Trim analysis by shooting and finite elements and Floquet eigenanalysis by QR and subspace iterations in helicopter dynamics

The trim analysis for the initial state and control inputs that satisfy response periodicity and flight conditions, and the Floquet eigenanalysis for a few largest eigenvalues of the Floquet transition matrix (FTM) are investigated. In the trim analysis, the convergence of Newton iteration is investigated in computing the periodic initial state and control inputs sequentially and in parallel. The trim analysis uses the shooting method and two h-versions of temporal finite element methods, one based on displacement formulation and the other on mixed formulation of displacements and momenta. In each method, both the sequential and in-parallel schemes are used, and the resulting nonlinear equations are solved by damped Newton iteration with an optimally selected damping parameter. The reliability of damped Newton iteration, including earlier-observed divergence problems, is quantified by the maximum condition number of the Jacobian matrices of the iterative scheme. For illustrative purposes, rigid flap-lag and flap-lag-torsion models based on quasisteady aerodynamics are selected. Demanding trim analysis conditions are included by considering advance ratios or dimensionless flight speeds twice as high as those of current helicopters. Concerning the Floquet eigenanalysis, the feasibility of using the Arnoldi-Saad method, one of the emerging subspace iteration methods, is explored as an alternative to the currently used QR method, which is not economical for partial eigenanalysis. The reliability of the Arnoldi-Saad method is quantified by the eigenvalue condition numbers and the residual errors of the eigenpairs. In the three trim analysis methods, while the optimally selected damping parameter provides almost global convergence, the in-parallel scheme requires much less machine time than the conventional sequential scheme; both schemes have comparable reliability of the Newton iteration without and with damping. The Arnoldi-Saad method takes much less machine time than the QR method with comparable reliability.