Imperfection Sensitivity of Compressed Circular Cylindrical Shells Under Periodic Axial Loads

2009 ◽  
Author(s):  
Francesco Pellicano
Keyword(s):  
Differential Equations ◽  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Excitation Frequency ◽  
Finite Amplitude ◽  
Axial Loads ◽  
Imperfection Sensitivity ◽  
Lagrange Equations ◽  
Circular Cylindrical Shells

In the present paper the dynamic stability of circular cylindrical shells is investigated; the combined effect of compressive static and periodic axial loads is considered. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration; Lagrange equations are used to reduce the nonlinear partial differential equations to a set of ordinary differential equations. The dynamic stability is investigated using direct numerical simulation and a dichotomic algorithm to find the instability boundaries as the excitation frequency is varied; the effect of geometric imperfections is investigated in detail. The accuracy of the approach is checked by means of comparisons with the literature.

scholarly journals Nonlinear Vibrations of Functionally Graded Cylindrical Shells: Effect of the Geometry

2012 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano ◽  
Antonio Zippo
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Linear Analysis ◽  
Finite Amplitude ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Displacement Fields

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. In the linear analysis, after spatial discretization, mass and stiff matrices are computed, natural frequencies and mode shapes of the shell are obtained. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions obtained by the linear analysis; specific modes are selected. The Lagrange equations reduce nonlinear partial differential equations to a set of ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response of the shell. A convergence analysis is carried out to determine the correct number of the modes to be used. The analysis is focused on determining the nonlinear character of the response as the geometry of the shell varies.

Stability of Empty and Fluid-Filled Circular Cylindrical Shells Subjected to Dynamic Axial Loads

2002 ◽  
Author(s):  
F. Pellicano ◽  
M. Amabili ◽  
M. P. Pai¨doussis
Keyword(s):  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Travelling Wave ◽  
Finite Amplitude ◽  
Axial Loads ◽  
Simply Supported ◽  
Mean Oscillation ◽  
Shell Motion ◽  
Circular Cylindrical Shells ◽  
The Mean

In the present study the dynamic stability of simply supported, circular cylindrical shells subjected to dynamic axial loads is analyzed. 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 large number of linear modes, including axisymmetric and asymmetric modes, and on 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 deflection of the mean oscillation with respect to the equilibrium position. The shell is simply supported and presents a finite length. Boundary conditions are considered in the model, which includes also the contribution of the external axial loads acting at the shell edges. The effect of a contained liquid is also considered. The linear dynamic stability and nonlinear response are analysed by using continuation techniques.

Snapping of Imperfect Thin-Walled Circular Cylindrical Shells of Finite Length

1971 ◽  
Vol 38 (1) ◽  
pp. 162-171 ◽  
Author(s):  
K. Y. Narasimhan ◽  
N. J. Hoff
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Finite Length ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Numerical Procedure ◽  
Maximal Load ◽  
Load Carrying ◽  
Circumferential Displacement ◽  
Circular Cylindrical Shells

The nonlinear partial differential equations of von Karman and Donnell governing the deformations of initially imperfect cylindrical shells are reduced to a consistent set of ordinary differential equations. A numerical procedure is then used to solve the equations together with the associated boundary conditions and to determine the number of waves at buckling as well as the load-carrying capacity of imperfect cylindrical shells of finite length subjected to uniform axial compression in the presence of a reduced restraint along the simply supported boundaries. It is found that details of the boundary conditions have little effect on the number of waves into which the shell buckles around the circumference. This number is determined essentially by the length-to-radius and radius-to-thickness ratios. The absence of an edge restraint to circumferential displacement reduces the classical value of the buckling load by a factor of about two. On the other hand, shells with these boundary conditions appear to be less sensitive to initial imperfections in the shape, and thus the maximal load supported in the presence of unavoidable initial deviations can be the same for shells with and without a restraint to circumferential displacements along the edges.

Nonlinear free vibration analysis of delaminated composite circular cylindrical shells

2020 ◽  
Vol 26 (19-20) ◽  
pp. 1697-1707
Author(s):  
Abbas Kamaloo ◽  
Mohsen Jabbari ◽  
Mehdi Yarmohammad Tooski ◽  
Mehrdad Javadi
Keyword(s):  
Differential Equations ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Free Vibration Analysis ◽  
Middle Surface ◽  
Free Vibrations ◽  
Wave Number ◽  
Number Of Layers ◽  
Circular Cylindrical Shells ◽  
Delamination Length

This study aims to present an analysis of nonlinear free vibrations of simply supported laminated composite circular cylindrical shells with throughout circumference delamination. Governing equations of motion are derived by applying energy methods; using Galerkin’s method reduced the nonlinear partial differential equations to a system of coupled nonlinear ordinary differential equations, which are subsequently solved using a numerical method. This research examines the effects of delamination on the oscillatory motion of delaminated composite circular cylindrical shells and then the effects of increase in delamination length, shell middle surface radius, number of layers, and orthotropy as changes in material properties on the nonlinearity of these types of shells. The results show that delamination leads to a decrease in frequency of oscillations and displacement. An increase in delamination length, shell middle surface radius, and orthotropy of layers decreases nonlinearity and displacement, whereas an increase in the number of layers increases nonlinearity and displacement. It is also observed that an increase in the circumferential wave number can decrease the effect of delamination.

Nonlinear forced vibration of rotating composite laminated cylindrical shells under hygrothermal environment

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiao Li ◽  
Wentao Jiang ◽  
Xiaochao Chen ◽  
Zhihong Zhou
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Dynamic Responses ◽  
Partial Differential ◽  
Moisture Concentration ◽  
Hygrothermal Environment ◽  
Laminated Cylindrical Shells

Abstract This work aims to study nonlinear vibration of rotating composite laminated cylindrical shells under hygrothermal environment and radial harmonic excitation. Based on Love’s nonlinear shell theory, and considering the effects of rotation-induced initial hoop tension, centrifugal and Coriolis forces, the nonlinear partial differential equations of the shells are derived by Hamilton’s principle, in which the constitutive relation and material properties of the shells are both hygrothermal-dependent. Then, the Galerkin approach is applied to discrete the nonlinear partial differential equations, and the multiple scales method is adopted to obtain an analytical solution on the dynamic response of the nonlinear shells under primary resonances of forward and backward traveling wave, respectively. The stability of the solution is determined by using the Routh–Hurwitz criterion. Some interesting results on amplitude–frequency relations and nonlinear dynamic responses of the shells are proposed. Special attention is given to the combined effects of temperature and moisture concentration on nonlinear resonance behavior of the shells.

Vibrations of Circular Cylindrical Shells With Complex Boundary Conditions

2006 ◽  
Author(s):  
Francesco Pellicano
Keyword(s):  
Boundary Conditions ◽  
Harmonic Functions ◽  
Nonlinear Vibrations ◽  
Cylindrical Shells ◽  
Rigid Bodies ◽  
Geometrically Nonlinear ◽  
Lagrange Equations ◽  
Displacement Fields ◽  
Complex Boundary ◽  
Circular Cylindrical Shells

In the present paper vibrations of circular cylindrical shells having different boundary conditions are analyzed. Sanders-Koiter theory is considered for shell modeling: both linear and nonlinear vibrations are analyzed. An energy approach based on Lagrange equations is considered; a mixed expansion of displacement fields, based on harmonic functions and Tchebyshev polynomials, is applied. Several boundary conditions are analyzed: simply supported, clamped-clamped, connection with rigid bodies. Comparisons with experiments and finite element analyses show that the technique is capable to model several and complex boundary conditions. Applications to geometrically nonlinear shells show that the technique is effective also in the case of nonlinear vibration: comparisons with the literature confirm the accuracy of the approach.

Dynamic stability of circular cylindrical shells under periodic compressive forces

1978 ◽  
Vol 58 (3) ◽  
pp. 425-441 ◽  
Author(s):  
K. Nagai ◽  
N. Yamaki
Keyword(s):  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Circular Cylindrical Shells

On the Parametrically Excited Response of the Quadratically Nonlinear Model of a Rotating Ring on an Elastic Hub

1993 ◽  
Author(s):  
Rick I. Zadoks ◽  
Charles M. Krousgrill
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Excitation Frequency ◽  
Normal Modes ◽  
Point Load ◽  
Tangential Displacement ◽  
Partial Differential

Abstract As a first approximation, a steel-belted radial tire can be modeled as a one dimensional rotating ring connected elastically to a moving hub. This ring can be modeled mathematically using a set of three nonlinear partial differential equations, where the three degrees of freedom are a radial displacement, a tangential displacement and a section rotation. In this study, only quadratic geometric nonlinearities are considered. The system is excited by a temporally harmonic point load f^(t) and a temporally harmonic hub motion z^(t) that have the same harmonic frequency. The point load f^(t) appears in the equations of motion as a single in-homogeneous term, while the hub motion z^(t) appears in inhomogeneous and parametric excitation terms. To simplifying the ensuing analysis, the rotation rate of the hub is assumed to be constant. The partial differential equations of motion are reduced to a set of four second-order ordinary differential equations by using two linear normal modes to approximate the spatial distribution of the displacements. A region of the parameter space, as defined by ranges of values of the excitation amplitude z and the excitation frequency ω (or detuning parameter σ), is identified, from a Strutt diagram, where the parametric excitation is expected to be dominant. In this region σ is varied to locate a secondary Hopf bifurcation that leads to a set of complex steady-state quasi-periodic solutions. These solutions contain two families of frequency components where the fundamental frequencies of these families are non-commensurate, and they are characterized by Poincaré sections with closed or nearly closed “orbits” as opposed to the distinct points displayed by periodic responses and the strange attractor sections displayed by chaotic solutions.

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