Coin Identification through NaturalFrequency Analysis

Emerson SteedBrigham Young University735 N 400 E Apt. 23, Provo, UT208.589.8906emersonsteed@gmail.com

Abstract

The natural frequency of an object is unique to each object like a fingerprint is to a person.Due to hardware limitations, the upper limit of natural frequencies studied was 20 kHz. Eachcoin was modeled using Computer Aided Design (CAD) software and then natural frequencieswere predicted using Finite Element Analysis software. The sound of each coin freely vibratingwas then recorded using LabVIEW and processed with the Fast Fourier Transform (FFT) todetermine the natural frequencies. Forty coins of each type were measured. All measured coinshad natural frequencies within the range of human hearing. The resulting spikes in the FFTgraph represented the natural frequency of each coin. This data was then used to create anotherLabVIEW program that successfully identified which coin was being dropped by recording thesound of the vibration and determining the natural frequency.

Introduction

Powerful singers have been known to sing so loudly andclearly that they break wine glasses. This comes as resultof the singer hitting the natural frequency of a wine glasswith enough power. Everything has a unique natural fre-quency. Essentially, the natural frequency is the vibrationof an object when it is hit or in some way given mechanicalenergy. This vibration produces sound waves that can berecorded and measured. The frequency of vibration is de-pendent on many factors such as mass, shape, density andmodulus of elasticity. Fortunately, breaking wine glassesis not the only potential use of this phenomenon.

The natural frequency of an object can be incredibly use-ful because it always stays the same. An object’s naturalfrequency can be composed of many different frequencymodes. Usually there is one main frequency and thenharmonics of that frequency. When these frequencies areidentified they can be used as a fingerprint to identify aspecific object. These fingerprint frequencies will stay thesame regardless of how mechanical energy is given to anobject. Coins, just like other objects, have specific mea-

surable natural frequencies. Currently vending machinesuse electromechanical systems to identify coins. This pa-per will examine a different approach to identifying indi-vidual coins from one another.

This experimental approach involved recording the soundproduced by coins freely vibrating and using the fre-quency domain to identify the different frequencies whichare present. The natural frequency of some objects isnot easily attained through sound recording. Some ma-terials have internal damping characteristics that preventthe object from vibrating in such a way that its naturalfrequency can be recorded easily recorded.(e.g. rubber)Generally, stiffer materials have clearer natural frequen-cies because vibration is more easily propagated throughthe material. To help get clean ’fingerprints’ of each coin,a consistent system will be used for measurement. Thecoins will then be able to be identified by a computer witha microphone.

This study took the principles of natural frequencies andapplied them to coins. Once the natural frequency ofeach coin was numerically modeled and measured, a nor-mal distribution was created of each coin’s natural fre-

Coin Identification 1 Emerson Steed

quency spectrum. This data was then used to identifycoins of certain value from others. This is an alternativeto the mechanical approach currently used by vendingmachines.

The natural frequency of an object is a useful piece ofinformation. The process that has been used to find thenatural frequencies of coins can be used for other objects.This technology has been used to check structural beamsfor cracking [1,2,3] and could have many other potentialengineering uses in the world today. It is important to beable to find the natural frequency of a mechanical systemor an object because other systems could have the samenatural frequency and could cause interference or damagewhen they operate together.

Methods

The natural frequencies of the coins were first predictedusing Finite Element Analysis (FEA) software. Twothings are needed to use FEA software; a CAD model andmaterial properties. The Autodesk Inventor CAD systemhas a built in frequency and modal analysis FEA tool.The coins were modeled as simple disks using dimensionsspecified by the U.S. Mint [4]. Determining the materialproperties of the coins was considerably more difficult. Aphone call to the U.S. Mint shed some light on why theywere so understandably tight lipped about the specificmaterials used in coins. There is a fear that people willattempt to fake coins for use in vending machines. Theonly information publicly given is the composition of thecoins by mass percentage.

Modern day coins are primarily composed of copper andnickel. The material properties of each coin was estimatedby using the publicly available percent composition fromthe mint and the properties of existing copper nickel al-loys, also known as ’cupronickel’. The modulus of elastic-ity of the cupronickel was estimated to be 118 GigaPascals(GPa). This is most definitely an estimate because it isnot known how the strain hardening of being stampedhas effect the modulus of elasticity of the coins. The ten-sile and yield strength were also estimated from existingalloys

As shown in Figure 1, FEA systems work by essentiallychopping an object into many small pieces that make up amesh. The more elements that are used for the model, themore accurate the model becomes. The natural frequen-cies of the coins were estimated using a mesh that con-tained approximately five thousand elements. The FEAsystem then modeled the interactions of each piece numer-ically to give an estimate of each coin’s natural frequencyand vibration modes.

Figure 1: A ’mesh’ created by the Autodesk In-ventor frequency and modal analysis tool. Thismesh shows the individual pieces that the coin iscut up into for natural frequency analysis. Thisparticular mesh contains 7,871 elements. In-creased mesh size will give more accurate results.

The vibrations of a coin after being dropped will berecorded using a standard computer microphone. Thiswill provide raw audio data as shown in Figure 2. Un-forturnately, this presents some limitations. These limi-tations come from the computer hardware that is beingemployed. A standard sound card is being used. Soundcards employ low-pass filters that block frequencies above20 kHz to condition audio signals and to prevent aliasing.This limited our natural frequency investigation by mi-crophone to the upper limit of human hearing or 20 kHz.Additionally, microphones have an upper limit on theirfrequency response between 16 and 20 kHz. This meansthat frequencies recorded closer to 20 kHz experiencedgreater attenuation and were harder to discern.

These coins also had additional harmonics that existabove the range of human hearing, but it was not practicalto capture them. To do so would have required an expen-sive microphone with an extraordinarily high frequencyresponse, an external processing device, and extremelyhigh sample rates.

The audio capture sample rate was set to 96,000 samplesper second or 96 kHz. This is the maximum sample rateallowed by the computer sound card. The Nyquist theo-rem tells us that to accurately capture a signal and avoidaliasing we must sample the signal at a rate that is atleast twice its frequency.

fNyquist = 2fSignal (1)

Since 20 kHz was the highest possible frequency captured,the audio was sampled at least 4.8 times the highest fre-quency that was be captured. The optimum sample rate

Coin Identification 2 Emerson Steed

would have been around ten times the highest frequency,but 4.8 was sufficient for natural frequency analysis. Ifthe audio was sampled below 40 kHz it would have beenpossible for aliasing to create erroneous signals.

Figure 2: Raw time domain data as acquired byLabVIEW. The amplitude of this graph is es-sentially voltage as detected by the microphone.This raw data contains little valuable informa-tion in the time domain. The Fast FourierTransform was used to retrieve useful data.

Each coin was dropped from the same height and landedthe same distance from the microphone each time. Tomake the interference predictable, a small tile partiallycomposed of quartz has been epoxied to a piece of woodas a landing pad. This is not absolutely necessary sincethe natural frequency of a coin stays the same even whenit may be dropped on different material. Through pre-liminary analysis it had been shown that the natural fre-quencies of the quartz, wood, and other common tablematerials had natural frequency components that wereconsistently lower than the frequencies of coins. This al-lowed for the lower end of the frequency spectrum in thefrequency domain to simply be thrown out. This lowfrequency interference region can be easily detected bydropping coins on different surfaces and comparing theresulting FFT graphs.

Figure 3: These peaks represent the natural fre-quencies of a half dollar coin. They are foundfrom the raw audio data by taking the FFT. TheFourier Transform takes the captured sound datafrom the time domain and puts it into the fre-quency domain. In this and other FFT outputgraphs frequencies below 7 kHz can be disregardedas noise frequencies captured by a dropped coincausing the quartz tile to vibrate. This noise flooris consistent across other surfaces.

A pipe was epoxied to another piece of wood to act asa rail and to ensure consistent drop height. This wasmounted to a hinge attached to the same piece of woodthat anchors the quartz tile. This system will allow all ofthe coins to fall into the same place from the same heightwith consistent kinetic energy.

Coins of the same monetary value were found to havesome variance in their natural frequencies. To account forthis, forty random coins of each denomination had theirnatural frequencies measured. The data from the dropswere compiled to produce a normal distribution of eachcoin’s natural frequency. The sample standard deviationfor each coin was also computed. This allows the exper-iment the ability to claim to be able to recognize 95%of coins dropped. This confidence interval represents allvalues within two standard deviations from the samplemean.

LabVIEW was used to record the sound data. Each timesound data was acquired, LabVIEW computed the FastFourier Transform (FFT) of the data.

Coin Identification 3 Emerson Steed

Xk =

n−1∑i=0

xn · e−j2πik/n (2)

The FFT took the sound data out of the time domain(x axis is time) and into the frequency domain (x axis isHz). Attempting to recognize unique frequencies in thetime domain would simply not be possible. The FourierTransform is what made this experiment possible becauseit showed what frequencies were recorded and how oftenthey occurred relative to one another. The sound datathat was processed was composed of hundreds of thou-sands of discrete samples. Therefore, the Discrete FourierTransform is the specific variation of the Fourier Trans-form that will be used. Through the use of this equa-tion and some logic programming, LabVIEW was ableto identify individual coins when certain amplitudes werereached in specific frequency ranges. Several of the coinnatural frequencies overlapped each other. This poten-tially would have been a problem for the LabVIEW pro-gram, but the use of boolean algebra made it possible toensure that only the correct coin is identified since thecoins have different natural frequency harmonics.

Results and Discussion

Penny

According the U.S. Mint the modern day penny is com-posed of 2.5% copper and 97.5% zinc. A penny is essen-tially zinc with copper plating. The material propertiesused for the FEA software were estimated from the prop-erties of zinc. This was used to then calculate the naturalfrequency modes of the penny. Surprising the first natu-ral frequency mode was found to exist at approximately13.1 kHz, a frequency well within human hearing. Thefirst mode of all the coins is the ’taco’ shape. If a highspeed camera with a high enough frame rate was used, itwould be possible to see the actual deflection of the coinin that shape. Figure 5 represents an exaggeration of thecoin deformation.

No usable sound data was captured that revealed the nat-ural frequency of the penny. There are a number of pos-sible causes for this. It is possible that because of thecomposition of the penny there is a significant amountof damping. It is also possible that the penny absorbs asignificant amount of energy as it bounces which preventsa significant amount of free vibration which produces thesound waves that are picked up by the microphone. Thepenny probably just isn’t large enough to produce soundwaves of high enough amplitude. Initially it was thoughtthat it was possible that the natural frequency of a penny

was higher than the range of human hearing, but the nu-merical model suggests otherwise.

Nickel

To create a numerical model of the natural frequencymodes of the nickel, a copper alloy of 25% nickel 75%was used per U.S. Mint specifications. The modulus ofthis material was approximated at 118 GPa. The FEAsoftware estimated that the first two modes natural fre-quency would land around 17.3 kHz. The third modeof natural frequency was calculated to be approximately29.2 kHz.

When the numbers from the numerical model are com-pared with the measured values it becomes apparent thatthere is a considerable amount of error. There are a num-ber of possible explanations for this erroneous behavior.The first is that the material used in the FEA could ac-tually be quite far from what the actual properties of anickel are. The material properties were estimated fromknown copper alloys, but when the coin was stamped atthe mint it is possible that it could have undergone a sig-nificant amount of strain hardening which would affect itsmodulus of elasticity significantly even though it is thesame material. Another possible source of error comesfrom the size and shape of the coin. The CAD model wasdesigned from diameter and thickness dimensions givenby the U.S. Mint, but that did not take into account thedetailed features that exist on the face of coins. The re-search showed that the natural frequency of a coin couldvary by several hundred Hz simply from the edges of acoin being worn away.

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Figure 4: This is the result of taking the Fouriertransform of raw audio capture data from a drop-ping nickel. It can be seen that this particularnickel has one natural frequency mode within therange of human hearing at about 16 kHz. Therewas a considerable amount of variation betweendifferent years of nickels.

Unlike the penny, sound data from the nickel was success-ful captured. As shown in Figure 4, the natural frequen-cies of forty nickels were measured. A sample mean andsample standard deviation were calculated.. The sam-ple mean was found to be around 12.7 kHz for the firsttwo natural frequency modes and 14.4 kHz for the thirdfrequency. The first two modes of natural frequency aregenerally very close together for coins, so much so thatthey can become indistinguishable in the FFT due to itsresolution. For the purpose of this experiment the firsttwo modes of natural frequency are generally averaged to-gether. The standard deviation was 797 Hz and 617 Hzrespectively.

Twenty-nine nickels were used to test the accuracy ofthe coin identifying program. All twenty-nine coins weresuccessfully identified by LabVIEW as nickels. This im-proves upon the 95% positive identification as was theo-rized.

Figure 5: This image shows the estimated firstmode of natural frequency of a nickel. It has thesame shape as the penny does because they areboth disk shaped objects. The deflection is some-what exaggerated to clearly show the deformationof the coin.

Dime

According to the U.S. Mint, dimes are composed of Nickeland Copper. The same material was used as was used toestimate the nickel’s natural frequencies. The numerical

model predicted the first two modes of natural frequencyto be approximately 15.2 kHz. This is a few kHz abovethe frequencies that were measured. The next mode ofnatural frequency was found to be around 26 kHz, welloutside the range of human hearing. As seen in Figure6, the modal deformation looks much like a drum. Usingthe same material as the nickel definitely added error tothe model since the nickel and dime have different weightcompositions of copper and nickel. However, the mode ofvibration is accurate. It shows that, unlike the first twomodes, the third mode of natural frequency comes fromthe center of the coin. It looks like a drum moving upand down.

Figure 6: This figure depicts the third mode ofthe natural frequency of the dime. The com-puted frequency from FEA is approximately 26kHz. This mode of vibration is through the de-formation much like beating a drum. The firsttwo modes are identical to the first two modes ofother disk shaped objects.

Capturing good data from the dime was somewhat dif-ficult. If the dime didn’t land close the microphone itusually didn’t pick up the vibrations. The copper nickelalloy that composes coins other than pennies has verygood vibrating qualities. Just as with the other coins, asample size of 40 was used to determine the sample meanand sample standard deviations.

Two natural frequency modes were found within 20 kHz.As seen in Figure 7, they were found to be at 12.3 kHzand 12.8 kHz. The sample standard deviations are around250 Hz for each frequency. This is a great improvementover the nickel. It is possible that since the nickel hasundergone several redesigns over the years that much ofthe variation could be blamed on this redesign. The dimeis a much tighter frequency range.

Coin Identification 5 Emerson Steed

Figure 7: Despite being smaller than the penny,the dime has two natural frequency peaks withinthe range of human hearing. These peaks arefairly consistent across other types of dimes be-cause of little redesigning done to dimes over theyears.

Once LabVIEW was calibrated to the frequency ranges ofthe dime, forty-one dimes were measured. Initially therewas some difficulty in getting a reliable result that wasnot confused with the nickel. The nickel and dime slightlyoverlap in their ranges. After some calibration 97.56% ofthe dime were identified correctly. This value is betterthan the predicted 95% confidence interval.

Quarter

The quarter was modeled using the same material as thedime and the nickel. The FEA software estimated its firstthree natural frequency modes to exist at 10.807 kHz,10.808 kHz, and 18.2 kHz with the same modal shapesas the other coins. The predicted first two modes of nat-ural frequency were actually close to the measured val-ues.

As seen in Figure 8, after measuring the natural frequen-cies of forty quarters the sample mean natural frequencieswere found to be at 9.1 kHz, 9.3 kHz, and 15.4 kHz. Thestandard deviation of each frequency was around 250 Hz.This is a similar standard deviation of the dime. Unlikethe nickel, the changes made to the quarter in the lasttwenty-five years have not had a significant of an impacton its natural frequency.

The LabVIEW program was able to predict the presenceof the quarter 97.62% of the time. The test was done withforty-two quarters. The program only missed one. This isbetter than the predicted 95% confidence interval.

Figure 8: This is the result of taking the Fouriertransform of raw audio capture data from a drop-ping quarter. It can be seen that the quarter hasthree natural frequency modes within the rangeof human hearing. Two around 9kHz and onearound 15 kHz.

Half Dollar

The half dollar was modeled in the same manner as theother coins using the same material that the quarter,dime, and nickel used. The FEA software predicted nat-ural frequency modes to exist at 8.3 kHz, 14.2 kHz, and18.8 kHz. As shown in Figure 9, the fourth mode of nat-ural frequency is a more complicated combination of thefirst and second modes.

The half dollar is by far the largest coin in circulation.Acquiring forty of them for statistical analysis was nota simple feat. As seen in Figure 3, the measured natu-ral frequencies were at 7.3 kHz, 13.3 kHz, and 16.5 kHz.These values are similar to the values predicted by theFEA software. The standard deviation for the half dollarfrequencies was around 200 Hz.

The LabVIEW program was able to successfully identify100% of the forty-three half dollars dropped. The largercoins have been easier to identify because they have at

Coin Identification 6 Emerson Steed

REFERENCES REFERENCES

least three distinct frequencies within the range of humanhearing across a broad spectrum.

Figure 9: This is the fourth natural frequencymode of the half dollar. The FEA software esti-mates that this mode occurs around 18 kHz.

Dollar Coin

The dollar coin was modeled as a disk but a soft yellowbrass was used to represent its composition. It is actuallycomposed of brass and manganese. Since the specific typeof alloy used by the mint to produce these coins couldn’tbe located, the effect of the manganese on the modulusof elasticity was left out. The FEA software predictednatural frequencies at 9.9 kHz, 9.9 kHz, and 17.1 kHz. Itwas shown to have the same modes of natural frequencyas the other coins.

Forty dollar coins had their natural frequencies measured.The frequencies were found to be at 8.4 kHz, 8.6 kHz, and14.4 kHz. The dollars coins are actually not a very pop-ular coin. Because of this, their sample standard devia-tions were very tight. The average standard devation wasless that a hundred hertz. This made the coins very easyto specifically identify. LabVIEW was able to correctlydetermine dollar coins one hundred percent of the time.The coins have had very little circulation and very littlewear.

Figure 10: This is the result of taking the Fouriertransform of raw audio capture data from a drop-ping dollar coin. It can be seen that the dimehas two natural frequency modes within the rangeof human hearing. One around 8 kHz and onearound 14 kHz.

Acknowledgements

I would like to thank Craig Bidstrup for helping me tocome up with ideas for this paper. I would also like tothank Dr. Tadd Truscott for advising me in in severaltopics covered in my paper.

Coin Identification 7 Emerson Steed

REFERENCES REFERENCES

References

[1] Ahmed, Ehsan & W.H. Wan, Journal of Engineer-ing Science and Technology, ’Evaluation of NaturalFrequency and Damping of Profiled Steel Sheet DryBoard and Composite Panel’, Volume 6, Issue 6, 695- 708, 2011.

[2] Glushkov, Evgeny & Glushkova, Natalia & Golub,Michael, AIP Conference Proceedings, Real naturalfrequencies and resonance wave localization in an elas-

tic waveguide with a horizontal crack, 160-166, May4, 2006.

[3] Ramm, A.G. Journal of Mathematical Physics, ’Cal-culating resonances (natural frequencies) and extract-ing them from transient fields’, 1012-1020, May 1,1985.

[4] United States Mint Current Coin Specificationshttp://www.usmint.gov/about the mint/?action=coin specifications

[5] Copper Development Association Websitehttp://www.copper.org/applications/cuni/txt properties.html

Coin Identification 8 Emerson Steed