Nonlinear modes of cylindrical panels with complex boundaries. <Emphasis Type="Italic">R</Emphasis>-function method

Nonlinear vibrations of cylindrical panels with complex base are analyzed. The Donnell-Mushtari-Vlasov equations with respect to displacements are used to study vibrations of shallow shell with geometrical nonlinearity. R-function method is applied to satisfy the panel boundary conditions. The Rayleigh-Ritz method is used to obtain the linear vibrations eigenmodes, which contain R-function. The nonlinear vibrations of panel are expanded by using these eigenmodes. The harmonic balance method and nonlinear normal modes are used to study the free nonlinear vibrations.     

Harmonic Vibrations of Shells with Cutouts Under Kinematic Conditions at Their Periphery

Problems on harmonic vibrations of shells with openings are considered for the case where displacements and angles of turn are specified at their periphery. A technique for determination of the natural frequencies and modes of vibrations is stated. The technique replaces the unknown peripheral forces and moments by a system of local statically equivalent loads. The intensity of these loads is found from the boundary conditions at the opening periphery. The solutions of the equations of shell theory for local loads are constructed by expanding the sought-for quantities into series in terms of the natural modes of the shell without openings. The performance of the approach developed is illustrated by calculating the natural frequencies and modes of shallow shells with a rectangular planform and various Gaussian curvatures     

Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells

A general approach, based on shearable shell theory, to predict the influence of geometric non-linearities on the natural frequencies of an elastic anisotropic laminated cylindrical shell incorporating large displacements and rotations is presented in this paper. The effects of shear deformations and rotary inertia are taken into account in the equations of motion. The hybrid finite element approach and shearable shell theory are used to determine the shape function matrix. The analytical solution is divided into two parts. In part one, the displacement functions are obtained by the exact solution of the equilibrium equations of a cylindrical shell based on shearable shell theory instead of the usually used and more arbitrary interpolating polynomials. The mass and linear stiffness matrices are derived by exact analytical integration. In part two, the modal coefficients are obtained, using Green's exact strain-displacement relations, for these displacement functions. The second- and third-order non-linear stiffness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. Comparison with available results is satisfactorily good.     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

Resonant many-mode periodic and chaotic self-sustained aeroelastic vibrations of cantilever plates with geometrical non-linearities in incompressible flow

The plates interacting with inviscid, incompressible, potential gas flow are analyzed. Many modes interaction is considered to describe self-sustained vibrations of plates. The singular integral equation is solved to obtain gas pressures acting on the plate. The Von Karman equations with respect to three displacements are used to describe the plate geometrical non-linear vibrations. The Galerkin method is applied to each partial differential equation to obtain the finite-degree-of-freedom model of the plate vibrations. Self-sustained vibrations, which take place due to the Hopf bifurcation, are investigated. These vibrations undergo the Naimark?CSacker bifurcation and the periodic motions are transformed into the almost periodic ones. If the stream velocity is increased, almost periodic motions are transformed into chaotic ones. As a result of the internal resonance, the saturation of the vibration mode is observed. The non-linear dynamics of low- and high-aspect-ratio plates is analyzed.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

Non-linear vibrations of doubly curved shallow shells

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain-displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.     

An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells

Using Donnell non-linear shallow shell equations in terms of the displacements and the potential flow theory, this work presents a qualitatively accurate low dimensional model to study the non-linear dynamic behavior and stability of a fluid-filled cylindrical shell under lateral pressure and axial loading. First, the reduced order model is derived taking into account the influence of the driven and companion modes. For this, a modal solution is obtained by a perturbation technique which satisfies exactly the in-plane equilibrium equations and all boundary, continuity, and symmetry conditions. Finally, the equation of motion in the transversal direction is discretized by the Galerkin method. The importance of each mode in the proposed modal expansion is studied using the proper orthogonal decomposition. The quality of the proposed model is corroborated by studying the convergence of frequency–amplitude relations, resonance curves, bifurcation diagrams, and time responses. The parametric analysis clarifies the influence of the lateral and axial loads on the non-linear vibrations and stability of the liquid-filled shell. Finally, the global response of the system is investigated in order to quantify the degree of safety of the shell in the presence of external perturbations through the use of bifurcation diagrams and basins of attraction. This allows one to evaluate the safety and dynamic integrity of the cylindrical shell in a dynamic environment.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

This paper discusses the derivation of discrete low-dimensional models for the non-linear vibration analysis of thin shells. In order to understand the peculiarities inherent to this class of structural problems, the non-linear vibrations and dynamic stability of a circular cylindrical shell subjected to dynamic axial loads are analyzed. This choice is based on the fact that cylindrical shells exhibit a highly non-linear behavior under both static and dynamic axial loads. Geometric non-linearities due to finite-amplitude shell motions are considered by using Donnell’s nonlinear shallow shell theory. A perturbation procedure, validated in previous studies, is used to derive a general expression for the non-linear vibration modes and the discretized equations of motion are obtained by the Galerkin method. The responses of several low-dimensional models are compared. These are used to study the influence of the modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses of the shell. It is shown that rather low-dimensional and properly selected models can describe with good accuracy the response of the shell up to very large vibration amplitudes.     

On effect of initial imperfections on parametric vibrations of cylindrical shells with geometrical non-linearity

The effect of initial imperfections on the parametric vibrations of cylindrical shells is analyzed. The shell has moderate amplitudes of vibrations; therefore, geometrically nonlinear theory is used. The shell vibrations are described by the Donnel equations. The interaction of three pairs of conjugate modes is considered in the analysis. Therefore, the shell vibrations are described by six-degrees-of-freedoms nonlinear dynamical system. The multiple scales method and the continuation technique are used to analyze the system dynamics. The role of initial imperfections in nonlinear dynamics of shell is discussed using frequency responses.     

Non-linear free periodic vibrations of variable stiffness composite laminated plates

Steady-state free vibrations, with large amplitude displacements, of variable stiffness composite laminated plates (VSCL) are analysed. The intentions of this research are: (1)?to find out how the natural frequencies and (mode) shapes evolve with the displacement amplitude in this new type of laminated composite material; (2)?to describe modal interactions in VSCL due to energy interchanges under the coupling induced by non-linearity; (3)?to compare the VSCL with traditional, constant stiffness, laminated plates. The VSCL of interest here have curvilinear fibres and the numerical analysis carried out is based on a recently developed p-version finite element with hierarchic basis functions. The element follows first-order shear deformation theory and considers Von Kármán??s non-linear terms. The time domain equations of motion are first reduced using the linear modes of vibration and then transformed to the frequency domain via the harmonic balance method. These frequency domain equations are solved by an arc-length continuation method.     

Finite amplitude vibrations of an orthotropic circular plate with an isotropic core

The free and forced non-linear vibrations of a fixed orthotropic circular plate, with a concentric core of isotropic material, are studied. Existence of harmonic vibrations is assumed and thus the time variable is eliminated by a Ritz-Kantorovich method. Hence, the governing non-linear partial equations for the axisymmetric vibration of the composite circular plate are reduced to a set of ordinary differential equations which form a non-linear eigen-value problem. Solutions are obtained by utilizing the related initial-value problems in conjunction with Newton's integration method. The results reveal the effects of finite amplitude and anisotropy of materials upon the dynamic responses. Further, the method developed in this paper, which is used to solve the title problem, is one of some generality. It can be applied to many differential eigenvalue problems with piecewise continuous functions.     

Second order hyperbolic equations with small non-linearities in the case of internal resonance

The monofrequent solutions of certain autonomous second order hyperbolic differential equations with weak non-linearities are found in the case when some of the natural frequencies of the generating equation are in integral ratio. The approach use is a development of the KrylovBogoliubov-Mitropolskii method. The solution found is applied to the case of the longitudinal vibrations of a rod for which the stress-strain relation contains a small non-linear term and to the case of the vibrations of a rod with small inhomogeneities of density and elastic modulus and with small damping.     

Spectral Problem for Shells with Fluid

Variational eigenvalue equations describing vibrations of orthotropic shells containing an ideal incompressible fluid are obtained. The vibration frequencies are assumed to be small, which makes it possible to use linear equations and to consider the boundary of the wet surface of the shell to be unchanged. The equations of anisotropic shells are based on the linear relations of multifield theory, which allows to obtain a more accurate model of anisotropic shells that satisfies the conditions of the finite-element method. The fluid flow is considered irrotational and is described using the Laplace equation. A finite-element algorithm is designed to determine the natural frequencies and modes of vibrations of an arbitrary multilayer orthotropic shell of revolution which is partially filled with an ideal incompressible fluid. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 128–135, November–December, 2005.     

Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

Nonlinear vibrations of cylindrical shells filled with a fluid and subjected to longitudinal and transverse periodic excitation

The paper proposes an approach to studying the nonlinear vibrations of thin cylindrical shells filled with a fluid and subjected to a combined transverse–longitudinal load. Methods of nonlinear mechanics are used to find and analyze periodic solutions of the system of equations that describes the dynamic behavior of the shell when the natural frequencies of the shell and the frequencies of both periodic forces are in resonance relations.     

Non-linear vibrations of imperfect free-edge circular plates and shells

Large-amplitude, geometrically non-linear vibrations of free-edge circular plates with geometric imperfections are addressed in this work. The dynamic analog of the von Kármán equations for thin plates, with a stress-free initial deflection, is used to derive the imperfect plate equations of motion. An expansion onto the eigenmode basis of the perfect plate allows discretization of the equations of motion. The associated non-linear coupling coefficients for the imperfect plate with an arbitrary shape are analytically expressed as functions of the cubic coefficients of a perfect plate. The convergence of the numerical solutions are systematically addressed by comparisons with other models obtained for specific imperfections, showing that the method is accurate to handle shallow shells, which can be viewed as imperfect plate. Finally, comparisons with a real shell are shown, showing good agreement on eigenfrequencies and mode shapes. Frequency-response curves in the non-linear range are compared in a very peculiar regime displayed by the shell with a 1:1:2 internal resonance. An important improvement is obtained compared to a perfect spherical shell model, however some discrepancies subsist and are discussed.     

Free vibrations of a viscoelastic noncircular cylindrical shell under homogeneous axial loading

Free low-frequency vibrations of a noncircular cylindrical shell subject to an axial static load are investigated. The case where the vibrations are localized near a certain generatrix, called the weakest, is considered. The complex WKB-method is used to determine the natural modes of vibrations that damp with time and drop down far from the weakest generatrix. An example on the vibration of a viscoelastic shell is considered with the relaxation rate function given and in the presence of a static axial force. State Polytechnic Academy of Belarus, Minsk. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 68–74, November, 1999.