In this paper we consider the case of a non-linear structure with cyclic symmetry which can be seen as approximation for bladed disk type structures. Using the multiple scale method, we show that soliton solutions are possible within such structure. The theoretical developments are illustrated with numerical simulations on a simple system with cubic non-linearities

Grolet, A.

Hoffmann, N.

Schwingshackl, C.

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SOLITONS IN NON-LINEAR CYCLIC SYSTEM
A. Grolet1, N. Hoffmann1 and C. Schwingshackl1
1 Department of Mechanical Engineering, Imperial College London
E-mail: a.grolet@imperial.ac.uk
Abstract: In this paper we consider the case of a non-linear structure with cyclic symmetry which can be seen as approximation
for bladed disk type structures. Using the multiple scale method, we show that soliton solutions are possible within such
structure. The theoretical developments are illustrated with numerical simulations on a simple system with cubic non-linearities.
Keywords: non-linear dynamics, cyclic system, travelling waves, soliton, multiple scales
1 Introduction
In aeronautics, cyclic structures are of particular interest especially for turbo machinery. Due to increased
flexibility, reduced clearance and other design imperatives, non-linear effects have to be taken into account in
order to model the system’s behaviour adequately.
In the linear case, it is well known that most of the eigen-modes are doublets and can be seen as two stationary
waves (or two travelling waves with with opposite velocities). In the non-linear case, mode shapes may be affected
by nonlinearity (non-similar modes) so that motion can localize on a small number of sectors Vakakis (1996);
Georgiades et al. (2009). Those stationary solutions are often referred to as intrinsic localized modes (ILM) or
discrete breathers (DB), and can be computed with numerical method such as the Harmonic Balance Method
(HBM). In contrast to those stationary solutions, travelling waves can also be computed using the HBM, and it
has been shown that such solution can bifurcate leading to localized travelling waves solutions (see eg Grolet and
Thouverez (2012)).
Following results of solid state physics, another type of travelling waves solution, composed of a carrier wave
modulated by an envelope, may also exist and are often referred to as solitons Remoissonet (1994). In mechanics,
there is a small number of studies about the numerical or experimental generation of solitons. Most of them focus
on simple systems such as discrete lattices Davies and Moon (1997); Konotop (1996). Here we aim at searching for
soliton-like solutions in simple non-linear cyclic structure. This constitutes an attempt to introduce the concept of
soliton in the structural mechanics community, and to show their relation with classical solutions such as obtained
with the harmonic balance method.
2 Description of the procedure
We consider a cyclic lattice with on site cubic nonlinearity (simple approximation for bladed disk structure
Vakakis (1993)). The equation of motion can be written as follow (with a = 2piNs ):
u¨n + ω
2
0un + ξu
3
n =
c20
a2
(un−1 + un+1 − 2un) (1)
linear analysis: Using cyclic symmetry condition leads to the following expression for the dispersion relation, the
group velocity cg = dwkdk and the group velocity dispersion (GVD) 2α =
d2wk
dk2 :
w2k = ω
2
0 + 2
c20
a2
(1− cos(ka)), cg = dwk
dk
=
1
wk
c20
a
sin(ka), 2α =
d2wk
dk2
=
1
wk
(c20 cos(ka)− c2g) (2)
soliton solution: Here we search for a solution of the cyclic lattice under the form of a travelling wave with
modulated amplitude Remoissonet (1994):
un = Ψn(X,T )e
i(k(n−1)a−ωkt) + c.c. = 2Re
(
Ψn(X,T )e
i(k(n−1)a−wkt)) (3)
where X = x, T = t and Ψn = Ψ(na) = Ψ(x). Using an order 2 Taylor expansion for Ψn+1 (continuum
approximation), keeping only one harmonic and supposing the non-linearity is of the order 2, we can show (see
e.g Remoissonet (1994)) that the envelope should verify the following Non-Linear Schroedinger Equation (NLSE):
i
dΨ
dτ
+ α
d2Ψ
dη2
+ β|Ψ|2 Ψ = 0 (4)
with τ = T = 2t, η = X − cgT = (x − cgt), α = 12 d
2wk
dk2 , β = − 12ωk 3ξ. There is a wide variety of solution
for the NLS equation (envelope and hole solitons, discrete breathers, ..., see eg Karjanto and Van Groesen (2007)).
Here we only consider envelope solitons which are known to exist for αβ ≥ 0 and which have the following
expression: Ψ(η, τ) = V sech(V
√
β
2αη)e
i β2 V
2τ . Going back to physical coordinates and dropping  gives :
un(t) = 2V sech
[
V
√
β
2α
((n− 1)a− cgt)
]
cos
[
k(n− 1)a− ωkt+ β
2
V 2t
]
(5)
3 Numerical results, Discussion
Here we apply the previous developments to a numerical example of Eq.(1) with the following parameters:
n = 100, ω20 = 1, c
2
0 = 1, ξ = 0.1. Using the expression in Eq.(5) one is able to compute initial conditions for
generating envelope soliton in the cyclic lattice (provided that the wave number k is such that αβ ≥ 0).
η
0 1 2 3 4 5 6
e
n
ve
lo
pe
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
time [s]
0 100 200 300 400 500 600 700
a
m
pl
itu
de
-0.1
-0.05
0
0.05
0.1
time [s]
0 100 200 300 400 500 600 700
a
m
pl
itu
de
-0.1
-0.05
0
0.05
0.1
η
0 1 2 3 4 5 6
e
n
ve
lo
pe
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time [s]
0 100 200 300 400 500 600 700
a
m
pl
itu
de
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
time [s]
0 100 200 300 400 500 600 700
a
m
pl
itu
de
-0.5
0
0.5
η
0 1 2 3 4 5 6
e
n
ve
lo
pe
0.2
0.4
0.6
0.8
1
1.2
1.4
time [s]
0 100 200 300 400 500 600 700
a
m
pl
itu
de
-1.5
-1
-0.5
0
0.5
1
1.5
time [s]
0 100 200 300 400 500 600 700
a
m
pl
itu
de
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Figure 1: Exemple of soliton solution for system (1). line 1: envelope evolution in the moving frame (solid line: non linear,
dash-dotted: linear). line 2: temporal evolution of nonlinear system. line 3: temporal evolution of linear system. (Results
obtained for k = 40 for different amplitude col.1: V = 0.1. col.2: V = 0.7. col.3: V = 1.5)
Typical results are computed for k = 40 for three different amplitudes and are depicted on Fig.1. For low
amplitude, the shape is not localized and non-linear results are close to the linear ones (the envelope behave almost
like a constant). For middle amplitude, one can see clearly the effect of dispersion in the linear case: the envelope
is flattening with time. In the non-linear case, the envelope does not change shape and is actually travelling with
velocity cg (envelope superimpose in the moving frame): it is a soliton type solution. For large amplitude, we can
see that the shape remains rather well approximated by the envelope soliton (the envelope does not change), even
if some error is made on the velocity (the envelope does not exactly superimpose in the moving frame).
The development presented here are valid as long as the continuum approximation holds (say forN ≥ 50). For
smaller number of sector, one would need to consider a discrete version of the non-linear Schroedinger equation
(DNLSE). Further works should address this problem and should also investigate the link between stationary
solution and soliton through variation of the group velocity.
References
M. Davies and F. Moon. Solitons, chaos and modal interaction in periodic structure. Nonlinear Dynamics: The
Richard Rand 50th Anniversary Volume: Series on Stability, Vibration and Control of Systems, Series B, 2:
119–143, 1997.
F. Georgiades, M. Peeters, G. Kerschen, and J. Golinval. Modal analysis of a nonlinear periodic structure with
cyclic symmetry. AIAA journal, 47(4):21014–1025, 2009.
A. Grolet and F. Thouverez. Free and forced vibration analysis of a nonlinear system with cyclic symmetry:
Application to a simplified model. Journal of sound and vibration, 331(12):2911–2928, 2012.
N. Karjanto and E. Van Groesen. Derivation of the nls breather solutions using displaced phase amplitude variables.
In 5th SEAMS-GMU International Conference on Mathematics and its Applications, pages 43–50, 2007.
V. V. Konotop. Small amplitude envelope soliton in nonlinear lattices. Wiley Interscience, 1996.
M. Remoissonet. Waves called solitons. Springer, 1994.
A. Vakakis. A multiple scale analysis of nonlinear localized modes in a cyclic periodic structure. Journal of
Applied Mechanics, 60:388–397, 1993.
A. Vakakis. Normal mode and localization in nonlinear systems. Physical review letters, 106(20):204502, 1996.


Solitons in non-linear cyclic system

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Solitons in non-linear cyclic system

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