Superconvergent Finite Element for Coupled Torsional-Flexural-Axial Vibration Analysis of Rotating Blades

摘要

A new two-noded, twelve degree of freedom finite element is developed for rotating blades. The shape functions are derived from the exact solutions of the governing static homogenous differential equations for the rotating blades. Such an approach leads to super-convergent elements. These differential equations include out-of-plane bending, in-plane bending, axial deformation, and torsion. The axial and torsion equations yield exact solutions and the flap and lag equations are solved by assuming a constant centrifugal force within the element. Differing from the conventional polynomial shape functions, the new shape functions account for the centrifugal stiffening effect as they depend upon the rotation speed, material properties, and the element position along the length of the blade. The finite element formulation is derived from the energy expressions using the Hamilton's principle. A convergence study for the natural frequencies is performed using the new shape functions and the polynomial shape functions for a coupled and an uncoupled blade. It is observed that the new shape functions lead to much more rapid convergence than the conventional polynomial shape functions for the first few modes at higher rotation speeds, where the effect of centrifugal stiffening is higher. The basis functions can also be used for finite element analysis of rotating rods and beams, and for energy methods.