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Report on Testing and Design of an Integrated Muffler_Catalytic

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					  Report on Testing and Design of an Integrated
  Muffler/Catalytic Converter for a Lawn Mower




  David Schwedler, Matt McQueen, Yun Guan, and Dr. Mohan Rao
         Michigan Technological University, Houghton, MI 49931
dlschwed@mtu.edu, mcmquee@mtu.edu, yuguan@mtu.edu, mrao@mtu.edu
Muffler/ Catalytic Converter                                                        Final Report


Abstract
The objective of this project was to test the effectiveness of a new integrated muffler for engines used in
lawn mowers and suggest ways of improving its design to make it at least as or more quiet than the
original muffler. Baseline sound pressure measurements were taken and analyzed comparing the lawn
mower’s stock muffler, the original muffler, and the new muffler. Measurements were also taken without
the muffler to provide insertion loss data. A mathematical model of the new and original muffler was
created using the Acoustical “Transfer Matrix Method” to establish the transmission loss of each muffler.
This model was then modified to create and compare a new design by using the measurement of the lawn
mower without a muffler and subtracting the transmission loss of the mufflers. Three modifiable design
parameters were established; placement of catalyst in the muffler and the lengths at which the inlet and
tailpipes protrude into the muffler cavity. A designed experiment was run on the model to find the
optimum design. The result is a design that is theoretically better than both the original muffler and the
new integrated muffler.

Introduction

VConverters, Inc., a small business engaged in the design and fabrication of an integrated muffler and
catalytic converter, is sponsoring the analysis and design of an integrated muffler/catalytic converter for
use on a single cylinder two-stroke engine for lawn mowers. The objective of this project was to test the
effectiveness of a new integrated muffler for engines used in lawn mowers and suggest ways of
improving its design to make it at least as or more quiet than the original muffler.




                                 Stock                                       Original Muffler
                                 Muffler




                                                      New Muffler


                                           Figure 1: Mufflers

The term Catalytic Converter covers the stainless steel box mounted in the exhaust system.
Inside the cover is the catalyst, a ceramic or metallic base with an active coating incorporating
alumina, ceria and other oxides and combinations of the precious metals platinum, palladium and
rhodium.

The regulated emissions due to imperfect combustion include the following:


Page 2 of 22
Muffler/ Catalytic Converter                                                       Final Report

Carbon monoxide (CO): A colorless, odorless gas. It is poisonous and extremely dangerous in confined
areas, building up slowly to toxic levels without warning if adequate ventilation is not available.

Hydrocarbons or volatile organic compounds (VOCs): Any chemical compound made up of hydrogen
and carbon.

Oxides of nitrogen (NOx): Chemical compounds of nitrogen, they combine with hydrocarbons to
produce smog.

These are the three main regulated emissions, and also the ones that catalytic converters are designed to
reduce.

There are two main types of structures used in catalytic converters:
The Reduction Catalyst
The reduction catalyst is the first stage of the catalytic converter. It uses platinum and rhodium to
help reduce the NOx emissions. When an NO or NO2 molecule contacts the catalyst, the catalyst
rips the nitrogen atom out of the molecule and holds on to it, freeing the oxygen in the form of
O2. The nitrogen atoms bond with other nitrogen atoms that are also stuck to the catalyst,
forming N2. For example:
                                 2NO => N2 + O2 or 2NO2 => N2 + 2O2
The Oxidization Catalyst
The oxidation catalyst is the second stage of the catalytic converter. It reduces the unburned
hydrocarbons and carbon monoxide by burning (oxidizing) them over a platinum and palladium
catalyst. This catalyst aids the reaction of the CO and hydrocarbons with the remaining oxygen
in the exhaust gas. For example:
                                          2CO + O2 => 2CO2



Experimental Setup for Muffler Sound Pressure Measurements

The original experimentation carried out by VConverters used a 2 stroke single cylinder engine that
was manufactured by AS Motor of Germany. A suitable 2 stroke single cylinder engine could not be
located so a lawn mower with a 4 stroke 2 cylinder engine manufactured by Briggs and Stratton was
located to carry out testing with the sponsor’s approval. Measurements to be taken included sound
pressure levels with no muffler, with the lawn mower’s stock muffler, the original muffler provided by
VConverters, and the new muffler provided by VConverters.

The initial testing measured sound pressure at 8, 16, and 32 inches from the muffler. A
measurement was also taken from the non-exhaust side of the mower in order to determine if
structure-borne noise would be considered. The results from the testing were very inconclusive.
It was noticed that a negative insertion loss was occurring in several octave bands for the original
and new muffler. Upon further inspection, it was deemed that the testing was flawed because the
measurement setup unintentionally created a reverberant field at close distances. The testing at
various distances was also deemed unnecessary because the measurements were being taken
outdoors away from large obstructions to avoid reverberant field measurements.




Page 3 of 22
Muffler/ Catalytic Converter                                              Final Report


There is no specific standard for conducting sound measurements of walk behind lawn mowers,
so ANSI S12.151992 (ASA 106-l 992) American National Standard for Acoustics - Portable
Electric Power Tools Stationary and Fixed Electric Power Tools, And Gardening Appliances
Measurement of Sound Emitted was used as a guide for measurements. The standard setup
appeared to be a five point measurement system from equal distances around the source. The
second test was carried out measuring from 64 inches above, behind, in front, from the exhaust
side, and from the non-exhaust side of the lawn mower (Figure 2). Time traces were taken of
each muffler configuration from those locations and were analyzed in the 1/3 octave band. From
this analysis insertion loss, sound power, and loudness were calculated.



                                         Above Position
          Front Position


                                                                  Right Position




               Exhaust Position                                 Back Position
                                                                [Operator Location]
                    Figure 2: Test 2 Microphone Locations

Data Analysis of Muffler Sound Pressure Results

The data from the testing was then analyzed. By using measurements taken of the surrounding
environment before the lawn mower was started the background noise A-weighting overall level
was calculated to be 61.6 dBA and the linear weighting overall level was 83.4 dB (Figure 3).
And the loudness for background noise is 15.47 Sones (67.5 dB equivalent SPL at 3150 Hz).
The sound pressure level results for each muffler at each location were over 10 dB larger than
the background noise so the noise effects of the background can be considered negligible.

Analysis was made for the 1/3 octave band for each muffler at the 5 locations. From these
measurements the average overall levels were calculated. Averages for each frequency at five
locations were found using the following equation:




Page 4 of 22
Muffler/ Catalytic Converter                                                                                                                                                                   Final Report


                                                           1 N 0.1*L pi
                             LPavg             10 log 10 ( *  10      )
                                                           N i 1
                             N = Number of locations here it is 5 locations.
                             LPi = SPL at location i


                                                                                                 Background test
                   90


                   80


                   70


                   60


                   50
              dB




                   40


                   30


                   20


                   10


                   0
                                                                                                                          1k



                                                                                                                                        1.6 k
                                                                                                                                                2k

                                                                                                                                                     2.5 k



                                                                                                                                                                      4k
                                                                                                                                                                           5k

                                                                                                                                                                                6.3 k
                                                                                                                                                                                        8k

                                                                                                                                                                                             10 k
                                                                                                                               1.25 k




                                                                                                                                                             3.15 k




                                                                                                                                                                                                         Lin*
                        20

                             25
                                  31.5

                                         40
                                              50

                                                   63
                                                        80

                                                             100
                                                                   125

                                                                         160
                                                                               200

                                                                                     250
                                                                                           315

                                                                                                 400
                                                                                                       500

                                                                                                             630
                                                                                                                    800




                                                                                                                                                                                                    A*
                                                                                     1/3 octave band


                                              Figure 3: Ambient Background Sound Pressure


From the averages the A weighted and linear weighted over all levels were calculated (Table 1).

                   Location                                                    Weighting                           No muffler                                Stock                      Original                New
 (64” to the center of the lawn mower)
 Exhaust Side                                                                  A*(dBA)                                          98.6                                  90.5                          90.7        91.5
                                                                               Lin*(dB)                                        102.6                                  96.5                          98.6        99.8
 Right Side                                                                    A*(dBA)                                          92.6                                  90.5                          90.7        91.5
                                                                               Lin*(dB)                                         99.1                                  96.5                          98.6        99.8
 Front                                                                         A*(dBA)                                          95.9                                  89.4                          89.2        89.7
                                                                               Lin*(dB)                                        100.6                                  95.3                          96.1        97.4
 Back                                                                          A*(dBA)                                          91.4                                  86.6                          86.9        87.6
                                                                               Lin*(dB)                                         97.4                                  91.9                          94.9        96.6
 Above                                                                         A*(dBA)                                          95.3                                  88.2                          88.6        88.5
                                                                               Lin*(dB)                                         99.3                                  94.4                          95.5        96.4
 Average SPL at 64”                                                            A*(dBA)                                          95.5                                  89.3                          89.4        90.0
                                                                               Lin*(dB)                                        100.2                                  95.2                          97.0        98.3

                                                             Table 1: Sound Pressure Levels


Page 5 of 22
Muffler/ Catalytic Converter                                                                               Final Report



After plotting the average sound pressure level from the 5 locations for each muffler
configuration there are several observations that can be made. It was observed that when no
muffler is present the sound pressure level is above the other sound pressure levels as would be
expected (Figure 4). The stock muffler also appears most effective around 200 Hz. At around
200 Hz the sound pressure level for new muffler is larger than that without muffler which could
explain VConverters perceived increase in loudness. The reason for this is shown in the testing
done on the catalyst that shows it provides a negative noise reduction at approximately the same
frequency; see Figure 8.

                                                           Average 1/3 Octave @ 64"

                          100

                          95

                          90

                          85                                                                               No Muffler
               SPL (dB)




                                                                                                           Stock
                          80
                                                                                                           Original
                          75                                                                               New

                          70

                          65

                          60
                                                      5

                                                             0

                                                                    5

                                                                           0

                                                                                     0
                            20



                                         50

                                              80




                                                                                     k



                                                                                               k
                                    .5




                                                                                               k



                                                                                                   k

                                                                                                       k
                                                   12

                                                          20

                                                                 31

                                                                        50

                                                                                  80

                                                                                  25



                                                                                            15
                                                                                             2



                                                                                                   5

                                                                                                       8
                                 31




                                                                               1.



                                                                                         3.




                                                                 Frequency (Hz)




               Figure 4: Average 1/3 Octave Sound Pressure Level from 64 Inches

The insertion loss was then calculated using the equation:

               Insertion Loss = LPavg (without muffler) - LPavg (with muffler)

Again, several trends become apparent from the insertion loss data (Figure 5). It was noticed
that the stock muffler is the most effective among the mufflers, with the original muffler being
second, and the new muffler is the third most effective. At low frequencies the new muffler is the
most effectual, then the original, and finally the stock muffler. At around 200 Hz for new
muffler and around 150 Hz for original muffler the insertion loss is negative. The possible reason
is again the same; The muffler is not tuned properly to the input exhaust noise, or the holes in the
baffle plate in the original muffler and the holes in the catalyst in the new muffler could be acting
as a sort of tuned whistle at those frequencies due to the forced exhaust air moving through them.




Page 6 of 22
Muffler/ Catalytic Converter                                                                                            Final Report


                                                                Insertion Loss Average @ 64"

                                     20


                                     15
               Insertion loss (dB)


                                     10

                                                                                                                        Stock
                                      5                                                                                 Original
                                                                                                                        New

                                      0
                                                                5

                                                                       0

                                                                              5

                                                                                     0

                                                                                            0
                                      20



                                                   50

                                                        80




                                                                                                k



                                                                                                            k
                                              .5




                                                                                                    k



                                                                                                                k

                                                                                                                    k
                                                             12

                                                                    20

                                                                           31

                                                                                  50

                                                                                         80

                                                                                               25



                                                                                                           15
                                                                                                    2



                                                                                                                5

                                                                                                                    8
                                           31




                                                                                            1.



                                                                                                        3.
                                      -5


                                     -10
                                                                            Frequency (Hz)




                                     Figure 5: Average Insertion Loss of Mufflers from 64 Inches

Sound power calculation:
The sound power was calculated assuming a free field. The equations used for sound power in a
free field were:

                 L w  L p  10 log 10 A 
               Assuming a hemi-spherical surface R = 1.626 m (64”).
               A = 2πR2 = 16.612m2
               10*log10(A) = 12.2dB so,
                  Lw  L p  12 .2dB

The sound power levels display a trend similar to the sound pressure levels (Figure 6). The
sound power level without muffler was above the levels of the mufflers. For the frequency range
between 800 Hz and 5K there is no noticeable difference for the three mufflers. Around the 200
Hz band the new muffler has a sound power spike that again may be attributed to muffler
possibly working as an amplifier for that frequency.




Page 7 of 22
Muffler/ Catalytic Converter                                                                                        Final Report


                                                                    Sound Power Levels

                                  110

                                  105

               Sound Power (dB)   100

                                   95                                                                              No muffler
                                                                                                                   Stock
                                   90
                                                                                                                   Original
                                   85                                                                              New

                                   80

                                   75

                                   70
                                                         5

                                                                0

                                                                       5

                                                                              0

                                                                                        0
                                      20



                                            50

                                                 80




                                                                                        k



                                                                                                  k
                                       .5




                                                                                                  k



                                                                                                       k

                                                                                                           k
                                                      12

                                                             20

                                                                    31

                                                                           50

                                                                                     80

                                                                                     25



                                                                                               15
                                                                                                2



                                                                                                      5

                                                                                                           8
                                    31




                                                                                  1.



                                                                                            3.
                                                                     Frequency (Hz)




                                                      Figure 6: Sound Power Levels

Applying the Steven’s Mark VII perceived loudness calculation the equivalent loudness at 3150
Hz is found (Table 2). From these values it can be seen that each muffler worked well to reduce
the perceived noise. It can also be noted that the stock muffler works more effectively than the
original and new mufflers. The perceivable difference between original muffler and new muffler
is only about 1 dB, much lower than what is appreciable by the human ear.

                                  Table 2: Loudness Results

                                                              No muffler                Stock         Original     New

                                  Loudness(Sones)                    213.27           107.05          118.92     125.31


                                  Equivalent SPL                      101.5                 92.3           94      94.8
                                  at 3150Hz (dB)



Impedance and Transmission Loss Testing of the Catalyst

An impedance measurement test was performed on the catalyst element using ASTM Standard
E1050-98. The Catalyst was cut to the proper length and into to different diameters to perform
high and low frequency tests using both a large and small impedance tube. Figure 7 shows how
the catalysts were cut. The real and imaginary parts of the impedance were found to give
complex impedance (Figure 8). This complex impedance was used to later model the
transmission loss of the catalyst.




Page 8 of 22
Muffler/ Catalytic Converter                                                                                  Final Report




                                              Figure 7: Catalyst as cut and prepared for Impedance Test



                                                               Material Acoustic Property Measurement
                                                               Two Microphone Acoustic Impedance (ASTM E1050-98)

                              Normalized Impedence                                                  Real and Imaginary Parts
                       2500
                                                                                                           Real Part -- --
                                                                                                           Imaginary Part -- --

                       2000



                       1500
 Impedance (kg/m2*s)




                       1000



                       500



                         0



                       -500

                         50   60     80       100        200            400      600   800 1000     2000                 4000     6400
                                                                        (log) Frequency, Hz
                        Material : catalyst
                        Comments :


                                                         Figure 8: Impedance Test Results


A test was also performed to determine the insertion loss caused by the catalyst. This was done
by placing the catalyst in the impedance tube with an open end (covered by foam to prevent the
introduction of room noise) and microphone on the opposite side from the source noise of the as


Page 9 of 22
Muffler/ Catalytic Converter                                              Final Report


shown in Figure 9. A measurement was taken with the tube empty, and then the catalyst was
added. The results of this test are shown in Figure 10 and 11. As stated above, the inser tion
gains seen in the test are also at the same frequency for the insertion gain in this test.




                               Figure 9: Noise Reduction Test setup




Page 10 of 22
Muffler/ Catalytic Converter                                                                Final Report


                                            Insertion Loss due to Catalyst Full Size

                                10


                                 8

                                 6
          Insertion Loss (dB)




                                 4

                                 2

                                 0
                                         20
                                         25

                                         40
                                         50
                                         63
                                         80
                                        100
                                        125
                                        160
                                        200
                                        250
                                        315
                                        400
                                        500
                                        630
                                        800
                                       31.5




                                         A*
                                       Lin*
                                        1k


                                        2k


                                        4k
                                        5k

                                        8k
                                      1.6 k

                                      2.5 k



                                      6.3 k

                                       10 k

                                       16 k
                                       20 k
                                     1.25 k



                                     3.15 k




                                     12.5 k
                                -2


                                -4

                                -6
                                                             Frequency (Hz)


                                     Figure 10: Noise Reduction due to Catalyst Full Size


                                             Insertion Loss due to Catalyst Small

                                8

                                7

                                6

                                5
          Insertion Loss (dB)




                                4

                                3

                                2

                                1

                                0
                                         20
                                         25

                                         40
                                         50
                                         63
                                         80
                                       31.5




                                        100
                                        125
                                        160
                                        200
                                        250
                                        315
                                        400
                                        500
                                        630
                                        800




                                         A*
                                       Lin*
                                        1k

                                      1.6 k
                                        2k
                                      2.5 k

                                        4k
                                        5k
                                      6.3 k
                                        8k
                                       10 k

                                       16 k
                                       20 k
                                     1.25 k




                                     3.15 k




                                     12.5 k




                                -1

                                -2

                                -3
                                                            Frequency (Hz)


                                      Figure 11: Noise Reduction due to Catalyst Small




Page 11 of 22
Muffler/ Catalytic Converter                                                                      Final Report


                                                     Modeling of Mufflers
The Original and New mufflers were modeled in Mat lab using the Transfer Matrix Method 1.
The components that were modeled are numbered for each of the mufflers below along with the
corresponding general form Transfer Matrix for each.

                      Original Muffler                   Original Muffler Transfer Matrices:
                                                                              cos(k * L)    j * y * sin(k * L)
                                                         T1  T3  T5  T7   j                               
              1         4                                                     * sin(k * L)     cos(k * L) 
                                                                             y                                
      2           2                                                      1          0
                              5                           T2            1           
                                                                j * y * cot(k * L) 1
              3                                                                      
                                         6           6                1      c*k 2
                                                                                         c * k * 14  1.7 * rhole    
                                                               1           *        j
                                                                                                                     
                                                                                                                     
                                                 7        T4       N hole  
                                                                                                 SAhole             
                                                               0
                                                                                          1                            
                                                                                                                        

                                                           1                            0
Figure 12: Original Muffler (not to scale) T6             1                             
                                                                                         1
                                                  j * y * cot(k * L)
                                                                                         

Note: Element 4 is a baffle plate with 5mm holes around the outside every 15º

                      New Muffler                           New Muffler Transfer Matrices:
          1            3
                                                                                  cos(k * L)        j * y * sin(k * L)
                                                           T1  T2  T4  T5   j                                     
                                                                                  * sin(k * L)         cos(k * L) 
          2
                                             4
                                                                                 y                                    
                                                                 1 Z catalyst 
                                                            T3  
                                                                 0      1    
                                                     5




Figure 13: New Muffler (not to scale)
Note: Element 3 is the catalyst

The transfer matrices were then applied to the following equations to determine the system
transmission loss.



1
    Transfer Matrix information taken from hand written notes from Michigan Tech.’s MEEM-5702


Page 12 of 22
Muffler/ Catalytic Converter                                                         Final Report


                                                                  B                 
                                                                A C* y  D        
                         A B                                    y                 
T1 * T2 * T3 .....* Tn                     TL  20 * log 10                      
                         C D                                 
                                                                     2
                                                                                     
                                                                                    
                                                                                    

Muffler Design Process

As established by VConverters, the only modifiable parts of the muffler could be the inlet and tail pipe
and the placement of the catalyst within the muffler. The team decided to tune the muffler by varying the
placement of the catalyst and setting a length at which the inlet and tail pipe would protrude into the
cavity of the muffler. A Mat lab model of this design was created as seen in Figure lead to the creation of
the muffler model in Figure 14.

           Designed Muffler                    Designed Muffler Transfer Functions:
                                                                    cos(k * L)      j * y * sin(k * L)
       1       4                             T1  T3  T5  T7   j                                   
                                                                    * sin(k * L)       cos(k * L) 
                                                                   y                                  
   2       2
                                                            1          0
                      5
                                             T2            1           
       3                                           j * y * cot(k * L) 1
                                 6       6
                                                                        

                                     7
                                                   1 Z catalyst 
                                              T4  
                                                   0       1   
                                                              1          0
       Figure 14: Designed Muffler            T6             1           
                                                     j * y * cot(k * L) 1
                                                                          
Note: Element 4 is the catalyst

A designed experiment was run using the three variables shown in Figure 15. To compare
different muffler designs the data collected from the measurements without the muffler were
inserted into the model, the transmission loss was applied for a 1/3 octave band, the frequency
bands were A-weighted and summed. The over all A-weighted Sound Pressure Level was then
used for comparison. The experiment consisted of running iterations for each variable at a 0x,
0.5x, 1x, 1.5x, and 2x test band to obtain a regression for each variable. From the regression an
optimal point was chosen and a narrower test band was used to narrow in on the optimal solution.
The last of 7 iterations is shown in Table 3. Figure 16 shows the per frequency theoretical
transmission loss between the original, new, and the designed mufflers. The large peek in the
original muffler is due to the fact that the impedance of the throat inlet and outlet is characterized
by and equation that includes the co-tangent, and for transmission loss requires that a shunted
element by applied as the inverse of the impedance. For the frequency of 4000Hz the inverse of
the impedance moves toward the tangent of π/2, or a transmission loss of infinity. In reality this
drastic transmission loss is not possible however it is this point that should be tuned to the most
critical frequency.



Page 13 of 22
Muffler/ Catalytic Converter                                                                                         Final Report



                                                                                                                                       SPL
                             X                                                                         x (mm)    y (mm)     z (mm)     (dBA)
                                                                                                            26        45         65     82.7059
                                                                                                            30        45         65     82.7054
                                                                                                            34        45         65     82.7052
                      Y



                                                                                                            26        50         65     83.1822
                                                                                                            30        50         65      83.181
                                                                                                            34        50         65     83.1798
                                                                                                            26        55         65     85.9931
                                                                                                            30        55         65     85.9815




                                                                               Z
                                                                                                            34        55         65     85.9711
                                                                                                            26        45         70     84.7377
                                                                                                            30        45         70     84.8279
                                                                                                            34        45         70     85.0006
                                                                                                            26        50         70     82.2736
                                                                                                            30        50         70     82.2736
Figure 15: Designed Muffler Variables
                                                                                                            34        50         70     82.2738
                                                                                                            26        55         70     83.1861
                                        Transmission Loss Muffler Comparison
            140                                                                                             30        55         70     83.1854
                                                                            Designed Muffler                34        55         70     83.1848
                                                                            Original Muffler
            120
                                                                            New Muffler                     26        45         75     82.7876
                                                                                                            30        45         75     82.7851
            100
                                                                                                            34        45         75     82.7726
                                                                                                            26        50         75     85.7822
            80
                                                                                                            30        50         75     85.7531
  TL [dB]




            60
                                                                                                            34        50         75     85.7702
                                                                                                            26        55         75     83.1929
            40                                                                                              30        55         75     83.1894
                                                                                                            34        55         75     83.1892
            20                                                                                              28       47.5       67.5    82.4054
                                                                                                            32       47.5       67.5     82.405
             0
                  0   1000       2000    3000   4000 5000 6000       7000      8000   9000 10000            28       52.5       67.5    82.9754
                                                  Frequency [Hz]
                                                                                                            32       47.5       72.5    85.5803
                  Figure 16: Transmission Loss Comparison                                                   28       52.5       67.5    82.9746
                                                                                                            32       47.5       72.5    85.6144
                                                                                                            28       52.5       72.5    82.6134
                                                                                                            32       52.5       72.5    82.6137
                                                                                                   Table 3: Designed Experiment Results




Page 14 of 22
Muffler/ Catalytic Converter                                                  Final Report



Conclusion

The suggestion of the team is to create a muffler similar to the new muffler, but with the
insertion of the inlet and tail pipe into the muffler. The front face of the Catalyst should be
placed 30 mm from the edge of the inlet pipe. The inlet pipe should protrude 50 mm into the
cavity while the tail pipe should protrude 70mm as shown in Figure 17. The team also suggests
further research into the possible perforation of the inlet and outlet pipes.




                                   Figure 17: Designed Muffler

In comparison to the original and new muffler the designed muffler provides an expected lower
theoretical A-weighted overall sound pressure level. As shown in Table 4, the designed muffler
theoretically should perform marginally better then the original, but exceptionally better then the
new muffler. Therefore it is the conclusion of the team that the design in Figure 15 should
perform as well if not better then the original muffler.

                               Modeled Muffler Theoretical Comparison
                                Original       New          Designed
                                Muffler       Muffler        Muffler
                               84.5 dBA      89.3 dBA       82.3 dBA
                                   Table 4: Muffler Comparison




Page 15 of 22
Muffler/ Catalytic Converter                                           Final Report


Mat lab code

Transmission Loss Comparison Code

clear all
%close all                                     dcan=88/1000;
                                               dpipe=32/1000;
%Designed Muffler
fstart=20; %frequency [Hz]                     d1=dpipe;
fstop=10000;                                   d2=dcan;
step=10;                                       d3=dcan;
f=fstart-step;                                 d6=dcan;
for i=1:1:((fstop-fstart)/step+1);             d7=dcan;
   f=f+step;                                   d8=dcan;
                                               d9=dpipe;
Tc=200; %Average Exhaust Temp. [degrees
C]                                             acan=pi/4*dcan^2;
c=331*(1+Tc/273)^.5; %speed of sound           apipe=pi/4*dpipe^2;
[m/s]                                          a1=apipe;
omega=2*pi*f; %angular frequency               a2=pi/4*(d2^2-d1^2);
k0=omega/c; %acoustic wave number              a3=acan;
                                               a6=acan;
Omega(i,1)=omega;                              a7=acan;
K0(i,1)=k0;                                    a8=pi/4*(d8^2-d9^2);
                                               a9=apipe;
%lengths
%Variables for Inlet and Outlet extension of   Y1=c/a1;
tailpipe                                       Y2=c/a2;
y=50;                                          Y3=c/a3;
z=70;                                          Y6=c/a6;
%Variables for Catalyst Length                 Y7=c/a7;
x=30;                                          Y8=c/a8;
q=43.9-x;                                      Y9=c/a9;

l1=(40+y)/1000; %inlet pipe                    Z2=j*Y2*cot(k0*l2);
l2=y/1000; %extension of inlet into first      Z6=339-113*j; %average impedance
chamber                                        Z8=-j*Y8*cot(k0*l8);
l3=(61+x)/1000; %first expansion
chamber                                        %Element 1: Inlet pipe [distributed]
l6=38.1/1000; %thickness of catalyst           A1=cos(k0*l1);
l7=(61+q)/1000; %second expansion              B1=j*Y1*sin(k0*l1);
chamber                                        C1=j/Y1*sin(k0*l1);
l8=z/1000; %extension of tailpipe into         D1=cos(k0*l1);
third chamber                                  T1=[[A1,B1];[C1,D1]];
l9=(40+z)/1000; %tail pipe



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Muffler/ Catalytic Converter                                         Final Report


%Element 2: Extension of inlet pipe into      Ttot=T1*T2*T3*T6*T7*T8*T9;
chamber [parallel]                            AT=Ttot(1,1);
A2=1;                                         BT=Ttot(1,2);
B2=0;                                         CT=Ttot(2,1);
C2=1/Z2;                                      DT=Ttot(2,2);
D2=1;
T2=[[A2,B2];[C2,D2]];                         Ytot=Y1;

%Element 3: Expansion chamber 1               TLd(i,1)=20*log10(abs(AT+BT/Ytot+CT*
[distributed]                                 Ytot+DT)/2);
A3=cos(k0*l3);                                F(i,1)=f;
B3=j*Y3*sin(k0*l3);                           end
C3=j/Y3*sin(k0*l3);                           %
D3=cos(k0*l3);                                %
T3=[[A3,B3];[C3,D3]];                         %
                                              %
%Element 6: Catalyst                          %Original Muffler
A6=1;                                         %
B6=Z6;                                        %
C6=0;                                         %
D6=1;                                         %
T6=[[A6,B6];[C6,D6]];                         fstart=20; %frequency [Hz]
                                              fstop=10000;
%Element 7: Expansion chamber 3               step=10;
[distributed]                                 f=fstart-step;
A7=cos(k0*l7);                                for i=1:1:((fstop-fstart)/step+1);
B7=j*Y7*sin(k0*l7);                              f=f+step;
C7=j/Y7*sin(k0*l7);
D7=cos(k0*l7);                                Tc=200; %Average Exhaust Temp. [degrees
T7=[[A7,B7];[C7,D7]];                         C]
                                              c=331*(1+Tc/273)^.5; %speed of sound
%Element 8: Extension of tailpipe back into   [m/s]
chamber [parallel]                            omega=2*pi*f; %angular frequency
A8=1;                                         k0=omega/c; %acoustic wave number
B8=0;
C8=1/Z8;                                      Omega(i,1)=omega;
D8=1;                                         K0(i,1)=k0;
T8=[[A8,B8];[C8,D8]];
                                              %lengths
%Element 9: Tailpipe [distributed]            l1=72/1000; %inlet pipe
A9=cos(k0*l9);                                l2=27/1000; %extension of inlet into first
B9=j*Y9*sin(k0*l9);                           chamber
C9=j/Y9*sin(k0*l9);                           l3=112/1000; %first expansion chamber
D9=cos(k0*l9);                                (Length to diffusion plate)
T9=[[A9,B9];[C9,D9]];                         l4=6.35/1000; %thickness of diffusion
                                              plate (.25")



Page 17 of 22
Muffler/ Catalytic Converter                                       Final Report


l5=112/1000; %second expansion             %Element 1: Inlet pipe [distributed]
chamber (Length between diffusion plate    A1=cos(k0*l1);
and catalyst)                              B1=j*Y1*sin(k0*l1);
l8=27/1000; %extension of tailpipe into    C1=j/Y1*sin(k0*l1);
second chamber                             D1=cos(k0*l1);
l9=72/1000; %tail pipe                     T1=[[A1,B1];[C1,D1]];

dcan=90/1000;                              %Element 2: Extension of inlet pipe into
dpipe=32/1000;                             chamber [parallel]
                                           A2=1;
d1=dpipe;                                  B2=0;
d2=dcan;                                   C2=1/Z2;
d3=dcan;                                   D2=1;
d4=dcan;                                   T2=[[A2,B2];[C2,D2]];
d5=dcan;
d8=dcan;                                   %Element 3: Expansion chamber 1
d9=dpipe;                                  [distributed]
                                           A3=cos(k0*l3);
acan=pi/4*dcan^2;                          B3=j*Y3*sin(k0*l3);
apipe=pi/4*dpipe^2;                        C3=j/Y3*sin(k0*l3);
a1=apipe;                                  D3=cos(k0*l3);
a2=pi/4*(d2^2-d1^2);                       T3=[[A3,B3];[C3,D3]];
a3=acan;
a4=acan;                                   %Element 4: Diffusion Plate
a5=acan;                                   A4=1;
a6=acan;                                   B4=Z4;
a7=acan;                                   C4=0;
a8=pi/4*(d8^2-d9^2);                       D4=1;
a9=apipe;                                  T4=[[A4,B4];[C4,D4]];

Y1=c/a1;                                   %Element 5: Expansion chamber 2
Y2=c/a2;                                   [distributed]
Y3=c/a3;                                   A5=cos(k0*l5);
Y4=c/a4;                                   B5=j*Y5*sin(k0*l5);
Y5=c/a5;                                   C5=j/Y5*sin(k0*l5);
Y6=c/a6;                                   D5=cos(k0*l5);
Y7=c/a7;                                   T5=[[A5,B5];[C5,D5]];
Y8=c/a8;
Y9=c/a9;
                                           %Element 8: Extension of tailpipe back into
dhole=5/1000;                              chamber [parallel]
ahole=pi/4*dhole^2;                        A8=1;
Z2=j*Y2*cot(k0*l2);                        B8=0;
Z4=1/24*(c*k0^2/pi+j*(c*k0*l4+1.7*dhole/   C8=1/Z8;
2)/ahole);                                 D8=1;
Z8=-j*Y8*cot(k0*l8);                       T8=[[A8,B8];[C8,D8]];



Page 18 of 22
Muffler/ Catalytic Converter                                      Final Report


%Element 9: Tailpipe [distributed]        l3=62/1000; %first expansion chamber
A9=cos(k0*l9);                            (Length to catalyst)
B9=j*Y9*sin(k0*l9);                       l6=38.1/1000; %thickness of catalyst
C9=j/Y9*sin(k0*l9);                       l7=103.9/1000; %second expansion
D9=cos(k0*l9);                            chamber (Length to catalyst)
T9=[[A9,B9];[C9,D9]];                     l9=46/1000; %tail pipe

Ttot=T1*T2*T3*T4*T5*T8*T9;                dcan=88/1000;
AT=Ttot(1,1);                             dpipe=32/1000;
BT=Ttot(1,2);
CT=Ttot(2,1);                             d1=dpipe;
DT=Ttot(2,2);                             d3=dcan;
                                          d6=dcan;
Ytot=Y1;                                  d7=dcan;
                                          d9=dpipe;
TLo(i,1)=20*log10(abs(AT+BT/Ytot+CT*
Ytot+DT)/2);                              acan=pi/4*dcan^2;
end                                       apipe=pi/4*dpipe^2;
%                                         a1=apipe;
%                                         a3=acan;
%                                         a6=acan;
%                                         a7=acan;
%New Muffler                              a9=apipe;
%
%                                         Y1=c/a1;
%                                         Y3=c/a3;
%                                         Y6=c/a6;
fstart=20; %frequency [Hz]                Y7=c/a7;
fstop=10000;                              Y9=c/a9;
step=10;
f=fstart-step;                            Z6=1534.9-1352.07*j;;
for i=1:1:((fstop-fstart)/step+1);
   f=f+step;                              %Element 1: Inlet pipe [distributed]
                                          A1=cos(k0*l1);
Tc=200; %Average Exhaust Temp. [degrees   B1=j*Y1*sin(k0*l1);
C]                                        C1=j/Y1*sin(k0*l1);
c=331*(1+Tc/273)^.5; %speed of sound      D1=cos(k0*l1);
[m/s]                                     T1=[[A1,B1];[C1,D1]];
omega=2*pi*f; %angular frequency
k0=omega/c; %acoustic wave number         %Element 3: Expansion chamber 1
                                          [distributed]
Omega(i,1)=omega;                         A3=cos(k0*l3);
K0(i,1)=k0;                               B3=j*Y3*sin(k0*l3);
                                          C3=j/Y3*sin(k0*l3);
%lengths                                  D3=cos(k0*l3);
l1=46/1000; %inlet pipe                   T3=[[A3,B3];[C3,D3]];



Page 19 of 22
Muffler/ Catalytic Converter                                           Final Report


%Element 6: Catalyst
A6=1;                                          Ytot=Y1;
B6=Z6;
C6=0;                                          TLn(i,1)=20*log10(abs(AT+BT/Ytot+CT*
D6=1;                                          Ytot+DT)/2);
T6=[[A6,B6];[C6,D6]];                          F(i,1)=f;

%Element 7: Expansion chamber 3                end
[distributed]
A7=cos(k0*l7);                                 figure
B7=j*Y7*sin(k0*l7);                            plot(F,TLd,F,TLo,F,TLn)
C7=j/Y7*sin(k0*l7);                            xlabel('Frequency [Hz]')
D7=cos(k0*l7);                                 ylabel('TL [dB]')
T7=[[A7,B7];[C7,D7]];                          title('Transmission Loss Muffler
                                               Comparison')
%Element 9: Tailpipe [distributed]             legend('Designed Muffler','Original
A9=cos(k0*l9);                                 Muffler','New Muffler')
B9=j*Y9*sin(k0*l9);                            %hold on
C9=j/Y9*sin(k0*l9);                            disp('done')
D9=cos(k0*l9);
T9=[[A9,B9];[C9,D9]];

Ttot=T1*T3*T6*T7*T9;
AT=Ttot(1,1);
BT=Ttot(1,2);
CT=Ttot(2,1);
DT=Ttot(2,2);

Designed Experiment Code

clear all                                      q=43.9-x;
%close all
                                               %1/3 Octave band
AA=[0;1;2;0;1;2;0;1;2;0;1;2;0;1;2;0;1;2;0;1;   fs=[20,25,31.5,40,50,63,80,100,125,160,200
2;0;1;2;0;1;2;.5;1.5;.5;.5;1.5;1.5;.5;1.5];    ,250,315,400,500,630,800,1000,1250,1600,2
BB=[0;0;0;1;1;1;2;2;2;0;0;0;1;1;1;2;2;2;0;0;   000,2500,3150,4000,5000,6300,8000,10000
0;1;1;1;2;2;2;.5;.5;1.5;.5;1.5;.5;1.5;1.5];    ];
CC=[0;0;0;0;0;0;0;0;0;1;1;1;1;1;1;1;1;1;2;2;      for i=1:1:28;
2;2;2;2;2;2;2;.5;.5;.5;1.5;.5;1.5;1.5;1.5];       f=fs(i);
for iii=1:1:35;
%Variables for Inlet and Outlet extension of   Tc=200; %Average Exhaust Temp. [degrees
tailpipe                                       C]
y=45+BB(iii)*5;                                c=331*(1+Tc/273)^.5; %speed of sound
z=65+CC(iii)*5;                                [m/s]
%Variables for Catalyst Length                 omega=2*pi*f; %angular frequency
x=4*AA(iii)+26;                                k0=omega/c; %acoustic wave number



Page 20 of 22
Muffler/ Catalytic Converter                                        Final Report


Omega(i,1)=omega;                           dhole=5/1000;
K0(i,1)=k0;                                 ahole=pi/4*dhole^2;
                                            Z2=j*Y2*cot(k0*l2);
%lengths                                    %1/3 Octave Complex Impedance of
                                            Catalyst
l1=(40+y)/1000; %inlet pipe                 Z6r=[36.5;36.5;233;415;191;-151.7;47;497;-
l2=y/1000; %extension of inlet into first   517;-
chamber                                     45;2393;657;461;452;415;328;272;432;364;
l3=(61+x)/1000; %first expansion            303;259;253;245;269.9;622.5;338.748;338.7
chamber                                     48;338.748]
l6=38.1/1000; %thickness of catalyst        Z6i=[-45;-45;94;-249;307;-
l7=(61+q)/1000; %second expansion           36;649;1636;1933;2113;-595;-504;-504;-
chamber                                     656;-517;-402;-316;-316;-455;-
l8=z/1000; %extension of tailpipe into      212;3;3;208;417;607;-
third chamber                               178;113.0769231;113.0769231]
l9=(40+z)/1000; %tail pipe                  Z6=Z6r(i)+j*Z6i(i)
                                            Z8=-j*Y8*cot(k0*l8);
dcan=88/1000;
dpipe=32/1000;                              %Element 1: Inlet pipe [distributed]
                                            A1=cos(k0*l1);
d1=dpipe;                                   B1=j*Y1*sin(k0*l1);
d2=dcan;                                    C1=j/Y1*sin(k0*l1);
d3=dcan;                                    D1=cos(k0*l1);
d6=dcan;                                    T1=[[A1,B1];[C1,D1]];
d7=dcan;
d8=dcan;                                    %Element 2: Extension of inlet pipe into
d9=dpipe;                                   chamber [parallel]
                                            A2=1;
acan=pi/4*dcan^2;                           B2=0;
apipe=pi/4*dpipe^2;                         C2=1/Z2;
a1=apipe;                                   D2=1;
a2=pi/4*(d2^2-d1^2);                        T2=[[A2,B2];[C2,D2]];
a3=acan;
a6=acan;                                    %Element 3: Expansion chamber 1
a7=acan;                                    [distributed]
a8=pi/4*(d8^2-d9^2);                        A3=cos(k0*l3);
a9=apipe;                                   B3=j*Y3*sin(k0*l3);
                                            C3=j/Y3*sin(k0*l3);
Y1=c/a1;                                    D3=cos(k0*l3);
Y2=c/a2;                                    T3=[[A3,B3];[C3,D3]];
Y3=c/a3;
Y6=c/a6;
Y7=c/a7;                                    %Element 6: Catalyst
Y8=c/a8;                                    A6=1;
Y9=c/a9;                                    B6=Z6;
                                            C6=0;



Page 21 of 22
Muffler/ Catalytic Converter                                          Final Report


D6=1;                                         Ytot=Y1;
T6=[[A6,B6];[C6,D6]];
                                              TLd(i,1)=20*log10(abs(AT+BT/Ytot+CT*
%Element 7: Expansion chamber 3               Ytot+DT)/2);
[distributed]                                 F(i,1)=f;
A7=cos(k0*l7);                                  end
B7=j*Y7*sin(k0*l7);
C7=j/Y7*sin(k0*l7);                           %Find projected new A-Weighted SPL
D7=cos(k0*l7);                                NM=[31.47;39.14;39.50;43.85;52.53;51.31;
T7=[[A7,B7];[C7,D7]];                         55.61;73.35;71.84;76.25;78.33;81.25;83.13;
                                              79.72;82.92;84.06;80.30;80.37;82.14;83.19;
%Element 8: Extension of tailpipe back into   81.76;82.46;84.52;83.15;85.39;86.10;84.00;
chamber [parallel]                            79.76];
A8=1;                                         AW=NM-TLd;
B8=0;                                         AW10=10.^(.1.*AW);
C8=1/Z8;                                      SPLA(iii,1)=10*log10(sum(AW10));
D8=1;                                         end
T8=[[A8,B8];[C8,D8]];
                                              %Regression analysis output
%Element 9: Tailpipe [distributed]            BO=[1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;
A9=cos(k0*l9);                                1;1;1;1;1;1;1;1;1;1;1;1;1;1;1];
B9=j*Y9*sin(k0*l9);                           XX=AA.*18.+2;
C9=j/Y9*sin(k0*l9);                           YY=BB.*35.+1;
D9=cos(k0*l9);                                ZZ=CC.*35.+1;
T9=[[A9,B9];[C9,D9]];                         X=[BO,XX,YY,ZZ,YY.*ZZ,XX.^2,YY.^2,
                                              ZZ.^2];
Ttot=T1*T2*T3*T6*T7*T8*T9;                    B=(X'*X)^-1*X'*SPLA
AT=Ttot(1,1);                                 SPLA
BT=Ttot(1,2);                                 disp('done')
CT=Ttot(2,1);
DT=Ttot(2,2);




Page 22 of 22