Semianalytical Sensitivity of Floquet Characteristic Exponents with Application to Rotary-Wing Aeroelasticity

The stability of a nonlinear dynamic system with periodic coefficients can be evaluated by linearizing the equations of motion of the system about a steady-state equilibrium position and then, using Floquet -Liapunov theory, by calculating the eigenvalues, or characteristi c multipliers, of the transition matrix at the end of one period, and finally by examining the real parts of the characteristic exponents corresponding to the characteristic multipliers. This article describes a methodology to calculate the sensitivity of the characteristic exponents to changes in system parameters, taken as design variables, using semianalytic derivatives rather than finite difference approximations. Linear and nonlinear contributions to such sensitivities are identified and algorithms for the calculation of each contribution are provided. The linear contribution to the sensitivity to each design variable can be calculated at a fraction of the cost of one analysis, with appropriate attention to the details of the computer implementation. The nonlinear contribution requires a fixed overhead cost of as many analyses as the number of variables that define the equilibrium position, regardless of the number of design variables with respect to which the sensitivities are sought. A potential problem in the calculation of finite difference gradients is identified and discussed. The linear portion alone is sufficiently accurate for the helicopter problem of the study.