varies, the box will resonate, much like an organ pipe, at various integer amounts of half the wavelength(1.8), in other words at 2*LB, 2*LB/2,2*LB/3,2*LB/4, etc.
From Eq. 8.5.1.2, this means at frequencies of A/2*LB, 2*A/2*LB, 3*A/2*LB, 4* A/2*LB, and so on. At these frequencies the silencer will provide a transmission loss of zero, i.e., no silencing effect at all, and such frequencies are known as the
"pass bands." It is also possible for the silencer to resonate in the transverse direction through the diametral dimension, DSB, and provide further pass-band frequencies at what would usually be a rather high frequency level.
There are many empirical equations in existence for the transmission loss of such a silencer, but Kato and Ishikawa(8.18) state that the theoretical solution of Fukuda(8.9) is found to be useful. The relationship of Fukuda(8.9) is as follows for the transmission loss of a diffusing silencer, TRd, in dB units:
TRd=10*log
10[Er2*f(k,L)]
2dB (8.5.1.3)
where, f(k,L)=(sin(k*LB)*sin(k*LT))/(cos(k*Ll)*cos(k*L2)} (8.5.1.4)
and, k=2*7t*FR/A (8.5.1.5) Not surprisingly, in such empirical relationships there are many correcting
factors offered for the modification of the basic relationship to cope with the effects of gas particle velocity, end correction effects for pipes which are reentrant or flush with the box walls, boxes which are lined with absorbent material but not suffi- ciently dense as to be called an absorption silencer, etc., etc. It will be left to the reader to pursue these myriad formulae which are to be found in the references.
To assist the reader with the use of the basic equations for design purposes, a simple computer program is included with this book in the Computer Program Appendix, Prog.8.1, "DIFFUSING SILENCER." This appears asProgList.8.1. The attenuation equations which are programmed are those seen above from Fukuda, Eqs. 8.5.1.3-5. '
To determine if such a design program is useful in a practical sense, an analysis
of SYSTEM 2 and SYSTEM 3, emanating from Coates(8.3), is attempted and the
results shown in Figs. 8.12 and 8.13. They show the computer screen output from
Prog.8.1, which are plots of the attenuation in dB as a function of noise frequency
up to a maximum of 4 kHz, beyond which frequency most experts agree the
diffusing silencer will have little silencing effect. The theory would continue to
predict some attenuation to the highest frequency levels, indeed beyond the upper
The Basic Design of Two-Stroke Engines
threshold of hearing. Also displayed on the computer screen is the input data for the geometry of that diffusing silencer according to the symbolism presented in Fig. 8.7 and in this section. However, the input data is in the more conventional length dimensions in mm units and temperature in deg. C values; they are converted within the program to strict SI units before being entered into the programmed equations for the attenuation of a diffusing silencer according to Fukuda(8.9), Eqs. 8.5.1.3-5.
The reader can check the input dimensions of SYSTEM 2 and SYSTEM 3 from Fig.
8J2.
The attenuation of SYSTEM 2, as predicted by the theory of Fukuda and shown in Fig. 8.12, has a first major attenuation of 14 dB at 320 Hz and the first two pass- band frequencies are at 550 and 1100 Hz. If one examines the measured noise fisquency spectrum in Fig. 8.4, it is to find that an attenuation hole of 12 dB is CK ated at a frequency of 400 Hz. Thus the correspondence with the theory of ftfkuda, with regard to this primary criterion, is quite good and gives some OB nfidence in its application for this particular function. There is also some evidence oftthe narrow pass band at 550 Hz and there is no doubt about the considerable pass- tar id frequency at 1100 Hz in the measured spectra. There is no sign of the predicted
' 5 0 DESIGN FOR A DIFFUSING SILENCER
4 0 -
30
20
10
CALCULATION BY Prog.3.1 DATA FROM Coates/Blair SYSTEM 2
INPUT DATA LB= 305 DSBr 76 D S U 28.6 DS2= 28.6 L1= 0 L2= 0 LT= 152 T2C= 10
320 760
1000 2000 3000
FREQUENCY, Hz 4000
K g. 8.12 Calculation by Prog.8.1 for the silencing characteristics of SYSTEM 2.
Chapter 8 - Reduction of Noise Emission from Two-Stroke Engines
DESIGN FOR A DIFFUSING SILENCER CALCULATION BY Prog.8.1 DATA FROM Coates/Blair SYSTEM 3
I I
2000 3000 4000 FREQUENCY, Hz
Fig. 8.13 Calculation by Prog.8.1 for the silencing characteristics of SYSTEM 3.
attenuation, nor the pass-band holes, at frequencies above 1.5 kHz in the measured spectrum. There is also no evidence from the theoretical solution of the reason for the attenuation in the measured spectrum of the fundamental pulsation frequency of 133 Hz, as commented on in Sect. 8.4.2. The general conclusion as far as SYSTEM 2 is concerned is that Prog.8.1 is a useful empirical design calculation method for a diffusing silencer up to a frequency of 1.5 kHz.
The attenuation of SYSTEM 3, as predicted by the theory of Fukuda and shown in Fig. 8.13, has a first major attenuation of 19 dB at 320 Hz, i.e., greater than that of SYSTEM 2. It is true that the measured silencing effect of SYSTEM 3 is greater than SYSTEM 2, but not so at the frequency of 400 Hz, where the measured attenuations are both at a maximum and are virtually identical. The theory would predict that the pass-band hole of SYSTEM 2 at 1100 Hz would not be so marked for SYSTEM 3, and that can be observed in the measured noise spectra. The theoretically predicted pass-band at 600 Hz is clearly seen in the measured noise diagram. Again, there is little useful correlation between theory and experiment after 1500 Hz.
The general conclusion to be drawn, admittedly from this somewhat limited
quantity of evidence, is that Prog.8.1 is a useful empirical design program for
diffusing silencers up to a frequency of about 1500 Hz.
8.5.2 The side-resonant type of exhaust silencer
The fundamental behavior of this type of silencer is to absorb a relatively narrow band of sound frequency by the resonance of the side cavity at its natural frequency.
The sketch in Fig. 8.8 shows a silencing chamber of length LB and area SB, or diameter DSB if the cross-section is circular. The exhaust pipe is usually located centrally in the silencer, is of area S3, or diameter DS3 if it is a round pipe, and has a pipe wall thickness TH. The connection to the cavity, whose volume is VB, is via holes, or a slit, or by a short pipe in some designs. The usual practice is to employ a number of holes, NH, of area SH, or diameter DSH if they are round holes for reasons of manufacturing simplicity. The volume of the resonant cavity is VB where, if all cross-sections are circular:
VB=LB*SB-[7t*LB*(DSB+2*TH)
2/4] (8.5.2.1)
According to Kato(8.18), the length, LH, that is occupied by the holes should not exceed the pipe diameter, DSB, otherwise the system should be theoretically treated as a diffusing silencer. The natural frequency of the side-resonant system, FRsr, is given by Davis(8.4) as:
FRsr=(A/2*Jc)*V(K/VB) (8.5.2.2) where K, the conductivity of the opening, is calculated from:
K=NH*SH/(TH+0.8*SH) (8.5.2.3) The attenuation or transmission loss in dB of this type of silencer, TRsr, is given
by Davis(8.4) as:
TRsr=10*log
10[l+Z
2] (8.5.2.4)
where the term Z is found from:
Z= {(V(K*VB))/(2*S3))/{(FR/FRsr)-(FRsr/FR)} (8.5.2.5) When the applied noise frequency is equal to the resonant frequency, FRsr, the
value of the term Z becomes infinite as does the resultant noise attenuation in Eq.
8.5.2.4. Clearly this is an impractical result, but it does give credence to the view that such a silencer has a considerable attenuation level in the region of the natural frequency of the side-resonant cavity and connecting passage.
To help the reader to use these acoustic equations for design purposes, a simple computer program is included with this book in the Computer Program Appendix, Prog.8.2, "SIDE-RESONANT SILENCER." This is listed as ProgList.8.2. The attenuation equations which are programmed are those discussed above from Davis, Eqs. 8.5.2.2-5.
Chapter 8 - Reduction of Noise Emission from Two-Stroke Engines To demonstrate the use of this computer design program, and to determine if the predictions emanating from it are of practical use to the designer of side-resonant elements within an exhaust muffler for a two-stroke engine, the geometrical and experimental data pertaining to SYSTEM 4 of Coates(8.3) are inserted as data and the calculation result is illustrated in Fig. 8.14. The information in that figure is the computer screen picture as seen by the user of Prog.8.2. The information displayed is the input data for, in this instance, SYSTEM 4 and the reader can check the methodology of data input from Fig. 8.2, and the output data which is the attenuation in dB over a frequency range up to 4 kHz. The reader may wonder about 10°C being the declared data value for exhaust temperature, but the simulation work of Coates(8.3) was conducted by a rotor valve delivering very realistic exhaust pulses, but in cold air!
The calculated sound attenuation of SYSTEM 4, as seen in Fig. 8.14, shows a large peak of 50 dB transmission loss at 600 Hz, with further attenuation stretching to 2.5 kHz. If one examines the measured noise spectrum of SYSTEM 4, and compares it to the unsilenced SYSTEM 1, it can be seen that there is considerable
DESIGN FOR A SIDE-RESONANT SILENCER
INPUT DATA LB= 305 DSB= 76 DS3= 28.6 TH= 1.5
NH= 40 DSH= 3.18 T^C= 10
CALCULATION BY Prog.8.2 DATA FROM Coates/Blair SYSTEM 4
- i — i — r — | — i — i — l — i — i — i — r -
1000 2000 FREQUENCY, Hz
Fig. 8.14 Calculation by Prog.8.2 for the silencing characteristics of SYSTEM 4.
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The Advanced Techniques in Two-Stroke Engine Design
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