XVI CONGRESSO BRASILEIRO DE ENGENHARIA MECÂNICA 16th BRAZILIAN CONGRESS OF MECHANICAL ENGINEERING ON THE STIFFENING EFFECT OF FLEXIBLE APPENDAGES IN LARGE ANGLE MANEUVER Marcelo A. Trindade Laborat´ orio de Dinˆ amica e Vibrac ¸˜ oes, Pontif´ ıcia Universidade Cat´ olica do Rio de Janeiro, Rio de Janeiro, RJ 22453-900 trindade@mec.puc-rio.br Rubens Sampaio Laborat´ orio de Dinˆ amica e Vibrac ¸˜ oes, Pontif´ ıcia Universidade Cat´ olica do Rio de Janeiro, Rio de Janeiro, RJ 22453-900 rsampaio@mec.puc-rio.br Abstract. Geometric stiffening of flexible beams has been largely discussed in the last two decades. Several methodologies have been proposed in the literature to account for the stiffening effect in the dynamics equations. This work aims first to present a brief review of the open literature on this subject. Then, a general non-linear model is formulated using a non-linear strain-displacement relation. This model is then used to deeply analyze simplified models arising in the literature. In particular, the assumption of steady-state values for the centrifugal load is analyzed and its consequences are discussed. Thereafter, four finite element models are proposed, one based on non-linear theory and the others on simplified linear theories. These models are then applied to the study of a flexible beam undergoing prescribed high speed large rotations. The analyses show that accounting for the geometric stiffening effect is necessary in order to obtain realistic results. Moreover, this must be done in association with the inclusion of axial displacements dynamics in the model. Keywords: beams, geometric stiffening, non-linear strain-displacement relations, large displacements 1 Introduction The dynamics and control of rotating beams has received great attention in the past decade due to their wide appli- cations in aerospace, aviation and robotic industries. However, the majority of the works reported in the open literature presents either only a refined model of the dynamics (Kane et al., 1987, Simo and Vu-Quoc, 1987) or a specific controller design applied to a simple dynamics model (Choi et al., 1999; G´ oes and Adade Filho, 1999). The main complexity in modeling and one of the most discussed topics on this literature is the geometric stiffening due to rotation, which was also referred to as dynamic, centrifugal or rotational stiffening. Moreover, as one should expect, controller design may be seriously ineffective if realistic models of the beam are not used. Recently, several methodologies for incorporating the stiffening effect into the dynamics were reviewed by Sharf (1995) and are still studied nowadays (Kuo and Lin, 2000). Kane et al. (1987) observed that previous multibody formula- tions did not account properly for the geometric stiffening effect. They proposed an alternative methodology using higher order strain measures and applied it to the dynamics of a cantilever beam attached to moving base under prescribed large translation and rotation. Padilla and von Flotow (1992) stated that the error of previous formulations were due to a pre- mature linearization of the displacement field. Later, Hanagud and Sarkar (1989) observed that the formulation proposed in Kane et al. (1987) was inconsistent and pointed out that, on the contrary to that stated by Kane et al., the stiffening effect can be accounted for using non-linear strain-displacement relations. In fact, the formulation used in (Kane et al., 1987) implicitly includes a non-linear strain-displacement relation, which is not apparent due to the choice of independent variables employed. Simo and Vu-Quoc (1987) applied their theory of geometrically non-linear beams to the case of a rotating beam. Their formulation accounted also for shear strains in the beam. However, their “consistent” linearization using the steady-state value for the axial internal force has led to equivalent equations as compared to those of transverse vibrations of beams subjected to a steady-state centrifugal load. Oguamanam and Heppler (1998) also accounted for shear strains to derive the equations of motion of a rotating Timoshenko beam with a tip mass. Wallrapp and Schwertassek (1991) preferred to account for the geometric stiffening through the use of a reference stress treated as an initial stress in the undeformed configuration. Recently, a generalization of the cantilever beam foreshortening to other structural members was proposed by Urruzola et al. (2000). A more complete review oriented to multibody systems may be found in (Sharf, 1995). In the next sections, a general non-linear model is formulated using a non-linear strain-displacement relation. Through a variational formulation, non-linear equations of motion are developed and used to derive the simplified models proposed in the literature above. Then, one non-linear and three linear simplified finite element models are presented and applied to the study of the dynamics of a flexible beam undergoing prescribed high speed large rotations. The results of the four models are compared in terms of representation of axial and transverse displacements dynamics behavior and the axial stress induced in the beam.