Inverse method predicting spinning modes radiated by a ducted fan from free-field measurements.
- Serge Lewy
- The Journal of the Acoustical Society of America
The bypass duct of an aircraft engine is a low-pass lter allowing some spinning modes to radiate outside the duct. The knowledge of the radiated modes can help in noise reduction, as well as the diagnosis of noise generation mechanisms inside the duct. We propose a nonintrusive technique using a circular microphone array outside the engine measuring the complex noise spectrum on an arc of a circle. The array is placed at various axial distances from the inlet or the exhaust of the engine. Using a model of noise radiation from the duct, an overdetermined system of linear equations is constructed for the complex amplitudes of the radial modes for a xed circumferential mode. This system of linear equations is generally singular, indicating that the problem is illposed. Tikhonov regularization is employed to solve this system of equations for the unknown amplitudes of the radiated modes. An application of our mode detection technique using measured acoustic data from a circular microphone array is presented. We show that this technique can reliably detect radiated modes with the possible exception of modes very close to cut-o . Senior Research Scientist, Aeroacoustics Branch, Aerodynamics, Aerothermodynamics and Acoustics Competency, AIAA Associate Fellow Computer Sciences Corporation, AIAA Member zResearch Scientist, Aeroacoustics Branch, Aerodynamics, Aerothermodynamics and Acoustics Competency, AIAA Senior Member Copyright c 2001 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner. List of Symbols a radius of circular microphone array A( ; ) a complex function de ned in equation (17) Aq the vector (Aq1; Aq2; : : : ; Aqn) T Aqn amplitude of mode (m;n),m = B+ qV B fan blade number BPF blade passage frequency (B =2 ) c speed of sound C(m;n; ) a parameter de ned in equation (12) dq the vector (Dq( 1); Dq( 2); : : : ; Dq( j)) T Dq a paramter de ned in equation (16) f frequency G a j n matrix, j n, see equation (23) G the Hermitian adjoint of G G the linear operator mapping X into Y whose matrix is G gvw element of the matrix G Jm(x) Bessel function of rst kind and order m k wavenumber, B =c ka(m;n) axial wavenumber of mode (m;n), see equation (22) kr(m;n) radial wavenumber of mode (m;n), see equation (19) m circumferential mode number M duct Mach number, M < 0 for inlet radiation n radial mode number, n = 1; 2; : : : NG the nullspace of linear operator G p(~x; t) acoustic pressure p̂(~x; f) temporal Fourier transform of p(~x; t) P (~x; B )see equation (14) P̂q Fourier transform of P in azimuthal variable , see equation (15) q a parameter de ning circumferential mode m in the relation m = B + qV; q = 0; 1; 2; : : :