Non linear N wave source impedance model

Regular paper

Deepesh Kumar Singh

Technical University Eindhoven

Monday 1 june, 2015, 11:00 - 11:20

0.7 Lisbon (47)

The N wave sound source also called as buzz-saw noise is produced when the blade tip mach number of the aero-engine fan exceeds unity and the resultant sound field is predominantly not only the first harmonic of the blade passing frequency but also contains several other harmonics. The amplitude of these other harmonics is inversely proportional to the blade passing frequency. Typically, we can describe N wave pressure field by the series <latex> \begin{equation} p'_{\mathrm{ex}} = F_{0} \sum_{n=1}^{\infty} \frac{\sin(n \omega t)}{n} \end{equation}</latex> where <latex>$\omega $</latex> and <latex>$F_0 $</latex> are the blade passing frequency, 1BPF and amplitude of excitation respectively. The liners are usually constructed corresponding to 1BPF. The present work aim to apply non linear corrections to a Helmholtz resonator type impedance based on a systematic asymptotic solution of the pertaining equations when the resonator is excited by N wave. The impedance values are obtained at the resonance frequency for various N wave harmonics and includes the non linear correction. We have the following equation <latex> \begin{equation} (1+2\epsilon \Delta)\frac{d^2 y}{d \tilde{\tau}^2}+\epsilon \frac{d y}{d \tilde{\tau}} \left|\frac{d y}{d \tilde{\tau}}\right| + \epsilon r \frac{d y}{d \tilde{\tau}} + y = \epsilon F_{0} \sum_{n=1}^{\infty} \frac{\sin n(\tilde{\tau}+\theta)}{n}. \end{equation} </latex> We wish to solve this equation asymptotically to obtain the normalized pressure amplitude <latex>$ y $</latex> which is used to obtain the pressure and velocity. The impedance at any harmonic of the natural frequency is obtained by the definition <latex> \begin{equation} Z(\eta = n\omega) = - \frac{\int_{- \infty}^{\infty}p'_{\mathrm{ex}} e^{-i \eta t}dt}{\int_{- \infty}^{\infty} u'_{\mathrm{ex}} e^{-i \eta t }dt}. \end{equation}</latex> A detailed derivation and some preliminary results can be found at the following link:

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