Design and Simulation of Two Stroke Engines

Chapter 8 • Reduction of Noise Emission from Two-Stroke Engines
For any sinusoidal variation, the mean square sound pressure level, p,^, of that nth frequency
component is given by:
n 2 = Pa.
Prms n _
(8.4.1)
Coates shows that the mean square sound pressure emanating from a sinusoidal efflux
from a pipe of diameter, d, is given by:
2 l
( \
(Prms) =-
{ 4 2
(f ) c
P
)
2Ji 7rfnd sin 0 '
a
0 J
7ifnd sin 0 (8.4.2)
where Ji is the notation for the first-order Bessel function of the terms within. It can be seen
from Eq. 8.4.2 that the combination of the second and fourth terms is the instantaneous mass
flow rate at the aperture of the exhaust, or intake, system. The final term, in the curly brackets,
indicates directivity of the sound, where 0 is the angle between the receiving microphone
from the centerline particle flow direction of the aperture. The variable, rm, is the distance of
the microphone from the pipe aperture to the atmosphere. From the theory in Sec. 2.2.3, and
Eq. 2.2.11 in particular, the instantaneous mass flow rate at the aperture to the atmosphere is
given by:
rh = G5a0p0
v4
G5>
(X; + Xr - 1) W (X, " Xf) (8.4.3)
where the Xj and Xr values are the time-related incident and reflected pressure amplitude
ratios, respectively, of the pressure pulsations at the aperture to the atmosphere. An engine
simulation of the type demonstrated in Chapter 5, using the theory of Chapter 2, inherently
computes the instantaneous mass flow rate at every section of the engine and its ducting,
including the inlet and exhaust apertures to the atmosphere. The instantaneous mass flow rate
at the aperture to the atmosphere is collected at each time step in the computation and a
Fourier analysis of the resulting periodic function is performed numerically to include all
harmonics within the audible range, giving a series of the form:
rht = cp0 + (cpal sin cot +