Free Vibration Analysis of a Nonlinear Structure with Cyclic Symmetry_Crimson Publishers

Mohammad Saleh Sadooghi*
Department of Mechanical Engineering, Iran
*Corresponding author: Mohammad Saleh Sadooghi, Department of Mechanical Engineering, Iran
Submission: October 10,2018; Published: November 01, 2018
Free Vibration Analysis of a Nonlinear Structure
with Cyclic Symmetry
Research Article
1/8Copyright © All rights are reserved by Mohammad Saleh Sadooghi.
Volume - 1 Issue - 3
Introduction
Generally structural systems are complex, and usually include
a large number of substructures acting as a single unit. Structural
systems with cyclic symmetry are an important class of engineering
applications, which are composed of some identical substructures.
They commonly use in large circular space antennas, bladed disk
assemblies, and magnetic storage devices. The cyclic symmetry of
the structure implies that most of the frequencies for the linear
model appear in pairwise Double-degenerate pairs with distinct
orthonormal normal modes [1].
In the linear case, a majority of double eigenfrequencies,
corresponding to distinct eigenforms reveals in study of the linear
normal modes (LNMs) [1,2]. These eigenforms are associated with
nodal diameter vibration modes. As these LNMs arise from an
eigenvalue problem, there are always as many LNMs as degrees
of freedom (dofs). In the case of nonlinear systems with cyclic
symmetry, it has been shown that the number of nonlinear normal
modes (NNMs) can exceed the number of dofs, the extra NNMs
being generated through bifurcations or internal resonances [3,4].
An example of this feature is localized nonlinear modes; they
correspond to a free motion in which only a few substructures
vibrate with noticeable amplitude and no counterpart can be found
for it in the linear theory [5,6].
As the nonlinearity is common in real-life applications, the
dynamics of nonlinear periodic structures have been examined
in several studies [7,8]. Wei and Pierre examined the effects
of dry friction on a nearly cyclic structure using the Harmonic
Balance Method (HBM) [9]. In a series of papers, Vakakis et al.
[10] demonstrated that, in contrast to the findings of linear theory,
nonlinear mode localization may occur in perfectly cyclic nonlinear
systems [6,10-12]. There are some studies dealing with mode
localization in nonlinear cyclic systems [13-15]. Samaranayake
& Bajaj [1] studied a weakly nonlinear cyclic symmetry structure
with multiple degrees of freedom. The system was consisted
of an identical particles of mass m that arranged in a ring and
interconnected by nonlinear extensional springs. All the masses
were assumed to be hinged to the ground by nonlinear torsional
springs. The asymptotic method of averaging was used to study
the nonlinear interactions between the pairs of modes with nearly
identical natural frequencies when the external excitation is nearly
three times the natural frequency of the modes being excited
[1]. Georgiades et al. [16] examined the nonlinear normal modes
and the bifurcations in cyclic periodic structures. The nonlinear
normal modes were computed using a numerical technique that
combines shooting and pseudo-arclength continuation which can
investigate strongly nonlinear regimes of motion. Their system is a
mathematical model of a bladed-disk assembly. The disk and blades
were modeled by lumped masses which are coupled by linear and
cubic springs.
Sarrouy et al. [17] studied both free and forced vibrations of a
nonlinear cyclic symmetry structure. The model was broken up into
n identical sectors. Each sector was modeled by a thin rectangular
plate, which was clamped to a fixed rigid disk. Consecutive plates
were coupled by a linear stiffness while nonlinearity is introduced
by taking into account their large deflection. Harmonic balance
method was used for solving the nonlinear equations.
Grolt & Thouverez [18] studied the nonlinear dynamics of a
system with cyclic symmetry. The nonlinearity was due to large
displacementofsubstructure(geometrictypenonlinearities)which
was composed of six identical blades (beam). The coupling between
beams was introduced via a linear stiffness running between two
Abstract
In this article free vibration of a nonlinear cyclic symmetry system is examined. The system is composed of six identical beams which are fixed at
the end. The coupling between beams is introduced via a nonlinear stiffness running between two consecutive beams. The equations of motion are as
a system of second order nonlinear differential equations which are coupled by cubic nonlinear terms. To solve the equations of motion, the numerical
methods are used, and the results are compared with those of the harmonic balance method. The effect of nonlinear stiffness on the backbone curves
is examined. The results show that the effect of nonlinear stiffness is increased as the diametrical mode number is increased from 1 to 3, whereas there
is no effect on zero-diameter mode.
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COJ Elec Communicat Copyright © Mohammad Saleh Sadooghi
2/8How to cite this article: Mohammad S S. Free Vibration Analysis of a Nonlinear Structure with Cyclic Symmetry. COJ Elec Communicat. 1(3). COJEC.000515.
2018. DOI: 10.31031/COJEC.2018.01.000515
consecutive substructures. The modeling leads to a system of
linearly coupled, second-order nonlinear differential equations, in
which nonlinearity appears via cubic terms. The harmonic balance
method was used for solving the differential equations of motion.
The authors were interested in examining the localized nonlinear
modes in the unforced system which arises from branching point
bifurcation at the certain vibration amplitudes. In the forced case,
these nonlinear modes give rise to a complex dynamical behavior.
In this paper, free vibration of a nonlinear cyclic symmetry
system is examined. The system is composed of six identical
beams which are fixed at the end. These beams are coupled to
each other using nonlinear stiffnesses. Also, the influence of the
nonlinear stiffness on the dynamics of the system is investigated.
In the previous studies, the geometrical nonlinearity of the beams
has been considered but the connectors are assumed with linear
stiffness. Nonlinear assumption of the connector stiffness leads to a
nonlinearcouplingbetweenthesecond-orderdifferentialequations
while otherwise there will be a linear coupling present. So, a
numerical method for solving the system of differential equations
is applied. The results of numerical method are compared to
those of the harmonic balance method when the linear stiffness is
introduced, and the accuracy of the numerical solution is validated.
It is shown that the nonlinearity in the stiffness of the connectors
has an important effect on the response of the system.
Modeling
Figure 1(a) represents a structure with cyclic symmetry which
is composed of a fixed disk and six identical blades. The blades are
considered as beams, which are clamped to the disk at one side
and axially restricted on the other side. The beams are connected
to each other by connectors with nonlinear stiffness. The system
is symmetrical; hence, one of the beams can be considered in the
modeling and the other beams would be integrated along with the
nonlinear coupling stiffening elements.
Figure 1(a): The schematic of the system with cyclic symmetry.
Figure 1(b): A beam with boundary conditions.
In Figure 1(b), one of the beams is shown. The boundary
conditions of the beam are as:
( ) ( ) ( ) ( )0, , 0 0 0,
v
u y u L y v
x
∂
= = = =
∂
(1)
M(L)=T(L)=0, (2)
where u, v, M, T, L are displacement in the x-direction,
displacement in the y-direction, bending moment, shear force, and
the length of the beam, respectively.
To derive the equations of motion, one needs to determine
the expressions for strain energy, kinetic energy and work of
external forces on a single beam, in terms of transverse and axial
displacements. Then the geometrical nonlinearity is considered in
the equations of motion by addition of a Von-Karman term in the
relation between displacements u, v, and the strain of the middle
line, ε, as:

2
1
,
2
u v
x x
ε
∂ ∂ 
= +  
∂ ∂ 
(3)
The total strain is:
.zykε ε= − (4)
where
2
2z
v
k
x
∂
=
∂
represents the change of curvature. The elastic
strain energy of the beam is:

0
1
,
2
L
A
U dA dxσε
 
=  
 
∫ ∫  (5)
where σ is the Kirchhoff stress and it is related to the Green
strain with the relation of Eσ ε=  which is the standard Hook’s low
(E is the Young’s modulus). By substituting the Hook’s low in Eq.
(5), the relation (6) is obtained for the strain energy:

22
2
2
0 0
1 1
.
2 2
L L
A
v
U E dA dx EI dx
x
ε
   ∂
= +   
∂  
∫ ∫ ∫
(6)
where A is the area of beam cross section. Consequently, the
kinetic energy (T) is as:
2 2
0 A
1
( ) A .
2
L
T u v d dxρ
 
= + 
 
∫ ∫   (7)
where ρ is the density of the beam material.
Using the Hamilton’s principle, the following relation for the
axial displacement u is obtained:
2
¨ 1
0 .
2
u v
Au ES
x x x
ρ
  ∂ ∂ ∂ 
+ + =    ∂ ∂ ∂    
(8)

3/8
How to cite this article: Mohammad S S. Free Vibration Analysis of a Nonlinear Structure with Cyclic Symmetry. COJ Elec Communicat. 1(3). COJEC.000515.
2018. DOI: 10.31031/COJEC.2018.01.000515
The axial force in the beam can be expressed as fa = EAε. Using
equations (3) and (8), and neglecting the effect of the inertia term,
the Eq. (8) is simplified to the following equation:
0 ,a af f cte
x
∂
= ⇒ =
∂
(9)
Integrating fa from 0 to L and Using Eq. (3) leads to:

2
0 0
1
,
2
L L
a a
u v
f dx Lf EA dx
x x
 ∂ ∂ 
= = +   ∂ ∂  
∫ ∫
(10)
The kinematic boundary condition in Eq. (1) eliminates the
term of
0
L
u
dx
x
∂ 
 
∂ 
∫ from Eq. (10). So, for the in-plane strain, ε, the
following expression is obtained:
2
0
1 1
,
2
L
v
dx
L x
ε
∂ 
=  
∂ 
∫
(11)
Substituting Eq. (11) into Eq. (6) leads to an expression for the
strain energy U which is only in terms of transverse displacement
v as follows:

2 22 2
2 2
0 0 0
1
.
8 2
L L L
ES v v
U dX EI dx
L x x
   ∂ ∂ 
= +     ∂ ∂    
∫ ∫ ∫
(12)
Also, by neglecting the rotary inertia, the kinetic energy T is
simplified as below:
2 2
0 A
1
( ) A .
2
L
T u v d dxρ
 
= + 
 
∫ ∫   (13)
Considering the cyclic symmetry
For a system of structures with cyclic symmetry which is
composed of n identical sectors, for 1≤ j ≤ n, vj denotes the
transverse displacement of the j-th beam, Uj and Tj are potential
and kinetic energies corresponding to j-th beam, respectively. So,
the total potential and kinetic energies of the system are:
1 0
,
n n
j j
j j
U U T T
= =
= =∑ ∑ (14)
In the present system, the connections between beams are
modeled using a set of nonlinear springs. The springs are set at
points xr
=L/4 on all beams. The relationship between force and
displacement of the nonlinear spring is:
3
1 1
.
4 4 4 4
j j j j
nl
L L L L
f k v v k v v+ +          
= − + −          
          
(15)
Due to cyclic symmetry of the system, when j is equal to n, it
can be written:
j+1=1,
The potential energy due to spring which is located between
beams j and j+1 is expressed as:
2 4
1 11 1
.
2 4 4 4 4 4
j j j j j
nl
L L L L
V k v v k v v+ +          
= − + −          
          
(16)
Also, the total potential energy V of all of springs is:

1
.
n
j
j
V V
=
= ∑ (17)
Equations of motion
The Assumed-mode method is used for discretization of the
model. The transverse displacements (vj) are interpolated using
the following expression:
( ) ( ) ( )
0
, , ,
N
j j
i i
i
V x y t t x yλ
=
= Φ∑ (18)
where they are kinematically admissible functions, the
participation of these functions in the displacement vj, and N the
number of functions selected for the interpolation step. Using this
method, the transformation is made from a continuous problem
to a discrete one with dimension nN where the unknowns are the
variables ( )1 ,1
j
i i N j n
λ
≤ ≤ ≤ ≤
. The Lagrange equations are used for deriving
the equation of motion, as follows:
0 1 ,1j j j
i i i
d T U V
for i N j n
dt λ λ λ
∂ ∂ ∂
+ + = ≤ ≤ ≤ ≤
∂ ∂ ∂  
(19)
Thenumberofbladeswaschosentobesetatn=6andN=1(only
one function) which leads to a nonlinear problem with 6 degrees
of freedom. The single function selected for this interpolation step
would be
2
Ö
x
L
 
=  
 
, which verifies the kinematic boundary conditions
given in Eq. (1). Applying Lagrange’s equations (Eq. (19)) for the
discretization
2
j
j
x
v
L
λ
 
=  
 
results in the following equation of motion
for the j-th beam:
( ) ( ) ( ) ( )
¨
3 3 3
1 1 1 12 2 0j j j j j j jc cλ α λ β γ λ λ λ γ λ λ+ − + −+ + + + − + − + =(20)
which is a system of nonlinearly coupled, second order
nonlinear differential equations.
By setting Y=(λj
)1≤ j ≤6
and Y 3
=(λ3
j
)1≤ j ≤6
, the matrix form of Eq.
(20) is given by the following:
[ ] [ ] [ ]
¨
3
0nlM Y K Y K Y+ + =(21)
where [M]=[I]6
and:
[ ]
2 0 0 0
2 0 0 0
0 2 0 0
(22)
0 0 2 0
0 0 0 2
0 0 0 2
c c c
c c c
c c c
K
c c c
c c c
c c c
α
α
α
α
α
α
+ − − 
 − + − 
 − + −
=  
− + − 
 − + −
 
− − + 
[ ]
2 0 0 0
2 0 0 0
0 2 0 0
(23)
0 0 2 0
0 0 0 2
0 0 0 2
nlK
β γ γ γ
γ β γ γ
γ β γ γ
γ β γ γ
γ β γ γ
γ γ β γ
+ − − 
 − + − 
 − + −
=  
− + − 
 − + −
 
− − + 
and the numeric value of the parameters are [18]:
α=8.7662×103
s-2
, c=148.36s-2
, β=4.6752×107
m-2
s-2
,
γ=3.6227×106
s-2
Free Response of a Six Degrees-of-Freedom System
In this section, the free responses of the system are obtained.
The linear analysis of the system is necessary for the nonlinear
analysis of the system. The nonlinear normal modes (NNM) would
be identified by means of the numerical method in the next step.

2018. DOI: 10.31031/COJEC.2018.01.000515
Study of the underlying linear system
According to Eq. (21), the equation for free undamped linear
system would be as:
[ ] [ ]
¨
0M Y K Y+ =(25)
To obtain the mode shapes, the following equation should be
solved:
[K]Φ=ω2
[M]Φ (26)
The eigenvalues are given by the relation below [1]:
ω2
p
=α+2c(1-cos(θp
)) with 2
p
p
n
π
θ = (27)
where p is the diametrical mode number, such that 0 ≤ p ≤ (n-
1)/2 if n is an odd number, and 0 ≤ p ≤ n/2 if n is an even number.
The linear mode shapes are given by:
Φ0
=[1,1,1,1,1,1]T
Φp
c
=[cos(θp
),cos(2θp
), …, cos(nθp
)]T
Φp
s
=[sin(θp
),sin(2θp
), …, sin(nθp
)]T
(28)
So, the frequencies and corresponding mode shapes are as
follows:
0 93.6278ω α= = [ ]0 1,1,1,1,1,1
T
Φ =
[ ]1 1,1,0, 1, 1,0
Tc
Φ= − − 1 94.4168cω α= + =
2 3 95.9753cω α= + = 1
T
s 1 1 1 1
1, , , 1, ,
2 2 2 2
 
Φ= − − −  
[ ]2 1, 1,0,1, 1,0
Tc
Φ = − −
2
T
s 1 1 1 1
1, , ,1, ,
2 2 2 2
 
Φ = − − − −  
4 96.7451
3
cω α= + = [ ]3 1, 1,1, 1,1, 1
T
Φ = − − − (29)
In this study, the mode shapes which contains just 0, 1, -1
components are studied. So, the solution vector of Y would have
one of the following forms:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ))1 0 1 1 2 1 2 3 1 3) (0 (1
( ) c c
p pp p
t y t or t y t or t y t or t y t= == =
= = = =YÖ Y Ö Y Ö Y Ö
(30)
Solution Method and Verification
When a linear stiffness is used for connecting the consecutive
blades, a system of linearly coupled, second order nonlinear
differential equation can be simplified into a single Duffing equation
for the one blade as the following [18]:
2 3
1 1 1 0py x xω β+ + =
Free response of this equation is obtained by harmonic balance
method and is verified with the exact solution (elliptic function)
which shows the high accuracy of the harmonic balance method in
this problem [18].
In this article, to obtain the frequency response curves the
numerical method is applied to the problem. For verification of the
numerical method, the problem of free response in reference [18]
in zero-diameter mode is considered and the accuracy of numerical
solution is evaluated by the results of harmonic balance method. In
Figure 2 the results of these two methods are compared. As it can
be seen, the numerical solution has the sufficient accuracy.
Figure 2: Verification of numerical method of Runge-Kutta (4,5) with HBM when a linear stiffness is used.
Free response of nonlinear system
It is possible now to study the effect of nonlinear stiffness on
the system. i.e. the system of nonlinear differential equations in Eq.
(21) can be solved accurately by the method of this article because
of the verification of the method in the previous section. Figure
3-5 show the amplitude-frequency relation of the free response of
the system of Eq. (21) in 1, 2 and 3-diameter modes, respectively.
In these figures the effect of the nonlinear stiffness on the system
behavior is clearly demonstrated. As it is expected, this effect is
more significant in the larger displacement. Also, this effect is
different in the variant modes. In Eq. (20) the 5th term and the last
two terms are the new nonlinear terms that are appeared due to
the new nonlinear stiffness. In zero-diameter mode, all components

5/8
2018. DOI: 10.31031/COJEC.2018.01.000515
of the mode shapes (Eq. (29)) are 1. So, the effects of these terms
are eliminated. But in 1, 2 and 3-diameter modes, these terms are
not removed. The 1-diameter mode displayed the most difference
and the 3-diameter mode displayed the least difference between
the systems with linear and nonlinear stiffness which is expected,
because of their mode components in Eq. (29).
Figure 3: Effect of nonlinear stiffness on the backbone for 1-diameter mode.

2018. DOI: 10.31031/COJEC.2018.01.000515
Also, the curves are tangent to the straight line that corresponds
totheeigenmodesofthelinearsystemforsmallvibrationamplitudes
[18], i.e. in each mode, the beginning of the nonlinear modal curve
on the horizontal axes, coincides with the linear eigenfrequency of
that mode. As the nonlinearity is dominant in real-life applications,
especially, when large deformation is occurred, this result is very
important for designing of such systems.
The frequency-energy consideration
The energy of the system is as [16]:
2
( ) n
n
E Y A= ∑
where An
are vectors of dimension six, which can be obtained
using the numerical solution. In Figure 6-8 the frequency-energy
curves of 1 to 3-diameter modes are shown. In each figure
the corresponding curves of linear and nonlinear stiffness are
compared. As it can be seen, in the higher frequency, the energy of
the system with nonlinear stiffness is lower than that of the system
with linear stiffness. It is clearly observed that in all modes, the
corresponding curves of the system with nonlinear stiffness have a
lower energy in the higher frequency, i.e. the nonlinearity is become
significant in the lower frequencies compared to the system with
linear stiffness.
Figure 6: Effect of nonlinear stiffness on the backbone curves in frequency-energy plan for 1-diameter mode.
Figure 7: Effect of nonlinear stiffness on the backbone curves in frequency-energy plan for 2-diameter mode.

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2018. DOI: 10.31031/COJEC.2018.01.000515
Figure 8: Effect of nonlinear stiffness on the backbone curves in frequency-energy plan for 3-diameter. mode.
Figure 9: Comparison of the backbone curves in frequency-energy plan for 1,2,3 and zero-diameter modes.
In Figure 9 the frequency-energy curves of 1 to 3-diameter
modes of the system with nonlinear stiffness and the corresponding
curve of zero-diameter mode in the system with linear stiffness
are depicted. As it can be seen, all curves are tangent to the linear
normal modes. Also, nonlinearity is more significant in 1, 2, 3 and
zero-diametermodes,respectively.Indeed,theeffectofnonlinearity
is of a hardening type.
Conclusion
In this study, the free vibration of a cyclic symmetry system
with geometric nonlinearity as well as nonlinearity in stiffness
of connector springs has been studied. The numerical method is
applied to the system of coupled nonlinear differential equations
with cubic nonlinearity. The results of numerical method are
verified with results of a similar system which was solved by the
harmonic balance method and exact solution (elliptic functions)
in another work. It can be concluded that the effect of nonlinear
stiffness on the system behavior is more significant in 1, 2, and
3-diameter modes, respectively due to their mode components
and the configuration of terms of differential equations, whereas
there is no difference in zero-diameter mode due to nonlinearity
in stiffness. Also, these modes have a lower energy in the higher

2018. DOI: 10.31031/COJEC.2018.01.000515
frequency in comparison to the system with the linear stiffness.
These results are important in designing of such systems when the
nonlinearity is dominant especially in large displacements.
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model. Journal of Sound and Vibration 331(12): 2911-2928.
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