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.     

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.     

On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space

In this paper, we obtain accurate analytic free vibration solutions of rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected.     

Buckling and vibration of cross-ply laminated circular cylindrical panels

Summary The free vibration problem of a thin cross-ply composite laminated circular cylindrical panel subjected to arbitrary combinations of axial and circumferential initial compressions is studied. The equations of motion are derived, in the framework of the Donnell-type theory, in terms of the panel middle surface displacements components. Closed form solutions are obtained, for simply-supported panels, and numerical results for both antisymmetric and unsymmetric laminated cross-ply panels are presented.

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Résumé On étudie le problème de libre vibration d'une coque cylindrique ouverte, composée de fines lamelles croisées, soumise aux combinaisons arbitraires des compressions initiales de la circonférence et de l'axe. Les équations de vibrations dérivent, en accord avec la théorie de Donnell, des composantes de déplacement de la surface moyenne de la coque ouverte. On obtient des solutions exactes pour les coques cylindriques ouvertes posées librement et on présente des résultats numériques pour les coques cylindriques ouvertes composées de fines lamelles croisées antisymétriques et non symétriques.

     

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.     

3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

Hamiltonian constants for several new entire solutions

Using the Hamiltonian identities and the corresponding Hamiltonian constants for entire solutions of elliptic partial differential equations, we investigate several new entire solutions whose existence were shown recently, and show interesting properties of the solutions such as formulas for contact angles at infinity of concentration curves.      

Small free vibrations of an infinite cylindrical shell rotating on rollers

Small free vibrations of an infinitely long rotating cylindrical shell being in contact with rigid cylindrical rollers are considered. A system of linear differential equations for the vibrations of such a shell is derived. By using the Fourier transform of the solutions in the circumferential coordinate, a system of algebraic equations for approximately determining the vibration frequencies and mode shapes is obtained. It is shown that, for any number n of uniformly distributed rollers, the approximate values of the first n frequencies and mode shapes can be found explicitly. On the basis of the orthogonal sweep method, an algorithm for numerically solving the boundary value eigenvalue problem describing the vibrations of a rotating shell is developed. Analytical and numerical results are compared. The obtained approximate formulas for frequencies and the numerical algorithm can be used to design centrifugal concentrators for ore enrichment.     

Computation of worst geometric imperfection profiles of composite cylindrical shell panels by minimizing the non-linear buckling load

In this article the “most unfavorable” shape of initial geometric imperfection profile for laminated cylindrical shell panel is obtained analytically by minimizing the limit point load. The partial differential equations governing the shell stability problem are reduced to a set of non-linear algebraic equations using Galerkin's technique. The non-linear equilibrium path is traced by employing Newton–Raphson method in conjunction with the Riks approach. A double Fourier series is used to represent the initial geometric imperfection profile for the cylindrical shell panel. The optimum values of these Fourier coefficients are determined by minimizing the limit point load using genetic algorithm. The results are determined for simply supported composite cylindrical shell panel. Numerical results show that more number of terms is needed in Fourier series representation to obtain the “worst” geometric imperfection profile which gives lower limit load compared to single term representation of imperfection. We have incorporated constraints on the shape of imperfection to avoid unrealistic limit point loads (due to imperfection shape) as we have assumed that the imperfection is due to machining/manufactuting.     

Boundary resonance in a cylindrical shell with free ends in the ring directions

It is shown that to describe the phenomenon of localization of oscillations in a neighborhood of the end-wall of a semi-infinite cylindrical shell with free end-walls in the ring direction we can use simplified equations, which makes it possible to obtain an approximate analytic solution.Translated from Dinamicheskie Sistemy, No. 8, pp. 45–46, 1989.     

Analytic-non-integrability of an integrable analytic Hamiltonian system

We introduce the polynomial Hamiltonian and we prove that the associated Hamiltonian system is Liouville-C^∞-integrable, but fails to be real-analytically integrable in any neighbourhood of an equilibrium point. The proof only uses power series expansions, and is elementary.     

Nonlinear vibration and instability of embedded double-walled boron nitride nanotubes based on nonlocal cylindrical shell theory

This paper investigates the nonlinear vibration and instability of the embedded double-walled boron nitride nanotubes (DWBNNTs) conveying viscous fluid based on nonlocal piezoelasticity cylindrical shell theory. The elastic medium is simulated as Winkler–Pasternak foundation, and adjacent layers interactions are assumed to have been coupled by van der Walls (vdW) force evaluated based on the Lennard–Jones model. The nonlinear strain terms based on Donnell’s theory are taken into account. The Hamilton’s principle is employed to obtain coupled differential equations, containing displacement and electric potential terms. Differential quadrature method (DQM) is applied to estimate the nonlinear frequency and critical fluid velocity for clamped supported mechanical and free electric potential boundary conditions at both ends of the DWBNNTs. Results indicated that some parameters including nonlocal parameter, elastic medium’s modulus, aspect ratio and vdW force have significant influence on the vibration and instability of the DWBNNT while the fluid viscosity effect is negligible. In addition, the low aspect ratio should be taken into account for DWBNNT in optimum design of nano/micro devices.     

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.     

Speed of Arnold diffusion for analytic Hamiltonian systems

For a convex, real analytic, ε-close to integrable Hamiltonian system with n≥5 degrees of freedom, we construct an orbit exhibiting Arnold diffusion with the diffusion time bounded by exp(Ce^{-\frac12(n-2)})\exp(C\epsilon^{-\frac{1}{2(n-2)}}). This upper bound of the diffusion time almost matches the lower bound of order exp(e^{-\frac12(n-1)})\exp(\epsilon ^{-\frac{1}{2(n-1)}}) predicted by the Nekhoroshev-type stability results. Our method is based on the variational approach of Bessi and Mather, and includes a new construction on the space of frequencies.     

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton's principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity, length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich structures conveying fluid flow are used as sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation.     

Periodic solutions for a class of new superquadratic second order Hamiltonian systems

A new superquadratic growth condition is introduced, which is an extension of the well-known superquadratic growth condition due to P.H. Rabinowitz and the nonquadratic growth condition due to Gui-Hua Fei. An existence theorem is obtained for periodic solutions of a class of new superquadratic second order Hamiltonian systems by the minimax methods in critical point theory, specially, a local linking theorem.     

Hamiltonian structures of the generalized cylindrical KdV equations

Hamiltonian structures of the cylindrical Korteweg-deVries equation and its higher order equations are found. The connection between the generalized C-KdV equation and nonisospectral problem is pointed out.Projects supported by the Science Fund of the Chinese Academy of Sciences.     

A new approach to free vibration analysis using boundary elements

A boundary element method for the analysis of free vibrations in solid mechanics is developed using a non-standard boundary integral approach. It is shown that, utilizing the statical fundamental solution and employing a special class of coordinate functions, the algebraic eigenvalue problem results. The method has been realized numerically for the two-dimensional elastodynamics, and a number of examples demonstrating its accuracy have been included.     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.     

Nonlinear free vibration analysis of SSMFG cylindrical shells resting on nonlinear viscoelastic foundation in thermal environment

In this paper, a semi-analytical method for the free vibration behavior of spiral stiffened multilayer functionally graded (SSMFG) cylindrical shells under the thermal environment is investigated. The distribution of linear and uniform temperature along the direction of thickness is assumed. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness. The cylindrical shell has three layers consist of ceramic, FGM, and metal in two cases. In the first model i.e. Ceramic-FGM-Metal (CFM), the exterior layer of the cylindrical shell is rich ceramic while the interior layer is rich metal and the functionally graded material is located between these layers and the material distribution is in reverse order in the second model i.e. Metal-FGM-Ceramic (MFC). The material constitutive of the stiffeners is continuously changed through the thickness. Using the Galerkin method based on the von Kármán equations and the smeared stiffeners technique, the problem of nonlinear vibration has been solved. In order to find the nonlinear vibration responses, the fourth order Runge–Kutta method is utilized. The results show that the different angles of stiffeners and nonlinear elastic foundation parameters have a strong effect on the vibration behaviors of the SSMFG cylindrical shells. Also, the results illustrate that the vibration amplitude and the natural frequency for CFM and MFC shells with the first longitudinal and third transversal modes (m = 1, n = 3) with the stiffeners angle θ = 30°, β = 60° and θ = β = 30° is less than and more than others, respectively.