ASSPIN Users' Manual - CAFE Foundation

the approximation of g in the neighborhood of the adjacent roots. Consequently, if theroots of the cubic near an extreme point of g are used as initial guesses, then Newton'smethod converges rapidly to the nearest roots of g. This technique ensures that thesearch for all roots of the ItTE is exhaustive.SpectralAnalysisFor a given observer point, 2', translating with the propeller, the periodic acousticsignature generated by a propeller wi_h N u blades is obtained by the superposition ofthe acoustic pressure signatures produced by the individual blades. If the blades areidentical and symmetrically spaced and if 1)_(2, t) is the signature of one of the blades,then the cumulative pressure signal is given by/V B'- (p(x, l=n= 1e,t+For each observer location, ASSPIN calculates pt1 at a discrete number of equally spacedtime points, and applies the above formula to obtain the combined acoustic signature.Before proceeding to the details of how p_ is calculated, the spectral analysis of p, isconsidered. Acoustic output from ASSPIN consists of discrete pressure signatures forthe loading, thickness, and combined noise, and the sound pressure levels as a functionof harmonic number for each noise component. The temporal Fourier series of theacoustic pressure can be writtenO(3p (x, t) = an(2',') cosa_nt + b,dx)sincont,n:Owhere COn = nNBw. The Fourier coefl:icients are given byT.n(x) = -__,(.,cos _'ntdt,(17a)0andTb,,(_) = _ 2 f p ,_. (., t)sin._,,tdt,0(17b)where T -27cN B w17