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The effect of geometric imperfections and viscous damping on the type of nonlinearity (i.e., the hardening or softening behaviour) of circular plates and shallow spherical shells with free edge is here investigated. The Von Kármán large-deflection theory is used to derive the continuous models. Then, nonlinear normal modes (NNMs) are used for predicting with accuracy the coefficient, the sign of which determines the hardening or softening behaviour of the structure. The effect of geometric imperfections, unavoidable in real systems, is studied by adding a static initial component in the deflection of a circular plate. Axisymmetric as well as asymmetric imperfections are investigated, and their effect on the type of nonlinearity of the modes of an imperfect plate is documented. Transitions from hardening to softening behaviour are predicted quantitatively for imperfections having the shapes of eigenmodes of a perfect plate. The role of 2:1 internal resonance in this process is underlined. When damping is included in the calculation, it is found that the softening behaviour is generally favoured, but its effect remains limited.

When continuous structures such as plates and shells
undergo large amplitude motions, the geometrical nonlinearity leads to a
dependence of free oscillation frequencies on vibration amplitude. The type of
nonlinearity describes this dependency, which can be of the hardening type (the
frequency increases with amplitude), or of the softening type (the frequency
decreases). A large amount of literature is devoted to predicting this type of
nonlinearity for continuous structures, and especially for structures with an
initial curvature such as arches or shells because the presence of the
quadratic nonlinearity makes the problem more difficult to solve. On the other
hand, the hardening behaviour of flat structures such as beams and plates is a
clearly established fact, on the theoretical as well as the experimental
viewpoint, (see, e.g., [

Among the available studies concerned with this
subject, quite all of them that were published before 1992 could not be
considered as definitive since they generally restrict to the case of a
single-mode vibration through Galerkin method, see, for example, [

However, these last studies are restricted to the case
of perfect structures, and the damping is neglected in the computations; and
both of them have an influence on the type of nonlinearity, so that a complete
and thorough theoretical study that could be applied to real structures need to
address the effect of imperfections and damping. The geometric imperfections
have a first-order effect on the linear as well as the nonlinear
characteristics of structures. A large amount of studies are available, where
the effect of imperfections on the eigenfrequencies and on the buckling loads
are generally addressed, see, for example, [

To the authors' knowledge, the role of the damping in
the prediction of the type of nonlinearity has been only recently detected as
an important factor that could change the behaviour from hardening to softening
type [

The article is organized as follows. In Section

A thin plate of diameter

(a) Top view and (b)
cross-section of an imperfect circular plate of radius

The equations are then written with nondimensional
variables, by introducing

The boundary conditions for the case of a free edge
write, in nondimensional form [

In order to discretize the PDEs, a Galerkin procedure
is used. As the eigenmodes cannot be computed analytically because the shape of
the imperfection is arbitrary, the eigenmodes of the perfect plate

In order to work with diagonalized linear parts, the
matrix of eigenvectors

The temporal equations (

Nonlinear oscillators differ from linear ones by the frequency dependence on vibration amplitude. The type of nonlinearity defines the behaviour, which can be of the hardening or the softening type.

As shown in [

Once the nonlinear change of coordinates operated,
proper truncations can be realized. In particular, keeping only the normal
coordinates

The asymptotic third-order approximation of the
dynamics onto the

Finally, the method used for deriving the type of
nonlinearity can be summarized as follows. For a geometric imperfection of a
given amplitude, the discretization leading to the nonlinear oscillator (

This section is devoted to numerical results about the
effect of typical imperfections on the type of nonlinearity of imperfect
plates. Two typical imperfections are selected. The first one is axisymmetric
and has the shape of mode (0,1), the second one has the shape of the first
asymmetric mode (2,0). Consequently, damping is not considered, so that in
each equation we have:

In this section, the particular case of an
axisymmetric imperfection having the shape of mode (0,1) (i.e., with
one nodal circle and no nodal diameter) is considered. The expression of the
static deflection writes

(a) Three-dimensional view and
(b) cross-section of the circular plate with geometric imperfection having the
shape of the first axisymmetric mode. As nondimensional quantities are used,

Figure

Nondimensional natural
frequencies

First, the effect of the imperfection on the
axisymmetric modes (0,1) and (0,2) is studied. In this case, the problem is
fully axisymmetric so that all the truncations can be limited to axisymmetric
modes only, which drastically reduces the numerical burden. The result for mode
(0,1) is shown in Figure

Type of nonlinearity for mode (0,1) with an axisymmetric imperfection having the shape of mode (0,1).

The numerical
result for mode (0,2) is shown in Figure

Type of nonlinearity for
mode (0,2) with an axisymmetric imperfection having the shape of mode (0,1).
2:1 internal resonances with modes (0,3), (0,4), and (0,5) occurs,
respectively, for

Finally, the effect of the imperfection on asymmetric
modes is shown in Figure

Type of nonlinearity for (a): mode (2,0), and (b): mode (4,0) with an axisymmetric imperfection having the shape of mode (0,1).

These results corroborate those obtained on shallow
spherical shells [

In this section, the effect of an imperfection having
the shape of mode (2,0) is studied. Due to the loss of symmetry, degenerated
modes are awaited to cease to exist : the equal eigenfrequencies of the

(a) 3D view, (b) top view,
and (c) cross-section along

The behaviour of the eigenfrequencies with the
imperfection is shown in Figure

Nondimensional natural frequencies

The numerical
results for type of nonlinearity relative to the two configurations (2,0,C) and
(2,0,S) are shown in Figure

Type of nonlinearity for (a): mode (2,0,C) and (b): (2,0,S); for an imperfection having the shape of mode (2,0,C).

Finally, the results for the first two axisymmetric
modes (0,1) and (0,2) are shown in Figure

Type of nonlinearity for (a): mode (0,1) and (b): (0,2); for an imperfection having the shape of mode (2,0,C).

In this section, the effect of viscous damping on the
type of nonlinearity is addressed. The particular case of the shallow spherical
shell is selected to establish the results. The equations of motion are first
briefly recalled. Then specific cases of damping are considered, hence
complementing the results of [

The local equations of motions for the shallow
spherical shell can be obtained directly, see [

A Galerkin expansion is used for discretizing the PDEs
of motion. As the eigenmodes

The type of nonlinearity can be inferred from (

Three cases are selected in order to derive results for a variety of damping behaviours:

For all

For all

For all

In the above cases,

The effect of increasing damping is shown for modes
(0,1) and (4,0), for Case

Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratio

Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratio

Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratio

Case

Finally, Case

The effect of geometric imperfections on the hardening/softening behaviour of circular plates with a free edge has been studied. Thanks to the NNMs, quantitative results for the transition from hardening to softening behaviour has been documented, for a number of modes and for two typical imperfections. Two general rules have been observed from the numerical results: for modes which eigenfrequency strongly depends on the imperfection, the type of nonlinearity changes rapidly, and softening behaviour occurs for a very small imperfection with an amplitude being a fraction of the plate thickness. On the other hand, some eigenfrequencies show a slight dependence with the considered imperfection. For these, 2:1 internal resonances are the main factor for changing the type of nonlinearity. In a second part of the paper, the effect of viscous damping on the type of nonlinearity of shallow spherical shells has been studied. It has been shown quantitatively that this effect is slight for usual damping values encountered in thin structures.