Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

Abstract

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies.The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.The expressions for the global equations of motion used in this study were developed by Andr6 [1] and Mazzilli [2]. They are very convenient for linear or nonlinear dynamic analysis of beams and frames with holonomic restraints under conservative or non-conservative loads and support motions. The holonomic restraints are such that they may be considered as escleronomic for a reference system fixed to the supports of the structure. Thus, the support motion acts on the structure as a rigid body motion, as it may be seen in Figure 1.Ox~x~ is considered as an inertial reference system and Osx~X b as a relative reference system associated to the structure. The parameters if(t) and qS(t) represent respectively the translation and rotation of the relative system to the inertial system. The points P0 and P represent respectively the position of a generic point of the structure before and after deformation. The vector z7 defines the displacement of P0. It is assumed that ff can be defined by a finite number of independent parameters, ql (t), q2 (t), ..., qN (t). These parameters are named generalized displacements, and are functions of time only.The vector position/~ of the point P with respect to the inertial system, can be expressed by: -=-{ S~(t) q-[X~i q-u~(ql (•),..., qN(t) )]T~ba (t)}aj,where S~(t) are the components of the translation vector S defined in the basis (51, h2) associated to the inertial reference system Oz~x~ (they are functions of time only); xbi are