In this paper, starting with the thin shell theory, the governing partial differential equation of motion for the transverse deflection of a rotating pre-twisted plate is derived. Straindisplacement relationships include the effect of warping of the cross-section due to twistbend coupling effect introduced as a result of pre-twist in the plate of non-circular (rectangular) cross-section. Then the equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil blade by considering it as a plate of an equivalent rectangular cross-section subjected to a quasi-static load due to centrifugal force field. The analytical derivation considers both the stress-stiffening as well as stress-softening effects of the centrifugal forces on the spinning airfoil. The partial differential equation governing the flexural motion of the plate is transformed into a matrix-eigenvalue form using a RayleighRitz technique. The plate deformations are represented by a set of 'admissible' sinusoidal trial functions, which fully satisfy all the clamped-end constrains as well as the free-edge boundary conditions. The results of the analytical model exhibit an excellent agreement with the previously published test data both for thin and thick plate geometries and even in highly twisted configurations. The results of the eigenvalue solution are presented in a non-dimensional form for plates of varying aspect ratios and different amounts of pre-twist in the plate. The numerical results are directly applicable in determining the static and running frequencies of typical blades used in turbo-machinery.
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