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

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 ®nite element procedures with a ``non-linearʹʹ 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 di€erential 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 presente