Nonlinear Vibrations Of Cylindrical Shells Research Articles

Nonlinear coupled multi-mode vibrations of simply-supported cylindrical shells: Comparison studies

In spite of extensive studies on the nonlinear vibrations of cylindrical shells, the significant influences of some effective modes and the in-plane nonlinearity on the nonlinear coupled multi-mode vibrations are not clear. In this work, which is based on Donnell’s nonlinear shell theory and Amabili–Reddy’s third-order shear deformation theory, the nonlinear differential equations of motion of both the thin-walled and moderately thick cylindrical shells under the simply-supported boundary condition are established via Lagrange equations. These high-dimensional differential equations with the quadratic and cubic nonlinearities are solved by using an iteration procedure that is a combination of incremental harmonic balance method, pseudo-arclength method and extrapolation techniques. Turning and bifurcation points of the system are determined with the help of the direct method, the stability of solution of the frequency–response is examined by using the multi-variable Floquet theory. The numerical responses obtained by adopting two present shell theories are compared to investigate the influence of ignoring the in-plane nonlinearity on nonlinear vibrations. In the coupled multi-mode vibrations with respect to the fundamental mode m,n, apart from the regular modes i×m,j×ni=1,3,5,…,andj=0,1,2,3,… that include the axisymmetric and asymmetric modes, some irregular modes are taken into account to study the frequency–responses. Results show that present iteration procedure is efficient and successful to get the frequency–responses of the coupled multi-mode vibrations, and the influence of the irregular mode on the coupled multi-mode vibration is dependent on the relationship between the natural frequency of the irregular mode and that of the fundamental mode.

Характер нелинейных колебаний цилиндрических оболочек, обтекаемых сверхзвуковым потоком газа

Характер нелинейных колебаний цилиндрических оболочек, обтекаемых сверхзвуковым потоком газа

Non-linear free vibration analysis of laminated cylindrical shells under static axial loading including accurate satisfaction of boundary conditions

The effect of a static preload on the non-linear vibration behaviour of laminated anisotropic shells is studied. Starting point of the analysis is a perturbation approach developed earlier to analyse the non-linear vibration behaviour of imperfect general structures under static preloading. This method is based on a perturbation expansion for both the frequency parameter and the dependent variables, and can be used to study the effects of geometric imperfections, a static fundamental state, and a non-trivial static state on the linearised and non-linear vibrations of cylindrical shells. The theory has been incorporated within a semi-analytical framework, based on Donnell-type governing equations.The approach is applied in the non-linear vibration analysis of laminated, anisotropic cylindrical shells. Special characteristics of the non-linear free vibration behaviour of statically, axially loaded cylindrical shells are shown for a specific composite shell. Internal resonances corresponding to interactions between first order modes and second order modes are found in the analysis. Novel aspect of the current work is that these phenomena are considered within a framework in which effects of various types of boundary conditions, including the effect of non-linear static state deformation due to the application of the static load, are accurately taken into account.

An analytical study of the non-linear vibrations of cylindrical shells

The primary resonance response of simply supported circular cylindrical shells is investigated using the perturbation method. Donnell's non-linear shallow-shell theory is used to derive the governing partial differential equations of motion. The Galerkin technique is then employed to transform the equations of motion into a set of temporal ordinary differential equations. Considering only the primary resonance case, the method of multiple scales is used to study the periodic solutions and their stability. The necessary and sufficient conditions for appearance of the so-called companion mode are also discussed. To this end, a range of the possible multi-mode solution is obtained for response and excitation amplitudes and also excitation frequency as a function of damping, geometry and material properties of the shell. Parametric studies are performed to illustrate the effect of different values of thickness, length and material composition on the possibility of the companion mode participation in primary resonance response.

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.

Nonlinear modes and traveling waves of parametrically excited cylindrical shells

Donnel's equations are used to predict nonlinear vibrations of cylindrical shells, which are excited by parametric dynamical load. A multi-degree-of-freedom dynamical system of cylindrical shells is derived. The nonlinear modes of the parametrically excited system are treated. The analyses have been carried out both with and without dissipation, using the Harmonic Balance Method. These nonlinear modes correspond to the standing waves in the shell. Traveling waves are also analyzed in detail. We come to the conclusion that the behavior of the nonlinear modes and the traveling waves are similar.

Nonlinear vibrations and imperfection sensitivity of a cylindrical shell containing axial fluid flow

The high imperfection sensitivity of cylindrical shells under static compressive axial loads is a well-known phenomenon in structural stability. On the other hand, less is known of the influence of imperfections on the nonlinear vibrations of these shells under harmonic axial loads. The aim of this work is to study the simultaneous influence of geometric imperfections and an axial fluid flow on the nonlinear vibrations and instabilities of simply supported circular cylindrical shells under axial load. The fluid is assumed to be non-viscous and incompressible and the flow to be isentropic and irrotational. The behavior of the thin-walled shell is modeled by Donnell's nonlinear shallow-shell equations. It is subjected to a static uniform compressive axial pre-load plus a harmonic axial load. A low-dimensional modal expansion, which satisfies the relevant boundary and continuity conditions, and takes into account all relevant nonlinear modal interactions observed in the past in the nonlinear vibrations of cylindrical shells with and without flow is used together with the Galerkin method to derive a set of eight coupled nonlinear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The shell is considered to be initially at rest, in a position corresponding to a pre-buckling configuration. Then, a harmonic excitation is applied and conditions for parametric instability and dynamic snap-through are sought. The results clarify the marked influence of geometric imperfections and fluid flow on the dynamic stability boundaries, bifurcations and basins of attraction.

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

The Problem of Forced Nonlinear Vibrations of Cylindrical Shells Completely Filled with Liquid

The forced nonlinear vibrations of a thin cylindrical shell completely filled with a liquid are studied. A refined mathematical model is used. The model takes into account the nonlinear terms up to the fifth power of the generalized displacement of the shell. The Bogolyubov’Mitropolsky averaging method is used to plot amplitude’frequency response curves for steady-state vibrations. The steady-state vibrations at the frequency of principal harmonic resonance are analyzed for stability

A comparison of analytical–numerical models for nonlinear vibrations of cylindrical shells

The nonlinear flexural vibration behaviour of cylindrical shells has received considerable attention to date. It is pointed out that, although in a well-known reference case there seems to be a reasonable agreement, there are unresolved discrepancies between the results obtained by different authors. In the present paper, the problem is studied using various analytical–numerical models with different levels of accuracy and complexity. The frequency–amplitude curves from the different analysis models developed are compared both for isotropic shells and for an orthotropic composite shell. Secondary modes can play an important role. In more complicated cases modal interactions may significantly influence the nonlinear vibration behaviour, and the results obtained strongly depend on the analysis model chosen.

Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method

The nonlinear (large-amplitude) response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of their lowest natural frequencies is investigated. The shell is assumed to be completely filled with an incompressible and inviscid fluid at rest. Donnell's nonlinear shallow-shell theory is used, and the solution is obtained by the Galerkin method. The proper orthogonal decomposition (POD) method is used to extract proper orthogonal modes that describe the system behaviour from time-series response data. These time series have been obtained via the conventional Galerkin approach (using normal modes as a projection basis) with an accurate model involving 16 degrees of freedom, validated in previous studies. The POD method, in conjunction with the Galerkin approach, permits a lower-dimensional model as compared to those obtainable via the conventional Galerkin approach. Different proper orthogonal modes computed from time series at different excitation frequencies are used and solutions are compared. Some of these sets of modes are capable of describing the system behaviour over the whole frequency range around the fundamental resonance with good accuracy and with only 3 degrees of freedom. They allow a drastic reduction in the computational effort, as compared to using the 16 degree-of-freedom model necessary when the conventional Galerkin approach is used.

Forced Nonlinear Vibrations of Cylindrical Shells Filled with a Liquid

Forced nonlinear vibrations of cylindrical shells partially filled with a liquid are studied. The wave motions of the free liquid surface are taken into account. Regular periodic deformation modes of the shell are studied, and frequency ranges where these modes do not occur are determined

Study of nonlinear vibrations of cylindrical shells with allowance for energy dissipation

The problem of induced nonlinear harmonic vibrations of a cylindrical shell is solved with allowance for energy dissipation by using a binomial approximation of the displacements. There are several distinctive features of the behavior of the shell due to interaction of two modes of vibration.

Resonance effects in the nonlinear vibrations of cylindrical shells containing a liquid

Resonance effects in the nonlinear vibrations of cylindrical shells containing a liquid

Effect of initial camber on natural nonlinear vibrations of cylindrical shells

Effect of initial camber on natural nonlinear vibrations of cylindrical shells

Resonance phenomena in nonlinear vibrations of cylindrical shells with initial imperfections

Resonance phenomena in nonlinear vibrations of cylindrical shells with initial imperfections

Non-linear vibrations of cylindrical shells of varying thickness in an incompressible fluid

Non-linear transverse vibrations of elastic, orthotropic cylindrical shells of linearly varying thickness and embedded in an incompressible fluid are analyzed by using the Rayleigh-Ritz procedure. Numerical results are presented for various types of boundary conditions and taper ratios.