Keywords: fan aero-acoustics, inverse problem, optimization.

The reconstruction of the forces responsible for the tonal noise in fans has been the subject of several works on the past years. The inversion of the tonal noise problem is done making use of the transfer function derived from the blade force equation. The condition number of this transfer function matrix plays an important role on the inverse problem reducing the influence of the measurement noise on the final result. This parameter has a direct influence of the problem geometry, and thus is important in relation to the measurement positions. With the objective of investigating the influence on the measurement positions and finding an optimal measurement grid for the tonal noise application, several analysis and optimization problems are performed. In this paper, the Morse-Ingard model [1] for the transfer function for tonal noise is used.

This paper has the objective to better understand the influence of the variables on the condition number for the tonal noise transfer function matrix. Different parameter optimization problems are performed in order to find the optimal microphone distribution that minimizes the condition number over the frequency band considered. A parametric analysis is performed in order to identify the influence of all the possible microphone position variables, with the objective to improve the optimization procedure. The optimized microphone positions will be compared to the classical hemisphere and arc microphone distribution presented by Gérard et al. [2], a drastic reduction of the conditional number is achieved.

Based on the optimization results, one can conclude that all theta_1..n must be equal and as close as possible to 90°, phi_1..n must be evenly distributed over the allowable range and all r_1..n must be equal. The microphones must be positioned and distributed on a circle with a chosen (fixed) radius. As a result of the optimization, the kappa_rms-value dropped from approximately 7.4x10⁷ to 1.15x10⁵ when compared to the hemispheric microphone distribution.

Based on the optimization results prssented in [3], the following generalized conclusions with respect to the optimized microphone positions can be drawn:

• The smallest achievable condition number rms-value equals kappa_rms=49.95, independent on the number of microphones. However, this value is not associated with a realistic microphone distribution, as all microphones coincide.
• When optimizing theta_1..n, all theta_1..n must be equal and the smallest kappa_rms-value is achieved for theta_1..n=90°.
• When optimizing phi_1..n, a uniform microphone distribution over the allowable phi-range yields the smallest kappa_rms-value. The smallest value is achieved when distributing the microphones evenly over a complete circle.
• When optimizing the microphone radius values r_1..n, all r_1..n must be equal, concluding that the microphones must be positioned and distributed on a circle with a chosen (fixed) radius. The actual value of the chosen radius does not have an influence on the kappa_rms-value, as long as all r_1..n are equal.

1

P.M. Morse, K.U. Ingard, "Theoretical Acoustics", McGraw-Hill, New York, 1968.

2

A. Gérard, A. Berry, P. Manson, "Control of Tonal Noise from Subsonic Axial Fan. Part 1: Reconstruction of Aeroacoustic Sources from Far-field Sound Pressures", Journal of Sound and Vibration, 288, 1049-1075, 2005. doi:10.1016/j.jsv.2005.01.023

3

F. Presezniak, G. Steenackers, P. Guillaume, "Microphone positioning optimization for conditioning inverse tonal fan noise", Mechanical Systems and Signal Processing, 24(6), 1682-1692, 2010. doi:10.1016/j.ymssp.2010.02.009

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