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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*Lpi 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: Lw Lp 10 log10 A Assuming a hemi-spherical surface R = 1.626 m (64”). A = 2πR2 = 16.612m2 10*log10(A) = 12.2dB so, Lw Lp 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 insertion 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 Method1. 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*k2 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) y 4 2 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 1 j * y * cot(k * L) 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 Page 16 of 22 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

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