Multiple scales analysis of nonlinear oscillations of a portal frame foundation for several machines

Abstract

An analytical study of the nonlinear vibrations of a multiple machines portal frame foundation is presented. Two unbalanced rotating machines are considered, none of them resonant with the lower natural frequencies of the supporting structure. Their combined frequencies is set in such a way as to excite, due to nonlinear behavior of the frame, either the first anti-symmetrical mode (sway) or the first symmetrical mode. The physical and geometrical characteristics of the frame are chosen to tune the natural frequencies of these two modes into a 1:2 internal resonance. The problem is reduced to a two degrees of freedom model and its nonlinear equations of motions are derived via a Lagrangian approach. Asymptotic perturbation solutions of these equations are obtained via the Multiple Scales Method

Nonlinear dynamics of a flexible portal frame under support excitation

“…Different authors studied dynamical system with nonideal excitation (Nayfeh & Mook, 1979;Balthazar et al, 2003;Dantas & Balthazar, 2007). Brasil (1999) performed an analytical study of the nonlinear vibrations of a portal frame excited by two ideal energy sources, consisting of unbalanced rotating machines. This same problem with two non-ideal sources was studied by Balthazar et al (2004), which focused on the self-synchronization of the energy sources.…”

Some Comments on the Nonlinear Dynamics of a Portal Frame under Base Excitation

“…The linear stiffness of the columns and the beam can be evaluated by a Raleigh-Ritz procedure from cubic trial functions. Moreover, geometric nonlinearities are introduced by considering the shortening due to columns and beams bending under inextensibility hypothesis as presented by Brasil [4] and shown in Eq. (2).…”

“…The device under consideration here consists of a Duffing electrical circuit coupled magnetically to a flexible portal frame. This system modeling was previously addressed by Brasil [4] and Balthazar et al [5].…”

“…The nonlinear vibrations of these systems have been investigated by several authors [1][2][3]. Brasil [4] presented the mathematical modeling of a portal frame excited by two ideal energy sources, consisting of unbalanced rotating machines, and performed an analytical study of its nonlinear vibrations. This same problem with two non-ideal sources was studied by Balthazar et al [5], which focused on the self-synchronization of the energy sources when excitation was provided due to unbalanced rotating machines.…”