Dynamic Optimization of Bearingless Helicopter Rotors

The paper outlines a beam non linear formulation capable of describing the three dimensional behaviours of composite blades and usable in an optimization system. The response couplings are all taken into account by the method used to characterize the beam cross section. Such a linear constitutive law represents the starting point for an updated Lagrangian formulation of the beam elastic problem. In connection with this approach, the use of Gibbs-Rodriguez parameters, to describe angular unknowns, greatly simplify the linearization of equations. The analysis code allows the evaluation of non linear situations, large displacements and rotations and centrifugal forces, and the eigenvalue extraction in a trimmed configuration. This code is employed by an optimization procedure, based on a SQPlike routine to supply analysis and sensitivities. dimensional finite element discretisation of the cross section, combined with one-dimensional generalized beam models. The modelisation of non uniform beams steps through the use of several elements along the span with both constant or variable properties. Some of these approaches make contemporary use of cross section warping, together with usual nodal displacement and rotation unknowns in the beam analysis, while other ones implicitly account for three-dimensional effects of crosssection strain, and no additional variables are required [3-10]. The beam approach is very efficient for the analysis but leads to some difficulties in managing optimization procedures. When a multi-level approach is chosen, using global design variables in the first level design phase such as stiffness and mass distribution, the second level optimization problem could give rise to structural solutions, that, even satisfying the design constraints resulting from the first step, present undesired couplings. The problem cannot be overcome neither by including in the design variable set, of the first phase, all the stiffness terms, nor by constraining the coupling effects in the inner optimization loop. On the other hand, whenever the design variables are related to detailed design parameters, such as laminate thickness, ply angle, and so on, undesired couplings can also arise during the design process: the evaluation of objective function and constraints could be biased by not taking into account these effects in an appropriate beam model. The consequence of the use of a full local stiffness matrix can be appreciated in Fig.1, showing the second in plane eigenvector of a composite rotating beam, resulting from the model proposed in this paper Examples presented are related to the validation of the analysis modules and to significant applications of the optimization procedure.