Marco Amabili

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Abstract Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleighs dissipation function. Four different non-linear thin shell theories, namely Donnells, Sanders–Koiter, Flugge–Lur’e-Byrne and Novozhilovs theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnells non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.

Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections

Abstract The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated. Donnells non-linear shallow-shell theory is used and the solution is obtained by the Galerkin method. Several expansions involving 16 or more natural modes of the shell are used. The boundary conditions on the radial displacement and the continuity of circumferential displacement are exactly satisfied. The effect of internal quiescent, incompressible and inviscid fluid is investigated. The non-linear equations of motion are studied by using a code based on the arclength continuation method. A series of accurate experiments on forced vibrations of an empty and water-filled stainless-steel shell have been performed. Several modes have been intensively investigated for different vibration amplitudes. A closed loop control of the force excitation has been used. The actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model. Several interesting non-linear phenomena have been experimentally observed and numerically reproduced, such as softening-type non-linearity, different types of travelling wave response in the proximity of resonances, interaction among modes with different numbers of circumferential waves and amplitude-modulated response. For all the modes investigated, the theoretical and experimental results are in strong agreement.

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnells non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.

Stability and vibration of empty and fluid-filled circular cylindrical shells under static and periodic axial loads

Abstract In the present study, the dynamic stability of simply supported, circular cylindrical shells subjected to dynamic axial loads is analysed. Geometric nonlinearities due to finite-amplitude shell motion are considered by using the Donnell’s nonlinear shallow-shell theory. The effect of structural damping is taken into account. A discretization method based on a series expansion involving a relatively large number of linear modes, including axisymmetric and asymmetric modes, and on the Galerkin procedure is developed. Axisymmetric modes are included; indeed, they are essential in simulating the inward deflection of the mean oscillation with respect to the equilibrium position and in describing the axisymmetric deflection due to axial loads. A finite length, simply supported shell is considered; the boundary conditions are satisfied, including the contribution of external axial loads acting at the shell edges. The effect of a contained liquid is investigated. The linear dynamic stability and nonlinear response are analysed by using continuation techniques and direct simulations.

Nonlinear supersonic flutter of circular cylindrical shells

The aeroelastic stability of simply supported, circular cylindrical shells in supersonic flow is investigated. Nonlinearities caused by large-amplitude shell motion are considered by using the Donnell nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied to the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear piston theory is applied to describe the fluid-structure interaction by using two different formulations, taking into account or neglecting the curvature correction term. The system is discretized by Galerkin projections and is investigated by using a model involving seven degrees of freedom, allowing for traveling-wave flutter of the shell and shell axisymmetric contraction. Results show that the system loses stability by standing-wave flutter through supercritical bifurcation; however, traveling-wave flutter appears with a very small increment of the freestream static pressure that is used as the bifurcation parameter. A very good agreement between theoretical and existing experimental data has been found for flutter amplitudes. The influence of internal static pressure has also been studied.

Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 1: Equations of Motion and Numerical Results

The response-frequency relationship in the vicinity of a resonant frequency, the occurrence of travelling wave response and the presence of internal resonances are investigated for simply supported, circular cylindrical shells. Donnells nonlinear shallow-shell theory is used. The boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied. The problem is reduced to a system of four ordinary differential equations by means of the Galerkin method. The radial deflection of the shell is expanded by using a basis of four linear modes. The effect of internal fluid is also investigated. The equations of motion are studied by using a code based on the Collocation Method. The present model is validated by comparison of some results with others available. A water-filled shell presenting the phenomenon of 1:1:1:2 internal resonances is investigated for the first time; it shows intricate and interesting dynamics.

Multimode Approach to Nonlinear Supersonic Flutter of Imperfect Circular Cylindrical Shells

The aeroelastic stability of simply supported, circular cylindrical shells in supersonic flow is investigated by using both linear aerodynamics (first-order piston theory) and nonlinear aerodynamics (third-order piston theory). Geometric nonlinearities, due to finite amplitude shell deformations, are considered by using the Donnells nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. The system is discretized by Galerkin method and is investigated by using a model involving up to 22 degrees-of-freedom, allowing for travelling-wave flutter around the shell and axisymmetric contraction of the shell. Asymmetric and axisymmetric geometric imperfections of circular cylindrical shells are taken into account. Numerical calculations are carried out for a very thin circular shell at fixed Mach number 3 tested at the NASA Ames Research Center. Results show that the system loses stability by travelling-wave flutter around the shell through supercritical bifurcation. Nonsimple harmonic motion is observed for sufficiently high post-critical dynamic pressure. A very good agreement between theoretical and existing experimental data has been found for the onset of flutter, flutter amplitude, and frequency. Results show that onset of flutter is very sensible to small initial imperfections of the shells. The influence of pressure differential across the shell skin has also been deeply investigated. The present study gives, for the first time, results in agreement with experimental data obtained at the NASA Ames Research Center more than three decades ago. ©2002 ASME

Non-linear dynamics and stability of circular cylindrical shells conveying flowing fluid

Abstract The non-linear dynamics and stability of simply supported, circular cylindrical shells containing inviscid, incompressible fluid flow is analyzed. Geometric non-linearities of the shell are considered by using the Donnells non-linear shallow shell theory. A viscous damping mechanism is considered in order to take into account structural and fluid dissipation. Linear potential flow theory is applied to describe the fluid–structure interaction. The system is discretized by Galerkins method and is investigated by using two models: (i) a simpler model obtained by using a base of seven modes for the shell deflection, and (ii) a relatively high-dimensional dynamic model with 18 modes. Both models allow travelling-wave response of the shell and shell axisymmetric contraction. Boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied. Stability, bifurcation and periodic responses are analyzed by means of the computer code AUTO for the continuation of the solution of ordinary differential equations. Non-stationary motions are analyzed with direct integration techniques. An accurate analysis of the shell response is performed by means of phase space representation, Fourier spectra, Poincare sections and their bifurcation diagrams. A complex dynamical behaviour has been found. The shell bifurcates statically (divergence) in absence of external dynamic loads by using the flow velocity as bifurcation parameter. Under harmonic load a shell conveying flow can give rise to periodic, quasi-periodic and chaotic responses, depending on flow velocity, amplitude and frequency of harmonic excitation.

Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 2: Perturbation Analysis

The nonlinear ordinary differential equations describing the dynamics of fluid filled circular cylindrical shell, obtained in Part 1 of the present study, is studied by using a second order perturbation approach and direct simulations. Strong modal interactions are found when the structure is excited with small resonant loads. Modal interactions arise in the whole range of vibration amplitude, showing that the internal resonance condition makes the system non-linearizable even for extremely small amplitudes of oscillation. Stationary and nonstationary oscillations are observed and the complex nature of modal interactions is accurately analyzed. No chaotic motion is observed in the case of 1:1:1:2 internal resonance studied.

Nonlinear Vibrations of Circular Cylindrical Shells with Different Boundary Conditions

Large-amplitude (geometrically nonlinear) vibrations of circular cylindrical shells with different boundary conditions and subjected to radial harmonic excitation in the spectral neighborhood of the lowest resonances are investigated. In particular, simply supported shells with allowed and constrained axial displacements at the edges are studied; in both cases, the radial and circumferential displacements at the shell edges are constrained. Elastic rotational constraints are assumed; they allow simulating any condition from simply supported to perfectly clamped, by varying the stiffness of this elastic constraint. Two different nonlinear, thin shell theories, namely Donnells and Novozhilovs theories, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. Geometric imperfections are taken into account. Comparison of calculations to experimental and numerical results available in the literature is performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The nonlinear equations of motion are studied by using a code based on arclength continuation method that allows bifurcation analysis.