Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

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.     

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.     

Large free vibrations of a beam carrying a moving mass

The focus of this work is to develop a technique to obtain numerical solution over a long range of time for non-linear multi-body dynamic systems undergoing large amplitude motion. The system considered is an idealization of an important class of problems characterized by non-linear interaction between continuously distributed mass and stiffness and lumped mass and stiffness. This characteristic results in some distinctive features in the system response and also poses significant challenges in obtaining a solution.

In this paper, equations of motion are developed for large amplitude motion of a beam carrying a moving spring–mass. The equations of motion are solved using a new approach that uses average acceleration method to reduce non-linear ordinary differential equations to non-linear algebraic equations. The resulting non-linear algebraic equations are solved using an iterative method developed in this paper. Dynamics of the system is investigated using a time-frequency analysis technique.     

Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

This paper presents an analytical approach to investigate the non-linear axisymmetric response of functionally graded shallow spherical shells subjected to uniform external pressure incorporating the effects of temperature. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. Equilibrium and compatibility equations for shallow spherical shells are derived by using the classical shell theory and specialized for axisymmetric deformation with both geometrical non-linearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is assumed and explicit expressions of buckling loads and load-deflection curves are determined due to Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of the shells.     

Global structural behaviour of thin and moderately thick monoclinic spherical shells using a mixed shear deformation model

Summary The static and dynamic responses of anisotropic spherical shells under a uniformly distributed transverse load are investigated. Analytical solutions using the mixed variational formulation are presented for spherical shells subjected to various boundary conditions. Numerical results of a refined mixed first-order shear deformation theory for natural frequencies, critical buckling, center deflections and stresses are compared with those obtained using the classical shell theory. A variety of simply-supported and clamped boundary conditions are considered and comparisons with the existing literature are made. The sample numerical results presented herein for global structural behaviour of monoclinic spherical shells should serve as references for future comparisons.     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

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 in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.     

Non-linear modal analysis of a rotating beam

The free non-linear vibration of a rotating beam has been considered in this paper. The von Karman strain-displacement relations are implemented. Non-linear equations of motion are obtained by Hamilton’s principle. Results are obtained by applying the method of multiple scales to a set of discretized ordinary differential equations which obtained by using the Galerkin discretization method. This set contains coupling between transverse and axial displacements as quadratic and cubic geometric non-linearities. Non-linear normal modes and non-linear natural frequencies with or without internal resonance are observed. In the internal resonance case, the internal resonance between two transverse modes and between one transverse and one axial mode are explored. Obtained results in this study are compared with those obtained from literature. The stability and some dynamic characteristics of the non-linear normal modes such as the phase portrait, Poincare section and power spectrum diagrams have been inspected. It is shown that, for the first internal resonance case, the beam has one stable or degenerate uncoupled mode and either: (a) one stable coupled mode, (b) one unstable coupled mode, (c) two stable and one unstable coupled modes, (d) three stable coupled modes, and (e) one stable coupled mode. On the other hand, for the second internal resonance case, the beam has one stable or unstable or degenerate uncoupled mode and either: (a) two stable coupled modes, (b) two unstable coupled modes, and (c) one stable coupled mode depending on the parameters.     

Transmural pressure effects on the stability of clamped cylindrical shells subjected to internal fluid flow: Theory and experiments

An experimental and theoretical parametric study is undertaken to investigate the effect of transmural pressure on the non-linear dynamics and stability of circular cylindrical shells with clamped ends subjected to internal fluid flow. The theoretical structural model is based on the Donnell non-linear shallow shell theory, and potential flow theory is employed to describe the fluid-structure interaction. It is found that, for low transmural pressures in the range investigated, the shell loses stability by static subcritical divergence, while for higher transmural pressures the loss of stability is supercritical. In addition, there are ranges of flow velocity in which the shell exhibits quasi-periodic or even chaotic behaviour.     

Experiments on the impact of cylindrical projectiles on thin spherical caps

The normal impact of a blunt cylindrical projectile on the apex of a thin spherical aluminium cap which caused large permanent displacements was experimentally investigated. Dynamic strain gauge measurements were carried out and the final deflection profile was measured. It was found that reversed plastic flow occurred throughout the deformation process.The project was supported by the Science Foundation of Shanxi Provice, China     

Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions

The effect of non-linear magnetic forces on the non-linear response of the shaft is examined for the case of superharmonic resonance in this paper. It is shown that the steady-state superharmonic periodic solutions lose their stability by either saddle-node or Hopf bifurcations. The system exhibits many typical characteristics of the behavior of non-linear dynamical systems such as multiple coexisting solutions, jump phenomenon, and sensitive dependence on initial conditions. The effects of the feedback gains and imbalance eccentricity on the non-linear response of the system are studied. Finally, numerical simulations are performed to verify the analytical predictions.     

Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses

The available accurate shell theories satisfy the interlaminar transverse stress continuity conditions based on linear strain-displacement relations. Furthermore, in majority of these theories, either influence of the transverse normal stress and strain or the transverse flexibility of the shell has been ignored. These effects remarkably influence the non-linear behavior of the shells especially in the postbuckling region. Furthermore, majority of the buckling analyses performed so far for the laminated composite and sandwich shells have been restricted to linear, static analysis of the perfect shells. Moreover, almost all the available shell theories have employed the Love-Timoshenko assumption, which may lead to remarkable errors for thick and relatively thick shells. In the present paper, a novel three-dimensional high-order global-local theory that satisfies all the kinematic and the interlaminar stress continuity conditions at the layer interfaces is developed for imperfect cylindrical shells subjected to thermo-mechanical loads.In comparison with the layerwise, mixed, and available global-local theories, the present theory has the advantages of: (1) suitability for non-linear analyses, (2) higher accuracy due to satisfying the complete interlaminar kinematic and transverse stress continuity conditions, considering the transverse flexibility, and releasing the Love-Timoshenko assumption, (3) less required computational time due to using the global-local technique and matrix formulations, and (4) capability of investigating the local phenomena. To enhance the accuracy of the results, compatible Hermitian quadrilateral elements are employed. The buckling loads are determined based on a criterion previously published by the author.     

The effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables

This study aims at comparing non-linear modal interactions in shallow horizontal cables with kinematically non-condensed vs. condensed modeling, under simultaneous primary external and internal resonances. Planar 1:1 or 2:1 internal resonance is considered. The governing partial-differential equations of motion of non-condensed model account for spatio-temporal modification of dynamic tension, and explicitly capture non-linear coupling of longitudinal/vertical displacements. On the contrary, in the condensed model, a single integro-differential equation is obtained by eliminating the longitudinal inertia according to a quasi-static cable stretching assumption, which entails spatially uniform dynamic tension. This model is largely considered in the literature. Based on a multi-modal discretization and a second-order multiple scales solution accounting for higher-order quadratic effects of a infinite number of modes, coupled/uncoupled dynamic responses and the associated stability are evaluated by means of frequency- and force-response diagrams. Direct numerical integrations confirm the occurrence of amplitude-steady or -modulated responses. Non-linear dynamic configurations and tensions are also examined. Depending on internal resonance condition, system elasto-geometric and control parameters, the condensed model may lead to significant quantitative and/or qualitative discrepancies, against the non-condensed model, in the evaluation of resonant dynamic responses, bifurcations and maximal/minimal stresses. Results of even shallow cables reveal meaningful drawbacks of the kinematic condensation and allow us to detect cases where the more accurate non-condensed model has to be used.     

Flexural vibrations and vibrational heating of a ring plate with thin piezoceramic pads under single-frequency electromechanical loading

The paper deals with the coupled problem of flexural vibrations and dissipative heating of a viscoelastic ring plate with piezoceramic actuators under monoharmonic electromechanical loading. The temperature dependence of the complex characteristics of passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoviscoelasticity is solved by an iterative method. At each iteration, orthogonal discretization is used to integrate the equations of elasticity and an explicit finite-difference scheme is used to solve the heat-conduction equation with a nonlinear heat source. The effect of the dissipative heating temperature, boundary conditions, and the thickness and area of the actuator on the active damping of the forced vibrations of the plate under uniform transverse harmonic pressure is examined __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 99–108, February 2008.     

Effect of static loading on the nonlinear vibrations of a three-time redundant portal frame: Analytical and numerical studies

An analytical study of the nonlinear vibrations in a three-time redundant portal frame is presented herewith, considering the effect of the axial forces caused by the static loading upon the first anti-symmetrical mode (sway) and the first symmetrical mode natural frequencies. It is seen that the axial forces may play an important role in tuning the sway mode and the first symmetrical mode into a 1:2 internal resonance. Harmonic support excitations resonant with the first symmetrical mode are then introduced and the amplitudes of nonlinear steady states are computed based upon a multiple scales solution. Comparisons with numerical analyses using a finite-element program developed by the authors show good qualitative agreement.     

Development of a generalised active vibration suppression strategy for a cantilever beam using internal resonance

In this paper, the potential to utilise modal coupling effects in the formulation of a generalised vibration suppression algorithm is investigated. The plant, a flexible cantilever beam undergoing first mode oscillation, is modelled by a second order differential equation with a spring constant and damping coefficient that are representative of the first mode flexibility and material damping of the beam, respectively.In order to establish an internal resonance condition, a second equation, designated the supplementary equation or controller, is appended to the plant to render a two-degree-of-freedom system. The objective is to generate an internally resonant pair. Upon successful completion of this task, a suppression technique is implemented whereby energy is removed from the plant via the supplementary system.The introduction of the supplementary system results in a set of design parameters which are employed to realise a state of internal resonance and to establish the desired dynamic response. The choice of 2:1 internal resonance models results in a unidirectional control torque making this technique particularly attractive for systems using thrusters or tendons as actuators. A similar structural configuration regulated under a PD (Proportional-Derivative) control law is compared to the proposed control scheme via simulation.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.