Proceedings of DETC’01 2001 ASME Design Engineering Technical Conferences September 9-12, 2001, Pittsburgh, Pennsylvania, USA DETC01/xxx-nnnn THE ROLE OF NONLINEAR STRAIN-DISPLACEMENT RELATION ON THE GEOMETRIC STIFFENING OF ROTATING FLEXIBLE BEAMS 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 Email: 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 Email: rsampaio@mec.puc-rio.br ABSTRACT Geometric stiffening of rotating flexible beams has been largely discussed in the last two decades. Several methodolo- gies have been proposed in the literature to account for the stiff- ening effect in the dynamics equations. This work aims first to present a brief review of the open literature on this subject. Then, a general nonlinear model is formulated using a nonlinear strain- displacement relation. This model is then used to deeply analyze simplified models arising in the literature. In particular, the as- sumption of steady-state values for the centrifugal load is ana- lyzed and its consequences are discussed. NOMENCLATURE A Area of the beam cross-section. β(x) Elastic line. E Lagrangian strain tensor. E Young’s modulus of the beam. ε xx Axial strain. F Deformation gradient. φ Beam cross-section rotation. H Strain energy function. h Beam thickness. I Moment of inertia of the beam cross-section. L Beam length. ϖ Natural frequency in rad/s. P Centrifugal load. R Rotation operator. ρ Beam mass density. T Kinetic energy function. U T Total displacement vector. u Mean axial displacement. u 0 Axial displacement. v Transverse displacement. X Position vector in the undeformed configuration. x Position vector in the deformed configuration. x Axial component of the undeformed position vector. z Transverse component of the undeformed position vector. INTRODUCTION Rotating beams have been the subject of several publica- tions in the last two decades. This was motivated by the study of practical applications like vibrations in turbomachine blades, rotorcraft flexbeams, robotic manipulator arms and satellite ap- pendages. One of the most discussed topics on this literature was the geometric stiffening due to rotation. This was also referred to as dynamic, centrifugal or rotational stiffening. For the au- thors’ knowledge, the first study of vibration of rotating beams was published by Schilhansl (Schilhansl, 1958) who analyzed the bending vibration, assuming steady-state revolution and negligi- ble Coriolis force. He derived a formula relating the fundamental bending eigenfrequency with the angular velocity of revolution. More recently, some authors have been interested in the geo- metric stiffening effect for applications in the dynamics of flex- ible multibody systems (Padilla and von Flotow, 1992; Sharf, 1995; Tadikonda and Chang, 1995; Wallrapp and Schwertassek, 1991). Hence, Kane et al. (Kane et al., 1987) observed that previ- 1 Copyright  2001 by ASME