal resonances. The modulation equations of the
amplitudes and phases are obtained and steady state solu-
tions are determined. The effects of some nonlinear pa-
rameters on the steady state response of the vibrating
cable leading to multi-valued solutions. From the analy-
sis the following may be concluded.
1) For the resonance case 2212
2, 2

 we note
that the steady state amplitude is increased to about
130% compared to basic case with multi-limit cycle, and
it is better to avoid this resonance case as working condi-
tions for the system.
2) The steady state amplitude of the system are in-
creasing for increasing external or parametric excitation
force, and for large values of the system become unsta-
ble.
3) Variation of α2, α3, β2, γ2, η2, γ3, η3, f2 leads to
multi-valued amplitudes and hence jump phenomena.
4) For increasing parametric excitation force f2 or
negative value of the nonlinear parameter γ3 we observe
that the steady state amplitudes of the two modes are
increasing with increasing instability solutions.
5) Increasing of the nonlinear parameters η3 or γ3 can
reduce the amplitude of the system and obtain the effect
of reduction of the amplitude.
6) Variation of the parameter α2 leads to multi-valued
amplitudes and hence to jump phenomena.
7) For increasing parametric excitation force f2 or de-
creasing nonlinear parameter α3 we show that the steady
state amplitudes of the two modes are increasing.
For increasing nonlinear parameter η3 we note that the
steady state amplitudes of the two modes are decreasing
with decrease of the stability solutions.
6. References
[1] H. N. Arafat and A. H. Nayfeh, “Non-Linear Responses
of Suspended Cables to Primary Resonance Excitations,”
Journal of Sound and Vibration, Vol. 266, No. 2, 2003,
pp. 325-354. doi:10.1016/S0022-460X(02)01393-7
[2] G. Rega, “Non-Linear Vibrations of Suspended Cables;
Part I: Modeling and Analysis,” Journal of Applied Me-
chanics Review, Vol. 57, No. 6, 2004, pp. 443-478.
doi:10.1115/1.1777224
[3] G. Rega, “Non-Linear Vibrations of Suspended Cables;
Part II: Deterministic Phenomena,” Journal of Applied
Mechanics Review, Vol. 57, No. 6, 2004, pp. 479-514.
doi:10.1115/1.1777225
[4] S. R. Nielsen and P. H. Kirkegaard, “Super and Combi-
natorial Harmonic Response of Flexible Inclined Cables
with Small Sag,” Journal of Sound and Vibration, Vol.
251, No. 1, 2002, pp. 79-102.
doi:10.1006/jsvi.2001.3979
[5] G. Zheng, J. M. Ko and Y. O. Ni, “Super-Harmonic and
Internal Resonances of a Suspended Cable with Nearly
Commensurable Natural Frequencies,” Nonlinear Dy-
namics, Vol. 30, No. 1, 2002, pp. 55-70.
doi:10.1023/A:1020395922392
[6] W. Zhang and Y. Tang, “Global Dynamics of the Cable
under Combined Parametrical and External Excitations,”
International Journal of Non-Linear Mechanics, Vol. 37,
No. 3, 2002, pp. 505-526.
doi:10.1016/S0020-7462(01)00026-9
[7] A. H. Nayfeh, H. Arafat, C. M. Chin and W. Lacarbonara,
“Multimode Interactions in Suspended Cables,” Journal
of Vibration and Control, Vol. 8, No. 3, 2002, pp. 337-
387. doi:10.1177/107754602023687
[8] H. Chen and Q. Xu, “Bifurcation and Chaos of an In-
clined Cable,” Nonlinear Dynamics, Vol. 57, No. 2-3,
2009, pp. 37-55. doi:10.1007/s11071-008-9418-3
[9] M. M. Kamel and Y. S. Hamed, “Non-Linear Analysis of
an Inclined Cable under Harmonic Excitation,” Acta
Mechanica, Vol. 214, No. 3-4, 2010, pp. 315-325.
doi:10.1007/s00707-010-0293-x
[10] A. Abe, “Validity and Accuracy of Solutions for Nonlin-
ear Vibration Analyses of Suspended Cables with One-
to-One Internal Resonance,” Nonlinear Analysis: Real
World Applications, Vol. 11, No. 4, 2010, pp. 2594-2602.
doi:10.1016/j.nonrwa.2009.09.006
[11] N. Srinil, G. Rega and S. Chucheepsakul, “Two-yo-One
Resonant Multi-Modal Dynamics of Horizontal/Inclined
Cables. Part I: Theoretical Formulation and Model Vali-
dation,” Nonlinear Dynamics, Vol. 48, No. 3, 2007, pp.
231-252. doi:10.1007/s11071-006-9086-0
[12] N. Srinil and G. Rega, “Two-To-One Resonant Multi-
Modal Dynamics of Horizontal/Inclined Cables. Part II:
Internal Resonance Activation Reduced-Order Models
and Nonlinear Normal Modes,” Nonlinear Dynamics, Vol.
48, No. 3, 2007, pp. 253-274.
doi:10.1007/s11071-006-9087-z
[13] R. Alaggio and G. Rega, “Characterizing Bifurcations
and Classes of Motion in the Transition to Chaos through
3D-Tori of a Continuous Experimental System in Solid
Mechanics,” Physica D, Vol. 137, No. 1, 2000, pp. 70-93.
doi:10.1016/S0167-2789(99)00169-4
[14] G. Rega and R. Alaggio, “Spatio-Temporal Dimensional-
Copyright © 2011 SciRes. AM
M. S. ABD ELKADER
Copyright © 2011 SciRes. AM
1478
ity in the Overall Complex Dynamics of an Experimental
Cable/Mass System,” International Journal of Solids and
Structures, Vol. 38, No. 10-13, 2001, pp. 2049-2068.
doi:10.1016/S0020-7683(00)00152-9
[15] A. Gonzalez-Buelga, S. A. Neild, D. J. Wagg and J. H. G.
Macdonald, “Modal Stability of Inclined Cables Sub-
jected to Vertical Support Excitation,” Journal of Sound
and Vibration, Vol. 318, No. 3, 2008, pp. 565-579.
doi:10.1016/j.jsv.2008.04.031
[16] N. C. Perkins, “Modal Interactions in the Non-Linear
Response of Inclined Cables under Parametric/External
Excitation,” International Journal of Non-linear Me-
chanics, Vol. 27, No. 2, 1992, pp. 233-250.
doi:10.1016/0020-7462(92)90083-J
[17] C. L. Lee and N. C. Perkins, “Nonlinear Oscillations of
Suspended Cables Containing a Two-to-One Internal
Resonance,” Nonlinear Dynamics, Vol. 3, 1992, pp. 465-
490.
[18] C. L. Lee and N. C. Perkins, “Three-Dimensional Oscil-
lations of Suspended Cables Involving Simultaneous In-
ternal Resonance,” Proceedings of ASME Winter Annual
Meeting AMD-14, 1992, pp. 59-67.
[19] M. Eissa and M. Sayed, “A Comparison be tween Passive
and Active Control of Non-Linear Simple Pendulum
Part-I,” Mathematical and Computational Applications,
Vol. 11, No. 2, 2006, pp. 137-149.
[20] M. Eissa and M. Sayed, “A Comparison be tween Passive
and Active Control of Non-Linear Simple Pendulum
Part-II,” Mathematical and Computational Applications,
Vol. 11, No. 2, 2006, pp. 151-162.
[21] M. Eissa and M. Sayed, “Vibration Reduction of a Three
DOF Non-Linear Spring Pendulum,” Communication in
Nonlinear Science and Numerical Simulation, Vol. 13,
No. 2, 2008, pp. 465-488.
doi:10.1016/j.cnsns.2006.04.001
[22] M. Sayed, “Improving the Mathematical Solutions of
Nonlinear Differential Equations Using Different Control
Methods,” Ph.D. Thesis, Menofia University, Egypt, No-
vember 2006.
[23] M. Sayed and Y. S. Hamed, “Stability and Response of a
Nonlinear Coupled Pitch-Roll Ship Model under Para-
metric and Harmonic Excitations,” Nonlinear Dynamics,
Vol. 64, No. 3, 2011, pp. 207-220.
doi:10.1007/s11071-010-9841-0
[24] M. Sayed and M. Kamel, “Stability Study a nd Control of
Helicopter Blade Flapping Vibrations,” Applied Mathe-
matical Modelling, Vol. 35, No. 6, 2011, pp. 2820-2837.
doi:10.1016/j.apm.2010.12.002
[25] M. Sayed and M. Kamel, “1:2 and 1:3 Internal Resonance
Active Absorber for Non-Linear Vibrating System,” Ap-
plied Mathematical Modelling, Vol. 36, No. 1, 2012, pp.
310-332. doi:10.1016/j.apm.2011.05.057
[26] Y. A. Amer and M. Sayed, “Stability at Principal Reso-
nance of Multi-Parametrically and Externally Excited
Mechanical System,” Advances in Theoretical and Ap-
plied Mechanics, Vol. 4, No. 1, 2011, pp. 1-14.
[27] M. Sayed, Y. S. Hamed and Y. A. Amer, “Vibration Re-
duction and Stability of Non-Linear System Subjected to
External and Parametric Excitation Forces under a Non-
linear Absorber,” International Journal of Contemporary
Mathematical Sciences, Vol. 6, No. 22, 2011, pp. 1051 -
1070.
[28] A. H. Nayfeh, “Non-Linear Interactions,” Wiley/Inter-
Science, New York, 2000.