Carlos E. N. Mazzilli

Nonlinear normal modes of planar frames discretised by the finite element method

Abstract This article describes the implementation of a computer program to calculate nonlinear normal modes of structural systems. The procedure follows the invariant manifold approach, adapted to handle equations of motion of systems discretised by finite element techniques. In its current version, it generates individual modes of planar framed structures exhibiting geometrically nonlinear behaviour. The program was tested in simple examples available in the literature, and showed very good results. Because finite element discretisations are not restrictive to the geometry of the structural system, it was also possible to generate, for the first time, nonlinear modes of framed structures.

Evaluation of non-linear normal modes for finite-element models

Abstract In this paper, an alternative technique for the evaluation of non-linear normal modes is presented and applied to finite-element models. It is based upon the method of multiple scales, so that non-linear normal modes are evaluated as asymptotic expansions starting with the linear damped vibration modes. Non-linear normal modes are characterized both by the time response of all generalised coordinates and by explicit non-linear relationships between them and the modal variables. For the sake of an example, a non-conservative model of a cantilever beam, discretised with Bernoulli–Euler finite elements, is analysed.

Reduction of finite-element models of planar frames using non-linear normal modes

Abstract This paper is an extended version of Mazzilli et al. (Mazzilli, C.E.N., Soares, M.E.S., Baracho Neto, O.G.P., 1999. Proceedings of the American Congress of Applied Mechanics, PACAM VI, vol. 8, pp. 1589–1592, Rio de Janeiro, Brazil) which presents a powerful reduction technique in non-linear dynamics based on the combination of finite element procedures with a “non-linear” Galerkin method (Zemann, J., Steindl, A., 1996. Proceedings of the 19th International Congress of Theoretical and Applied Mechanics, Kyoto, Japan) and non-linear normal modes (Shaw, S.W., Pierre, C., 1993. Journal of Sound and Vibration 164 (1), 85–124). Its implementation, in the form of a symbolic computation code, was carried out for planar framed structures under assumptions of linear elasticity and geometrical non-linearity, according to the Bernoulli–Euler rod theory (Brasil, R.M.L.R.F., Mazzilli, C.E.N., 1993. Applied Mechanics Reviews 46 (11), S110–S117). To obtain the desired drastic reduction of degrees of freedom and the corresponding set of differential equations of motion in explicit form, it is necessary to supply as input data the displacement components of the pre-selected non-linear normal modes. Validation tests for non-linear free-vibration problems are shown, considering reduced models of higher hierarchy and their ability to supply accurate regenerated non-linear normal modes. For non-linear forced vibration problems, a brief outlook of what is intended to be done is presented.

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.

An analysis of parametric instability of risers

In this work, three catenary riser models subjected to harmonic oscillations are studied. Two are finite-element models, one studied with Orcaflex, an offshore marine system analysis software, and another one with Abaqus, a generalist structural analysis software. The third model is an analytical reduced-order model that represents only the touch-down zone. The aim of this study is to discuss the feasibility, potentialities and limitations of the analytical model in confrontation with the specialist and the generalist softwares for the analysis of risers, under conditions of parametric excitation and unilateral contact at the seabed.

Non-Linear Dynamic Analysis of a Cable-Stayed Beam

The dynamic behaviour of a simple clamped beam suspended at the other end by an inclined cable stay is surveyed in this paper. The sag due to the cable weight, as well as the non-linear coupling between the cable and the beam motions are taken into account. The formulation for in-plane vibration follows closely that of Gattulli et al. [1] and confirms their findings for the overall features of the equations of motion and the system modal properties. A reduced non-linear mathematical model, with two degrees of freedom, is also developed, following again the steps of Gattulli and co-authors [2,3]. Hamilton’s Principle is evoked to allow for the projection of the displacement field of both the beam and the cable onto the space defined by the first two modes, namely a “global” mode (beam and cable) and a “local” mode (cable). The method of multiple scales is then applied to the analysis of the reduced equations of motion, when the system is subjected to the action of a harmonic loading. The steady-state solutions are characterised in the case of internal resonance between the local and the global modes, plus external resonance with respect to either one of the modes considered. A numerical application is presented, for which multiple-scale results are compared with those of numerical integration. A reasonable qualitative and quantitative agreement is seen to happen particularly in the case of external resonance with the higher mode. Discrepancies should obviously be expected due to strong non-linearities present in the reduced equations of motion. That is specially the case for external resonance with the lower mode.Copyright