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Non-linear behaviour of free-edge shallow spherical
shells: Eect of the geometry
Cyril Touzé, Olivier Thomas
To cite this version:
Cyril Touzé, Olivier Thomas. Non-linear behaviour of free-edge shallow spherical shells: Eect of
the geometry. International Journal of Non-Linear Mechanics, Elsevier, 2006, 41 (5), pp.678-692.
�10.1016/j.ijnonlinmec.2005.12.004�. �hal-00838885�
Non-linear behaviour of free-edge shallow spherical shells:
Effect of the geometry
C. Touzé
a,∗
, O. Thomas
b
a
ENSTA-UME, Chemin de la Hunière, 91761 Palaiseau Cedex, France
b
Structural Mechanics and Coupled Systems Laboratory, CNAM, 2 rue Conté, 75003 Paris, France
Abstract
Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening
behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán’s theory for large
deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the
partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism
allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to
the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as
asymmetric modes. Plates (obtained as R →∞) are known to display a hardening behaviour, whereas shells generally behave in a softening
way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process
is clarified.
Keywords: Shallow spherical shells; Hardening/softening behaviour; Non-linear normal modes; Internal resonance
1. Introduction
Geometrically non-linear vibrations of shallow shells is a
problem of widespread relevance, with a range of applications
from aerospace industry to musical acoustics [1–3]. Despite nu-
merous studies, some important features still remain partially or
completely unsolved, due to the non-linear nature of the prob-
lem. One of the most common property of non-linear oscilla-
tions is the dependence of the frequency of free oscillation on
vibration amplitude, which can be of the hardening or soften-
ing type. It is a well-known fact that flat plates display a hard-
ening behaviour, as it has been shown both theoretically and
experimentally (see e.g. [4–9]). Introducing an initial curvature
in the middle surface of the structure creates a quadratic non-
linearity, which, in turn, may change the non-linear behaviour
to softening type, depending on the balance of the magnitude
of quadratic and cubic terms [6,10,11]. It is thus a legitimate
question to determine the correct non-linear behaviour of shal-
low
spherical shells, and more precisely, the transition from
the hardening (flat plate) behaviour to the softening one, as the
curvature increases.
Among the available studies concerned with this question,
almost all of them restrict to the case of axisymmetric vibra-
tions. Evensen and Evan-Iwanowski [12] found a softening
behaviour with the harmonic balance method, without studying
the transition from hardening to softening behaviour. Many in-
vestigators used a single-mode approach to study the effect of
geometry on the non-linear behaviour: Grossman et al. [13] in-
vestigated different type of boundary conditions and mentioned
the transition from hardening to softening behaviour as the rise-
to-thickness ratio increases, Yasuda and Kushida [14] found
the first mode to be softening for a little curvature whereas
the second stays of the hardening type. Singh et al. [15] and
Sathyamoorthy [16] studied the influence of transverse shear
deformation and rotatory inertia in the case of a moderately
thick shell. Some slight improvements were proposed in order
to overcome the single-mode truncation. Pandalai and Varadan
1
[17] used a two-mode expansion, but for the precise transition
from hardening to softening behaviour, restrict once again to a
single-mode expansion [10].Li[18] proposed a time-mode ap-
proach to solve the axisymmetrical vibrations, and studied the
effect of orthotropy. Gonçalves [19] used three modes to de-
rive free vibration results, but took only a few discrete values
for the geometry, and didn’t mention internal resonances. Other
geometrical parameters have been studied, as the effect of geo-
metrical imperfections on the type of non-linearity [20]. Leissa
and Kadi [21] studied the transition for a shallow shell hav-
ing a rectangular boundary. Doubly-curved shallow shells have
also been investigated, by Shin [22] with the assumption of a
single-mode, recently by Alhazza [23] with the direct method
proposed by Nayfeh [24], and by Amabili [25] for a number of
different geometries.
During the last decade, a number of different proofs showed
that too severe truncations lead to erroneous results in the pre-
diction of the trend of non-linearity. Nayfeh et al. [26] seem to
be the first to point out the shortcomings of using a single-mode
approach, which could lead to quantitative as well as qualita-
tive erroneous results. The most direct solution is to keep a
sufficient number of modes in the analysis, which renders ana-
lytical expressions almost intractable and leads to intensive nu-
merical computations. This approach has been used by Amabili
et al. [27–29] for the non-linear behaviour of circular cylindri-
cal shells in order to clarify controversial results upon the trend
of non-linearity (see e.g. [27,30–33]). They used a model with
up to 23 degrees of freedom (dofs) [29], and highlight the fun-
damental role played by axisymmetrical contractions in asym-
metrical vibrations. Moreover, Pellicano et al. [29] propose a
map of non-linearity, showing the trend of non-linearity as a
function of the two independent geometrical parameters of the
shell, with a severely reduced models composed of three modes.
In order to avoid the main drawbacks associated to the large
number of modes retained, significant efforts have been made
toward definitions of reduced-order models (ROMs), able to
predict the correct non-linear behaviour with a limited number
of equations. A complete review of the available mathemati-
cal methods is provided by Steindl and Troger [34]. Strategies
based upon the application of the multiple scales method di-
rectly into the PDE have been proposed by Nayfeh and cowork-
ers [24,35], and has been successfully applied to the cases of
non-linear vibrations of buckled beams [36,37], shallow sus-
pended cables [38] and doubly-curved cross-ply shallow shells
[23]. Non-linear normal modes (NNMs), defined as invariant
manifolds in phase space [39], state a proper framework to em-
bed the influence of all linear modes into a single NNM. It has
been shown that the motion onto the invariant manifold, de-
fined by a single oscillator equation, predicts the correct trend
of non-linearity [40].
The objective of this paper is to derive the correct trend
of non-linearity for axisymmetric as well as asymmetric non-
linear vibrations of shallow spherical shells, by using the frame-
work of NNMs, defined through real normal form theory, as
proposed in [40]. The paper is divided into three main parts:
first, a non-linear model of the shell is briefly presented, re-
lying on the analog for thin shallow shells of von Kármán’s
theory for large deflection of plates. A thorough presentation
of the model as well as experimental validations are available
in [41,42]. Then the framework of NNMs is presented and
the analytical coefficient governing the trend of non-linearity
is derived. Important comments from the analytical formula,
with respect to modal truncation and 2:1 internal resonances,
are given. Finally, results are presented for purely asymmet-
ric modes, axisymmetric and mixed modes (asymmetric modes
with at least one nodal circle).
2. Governing equations
The aim of this section is to provide the PDEs of motion
for a shallow spherical shell. Only the main results are given,
the interested reader is referred to [41] for more details on the
non-linear model.
2.1. Local equations
A spherical shell of thickness h, radius of curvature R and
outer diameter 2a, made of a homogeneous isotropic material
of density , Poisson ratio and Young’s modulus E, is con-
sidered (see Fig. 1). Large transverse deflections and moderate
rotations are considered, so that the model is a generalization
of von Kármán’s theory for large deflection of plates. The main
geometrical hypotheses, which are relevant for this study are
the following:
• the shell is thin: h/a>1 and h/R>1;
• the shell is shallow: a/R>1.
Other assumptions are classical for large deflection von Kár-
mán’s type models [41]. Finally, as this study is concerned with
the trend of non-linearity, which is dictated by the conservative
problem, damping and external forces are not taken into ac-
count. The equations of motion are given in terms of the trans-
verse displacement w along the normal to the mid-surface and
the Airy stress function F , for all time t:
Dw +
1
R
F + h ¨w = L(w, F ), (1a)
F −
Eh
R
w =−
Eh
2
L(w, w), (1b)
where D = Eh
3
/12(1 −
2
) is the flexural rigidity, ¨w is the
second partial derivative of w with respect to time, is the
laplacian and L is a bilinear quadratic operator. With the as-
sumption of a shallow shell fulfilled, the spatial operators are
written in polar coordinates, and thus reads:
(·) = (·)
,rr
+
1
r
(·)
,r
+
1
r
2
(·)
,
, (2)
and
L(w, F ) = w
,rr
F
,r
r
+
F
,
r
2
+ F
,rr
w
,r
r
+
w
,
r
2
− 2
w
,r
r
−
w
,
r
2
F
,r
r
−
F
,
r
2
. (3)
2
a
Rh
H
M
r
w
Fig. 1. Geometry of the shell: three-dimensional sketch and cross section.
Free-edge boundary conditions are considered:
F and w are bounded at r = 0, (4a)
F
,r
+
1
a
F
,
= 0,F
,r
+
1
a
F
,
= 0, at r = a, (4b)
w
,rr
+
a
w
,r
+
a
2
w
,
= 0, at r = a, (4c)
w
,rrr
+
1
a
w
,rr
−
1
a
2
w
,r
+
2 −
a
2
w
,r
−
3 −
a
3
w
,
= 0, at r = a. (4d)
The above equations stem from the vanishing of the external
force at the edge: Eqs. (4b) are related to the membrane forces,
Eq. (4c) to the bending moment and Eq. (4d) to the twisting
moment and transverse shear force.
2.2. Dimensionless form of the equations
Dimensionless variables are introduced by
r = a ¯r, t = a
2
h/D
¯
t, w =h ¯w, F =Eh
3
¯
F . (5)
Substituting the above definitions in equations of motion,
Eq. (1a,b), and dropping the overbars in the results, one obtains:
w + ε
q
F +¨w = ε
c
L(w, F ), (6a)
F −
√
w =−
1
2
L(w, w), (6b)
where the aspect ratio of the shell has been introduced:
=
a
4
R
2
h
2
. (7)
As it will be shown next, for a fixed value of the Poisson ratio
, all the linear results (eigenfrequencies and mode shapes), as
well as the trend of non-linearity, only depend on , which is
the only free parameter related to the geometry of the shell. The
two other parameters ε
q
and ε
c
appearing in Eq. (6) are equal to
ε
q
=12
1−
2
a
2
Rh
=12
1−
2
√
,ε
c
=12
1−
2
. (8)
Their subscripts comes from the fact that they balance, re-
spectively, the quadratic and the cubic terms in the non-linear
ordinary differential equations governing the dynamics of the
problem (see Eq. (11)).
2.3. Linear analysis
The linearized equations of motion are analyzed to derive the
eigenmodes and eigenfrequencies of the problem, as a function
of the geometry. The eigenmodes are the solutions of:
+ −
2
= 0, (9a)
= , (9b)
where refers to the eigenmodes of the transverse motion
and to those of the membrane motion. The coefficient =
12(1−
2
) is the only parameter of the linear problem. All the
study could have been realized by taking as the geometrical
parameter, as it is sometimes done by various authors [12,19].
However, the results will be presented as functions of ,in
order to set apart the material property which appear through
the Poisson ratio in the expression of . In the remainder of
this study, is kept constant at = 0.33.
Transverse and membrane mode shapes are numbered
(k,n)
and
(k,n)
where k is the number of nodal diameters and n
the number of nodal circles. Axisymmetric modes are such that
k =0. For k 1 (asymmetric modes), the associated eigenvalue
has a multiplicity of two, so that for each eigenfrequency, there
are two independent mode shapes, called preferential configu-
rations or companion modes. Among these modes, purely asym-
metric modes (such that k 2 and n =0) are distinguished from
mixed modes (such that k 1 and n 1).
The linear analysis provided in Ref. [41] shows that all de-
formed shapes, except membrane mode shapes for purely asym-
metric modes, have a negligible dependence on the geometry.
On the contrary, the eigenfrequencies dependence on the as-
pect ratio , represented on Fig. 2, shows different behaviour,
which leads to classify the modes into two families.
The first family contains the purely asymmetric modes, since
their eigenfrequencies display a slight dependence on curvature.
The second family contains axisymmetric and mixed modes.
They show a huge eigenfrequency dependence on curvature.
Due to these different behaviours, the study of the trend of non-
linearity will distinguish: purely asymmetric modes, axisym-
metric and mixed modes.
2.4. Modal expansion
The complete non-linear equations of motion (6) are pro-
jected onto the natural basis of the transverse eigenmodes.
3
0 20 40 60 80 100 120 140 160 180 200
0
500
1000
1500
2000
2500
3000
3500
Angular frequency - ω
kn
[adim]
Aspect ratio:κ [adim]
3.0 4.0 5.0 6.0 7.0 9.08.0 10.0
0.1
11.0 12.0
2.1
5.2
8.1
3.3
2.0 1.1
0.2
3.1
1.2
4.1
6.1
1.3
3.2
5.1
2.2
0.3
4.2
7.1
0.4
2.3
Fig. 2. Dimensionless natural frequencies
kn
of the shell as a function of the aspect ratio , for = 0.33.
The displacement is thus written as:
w(r, ,t)=
+∞
p=1
X
p
(t)
p
(r, ), (10)
where the subscript p refers to a specific mode of the shell,
defined by a couple (k, n) and, if k = 0, a binary vari-
able which indicates the preferential configuration consid-
ered (sine or cosine companion mode). The modal displace-
ments X
p
are the unknowns, and their dynamics is governed
by ∀p 1:
¨
X
p
+
2
p
X
p
+ ε
q
+∞
i=1
+∞
j=1
p
ij
X
i
X
j
+ ε
c
+∞
i=1
+∞
j=1
+∞
k=1
p
ij k
X
i
X
j
X
k
= 0. (11)
The expression of the non-linear coefficients are
p
ij
=−
S
⊥
p
L(
i
,
j
) dS −
1
2
+∞
b=1
1
4
b
×
S
⊥
L(
i
,
j
)Υ
b
dS
S
⊥
p
Υ
b
dS, (12)
p
ij k
=
1
2
+∞
b=1
1
4
b
S
⊥
L(
i
,
j
)Υ
b
dS
×
S
⊥
p
L(
k
,Υ
b
) dS. (13)
The Υ
n
, as well as its associated zero
n
, are defined in [41].
S
⊥
is the domain defined by (r, ) ∈[01]×[02].
The temporal equations (11) describe the dynamics of the
shell. In particular, the trend of non-linearity can be inferred
from these equations. As shown by various authors (see e.g.
