# Effect of Imperfections and Damping on the Type of Nonlinearity of Circular Plates and Shallow Spherical Shells

Authors: Cyril Touzé; Cédric Camier; Gaël Favraud; Olivier Thomas
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

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.


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## Body

## 1. Introduction

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., [1–6]). The presence of the quadratic nonlinearity may
change the behaviour from hardening to softening type, depending on the
relative magnitude of quadratic and cubic nonlinear terms.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, [7–9]
for shallow spherical shells, or [10] for imperfect circular plates. Unfortunately, it has
been shown by a number of more recent investigations that too severe truncations
lead to erroneous results in the prediction of the type of nonlinearity,
see, for example, [11, 12], or the abundant literature on circular cylindrical
shells, where the investigators faced this problem for a long time [13–18]. As a consequence, a large
number of modes must mandatory be kept in the truncation of the partial
differential equations (PDEs) of motion, in order to accurately predict the
type of nonlinearity. Recent papers are now available where a reliable
prediction is realized, for the case of buckled beams [19], circular cylindrical
shells [20], suspended
cables [21], and
shallow spherical shells [22].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, [23–28] for the
case of circular cylindrical shells,
[29] for shallow
cylindrical panels, and [30] for the case of rectangular plates. Nonlinear frequency-responses
curves are shown in [31, 32] for clamped circular plates, [33–35] for rectangular plates,
[36] for circular
cylindrical shells, and [37] for circular cylindrical panels. Even though the
presence of geometric imperfection has been recognized as a major factor that
could make the hardening behaviour of the flat plate turn to softening
behaviour for an imperfection amplitude of a fraction of the plate thickness
[10, 38], a quantitative study,
which is not restricted to axisymmetric modes and that does not perform too
crude truncations in the Galerkin expansion, is still missing.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 [39]. In
particular, it is shown in [39] on a simple two degrees-of-freedom (dofs) system,
that the damping generally favours the softening behaviour. The aim of the
present study is thus to apply this theoretical result to the practical case of
a damped shallow spherical shell, so as to quantitatively assess the effect of
structural damping of the viscous type on the type of nonlinearity of a
two-dimensional vibrating structure.The article is organized as follows. In Section2,
local equations and boundary conditions for an imperfect circular plate with
free edge are given. Then the method used for computing the type of
nonlinearity is explained. Section 3 investigates how typical imperfections may
turn the hardening behaviour of flat plates to softening behaviour.
Quantitative results are given for selected imperfections having the shape of
eigenmodes of the perfect structure. Section 4 is devoted to the effect of
viscous damping. The particular case of a spherical imperfection is selected,
and the results are shown for three different damping dependances on frequency.


## 2. Theoretical Formulation

### 2.1. Local Equations and Boundary Conditions

A thin plate of diameter2a and uniform
thickness h is considered,
with h≪a, and free-edge boundary condition. The local
equations governing the large-amplitude displacement of a perfect plate,
assuming the nonlinear Von Kármán strain-displacement relationship and neglecting
in-plane inertia, are given, for example, in [5, 40]. An initial imperfection, denoted by w0(r,θ) and associated
with zero inital stresses is also considered, see Figure 1. The shape of this
imperfection is arbitrary, and its amplitude is small compared to the diameter
(shallow assumption): w0(r,θ)≪a. The local equations
for an imperfect plate deduce from the perfect case [18, 41, 42]. With w(r,θ,t) being the transverse
displacement from the imperfect position at rest, the equations of motion write
(2.1a)DΔΔw+ρhw¨=L(w,F)+L(w0,F)−cw˙,(2.1b)ΔΔF=−Eh2[L(w,w)+2L(w,w0)],where D=Eh3/12(1−ν2) is the flexural
rigidity, Δ stands for the
laplacian operator, c accounts for
structural damping of the viscous type, F is the Airy
stress function, and L is a bilinear
operator, whose expression in polar coordinates reads(2.2)L(w,F)=w,rr(F,rr+F,θθr2)+F,rr(w,rr+w,θθr2)−2(w,rθr−w,θr2)(F,rθr−F,θr2).(a) Top view and (b)
cross-section of an imperfect circular plate of radiusa and thickness h. (c) The particular case of a spherical imperfection,
with radius of curvature R.
(a)(b)(c)The equations are then written with nondimensional
variables, by introducing(2.3)r=ar¯,t=a2ρh/Dt¯,w=hw¯,w0=hw¯0,F=Eh3F¯,c=[Eh3/a2]ρh/Dc¯.As nondimensional equations will
be used in the remainder of the study, overbars are now omitted in order to
write the dimensionless form of the equations of motion
(2.4a)ΔΔw+w¨=ε[L(w,F)+L(w0,F)−cw˙],(2.4b)ΔΔF=−12[L(w,w)+2L(w,w0)],where ε=12(1−ν2).The boundary conditions for the case of a free edge
write, in nondimensional form [5]
(2.5a)F,r+F,θθ=0,F,rθ+F,θ=0,atr=1,(2.5b)w,rr+νw,r+νw,θθ=0,atr=1,(2.5c)w,rrr+w,rr−w,r+(2−ν)w,rθθ−(3−ν)w,θθ=0,atr=1.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 plateΨp(r,θ) are selected as
basis functions. Analytical expressions of Ψp(r,θ) involve Bessel
functions and can be found in [5]. The unknown displacement is expanded
with(2.6)w(r,θ,t)=∑p=1+∞qp(t)Ψp(r,θ),where the time functions qp are now the
unknowns. In this expression, the subscript p refers to a
specific mode of the perfect plate, defined by a couple (k,n), where k is the number
of nodal diameters and n the number of
nodal circles. If k≠0, a binary variable is added, indicating the
preferential configuration considered (sine or cosine companion mode). Inserting the
expansion (2.6) into
(2.4a) and
(2.4b) and using the orthogonality properties of the expansion
functions, the dynamical equations are found to be, for all p=1…N,(2.7)q¨p+2ξpωpq˙p+ε[∑i=1+∞αipqi+∑i,j=1+∞βijpqiqj+∑i,j,k=1+∞Γijkpqiqjqk]=0.Linear coupling terms between
the oscillator equations are present, as the natural modes have not been used
for discretizing the PDEs. Analytical expressions of the coupling coefficients (αip,βijp,Γijkp) are given in
[42]. The generic
viscous damping term c of (2.4a) has
been specialized in the discretized equations so as to handle the more general
case of a modal viscous damping term of the form 2ξpωpq˙p, where ξp is the damping
factor and ωp the
eigenfrequency of mode p. On the other hand, external forces have been
cancelled, as the remainder of the study will consider free vibrations only.In order to work with diagonalized linear parts, the
matrix of eigenvectorsP of the linear
part L=[αip]p,i is numerically
computed. A linear change of coordinates is processed, q=PX, where X=[X1…XN]T is, by
definition, the vector of modal coordinates, and N is the number
of expansion function kept in practical application of the Galerkin's method.
Application of P makes the
linear part diagonal, so that the discretized equations of motion finally
writes, ∀p=1…N,(2.8)X¨p+2ξpωpX˙p+ωp2Xp+ε[∑i,j=1NgijpXiXj+∑i,j,k=1NhijkpXiXjXk]=0.The temporal equations (2.8) describe the dynamics of an
imperfect circular plate. The type of nonlinearity can be inferred from these
equations. Unfortunately, too severe truncations in (2.8),
for example, by keeping only one degree-of-freedom (dof) (N=1) when
studying the nonlinear behaviour of the pth mode, lead
to incorrect predictions. Nonlinear normal modes (NNMs) offer a clean framework
for deriving a single oscillator equation capturing the correct type of
nonlinearity [12].
This is recalled in Section 3, where the analytical expression of the
coefficient dictating the type of nonlinearity is given.


### 2.2. Type of Nonlinearity

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 [12], NNMs provide an efficient framework for properly
truncating nonlinear oscillator equations like (2.8) and predict the type of
nonlinearity (hardening or softening behaviour). The method has already been
successfully applied to the case of undamped shallow spherical shells in
[22]. The main idea is
to derive a nonlinear change of coordinates, allowing one to pass from the modalXp coordinates to
new-defined normal coordinates Rp, describing the motions in an invariant-based span of
the phase space. The nonlinear change of coordinates is computed from Poincaré and Poincaré-Dulac's theorems, by successive elimination of
nonessential coupling terms in the nonlinear oscillator equations. Formally,
it reads(2.9a)Xp=Rp+∑i=1N∑j≥iN(aijpRiRj+bijpSiSj)+∑i=1N∑j=1NcijpRiSj+∑i=1N∑j≥iN∑k≥jN(rijkpRiRjRk+sijkpSiSjSk)+∑i=1N∑j=1N∑k≥jN(tijkpSiRjRk+uijkpRiSjSk),(2.9b)Yp=Sp+∑i=1N∑j≥iN(αijpRiRj+βijpSiSj)+∑i=1N∑j=1NγijpRiSj+∑i=1N∑j≥iN∑k≥jN(λijkpRiRjRk+μijkpSiSjSk)+∑i=1N∑j=1N∑k≥jN(νijkpSiRjRk+ζijkpRiSjSk).A third-order approximation of the complete change of coordinates is
thus computed. The analytical expressions of the introduced coefficients {aijp,bijp,cijp,rijkp,sijkp,tijkp,uijkp,} and
{αijp,βijp,γijp,λijkp,μijkp,νijkp,ζijkp,} are not given
here for the sake of brevity. The interested reader may find them in [12] for the undamped case, and
in [39] for the damped
case.Once the nonlinear change of coordinates operated,
proper truncations can be realized. In particular, keeping only the normal
coordinatesRp allows prediction
of the correct type of nonlinearity for the pth mode. The
dynamics onto the pth  NNM
reads(2.10)R¨p+ωp2Rp+2ξpωpR˙p+(εhpppp+Apppp)Rp3+BppppRpR˙p2+CppppRp2R˙p=0,where Apppp, Bpppp, and Cpppp are new
coefficients coming from the change of coordinates. Their expressions involve
the quadratic coefficients {gijp} only, together
with some of the transformation coefficients, {aijp,bijp,cijp} from (2.9a) and
(2.9b)
[39]:(2.11a)Apppp=ε[∑l≥iNgplpappl+∑l≤iglppappl],(2.11b)Bpppp=ε[∑l≥iNgplpbppl+∑l≤iglppbppl],(2.11c)Cpppp=ε[∑l≥iNgplpcppl+∑l≤iglppcppl].The asymptotic third-order approximation of the
dynamics onto thepth NNM given by
(2.10) allows one to accurately predict the type of nonlinearity of mode p. A first-order perturbative development from (2.10)
gives the dependence of the nonlinear oscillation frequency ωNL on the
amplitude of vibration a by the
relationship:(2.12)ωNL=ωp(1+Tpa2),where ωp is the natural
angular frequency. In this expression, Tp is the
coefficient governing the type of nonlinearity. If Tp>0, then hardening behaviour occurs, whereas Tp<0 implies
softening behaviour. The analytical expression for Tp writes
[12, 22](2.13)Tp=18ωp2[3(Apppp+εhpppp)+ωp2Bpppp].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 (2.8) is
first computed. The numerical effort associated to this operation is the most
important but remains acceptable on a standard computer. Then the nonlinear
change of coordinates is computed, which allows derivation of the Apppp and Bpppp terms occuring
in  (2.13), the sign of which determines the type of nonlinearity. Numerical
results are given in Section 3 for specific imperfections.


## 2.1. Local Equations and Boundary Conditions

A thin plate of diameter2a and uniform
thickness h is considered,
with h≪a, and free-edge boundary condition. The local
equations governing the large-amplitude displacement of a perfect plate,
assuming the nonlinear Von Kármán strain-displacement relationship and neglecting
in-plane inertia, are given, for example, in [5, 40]. An initial imperfection, denoted by w0(r,θ) and associated
with zero inital stresses is also considered, see Figure 1. The shape of this
imperfection is arbitrary, and its amplitude is small compared to the diameter
(shallow assumption): w0(r,θ)≪a. The local equations
for an imperfect plate deduce from the perfect case [18, 41, 42]. With w(r,θ,t) being the transverse
displacement from the imperfect position at rest, the equations of motion write
(2.1a)DΔΔw+ρhw¨=L(w,F)+L(w0,F)−cw˙,(2.1b)ΔΔF=−Eh2[L(w,w)+2L(w,w0)],where D=Eh3/12(1−ν2) is the flexural
rigidity, Δ stands for the
laplacian operator, c accounts for
structural damping of the viscous type, F is the Airy
stress function, and L is a bilinear
operator, whose expression in polar coordinates reads(2.2)L(w,F)=w,rr(F,rr+F,θθr2)+F,rr(w,rr+w,θθr2)−2(w,rθr−w,θr2)(F,rθr−F,θr2).(a) Top view and (b)
cross-section of an imperfect circular plate of radiusa and thickness h. (c) The particular case of a spherical imperfection,
with radius of curvature R.
(a)(b)(c)The equations are then written with nondimensional
variables, by introducing(2.3)r=ar¯,t=a2ρh/Dt¯,w=hw¯,w0=hw¯0,F=Eh3F¯,c=[Eh3/a2]ρh/Dc¯.As nondimensional equations will
be used in the remainder of the study, overbars are now omitted in order to
write the dimensionless form of the equations of motion
(2.4a)ΔΔw+w¨=ε[L(w,F)+L(w0,F)−cw˙],(2.4b)ΔΔF=−12[L(w,w)+2L(w,w0)],where ε=12(1−ν2).The boundary conditions for the case of a free edge
write, in nondimensional form [5]
(2.5a)F,r+F,θθ=0,F,rθ+F,θ=0,atr=1,(2.5b)w,rr+νw,r+νw,θθ=0,atr=1,(2.5c)w,rrr+w,rr−w,r+(2−ν)w,rθθ−(3−ν)w,θθ=0,atr=1.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 plateΨp(r,θ) are selected as
basis functions. Analytical expressions of Ψp(r,θ) involve Bessel
functions and can be found in [5]. The unknown displacement is expanded
with(2.6)w(r,θ,t)=∑p=1+∞qp(t)Ψp(r,θ),where the time functions qp are now the
unknowns. In this expression, the subscript p refers to a
specific mode of the perfect plate, defined by a couple (k,n), where k is the number
of nodal diameters and n the number of
nodal circles. If k≠0, a binary variable is added, indicating the
preferential configuration considered (sine or cosine companion mode). Inserting the
expansion (2.6) into
(2.4a) and
(2.4b) and using the orthogonality properties of the expansion
functions, the dynamical equations are found to be, for all p=1…N,(2.7)q¨p+2ξpωpq˙p+ε[∑i=1+∞αipqi+∑i,j=1+∞βijpqiqj+∑i,j,k=1+∞Γijkpqiqjqk]=0.Linear coupling terms between
the oscillator equations are present, as the natural modes have not been used
for discretizing the PDEs. Analytical expressions of the coupling coefficients (αip,βijp,Γijkp) are given in
[42]. The generic
viscous damping term c of (2.4a) has
been specialized in the discretized equations so as to handle the more general
case of a modal viscous damping term of the form 2ξpωpq˙p, where ξp is the damping
factor and ωp the
eigenfrequency of mode p. On the other hand, external forces have been
cancelled, as the remainder of the study will consider free vibrations only.In order to work with diagonalized linear parts, the
matrix of eigenvectorsP of the linear
part L=[αip]p,i is numerically
computed. A linear change of coordinates is processed, q=PX, where X=[X1…XN]T is, by
definition, the vector of modal coordinates, and N is the number
of expansion function kept in practical application of the Galerkin's method.
Application of P makes the
linear part diagonal, so that the discretized equations of motion finally
writes, ∀p=1…N,(2.8)X¨p+2ξpωpX˙p+ωp2Xp+ε[∑i,j=1NgijpXiXj+∑i,j,k=1NhijkpXiXjXk]=0.The temporal equations (2.8) describe the dynamics of an
imperfect circular plate. The type of nonlinearity can be inferred from these
equations. Unfortunately, too severe truncations in (2.8),
for example, by keeping only one degree-of-freedom (dof) (N=1) when
studying the nonlinear behaviour of the pth mode, lead
to incorrect predictions. Nonlinear normal modes (NNMs) offer a clean framework
for deriving a single oscillator equation capturing the correct type of
nonlinearity [12].
This is recalled in Section 3, where the analytical expression of the
coefficient dictating the type of nonlinearity is given.


## 2.2. Type of Nonlinearity

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 [12], NNMs provide an efficient framework for properly
truncating nonlinear oscillator equations like (2.8) and predict the type of
nonlinearity (hardening or softening behaviour). The method has already been
successfully applied to the case of undamped shallow spherical shells in
[22]. The main idea is
to derive a nonlinear change of coordinates, allowing one to pass from the modalXp coordinates to
new-defined normal coordinates Rp, describing the motions in an invariant-based span of
the phase space. The nonlinear change of coordinates is computed from Poincaré and Poincaré-Dulac's theorems, by successive elimination of
nonessential coupling terms in the nonlinear oscillator equations. Formally,
it reads(2.9a)Xp=Rp+∑i=1N∑j≥iN(aijpRiRj+bijpSiSj)+∑i=1N∑j=1NcijpRiSj+∑i=1N∑j≥iN∑k≥jN(rijkpRiRjRk+sijkpSiSjSk)+∑i=1N∑j=1N∑k≥jN(tijkpSiRjRk+uijkpRiSjSk),(2.9b)Yp=Sp+∑i=1N∑j≥iN(αijpRiRj+βijpSiSj)+∑i=1N∑j=1NγijpRiSj+∑i=1N∑j≥iN∑k≥jN(λijkpRiRjRk+μijkpSiSjSk)+∑i=1N∑j=1N∑k≥jN(νijkpSiRjRk+ζijkpRiSjSk).A third-order approximation of the complete change of coordinates is
thus computed. The analytical expressions of the introduced coefficients {aijp,bijp,cijp,rijkp,sijkp,tijkp,uijkp,} and
{αijp,βijp,γijp,λijkp,μijkp,νijkp,ζijkp,} are not given
here for the sake of brevity. The interested reader may find them in [12] for the undamped case, and
in [39] for the damped
case.Once the nonlinear change of coordinates operated,
proper truncations can be realized. In particular, keeping only the normal
coordinatesRp allows prediction
of the correct type of nonlinearity for the pth mode. The
dynamics onto the pth  NNM
reads(2.10)R¨p+ωp2Rp+2ξpωpR˙p+(εhpppp+Apppp)Rp3+BppppRpR˙p2+CppppRp2R˙p=0,where Apppp, Bpppp, and Cpppp are new
coefficients coming from the change of coordinates. Their expressions involve
the quadratic coefficients {gijp} only, together
with some of the transformation coefficients, {aijp,bijp,cijp} from (2.9a) and
(2.9b)
[39]:(2.11a)Apppp=ε[∑l≥iNgplpappl+∑l≤iglppappl],(2.11b)Bpppp=ε[∑l≥iNgplpbppl+∑l≤iglppbppl],(2.11c)Cpppp=ε[∑l≥iNgplpcppl+∑l≤iglppcppl].The asymptotic third-order approximation of the
dynamics onto thepth NNM given by
(2.10) allows one to accurately predict the type of nonlinearity of mode p. A first-order perturbative development from (2.10)
gives the dependence of the nonlinear oscillation frequency ωNL on the
amplitude of vibration a by the
relationship:(2.12)ωNL=ωp(1+Tpa2),where ωp is the natural
angular frequency. In this expression, Tp is the
coefficient governing the type of nonlinearity. If Tp>0, then hardening behaviour occurs, whereas Tp<0 implies
softening behaviour. The analytical expression for Tp writes
[12, 22](2.13)Tp=18ωp2[3(Apppp+εhpppp)+ωp2Bpppp].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 (2.8) is
first computed. The numerical effort associated to this operation is the most
important but remains acceptable on a standard computer. Then the nonlinear
change of coordinates is computed, which allows derivation of the Apppp and Bpppp terms occuring
in  (2.13), the sign of which determines the type of nonlinearity. Numerical
results are given in Section 3 for specific imperfections.


## 3. Effect of Imperfections

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:∀p=1…N,ξp=0. The study of the effect of damping will be done
separately and is postponed to Section 4.


### 3.1. Axisymmetric Imperfection

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(3.1)w0(r)=a(0,1)Ψ(0,1)(r),where Ψ(0,1)(r) is the mode
shape, depending only on the radial coordinate r as a
consequence of axisymmetry, and a(0,1) the considered
amplitude. The mode shape Ψ(0,1)(r) depends on
Bessel function [5],
and is shown in Figure 2. The
eigenmode is normalized so that ∫01Ψ(0,1)2(r)dr=1.(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,a=1 and the
amplitude a(0,1) of the
imperfection is made nondimensional with respect to the thickness h.
(a)(b)Figure3 shows the effect of the imperfection on the
eigenfrequencies, for an imperfection amplitude from 0 (perfect plate) to 10 h. It is observed that the purely asymmetric modes (k,0), having no nodal circle and k nodal
diameters, are marginally affected by the axisymmetric imperfection. The
computation has been done by keeping 51 basis functions: purely asymmetric
modes from (2,0) to (10,0), purely axisymmetric modes from (0,1) to (0,13); and
mixed modes from (1,1) to (6,1), (1,2), (2,2), (3,2) and (1,3). More details
and comparisons with a numerical solution based on finite elements are provided
in [42, 43]. The slight dependence of
purely asymmetric eigenfrequencies on an axisymmetric imperfection has already
been observed in [44]
with the case of the shallow spherical shell.Figure 3
Nondimensional natural
frequenciesω(k,n) of the
imperfect plate versus the amplitude of the imperfection having the shape of
mode (0,1).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 Figure4. It is observed
that the huge variation of the eigenfrequency with respect to the amplitude of
the imperfection results in a quick turn of the behaviour from the hardening to
the softening type, occuring for an imperfection amplitude of a(0,1) = 0.38 h. This small value has direct implication for the
case
of real plates. As the behaviour changes for a fraction of the plate thickness,
it should not be intriging to measure a softening behaviour with real plates
having small imperfections. This result can also be compared to an earlier
result obtained by Hui [10]. Although Hui did not study free-edge boundary
condition, he reported a numerical result for the case of simply supported
boundary conditions, where the behaviour changes for an imperfection amplitude
of 0.28  h. The
second main observation inferred from Figure 4 is the occurrence of 2:1
internal resonance between eigenfrequencies, leading to discontinuities in the
coefficient T(0,1) dictating the
type of nonlinearity. This fact has already been observed and commented for the
case of shallow spherical shells in [22].
It has also been observed for buckled beams and suspended cables [19, 21]. This is a small
denominator effect typical of internal resonance, that is, when the frequency of the studied
mode (0,1) exactly fulfills the relationship 2ω(0,1)=ω(0,n) with another
axisymmetric mode. 2:1 resonance arises here with mode (0,2) at 1.85 h and with mode
(0,3) at 5.66 h. On a practical point of view, one must bear in mind
that when   2:1  internal resonance occurs, single-mode solution does not exist anymore,
only coupled solutions are possible. Hence the concept of the type of
nonlinearity, intimately associated with a single
dof behaviour, loses its meaning in a narrow
interval around the resonance.Figure 4
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 Figure5. Once again, the geometric effect is
important and leads to a change of behaviour occurring at a(0,1) = 0.75 h, that is, for a small level of imperfection. 2:1 internal resonance also occurs, thus
creating narrow region where hardening behaviour could be observed. This result
extends Hui's analysis since only mode (0,1) was studied. Moreover, as a
single-mode truncation was used in [10], 2:1 resonances were missed.Figure 5
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, fora(0,1)/h = 1.74, 5.43,
and 9.92.Finally, the effect of the imperfection on asymmetric
modes is shown in Figure6 for modes (2,0) and (4,0). The very slight variation
of the eigenfrequencies of these modes versus the axisymmetric imperfection
results in a very slight effect of the geometry. It is observed that before the
first 2:1  internal resonance, the type of nonlinearity shows small variations.
Hence it is the behaviour of the other eigenfrequencies and the occurrence of
2:1 internal resonance that makes, in these
cases, the behaviour turn from
hardening to softening behaviour. For mode (2,0), this occurs for an
imperfection amplitude of a(0,1) = 0.44  h, where  2:1
resonance with mode (0,1) is observed. For mode (4,0), the first  2:1 resonance
occurs with mode (0,2) at a(0,1) = 1.39 h, but do not change the behaviour. It is the resonance
with mode (0,1) at a(0,1) = 4 h which makes the
behaviour turn from hardening to softening.Type of nonlinearity
for (a): mode (2,0), and (b): mode (4,0) with an axisymmetric imperfection
having the shape of mode (0,1).
(a)(b)These results corroborate those obtained on shallow
spherical shells [22].
The fundamental importance of axisymmetric modes in the study of asymmetric
ones is confirmed, showing once again that reduction to single mode has no
chance to deliver correct results. The behaviour of purely asymmetric modes is
found to be of the hardening type until the  2:1 internal resonance with mode
(0,1) occurs. However, a specificity of mode (2,0) with regard to all the other
purely asymmetric modes is that after this resonance, hardening behaviour
(though with a very small value of T(2,0)) is observed.
This was also the case for shallow spherical shells [22]. Finally, for very large
values of the imperfection, the behaviour tends to be neutral.


### 3.2. Asymmetric Imperfection

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 thesine and cosine configuration of degenerated modes
split. In the following, distinction is made systematically between the sine or
cosine configuration of companion modes, for example, mode (2,0,C) (resp., (2,0,S))
refers to the cosine (resp., sine) configuration. More precisely, the
imperfection has the shape of (2,0,C) and is shown in Figure 7.(a) 3D view, (b) top view,
and (c) cross-section alongθ=0 for the plate
with imperfection having the shape of mode (2,0,C).
(a)(b)(c)The behaviour of the eigenfrequencies with the
imperfection is shown in Figure8. As expected, the variation of the
eigenfrequency corresponding to (2,0,C) is huge, whereas (2,0,S) keep quite a
constant value. The symmetry is not completely broken. One can show that only
eigenmodes of the type (2k,n) split. On the other hand, as shown in Figure 8,
modes (3,0), (5,0), and (1,1) are still degenerated.Figure 8
Nondimensional natural frequenciesω(k,n) of the
imperfect plate versus the amplitude of the imperfection having the shape of
mode (2,0,C).The numerical
results for type of nonlinearity relative to the two configurations (2,0,C) and
(2,0,S) are shown in Figure9. The natural frequency of mode (2,0,C) undergoes
a huge variation, which results in a quick change of behaviour, occurring at
0.54 h. Then, a 2:1 internal resonance with (0,2) is noted,
but without a noticeable change in the type of nonlinearity, as the interval
where the discontinuity present is very narrow. In this case, the behaviour of T(2,0,C) looks like the
one observed in the precedent case, that is, the variation of T(0,1) versus an
imperfection having the same shape. On the other hand, the eigenfrequency of
mode (2,0,S) remains quite unchanged, so that the behaviour of T(2,0,S) is not much
affected by the imperfection until the 2:1 internal resonance is encountered.
In that case, the resonance occurs with the other configuration, that is, mode (2,0,C).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).
(a)(b)Finally, the results for the first two axisymmetric
modes (0,1) and (0,2) are shown in Figure10. Mode (0,1) shows a very slight
variation of its eigenfrequency with respect to the asymmetric imperfection
(2,0,C). Consequently, the type of nonlinearity is not much affected, until the
eigenfrequency of (2,0,C) comes to two times ω(0,1): 2:1 internal resonance occurs, and the behaviour
becomes softening. On the other hand, the eigenfrequency of (0,2) is more
affected by the imperfection. This result in an important decrease of T(0,2) while still
remaining positive. A  2:1  internal resonance with (0,3) is encountered for 3.51 h, and two
others 2:1  resonance, with (0,4) and (0,5), occur around 8 h. However, the interval on which the type of
nonlinearity changes its sign is so narrow that it can be neglected. The
behaviour is thus mainly of the hardening type for (0,2).Type of nonlinearity
for (a): mode (0,1) and (b): (0,2); for an imperfection having the shape of
mode (2,0,C).
(a)(b)


## 3.1. Axisymmetric Imperfection

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(3.1)w0(r)=a(0,1)Ψ(0,1)(r),where Ψ(0,1)(r) is the mode
shape, depending only on the radial coordinate r as a
consequence of axisymmetry, and a(0,1) the considered
amplitude. The mode shape Ψ(0,1)(r) depends on
Bessel function [5],
and is shown in Figure 2. The
eigenmode is normalized so that ∫01Ψ(0,1)2(r)dr=1.(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,a=1 and the
amplitude a(0,1) of the
imperfection is made nondimensional with respect to the thickness h.
(a)(b)Figure3 shows the effect of the imperfection on the
eigenfrequencies, for an imperfection amplitude from 0 (perfect plate) to 10 h. It is observed that the purely asymmetric modes (k,0), having no nodal circle and k nodal
diameters, are marginally affected by the axisymmetric imperfection. The
computation has been done by keeping 51 basis functions: purely asymmetric
modes from (2,0) to (10,0), purely axisymmetric modes from (0,1) to (0,13); and
mixed modes from (1,1) to (6,1), (1,2), (2,2), (3,2) and (1,3). More details
and comparisons with a numerical solution based on finite elements are provided
in [42, 43]. The slight dependence of
purely asymmetric eigenfrequencies on an axisymmetric imperfection has already
been observed in [44]
with the case of the shallow spherical shell.Figure 3
Nondimensional natural
frequenciesω(k,n) of the
imperfect plate versus the amplitude of the imperfection having the shape of
mode (0,1).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 Figure4. It is observed
that the huge variation of the eigenfrequency with respect to the amplitude of
the imperfection results in a quick turn of the behaviour from the hardening to
the softening type, occuring for an imperfection amplitude of a(0,1) = 0.38 h. This small value has direct implication for the
case
of real plates. As the behaviour changes for a fraction of the plate thickness,
it should not be intriging to measure a softening behaviour with real plates
having small imperfections. This result can also be compared to an earlier
result obtained by Hui [10]. Although Hui did not study free-edge boundary
condition, he reported a numerical result for the case of simply supported
boundary conditions, where the behaviour changes for an imperfection amplitude
of 0.28  h. The
second main observation inferred from Figure 4 is the occurrence of 2:1
internal resonance between eigenfrequencies, leading to discontinuities in the
coefficient T(0,1) dictating the
type of nonlinearity. This fact has already been observed and commented for the
case of shallow spherical shells in [22].
It has also been observed for buckled beams and suspended cables [19, 21]. This is a small
denominator effect typical of internal resonance, that is, when the frequency of the studied
mode (0,1) exactly fulfills the relationship 2ω(0,1)=ω(0,n) with another
axisymmetric mode. 2:1 resonance arises here with mode (0,2) at 1.85 h and with mode
(0,3) at 5.66 h. On a practical point of view, one must bear in mind
that when   2:1  internal resonance occurs, single-mode solution does not exist anymore,
only coupled solutions are possible. Hence the concept of the type of
nonlinearity, intimately associated with a single
dof behaviour, loses its meaning in a narrow
interval around the resonance.Figure 4
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 Figure5. Once again, the geometric effect is
important and leads to a change of behaviour occurring at a(0,1) = 0.75 h, that is, for a small level of imperfection. 2:1 internal resonance also occurs, thus
creating narrow region where hardening behaviour could be observed. This result
extends Hui's analysis since only mode (0,1) was studied. Moreover, as a
single-mode truncation was used in [10], 2:1 resonances were missed.Figure 5
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, fora(0,1)/h = 1.74, 5.43,
and 9.92.Finally, the effect of the imperfection on asymmetric
modes is shown in Figure6 for modes (2,0) and (4,0). The very slight variation
of the eigenfrequencies of these modes versus the axisymmetric imperfection
results in a very slight effect of the geometry. It is observed that before the
first 2:1  internal resonance, the type of nonlinearity shows small variations.
Hence it is the behaviour of the other eigenfrequencies and the occurrence of
2:1 internal resonance that makes, in these
cases, the behaviour turn from
hardening to softening behaviour. For mode (2,0), this occurs for an
imperfection amplitude of a(0,1) = 0.44  h, where  2:1
resonance with mode (0,1) is observed. For mode (4,0), the first  2:1 resonance
occurs with mode (0,2) at a(0,1) = 1.39 h, but do not change the behaviour. It is the resonance
with mode (0,1) at a(0,1) = 4 h which makes the
behaviour turn from hardening to softening.Type of nonlinearity
for (a): mode (2,0), and (b): mode (4,0) with an axisymmetric imperfection
having the shape of mode (0,1).
(a)(b)These results corroborate those obtained on shallow
spherical shells [22].
The fundamental importance of axisymmetric modes in the study of asymmetric
ones is confirmed, showing once again that reduction to single mode has no
chance to deliver correct results. The behaviour of purely asymmetric modes is
found to be of the hardening type until the  2:1 internal resonance with mode
(0,1) occurs. However, a specificity of mode (2,0) with regard to all the other
purely asymmetric modes is that after this resonance, hardening behaviour
(though with a very small value of T(2,0)) is observed.
This was also the case for shallow spherical shells [22]. Finally, for very large
values of the imperfection, the behaviour tends to be neutral.


## 3.2. Asymmetric Imperfection

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 thesine and cosine configuration of degenerated modes
split. In the following, distinction is made systematically between the sine or
cosine configuration of companion modes, for example, mode (2,0,C) (resp., (2,0,S))
refers to the cosine (resp., sine) configuration. More precisely, the
imperfection has the shape of (2,0,C) and is shown in Figure 7.(a) 3D view, (b) top view,
and (c) cross-section alongθ=0 for the plate
with imperfection having the shape of mode (2,0,C).
(a)(b)(c)The behaviour of the eigenfrequencies with the
imperfection is shown in Figure8. As expected, the variation of the
eigenfrequency corresponding to (2,0,C) is huge, whereas (2,0,S) keep quite a
constant value. The symmetry is not completely broken. One can show that only
eigenmodes of the type (2k,n) split. On the other hand, as shown in Figure 8,
modes (3,0), (5,0), and (1,1) are still degenerated.Figure 8
Nondimensional natural frequenciesω(k,n) of the
imperfect plate versus the amplitude of the imperfection having the shape of
mode (2,0,C).The numerical
results for type of nonlinearity relative to the two configurations (2,0,C) and
(2,0,S) are shown in Figure9. The natural frequency of mode (2,0,C) undergoes
a huge variation, which results in a quick change of behaviour, occurring at
0.54 h. Then, a 2:1 internal resonance with (0,2) is noted,
but without a noticeable change in the type of nonlinearity, as the interval
where the discontinuity present is very narrow. In this case, the behaviour of T(2,0,C) looks like the
one observed in the precedent case, that is, the variation of T(0,1) versus an
imperfection having the same shape. On the other hand, the eigenfrequency of
mode (2,0,S) remains quite unchanged, so that the behaviour of T(2,0,S) is not much
affected by the imperfection until the 2:1 internal resonance is encountered.
In that case, the resonance occurs with the other configuration, that is, mode (2,0,C).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).
(a)(b)Finally, the results for the first two axisymmetric
modes (0,1) and (0,2) are shown in Figure10. Mode (0,1) shows a very slight
variation of its eigenfrequency with respect to the asymmetric imperfection
(2,0,C). Consequently, the type of nonlinearity is not much affected, until the
eigenfrequency of (2,0,C) comes to two times ω(0,1): 2:1 internal resonance occurs, and the behaviour
becomes softening. On the other hand, the eigenfrequency of (0,2) is more
affected by the imperfection. This result in an important decrease of T(0,2) while still
remaining positive. A  2:1  internal resonance with (0,3) is encountered for 3.51 h, and two
others 2:1  resonance, with (0,4) and (0,5), occur around 8 h. However, the interval on which the type of
nonlinearity changes its sign is so narrow that it can be neglected. The
behaviour is thus mainly of the hardening type for (0,2).Type of nonlinearity
for (a): mode (0,1) and (b): (0,2); for an imperfection having the shape of
mode (2,0,C).
(a)(b)


## 4. Effect of Damping

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 [22], that were limited to the undamped shell.


### 4.1. The Shallow Spherical Shell Equations

The local equations of motions for the shallow
spherical shell can be obtained directly, see [44] for a thorough
presentation. They can also be obtained from (2.4a) and
(2.4b), by selecting an imperfection
having a spherical shape, as shown in Figure 1(c), see [42]. With R, the radius of
curvature of the spherical shell (R≫a to fulfill the
shallow assumption), the local equations write [44]
(4.1a)ΔΔw+εqΔF+w¨=ε[L(w,F)−cw˙+p(r,θ,t)],(4.1b)ΔΔF−κΔw=−12L(w,w),where the aspect ratio κ of the shell
has been introduced:(4.2)κ=a4R2h2,and εq=12(1−ν2)κ. The boundary conditions for the case of the
spherical shell with free edge write exactly as in the case of the imperfect
circular plates so that (2.5a),
(2.5b) and
(2.5c)
are still fulfilled [42, 44]. A peculiarity of the
spherical shell is that all the involved quantities, linear (eigenfrequencies
and mode shapes), and nonlinear (coupling coefficients and type of
nonlinearity) only depends on κ, which is the only free-geometric parameter. Hence
all the results will be presented as functions of κ.A Galerkin expansion is used for discretizing the PDEs
of motion. As the eigenmodesΦp(r,θ) are known
analytically [44],
they are used for expanding the unknown transverse
displacement:(4.3)w(r,θ,t)=∑p=1+∞Xp(t)Φp(r,θ).The modal displacements Xp are the
unknowns, and their dynamics are governed by, ∀p≥1(4.4)X¨p+2ξpωpX˙p+ωp2Xp+εq∑i,j=1+∞g˜ijpXiXj+ε∑i,j,k=1+∞h˜ijkpXiXjXk=0.The analytical expressions for
the quadratic and cubic coupling coefficients (g˜ijp,h˜ijkp) involve integrals
of products of eigenmodes on the surface, they can be found in [22, 44]. As in
Section 3, a modal
viscous damping term of the form 2ξpωpX˙p is considered,
whereas external forces have been cancelled as only free responses are studied.The type of nonlinearity can be inferred from (4.4) by
using the NNM method. The results for an undamped shell have already been
computed and are presented in [22]. However, an extension of the NNM-method, taking into
account the damping term, has been proposed in [39]. Amongst other things, it
has been shown on a simple two dofs system of coupled oscillators, that the
type of nonlinearity depends on the damping. The aim of this section is thus to
complement the results presented in [22] for documenting the dependence of a shell on viscous
damping and for assessing its effect.


### 4.2. Numerical Results

Three  cases are selected in order to derive results
for a variety of damping behaviours:Case 1.
For allp=1…N,ξp=ξ/ωp;Case 2.
For allp=1…N,ξp=ξ;Case 3.
For allp=1…N,ξp=ξωp;In the above cases,ξ is a constant
value, ranging from 0 to 0.3. Case 1 corresponds to a decay factor (2ξpωp=2ξ) that is
independent from the frequency, that
is, with a constant 2ξ value for any mode.
With a small value of ξ, it may model the low-frequency (i.e., below
the critical frequency) behaviour of thin metallic structures such as aluminium
plates [45, 46]. Case 2 describes a decay
factor that is linear with the frequency, and may model, for instance, damped
structures as glass plates in the low-frequency range [45]. Finally, Case 3
accounts for a strongly damped structure, with a center manifold limited to a
few modes.The effect of increasing damping is shown for modes
(0,1) and (4,0), for Case1 in Figure 11,
Case 2 in
Figure 12, and Case
3 in Figure 13. Mode (0,1) undergoes a rapid change of behaviour: the
transition from hardening to softening type nonlinearity occurs at κ = 1.93. Then
2:1 internal resonance with mode (0,2) occurs at κ = 36, but the
behaviour remains of the softening type. Mode (4,0) displays a hardening
behaviour until the 2:1 resonance with mode (0,1) at κ=174.1. The
first resonance with (0,2) at κ=36.9 does not
change the behaviour on a large interval. Adding the damping of Case 1 shows
that the discontinuity ocurring at 2:1 internal resonance is smoothened.
However, it happens for a quite large amount of damping in the structure.
Damping values of 0, 1e-4, 1e-3, and 1e-2 have been tested and give exactly the
same behaviour so that only one curve is visible in Figure 11. Large values of
the damping term ξ, namely, 0.1 and 0.3 (which correspond to strongly
damped structures) must be selected to see the discontinuity smoothened.
Moreover, outside the narrow intervals where 2:1 resonance occurs, the effect
of damping is not visible. As a conclusion for Case 1, it appears that this
kind of damping has a really marginal effect on the type of nonlinearity, so
that undamped results can be estimated as reliable for lightly damped
structures with modal damping factor below 0.1.Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratioκ of a shallow
spherical shell. Increasing values of damping for Case 1(∀p=1…N,ξp=ξ/ωp) are shown,
with ξ = 0 and 0.01
(red), 0.1 (cyan) and 0.3 (violet).
(a)(b)Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratioκ. Increasing values of damping for Case 2(∀p=1…N,ξp=ξ) are shown,
with ξ = 0 and 0.01
(red), 0.1 (cyan), and 0.3 (violet).
(a)(b)Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratioκ. Increasing values of damping for Case 3(∀p=1…N,ξp=ξωp) are shown,
with ξ = 0 and 1e-4
(black), 1e-3 (magenta), and 1e-2 (red).
(a)(b)Case2 corresponds to a more damped structure than
Case 1. However, it is observed in Figure 12 that the discontinuity is not
smoothened at the 2:1 internal resonance. Inspecting back the analytical
results shows that this is a natural consequence of the expression of the
coefficients of the nonlinear change of coordinates for asymptotic NNMs. When
the specific Case of constant damping factors is selected, small denominators
remain present. On the other hand, outside the regions of 2:1 resonance, the
effect of damping is pronounced and enhances the softening behaviour. But once
again, very large values of damping factors such as 0.3 must be reached to see
a prominent influence.Finally, Case3 depicts the case of a rapidly
increasing decay factor with respect to the frequency. As the overall damping
in the structure is thus larger, smaller values of ξ have been
selected, namely, 1e-4, 1e-3, and 1e-2. ξ = 1e-4 gives
quite coincident results with ξ = 0. But from ξ = 1e-3, the
effect of the damping is very important: the discontinuities are smoothened,
except the larger one occurring for mode (4,0) with mode (0,1). For ξ = 1e-2, 2:1
resonance are not visible anymore. A particular result with this value is for
mode (4,0): the smoothening effect is so important that the nonlinearity
remains of the hardening type. Finally, the fact that the damping generally
favours the softening behaviour cannot be declared as a general rule, as one
counterexample has been exhibited here. From these results, it can be inferred
that the damping has little incidence on the type of nonlinearity for thin
structures, until very large values are attained. It is observed that the
damping generally favours the softening behaviour, but this rule is not true in
general. In particular when the transition from hardening to softening type
nonlinearity is due to a 2:1 internal resonance and is not the direct effect of
the change of geometry, a large value of damping may favours hardening
behaviour, as observed here for mode (4,0) in Case 3.


## 4.1. The Shallow Spherical Shell Equations

The local equations of motions for the shallow
spherical shell can be obtained directly, see [44] for a thorough
presentation. They can also be obtained from (2.4a) and
(2.4b), by selecting an imperfection
having a spherical shape, as shown in Figure 1(c), see [42]. With R, the radius of
curvature of the spherical shell (R≫a to fulfill the
shallow assumption), the local equations write [44]
(4.1a)ΔΔw+εqΔF+w¨=ε[L(w,F)−cw˙+p(r,θ,t)],(4.1b)ΔΔF−κΔw=−12L(w,w),where the aspect ratio κ of the shell
has been introduced:(4.2)κ=a4R2h2,and εq=12(1−ν2)κ. The boundary conditions for the case of the
spherical shell with free edge write exactly as in the case of the imperfect
circular plates so that (2.5a),
(2.5b) and
(2.5c)
are still fulfilled [42, 44]. A peculiarity of the
spherical shell is that all the involved quantities, linear (eigenfrequencies
and mode shapes), and nonlinear (coupling coefficients and type of
nonlinearity) only depends on κ, which is the only free-geometric parameter. Hence
all the results will be presented as functions of κ.A Galerkin expansion is used for discretizing the PDEs
of motion. As the eigenmodesΦp(r,θ) are known
analytically [44],
they are used for expanding the unknown transverse
displacement:(4.3)w(r,θ,t)=∑p=1+∞Xp(t)Φp(r,θ).The modal displacements Xp are the
unknowns, and their dynamics are governed by, ∀p≥1(4.4)X¨p+2ξpωpX˙p+ωp2Xp+εq∑i,j=1+∞g˜ijpXiXj+ε∑i,j,k=1+∞h˜ijkpXiXjXk=0.The analytical expressions for
the quadratic and cubic coupling coefficients (g˜ijp,h˜ijkp) involve integrals
of products of eigenmodes on the surface, they can be found in [22, 44]. As in
Section 3, a modal
viscous damping term of the form 2ξpωpX˙p is considered,
whereas external forces have been cancelled as only free responses are studied.The type of nonlinearity can be inferred from (4.4) by
using the NNM method. The results for an undamped shell have already been
computed and are presented in [22]. However, an extension of the NNM-method, taking into
account the damping term, has been proposed in [39]. Amongst other things, it
has been shown on a simple two dofs system of coupled oscillators, that the
type of nonlinearity depends on the damping. The aim of this section is thus to
complement the results presented in [22] for documenting the dependence of a shell on viscous
damping and for assessing its effect.


## 4.2. Numerical Results

Three  cases are selected in order to derive results
for a variety of damping behaviours:Case 1.
For allp=1…N,ξp=ξ/ωp;Case 2.
For allp=1…N,ξp=ξ;Case 3.
For allp=1…N,ξp=ξωp;In the above cases,ξ is a constant
value, ranging from 0 to 0.3. Case 1 corresponds to a decay factor (2ξpωp=2ξ) that is
independent from the frequency, that
is, with a constant 2ξ value for any mode.
With a small value of ξ, it may model the low-frequency (i.e., below
the critical frequency) behaviour of thin metallic structures such as aluminium
plates [45, 46]. Case 2 describes a decay
factor that is linear with the frequency, and may model, for instance, damped
structures as glass plates in the low-frequency range [45]. Finally, Case 3
accounts for a strongly damped structure, with a center manifold limited to a
few modes.The effect of increasing damping is shown for modes
(0,1) and (4,0), for Case1 in Figure 11,
Case 2 in
Figure 12, and Case
3 in Figure 13. Mode (0,1) undergoes a rapid change of behaviour: the
transition from hardening to softening type nonlinearity occurs at κ = 1.93. Then
2:1 internal resonance with mode (0,2) occurs at κ = 36, but the
behaviour remains of the softening type. Mode (4,0) displays a hardening
behaviour until the 2:1 resonance with mode (0,1) at κ=174.1. The
first resonance with (0,2) at κ=36.9 does not
change the behaviour on a large interval. Adding the damping of Case 1 shows
that the discontinuity ocurring at 2:1 internal resonance is smoothened.
However, it happens for a quite large amount of damping in the structure.
Damping values of 0, 1e-4, 1e-3, and 1e-2 have been tested and give exactly the
same behaviour so that only one curve is visible in Figure 11. Large values of
the damping term ξ, namely, 0.1 and 0.3 (which correspond to strongly
damped structures) must be selected to see the discontinuity smoothened.
Moreover, outside the narrow intervals where 2:1 resonance occurs, the effect
of damping is not visible. As a conclusion for Case 1, it appears that this
kind of damping has a really marginal effect on the type of nonlinearity, so
that undamped results can be estimated as reliable for lightly damped
structures with modal damping factor below 0.1.Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratioκ of a shallow
spherical shell. Increasing values of damping for Case 1(∀p=1…N,ξp=ξ/ωp) are shown,
with ξ = 0 and 0.01
(red), 0.1 (cyan) and 0.3 (violet).
(a)(b)Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratioκ. Increasing values of damping for Case 2(∀p=1…N,ξp=ξ) are shown,
with ξ = 0 and 0.01
(red), 0.1 (cyan), and 0.3 (violet).
(a)(b)Type of nonlinearity for
(a): mode (0,1) and (b): (4,0) versus the aspect ratioκ. Increasing values of damping for Case 3(∀p=1…N,ξp=ξωp) are shown,
with ξ = 0 and 1e-4
(black), 1e-3 (magenta), and 1e-2 (red).
(a)(b)Case2 corresponds to a more damped structure than
Case 1. However, it is observed in Figure 12 that the discontinuity is not
smoothened at the 2:1 internal resonance. Inspecting back the analytical
results shows that this is a natural consequence of the expression of the
coefficients of the nonlinear change of coordinates for asymptotic NNMs. When
the specific Case of constant damping factors is selected, small denominators
remain present. On the other hand, outside the regions of 2:1 resonance, the
effect of damping is pronounced and enhances the softening behaviour. But once
again, very large values of damping factors such as 0.3 must be reached to see
a prominent influence.Finally, Case3 depicts the case of a rapidly
increasing decay factor with respect to the frequency. As the overall damping
in the structure is thus larger, smaller values of ξ have been
selected, namely, 1e-4, 1e-3, and 1e-2. ξ = 1e-4 gives
quite coincident results with ξ = 0. But from ξ = 1e-3, the
effect of the damping is very important: the discontinuities are smoothened,
except the larger one occurring for mode (4,0) with mode (0,1). For ξ = 1e-2, 2:1
resonance are not visible anymore. A particular result with this value is for
mode (4,0): the smoothening effect is so important that the nonlinearity
remains of the hardening type. Finally, the fact that the damping generally
favours the softening behaviour cannot be declared as a general rule, as one
counterexample has been exhibited here. From these results, it can be inferred
that the damping has little incidence on the type of nonlinearity for thin
structures, until very large values are attained. It is observed that the
damping generally favours the softening behaviour, but this rule is not true in
general. In particular when the transition from hardening to softening type
nonlinearity is due to a 2:1 internal resonance and is not the direct effect of
the change of geometry, a large value of damping may favours hardening
behaviour, as observed here for mode (4,0) in Case 3.


## 5. Conclusion

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.



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*Source: 678307-2008-05-14.xml*