Non linear N wave source impedance model

Regular paper

Technical University Eindhoven

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

0.7 Lisbon (47)

Abstract:

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:
http://www.win.tue.nl/~dsingh/images/euronoise.pdf