Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.     

Free vibrations of circular cylindrical shells

The finite element method is used to predict the dynamic behaviour of circular cylindrical shells in free vibrations. A suitable shape function for the circumferential displacement distribution has been proposed. This reduces the three-dimensional character of the problem to a two-dimensional one. The simultaneous iteration method to determine the eigen-frequencies and eigenvectors is utilised for solving the eigenvalue problem. The accuracy of the method has been checked by verifying the results of known cases. Finally an experimental shell structure containing elastic rings welded at the ends has also been analysed and the experimental results compared with the theoretical ones.     

Vibration analysis of laminated cross-ply oval cylindrical shells

Here, free vibrations and transient dynamic response analyses of laminated cross-ply oval cylindrical shells are carried out. The formulation is based on higher order theory that accounts for the transverse shear and the transverse normal deformations, and includes zig-zag variation in the in-plane displacements across the thickness of the multi-layered shells. The contributions of inertia effect due to in-plane and rotary motions, and the higher order function arising from the assumed displacement models are included. The governing equations obtained using Lagrangian equations of motion are solved through finite element approach. A detailed parametric study is conducted to bring out the influence of different shell geometry, ovality parameter, lay-up and loading environment on the vibration characteristics related to different modes of vibrations of oval shell.     

Nonlinear vibrations of clamped-free circular cylindrical shells

Only experimental studies are available on large-amplitude vibrations of clamped-free shells. In the present study, large-amplitude nonlinear vibrations of clamped-free circular cylindrical shell are numerically investigated for the first time. Shells with perfect and imperfect shape are studied. The Sanders-Koiter nonlinear shell theory is used to calculate the elastic strain energy. Shell displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series, i.e. harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. All boundary conditions are satisfied. The system is discretized by using natural modes of the shell and Lagrange equations by an energy approach, retaining damping through Rayleigh's dissipation function. Different expansions involving from 18 to 52 generalized coordinates are used to study the convergence of the solution. The nonlinear equations of motion are numerically studied by using arclength continuation method and bifurcation analysis. Numerical responses to harmonic radial excitation in the spectral neighborhood of the lowest natural frequency are compared with experimental results available in literature. The effect of geometric imperfections and excitation amplitude are numerically investigated and fully explained.     

Dynamic analysis of ring-stiffened circular cylindrical shells

A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.     

Natural frequencies of clamped-free circular cylindrical shells

Some experimental studies of the circumferential mode vibration characteristics of clamped-free circular cylindrical shells are reported and the results compared with some available theoretical predictions. Good agreement has been obtained for the natural frequencies for configurations typical of unstiffened steel stacks.     

The forced motion of viscoelastic thick cylindrical shells

This paper presents a theoretical analysis of a dynamic boundary value problem of the axially-symmetric motion of isotropic, homogeneous, linearly-viscoelastic, thick, cylindrical shells subjected to time-dependent surface tractions and/or time-dependent boundary conditions. Williams' modal-acceleration method has been used to treat the time-dependent boundary conditions. Two forms of the correspondence principle are used to obtain the governing differential equations and the quasi-static solutions. A numerical example is given to study the transient response of a cylindrically hollow rod subject to longitudinal impacts.     

Vibrations of circular cylindrical shells with cutouts

An experimental and analytical study was carried out to examine the effect of circular cutouts on the resonant frequencies of thin cylindrical shells. The experimental results were obtained from tests performed on clamped-free aluminum cylinders and clamped ring-stiffened tri-acetyl cellulose shells with a lap-joint seam. The analytical solution was a simplified Rayleigh-Ritz type approximation. For the beam type mode, the circular cutouts had a significant influence on the frequency. For the mode with higher numbers of circumferential waves, however, the cutouts had a relatively small effect on the frequency spectra.     

Free vibration analysis of circular cylindrical shells

A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders' shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTP). The numerical results obtained are compared with finite element method solutions.     

Forced axisymmetric motion of circular,viscoelastic plates

Williams' method for forced motion of elastic systems is applied to circular, viscoelastic plates where the effects of rotatory inertia, transverse shear and time-dependent boundary conditions are included. The viscoelastic material is assumed to have a constant Poisson's ratio. A particular problem is solved for a symmetrically loaded, completely free plate. The material used is vulcanized rubber where the viscoelastic behavior in shear is used in specifying the material parameters of a three-element solid.     

Non-linear vibrations of laminated cylindrical shallow shells under thermomechanical loading

The geometrically non-linear vibrations of linear elastic composite laminated shallow shells under the simultaneous action of thermal fields and mechanical excitations are analysed. For this purpose, a model based on a very efficient p-version first-order shear deformation finite element, with hierarchical basis functions, is employed. The equations of motion are solved in the time domain by a Newmark implicit time integration method. The model and code developed are partially validated by comparison with published data. Parametric studies are carried out in order to study the influence of temperature change, initial curvature, panel thickness and fibre orientation on the shells’ dynamics.     

Calculation of natural frequencies of fixed-free circular cylindrical shells

A theoretical analysis is discussed here in detail which is used to investigate the natural frequencies of fixed-free circular cylindrical shells. The results are compared with the experimental observations of other workers. Analytical results are, in general, in reasonably good agreement with the experimental results.The effect of the simplifying assumption of zero hoop and shear strains has been studied carefully. It is shown that for shells of large length-to-radius ratio this assumption can be incorporated resulting in a major simplification of the analysis involved without significantly affecting the accuracy of the results, whereas for shells of small length-to-radius ratio and for higher axial modes the simplified analysis must be used with care, especially for the low circumferential wave numbers because the effect there is to significantly increase the natural frequencies. In an attempt to understand the physical significance of this assumption, calculations were made to estimate the proportion of strain energy due to bending and stretching actions for various modal arrangements with and without this assumption. It was found that the effect of the assumption is to increase considerably the strain energy due to extension with little effect on the strain energy due to bending and thus increase the total strain energy, substantially in case of the shells with small length-to-radius ratio and for higher axial modes with consequent increase in the natural frequencies. In such cases the analysis without making this assumption is to be used and it is shown that this can quite easily be handled.The integrals involving characteristic beam functions have been derived in closed form.     

Mathematical modeling of the processes of projection of plates and cylindrical shells

The equation of state was developed for materials subjected to the polymorphic transformation upon explosive loading with an explosive charge. The formula obtained was used in the numerical modeling of the explosive projection of plates and cylindrical shells fabricated from St 20 steel. The theoretically estimated kinematic parameters of a projected body were compared with the experimental data.     

Axially symmetric vibrations of fluid-filled poroelastic circular cylindrical shells

Employing Biot's theory of wave propagation in liquid saturated porous media, axially symmetric vibrations of fluid-filled and empty poroelastic circular cylindrical shells of infinite extent are investigated for different wall-thicknesses. Let the poroelastic cylindrical shells are homogeneous and isotropic. The frequency equation of axially symmetric vibrations each for a pervious and an impervious surface is derived. Particular cases when the fluid is absent are considered both for pervious and impervious surfaces. The frequency equation of axially symmetric vibrations propagating in a fluid-filled and an empty poroelastic bore, each for a pervious and an impervious surface is derived as a limiting case when ratio of thickness to inner radius tends to infinity as the outer radius tends to infinity. Cut-off frequencies when the wavenumber is zero are obtained for fluid-filled and empty poroelastic cylindrical shells both for pervious and impervious surfaces. When the wavenumber is zero, the frequency equation of axially symmetric shear vibrations is independent of nature of surface, i.e., pervious or impervious and also it is independent of presence of fluid in the poroelastic cylindrical shell. Non-dimensional phase velocity for propagating modes is computed as a function of ratio of thickness to wavelength in absence of dissipation. These results are presented graphically for two types of poroelastic materials and then discussed. In general, the phase velocity of an empty poroelastic cylindrical shell is higher than that of a fluid-filled poroelastic cylindrical shell.The phase velocity of a fluid-filled bore is higher than that of an empty poroelastic bore. Previous results are shown as a special case of present investigation. Results of purely elastic solid are obtained.     

The dynamic analysis of circular plates and shallow spherical shells

In the investigation reported here an attempt has been made to study the influence of Berger's approximation on the non-linear transient response of circular plates and shallow spherical shells. The governing equations of motion obtained from Berger's approximation are solved by using the rapidly converging Chebyshev series spacewise and the Houbolt scheme for integration in the time domain. Results calculated when using Berger's approximation are compared with exact results. It is shown that Berger's method yields very accurate values for plates and shells under transient loading, in the case of immovable edge conditions.     

Free vibrations of laminated composite cylindrical shells with an interior rectangular plate

The free vibration analysis of a laminated composite cylindrical shell with an interior rectangular plate is performed by the analytical and experimental methods. The frequency equations of vibration of the shell including the plate are formulated by using the receptance method. To obtain the free vibration characteristics before the combination of two structures, the energy principle based on the classical plate theory and Love's thin shell theory is adopted. The numerical results are compared with the results from an experiment, as well as a finite element analysis, to validate the current formulation. The influences of the length-to-radius ratio (L_S/a) and radius-to-thickness ratio (a/h_S) of the shell and fiber orientation angles (Θ) of symmetric cross- and angle-ply composite materials on the natural frequencies of a cylindrical laminated combined shell are also discussed in details.     

Nonlinear dynamics of harmonically excited circular cylindrical shells containing fluid flow

In the present study, the geometrically nonlinear vibrations of circular cylindrical shells, subjected to internal fluid flow and to a radial harmonic excitation in the spectral neighbourhood of one of the lowest frequency modes, are investigated for different flow velocities. The shell is modelled by Donnell's nonlinear shell theory, retaining in-plane inertia and geometric imperfections; the fluid is modelled as a potential flow with the addition of unsteady viscous terms obtained by using the time-averaged Navier-Stokes equations. A harmonic concentrated force is applied at mid-length of the shell, acting in the radial direction. The shell is considered to be immersed in an external confined quiescent liquid and to contain a fluid flow, in order to reproduce conditions in previous water-tunnel experiments. For the same reason, complex boundary conditions are applied at the shell ends simulating conditions intermediate between clamped and simply supported ends. Numerical results obtained by using pseudo-arclength continuation methods and bifurcation analysis show the nonlinear response at different flow velocities for (i) a fixed excitation amplitude and variable excitation frequency, and (ii) fixed excitation frequency by varying the excitation amplitude. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented, as well as the maximum Lyapunov exponent, in order to classify the system dynamics. In particular, periodic, quasi-periodic, sub-harmonic and chaotic responses have been detected. The full spectrum of the Lyapunov exponents and the Lyapunov dimension have been calculated for the chaotic response; they reveal the occurrence of large-dimension hyperchaos.