Paper proposal to IFToMM 7th International Conference on Rotor Dynamics, Vienna, Austria, September 25-28, 2006 Cross Coupled Bending Vibrations of Rotating Shaft due to a Transverse Breathing Crack A. C. Chasalevris and C. A. Papadopoulos Department of Mechanical Engineering and Aeronautics University of Patras Patras-26500-Greece Fax: +30.261.099.6258 chasalev@mech.upatras.gr chris.papadopoulos@upatras.gr KEY WORDS: Transverse Crack, Rotor, Coupling, Rotor Dynamics, Breathing ABSTRACT In the present paper the free bending vibration of a rotating shaft having a uniform circular cross-section and a transverse crack is calculated. A disk is adopted in the midspan of the rotor and the crack, which is positioned also in the midspan, is supposed to open and close under the domination of the gravity on the vibration amplitude. The equations of motion of the continuous and isotropic model of the shaft follow the Timoshenko theory and take into account the gyroscopic effect, the shear deformation as long as the torsion that the shaft takes over due to the supposed power transmission. The governing equations are coupled and the partial solution is obtained by solving a linear system of equations. The rotation of the transverse crack changes the shaft stiffness, which is calculated at each angle of rotation analytically in a new way. The existence of the crack introduces the coupling between vibrations in the vertical and horizontal planes and the cross coupling terms of the local compliance matrix due to the crack are also calculated analytically as functions of the rotational angle. The local compliance matrix due to the bending is calculated using the well-known method of integration of the strain energy density function over the crack surface. The boundary conditions consider for simplicity reasons rigid bearings at both ends of the shaft and the suitable continuity conditions at the crack position. The coupled linear system of equations with periodically varying boundary conditions becomes solvable for every rotational angle. The characteristic equation can be calculated for every value of rotational speed and the natural frequencies are obtained. The natural frequencies follow the variation of the stiffness coefficients due to the crack. The frequency response and the orbit of the shaft in the cross section of the disk are also calculated for variable crack depths. The aim is to show the differences in the dynamic properties of the model due to the existence of the coupling phenomenon the crack provokes and also the effect of a deeper crack in the amplitude of vibration under the action of coupling. The resulting diagrams of response and natural frequency as functions of crack depth present the dynamic behavior of this continuous cracked rotating shaft model. 1. INTRODUCTION The vibration of cracked rotors is an issue gradually investigated since 1970 due to a case of a turbine rotor failure due to a fatigue crack observed by Dimarogonas in [4, 5] and Pafelias in [18] at the Turbine department of the General Electric Company in Schenectady. Since that time many researchers have investigated the dynamic behavior of cracked rotors that as phenomenon is close enough to the case of rotors with dissimilar moments of inertia as Dimentberg and Tondl has treated extensively in [6] and in [25] respectively. He identified higher harmonics and subharmonics which Dimarogonas suggested as a potential method for crack detection. It is known that when a cracked shaft rotates the stiffness in a fixed direction changes with time due to the local compliance the crack introduces. The precise computing of local flexibility was confronted by Dimarogonas in [4, 5] and he gave functions for the change of local compliance during rotation. Chondros and Dimarogonas in [1] modeled a Paper ABC-123 1