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 strateg...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.展开更多
The goal of this study is to propose a method of estimation of bounds for roots of polynomials with complex coefficients. A well-known and easy tool to obtain such information is to use the standard Gershgorin’s theo...The goal of this study is to propose a method of estimation of bounds for roots of polynomials with complex coefficients. A well-known and easy tool to obtain such information is to use the standard Gershgorin’s theorem, however, it doesn’t take into account the structure of the matrix. The modified disks of Gershgorin give the opportunity through some geometrical figures called Ovals of Cassini, to consider the form of the matrix in order to determine appropriated bounds for roots. Furthermore, we have seen that, the Hessenbeg matrices are indicated to estimate good bounds for roots of polynomials as far as we become improved bounds for high values of polynomial’s coefficients. But the bounds are better for small values. The aim of the work was to take advantages of this, after introducing the Dehmer’s bound, to find an appropriated property of the Hessenberg form. To illustrate our results, illustrative examples are given to compare the obtained bounds to those obtained through classical methods like Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds.展开更多
基金supported by the National Natural Science Foundation of China(Nos.51576097,51976089)the Funding for Outstanding Doctoral Dissertation in Nanjing University of Aeronautics and Astronautics,China(No.BCXJ24-05)the Aeronautical Science Foundation of China(No.2023L060052001).
文摘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.
文摘The goal of this study is to propose a method of estimation of bounds for roots of polynomials with complex coefficients. A well-known and easy tool to obtain such information is to use the standard Gershgorin’s theorem, however, it doesn’t take into account the structure of the matrix. The modified disks of Gershgorin give the opportunity through some geometrical figures called Ovals of Cassini, to consider the form of the matrix in order to determine appropriated bounds for roots. Furthermore, we have seen that, the Hessenbeg matrices are indicated to estimate good bounds for roots of polynomials as far as we become improved bounds for high values of polynomial’s coefficients. But the bounds are better for small values. The aim of the work was to take advantages of this, after introducing the Dehmer’s bound, to find an appropriated property of the Hessenberg form. To illustrate our results, illustrative examples are given to compare the obtained bounds to those obtained through classical methods like Cauchy’s bounds, Montel’s bounds and Carmichel-Mason’s bounds.
