Lagrangian formulation for the rapid estimation of helicopter rotor blade vibration characteristics

Abstract

Abstract This paper presents a numerical formulation targeting the rapid estimation of natural vibration characteristics of helicopter rotor blades. The proposed method is based on application of Lagrange’s equation of motion to the kinematics of blade flap/lag bending and torsion. Modal properties obtained from Bernoulli-Euler beam and classical torsional vibration theory, are utilised as assumed deformation functions in order to estimate the time variations of strain and kinetic energy for each degree of freedom. Integral expressions are derived, describing the generalised centrifugal force and torsional moment acting on the blade in terms of normal coordinates, for flap/lag transverse displacement and torsional deformation. Closed form expressions are provided for the direct analysis of hingeless, freely-hinged and spring-hinged articulated rotor blades. Results are presented in terms of natural frequencies and mode shapes for two small-scale rotor blade models. Extensive comparisons are carried out with experimental measurements and nonlinear finite element analysis. Predictions of resonant frequencies are also presented for two full-scale rotor blade models and the results are compared with established multi-body dynamics analysis methods. It is shown that, the proposed approach exhibits excellent numerical behaviour with low computational cost and definitive convergence characteristics. The comparisons suggest very good and in some cases excellent accuracy levels, especially considering the method’s simplicity, computational efficiency, and ease of implementation.