scholarly journals Vibrations of an elastic cylindrical shell near the lowest cut-off frequency

2016 ◽  
Vol 472 (2189) ◽  
pp. 20150753 ◽  
Author(s):  
J. Kaplunov ◽  
L. I. Manevitch ◽  
V. V. Smirnov
Keyword(s):  
Cylindrical Shell ◽  
Dispersion Relation ◽  
Ad Hoc ◽  
Circular Cylindrical Shell ◽  
Dynamic Modelling ◽  
Elastic Cylindrical Shell ◽  
Two Dimensional ◽  
Membrane Theory ◽  
Thin Walled Structures ◽  
Shell Equations

A new asymptotic approximation of the dynamic equations in the two-dimensional classical theory of thin-elastic shells is established for a circular cylindrical shell. It governs long wave vibrations in the vicinity of the lowest cut-off frequency. At a fixed circumferential wavenumber, the latter corresponds to the eigenfrequency of in-plane vibrations of a thin almost inextensible ring. It is stressed that the well-known semi-membrane theory of cylindrical shells is not suitable for tackling a near-cut-off behaviour. The dispersion relation within the framework of the developed formulation coincides with the asymptotic expansion of the dispersion relation originating from full two-dimensional shell equations. Asymptotic analysis also enables refining the geometric hypotheses underlying various ad hoc set-ups, including the assumption on vanishing of shear and circumferential mid-surface deformations used in the semi-membrane theory. The obtained results may be of interest for dynamic modelling of elongated cylindrical thin-walled structures, such as carbon nanotubes.

Cylindrical Shells Under Line Load

1957 ◽  
Vol 24 (4) ◽  
pp. 553-558
Author(s):  
R. M. Cooper
Keyword(s):  
Cylindrical Shell ◽  
Radial Displacement ◽  
Circular Cylindrical Shell ◽  
Transverse Shear Deformation ◽  
System Of Equations ◽  
Principle Of Superposition ◽  
Line Load ◽  
Simply Supported ◽  
Shell Equations ◽  
Shell Geometry

Abstract The problem of a line load along a segment of a generator of a simply supported circular cylindrical shell is treated using shallow cylindrical shell equations which include the effect of transverse-shear deformation. The line load is first treated as a sinusoidally-varying edge load over the length of the shell, with boundary conditions prescribed along the loaded generator such that the continuity of the shell is maintained. The solution for the problem of a uniform line load over a segment of a generator is obtained from the preceding solution, using the principle of superposition. By means of a numerical example it is shown that the results predicted by the Donnell equations for the stresses are in excellent agreement with those obtained from the system of equations employed here. However, the radial displacement predicted by the Donnell equations is in error by as much as 20 per cent in the range of shell geometry considered.

Nonlinear Response of an Elastic Cylindrical Shell to a Transient Acoustic Wave

1981 ◽  
Vol 48 (1) ◽  
pp. 15-24 ◽  
Author(s):  
T. L. Geers ◽  
C.-L. Yen
Keyword(s):  
Cylindrical Shell ◽  
Acoustic Wave ◽  
Nonlinear Response ◽  
Circular Cylindrical Shell ◽  
Elastic Cylindrical Shell ◽  
Fluid Medium ◽  
Energy Functionals ◽  
Potential Formulation ◽  
Rigorous Treatment ◽  
Residual Potential

Governing equations are developed for the nonlinear response of an infinite, elastic, circular cylindrical shell submerged in an infinite fluid medium and excited by a transverse, transient acoustic wave. These equations derive from circumferential Fourier-series decomposition of the field quantities appearing in appropriate energy functionals, and from application of the “residual potential formulation” for rigorous treatment of the fluid-structure interaction. Extensive numerical results are presented that provide understanding of the phenomenology involved.

Excitation of an Elastic Cylindrical Shell by a Transient Acoustic Wave

1969 ◽  
Vol 36 (3) ◽  
pp. 459-469 ◽  
Author(s):  
T. L. Geers
Keyword(s):  
Cylindrical Shell ◽  
Acoustic Wave ◽  
Circular Cylindrical Shell ◽  
Elastic Cylindrical Shell ◽  
Sandwich Shell ◽  
Governing Equations ◽  
Fluid Medium ◽  
Method Of Solution ◽  
Plane Step ◽  
Strain Responses

An infinite, elastic, circular cylindrical shell submerged in an infinite fluid medium is engulfed by a transverse, transient acoustic wave. The governing equations for modal shell response are reduced through the application of a new method of solution to two simultaneous equations in time; these equations are particularly amenable to solution by machine computation. Numerical results are presented for the first six modes of a uniform sandwich shell submerged in water and excited by a plane step-wave. These results are then used to evaluate the accuracy of a number of approximations which have been employed previously to treat this and similar problems. The results are also used to compute displacement, velocity, and flexural strain responses at certain points in the sandwich shell.

Longitudinal Impact of a Semi-Infinite Elastic Cylindrical Shell

1963 ◽  
Vol 30 (3) ◽  
pp. 347-354 ◽  
Author(s):  
H. M. Berkowitz
Keyword(s):  
Cylindrical Shell ◽  
Asymptotic Expansion ◽  
Elastic Cylindrical Shell ◽  
Longitudinal Impact ◽  
Axially Symmetric ◽  
Membrane Theory ◽  
Pressure Signal ◽  
Rigid Obstacle ◽  
Leading Term

The axially symmetric pressure signal produced by the longitudinal impact of a semi-infinite elastic cylindrical shell upon a rigid obstacle is determined by using membrane theory. Using this simplified theory, it is possible to obtain not only the leading term of an asymptotic expansion for large values of time, but also additional terms.

Plane-Strain Dynamic Response of a Cylindrical Shell—A Comparison Study of Three Different Shell Theories

1968 ◽  
Vol 35 (2) ◽  
pp. 297-305 ◽  
Author(s):  
H. Reismann ◽  
P. S. Pawlik
Keyword(s):  
Cylindrical Shell ◽  
Dynamic Response ◽  
Plane Strain ◽  
Analytical Study ◽  
Circular Cylindrical Shell ◽  
Response Prediction ◽  
Comparison Study ◽  
Membrane Theory ◽  
Rotatory Inertia ◽  
Initial Motion

An analytical study of the plane-strain dynamic response of a circular, cylindrical shell is presented. The shell is subjected to a radially directed concentrated impulse acting on its surface. Solutions are presented within the framework of (a) membrane theory, (b) Flu¨gge theory, and (c) improved theory (including shear deformation and rotatory inertia). A quantitative study of the initial motion of the shell indicates major differences in response prediction of the three theories. An explanation of these differences is offered.

Stresses in Eccentric Nonuniform Rings Reinforcing Cylindrical Shells

1967 ◽  
Vol 11 (02) ◽  
pp. 73-88
Author(s):  
Arnold Kempner ◽  
Joseph Kempner
Keyword(s):  
Cylindrical Shell ◽  
Hydrostatic Pressure ◽  
Circular Cylindrical Shell ◽  
Cylindrical Shells ◽  
Approximate Solutions ◽  
Ring Theory ◽  
Stress Distributions ◽  
Membrane Stress ◽  
Shell Equations ◽  
Interaction Problem

Bending and membrane stresses are determined in nonuniform frames of an infinitely long reinforced circular cylindrical shell subjected to hydrostatic pressure. The Donnell shell equations and deep-ring theory are used to solve the interaction problem. The frames, periodically spaced along the shell, are composed of two uniform but different sections. Each section of each frame has a different centroidal radius. Analyses of bending and membrane stress distributions in the frames are presented. Approximate solutions of different degrees of simplicity and accuracy are also given.

Mathematical modelling of linearly elastic shells

Acta Numerica ◽  
2001 ◽  
Vol 10 ◽  
pp. 103-214 ◽  
Author(s):  
Philippe G. Ciarlet
Keyword(s):  
Ad Hoc ◽  
Asymptotic Methods ◽  
Three Dimensional ◽  
Middle Surface ◽  
Elastic Membrane ◽  
Two Dimensional ◽  
Mathematical Basis ◽  
Elastic Shells ◽  
Applied Forces ◽  
Shell Equations

The objective of this article is to lay down the proper mathematical foundations of the two-dimensional theory of linearly elastic shells. To this end, it provides, without any recourse to any a priori assumptions of a geometrical or mechanical nature, a mathematical justification of two-dimensional linear shell theories, by means of asymptotic methods, with the thickness as the ‘small’ parameter.A major virtue of this approach is that it naturally leads to precise mathematical definitions of linearly elastic ‘membrane’ and ‘flexural’ shells. Another noteworthy feature is that it highlights in particular the role played by two fundamental tensors, each associated with a displacement field of the middle surface, the linearized change of metric and linearized change of curvature tensors.More specifically, under fundamentally distinct sets of assumptions bearing on the geometry of the middle surface, on the boundary conditions, and on the order of magnitude of the applied forces, it is shown that the three-dimensional displacements, once properly scaled, converge (in H1, or in L2, or in ad hoc completions) as the thickness approaches zero towards a ‘two-dimensional’ limit that satisfies either the linear two-dimensional equations of a ‘membrane’ shell (themselves divided into two subclasses) or the linear two-dimensional equations of a ‘flexural’ shell. Note that this asymptotic analysis automatically provides in each case the ‘limit’ two-dimensional equations, together with the function space over which they are well-posed.The linear two-dimensional shell equations that are most commonly used in numerical simulations, namely Koiter's equations, Naghdi's equations, and ‘shallow’ shell equations, are then carefully described, mathematically analysed, and likewise justified by means of asymptotic analyses.The existence and uniqueness of solutions to each one of these linear two-dimensional shell equations are also established by means of crucial inequalities of Korn's type on surfaces, which are proved in detail at the beginning of the article.This article serves as a mathematical basis for the numerically oriented companion article by Dominique Chapelle, also in this issue of Acta Numerica.

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