NONLINEAR EQUATIONS OF FLEXURAL–FLEXURAL–TORSIONAL OSCILLATIONS OF ROTATING BEAMS WITH ARBITRARY CROSS-SECTION K. V. Avramov 1 , C. Pierre 2 , and N. V. Shyriaieva 1 UDC 539.374 A system of three nonlinear partial differential equations describing the flexural–flexural–torsional vibrations of a rotating slender cantilever beam of arbitrary cross-section is derived using Hamilton’s principle. It is assumed that the center of gravity and the shear center are at different points. The interaction between flexural and torsional vibrations is accounted for in the linear and nonlinear parts of model Keywords: flexural–flexural–torsional vibrations, rotating beam, cross-section 1. Introduction. Pretwisted rotating beams are elements of wind-driven equipment, helicopter rotor blades, and robot’s arms. While operating, these structures often undergo intensive oscillations. Turbine and helicopter rotor blades are designed to avoid resonance vibrations. Experimental investigations testify that the deformations of helicopter rotor blades during vibrations are geometrically nonlinear [7, 9]. We stress that most of such systems have internal resonances [9]. The non-symmetry of the blade cross-section is often taken into account. In this case, the center of gravity and the shear center are considered as different points [10]. Timoshenko [10] derived the equations of linear flexural–flexural–torsional vibrations of beams with nonsymmetric cross-sections considering that the center of gravity and the shear center are at different points. Filippov [6] obtained a linear model of flexural–flexural–torsional oscillations of beams with arbitrary cross-sections taking shear strains and rotary inertia into account. Dzhanelidze [5] suggested equations of flexural–torsional vibrations of thin-walled rods. Hodges and Dowell [8] derived a system of nonlinear partial differential equations for geometrically nonlinear flexural–flexural–torsional–extensional vibrations of rotating beams assuming that the center of gravity and the shear center are at the same point. Crespo da Silva and Glynn [1, 2] considered equations of flexural–flexural–torsional vibrations of inextensible beam assuming that the center of gravity and the shear center are at the same point. We will derive a system of three nonlinear partial differential equations of flexural–flexural–torsional vibrations of a slender rotating beam with arbitrary cross-section, assuming that the center of gravity and the shear center of the cross-section are at different points. Hamilton’s principle is used to obtain the equations of motion. We stress that in this case the interaction between flexural and torsional vibrations takes place in both the linear and nonlinear parts of model. 2. Main Geometrically Nonlinear Relations. It is suggested that a twisted cantilever beam is rotating with constant angular velocity. The Euler–Bernoulli beam model is considered, i.e., the cross-sections of the beams remain planar during nonlinear oscillations [3]. The coordinate system ( $ , $ , $ ) xhV is attached to the beams’ cross-section to predict their motions (Fig. 1). The origin of this coordinate system is at the center of gravity of the cross-section. The axes $ h and $ V are principal. The motions of the coordinate system ( $ , $ , $ ) xhV with respect to the global coordinate system ( $ , $ , $ ) xyz (Fig. 1) are analyzed to study the motions of the cross-section. The oscillations of the cross-sections are described by displacements u, v, and w along $ x , $ y , and $ z and by three successive rotations. We now consider a scheme of rotations of the coordinate system. As a result of these rotations, the coordinate system ( $ , $ , $ ) xhV transforms into ( $ , $ , $ ) xyz . The first rotation q z about the $ z-axis takes the system ( $ , $ , $ ) xyz to ( $ x 1 , $ h 1 , $ V 1 ) International Applied Mechanics, Vol. 44, No. 5, 2008 582 1063-7095/08/4405-0582 ©2008 Springer Science+Business Media, Inc. 1 National Technical University “KhPI,” Karkov, Ukraine. 2 University of Montreal, Canada. Published in Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 123–132, May 2008. Original article submitted November 3, 2006.