A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.
Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions
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Zu, J. W., and Han, R. P. S. (June 1, 1992). "Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions." ASME. J. Appl. Mech. June 1992; 59(2S): S197–S204. https://doi.org/10.1115/1.2899488
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