Angelo Luongo

Planar non-linear free vibrations of an elastic cable

Continuum non-linear equations of free motion of a heavy elastic cable about a deformed initial configuration are developed. Referring to an assumed mode technique one ordinary equation for the cable planar motion is obtained via a Galerkin procedure, an approximate solution of which is pursued through a perturbation method. Suitable nondimensional results are presented for the vibrations in the first symmetric mode with different values of the cable properties. Which procedure is the proper one to account consistently for the non-linear kinematical relations of the cable in one ordinary equation of motion is discussed.

A complete dynamic approach to the Generalized Beam Theory cross-section analysis including extension and shear modes

A simple and efficient ‘complete dynamic approach’ is proposed, and named GBT-D, to evaluate a suitable basis of modes for the elastic analysis of thin-walled members in the framework of Generalized Beam Theory (GBT). The basis includes conventional and non-conventional modes, the latter accounting for transverse extension and membrane shear strain of the plate elements forming the cross-section, which identically vanish in the former set. The method relies on the solution of two distinct eigenvalue problems, governing the in-plane and the out-of-plane free oscillations of a segment of a thin-walled beam. Both the eigenvalue problems, differential in origin, and defined on a one-dimensional spatial domain, are transformed into an algebraic problem by means of a discretization carried out at the cross-section middle line. Numerical examples are then presented to outline the ease of use of the proposed method considering a single plate, an open cross-section and a partially closed one. Member analyses are also performed for the simplest boundary conditions, to validate the accuracy of the proposed GBT-D approach against finite element method results and analytical solutions, highlighting the importance in including the non-conventional modes.

Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε3-order in the 1:3 case, they already arise at theε2-order in the 1:2 case. Thus, by truncating theanalysis at the ε3-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.

Mode Localization in Dynamics and Buckling of Linear Imperfect Continuous Structures

Localization phenomena in one-dimensional imperfect continuous structures are analyzed, both in dynamics and buckling. By using simple models, fundamental concepts about localization are introduced and similarities between dynamics and buckling localization are highlighted. In particular, it is shown that strong localization of the normal modes is due to turning points in which purely imaginary characteristic exponents assume a non zero real part; in contrast, if turning points do not occur, only weak localization can exist. The possibility of a disturbance propagating along the structure is also discussed. A perturbation method is then illustrated, which generalizes the classical WKB method; this allows the differential problem to be transformed into a sequence of algebraic problems in which the spatial variable appears as a parameter. Applications of the method are worked out for beams and strings on elastic soil. All these structures are found to have nearly-defective system matrices, so their characteristic exponents are highly sensitive to imperfections.

On Nonlinear Dynamics of Planar Shear Indeformable Beams

The planar forced oscillations of shear indeformabie beams with either movable or immovable supports are studied through a unified approach. An exact nonlinear beam model is referred to and a consistent procedure up to order three nonlinearities is followed. By eliminating the longitudinal displacement component through a constraint condition and assuming one mode, the problem is reduced to one nonlinear differential equation. A perturbational solution in the neighborhood of the resonant frequency is determined and the stability of the steady-state solutions is studied. The dependence of the phenomenon on the geometrical and mechanical characteristics of the system is put into light and the frequency-response curves for different boundary conditions are furnished.

Aeroelastic instability analysis of NES-controlled systems via a mixed multiple scale/harmonic balance method

The issue of passively controlling aeroelastic instability of general nonlinear multi-degree-of-freedom systems, suffering from Hopf bifurcation, is addressed. The passive device consists of an essentially nonlinear oscillator (nonlinear energy sink [NES]), having the task of absorbing energy from the main structure. The mathematical problem is attacked by a new algorithm, based on a suitable combination of the multiple scale and the harmonic balance methods. The procedure is able to furnish the reduced amplitude modulation equations, which govern the slow flow around a critical manifold, on which the equilibrium points lie. The method is applied to a sample structure already studied in literature, consisting of a two-degree-of-freedom rigid airfoil under steady wind. It is shown that NES, under suitable conditions, can shift forward the bifurcation point and, moreover, it can reduce the amplitude of the limit cycles. Relevant asymptotic results are compared, for validation purposes, with numerical simulations.

Asymmetric interactive buckling of thin-walled columns with initial imperfections

Abstract In this paper the effect of the interaction between two or more simultaneous buckling modes on the postbuckling behaviour of uniformly compressed thin-walled members (TWM) is analysed by means of the general theory of elastic stability. The analysis is restricted to third-order terms of the energy expansion and therefore can be fruitfully applied to the investigation of structures with asymmetric postbuckling behaviour only. Initial imperfection effect is taken into account. A simplified procedure is suggested for solving the nonlinear equations relative to the evaluation of the bifurcated paths. By using the finite strip method an extensive parametric analysis is performed. It is found that when the flexural-torsional (FT) buckling interacts with a local symmetric and antisymmetric mode, sensitivity to initial inperfections is remarkable and is comparable to the one arising from the interaction between the Euler (E) and any local buckling.

Perturbation Methods for Bifurcation Analysis from Multiple Nonresonant Complex Eigenvalues

It is shown that the logical bases of the static perturbation method, which is currently used in static bifurcation analysis, can also be applied to dynamic bifurcations. A two-time version of the Lindstedt–Poincaré Method and the Multiple Scale Method are employed to analyze a bifurcation problem of codimension two. It is found that the Multiple Scale Method furnishes, in a straightforward way, amplitude modulation equations equal to normal form equations available in literature. With a remarkable computational improvement, the description of the central manifold is avoided. The Lindstedt–Poincaré Method can also be employed if only steady-state solutions have to be determined. An application is illustrated for a mechanical system subjected to aerodynamic excitation.

On the effect of the local overall interaction on the postbuckling of uniformly compressed channels

Abstract On the basis of the general theory of elastic stability due to Koiter, postbuckling analysis of simply supported channels under uniform compression is performed. Attention is essentially focused on local/eularian and local/flexural-torsional simultaneous bucklin modes interaction. The column is treated as a plate assemblage. Linearized expressions for the displacement field are employed while assuming strain-displacements relationships that are linear for the curvatures and up to second order terms for the in-plane strains. The total potential energy is hence written up to third order terms in order to investigate asymmetric buckling phenomena. A discrete model is developed through an automatic procedure of algebraic manipulation, and an extensive parametric analysis is performed. After determining the range of geometric parameters which characterize different types of interaction, it is found that in the postbuckling range the local/eulerian interaction is more dangerous than the local/flexural-torsional one, due to the column higher imperfection sensitivity.

Dynamics of the pendulum with periodically varying length

Dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied. This is the pendulum with periodically varying length which is also treated as a simple model of a childs swing. Asymptotic expressions for boundaries of instability domains near resonance frequencies are derived. Domains for oscillation, rotation, and oscillation-rotation motions in parameter space are found analytically and compared with a numerical study. Chaotic motions of the pendulum depending on problem parameters are investigated numerically.