Computational nonlinear vibration analysis for distributed geometrical nonlinearities in structural dynamics

Abstract

The demand to reduce the impact of aviation on the environment is leading jet engine manu- facturers to increase the fuel and propulsion efficiency of the engines. This in turn is pushing materials to their physical limits by undergoing increasingly higher thermo-mechanical loads. In this regime, blades and other engine components are subjected to large deforma- tions generating nonlinearities that activate new failure mechanisms not dealt with before. Therefore, vibration analysis is essential to develop new methodologies for the accurate prediction of components’ behaviour. This research focuses on investigating the effect of the distributed geometric nonlinearities and rotational speed on the dynamic behaviour of three-dimensional structures. The Green-Lagrange strain measures are employed in this research to express the nonlinear relationship between the displacement and the strain. The nonlinear algorithms used for the numerical simulations are developed based on the Finite Element Method combined with the Harmonic Balance method. The complex geometries are discretised by using the geometric exact three-dimensional solid elements. The forced response functions and the backbone curves for the steady-state response of the nonlinear system can be computed. The research aims to develop and validate methodologies for the identification and control of undesired vibration modes which will inform new design choices. Finite element modelling of the blades generally involves an immense number of degree-of-freedoms, which could be infeasible to compute. The reduced order modelling (ROM) techniques are crucial for achieving an accurate prediction of the nonlinear behaviour in an efficient way. Detailed computation strategies for the intrusive ROM methods are delivered. ROM techniques based on the linear and nonlinear mapping between the full model and the reduced basis are presented. The capabilities and limitations of both methods are assessed. The projection method based on the linear eigenmodes only has a slow converge to the full system. On the other hand, the quadratic manifold method with the static modal derivatives involved in the reduced coordinates provides a fast convergence.