Frequency Domain gappy-POD for Rotor Acoustic Measurement Optimization

This paper describes a mathematical framework for determining the optimal sensor set location for adequately capturing the sound generated by rotors. The approach leverages the gappy-POD method proposed by Everson and Sirovich [J. Opt. Soc. Am., Vol. 12, 1995, pp. 1657-1664], which first identifies the various mode constituents that make up the first few rotor blade-pass frequency harmonics of the sound-field. The algorithm is developed using a covariance matrix for the POD problem comprising auto- and cross-spectral densities of spatially and temporally resolved sound waves captured by an array of microphones oriented parallel to the axis of a laboratory-scale hovering rotor. Three different forms of the technique are developed and compared. These comprise a homogeneous form and two heterogeneous forms; the heterogeneous forms are referred to as XX-topos and XX-chronos and depends on which term in the error minimization equation is assigned the gappy sensor set. A greedy algorithm is then employed to determine the optimal location of the limited sensor set. The findings are analyzed for different combinations of POD modes and blade-pass frequency harmonics of the sound generated by the hovering rotor.