Report Muffler

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Design Project

Vibration Analysis on an Automobile Muffler

April 23, 2003

Amanda Frederick

Sarah Brady

EGR 315: Mechanical VibrationsDr. Amit Shukla

Department of Mechanical and Manufacturing Engineering

School of Engineering and Applied Science

Miami University



1

Executive Summary

A muffler is a part of the exhaust system on an automobile that plays a vital role. It needs to

have modes that are located away from the frequencies that the engine operates at, whether the

engine be idling or running at the maximum amount of revolutions per second. The purpose of

the design project performed was to determine which modes are very high and may affect the

automobile adversely while in operation. A muffler that affects an automobile in a negative way

is one that causes noise or discomfort while the car engine is running. In order to determine the

modes most at risk of adversely affecting an automobile, an impact test was conducted.

Research was performed prior to the test to determine which frequencies to look for modes at. It

was determined to conduct the experiment so data from 0 Hz to 1000 Hz could be collected. The

force was caused manually by a hammer with a hard head. After collecting the data, the transfer

functions were plotted using Matlab. To ensure correct analyzation, the transfer function

equivalent graphs were also plotted. The data was put into an Excel table and analyzed. Six

points on the muffler were chosen, after looking at the data, and determined to be under damped.

Therefore, our design study suggests to increase the mass, increase the damping, or provide a

negative stiffness to make the muffler more damped and to lower the modes of the transfer

function. Once at least one of these tasks is performed, the damping will be lowered and there

ought to be no more dangerous modes that could lead to excess noise and vibration.



2

Table of Contents

Abstract or Executive Summary…………………………………………………………………1

Introduction and Background……………………………………………………………………3

Literature Review…………………………………………………………………………..........4

Figure 1: CAD rendition of a muffler system…………………………………………...5

Research Approach…………………………………………………………………………...…5Experimental Details……………………………………………………………………….........6

Figure 2: Experimental muffler with data points labeled………………………………..7

Data Analysis and Observations………………………………………………………………..10

Figure 3a: Example Transfer Function Graph…………………………………….........11

Figure 3b: Example Transfer Function Equivalent Graph……………………………...11Figure 4: Aliasing Error…………………………………………………………………12

Design Studies……………………………………………………………………………..........13

Figure 5: Experimental muffler with improvement data points labeled………………...14Conclusions and Future Work……………………………………………………………..........14

References………………………………………………………………………………….........16

Appendices

A. Graphs of transfer function and transfer function equivalent for each

point on the muffler

B. List of resonance peaks for the transfer function and transfer function equivalentgraphs of each point on the muffler

C. List of frequency of each peak and occurrences of each point for the

peaks selected



3

Introduction and Background

The muffler in an automobile plays an integral role in the sound of the automobile as well as the

ride itself. In order to maintain a desired noise and comfortable ride, the modes of a muffler

need to be analyzed. Any modes that occur near to a frequency that the car engine operates at

should be considered dangerous because they could cause harmonic oscillations. The term

dangerous refers to the fact that if a high transfer value occurs at an operating frequency of the

engine, then the noise and ride comfort are at risk of non-optimal conditions. The typical

frequency that a car operates is at 315 Hz, which is equivalent to 3000 revolutions per minute

(rpm). The maximum frequency of a car is 838 Hz, which is equivalent to 8000 rpm. The

average idling frequency of car engines is at 73 Hz, which is equivalent to 700 rpm. With this

data in mind, the experimental data collected will be in the range of 50 Hz through 800 Hz.

In order to figure out the modes of the muffler, an impact test ought to be performed.

Impact testing is a simple and fast technique for obtaining good approximations of a systems

modal properties and frequency response information. “The usefulness of the impulse technique

lies in the fact that the energy in an impulse is distributed continuously in the frequency domain

rather than occurring at discrete spectral lines as in the case of period signals. Thus, an impulse

force will excited all resonance frequencies within its useful frequency range.” (Halvorsen)

One of the key characteristics of impact testing that makes it such a useful technique is

the reciprocity property. This property basically states that the properties of X/F are the same as

those for F/X. Consequently, in order to obtain data from all the points in the system, each point

must either be hit by the impact hammer or have an accerlometer on it, but not both. This greatly

reduces the amount of data that needs to be taken and analyzed, making time efficiency one of

the primary benefits of impact testing.



4

Literature Review

The area of vibrations testing on mufflers is very extensive. While review the various journals

and publications on impact testing of mufflers, three of those journals and publications were of

particular interest. The first was the Characterization of Rasping Noise in Automotive Engine

Exhaust Ducts journal, published by M. Ayadi, S. Frikha, and P.-Y. Hennion. In this journal, the

details of the problem, experimental set up, data collection, analysis, interpretation, and

conclusions were explained. The main idea behind this experiment was to discover the reason

behind the “rasping noise” heard in exhaust systems when an automobile abruptly accelerates

when it is subjected to cold conditions. The experimental set up included a shaker to input the

force as well as piezoelectric transducers to measure the acceleration. After the data was

collected, it was analyzed and then transformed into wavelets. These wavelets allowed for the

analysts to see if any abrupt changes in non-stationary phenomena took place. In conclusion, it

was proven that the rasping noise was due to a “balance between dissipative effects and non-

linear coupling between exhaust system resonance modes which makes them sustain each other.”

(Ayadi)

The second literature review of interest was of a case study by Lusas Analysts. This case

study explored faster vibration analysis of automotive exhaust systems. It utilized the Arvin

Modal Analysis Program (AMAP) which was produced by Arvin Exhaust R&D, based in

Warton, England. This program, which commissioned FEA Ltd’s consultancy department to

create it, has become widely used throughout the automotive industry. See Figure 1 for an

example of a type of model it can produce.



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Figure 1

In this figure is a muffler with some of the exhaust duct attached to it that is found underneath an

automobile. This software allows for man-time reductions of 50%. It can be used in conjunction

with the LUSAS analysis run, which provides a number of different eigensolvers for natural

frequency analysis. These programs allow for proposed exhaust systems to be analyzed much

more quickly than ever before.

The third literature review of interest was the article titled Development of the 2001 Year

Model Civic by Chitoshi Yokota, Yoshinori Nakamura, Shigeru Yada, Kouichi Funatsu, Fumio

Kubo, Tsuyoshi Ohkubo, and Tadashi Watanabe. The article opens up with the introduction,

goes into the concept and development objectives, discusses the different portions of the car and

improvements that will be made, and then conclusions. In the introduction, the issue of Honda’s

specific approach to designing the 2001 Civic in such a way that is was optimal in all aspects,

including stylishness, efficiency, and comfort is introduced. The goals for comfort, efficiency,

and stylishness are then given with very specific guidelines to be met. The article then steps

through the different car portions, naming specific areas that are going to be improved and how

the improvement will take place. To conclude, the article sums up the fact that the goal for the

car to be optimized to meet people’s standards has been met.

Research Approach

In order to conduct an impact test on the muffler to gather data, research had to be conducted.

The method in which the research was performed can be viewed in a hierarchical manner. The



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first step was to define our problem at hand, which was to determine any high transfer function

values occurring during the operating frequencies of a car. Once these values are determined,

design studies will be conducted to move the modes to higher frequencies so they are out of a

car’s operating range of frequencies. In order to know which frequencies ought to be explore for

any mode existences, web-based research as well as journal article research was conducted.

From reading various sources, the frequencies determined to be analyzed ranged from 73 Hz up

through 838 Hz. At 73 Hz, a car is idle. At 838 Hz, a car is running at 160+ miles per hour.

Therefore, the experiment was broadened to analyze data in the range of 0 Hz to 1000 Hz. After

gathering this constraint data on the frequencies, methods of actually conducting the experiment

had to be decided upon. Impact testing was chosen based upon its availability as well as

familiarity to those performing the experiment. After deciding upon impacting testing, the type

of hammer head to be used was chosen to be the metal head rather than the plastic head. The

reason for this was based upon findings of the fact that mufflers, in general, are very damped to

begin with, so a stronger force is required to achieve good vibration results. “Harder tips will

deform less than softer tips during impact” (GlobalSpec). Since less deformation results in more

vibration, the harder hammer head was chosen. To actually read the acceleration data, Microsoft

Excel was used in conjunction with a piezoelectric accelerometer. The piezoelectric crystal

accelerometer was chosen based upon the availability of it. While visiting ArvinMeritor, it was

learned that the accelerometer is positioned using a wax-based material, so that was also

employed in this experiment.

Experimental Details

Before taking experimental data, one needs to establish coordinates for the geometry of

their system. In performing an impact test it is important to take enough points that one has an



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accurate display of all the significant modes and resonant frequencies for the system, but also not

to take so many data points that the acquisition becomes impossible, meaningless, or redundant

in its analysis.

When labeling the geometry of the system it is important to be as accurate as possible.

This is primarily so that if multiple tests need to be done on the system, the person performing

the experiment can be sure that they are taking data from the exact same place on the system

every time they do an analysis. In an ideal setting, the person performing the experiment would

want to do several hits on the system and average them in order to average out the effect of not

hitting the same place exactly every time. However, for the time period of this experiment, the

averaging technique is not a realistic approach. The analyst also wants to make sure that he or

she keeps careful track of which experimental data goes with which point on the system. This is

key for obtaining accurate models of the modes properties and shapes of the system.

In this experiment, twelve data points were taken to represent the experimental geometry.

The reason was that, for the purposes of this experiment, this would provide sufficient data in

order to obtain an accurate model of the system. It was also thought that this was a practical

number of points to analyze in terms of interpreting data. A picture of the data points labeled on

the muffler is shown below in Figure 2.

Figure 2



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Before conducting the test, it was necessary to choose certain testing parameters within

the program. One of the most important of these was the sampling rate. Choosing an

appropriate sampling rate is important because the sampling rate must be sufficiently large to

prevent alias from occurring. A larger sampling rate increases the resolution, which improves

the accuracy of the data because there is a smaller step size in between the data points taken. A

sampling rate of 2000 Hz was used for this experiment.

While the data was being taken, line graphs were plotted for each set of data along the

way. Doing so provides the analyst with a good initial perspective on if the data was relatively

good or bad, allowing them to know whether to keep the set or take another reading.

One of the first problems encountered in testing the system was ensuring proper system

isolation. When performing an impact test, one wants to isolate the system correctly to make

certain that all of the acceleration that the accelerometer is experiencing is due to the vibrations

from the impact test and not from any outside source. This is the reason why more advanced

vibrations testing facilities, such as the one used at ArvinMeritor, feature a specially isolated

testing floor which has been constructed to eliminate the effects of any outside forces. For the

purpose of this experiment, foam was used as a way to damp out the vibrations from the

surrounding environment. Another option would have been to suspend the muffler system from

bungee chords so that it would not be resting on any surface that could have an impact on the

test.

One reason why it was so difficult to obtain accurate results from the impact test was that

the muffler system is a very highly damped system. Noise interference is particularly inherent in

heavily damped systems. Although it is not possible to eliminate this problem, it was hoped that



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using the Transfer Function Equivalent graph as one of the analysis tools would remove some of

the effects from the analysis.

It was decided that for the purpose of this experiment use of a single accelerometer in our

testing was the best option to keep the data collection process relatively simple. The original

idea was just to put the accerlometer at the point we had labeled 1 on the muffler. However, this

was a poor idea due to the fact that the point was on an angle and consequently it would be

difficult to make sure that the accelerometer was facing exactly towards the z-axis. As a result,

the accelerometer was instead placed on the point labeled 6 since this point was parallel with the

ground and would allow the accelerometer to be facing directly vertically.

Once the data collection began, a problem was encountered due to a loose connection

between the accelerometer and the connection cord to the computer. This would not have

allowed for obtaining readings out of the system. Thus it is important for the person setting up

the experiment to be very careful about how they set up their system when taking data.

It is extremely important when performing impact testing to make sure that the impact

hammer is used in an appropriate manner. Poor impact hits can result in inaccurate data. One of

the primary problems encountered during this experiment was that the hammer was striking the

system twice when the system would vibrate. The tester needs to ensure that the system is only

encountering a single impact; otherwise, the data will be a summation of multiple impacts on the

system. Another problem that was faced involved making sure that the hammer was hitting the

system directly along the Z-axis. If the hammer was not striking the system directly vertically

then we could be altering the vibration properties in each direction and not obtaining the proper

results. The accelerometer that was used was only capable of measuring vibrations along one

axis at a time. In order to ensure maximum vibration and accurate resonant frequencies of



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impacts along this axis, we needed to make sure that we were hitting the system in this direction

only.

Data Analysis and Observation

The first goal in being able to interpret the data from the impact testing was to establish

transfer functions for each data point on the muffler. Once these transfer functions had been

graphed, it would be possible to distinguish the resonant frequencies that occurred at each spot.

The transfer function plot is a graph of input over output with respect to frequency. The input of

this impact testing was the force exerted on the system by the impact hammer. The output of the

impact testing was the acceleration of the muffler system, as measured by the accelerometer.

The Excel data acquisition system used to compile the data collected this experiment is in the

form of two columns, one for the hammer data and the other for the accelerometer data.

In order to make this data usable in Matlab, each pair of data points were saved as text

files and then imported directly into Matlab. Next, a time vector was created that accurately

represented this vibration test’s experimental time range and was equal to the length of the vector

created for the transfer function data. Once these steps were complete, it was possible to obtain

graphs of the force and acceleration with respect to time. When the graphs were plotted in

Matlab, it was necessary to divide the accelerometer data by the calibration constant (110.3

mV/g) of the accelerometer used in the experiment in order to have the data displayed on an

accurate scale. Since the hammer had a calibration constant of 1, this did not need to be

accounted for in the data interpretation.

In plotting these graphs, peaks were found at all multiples of 60 Hz. These peaks were

actually due to an electrical frequency in the surrounding environment, making these individual

graphs very hard to interpret. Fortunately, since these peaks occurred in both the hammer and



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accelerometer data at exactly the same points, when the transfer function was graphed with

respect to frequency, the affects of the electrical peaks cancelled each other out.

To obtain the plots of the transfer function, it is necessary to convert the readings from

time domain to frequency domain. Numerically this is done by taking the Fourier transfer of the

input divided by the Fourier transfer of the output. In this experiment, a Matlab function was

used to obtain this desired result and create plots of the accelerometer / hammer with respect to

frequency, giving us our desired graph of output / input with respect to frequency. We also used

a semi log plot of this same data in order to determine what points were of interest to our

analysis.

The Transfer Function Equivalent graph was used as a second source for interpreting the

data. This graph is useful because it eliminates some of the effects of noise, allowing for a

clearer display of frequencies due to the force of the impact. This is particularly useful for

observing the peaks that occur at lower frequency ranges.

Examples of the transfer function graph and the transfer function equivalent graph are

shown below in Figure 3a and Figure 3b. These graphs are for the first point on the muffler.

The graphs for all of the points are given in Appendix A.

Transfer Function Graph Transfer Function Equivalent Graph

Figure 3a Figure 3b



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One of the problems inherent in an impact testing is aliasing error. Shannon’s sampling

theorem, also known as the Nyquist criterion, states that the sampling frequency for a system

needs to be at least twice the maximum frequency to be measured; otherwise, aliasing error will

occur. When alias occurs, an inaccurate graph of the sample data will be obtained, as is shown

in the figure below in Figure 4.

Figure 4

Another error that occurs during an impact test is leakage error. Leakage error occurs

when the analyst stops taking data before the energy of the system dies out. This can result in

inaccuracy in the transfer function. Vibrations testing for all systems will have at least a minimal

amount of leakage error occurring naturally: the goal is to minimize it. This is another reason for

isolating the muffler system with the surrounding foam.

It was decided early on that the range of 0 to 1000 Hz was appropriate for analyzing the

frequencies that are typically encountered by a vehicle. In order to see what frequencies were

exciting peaks in the transfer function the most frequently, a spreadsheet was compiled that listed

the peaks occurring in both transfer function graphs for each point on the muffler. See appendix

B for a copy of this spreadsheet. Then, another spreadsheet was made that showed the most

Alias

t

Amplitude

∆t>T/2

T



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frequently occurring peaks and at how many data points that peak occurred. See appendix C for

a copy of this spreadsheet.

One goal in selecting the most important resonance frequencies was to select the peaks

that were occurring the most often. Another goal was to select peaks over a wide range of

frequencies in order to eliminate a good range of problem spots. Finally, it was desirable to

eliminate those peaks that were often the highest in the graphs, since those were the ones which

were most likely to cause problems in the system. Although our testing range went from 0 to

1000 Hz, we were mostly concerned with points below about 800 Hz since these were the ones

most likely to be reached by the system. With these criteria in mind, five resonance frequency

peaks were selected as those to be removed from the system. These frequency peaks occur at

120, 420, 560, 730, and 880. Although the 880 is slightly out of our range of interest, there was

a large number of peaks at this point and since it was fairly close to the range of interest, it

remained a one of the peaks targeted.

For this experiment, because there was such a large amount of noise, it was not possible

to obtain meaningful data and graphs that could be used in a software program such as diamond

in order to analyze the modal shapes. If the mode shapes had been able to be graphed, this

would have provided an indication of how the system was moving when it was stuck by the

hammer. The deflection of these shapes would have also helped in determining the locations

where damping could have been added to the system for the most effective deflection resistance.

Design Studies

According to the criteria listed above, the resonance peaks of 120, 420, 560, 730, and 880

Hz are those that have been targeted to be removed from the system. To decrease these peaks,

one can look at what type of affect changing the M, C, K properties of the system will have on



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the transfer function. This can be done by using Matlab to plot graphs of the transfer function

with respect to w, similar to what was done during EGR 315 Homework #9. In doing so, one

will find that increasing the mass, increasing the damping, or providing a negative stiffness will

all have the effect of decreasing the height and flattening the peaks of the transfer function plot.

In order to determine which would be the best locations for conducting these system

changes, one can count the number of times that each location on the muffler was excited at our

peaks of interest. This information is available in Appendix C. Through this analysis, points 1,

2, 3, 4, 7, and 11 were selected as the most important. Their location on the muffler is shown by

a red dot on the following graph:

Figure 5

Our final design suggestion is thus to increase the damping inside the muffler, specifically at

these points of interest.

Conclusions and Future Work

The purpose of this experiment was to conduct an impact test on a muffler system in

order to determine the resonant frequencies of the system and suggest changes in the system

design. For this impact test, an impact hammer was used to provide the input force and an

accelerometer was used to measure the output response. In order to determine the resonance

frequencies, Matlab was used to graph the transfer functions for each data point labeled on the

muffler, specifically, the transfer function and transfer function equivalent graphs. Spreadsheets



15

were then compiled to determine which peaks were the most significant for the system. From

the data, five peaks were selected as potential resonance frequencies to remove from the muffler

system. These peaks occurred at 120, 240, 560, 730, and 880 Hz.

In order to minimize the effects of these resonance frequencies, the suggested design

improvement is to add damping to the system. By creating a spreadsheet counting the number of

times that each point on the muffler was excited at each of these peaks, the locations for adding

the damping have been determined to be points 1, 2, 3, 4, 7, and 11 on the muffler.

In terms of future research, a good deal more analysis could be done on this system. If a

similar experiment would be done in hopes of eliminating some of the noise, the analyst could

attempt to isolate the system better, perhaps using the bungee chord technique. If more accurate

data could be obtained, then the analyst could observe the modal shapes of the system to help in

determining appropriate locations for damping. Finally, a Matlab model of the system could be

constructed that would allow the user to change the M, C, K properties of the system in order to

see the result which the changes have on the system without actually altering the mufflers

physical properties in person. Although it would be difficult to obtain an accurate virtual model,

having one would be extremely valuable in judging the value of design improvement

suggestions.



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References

Ayadi, M.,Frikha, S., Hennion, P.-Y. Characterization of Rasping Noise in Automotive Engine

Exhaust Ducts. Journal of Sound and Vibration. 2001.

www.camcraftcams.com/Archives/monthlynews0702.htm

Fujikawa, Kenji. Analysis of Steering Column Vibration. Motion & Control. 1998.

Halvorsen, William, Brown, David. Impulse Technique for Structural Frequency Response

Testing. Sound and Vibration. November 1977.

www.lusas.com/case/analyst/arvin.html

www.lydallautomotive.com/images/brochure/lydallbrochure.pdf

www.scientific-computing.co.uk/acoust/software.htm

test-equipment.globalspec.com/ProductGlossary/Test_Measurement/Vibration_Acceleration_Sensing

www.users.muohio.edu/mortonyt/EGR303/EGR303MainFramSet.html

Yokota, Chitoshi, Nakamura, Yoshinori, Yada, Shigeru, Kouichi, Funatsu, Kubo, Fumio,

Ohkubo, Tsuyoshi, Watanabe, Tadashi. Development of the 2001 year model Civic.Elsevier. January 2002.



Appendix A

Transfer function and transfer function equivilent graphs

Point 1

Point 2

Point 3



Point 4

Point 5

Point 6

Where the accelerometer was placed

Point 7



Point 8

Point 9

Point 10



Point 11

Point 12



Appendix B

Data Set 1 Data Set 2 Data Set 3 Data Set 4 Data Set 5

Mode M1 M2 M1 M2 M1 M2 M1 M2 M1 M2

1 520 70 240 240 5 8 8 8 250

2 700 120 290 440 80 120 110 70 395

3 820 275 300 515 97 180 120 120 520

4 880 340 310 590 102 240 130 180 750

5 940 375 350 650 120 270 160 230 930

6 990 470 550 680 270 325 200 290 1000

7 495 730 760 535 460 220 310

8 575 790 880 650 610 300 320

9 675 990 980 710 680 310 350

10 730 1000 730 320 375

11 840 760 360 420

12 940 800 390 445

13 990 840 420 480

14 890 455 500

15 920 460 54016 960 600 560

620 600

680 620

780 680

M1=Transfer Function M2=TFE 880 720

970 780

810

870

900

920

960

980



Data Set 8 Data Set 9 Data Set 10 Data Set 11 Data Set 12

M1 M2 M1 M2 M1 M2 M1 M2 M1 M2

10 50 255 60 420 420 520 30 325 125

440 70 275 115 585 585 580 120 360 410

640 100 400 185 995 885 640 165 420 580

920 180 700 235 910 700 240 850 635

930 220 760 295 975 760 330 860 670

940 290 880 405 820 560 990 800

950 310 940 445 880 595 995 865

960 370 600 940 725 890

970 410 710 1000 930 990

990 560 745 1000

750 880

830 930

900

960

1000



Appendix C

* indicates appears in both TF and TFE

red indicates importance

frequency points on muffler # times points on # of occurances at70 1,4,8 3 muffler resonance frequencies

120 2,3*,4*,7*,11 5 1 2

240 2*,3,11 3 2 3

420 10*,12,4*,7 4 3 2

560 11,8,7*,4 4 4 4

600 9,4*,7* 3 5 0

730 1,2,3,7* 4 6 0

760 9,11,2,3 4 7 4

820 1,11,5,7 4 8 1

880 1,9*,11,2,4 5 9 1

920 8,3,4,7 4 10 1

930 5*,8,11,9 4 11 3940 1*,8,9,11,7 5 12 1

960 8*,4,7,3 4

990 1*,12*,2,8,7 5

1000 5,11*,3,8 4

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