Predicting and minimizing twin-propeller noise: Hessian matrix and Fourier-Frobenius matrix analysis in improved propeller signatures theory

Quick and accurate determination of the optimal synchrophase angle is crucial for synchrophasing control of multi-propeller aircraft with low noise. This paper proposes a novel noise prediction and optimization strategy, developing a continuous and accurate noise prediction model and obtaining its minimum by solving the Hessian matrix and Fourier-Frobenius matrix. Firstly, a novel propeller noise prediction method uses acoustic simulation pressure signals and improved propeller signatures theory to accurately estimate noise for all synchrophase angles and receiving points. Secondly, a novel optimization approach is proposed to solve the analytical solution of the minimum propeller noise: (A) A noise objective function is established, and use its first derivatives’ zeros and Hessian matrix to determine the function minimum. (B) A novel Euler formula transform method is proposed to convert trigonometric polynomials into algebraic polynomials, changing the zeros of the former into those of the latter. (C) Utilize the Fourier-Frobenius matrix method to solve the zeros of algebraic polynomials. To assess the computation time and accuracy, a turboprop aircraft with two six-bladed propellers was analyzed using the computational fluid dynamics and acoustic analogy method, providing acoustic pressure signals at 20 receivers for noise prediction and optimization. The Durand-Kerner and Fourier-Frobenius matrix methods were compared. Results demonstrate that improved propeller signatures theory is more accurate, and the Hessian matrix + Fourier-Frobenius matrix method is faster and more precise than the Hessian matrix + Durand-Kerner method.