One-dimensional models in turbine blades dynamics

Abstract

Turbine blades are considered as straight naturally twisted rods. Two models are discussed: Bernoulli-Euler beam and Cosserat rod. Linear theories with small displacements, rotations and loads are used. The equations of dynamics taking into account bending, twisting, axial and shear deformations and cross links between them are derived. The stiffness coefficients in elasticity relations are defined. In the case of harmonic oscillations, we have for amplitudes the ordinary differential equations solved by means of computer mathematics (Mathcad). As a result the normal modes, the natural frequencies and also the amplitudes of forced oscillations are obtained. For the Bernoulli-Euler beam the Lagrange-Ritz-Kantorovich variational approach with approximations of deflections is proposed. The unknown coefficients of approximation depending on time are found by the numerical integration of the Lagrange system of equations. Proposed methods are applied to calculation of the real turbine blade.