Keywords: nonlinear dynamics, structural dynamics, earthquake engineering, geometric nonlinearities, framed structures.

We present a study of the nonlinear dynamic behaviour of a simple portal frame excited by ground motion. The structure is composed of two vertical columns clamped at their bases supporting a horizontal beam pinned in both ends. Nonlinearity is introduced by considering the elements shortening due to bending. A two-degree-of-freedom mathematical model is derived using Lagrange's formulation leading to a set of two second order differential equations with quadratic nonlinearities. The geometry and mass distribution is chosen in such a way as to render a 1:2 relationship between the frequency of the first mode (sway mode), related with the lateral motion of the vertical columns, and the frequency of the second mode (symmetrical mode), related to the vertical motions of the horizontal beam axis. This ratio may cause internal resonance between those modes with possible energy transfer. Further, we impose harmonic support motions in the vertical and horizontal directions in near resonance with the modes.

Several interesting nonlinear phenomena are observed. In one case, when the support excitation frequency is near the second frequency of the structure, as its amplitude is increased the vertical motion of the horizontal beam axis will increase accordingly up to a point when it stops growing, that is, saturates. At this point, the energy pumped into the system via the second mode is transferred to the first mode, not directly excited, causing large amplitude sway motions, which are potentially dangerous. These are the saturation and energy transference phenomena. In another case, when the support excitation frequency is near the first frequency of the structure, as its amplitude is increased again energy transfer between the modes occurs in an intermittent fashion.

Nayfeh and Mook [1] mathematically studied similar equations using the multiple scales perturbations method. The research group the first author belongs has studied several framed structures subject to excitation from machines that the structure supports (Brasil and Mazzilli [2]; Balthazar et al. [3]). Here we present a similar frame excited by ground motions.

1

A.H. Nayfeh, D.T. Mook, "Nonlinear Oscillations", John Wiley, New York, 1979.

2

R.M.L.R.F. Brasil, C.E.N. Mazzilli, "Vibracoes Nao-Lineares Em Fundacoes Aporticadas de Maquinas", Rev. Internacional de Met. Numericos p Calculo y Diseno en Ingenieria, 6, 147-158, 1990.

3

J.M. Balthazar, J.L.F.P. Felix, R.M.L.R.F. Brasil, "On nonlinear dynamics and control of a particular portal frame excited by a non-ideal motor", Materials Science Forum, 440, 371-378, 2003. doi:10.4028/www.scientific.net/MSF.440-441.371

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