The vibrations of a rotating thin piezoceramic cylindrical shell

G. H. Bryan [8] discovered and studied the dynamic phenomena that arise during the vibrations of thinwalled shells in natural modes while rotating uniformly about an axis of symmetry. In the present article a more detailed mathematical analysis has been conducted for the solution of the problem of the vibrations of a ring rotating with constant angular velocity under feasible conditions of perturbation of the vibrations and realistic mechanical properties of ceramic materials. It is established that two slowly precessing waves arise during the vibration of the ring, one of which (the one discovered by Bryan) is an inverse precession wave having an angular velocity of precession less than the angular velocity of rotation of the ring, while the second is a direct precession wave having angular velocity larger than the angular velocity of rotation of the ring. Under the actual working conditions for piezoceramic resonators the amplitudes of the direct and inverse wave become comparable. Consequently the vibrations present as a modulated standing wave, that is, the phenomenon of beats in the mode of steady-state normal vibration occurs. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 7–18.     

Axisymmetric vibrations of an orthotropic non-uniform cylindrical shell

The axisymmetric vibrations of a circular cylindrical shell of constant thickness, orthotropic along the principal geometrical directions and non-uniform along the generatrices, are considered. It is shown, using a particular problem as an example, that the non-uniform physical-mechanical characteristics of the shell material may lead to qualitatively new results in problems of shell dynamics.     

Radial vibrations of an infinite medium with a cylindrical cavity

Zusammenfassung Die vorliegende Arbeit bringt die vollständige theoretische Berechnung der Radialschwingungen eines unendlichen elastischen Mediums mit zylindrischem Ausschnitt. Die zylindrische Körperwand steht unter einem mit der Zeit allgemein veränderlichen Druck, und am Anfang sind alle Körperteile frei von Verschiebungen und Geschwindigkeiten.     

The vibrations of a thin elastic orthotropic circular cylindrical shell with free and hinged edges

The problem of the existence of natural oscillations of a thin elastic orthotropic circular closed cylindrical shell with free and hinge-mounted ends and of an open cylindrical shell with free and hinge-mounted edges, when the two boundary generatrices are hinge-mounted is investigated. Dispersion equations and asymptotic formulae for finding the natural frequencies of possible vibration modes are obtained using the system of equations corresponding to the classical theory of orthotropic cylindrical shells. A mechanism is proposed by means of which the vibrations can be separated into possible types. Approximate values of the dimensionless characteristic of the natural frequency and the attenuation characteristic of the corresponding vibration modes are obtained using the examples of closed and open orthotropic cylindrical shells of different lengths.     

Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell

In this paper, buckling and free vibration behavior of a piezoelectric rotating cylindrical carbon nanotube-reinforced (CNTRC) shell is investigated. Both cases of uniform distribution (UD) and FG distribution patterns of reinforcements are studied. The accuracy of the presented model is verified with previous studies and also with those obtained by Navier analytical method. The novelty of this study is investigating the effects of critical voltage and CNT reinforcement as well as satisfying various boundary conditions implemented on the piezoelectric rotating cylindrical CNTRC shell. The governing equations and boundary conditions have been developed using Hamilton's principle and are solved with the aid of Navier and generalized differential quadrature (GDQ) methods. In this research, the buckling phenomena in the piezoelectric rotating cylindrical CNTRC shell occur as the natural frequency is equal to zero. The results show that, various types of CNT reinforcement, length to radius ratio, external voltage, angular velocity, initial hoop tension and boundary conditions play important roles on critical voltage and natural frequency of piezoelectric rotating cylindrical CNTRC shell.     

Parametric vibrations of an orthotropic cylindrical shell with an elastic core. 2. Some numerical results

It is shown that the operator of the problem belongs to the class of oscillation operators. The first five regions of dynamic instability of an elastic orthotropic cylindrical shell with an elastic isotropic core are determined in the first approximation. The effect of the core and transverse shear in the shell on the width and location of these regions of dynamic instability of the system is determined. The effect of transverse shear on the natural frequencies of the empty shell is established.     

Analysis of free damped vibrations of laminated composite cylindrical shells

Damped free vibrations of multilayered composite cylindrical shells are investigated. Vibration and damping analysis of cylindrical shells is performed by using the first-order shear deformation theory (FOSDT). Based on other researchers' works, two damping models are developed, i.e., the energy method (EM), and the method of complex eigenvalues (MCE). Several numerical examples of the damped free vibration problem of laminated composite cylindrical shells have been solved and comparison has been made with the results of other authors.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 646–659, September–October, 1995.     

Theoretical-experimental investigation of frequencies of free vibrations of circular cylindrical shells

We consider a holographic interferometry technique for the determination of frequencies of free vibrations of isotropic circular cylindrical shells and compare the experimental results with results of calculation of frequencies of these shells on the basis of the spline collocation technique.     

Nonlinear free dynamic response of laminated compressible cylindrical shell panels

The governing equations for a free dynamic response of a symmetrically laminated composite shell are used to analyze a nonlinear differential panel. The FEM and the Lindstedt–Poincare perturbation technique are invoked to construct a uniform asymptotic expansion of the solution to a nonlinear differential equation ofmotion. A comparison between numerical and finite-element methods for analyzing a symmetrically laminated graphite/epoxy shell panel is performed to show that the nonlinearities are of hardening type and are more repeated for smaller opening angles. It is also shown that large-amplitude motions are dominated by lower modes.     

Barodiffusion in a rotating binary mixture over an infinite rotating disk

Barodiffusion (diffusion of species brought about by pressure-gradients) in an isothermal, incompressible, Newtonian binary mixture in a steady, laminar, axisymmetric, rotating motion over an infinite, rotating, impermeable disk is considered. In view of some potential applications the simplifying features of isotopic mixtures are taken into consideration in the formulation of the problem. An exact solution for the barodiffusion problem analogous to the von Karman flow solutions is given. Results clearly identifying the small but significant separative action of the pressure-gradients in this configuration are presented for Schmidt numbers of order unity (typical of gaseous mixtures).     

Chaotic vibrations of flexible infinite length cylindrical panels using the Kirchhoff–Love model

Treated as continuous deformable systems with an infinite number of degrees of freedom, flexible infinite length cylindrical panels subject to harmonic load are studied. Using the finite difference method with respect to spatial coordinates, the continuous system is reduced to lumped one governed by ordinary differential equations. These equations are transformed to a normal form and then solved numerically using the fourth order Runge–Kutta method. In order to trace and explain vibrational behaviour, dependencies w_max(q₀) and Lyapunov exponents are calculated for panels with parameter value k_x = 48. The corresponding charts of the control parameters {q₀, ω_q} are also reported. Novel scenarios yielding chaotic dynamics exhibited by cylindrical panels are illustrated and discussed.     

Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels

This paper deals with the classical challenging free vibration problems of non-Lévy-type cylindrical shell panels, i.e., those without two opposite edges simply supported, by a Hamiltonian system-based symplectic superposition method. The governing equations of a vibrating cylindrical panel are formulated within the Hamiltonian system framework such that the symplectic eigen problems are constructed, which yield analytic solutions of two types of fundamental problems. By the equivalence between the superposition of the fundamental problems and the original problem, new analytic frequency and mode shape solutions of the panels with four different combinations of boundary conditions are derived. Comprehensive benchmark results are tabulated and plotted, which are useful for validation of other numerical/approximate methods. The primary advantage of the developed approach that no pre-determination of solution forms is needed enables one to pursue more analytic solutions of intractable shell problems.     

Radial vibrations of an infinite medium with a spherical cavity

Zusammenfassung Die vorliegende Arbeit enthält die vollständige theoretische Berechnung der Radialschwingungen eines unendlichen elastischen Mediums mit kugelförmigem Hohlraum. Die innere Oberfläche des Körpers ist einem mit der Zeit allgemein veränderlichen Druck ausgesetzt. Am Aufang sind alle Körperteile frei von Verschiebungen und Geschwindigkeiten.     

The numerical analysis of an orthotropic cylindrical shell

This paper is devoted to an orthotropic cylindricall simply supported shell subjected to external pressure. This analysis is focused on determining geometrical shape of shell. Main subject of investigation is to maximize the stability of the shell. Mathematical model of double layer shell was implemented. Numerical solution was obtained for particular class of shells by using the finite element code ABAQUS. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dual problem of optimizing an orthotropic cylindrical shell

The problem of optimizing the weight of an orthotropic cylindrical shell is examined, using the duality theory of geometrical programing.     

Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness

Based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, this paper presents the vibration behavior of an isotropic and orthotropic oval cylindrical shell with parabolically varying thickness along its circumference. The governing equations of motion of the shell, which have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling, which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to eight first‐order differential equations in the circumferential coordinate and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different type modes of vibration is investigated. Copyright © 2011 John Wiley & Sons, Ltd.