Direct application of the transfer matrix method to solve nonlinear autonomous boundary value problems with multiple branches

A direct transfer matrix method is developed to analyze the nonlinear, branched and one-dimensional autonomous boundary value problems. The method is applied to determine (1) the natural frequencies and modes about the initial state (vacuum), (2) the nonlinear steady-state deflections, (3) the natural frequencies and modes about the trimmed state, and (4) the aeroelastic stability of branched blades in hover. A quasi-steady strip theory is employed for calculation of the aerodynamic forces. A Newton-Raphson method based on a quasi-linearization of a nonlinear distributed boundary value problem is developed to solve the steady-state deflections of the blade. The formulation is validated by comparing the results with those obtained by other methods in the literature.