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.     

Nonlinear vibrations of a sandwich piezo-beam system under piezoelectric actuation

The paper describes the use of active structures technology for deformation and nonlinear free vibrations control of a simply supported curved beam with upper and lower surface-bonded piezoelectric layers, when the curvature is a result of the electric field application. Each of the active layers behaves as a single actuator, but simultaneously the whole system may be treated as a piezoelectric composite bender. Controlled application of the voltage across piezoelectric layers leads to elongation of one layer and to shortening of another one, which results in the beam deflection. Both the Euler–Bernoulli and von Karman moderately large deformation theories are the basis for derivation of the nonlinear equations of motion. Approximate analytical solutions are found by using the Lindstedt–Poincaré method which belongs to perturbation techniques. The method makes possible to decompose the governing equations into a pair of differential equations for the static deflection and a set of differential equations for the transversal vibration of the beam. The static response of the system under the electric field is investigated initially. Then, the free vibrations of such deformed sandwich beams are studied to prove that statically pre-stressed beams have higher natural frequencies in regard to the straight ones and that this effect is stronger for the lower eigenfrequencies. The numerical analysis provides also a spectrum of the amplitude-dependent nonlinear frequencies and mode shapes for different geometrical configurations. It is demonstrated that the amplitude–frequency relation, which is of the hardening type for straight beams, may change from hard to soft for deformed beams, as it happens for the symmetric vibration modes. The hardening-type nonlinear behaviour is exhibited for the antisymmetric vibration modes, independently from the system stiffness and dimensions.

     

Chaotic vibrations of spherical and conical axially symmetric shells

Summary Chaotic vibrations of deterministic, geometrically nonlinear, elastic, spherical and conical axially summetric shells, subject to sign-changing transversal load using the variational principle, are analysed. The paper is motivated by an observation that variational equations of the hybrid type are suitableto solve many dynamical problems of the shells theory. It is assumed that the shell material is isotropic, and the Hook's principle holds. Intertial forces in directions tangent to mean shell surface and rotation inertia of a normal shell cross section are neglected. A transition form PDEs to ODEs (the Cauchy problem) is realized through the Ritz procedure. Next, the Cauchy problem is solved using the fourth-order Runge-Kutta method. Qualitative and quantitative analysis is carried out in the frame of both nonlinear dynamics and quantitative theory of differential equations. New scenarios from harmonic to chaotic dynamics are detected. Various vibration forms development versus control parameters (rise of arc; amplitude and frequency of the exciting force and number of vibrational modes accounted) are illustrated and discussed.     

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 with initial imperfections in a supersonic flow

The paper studies the dynamics of nonlinear elastic cylindrical shells using the theory of shallow shells. The aerodynamic pressure on the shell in a supersonic flow is found using piston theory. The effect of the flow and initial deflections on the vibrations of the shell is analyzed in the flutter range. The normal modes of both perfect shells in a flow and shells with initial imperfections are studied. In the latter case, the trajectories of normal modes in the configuration space are nearly rectilinear, only one mode determined by the initial imperfections being stable __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 63–73, September 2007.     

Natural vibrations of corrugated orthotropic shells of revolution

The torsional and longitudinal–flexural vibrations of corrugated orthotropic shells are investigated. Relations, including the equations of motion in forces and moments and Hooke’s relations, are obtained using the Kirchhoff–Love hypotheses. The influence of the geometric parameters of the shell (corrugation amplitude and length) on the eigenfrequencies and natural vibrations modes is studied for fixed-end shells. It is found that during torsional vibrations, increasing the corrugation amplitude and increasing the number of corrugations leads to a decrease in the resonant frequencies. In the case of torsional and longitudinal–flexural vibrations, the influence of the corrugation amplitude on the natural vibration modes is investigated.     

圆柱壳的非线性自由振动分析

本文分析了各向同性封闭圆柱壳的非线性自由振动。文中采用经典的非线性弹性力学方法推导了圆柱壳的大振幅运动方程,这些方程的静态形式与冯·卡门的板理论方程具有同样的精度。文中讨论了四种基本振动模态,并且还以数学公式的形式给出了一般的最终结果,一些例子以曲线给出结果,并进行了比较。结果还表明线性振动可以作为非线性振动的一种特例。     

Natural vibrations of truncated conical shells with free edges

Conclusions A truncated conical shell with free edges can vibrate at resonance both without and with one half-wave along the generatrix. For m=0, the amplitude decreases from the large end faces of the shell toward the small end faces. For m=1, the vibration amplitude decreases from the end faces toward the central portion of the shell. Overlapping of the natural frequencies of various mode shapes of vibration leads to a complex pattern of the node distribution over the shell surface.The author agrees with Watkins and Clary's [5] remarks concerning the need for a new analytical method of solving dynamic problems for cylindrical and conical shells with allowance for the peculiar features of vibrations for various boundary conditions.Kazan Aviation Institute. Translated from Prikladnaya Mekhanika, Vol. 5, No. 1, pp. 113–117, January, 1969.     

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.     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.     

Nonlinear vibrations of orthotropic shallow shells of revolution

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated.following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.     

Large flexural vibrations of thermally stressed layered shallow shells

A dynamic nonlinear theory for layered shallow shells is derived by means of the von Karman-Tsien theory, modified by the generalized Berger-approximation. Moderately thick shells with polygonal planform composed of multiple perfectly bonded layers are considered. The shell edges are assumed to be prevented from in-plane motions and are simply supported. A distributed lateral force loading is applied to the structure, and additionally, the influence of a static thermal prestress, corresponding to a spatial distribution of cross-sectional mean temperature, is taken into account. In the special case of laminated shells made of transversely isotropic layers with physical properties symmetrically distributed about the middle surface, a correspondence to moderately thick homogeneous shells is found. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic as well as quadratic nonlinearities. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the perturbation method of multiple scales. A unifying non-dimensional representation of the nonlinear frequency response function is presented that is independent of the special shell planform.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

Generalized non-linear free vibrations of prestressed plates and shells

This paper investigates the non-linear free vibration of prestressed plates and shells in a general form. The analysis includes the effects of in-plane inertia. The analysis is based on the non-linear equations of motion and uses a perturbation procedure. No assumption is made a priori for the form of the time or space mode. The boundary conditions are treated in a general manner including boundary conditions where non-linear stress resultants are specified. The method is illustrated by three examples.     

Forced axisymmetric vibrations and self-heating of thermoviscoelastic cylindrical shells with piezoelectric actuators

The coupled problem of the forced axisymmetric vibrations and self-heating of electrothermoviscoelastic cylindrical shells with piezoceramic actuators under monoharmonic electromechanical loading is solved. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The coupled nonlinear problem of electrothermoelasticity is solved by using a time-marching method with discrete orthogonalization at each time step (to integrate the equations of elasticity) and an explicit finite-difference method (to solve the heat-conduction equations). An analysis is made of the effect of the boundary conditions at the shell ends, the dimensions of the piezoactuator, and the self-heating temperature on the actuator voltage and the effectiveness of active damping of the forced vibrations of the shell under uniform transverse monoharmonic pressure     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

大跨度双层网壳的非线性动态响应

网壳结构在大跨度结构中得到广泛应用．在建立了双层网格扁球壳的非线性强迫振动微分方程的基础上,研究了在边缘滑动固定的边界条件下,双层网格扁球壳的非线性动态响应问题．用突变理论建立了该网壳的尖点突变模型,得出了突变的临界方程,并阐述了网壳参数对该结构动态屈曲的影响．     

Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads

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