Global and bifurcation analysis of a structure with cyclic
symmetry
Emmanuelle Sarrouy, Aurélien Grolet, Fabrice Thouverez
To cite this version:
Emmanuelle Sarrouy, Aurélien Grolet, Fabrice Thouverez. Global and bifurcation analysis of
a structure with cyclic symmetry. International Journal of Non-Linear Mechanics, Elsevier,
2011, 46 (5), pp.727–737. .
HAL Id: hal-00623630
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Submitted on 14 Sep 2011
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Global and bifurcation analysis of a structure with cyclic symmetry
E. Sarrouya,∗, A. Groleta , F. Thouvereza
a Ecole
Centrale de Lyon, Laboratoire LTDS, Bat. E6, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France
Abstract
This article combines the application of a global analysis approach and the more classical continuation,
bifurcation and stability analysis approach of a cyclic symmetric system. A solid disc with four blades,
linearly coupled, but with an intrinsic non-linear cubic stiffness is at stake. Dynamic equations are turned
into a set of non-linear algebraic equations using the Harmonic Balance Method. Then periodic solutions are
sought using a recursive application of a global analysis method for various pulsation values. This exhibits
disconnected branches in both the free undamped case (non-linear normal modes, NNMs) and in a forced
case which shows the link between NNMs and forced response. For each case, a full bifurcation diagram is
provided and commented using tools devoted to continuation, bifurcation and stability analysis.
Keywords: Global analysis, Non-linear, Homotopy, Rotordynamics, Cyclic symmetry
1. Introduction
In this paper, both free and forced vibrations of
a non-linear cyclic symmetric structure are studied.
The structure is composed of nb identical substructures which undergo large strains. This system is
typical when one studies bladed disks [2, 27]. After
forms [22]. 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). Moreover, in the free or forced case, only one solution
exists for a given frequency.
modeling the system, a set of coupled non-linear dif-
In the case of non-linear systems with cyclic sym-
ferential equations in which non-linearity appears
metry, it has been shown that the number of non-
by cubic terms is obtained .
linear normal modes (NNMs, [7]) can exceed the
In the linear case, the study of the linear nor-
number of dofs, the extra NNMs being generated
mal modes (LNMs) reveals a majority of double
through bifurcations or internal resonances [7, 28].
eigenfrequencies, corresponding to distinct eigen-
Localized non-linear modes are an example of this
property; they correspond to a free motion in which
∗ Corresponding
author. Tel: +33 681393569/Fax:+33
144246468
Email addresses: [email protected] (E.
only a few substructures vibrate with non negligible amplitude and they have no counterpart in the
Sarrouy), [email protected] (A.
linear theory [2, 29]. In the forced case, these ad-
Grolet), [email protected] (F. Thouverez)
ditional NNMs give rise to a number of additional
Preprint submitted to International Journal of Non-Linear Mechanics
December 8, 2010
resonances leading to multiple solutions as in [29]
2. Studied system
where Vakakis showed a very complicated struc2.1. General description and dynamical equations
ture of resonance for a non-linear system with cyclic
symmetry by using the multiple scales method. He
The studied structure has a cyclic symmetry
showed that for a given frequency many solutions
property and can therefore be broken up in nb iden-
can coexist, some of them being stable.
tical sectors (Fig.
1).
Each sector is modeled
by a thin rectangular plate clamped to the rigid
Not only can multiple solutions coexist but also
disk which itself is fixed (Fig. 2). Consecutive
can they be disconnected from each other. In this
plates are coupled by a linear stiffness while non-
latter case, classical methods based on continuation
linearity is introduced by taking into account their
and bifurcation analysis fail at finding the discon-
large deflection. Plane stress assumption as well
nected branches of solutions. If one wants to di-
as the Love-Kirchhoff hypothesis (cross-sections ex-
mension a structure properly - by considering all the
hibit solid body motion and remain perpendicular
possible solutions - one then needs to adopt a global
to the deformed surface of the middle sheet) are
analysis (GA) approach. Several solutions are men-
made. Then, plates displacements are entirely pa-
tioned in the literature (see [23] for an overview)
rameterized by their middle sheet transverse dis-
with a common drawback even for small systems
placement wj , 1 ≤ j ≤ nb . Moreover the mate-
which is the computation cost. The GA method
rial is assumed to follow a standard bi-dimensional
proposed in this paper takes advantage of the cubic
Hooke’s law leading to the following expression for
form of the non-linearity combined with a reformu-
the strain energy of plate j:
lation of equations through the harmonic balance
2
Z Lx Z Ly /2 "
1 Eh
1 ∂wj 2
Uj =
(
)
2 (1 − ν 2 ) x=0 y=−Ly /2
2 ∂x
2
1 ∂wj 2 1 ∂wj 2
1 ∂wj 2
+ 2ν
(
)
(
) (
)
+
2 ∂y
2 ∂x 2 ∂y
#
2
1 − ν ∂wj ∂wj
+
dxdy
2
∂x ∂y
Z Lx Z Ly /2 " 2 2
∂ wj
Eh3
1
+
2
2 12(1 − ν ) x=0 y=−Ly /2
∂x2
2 2
2
∂ wj
∂ wj ∂ 2 wj
+
+
2ν
∂y 2
∂x2 ∂y 2
2 2 #
1−ν
∂ wj
+
2
dxdy
2
∂x∂y
method (HBM); the resulting systems in the free
undamped case for searching NNMs as well as in a
forced case can then be solved (globally) in a reasonable amount of time.
Section 2 describes the system and its dynamical
and HBM equations. The global analysis principle is then explained in section 3.1; this is followed
by recalls on continuation methods, bifurcation and
stability analysis in section 3.2. Finally these meth-
(1)
ods are applied to the undamped free system in order to find NNMs in section 4 and to a forced case
where E is the Young’s modulus and ν is the Pois-
which exhibits a very rich response in section 5.
son’s ratio.
2
The energy V j of the linear stiffness localized at
x = 0) and leads to the following interpolation for
(xr , yr ) = (Lx /4, 0) between plates j and j + 1 is
transverse displacement of the j-th blade:
given by:
Vj =
1
2 k(wj (xr , yr )
wj (x, y) = qj Φ(x, y) for 1 ≤ j ≤ nb
− wj+1 (xr , yr ))
2
(5)
(2)
By applying Lagrange’s equations, and by adding
for 1 ≤ j ≤ nb with convention j + 1 = 1 if j = nb .
a damping term, the following motion equations are
By neglecting rotary inertia the kinetic energy Tj
obtained:
of plate j is given by :
1
Tj = ρh
2
Z
Lx
x=0
Z
[M]Ẍ + [C]Ẋ + [K]X + βX3 = Fe (t)
Ly /2
2
ẇj dxdy
(6)
(3)
with the notations X = (qj )1≤j≤n , and X3 =
y=−Ly /2
In this paper, only an harmonic force, orthogonal
(qj3 )1≤j≤n . The vector Fe = Fe0 cos(ωt) stands for
to the plate, localized in (xf , yf ) = (Lx , 0) is con-
the external forces amplitude, [M] = [I] is the mass
sidered. The work Wj due to such an excitation on
matrix, [C] = δ[I] is the damping matrix, [K] is the
plate j is given by:
stiffness matrix given by:

α + 2c
−c


 −c
α + 2c
[K] = 


0
−c

−c
0
Wj = wj (Lx , 0)Fej
(4)
The total energies U , T , V and W are then given
by the sum over the number of plates nb of the different local energies Uj , Tj , Vj and Wj . Equations
0
−c
−c
0
α + 2c
−c
−c
α + 2c




 (7)



and β is the non-linear stiffness coefficient. The
of motions are finally derived by using Lagrange’s
definitions of α, β, c, and δ can be found in A along
equations along with a Rayleigh-Ritz approxima-
with their numerical values.
tion [16].
2.2. Harmonic balance method
[Figure 1 about here.]
The harmonic balance method (HBM) is widely-
[Figure 2 about here.]
used for the study of non-linear systems. Numerous
The remainder of this article will be devoted to
applications can be found in the literature, showing
studying the case of a system composed of nb = 4
its ability to treat strongly non-linear systems like
identical sectors, whose displacements are interpo-
friction between blades and casing [11, 10] or ge-
lated by a single Ritz function approximating the
ometric non-linearities [14, 21]. One major advan-
first bending mode thus yielding a non-linear prob-
tage of the method is that it requires no assumption
lem with n = 4 dofs that will serve as an exam-
about the non-linearities magnitudes and uses the
ple for the present work.
same procedure for strongly and weakly non-linear
Φ(x, y) =
( Lxx )2
The Ritz function is
models.
(consistent with the clamping at
3


 A 
is the unknown in the frewhere X̃ =
 B 
The HBM consists in a decomposition of the solution X in a truncated Fourier series:
X(t) = A0 +
Nh
X
quency domain; it follows that ñ = 4 × 2 = 8. [Hl ]
Ak cos(kωt) + Bk sin(kωt) (8)
corresponds to the linear part of the HBM:


[K] − ω 2 [M]
ω[C]

[Hl ] = 
−ω[C]
[K] − ω 2 [M]
k=1
Injecting
(6),
this
and
by
development
projecting
[1, (cos(kωt), sin(kωt))1≤k≤Nh ]
following scalar product
Z
hf, gi =
(8)
in
equation
equations
basis
on
the
using
the
(12)
He corresponds to the constant part related to the
excitation
2π
ω
f (t)g(t)dt


 F 
e0
He =
 0 
(9)
0
one gets a system of ñ = n × (2Nh + 1) non-linear
(13)
and Hnl corresponds to the non-linear part
algebraic equations with n×(2Nh +1)+1 unknowns
Ak , Bk and ω.
Hnli (X̃) = 43 β(A3i + Ai B2i )
The number of harmonics retained Nh is a very
Hnl n+i (X̃) = 34 β(B3i + Bi A2i )
for 1 ≤ i ≤ n
important parameter. Generally, the higher Nh is,
(14)
the better the solution. However, in the case where
Notice that in this case the solution frequency ω
the number of harmonics selected is high, the so-
is taken equal to the one of the excitation force.
lution procedure can quickly become difficult and
Free Case
time consuming. Fortunately, in most cases the se-
The equation to be solved in the free undamped
ries converges fast enough and leads to systems with
case is given by:
reasonable dimensions. In this article only the first
harmonic is going to be retained (Nh = 1) and the
[M]Ẍ + [K]X + βX3 = 0
(15)
constant term A0 is dropped due to the symmetry
and the solution X is sought of the following form:
of the system. Depending on whether the system
is free or forced, different formulations can be ob-
X(t) = A cos(ωt)
tained.
(16)
The fact that only the cosine terms are retained
Forced case
corresponds to a phase condition in which all initial
The equation to be solved in the forced case is
velocities are set to zero. With this approximation
given by (6). The solution X is sought of the fol-
for the solution X, ñ = 4 and the following set of
lowing form:
algebraic equations is obtained:
X(t) = A cos(ωt) + B sin(ωt)
(10)
3
([K] − ω 2 [M])A + βA3 = 0
4
which leads to the following set of 2n algebraic
equations:
(17)
As there is no excitation force, the free solution
[Hl ]X̃ + Hnl (X̃) = He
(11)
frequency ω is also an unknown.
4
In order to be able to guarantee that all P zeros
3. Theory
will be reached, one has to respect a few rules: first
This section is devoted to theory on both global
of all, as each non-singular zero of P must be linked
analysis (GA) aspects and continuation coupled to
to a non-singular zero of Q, the initial polynomial
stability and bifurcation analysis.
must have at least as many zeros as P. Then, the
paths linking Q zeros to P zeros must be smooth
3.1. Global analysis
(undergo no bifurcation when λ slides from 1 to
When aiming at finding all the solutions of a
0) which implies that both polynomials must share
nonlinear system, looking at the whole state space
structural properties [25, Chap. 8]. A basic poly-
seems the right way to proceed. This is what most
nomial which has easy to find zeros and respects
global analysis methods such as cell-mapping [5],
these rules is


X̃d1 − 1


 1
..
Q(X̃) = γ
.



 X̃dñ − 1
ñ
cell exclusion algorithm [32] or algorithms based on
interval analysis [4] do. The main drawback of all
these methods is their huge cost with regards to
their computational resource consumption (see [23]
for a comparison). In order to bypass this problem,









, γ ∈ C, di = deg(Pi )
Such a polynomial has N =
one can take advantage of the algebraic form of our
(19)
Qñ
i=1
di zeros, that
is it has the maximum number of zeros a polyno-
problem formulated in the frequency space and use
mial with such total degrees (maximum degree di
a dedicated method to solve it globally relying on
among its monomials for each component Pi ) can
homotopy [25, 23, 24, 15]. Indeed, the nonlinearity
have. A homotopy process with this kind of ini-
introduced being cubic, equations (11) or (17) can
tial polynomial is called a total degree homotopy.
be considered as polynomials in X̃ variable. We will
The total degree homotopy theorem demonstrated
now denote P the corresponding polynomial.
in [25, chap. 8, p. 123] ensures that the N zeros
The main idea of this method is to embed the
of Q will lead to all the non-singular zeros of P. In
polynomial P to solve in a space whose parameters
our frame, only zeros with real components will be
are the polynomial coefficients. Then, starting from
considered.
a polynomial Q whose zeros (in Cñ ) are easy to
In many cases, P has only a few complex ze-
compute, a continuation scheme is applied to slide
ros. Starting with a lot more means a waste of
smoothly from these zeros to the ones of interest
time following paths that will mostly diverge to-
by substituting slowly P to Q which is, namely,
wards infinity (as a homogeneization technique is
applying a homotopy method. The system solved
used, they will remain numerically bounded, see
is
[25, 31]). That is why, in most cases it is interesting to work on the starting polynomial to reduce
R(X̃, λ) = λQ(X̃) + (1 − λ)P(X̃), λ ∈ [0, 1] (18)
the number of starting zeros while still respecting
with λ varying from 1 to 0.
the rules enunciated previously. Most current work
5
aims at exhibiting such systems and automatizing
next point to rely at a given distance from the pre-
their construction ([17, 18, 30, 25, 6, 13]). As will
vious one. The prediction step is performed us-
be shown later in this paper, such work is useless
ing a tangent predictor and the correction one uses
in the studied case. Using polynomial Q defined
a Newton-Raphson algorithm. More details about
in (19), 38 = 6561 starting zeros will be generated
continuation techniques can be found in [1].
in the forced case (HBM equation (11)) and only
34 = 81 paths will be followed in the free one (HBM
3.2.2. Bifurcation detection
equation (17)).
In this paragraph, we will focus on turning points
and branching points which are the main bifurca-
3.2. Continuation, bifurcation and stability
tions encountered in this study. The detection of
In this section, the classical tools to perform con-
these bifurcation points is done by monitoring the
∂G
∂Y
tinuation and bifurcation analysis of fixed points
determinant of the Jacobian matrix [JY ] =
(solutions in frequency space are constant) as well
(20): this determinant is null for such bifurcation
as the stability analysis of periodic solutions rely-
points [8]. A Newton like algorithm, described in
ing on Floquet coefficients evaluation are briefly re-
[19], is used to determine accurately the solutions
called.
(Yb , ωb ) such that
of
G(Yb , ωb ) = 0
3.2.1. Continuation algorithm
∂G
∂Y (Yb , ωb )T
A general system of equations used by a contin-
=0
(21)
kTk = 1
uation algorithm is
G(Y, µ) = 0
where T is the tangent direction.
(20)
The type of bifurcation is then determined by
where G is a non-linear function, Y is the vector of
estimating the rank of the Jacobian matrix [J] =
unknowns and µ is the continuation parameter. In
[JY Jω ] (where [Jω ] =
the present case Y = X̃, the HBM unknowns and
(ñ + 1). This rank is exactly ñ for turning points
µ = ω, the free or forced frequency. The contin-
and no further work has to be done since the arc-
uation procedure builds the path solution of (20)
length continuation scheme is able to handle such
for some µ range by iterating two steps. First, a
points. In the case of branching point bifurcations,
point is predicted using one or more of the previ-
this rank is at most ñ − 1 and several linearly inde-
ous points of the path, yielding an estimation of
pendent tangents Tj respecting [JY ]Tj = 0 can be
the next point. This predicted solution is then cor-
exhibited. Each of them corresponds to a prediction
rected iteratively until convergence is achieved.
direction to explore. In this study, the prediction
∂G
∂ω )
which is of size ñ ×
In this study an arc-length continuation is used:
directions are computed using the eigenvectors Ψi
G is obtained by augmenting the HBM set of equa-
of [JY ] associated with the zero eigenvalue at the
tions (11) or (17) with an equation constraining the
bifurcation point [20].
6
authors confronted two methods. First a compu-
3.2.3. Stability using Floquet theory
The stability of solutions is determined by using
tation using Floquet basic theory relying on time
Floquet’s theory. In this study, just the primary
integrations; the cycle U∗ is then rebuilt using the
results of this method will be recalled; a more de-
HBM coefficients. Second, the perturbation tech-
tailed presentation is available in [19].
nique proposed by D. Laxalde in his PhD work [9]
was implemented. Each time, both methods gave
To do so, the first order formulation of the dy-
the same results.
namic system (6) is considered:
U̇ = f (U, t)
(22)
4. Application to backbone curves
with U a vector combining X and Ẋ.
If U∗ (t) is a periodic solution of (22) with period
The aim of this section is to build the NNMs
T ∗ and z(t) is a small perturbation added to U∗ ,
of the undamped structure (17). The results will
the equation governing its evolution is derived from
be displayed mainly in the Frequency Energy Plot
(22):
(FEP) which is one of the most appropriate ways
∂f
ż =
(U∗ , t)z
∂U
to depict the frequency-energy dependence of such
(23)
modes [7, 3]. First, a brief recall of the linear study
which is linear with periodic coefficients. There-
results is proposed. Then, a global analysis step is
fore, any solution z(t) corresponding to any initial
performed and exhibits many solutions which are
perturbation is a linear combination of m(i) (t), so-
grouped with respect to their energy, creating en-
lutions of (23) at time t having all its components
ergy branches. Each solution branch is finally fol-
(i)
but the i-th null at time t = 0: mj (0) = δij ,
lowed and analyzed using tools of section 3.2. One
1 ≤ i, j ≤ 2n. The growth of z is then directly re-
of the branches and its bifurcations is analyzed in
lated to the one of m(i) which is in turn evaluated
a detailed way before the drawing of the full bifur-
via the eigenvalues of the monodromy matrix
h
i
[Mon] = m(1) (T ∗ ), . . . , m(2n) (T ∗ )
(24)
cation diagram.
4.1. Linear system
This was demonstrated and stated properly in the
Floquet’s theorem (1883). Eigenvalues of the mon-
The linear system has the following pair of eigen-
odromy matrix are usually complex and are called
vector and eigenvalues (eigenvectors normalized
Floquet coefficients. They give us information on
with respect to mass matrix), i.e. linear normal
the stability of the solution (U∗ is unstable as soon
modes (LNM) :
as one of them has a magnitude greater than 1)
ω1 = 93.63 rad.s−1 , L1 = {1, 1, 1, 1}t /2
√
ω2 = 95.20 rad.s−1 , L2 = {1, 0, −1, 0}t / 2
√
ω2 = 95.20 rad.s−1 , L3 = {0, −1, 0, 1}t / 2
and on the type of bifurcation (through the way
they cross the unit circle in complex plane) [19, 8].
Some other techniques [12, 9] exploit the HBM re-
ω4 = 96.75 rad.s−1 , L4 = {−1, 1, −1, 1}t /2
sults directly to determine the cycle stability. The
7
(25)
Using classical denominations [26], one can state
method with a drastic 10−12 criterion on residual;
that
no difference between raw GA results and refined
ones appeared. We then ensured that each solution
• L1 is a “mode 0” were all the coordinates have
kept by the global analysis algorithm was different
the same amplitude. This is a non-degenerate
from the others from each step and counted them:
mode with no nodal diameter.
there are up to 81 solutions for pulsations higher
• L2 and L3 are degenerate “cos-1” and “sin-1”
than 99.4 rad.−1 ! This justifies the use of the ba-
modes with 1 nodal diameter. According to
sic total degree homotopy method (same number of
[26, p. 202], the corresponding NNMs should
zeros for Q and P). This step also exhibited discon-
have the same backbone (n = 4 × p × 1).
nected solutions. To have a better understanding of
these numerous solutions, they were grouped with
• L4 is a “mode n/2” that exists because the sys-
respect to their energy level. These energy levels
tem has an even number of blades. All coor-
are represented in a FEP, drawing branches that
dinates vibrate with the same amplitudes but
are named, as shown on Fig. 4. Each branch is
opposite phases, generating 2 nodal diameters.
named n.p1 .p2 . n denotes the number of nodal di-
These notations will be useful for backbone analysis
ameters of the main branch and a .pi extension is
in section 4.3.
added each time a bifurcated branch occurs. d1 and
d2 refers to the disconnected branches and NS to
4.2. Global analysis
the null solution. Symbols on this figure refer to
the number of solutions that have the same energy
A global analysis method as explained in 3.1
level for a given pulsation ω.
was applied to the free system (17) in order to
build backbone curves. As the system is already
[Figure 3 about here.]
in a polynomial form, no further work was done on
equations. The total degree homotopy method was
applied recursively for ω between 93.0 rad.s
−1
[Figure 4 about here.]
and
4.3. Branches analysis using continuation
100.0 rad.s−1 with a 0.1 rad.s−1 step. For each step
34 = 81 paths are followed. This computation took
Every solution corresponding to each energy
758 s (about 12 min 38 s using an Intel Core 2 Duo
branch has been continued and analyzed with re-
E8400 (3Go RAM)) that is about 11 s per step and
gards to its stability and bifurcations using tools
led to a great amount of solutions as depicted in Fig.
of section 3.2. As the previous global analysis step
3. Besides the great number of solutions, one can
provided startpoints for each solution branch, the
see that plots for each Ai look similar which is in ac-
structure symmetry was not used to reduce the
cordance with the cyclic symmetry of the structure.
number of solutions to study; this would indeed
Each solution returned by the global analysis algo-
have required a fine bifurcation analysis in conjunc-
rithm was refined using a local Newton-Raphson
tion with the symmetry consideration to determine
8
the number of solutions appearing at each bifurca-
of the 4 solutions underlying energy branch 1’ bi-
tion point. As discussing each branch would be too
furcates and gives birth to two half branches that
long, one proposes to focus on branch 1’ and its bi-
match the branch 1’.1 energy level.
furcated branches. The same analysis was carried
[Figure 5 about here.]
on for each branch and the results are summed up
Branch 1’.1 matches 8 solutions that come from
in subsection 4.4.
branch 1’ bifurcation. These solutions include a
For each energy branch 1’, 1’.1 and 1’.1.1, one
LNM L1 contribution (Fig. 6 (b)) that becomes
of the solutions matching this energy evolution is
greater as ω increases. This indicates that this is
picked up as a representative of all the other solu-
not a similar mode. This branch is first unstable
tions with the same energy evolution. Each time,
and becomes stable for ω4 = 96.75 rad.s−1 , which
a figure divided in four frames presents its HBM
is the frequency at which branch 2 arises. It then
components evolution along with ω in frame (a);
gives birth to a new branch (branch 1’.1.1) that
in frame (b) the evolution of its decomposition on
“takes advantage” on the existence of this new en-
LNM in percent is drawn: the solution components
ergy level (of branch 2) as explained in the next
are projected on LNM of subsection 4.1 and these
paragraph. In the state space, each of the 8 half
“modal coordinates” are brought back to percent
branches matching branch 1’.1 energy branch bifur-
by dividing by the sum of their absolute value and
cates once and gives birth to 2 new half branches.
multiplying this by 100 . This will let us conclude
Solutions underlying this energy branch 1’.1 will bi-
on similar or non-similar property of the mode. Fi-
furcate once again for ω8 = 99.36 rad.s−1 , that is
nally, Floquet coefficients are plotted in frames (c)
for the same pulsation at which branches d1 and d2
and (d): frame (c) shows their norm along ω while
arise.
frame (d) represents them in the complex plane su-
[Figure 6 about here.]
perimposed on the unit circle. In order to get accurate Floquet coefficients, solutions initially com-
Branch 1’.1.1 matches 16 solutions bifurcated
puted using only the first harmonic A1 were refined
from branch 1’.1 solutions.
using A1 , A3 and A5 components.
These solutions in-
volve a new LNM L4 contribution that increases as
Branch 1’ matches 4 different solutions with the
the L1 contribution introduced in branch 1’.1 de-
same energy that contain both L2 and L3 contri-
creases. Fig. 7 (b) let us state this is a non-similar
butions with equal weight (Fig. 5 (a)). These 4
mode. This branch is always unstable.
solutions are identical with respect to a circular
shift of tha Ai components.
Fig.
[Figure 7 about here.]
5 (b) shows
5 (c) and (d)
Fig. 8 recapitulates branch 1’ bifurcations by
show that this mode bifurcates and become un-
showing the four initial solutions that match this
stable for ω3 = 95.98 rad.s−1 generating one new
energy branch and the subsequent bifurcated solu-
energy branch, namely branch 1’.1. In fact, each
tion branches. It is obvious via this figure that each
that it is a similar mode.
Fig.
9
is defined by Fe0 = af {1, 1, −1, −1}t . Simulations
initial solution and its bifurcated solutions (each
are carried out for af = 0.25 N.kg−1 .
line of plots in the figure) is equal to the others
First, the principal non-linear response has been
with respect to a circular shift of its components.
computed by continuation. This solution, called
[Figure 8 about here.]
Sol-A, corresponds to a motion with the 1’ mode
shape.
4.4. Conclusion
The solution and its stability are repre-
sented in a FEP on Fig. 10.
Table 1 and Fig. 9 summarize the results of the
By monitoring the determinant of the Jacobian
previous analysis carried out on all other branches.
matrix, 4 bifurcation points have been computed for
Fig. 9 exhibits branches similar to Fig. 4 but this
Sol-A, they are also represented on Fig. 10 labeled
time, points are linked continuously thanks to the
from A1 to A4. By studying the rank of the Jaco-
continuation step. Moreover, this figure provides
bian matrix, one can tell that points A1 and A3 are
information on stability and not on the number of
branching points (rank([J]) = 7) and points A2 and
solutions sharing the same energy branch anymore.
A4 are not only turning points but also branching
This simple structure exhibits in fact a very
points (rank([J]) = 6). Stable parts of the energy
rich modal behavior: a great number of solutions
branch are denoted by a thick (blue) line on Fig. 10
branches can be captured using continuation and
while unstable ones are represented by a thin (red)
bifurcation analysis. Due to cyclic symmetry, they
one. Underlying NNMs are plotted using dashed
draw a less important but still great number of
dot (black) lines. Squared points denote branching
energy branches. In addition to these numerous
points while circled ones indicate Hopf bifurcations.
branches, the use of a GA method lets us intercept
The same representation code is used for the follow-
two disconnected branches d1 and d2 that would
ing figures (Fig. 11 to Fig. 13). One can then see
have been missed otherwise.
on Fig. 10 that sol-A is unstable even for relatively
low energy levels.
[Table 1 about here.]
[Figure 10 about here.]
[Figure 9 about here.]
Starting from A2 (or A4), the two bifurcated
branches of solution have been computed (Sol-L2
5. Application to a forced case
and Sol-L3). The representation of these solutions
The previous section exhibited numerous NNMs
in the amplitude frequency diagram is not very con-
which will lead to complicated structures of reso-
venient because they superimpose themselves with
nance in forced cases. The aim of this section is to
a part of Sol-A, the only difference being that curves
illustrate this complexity through an example.
are not covered in the same way. To overcome this
The system is therefore excited with a force tak-
issue, we chose to represent these solutions in a
ing the branch 1’ shape, thus vector Fe0 of Eq. (13)
FEP in Fig. 11. In this FEP, Sol-L2 and Sol-L3
10
are superimposed, but one can see that these solu-
All the solutions are also depicted in a frequency-
tions possess stable parts and are positioned around
amplitude plot in Fig. 14 and Fig. 15. Two differ-
backbone curves of modes with 1 nodal diameter
ent figures were used for the sake of clarity. Only
(branch 1), therefore one can conclude that these
the amplitude of the first sector is considered.
two solutions correspond to resonances around non-
[Figure 14 about here.]
linear modes with shape L2 and L3.
[Figure 15 about here.]
[Figure 11 about here.]
In order to fully validate the stability analysis, di-
Starting from A1 (or A3), a bifurcated branch has
rect numerical integrations have been performed.
been computed (Sol-B), and its representation in
Initial conditions were provided by HBM solutions
the FEP is given on Fig. 12. Sol-B is positioned
and simulations were carried out over 1000 periods.
around the backbone curve branch 1’.1. A stability
Results are plotted only for the five last periods.
analysis reveals that Sol-B possesses stable parts
Fig. 16 (resp. Fig. 17) shows the temporal evo-
and also unstable parts generated through Hopf bi-
lution of both numerical integration and HBM so-
furcation. Several branching points have been de-
lution for a stable point of Sol-B (resp. Sol-E) at
tected for Sol-B, namely B1, B2 and B3 (Fig. 12).
ω = 99.53 (resp. ω = 98.84). A good agreement
B2 (or B3) leads to an unstable bifurcated re-
can be observed for both simulations, thus confirm-
sponse, Sol-C, positioned around the backbone
ing the validity of the proposed approach.
curve of branch 1’.1.1 (Fig. 13), and B1 leads to an
unstable bifurcated response, Sol-D, which seems to
[Figure 16 about here.]
be positioned close to the backbone curve of branch
[Figure 17 about here.]
1’ (Fig. 13).
This example exhibits a very complicated struc-
[Figure 12 about here.]
ture of resonance which is closely related to the
Finally, global analysis reveals another kind of so-
backbone curves previously presented. The FEP
lution, disconnected from the other. They are rep-
appears to be a convenient tool for identifying non-
resented by closed curves in the FEP. This solution
linear modes involved in non-linear resonances.
family, termed Sol-E, is positioned near the back-
Applied to this case, global analysis has two virtues:
bone branch 2.1 and is depicted on Fig. 13. Using
first of all, finding the disconnected solutions, that
formulation (11), 6561 paths are followed for each
is guarantying that all the possible stable states
tested ω. This takes approximately 1 h 15 min per ω
will be considered when dimensioning the structure.
value using an Intel Core 2 Duo E8400 (3Go RAM)
The second one is not theoretical, but practical: if
which seems reasonable with regard to the informa-
one can detect all the bifurcation points by carry-
tion it provides.
ing out a bifurcation analysis along the continuation
process, it is easy to skip such points. For example,
[Figure 13 about here.]
11
bifurcation points A2 and A4 being turning points
Z
Φ2 dP =
ρhLx Ly
5
with two additional tangent for each, the determi-
m=ρ
nant of the Jacobian gets null but does not change
sign. An automated script based on this criterion
1 k
k
c = Φ(xr , yr )2 =
m
256 m
had therefore missed Sol-L2 and Sol-L3 branches
Fe j = Φ(xf , yf )
and they were in fact found by the GA approach.
α=
β=
6. Conclusion
P
f j (t)
f j (t)
=
m
m
5
Eh2
3 ρL4x (1 − ν 2 )
8E
ρL4x (1 − ν 2 )
The damping coefficient δ is defined as:
√
α
δ=
200
This paper applies both a classical continuation
method and an original global analysis approach
(26)
(27)
Here are the numerical values used in this study:
to a cyclic symmetric structure. These methods
the geometric and physical parameters are affected
enable the drawing of a complete bifurcation di-
by the following set of values:
agram in a reasonable amount of time even in
the case of numerous bifurcations and existence
Lx = 1.5 m, Ly = 0.3 m, h = 0.03 m,
of disconnected solutions. Moreover it can inter-
xr = Lx /4, yr = 0, xf = Lx , yf = 0,
cept branches that would have been missed by the
E = 210 GPa, ν = 0.3, ρ = 7800 kg.m−3 ,
continuation and bifurcation analysis scheme. The
k = 8 · 105 N.m−1
forced case example also emphasizes the usefulness
(28)
which correspond to the following values for the dif-
of NNMs to analyze a non-linear structure.
ferent parameters in equation (6) :
The ability of the proposed global analysis
α = 8.7662 · 103 s−2 , c = 148.36 s−2 ,
method to exhibit disconnected solutions being
β = 4.6752 · 107 m−2 .s−2
proved, further work now should focus on devel-
(29)
opments allowing its application to larger systems
and to quasi-periodic solutions search.
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14
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Bladed wheel model . . . . . . . . . 16
Rectangular plate diagram . . . . . . 17
GA applied to free system . . . . . . 18
GA for NNMs: energy branches . . . 19
Branch 1’ analysis . . . . . . . . . . 20
Branch 1’.1 analysis . . . . . . . . . 21
Branch 1’.1.1 analysis . . . . . . . . 22
Energy branch 1’ sum up . . . . . . 23
Backbones FEP with stability . . . . 24
Forced case: Sol-A . . . . . . . . . . 25
Forced case: Sol-A, Sol-L2 and Sol-L3 26
Forced case: Sol-A and Sol-B . . . . 27
Forced case: Sol-B, Sol-D and Sol-E
28
Forced case: sector 1 magnitude (1/2) 29
Forced case: sector 1 magnitude (2/2) 30
Forced case: Comparison between HBM and numerical integration (1/2) 31
Forced case: Comparison between HBM and numerical integration (2/2) 32
15
Linear stiffness
Sectors (Thin
rectangular plates)
Rigid disk
Figure 1: Bladed wheel model used for establishing equations
16
z
y
w
v
Ly
u
x
h
Lx
Figure 2: Diagram of a rectangular plate and corresponding
coordinate system
17
A1 (mm)
6
4
4
2
2
0
0
−2
−2
−4
−4
−6
92
94
6
96
98
ω (rad.−1 )
A3 (mm)
100
−6
92
4
2
2
0
0
−2
−2
−4
−4
94
96
98
ω (rad.−1 )
94
ω (rad.−1 )
A4 (mm)
94
ω (rad.−1 )
6
4
−6
92
A2 (mm)
6
100
−6
92
96
98
100
96
98
100
Figure 3: GA applied to free system: solutions components
18
d2
d1
0
1’
1’.1.1
2
2.2
2.3 1
1’.1
2.3.1
2.1
NS
101
100
ω (rad.s−1 )
99
98
97
96
95
94
93
0
0.1
1 sol.
0.2
2 sol.
0.3
0.4
Energy (J)
4 sol.
0.5
8 sol.
0.6
0.7
16 sol.
Figure 4: GA for NNMs: energy branches
19
(a) Ai (mm)
(b) Modal Decompo. (%)
50
1,2
5
0
0
−5
95
96
97
98
99
−50
95
3,4
100 101
(rad.s−1 )
ω
(c) Floquet: norm
0.2
1.1
0.1
Im
1.2
1
L1 ,L4
96
97
98
99
L3
100 101
(rad.s−1 )
ω
(d) Floquet: C plane
0
−0.1
0.9
−0.2
0.8
95
L2
96
97
98
99
100
ω (rad.s−1 )
101
0.8
1
Re
1.2
Figure 5: Branch 1’ analysis
20
(a) Ai (mm)
(b) Modal Decompo. (%)
6
1,4
50
4
40
L1
L2 ,L3
30
2
20
0
10
2,3
−2
95
96
97
98
99
100
ω (rad.s−1 )
(c) Floquet: norm
0
95
101
0.3
1.02
0.2
1.01
0.1
Im
1.03
1
97
98
99
L4
100 101
ω (rad.s−1 )
(d) Floquet: C plane
0
0.99
−0.1
0.98
−0.2
−0.3
0.97
95
96
96
97
ω
98
99
100
(rad.s−1 )
101
0.6
0.8
1
Re
1.2
1.4
Figure 6: Branch 1’.1 analysis
21
(a) Ai (mm)
6
(b) Modal Decompo. (%)
1
2
L2
40
4
20
2
0
−2
−20
−4
96
L1
L4
0
4
3
97
98
99
100
ω (rad.s−1 )
(c) Floquet: norm
L3
−40
96
101
1.2
97
98
99
100
101
ω (rad.s−1 )
(d) Floquet: C plane
0.3
0.2
Im
1.1
1
0.1
0
−0.1
0.9
0.8
96
−0.2
−0.3
97
98
99
100
ω (rad.s−1 )
101
0.6
0.8
1
Re
1.2
1.4
Figure 7: Branch 1’.1.1 analysis
22
0
−6
−6
ω2
ω3
ω4
0
ω8
−4
ω2
ω3
ω4
−4
ω8
2
ω2
ω3
ω4
2
ω8
−2
4
−2
−4
2
2
−4
−6
0
0
−6
ω2
ω3
ω4
4
ω8
−2
ω2
ω3
ω4
0
ω8
6
ω2
ω3
ω4
6
ω8
0
4
4
−4
−4
2
2
−6
−6
0
0
−2
4
2
−4
−4
2
0
−6
−6
0
1’
Stable
1’.1
Unstable
ω2
ω3
ω4
−2
ω8
4
ω2
ω3
ω4
6
ω8
0
ω2
ω3
ω4
0
ω8
6
ω2
ω3
ω4
Solution 4
ω2
ω3
ω4
−2
ω8
−2
ω2
ω3
ω4
6
ω8
6
ω2
ω3
ω4
0
ω8
0
A4
ω8
0
ω8
A3
ω8
0
ω8
A2
−2
ω2
ω3
ω4
Solution 2
6
4
ω2
ω3
ω4
Solution 3
A1
4
ω2
ω3
ω4
Solution 1
6
1’.1.1
Figure 8: Solutions matching energy branch 1’ and their
bifurcations
23
0
1’
d2
d1
2.3
1
1’.1
2.3.1
2
1’.1.1
2.2
2.1
ω (rad.s−1 )
99.36
98.45
98.27
97.5
96.75
95.98
95.2
93.63
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Energy (J)
Stable
Unstable
Figure 9: Backbones FEP with stability
24
1’
101
A2
100
ω (rad.s−1 )
99
98
A3
97
A4
96
95
A1
94
93
92
0
0.1
0.2
0.3
0.4
0.5
0.6
Energy (J)
Figure 10: Forced case: Sol-A (positioned around branch 1’)
and its bifurcation points
25
1’
1
101
100
A2
99
ω (rad.s−1 )
Sol−L2
Sol−L3
98
A3
97
A4
96
95
A1
94
Sol−A
93
92
0
0.1
0.2
0.3
0.4
0.5
0.6
Energy (J)
Figure 11: Forced case: Sol-A and its bifurcated branch SolL2 and Sol-L3 (positioned around branch 1)
26
1’.1
1’
101
100
A2
99
ω (rad.s−1 )
Sol−B
98
B3
97
A3
A4
Sol−A
96
B2
95
A1
B1
94
93
92
0
0.1
0.2
0.3
0.4
0.5
0.6
Energy (J)
Figure 12: Forced case: Sol-A and its bifurcated branch SolB (positioned around branch 1’.1)
27
100
1’
2
1’.1.1
1’.1
2.1
101
Sol−C
Sol−E
99
ω (rad.s−1 )
Sol−B
98
Sol−D
B3
97
A3
96
B2
B1
95
A1
94
93
92
0
0.1
0.2
0.3
0.4
0.5
0.6
Energy (J)
Figure 13: Forced case: Sol-B and its bifurcated branches
Sol-C (positioned around branch 1’.1.1) and Sol-D (positioned near branch 1’) in addition with the disconnected
solution Sol-E (positioned near branch 2.1)
28
−3
6
x 10
Sol−B
A2
5
q
A21 + B21 (m)
Sol−L2
Sol−L3
(Superimposed)
4
−3
B2
1.3
B1
3
x 10
Sol−A
B3
A3
1.2
Sol−E
1.1
2
A1
1
B3
A4
98.4 98.6 98.8
99
1
B1 B2
0
94
95
96
97
98
99
100
101
102
ω (rad.s−1 )
Figure 14: Solutions of the forced case in the frequencyamplitude plot for sector 1: (−) Sol-A, (− ·) Sol-B, (- -)
Sol-L2/Sol-L3, (-) Sol-E
29
−3
6
x 10
Sol−C
q
A21 + B21 (m)
5
4
B2
Sol−B
B3
B1
3
A3
Sol−D
2
B3
A1
1
B1
B2
0
94
95
96
97
98
99
100
101
102
ω (rad.s−1 )
Figure 15: Solutions of the forced case in the frequencyamplitude plot for sector 1: (− ·) Sol-B, (- -) Sol-C, (-) Sol-D
30
−4
−4
x 10
x 10
5
q2
q1
5
0
−5
0
−5
0
0.1
0.2
0.3
0
0.1
time (s)
0.2
0.3
5
q4
q3
0.3
−3
x 10
5
0
−5
0
0.2
time (s)
−3
x 10
0.1
0.2
time (s)
0.3
0
−5
0
0.1
time (s)
Figure 16: Comparison between HBM and numerical integration for a stable point of Sol-B at ω = 99.53: (·) Numerical integration, (◦) HBM solution
31
−3
−4
x 10
x 10
1
5
q2
q1
0.5
0
0
−0.5
−5
−1
0
0.1
0.2
time (s)
0.3
0
0.3
0.1
0.2
time (s)
0.3
4
2
q4
0.5
q3
0.2
time (s)
x 10
1
0
−0.5
0
−2
−1
0
0.1
−3
−3
x 10
0.1
0.2
time (s)
0.3
−4
0
Figure 17: Comparison between HBM and numerical integration for a stable point of Sol-E at ω = 98.84: (·) Numerical integration, (◦) HBM solution
32
List of Tables
1
Backbones branches characterization sum up 34
33
Branch
0
1
1’
1’.1
1’.1.1
2
2.1
2.2
2.3
2.3.1
d1
d2
Nsol
2
4
4
8
16
2
8
8
4
8
8
8
ωi (rad.s−1 )
ω1 = 93.63
ω2 = 95.20
ω2 = 95.20
ω3 = 95.98
ω4 = 96.75
ω4 = 96.75
ω5 = 97.50
ω5 = 97.50
ω6 = 98.27
ω7 = 98.45
ω8 = 99.36
ω8 = 99.36
Sim/Non-sim
Similar
Similar
Similar
Non-similar
Non similar
Similar
Non similar
Non similar
Non similar
Non similar
Non similar
Non similar
Table 1: Backbones branches characterization sum up
34