Vibration analysis of composite airfoil blade using orthotropic thin shell bending theory

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 airfoil is derived. Strain-displacement relationships include the effect of warping of the cross-section due to twist-bend coupling effect introduced as a result of varying stagger angle and camber radius of the blade. The equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil by considering it as an anisotropic shell in full curvilinear coordinates subjected to a 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 fourth-order partial differential equation characterizing the flexural motion of the airfoil is transformed into a matrix- eigenvalue form using a Rayleigh-Ritz technique. The blade deformations are represented by a set of “admissible” sinusoidal trial functions, which fully satisfy all the clamped-end constraints as well as the free-edge boundary conditions. The numerical results presented in a non-dimensional parametric form are directly applicable in determining the static and running frequencies of typical composite blades used in the fan module of an aeroengine.