Ac31181207

P.Kiran Kumar, J.V.Subrahmanyam, P.RamaLakshmi / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 1, January-February 2013, pp.181-207
A Review on Non-Linear Vibrations of Thin Shells
P.Kiran Kumar 1, J.V.Subrahmanyam 2, P.RamaLakshmi 3
1,3 Assistant Professor, Department of Mechanical Engineering, Chaitanya Bharathi Institute of Technology,
Gandipet, Hyderabad – 500 075,
2 Professor, Department of Mechanical Engineering, Vasavi College of Engineering,
Ibrahimbagh, Hyderabad – 500 0075,

Abstract
This review paper aims to present an computational facilities have enabled researchers to
updated review of papers, conference papers, abandon the linear theories in favor of non-linear
books, dissertations dealing with nonlinear methods of solutions.
vibrations of circular cylindrical thin shells. This
paper surveyed mathematically, experimentally, 2. Literature Review
analytically, numerically analyzed vibrations of 2.1. GEOMETRICALLY NONLINEAR SHELL
cylindrical shells. This includes shells of open type, THEORIES
closed type, with and without fluid interactions; A short overview of some theories for
shells subjected to free and forced vibrations, geometrically nonlinear shells will now be given.
radial harmonic excitations, seismically
excitations; perfect and imperfect shell structures
of various materials with different boundary
conditions. This paper presented 210 reference
papers in alphabetical order.
This paper is presented as Geometrically
nonlinear shell theories, Free and forced
vibrations under radial harmonic excitation,
Imperfect shells, shells subjected to seismic
excitations, References.

1. INTRODUCTION
Most of the structural components are Figure 2.1 shows a circular cylindrical shell with the
generally subjected to dynamic loadings in their co-ordinate system and the displacements of the
working life. Very often these components may have middle surface.
to perform in severe dynamic environment where in
the maximum damage results from the resonant Shell geometry and coordinate system
vibrations. Susceptibility to fracture of materials due Donnell (1934) established the nonlinear
to vibration is determined from stress and frequency. theory of circular cylindrical shells under the
Maximum amplitude of the vibration must be in the simplifying shallow-shell hypothesis. Due to its
limited for the safety of the structure. Hence vibration relative simplicity and practical accuracy, this theory
analysis has become very important in designing a has been widely used. The most frequently used form
structure to know in advance its response and to take of Donnell‟s nonlinear shallowshell theory (also
necessary steps to control the structural vibrations referred as Donnell-Mushtari-Vlasov theory)
and its amplitude.The non-linear or large amplitude introduces a stress function in order to combine the
vibration of plates has received considerable attention three equations of equilibrium involving the shell
in recent years because of the great importance and displacements in the radial, circumferential and axial
interest attached to the structures of low flexural directions into two equations involving only the
rigidity. These easily deformable structures vibrate at radial displacement w and the stress function F:
large amplitudes. The solution obtained based on the The fundamental investigation on the
lineage models provide no more than a first stability of circular cylindrical shells is due to Von
approximation to the actual solutions. The increasing Karman and Tsien (1941), who analyzed the static
demand for more realistic models to predict the stability (buckling) and the postcritical behavior of
responses of elastic bodies combined with the axially loaded shells. In this study, it was clarified
availability of super that discrepancies between forecasts of linear models
and experimental results were due to the intrinsic

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simplifications of linear models; indeed, linear hardening or softening type, depending on the
analyses are not able to predict the actual buckling geometry of the lobe. Almost at the same time,
phenomenon observed in experiments; conversely, Grigolyuk (1955) studied large-amplitude free
nonlinear analyses show that the bifurcation path is vibrations of circular cylindrical panels simply
strongly subcritical, therefore, safe design supported at all four edges. He used the same shell
information can be obtained with a nonlinear theory as Reissner (1955) and a two-mode expansion
analyses only. After this important contribution, for the flexural displacement involving the first and
many other studies have been published on static and third longitudinal modes. He also developed a single
dynamic stability of shells. mode approach. Results show a hardening type
Mushtari and Galimov (1957) presented nonlinearity. Chu (1961) continued with Reissner‟s
nonlinear theories for moderate and large work, extending the analysis to closed cylindrical
deformations of thin elastic shells in their book. In shells. He found that nonlinearity in this case always
the book of Vorovich (1999) the nonlinear theory of leads to hardening type characteristics, which, in
shallow shells is also discussed. some cases, can become quite strong.
Sanders (1963) and Koiter (1966)
developed a more refined nonlinear theory of shells, Cummings (1964) confirmed Reissner‟s
expressed in tensorial form; the same equations were analysis for circular cylindrical panels simply
obtained by them around the same period, leading to supported at the four edges; he also investigated the
the designation of these equations as the Sanders- transient response to impulsive and step functions, as
Koiter equations. Later, this theory has been well as dynamic buckling. Nowinski (1963)
reformulated in lines-of-curvature coordinates, i.e. in confirmed Chu‟s results for closed circular shells. He
a form that can be more suitable for applications; see used a single-degree-of-freedom expansion for the
Budiansky (1968). According to the Sanders-Koiter radial displacement using the linear mode excited,
theory, all three displacements are used in the corrected by a uniform displacement that was
equations of motion. introduced to satisfy the continuity of the
Changes in curvature and torsion are linear circumferential displacement. All of the above
according to both the Donnell and the Sanders-Koiter expansions for the description of shell deformation,
nonlinear theories (Yamaki 1984). The Sanders- except for Grigolyuk‟s, employed a single mode
Koiter theory gives accurate results for vibration based on the linear analysis of shell vibrations.
amplitudes significantly larger than the shell
thickness for thin shells. Evensen (1963) proved that Nowinski‟s
analysis was not accurate, because it did not maintain
Details on the above-mentioned nonlinear a zero transverse deflection at the ends of the shell.
shell theories may be found in Yamaki‟s (1984) Furthermore, Evensen found that Reissner‟s and
book, with an introduction to another accurate Chu‟s theories did not satisfy the continuity of in-
theory, called the modified Flügge nonlinear theory plane circumferential displacement for closed circular
of shells, also referred to as the Flügge-Lur‟e-Byrne shells. Evensen (1963) noted in his experiments that
nonlinear shell theory (Ginsberg 1973). The Flügge- the nonlinearity of closed shells is of the softening
Lur‟e-Byrne theory is close to the general large type and weak, as also observed by Olson
deflection theory of thin shells developed by (1965). Indeed, Olson (1965) observed a slight
Novozhilov (1953) and differs only in terms for nonlinearity of the softening type in the experimental
change in curvature and torsion. Additional nonlinear response of a thin seamless shell made of copper; the
shell theories were formulated by Naghdi and measured change in resonance frequency was only
Nordgren (1963),using the Kirchhoff hypotheses, and about 0.75 %, for a vibration amplitude equal to 2.5
Libai and Simmonds (1988). times the shell thickness. The shell ends were
attached to a ring; this arrangement for the boundary
2.2. Free and Forced (Radial Harmonic conditions gave some kind of constraint to the axial
Excitation) Vibrations of Shells displacement and rotation. Kaña et al. (1966; see also
The first study on vibrations of circular Kaña 1966) also found experimentally a weak
cylindrical shells is attributed to Reissner (1955), softening type response for a simply supported thin
who isolated a single half-wave (lobe) of the circular cylindrical shell.
vibration mode and analyzed it for simply supported
shells; this analysis is therefore only suitable for To reconcile this most important
circular panels. By using Donnell‟s nonlinear discrepancy between theory and experiments,
shallow-shell theory for thin-walled shells, Reissner Evensen (1967) used Donnell‟s nonlinear shallow-
found that the nonlinearity could be either of the shell theory but with a different form for the assumed

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flexural displacement w, involving more modes. approach and shows that nonperiodic (more
Specifically, he included the companion mode in the specifically, quasiperiodic) motion is obtained for
analysis, as well as an axisymmetric contraction free nonlinear vibrations. In their analysis based on
having twice the frequency of the mode excited. Donnell‟s theory, the same expansion introduced by
Evensen, without the companion mode, was initially
Evensen‟s assumed modes are not moment- applied; the backbone curves (pertaining to free
free at the ends of the shell, as they should be for vibrations) of Evensen were confirmed almost
classical simply supported shells, and the exactly. A second expansion with an additional
homogeneous solution for the stress function is degree of freedom was also applied for shells with
neglected; however, the continuity of the many axial waves; finally, an expansion with more
circumferential displacement is exactly satisfied. axisymmetric terms was introduced, but the
Evensen studied the free vibrations and the response corresponding backbone curves were not reported. In
to a modal excitation without considering damping their analysis based on the Sanders-Koiter theory, an
and discussed the stability of the response curves. His original expansion for the three shell displacements
results are in agreement with Olson‟s (1965) (i.e. radial, circumferential and axial) was used,
experiments. Evensen‟s study is an extension of his involving 7 degrees of freedom. However, only one
very well-known study on the vibration of rings term was used for the flexural displacement, and the
(1964, 1965, 1966), wherein he proved theoretically axisymmetric radial contraction was neglected;
and experimentally that thin circular rings display a axisymmetric terms were considered for the in-plane
softening-type nonlinearity. The other classical work displacements. The authors found that the backbone
on the vibration of circular rings is due to Dowell curves for a mode with two circumferential waves
(1967a) who confirmed Evensen‟s results and predict a hardening-type nonlinearity which increases
removed the assumption of zero mid-surface with shell thickness.
circumferential strain. Evensen (1968) also extended
his work to infinitely long shells vibrating in a mode Matsuzaki and Kobayashi (1969a, b) studied
in which the generating lines of the cylindrical theoretically simply supported circular cylindrical
surface remain straight and parallel, by using the shells (1969a), then studied theoretically and
method of harmonic balance and without considering experimentally clamped circular cylindrical shells
the companion mode. Evensen (2000) published a (1969b). Matsuzaki and Kobayashi (1969a) based
paper, in which he studied the influence of pressure their analysis on Donnell‟s nonlinear shallow-shell
and axial loading on large-amplitude vibrations of theory and used the same approach and mode
circular cylindrical shells by an approach similar to expansion as Evensen (1967), with the opposite sign
the one that he used in his previous papers, but to the axisymmetric term because they assumed
neglecting the companion mode; consequently only positive deflection outwards. In addition, they
backbone curves, pertaining to free vibrations, were discussed the effect of structural damping and studied
computed. in detail travelling-wave oscillations. In the paper,
considering clamped shells, Matsuzaki and
It is important to note that, in most studies, Kobayashi (1969b) modified the mode expansion in
the assumed mode shapes (those used by Evensen, order to satisfy the different boundary conditions and
for example) are derived in agreement with the retained both the particular and the homogeneous
experimental observation that, in large amplitude solutions for the stress function. The analysis found
vibrations, (i) the shell does not spend equal time- softening type nonlinearity also for clamped shells, in
intervals deflected outwards and deflected inwards, agreement with their own experimental results. They
and (ii) inwards maximum deflections, measured also found amplitude-modulated response close to
from equilibrium, are larger than outwards ones. resonance and identified it as a beating phenomenon
However, it is important to say that these differences due to frequencies very close to the excitation
are very small if compared to the vibration amplitude. frequency.
Hence, the original idea for mode expansion was to
add to asymmetric linear modes an axisymmetric Dowell and Ventres (1968) used a different
term (mode) giving a contraction to the shell. expansion and approach in order to satisfy exactly the
out-of-plane boundary conditions and to satisfy “on
Mayers and Wrenn (1967) analyzed free the average” the in-plane boundary conditions. They
vibrations of thin, complete circular cylindrical studied shells with restrained in-plane displacement
shells. They used both Donnell‟s nonlinear shallow- at the ends and obtained the particular and the
shell theory and the Sanders-Koiter nonlinear theory homogeneous solutions for the stress function. Their
of shells. Their analysis is based on the energy interesting approach was followed by Atluri (1972)

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who found that some terms were missing in one of and Cummings (1964); the converse was found for the
the equations used by Dowell and Ventres; Dowell et hyperbolic paraboloid. The analysis shows that for all
al. (1998) corrected these omissions. The boundary shells, excluding the hyperbolic paraboloid but including
conditions assumed both by Dowell and Ventres and the circular cylindrical panel, a softening nonlinearity is
by Atluri constrain the axial displacement at the shell expected, changing to hardening for very large
extremities to be zero, so that they are different from deflections (in particular for vibration amplitude
the classical constraints of a simply supported shell equal to 15 times the shell thickness for a circular
(zero axial force at both ends). Atluri (1972) found cylindrical panel).
that the nonlinearity is of the hardening type for a
closed circular cylindrical shell, in contrast to what El-Zaouk and Dym (1973) investigated free
was found in experiments. Varadan et al. (1989) and forced vibrations of closed circular shells having
showed that hardening-type results based on the a curvature of the generating lines by using an
theory of Dowell and Ventres and Atluri are due to extended form of Donnell‟s nonlinear shallow-shell
the choice of the axisymmetric term. More recently, theory to take into account curvature of the
Amabili et al. (1999b, 2000a) showed that at least the generating lines and orthotropy. They also obtained
first and thirdaxisymmetric modes (axisymmetric numerical results for circular cylindrical shells; for this
modes with an even number of longitudinal half- special case, their study is almost identical to Evensen‟s
(1967), but the additional effects of internal pressure and
waves do not give any contribution) must be included
orthotropy are taken into account. Their numerical
in the mode expansion (for modes with a single
results showed a softening type nonlinearity, which
longitudinal half-wave), as well as using both the
becomes hardening only for very large
driven and companion modes, to correctly predict the
displacements, at about 30 times the shell thickness.
trend of nonlinearity with sufficiently good accuracy.
Stability of the free and forced responses was also
investigated.
Sun and Lu (1968) studied the dynamic
Ginsberg used a different approach to solve
behaviour of conical and cylindrical shells under
the problem of asymmetric (Ginsberg 1973) and
sinusoidal and ramp-type temperature loads. The
axisymmetric (Ginsberg 1974) large-amplitude
equations of motion are obtained by starting with
vibrations of circular cylindrical shells. In fact, he
strain-displacement relationships that, for cylindrical
avoided the assumptions of mode shapes and
shells, are those of Donnell‟s nonlinear shallow-shell
employed instead an asymptotic analysis to solve the
theory. All three displacements were used, instead of
nonlinear boundary conditions for a simply supported
introducing the stress function F. Therefore, the three
shell. The solution is reached by using the energy of
displacements were expanded by using globally six
the system to obtain the Lagrange equations, which are
terms, without considering axisymmetric terms in the
then solved via the harmonic balance method. Both
radial displacement; these six terms were related softening and hardening nonlinearities were found,
together to give a single-degree-of-freedom system depending on some system parameters. Moreover,
with only cubic nonlinearity, and the shell thus Ginsberg used the more accurate Flügge-Lur‟e-Byrne
exhibited hardening type nonlinearity. nonlinear shell theory, instead of Donnell‟s nonlinear
Leissa and Kadi (1971) studied linear and shallow-shell theory which was used in all the work
nonlinear free vibrations of shallow shell panels, discussed in the foregoing (excluding the two papers
simply supported at the four edges without in-plane allowing for curvature of the generating lines). The
restraints. Panels with two different. curvatures in Flügge-Lur‟e-Byrne theory is the same referred as the
orthogonal directions were studied. This is the first modified Flügge theory by Yamaki (1984). Ginsberg
study on the effect of curvature of the generating (1973) studied the damped response to an excitation
lines on large-amplitude vibrations of shallow shells. and its stability, considering both the driven and
Donnell‟s nonlinear shallow-shell theory was used in companion modes.
a slightly modified version to take into account the
meridional curvature. A single mode expansion of the Chen and Babcock (1975) used the
transverse displacement was used. The Galerkin perturbation method to solve the nonlinear equations
method was used; the compatibility equation was obtained by Donnell‟s nonlinear shallow-shell theory,
exactly satisfied, and in-plane boundary conditions without selecting a particular deflection solution.
were satisfied on the average. Results were obtained They solved the classical simply supported case and
by numerical integration and non-simple harmonic studied the driven mode response, the companion
oscillations were found. For cylindrical and spherical mode participation, and the appearance of a travelling
panels, phase plots show that, during vibrations, wave. A damped response to an external excitation
inward deflections are larger that outward was found. The solution involved a sophisticated
deflections, as previously found by Reissner (1955)

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mode expansion, including boundary layer terms in Raju and Rao (1976) employed the Sanders-
order to satisfy the boundary conditions. They also Koiter theory and used the finite element method to
presented experimental results in good agreement study free vibrations of shells of revolution. They
with their theory, showing a softening nonlinearity. found hardening-type results for a closed circular
Regions with amplitude-modulated response were cylindrical shell, in contradiction with all
also experimentally detected. experiments available. Their paper was discussed by
Evensen (1977), Prathap (1978a, b), and then again
It is important to state that Ginsberg‟s by Evensen (1978a, b). In particular, Evensen (1977)
(1973) numerical results and Chen and Babcock‟s commented that the authors ignored the physics of
(1975) numerical and experimental results constitute the problem: i.e., that thin shells bend more readily
fundamental contributions to the study of the than they stretch.
influence of the companion mode on the nonlinear
forced response of circular cylindrical shells. Ueda (1979) studied nonlinear free
Moreover, in these two studies, a multi-mode vibrations of conical shells by using a theory
approach involving also modes with 2n equivalent to Donnell‟s shallow-shell theory and a
circumferential waves was used. However, Ginsberg finite element approach. Each shell element was
(1973) and Chen and Babcock (1975), in order to reduced to a nonlinear system of two degrees of
obtain numerical results, condensed their models to a freedom by using an expansion similar to Evensen‟s
two-degree-of-freedom one by using a perturbation (1967) analysis but without companion mode
approach with some truncation error. participation. Numerical results were carried out also
for complete circular cylindrical shells and are in
Vol‟mir et al. (1973) studied nonlinear good agreement with those obtained by Olson (1965)
oscillations of simply supported, circular cylindrical and Evensen (1967).
panels and plates subjected to an initial deviation
from the equilibrium position (response of the panel Yamaki (1982) performed experiments for
to initial conditions) by using Donnell‟s nonlinear large-amplitude vibrations of a clamped shell made
shallow-shell theory. Results were calculated by of a polyester film. The shell response of the first
numerical integration of the equations of motion eight modes is given and the nonlinearity is of
obtained by Galerkin projection, retaining three or softening type for all these modes with a decrease of
five modes in the expansion. The results reported do natural frequencies of 0.6, 0.7, 0.9 and 1.3 % for
not show the trend of nonlinearity but only the time modes with n = 7, 8, 9 and 10 circumferential waves
response. (and one longitudinal half-wave), respectively, for
vibration amplitude equal to the shell thickness (rms
Radwan and Genin (1975) derived nonlinear amplitude equal to 0.7 times the shell thickness). He
modal equations by using the Sanders- Koiter also proposed two methods of solution of the
nonlinear theory of shells and the Lagrange equations problem by using the Donnell‟s nonlinear shallow-
of motion, taking into account imperfections. shell theory: one is a Galerkin method and the second
However, the equations of motion were derived only is a perturbation approach. No numerical application
for perfect, closed shells, simply supported at the was performed.
ends. The nonlinear coupling between the linear
modes, that are the basis for the expansion of the Maewal (1986a, b) studied large-amplitude
shell displacements, was neglected. As previously vibrations of circular cylindrical and axisymmetric
observed, this singlemode approach gives the wrong shells by using the Sanders-Koiter nonlinear shell
trend of nonlinearity for a closed circular shell. The theory. The mode expansion used contains the driven
numerical results give only the coefficients of the and companion modes, axisymmetric modes and
Duffing equation, obtained while solving the terms with twice the number of circumferential
problem. Radwan and Genin (1976) extended their waves of the driven modes. However, it seems that
previous work to include the nonlinear coupling with axisymmetric modes with three longitudinal half-
axisymmetric modes; however, only an approximate waves were neglected; as previously discussed, these
coupling was proposed. Most of the numerical results modes are essential for predicting accurately the
still indicated a hardening type nonlinearity for a trend of nonlinearity. It is strongly believed that the
closed circular shell; however, a large reduction of difference between the numerical results of Maewal
the hardening nonlinearity was obtained relative to (1986b) and those obtained by Ginsberg (1973) may
the case without coupling. be attributed to the exclusion of the axisymmetric
modes with three longitudinal half-waves in the
analysis. Results were obtained by asymptotic

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analysis and by a speciallydeveloped finite element however, here forced vibrations were studied. It
method. should be recognized that travelling-wave responses
were also obtained in all the other previous work that
Nayfeh and Raouf (1987a, b) studied included the companion mode; however, the
vibrations of closed shells by using plane-strain expansion used was different.
theory of shells and a perturbation analysis; thus,
their study is suitable for rings but not for supported Andrianov and Kholod (1993) used an
shells of finite length. They investigated the response analytical solution for shallow cylindrical shells and
when the frequency of the axisymmetric mode is plates based on Bolotin‟s asymptotic method, but
approximately twice that of the asymmetric mode they obtained results only for a rectangular plate. A
(two-to-one internal resonance). The phenomenon of modified version of the nonlinear Donnell shallow-
saturation of the response of the directly excited shell equations was used. Large amplitude vibrations
mode was observed. Raouf and Nayfeh (1990) of thin, closed circular cylindrical shells with wafer,
studied the response of the shell by using the same stringer or ring stiffening were studied by Andrianov
shell theory previously used by Nayfeh and Raouf et al. (1996) by using the Sanders-Koiter nonlinear
(1987a, b), retaining both the driven and companion shell theory. The solution was obtained by using an
modes in the expansion, finding amplitude- asymptotic procedure and boundary layer terms to
modulated and chaotic solutions. Only two degrees of satisfy the shell boundary conditions. Only the free
freedom were used in this study; an axisymmetric vibrations (backbone curve) were investigated. The
term and terms with twice the number of single numerical case investigated showed a
circumferential waves of the driven and companion softening type nonlinearity.
modes were obtained by perturbation analysis and
added to the solution without independent degrees of Chiba (1993a) studied experimentally large-
freedom. The method of multiple scales was applied amplitude vibrations of two cantilevered circular
to obtain a perturbation solution from the equations cylindrical shells made of polyester sheet. He found
of motion. By using a similar approach, Nayfeh et al. that almost all responses display a softening
(1991) investigated the behaviour of shells, nonlinearity. He observed that for modes with the
considering the presence of a two-to-one internal same axial wave number, the weakest degree of
resonance between the axisymmetric and asymmetric softening nonlinearity can be attributed to the mode
modes for the problem previously studied by Nayfeh having the minimum natural frequency. He also
and Raouf (1987a, b). found that shorter shells have a larger softening
nonlinearity than longer ones. Travelling wave modes
Gonçalves and Batista (1988) conducted an were also observed.
interesting study on fluid-filled, complete circular
cylindrical shells. A mode expansion that can be Koval‟chuk and Lakiza (1995) investigated
considered a simple generalisation of Evensen‟s experimentally forced vibrations of large amplitude
(1967) was introduced by Varadan et al. (1989), in fiberglass shells of revolution. The boundary
along with Donnell‟s nonlinear shallow-shell theory, conditions at the shell bottom simulated a clamped
in their brief note on shell vibrations. This expansion end, while the top end was free (cantilevered shell).
is the same as that used by Watawala and Nash One of the tested shells was circular cylindrical. A
(1983); as previously observed, it is not moment-free weak softening nonlinearity was found, excluding the
at the ends of the shell. The results were compared beam-bending mode for which a hardening
with those obtained by using the mode expansion nonlinearity was measured. Detailed responses for
proposed by Dowell and Ventres (1968) and Atluri three different excitation levels were obtained for the
(1972). Varadan et al. (1989) showed that the mode with four circumferential waves (second mode
expansion of Dowell and Ventres and Atluri gives of the shell). Travelling wave response was observed
hardening-type results, as previously discussed; the around resonance, as well as the expected weak
results obtained with the expansion of Watawala and softening type nonlinearity.
Nash correctly display a softening type nonlinearity.
Kobayashi and Leissa (1995) studied free
Koval‟chuk and Podchasov (1988) vibrations of doubly curved thick shallow panels;
introduced a travelling-wave representation of the they used the nonlinear first-order shear deformation
radial deflection, in order to allow the nodal lines to theory of shells in order to study thick shells. The
travel in the circumferential direction over time. This rectangular boundaries of the panel were assumed to
expansion is exactly the same as that introduced by be simply supported at the four edges without in-
Kubenko et al. (1982) to study free vibrations; plane restraints, as previously assumed by Leissa and

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Kadi (1971). A single mode expansion was used for Evensen (1977) on Nowinski‟s and Kanaka Raju and
each of the three displacements and two rotations Venkateswara Rao‟s results should be recalled to
involved in the theory; in-plane and rotational inertia explain the hardening type of nonlinearity. Lakis et
were neglected. The problem was then reduced to one al. (1998) presented a similar solution for closed
of a single degree of freedom describing the radial circular cylindrical shells.
displacement. Numerical results were obtained for
circular cylindrical, spherical and paraboloidal A finite element approach to nonlinear
panels. Except for hyperbolic paraboloid shells, a dynamics of shells was developed by Sansour et al.
softening behaviour was found, becoming hardening (1997, 2002). They implemented a finite shell
for vibration amplitudes of the order of the shell element based on a specifically developed nonlinear
thickness. However, increasing the radius of shell theory. In this shell theory, a linear distribution
curvature, i.e. approaching a flat plate, the behaviour of the transverse normal strains was assumed, giving
changed and became hardening. The effect of the rise to a quadratic distribution of the displacement
shell thickness was also investigated. field over the shell thickness. They developed a time
integration scheme for large numbers of degrees of
Thompson and de Souza (1996) studied the freedom; in fact, problems can arise in finite element
phenomenon of escape from a potential well for two- formulations of nonlinear problems with a large
degree-of-freedom systems and used the case of number of degrees of freedom due to the instability
forced vibrations of axially compressed shells as an of integrators. Chaotic behaviour was found for a
example. A very complete bifurcation analysis was circular cylindrical panel simply supported on the
performed. straight edges, free on the curved edges and loaded
by a point excitation having a constant value plus a
Ganapathi and Varadan (1996) used the harmonic component. The constant value of the load
finite element method to study large-amplitude was assumed to give three equilibrium points (one
vibrations of doubly- curved composite shells. unstable) in the static case.
Numerical results were given for isotropic circular
cylindrical shells. They showed the effect of Amabili et al. (1998) investigated the
including the axisymmetric contraction mode with nonlinear free and forced vibrations of a simply
the asymmetric linear modes, confirming the supported, complete circular cylindrical shell, empty
effectiveness of the mode expansions used by many or fluid-filled. Donnell‟s nonlinear shallowshell
authors, as discussed in the foregoing. Only free theory was used. The boundary conditions on the
vibrations were investigated in the paper, using radial displacement and the continuity of
Novozhilov‟s theory of shells. A four-node finite circumferential displacement were exactly satisfied,
element was developed with five degrees of freedom while the axial constraint was satisfied on the
for each node. Ganapathi and Varadan also pointed average. Galerkin projection was used and the mode
out problems in the finite element analysis of closed shape was expanded by using three degrees of
shells that are not present in open shells. The same freedom; specifically, two asymmetric modes (driven
approach was used to study numerically laminated and companion modes), plus an axisymmetric term
composite circular cylindrical shells (Ganapathi and involving the first and third axisymmetric modes
Varadan 1995). (reduced to a single term by an artificial constraint),
were employed. The time dependence of each term of
Selmane and Lakis (1997a) applied the finite the expansion was general. Different tangential
element method to study free vibrations of open and constraints were imposed at the shell ends. An
closed orthotropic cylindrical shells. Their method is inviscid fluid was considered. Solution was obtained
a hybrid of the classical finite element method and both numerically and by the method of normal forms.
shell theory. They used the refined Sanders-Koiter Numerical results were obtained for both free and
nonlinear theory of shells. The formulation was forced vibrations of empty and water-filled shells.
initially general but in the end, to simplify the Some additions to this paper were given by Amabili
solution, only a single linear mode was retained. As et al. (1999a). Results showed a softening type
previously discussed, this approximation gives nonlinearity and travelling-wave response close to
erroneous results for a complete circular shell. In resonance. Numerical results are in quantitative
fact, numerical results for free vibrations of the same agreement with those of Evensen (1967), Olson
closed circular cylindrical shell, simply supported at (1965), Chen and Babcock (1975) and Gonçalves and
the ends, investigated by Nowinski (1963) and Batista (1988).
Kanaka Raju and Venkateswara Rao (1976) showed a
hardening type nonlinearity. The criticism raised by

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In a series of four papers, Amabili, Pellicano number of modes can be retained. Numerical results
and Païdoussis (1999b, c, 2000a, b) studied the with up to 23 modes were obtained and the
nonlinear stability and nonlinear forced vibrations of convergence of the solution discussed. In particular,
a simply supported circular cylindrical shell with and it is shown that the model developed by Amabili et
without flow by using Donnell‟s nonlinear shallow- al. (2000a) is close to convergence. In that study a
shell theory. The Amabili et al. (1999c, 2000a) map is also given, for the first time, showing whether
papers deal with large-amplitude vibrations of empty the nonlinearity is softening or hardening as a
and fluid-filled circular shells, also investigated by function of the shell geometry.
Amabili et al. (1998), but use an improved model for
the solution expansion. In Amabili et al. (1999c) If the shell is not very short (L/R > 0.5) and
three independent axisymmetric modes with an odd not thick (h/R < 0.045) the nonlinearity is of the
number of longitudinal half-waves were added to the softening type, excluding the case of long (L/R > 5)
driven and companion modes. Therefore, the model thin shells. It is interesting to note that the boundary
can be considered to be an extension of the three- between softening and hardening regions has been
degree-of-freedom one developed by Amabili et al. found to be related to internal resonances (i.e. an
(1998), in which the artificial kinematic constraint integer relationship between natural frequencies)
between the first and third axisymmetric modes, between the driven mode and axisymmetric modes.
previously used to reduce the number of degrees of
freedom, was removed. Results showed that the first Large-amplitude (geometrically nonlinear)
and third axisymmetric modes are fundamental for vibrations of circular cylindrical shells with different
predicting accurately the trend of nonlinearity and boundary conditions and subjected to radial harmonic
that the fifth axisymmetric mode only gives a small excitation in the spectral neighbourhood of the lowest
contribution. resonances have been investigated by Amabili
(2003b). In particular, simply supported shells with
Driven modes with one and two longitudinal allowed and constrained axial displacements at the
half-waves were numerically computed. Periodic edges have been studied; in both cases the radial and
solutions were obtained by a continuation technique circumferential displacements at the shell edges are
based on the collocation method. Amabili et al. constrained. Elastic rotational constraints have been
(2000a) added, to the expansion of the radial assumed; they allow simulating any condition from
displacement of the shell, modes with twice the simply supported to perfectly clamped, by varying
number of circumferential waves vis-à-vis the driven the stiffness of this elastic constraint. The Lagrange
mode and up to three longitudinal half-waves, in equations of motion have been obtained by energy
order to check the convergence of the solution with approach, retaining damping through the Rayleigh‟s
different expansions. Results for empty and water- dissipation function. Two different nonlinear shell
filled shells were compared, showing that the theories, namely Donnell‟s and Novozhilov‟s
contained dense fluid largely enhances the weak theories, have been used to calculate the elastic strain
softening type nonlinearity of empty shells; a similar energy. The formulation is valid also for orthotropic
conclusion was previously obtained by Gonçalves and symmetric cross-ply laminated composite shells;
and Batista (1988). Experiments on a water-filled geometric imperfections have been taken into
circular cylindrical shell were also performed and account. The large-amplitude response of circular
successfully compared to the theoretical results, cylindrical shells to harmonic excitation in the
validating the model developed. Experiments showed spectral neighbourhood of the lowest natural
a reduction of the resonance frequency by about 2 % frequency has been computed for three different
for a vibration amplitude equal to the shell thickness. boundary conditions. Circular cylindrical shell with
These experiments are described in detail in Amabili, restrained axial displacement at the shell edges
Garziera and Negri (2002), where additional display stronger softening type nonlinearity than
experimental details and results are given. It is simply supported shells. Both empty and fluid-filled
important to note that, in this series of papers shells have been investigated by using a potential
(Amabili et al. 1999b, c; 2000a, b), the continuity of fluid model.
the circumferential displacements and the boundary A problem of internal resonance (one-to-
conditions for the radial deflection were exactly one-to-one-to-two) was studied by Amabili, Pellicano
satisfied. and Vakakis (2000c) and Pellicano et al. (2000) for a
water-filled shell. In these papers, both modal and
Pellicano, Amabili and Païdoussis (2002) point excitations were used.
extended the previous studies on forced vibrations of
shells by using a mode expansion where a generic

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Autoparametric resonances for free flexural (elastic support), becoming hardening for a vibration
vibrations of infinitely long, closed shells were amplitude of the order of the shell thickness; in-plane
investigated by McRobie et al. (1999) and Popov et constraints reduce the softening nonlinearity, which
al. (2001) by using the energy formulation previously turns to hardening for smaller vibration amplitudes.
developed by the same authors (Popov et al. 1998a). The objective of this study was to investigate regions
A simple two-mode expansion, derived from Evensen of chaotic motion; these regions were identified by
(1967) but excluding the companion mode, was used. means of Poincare maps and Lyapunov exponents. It
An energy approach was applied to obtain the was found that, when approaching the static
equations of motion, which were transformed into instability point (due to the constant acceleration
action-angle co-ordinates and studied by averaging. load), chaotic shell behaviour may be observed. In a
Dynamics and stability were extensively investigated previous study, Nagai and Yamaguchi (1995)
by using the methods of Hamiltonian dynamics, and investigated shallow cylindrical panels without in-
chaotic motion was detected. Laing et al. (1999) plane elastic support, via an approach similar to that
studied the nonlinear vibrations of a circular used by Yamaguchi and Nagai (1997). Also in this
cylindrical panel under radial point forcing by using case, an accurate investigation of the regions of
the standard Galerkin method, the nonlinear Galerkin chaotic motion was performed.
method and the post-processed Galerkin method. The
system was discretized by the finite-difference Using a similar approach, Yamaguchi and
method, similarly to the study of Foale et al. (1998). Nagai (2000) studied oscillations of a circular
The authors conclude that the post-processed cylindrical panel, simply supported at the four edges,
Galerkin method is often more efficient than either having an elastic radial spring at the center. The
the standard Galerkin or nonlinear Galerkin panel was excited by an acceleration with a constant
methods. term plus a harmonic component, and regions of
chaotic motion were identified. Nagai et al. (2001)
Kubenko and Koval‟chuk (2000b) used also studied theoretically and experimentally chaotic
Donnell‟s nonlinear shallow-shell theory with vibrations of a simply supported circular cylindrical
Galerkin projection to study nonlinear vibrations of panel carrying a mass at its center. A single-mode
simply supported circular cylindrical shells. Driven approximation was used by Shin (1997) to study free
and companion modes were included, but vibration of doubly-curved, simply supported panels.
axisymmetric terms were neglected, giving a Backbone curves and response to initial conditions
hardening type response. Interaction between two were studied. Free vibrations of doubly-curved,
modes with different numbers of circumferential laminated, clamped shallow panels, including circular
waves was discussed; this kind of interaction arises in cylindrical panels, were studied by Abe et al. (2000).
correspondence to internal resonances. Results show Both first-order shear deformation theory and
interesting interactions between these modes. classical shell theory (analogous to Donnell‟s theory)
Another study considering the interaction between were used. Results obtained neglecting in-plane and
two modes with different numbers of circumferential rotary inertia are very close to those obtained
waves was developed by Amiro and Prokopenko retaining these effects. Only two modes were
(1999), who also did not consider axisymmetric considered to interact in the nonlinear analysis. Free
terms in the mode expansion. vibrations and parametric resonance of complete,
simply supported circular cylindrical shells were
Yamaguchi and Nagai (1997) studied studied by Mao and Williams (1998a, b). The shell
vibrations of shallow cylindrical panels with a theory used is similar to Sanders‟, but the axial
rectangular boundary, simply supported for inertia is neglected. A single-mode expansion was
transverse deflection and with in-plane elastic used without considering axisymmetric contraction.
support at the boundary. The shell was excited by an Han et al. (1999) studied the axisymmetric motion of
acceleration having a constant value plus a harmonic circular cylindrical shells under axial compression
component. Donnell‟s nonlinear shallow-shell theory and radial excitation. A region of chaotic response
was utilized with the Galerkin projection along with was detected.
a multi-mode expansion of flexural displacement.
Initial deflection (imperfection) was taken into Extensive experiments on large-amplitude
account in the theoretical formulation but not in the vibrations of clamped, circular cylindrical shells were
calculations. The harmonic balance method and performed by Gunawan (1998). Two almost perfect
direct integration were used. The response of the aluminum shells and two with axisymmetric
panel was of the softening type over the whole range imperfection were tested. Contact-free acoustic
of possible stiffness values of the in-plane springs excitation and vibration measurement were used.

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Mostly of the experiments were performed for the Numerical results display a softening type response
mode with 11 and 9 circumferential waves for perfect with traveling wave in the vicinity of resonance. The
and imperfect shells, respectively. These modes were narrow frequency range in which there are amplitude
selected in order to have a good frequency separation modulations in the response was deeply investigated.
from other modes and avoid nonlinear interaction In another paper, Jansen (2001) presented a
among different modes. The effect of axial load on perturbation approach for both the frequency and the
the shell response was investigated. Softening type shell coordinates. The resulting boundary value
nonlinearity and traveling wave response were problem was numerically approached by using a
observed for all the shells. Nonstationary responses parallel shooting method. Results are in good
appeared at relatively large vibration amplitudes. agreement with those obtained by the same author
(Jansen 2002) via Galerkin‟s method.
Mikhlin (2000) studied vibrations of circular
cylindrical shells under a radial excitation and an Amabili et al. (2003) experimentally studied
axial static load, using Donnell‟s nonlinear shallow- large-amplitude vibrations of a stainless steel
shell theory with Galerkin projection and two circular cylindrical panels supported at the four
different two-mode expansions. One included the edges. The nonlinear response to harmonic excitation
first circumferential harmonic of the driven mode and of different magnitude in the neighbourhood of three
the other was derived from Evensen‟s (1967) resonances was investigated. Experiments showed
expansion; in both cases, the companion mode was that the curved panel tested exhibited a relatively
not considered. Stability and energy “pumping” from strong geometric nonlinearity of softening type. In
one mode to the other were discussed. Lee and Kim particular, for the fundamental mode, the resonance
(1999) used a single-mode approach to study was reached at a frequency 5 % lower than the
nonlinear free vibration of rotating cylindrical shells. natural (linear) frequency, when the vibration
The equation of motion shows only a cubic amplitude was equal to 0.55 times the shell thickness.
nonlinearity, thus explaining the appearance of This is particularly interesting when these results are
hardening nonlinearity. Moussaoui et al. (2000) compared to the nonlinearity of the fundamental
studied free, large-amplitude vibrations of infinitely mode of complete (closed around the circumference),
long, closed circular cylindrical shells, neglecting the simply supported, circular cylindrical shells of
motion in the longitudinal direction and assuming similar length and radius, which display much
that the generating lines of the shell remain straight weaker nonlinearity.
after deformation. Thus, their model is suitable for
rings but it is not adequate for real shells of finite Finite-amplitude vibrations of long shells in
length. The system was discretised by using a multi- beam-bending mode were studied by Hu and Kirmser
mode expansion that excluded all axisymmetric (1971). Birman and Bert (1987) and Birman and
terms, which are fundamental. Therefore, an Twinprawate (1988) extended the study to include
erroneous hardening type nonlinearity was obtained. elastic axial constraints, shells on an elastic
Nayfeh and Rivieccio (2000) used a single-mode foundation, and uniformly distributed static loads.
expansion of the displacements to study forced Single-mode and two-mode approximations were
vibrations of simply supported, composite, complete used. Results show a softening nonlinearity for long
circular cylindrical shells. This approach, as shells without restrained axial displacements at the
previously discussed, is inadequate for predicting the ends, which becomes hardening for long shells with
correct trend of nonlinearity; in fact, hardening type restrained displacements; other boundary conditions
results were erroneously obtained and only a cubic are those of a clamped shell.
nonlinearity was found in the equation of motion.
However, due to the simple expansion, the equation Circular cylindrical shells carrying a
of motion was obtained in closed-form. concentrated mass were investigated by Likhoded
(1976). Large-amplitude free vibrations of non-
Jansen (2002) studied forced vibrations of circular cylindrical shells on Pasternak foundations
simply supported, circular cylindrical shells under were studied by Paliwal and Bhalla (1993) by using
harmonic modal excitation and static axial load. the approach developed by Sinharay and Banerjee
Donnell‟s shallow-shell theory was used, but the in- (1985, 1986).
plane inertia of axisymmetric modes was taken into
account in an approximate way. In-plane boundary 2.3 Imperfect shells
conditions were satisfied on the average. A four- A point that was still waiting for an answer
degree-of-freedom expansion of the radial was the accuracy of the different nonlinear shell
displacement was used with Galerkin‟s method. theories available. For this reason, Amabili (2003a)

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computed the large-amplitude response of perfect and mode expansion involving two degrees of freedom:
imperfect, simply supported circular cylindrical the driven mode and an axisymmetric term having
shells subjected to harmonic excitation in the spectral the same spatial form of the one introduced by
neighbourhood of the lowest natural frequency, by Evensen. Imperfections having the same shape as this
using five different nonlinear shell theories: (i) expansion were assumed. Donnell‟s nonlinear
Donnell‟s shallow-shell, (ii) Donnell‟s with in-plane shallow-shell theory was used. Softening and
inertia, (iii) Sanders-Koiter, (iv) Flügge-Lur‟e-Byrne hardening nonlinearity were found depending on
and (v) Novozhilov‟s theories. Except for the first shell geometry. Results show, in contrast with other
theory, the Lagrange equations of motion were studies, that initial imperfections change the linear
obtained by an energy approach, retaining damping frequency only in the presence of an axial load.
through Rayleigh‟s dissipation function. The Koval‟chuk and Krasnopol‟skaia (1980) studied
formulation is also valid for orthotropic and forced vibrations of closed, simply supported circular
symmetric cross-ply laminated composite shells. cylindrical shells using Donnell‟s nonlinear shallow-
Both empty and fluid-filled shells were investigated shell theory. The same expansion of the radial
by using a potential fluid model. The effect of radial displacement introduced by Evensen (1967) was
pressure and axial load was also studied. Results used. Geometric imperfections having the same shape
from the Sanders-Koiter, Flügge-Lur‟e-Byrne and as the driven mode were considered. Both radial and
Novozhilov‟s theories were extremely close, for both longitudinal (parametric) excitations were examined.
empty and water-filled shells. For the thin shell Some unstable zones with nonstationary vibrations
numerically investigated by Amabili (2003a), for and travelling waves were detected, due to
which h/R 288, there is almost no difference differences between the driven- and companion-mode
among them. Appreciable difference, but not natural frequencies, caused by the geometric
particularly large, was observed between the previous imperfections. The authors found softening radial
three theories and Donnell‟s theory with in-plane responses. Kubenko et al. (1982) used a two-mode
inertia. On the other hand, Donnell‟s nonlinear travellingwave expansion, taking into account
shallow-shell theory turned out to be the least axisymmetric and asymmetric geometric
accurate among the five theories compared. It gives imperfections; the rest of the model for the complete
excessive softening type nonlinearity for empty circular shell was similar to those proposed by
shells. However, for water-filled shells, it gives Koval‟chuk and Krasnopol‟skaia (1980). Only free
sufficiently precise results, also for quite large vibrations were studied, but the effect of the number
vibration amplitude. The different accuracy of the of circumferential waves on the nonlinear behaviour
Donnell‟s nonlinear shallow-shell theory for empty was investigated. In particular, the results showed
and water-filled shell can easily be explained by the that the nonlinearity is of the softening type and that
fact that the in-plane inertia, which is neglected in it increases with the number of circumferential
Donnell‟s nonlinear shallowshell theory, is much less waves. In the studies of Koval‟chuk and
important for a water-filled shell, which has a large Krasnopol‟skaia (1980) and Kubenko et al. (1982)
radial inertia due to the liquid, than for an empty the effect of axisymmetric and asymmetric
shell. Contained liquid, compressive axial loads and imperfections was to increase the natural frequency;
external pressure increase the softening type this is in contrast with results obtained in other
nonlinearity of the shell. A minimum mode studies.
expansion necessary to capture the nonlinear
response of the shell in the neighbourhood of a Watawala and Nash (1983) studied the free
resonance was determined and convergence of the and forced conservative vibrations of closed circular
solution was numerically investigated. cylindrical shells using the Donnell‟s nonlinear
shallow-shell theory. Empty and liquidfilled shells
Geometric imperfections of the shell with a free-surface and a rigid bottom were studied.
geometry (e.g. out-of-roundness) were considered in A mode expansion that may be considered as a
buckling problems since the end of the 1950s; e.g., simple generalization of Evensen‟s (1967) was
see Agamirov and Vol‟mir (1959), Koiter (1963) and introduced; therefore, it is not moment-free at the
Hutchinson (1965). However, there is no trace of ends of the shell. A single-term was used to describe
their inclusion in studies of largeamplitude vibrations the shell geometric imperfections. Cases of (i) the
of shells until the beginning of the ‟70s. Research on circumferential wave pattern of the shell response
this topic was probably introduced by Vol‟mir being the same as that of the imperfection and (ii) the
(1972). Kil‟dibekov (1977) studied free vibrations of circumferential wave pattern of the response being
imperfect, simply supported, complete circular shells different from that of the imperfection were analyzed.
under pressure and axial loading by using a simple Numerical results showed softening type

191 | P a g e

nonlinearity. The imperfections lowered the linear softening nonlinearity, which becomes hardening
frequency of vibrations when the circumferential only for very large vibrations, as expected. The
wave pattern of shell response was the same as that equations of motion are obtained by Gakerkin‟s
of the imperfection, affecting the observed method and are studied by using the harmonic
nonlinearity. Results for forced vibrations were balance method. Only the backbone curves are given.
obtained only for beam-bending modes, for which Iu and Chia (1988a) studied antisymmetrically
Donnell‟s nonlinear shallow-shell theory is not laminated cross-ply circular cylindrical panels by
appropriate; these results indicated a hardening ype using the Timoshenko-Mindlin kinematic hypothesis,
nonlinearity. Hui (1984) studied the influence of which is an extension of Donnell‟s nonlinear
imperfections on free vibrations of simply supported shallow-shell theory. Effects of transverse shear
circular cylindrical panels; he used Donnell‟s deformation, rotary inertia and geometrical
nonlinear shallow-shell theory and a single-mode imperfections were included in the analysis. The
expansion; the imperfection was assumed to have the solution was obtained by the harmonic balance
same shape as his single-mode expansion. These method after Galerkin projection. Fu and Chia (1989)
imperfections changed the linear vibration extended this analysis to include a multi-mode
frequency (increase or decrease according with the approach. Iu and Chia (1988b) used Donnell‟s
number of circumferential half-waves) and nonlinear shallow-shell theory to study free
influenced the nonlinearity of the panel. vibrations and post-buckling of clamped and simply
supported, asymmetrically laminated cross-ply
Jansen (1992) studied large-amplitude circular cylindrical shells. A multi-mode expansion
vibrations of simply supported, laminated, complete was used without considering companion mode, as
circular cylindrical shells with imperfections. He only free vibrations were investigated; radial
used Donnell‟s nonlinear shallowshell theory and the geometric imperfections were taken into account. The
same mode expansion as Watawala and Nash (1983); homogeneous solution of the stress function was
the boundary conditions at the shell ends were not retained, but the dependence on the axial coordinate
fully satisfied. The results showed a softening-type was neglected, differently to what was done by Fu
nonlinearity becoming hardening only for very large and Chia (1989). The equations of motion were
amplitude of vibrations (generally larger than ten obtained by using the Galerkin method and were
times the shell thickness). Imperfections having the studied by the harmonic balance method. Three
same shape as the asymmetric mode analyzed gave a asymmetric and three axisymmetric modes were used
less pronounced softening behaviour, changing to in the numerical calculations. Numerical results were
hardening for smaller amplitudes. Moreover, the compared to those obtained by El-Zaouk and Dym
linear frequency of the imperfect shell may be (1973) for a simply supported, glass-epoxy
considerably lower than the frequency of the perfect orthotropic circular cylindrical shell, showing some
shell. In a subsequent study Jansen (2002) developed differences. In a later paper, Fu and Chia (1993)
his model further by using a four-degree-of-freedom included in their model nonuniform boundary
expansion. However, numerical results are presented conditions around the edges. Softening or hardening
only for a perfect shell. In a third paper by the same type nonlinearity was found, depending on the
author (Jansen 2001), results for composite shells radius-to-thickness ratio. Only undamped free
with axisymmetric and asymmetric imperfections are vibrations and buckling were investigated in all this
given. series of studies.

Chia (1987a, b) studied nonlinear free Elishakoff et al. (1987) conducted a
vibrations and postbuckling of symmetrically and nonlinear analysis of small vibrations of an imperfect
asymmetrically laminated circular cylindrical panels cylindrical panel around a static equilibrium position.
with imperfections and different boundary Librescu and Chang (1993) investigated
conditions. Donnell‟s nonlinear shallow-shell theory geometrically imperfect, doubly-curved, undamped,
was used. A single-mode analysis was carried out, laminated composite panels. The nonlinear theory of
and the results showed a hardening nonlinearity. In a shear-deformable shallow panels was used. The
subsequent study Chia (1988b) also investigated nonlinearity was due to finite deformations of the
doubly-curved panels with rectangular base by using panel due to in-plane loads and imperfections. Only
a similar shell theory, and a single-mode expansion in small-amplitude free vibrations superimposed on this
all the numerical calculations, for both vibration finite initial deformation were studied. A single-mode
shape and initial imperfection. However, in this expansion was used to describe the free vibrations
study, numerical results for circular cylindrical panels and the initial imperfections. The authors found that
and doubly curved shallow shells generally show a imperfections of the panels, of the same shape as the

192 | P a g e

mode investigated, lowered the vibration frequency manufacturing imperfections, changes the traveling
significantly. They also described accurately the post- wave response. Good agreement betweentheoretical
buckling stability. In fact, curved panels are and experimental results was obtained. All the modes
characterized by an unstable post-buckling investigated show a softening type nonlinearity,
behaviour, in the sense that they are subject to a snap- which is much more accentuated for the water-filled
through-type instability. Librescu et al. (1996), shell, except in a case displaying internal resonance
Librescu and Lin (1997a) and Hause et al. (1998) (one-to-one) among modes with different number of
extended this work to include thermomechanical circumferential waves. Travelling wave and
loads and nonlinear elastic foundations (Librescu and amplitude-modulated responses were observed in the
Lin 1997b, 1999). Cheikh et al. (1996) studied experiments.
nonlinear vibrations of an infinite circular cylindrical Results for imperfect shells obtained by
shell with geometric imperfection of the same shape Donnell‟s nonlinear shell theory retaining
as the mode excited. Plane-strain theory of shells was inplaneinertia and Sanders-Koiter nonlinear shell
assumed, which is suitable only for rings (generating theory are given by Amabili (2003a).
lines remain straight and parallel to the shell axis)
and a three-degree-of-freedom model, subsequently Kubenko and Koval‟chuk (1998) published
reduced to a single-degree-of-freedom one, was an interesting review on nonlinear problems of shells,
developed. Results exhibit a hardening type where several results were reported about parametric
nonlinearity, which is not reasonable for the shell vibrations; in such review the limitations of reduced
geometry investigated. order models were pointed out. Babich and
Khoroshun (2001) presented results obtained at the
S.P. Timoshenko Institute of Mechanics of the
National Academy of Sciences of Ukraine over 20
years of research; the authors focused the attention on
the variational–difference methods; more than 100
Amabili (2003c) included geometric papers were cited.
imperfections in the model previously developed by
Pellicano et al. (2002) and performed in depth Kubenko and Koval‟chuk (2004) analyzed
experimental investigations on large-amplitude the stability and nonlinear dynamics of shells,
vibrations of an empty and water-filled, simply following the historical advancements in this field,
supported circular cylindrical shell subject to about 190 papers were deeply commented; they
harmonic excitations. The effect of geometric suggested, among the others, the effect of
imperfections on natural frequencies was imperfections as an important issue to be further
investigated. In particular, it was found that: (i) investigated.
axisymmetric imperfections do not split the
double natural frequencies associated with each The fundamental investigation on the
couple of asymmetric modes; outward (with respect stability of circular cylindrical shells is due to Von
with the center of curvature) axisymmetric Karman and Tsien (1941), who analyzed the static
imperfections increase natural frequencies; small stability (buckling) and the postcritical behavior of
inward (with respect with the center of curvature) axially loaded shells. In this study, it was clarified
axisymmetric imperfections decrease natural that discrepancies between forecasts of linear models
frequencies; (ii) ovalisation has a small effect on and experimental results were due to the intrinsic
natural frequencies of modes with several simplifications of linear models; indeed, linear
circumferential waves and it does not split the double analyses are not able to predict the actual buckling
natural frequencies; (iii)imperfections of the same phenomenon observed in experiments; conversely,
shape as the resonant mode decrease both nonlinear analyses show that the bifurcation path is
frequencies, but much more the frequency of the strongly subcritical, therefore, safe design
mode with the same angular orientation; (iv) information can be obtained with a nonlinear
imperfections with the twice the number of analyses only. After this important contribution,
circumferential waves of the resonant mode decrease many other studies have been published on static and
the frequency of the mode with the same angular dynamic stability of shells.
orientation and increase the frequency of the other
mode; they have a larger effect on natural frequencies 2.4 Shells Subjected to Periodic axial loads
than imperfections of the same shape as the resonant Koval (1974) used the Donnell‟s nonlinear
mode. The split of the double natural frequencies, shallow-shell theory to study the effect of a
which is present in almost all real shells due to longitudinal resonance in the parametric transversal

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instability of a circular shell. He found that, the dynamic stability of layered anisotropic circular
combined parametric resonances give rise to complex cylindrical shells.
regions of parametric instability. However, this kind
of phenomenon was found at very high frequency Popov et al. (1998) analyzed the parametric
and no damping was included in the analysis. stability and thepostcritical behavior of an infinitely
Hsu (1974) used the Donnell‟s linear long circular cylindrical shell, dropping the boundary
shallow-shell theory to analyzethe parametric conditions. A three-mode expansion was used,
instability of a circular cylindrical shell: a uniform without the inclusion of the companion mode.
pressure load and an axial dynamic load were Membrane theory was used to evaluate the in-plane
considered, the former was added in order to stresses due to the axial load. The effect of internal
eliminate the Poisson‟s effect in the in-plane stresses. resonances between asymmetric modes was analyzed
The same problem was studied by Nagai and Yamaki in detail.
(1978) using the Donnell‟s linear shallow-shell
theory, considering different boundary conditions; in Goncalves and Del Prado (2000, 2002)
this work, the effect of axisymmetric bending analyzed the dynamic buckling of a perfect circular
vibrations, induced by the axial load and essentially cylindrical shell under axial static and dynamic loads.
due to the Poisson‟s effect, was considered. The The Donnell‟s nonlinear shallow-shell theory was
classical membrane approach for the in-plane stresses used and the membrane theory was considered to
was found inaccurate when the vibration amplitude of evaluate the in-plane stresses. The partial differential
axisymmetric modes is not negligible. operator was discretized through the Galerkin
technique, using a relatively large modal expansion.
Koval‟chuck and Krasnopol‟skaya (1979) However, no companion mode participation was
considered the role of imperfections in the resonances considered and the boundary conditions were
of shells subjected to combined lateral and axial dropped by assuming an infinitely long shell. Escape
loads. They used the Donnell‟s nonlinear shallow from potential well was analyzed in detail and a
shell theory and a three terms expansion for correlation of this phenomenon with the parametric
investigating the onset of travelling waves and resonance was given.
nonstationary response; moreover, they gave an
initial explanation about discrepancies between In Pellicano and Amabili (2003), parametric
theoretical stability boundaries and experiments, instabilities of circular shells (containing an internal
addressing such discrepancies to the geometrical fluid) due to combined static and dynamic axial loads
imperfections. A further contribution toward such were studied by means of the Donnell‟s nonlinear
discrepancies is due to Koval‟chuck et al. (1982) shallow shell theory for the shell and potential flow
using the same theory and similar modeling; they theory for the fluid. Among the other findings it was
claimed that: the theoretical analyses give narrower found that the membrane stress assumptions can lead
instability region with respect to experiments if to wrong forecasting of the dynamics when the
imperfections are not considered. Imperfections dynamics of axisymmetric modes cannot be
cause splitting of conjugate mode frequencies and neglected (this is in agreement with Nagai and
give rise to doubling of the instability regions. Yamaki (1978)), the presence of an internal fluid
greatly changes the dynamic response.
Bert and Birman (1988) studied the
parametric instability of thick circular shells, they In Catellani et al. (2004) a multimode
developed a special version of the Sanders– Koiter modelling of shells subjected to axial static and
thin-shell theory for thick shells. A single mode periodic loads was proposed considering the
analysis, giving rise to a Mathieu type equation of Donnell‟s nonlinear shallow shell theory. The effect
motion, allowed to determine the instability regions. of combined static and periodic loads was analyzed
in detail considering geometric imperfections. It was
Argento (1993) used the Donnell‟s linear found a great influence of imperfections on the
theory and the classical lamination theory to study instability boundaries and low influence on the
the dynamic stability of composite circular clamped– postcritical dynamic behaviour for moderate
clamped shells under axial and torsional loading. The imperfections.
linear equations of motion, obtained from
discretization, were analyzed by means of the Goncalves and Del Prado (2005) presented a
harmonic balance method and the linear instability low order model based on the Donnell‟s nonlinear
regions were found. Argento and Scott (1993) studied shallow shell theory, the nonlinear dynamics and
stability of circular cylindrical shells was analyzed.

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They pointed out that suitable reduced order models lead the shell to large amplitude of vibration and the
could be able to reproduce correctly the shell response is highly sensitive to geometric
response. An analytical simplified approach was imperfections. In Pellicano (2009) the dynamic
developed by Jansen (2005), in order to simulate stability of strongly compressed shells, subjected to
dynamic step and periodic axial loads acting on periodic axial loads, having geometric imperfections,
isotropic and anisotropic shells; he showed that was investigated using the nonlinear Sanders–Koiter
simple periodic responses can be simulated through theory, a multimode expansion was considered in the
low dimensional models. modelling. Such paper clarifies that, the role of
imperfections becomes important in the case of
The effect of in-plane inertia was included combined axial compression and periodic loads;
in the Donnell-type equations: it was found that, indeed, dynamic instabilities interacts with the static
neglecting the in-plane inertia gives rise to a potential well. Kochurov and Avramov (2010)
moderate underestimation of the instability region. presented an analytical study based on the Donnell‟s
The nonlinear dynamics and chaos of shells subjected nonlinear shallow shell theory, a low order Galerkin
to an axial load with or without the presence of a approach and a perturbation procedure. They proved
contained fluid was presented in Pellicano and the coexistence of nonlinear normal modes and
Amabili (2006); both nonlinear Donnell‟s shallow travelling waves. They pointed out the high modal
shell and nonlinear Sanders–Koiter theories were density of shell like structures and the need of multi-
used; a multimode expansion, able to model both mode models for an accurate analysis.
postcritical buckling and nonlinear vibrations, was Ilyasov (2010) developed a theoretical
considered for reducing the original nonlinear partial model for analyzing the dynamic stability of shells
differential equations (PDE) to a set of ordinary having viscoelastic behavior. This kind of models can
differential equations (ODE). The effect of a be particularly useful in the case of polymeric
contained fluid and as analyzed and an accurate materials.
analysis of the postcritical dynamics was carried out,
including a deep study on the chaotic properties of 2.5 Shells Subjected to Seismic excitation
the response. Vijayarachavan and Evan-Iwanowski (1967)
analyzed, both analytically and experimentally, the
Goncalves et al. (2007) studied steady state parametric instabilities of a circular shell under
and transient instabilityof circular shells under seismic excitation. The cylinder position was vertical
periodic axial loads using the Donnell‟s nonlinear and the base was axially excited by using a shaker. In
shallow shell theory and a low dimensional Galerkin this problem, the in-plane inertia is variable along the
model. The study clarified the complexity of the shell axis and, when the base is harmonically excited,
basins of attraction of low vibration responses and it gives rise to a parametric excitation. Instability
the escape from the potential well due to the regions were found analytically and compared with
combination of periodic and static compressive loads; experimental results.
moreover, they pointed out that shells are highly Bondarenko and Galaka (1977) investigated
sensitive to geometric imperfections. the parametric instabilities of a composite shell under
base seismic motion and free top end. They identified
Darabi et al. (2008) analyzed the dynamic principal instability regions of several modes and also
stability of functionally graded circular cylindrical secondary regions; they observed that, the transition
shells using Donnell shallow shell theory. They used from stable to unstable regions is accompanied by a
a displacement expansion without axisymmetric „„bang‟‟ that can lead the shell to the collapse.
modes, the Galerkin approach for reducing the PDE Bondarenko and Telalov (1982) studied
to ODE and the Bolotin method for solving the latter experimentally the dynamic instability and nonlinear
one. Only hardening results were obtained, probably oscillations of shells; the frequency response was
due to neglect axisymmetric modes. A detailed hardening in the region of the main parametric
bibliographic analysis can be found in such paper. resonance for circumferential wave number n = 2
and softening for n > 2.
Del Prado et al. (2009) analyzed the Trotsenko and Trotsenko (2004), studied
instabilities occurring to a shell conveying an internal vibrations of circular cylindrical shells, with attached
heavy fluid and loaded with compressive and rigid bodies, by means of a mixed expansion based
periodic axial loads. The Donnell‟s nonlinear shallow on trigonometric functions and Legendre
shell theory was used in combination with a low polynomials; they considered only linear vibrations.
order Galerkin approach. They showed that the Pellicano (2005) presented experimental
combination of a flowing fluid with axial loads can results about violent vibration phenomena appearing

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in a shell with base harmonic excitation and carrying found, the conjecture made by such scientists was
a rigid mass on the top. When the first axisymmetric that the disagreement was due to shells geometric
mode is in resonance conditions the top mass imperfections. Such paper summarizes some of
undergoes to large amplitude of vibration and a huge experimental results published on a previous book
out of plane shell vibration is detected (more than (Kubenko et al., 1984)
2000g), with a relatively low base excitation (about Mallon et al. (2010) studied circular
10g). cylindrical shells made of orthotropic material, the
Pellicano and Avramov (2007) published a Donnell‟s nonlinear shallow shell theory was used
paper concerning the nonlinear dynamics of a shell with a multimode expansion for discretization (PDE
with base excitation and a top disk. The work was to ODE). They presented also experimental results.
mainly theoretical, i.e. only some experimental The theoretical model considered also the shaker-
results concerning the linear dynamics were shell interaction (it is to note that one of the first
presented; the shell was modelled using the nonlinear works concerning the interaction between an
Sanders–Koiter theory and a reduced order model electromechanical shaker and a mechanical system is
was used; the analysis was mainly focused on the due to Krasnopol‟skaya (1976)). Mallon et al. (2010)
principal parametric resonance due to high frequency investigated the imperfections sensitivity and found a
excitations.A similar analysis was presented also in reduction of the instability threshold. No good
Avramov and Pellicano (2006). quantitative match between theory and experiments
was found: saturation phenomena, beating and chaos
Pellicano (2007) developed a new method, were found numerically. However, this can be
based on the nonlinear Sanders–Koiter theory, considered a seminal work due to the intuition that
suitable for handling complex boundary conditions of some complex phenomena can be due to the shaker
circular cylindrical shells and large amplitude of shell interaction.
vibrations. The method was based on a mixed In Pellicano(2011), experiments are carried
expansions considering orthogonal polynomials and out on a circular cylindrical shell, made of a
harmonic functions. Among the others, the method polymeric material (P.E.T.) and clamped at the base
showed good accuracy also in the case of a shell by gluing its bottom to a rigid support. The axis of
connected with a rigid body; this method is the the cylinder is vertical and a rigid disk is connected
starting point for the model developed in the present to the shell top end. In Pellicano (2007) this problem
research. was fully analyzed from a linear Point ofview. Here
nonlinear phenomena are investigated by exciting the
Mallon et al. (2008) studied isotropic shell using a shaking table and a sine excitation.
circular cylindrical shells under harmonic base Shaking the shell from the bottom induces a vertical
excitation; they used the Donnell‟s nonlinear shallow motion of the top disk that causes axial loads due to
shell theory; a multimode expansion, suitable for inertia forces. Such axial loads generally give rise to
analyzing the nonlinear resonance and dynamic axial-symmetric deformations; however, in some
instability of a specific mode, was used; they conditions it is observed experimentally that a violent
presented comparisons between the semianalytical resonant phenomenon takes place, with a strong
procedure and a FEM model in the case of linear energy transfer from low to high frequencies and
vibrations or buckling. Using the nonlinear semi- huge amplitude of vibration. Moreover, an interesting
analytical model they found beating and chaotic saturation phenomenon is observed: the response of
responses, severe out of plane vibration and the top disk was completely flat as the excitation
sensitivity to imperfections. frequency was changed around the first axisymmetric
mode resonance.
Kubenko and Koval‟chuk (2009) published Farshidianfar A. and Farshidianfar
an experimental work focused on shells made of M.H.(2011) were carried out both theoretical and
composite materials; they pointed out the experimental analyses on a long circular cylindrical
inadequateness of the linear viscous damping models. shell. A total of 18 modes, consisting of all the three
Axial loads (base excitation, and free top end of the main mode groups (axisymmetric, beam-like and
shell) and combined loads were considered. The asymmetric) were found under a frequency range of
analysis of the principal parametric instability was 0–1000 Hz, by only applying acoustical excitation.
carried out (probably with sub harmonic response, Acoustical excitation results were compared with
but no information are give about). Dynamic those obtained from contact excitation. It was
instability regions were determined experimentally: a discovered that if one uses contact methods, several
disagreement between previous theoretical models exciting points are required to obtain all modes;
(narrower region) and experiments (wider) was whereas with acoustical excitation only one

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