Analytical Linearization of a State-Space Viscous Vortex Particle Method for Rotary-Wing Simulations

This work presents an extension of previous studies on the analytical linearization of state-space vortex particle methods (VPM) by incorporating viscous effects. The developed framework couples a panel method to capture blade surface and near-wake aerodynamics with a viscous vortex particle method (VVPM) for modeling the far-wake. The resulting formulation yields a nonlinear time-periodic (NLTP) system described by ordinary differential equations (ODEs) in first-order form. To enable linear analysis, the NLTP system is linearized into a linear time-periodic (LTP) representation using two techniques: finite differencing and a novel analytical linearization approach. Harmonic decomposition is then applied to transform the LTP system into a higher-order linear time-invariant (LTI) model, enabling the use of time-invariant analysis tools. The methodology is implemented in MATLAB® and applied to a representative utility helicopter rotor blade. Validation is performed against experimental data, and the linearized models are evaluated through time and frequency domain comparisons with the nonlinear system. Results demonstrate that the linearized models accurately capture unsteady wake dynamics, particularly under low to mid-frequency excitation. Moreover, the analytical linearization method significantly reduces computational cost – achieving an efficiency improvement of O(n2), where n is the number of system states – compared to finite-difference approaches.