1D finite element for modelling of turbine blade vibration in the field of centrifugal forces

Abstract

The paper deals with the modelling of turbine blade vibrations by means of a novel 1D finite element that has only 16 degrees of freedom. Assuming linear elastic behaviour of the blade material and considering small displacements and strains, the derived blade finite element takes into account the effects of tension, torsion and bending in accordance with the Bernoulli’s hypothesis. Additionally, the finite element interlinks bending and torsion, and respects membrane forces acting on the blade. The derivation of matrices and vectors describing the blade finite element is provided in detail by using the Lagrange’s equations while the effect of membrane forces is included via the virtual work principle. For modelling purposes, the mathematical model of a turbine blade requires only the knowledge of cross-section contour points at several selected sections along the turbine blade axis. On the basis of these points, cross section characteristics including the warping function ar e approximated along the blade axis by means of cubic splines. The advantage of this approach lies in the fact that all the blade cross section parameters are identified before running numerical simulations. The warping function introduced in this paper and derived by variational principle describes cross section warping caused just by torsion of a prismatic rod.
     For the verification of the proposed 1D finite element, an analysis of modal properties of the turbine blade M6 L-1 manufactured by ŠKODA-Doosan Pilsen is performed. This is achieved by comparing the lowest natural frequencies and corresponding mode shapes computed by the 1D and 3D models for a standing blade. The results revealed good agreement between both models despite the significant difference in their degrees of freedom. The applicability of the 1D finite element is further demonstrated by analyzing the dependence of natural frequencies on rotor speed.