Modal Analysis of Guitar Bodies

Adam KemptonDepartment of Physics, University of Illinois at Urbana-Champaign

1010 West Green Street, Urbana, IL 61801-3080, USA

ABSTRACTResonant frequencies of guitar bodies were investigated to further theunderstanding of the tonal qualities of guitars. To do this a DataAcquisition (DAQ) setup using Lab Windows and an HP3562A DynamicSignal Analyzer were used to discover resonant frequencies. Resonantfrequencies of a 1968 Gibson J-45 acoustic guitar were located at 90Hz,180Hz, and 786Hz and 3-dimensional plots were developed at eachfrequency. The method used does provide some insight as to thevibrations of guitar bodies.

I. Introduction

Modal analysis is an investigationinto the resonant frequencies or modesof vibration of solid objects. All objectshave frequencies at which they vibrateafter an excitation has occurred, i.e. apiece of wood or a piece of metal. In thecase of guitar bodies, the vibrations arewaves creating in the body of the guitaroscillation from an equilibrium position.In acoustic guitars, the vibrating stringtransfers energy to the bridge, whichtransfers energy to the air cavity andfront and back plates of the guitar,inducing vibration1. Anyone who playsguitar feels these vibrations all the time.The difficulty of grasping the individualmodes of a guitar is due to thecomplexity of the modes of vibration.Each resonant frequency occupiesdifferent locations of the guitar, and eachresonance characteristically has nodes(points of zero displacement) and anti-nodes (points of maximumdisplacement) at various locations on thebody of the guitar. In addition, somemodes exhibit both twisting and turning,adding to the difficulty of analyzingguitar modes. Quite simply, full

understanding and knowledge of theactions of guitar bodies is not entirelyfeasible. Nonetheless, thorough analysiscan be conducted and can be beneficial.The reason for conducting the researchwas to gain a better understanding of thetonal qualities of guitars, orunderstanding why certain guitars, suchas Fender and Gibson guitars, sound theway they do. In addition, by gaining afeel for vibration modes, one couldtheoretically shape the tonal qualities ofa guitar according to his or her personalpreference. Both acoustic and electricguitars can be studied, but more actionwill be seen in the bodies (neglecting theneck) of acoustic guitars than solid bodyelectrics.

II. Background

Previous Research

Research of modal analysis of guitarshas been done in the past. Dr. DanRussell and Paul Pedersen of KetteringUniversity investigated modes ofvibration of various guitars2. To do this,they made a grid out of the body of theguitar and excited it by hitting each

location with a force hammer that had atransducer located on the tip. Anaccelerometer located on the wing of theguitar converted the guitar’s accelerationinto voltage. The accelerometer andforce transducer were fed into an FFT(Fast Fourier Transform) Analyzer andthe peak frequencies corresponding toresonant frequencies were determined.Animated movies displaying the modeswere made using modal analysissoftware STAR Modal from SpectralDynamics.

Piezoelectric Transducer

The heart of our method is thepiezoelectric transducer, a ceramiccrystalline structure. It can convertelectrical energy into mechanical energy,and it can convert mechanical energyinto electrical energy. When a voltage isapplied to the transducer, the crystallinestructure expands and compressesproportionally to the applied voltage.Conversely, when mechanical pressureis applied, a voltage is induced acrossthe transducer proportional to thepressure so as to oppose the change inmechanical structure. A largemechanical pressure induces a largevoltage. This two-way flexibility of thetransducer makes it very valuable to us.In our experimental setup, we used twoidentical transducers on the surface ofthe guitar- one to excite the guitar intovibration (transmit transducer) wheremechanical compression is induced via avoltage, and one to pick up the vibration(receive transducer) where a voltage isinduced via mechanical compression.The pure sine wave signal from afunction generator applied to thetransmit transducer on the body of theguitar will excite the guitar at the samefrequency as the input sine wave. A

resonance can thus be located byviewing the output voltage peaks of thereceive transducer. A peak voltageshould correspond to a resonantfrequency. If the receive is at a node ofa particular resonance, then very littlevoltage will be induced. Conversely, ifthe receive is at an anti-node of aresonance, a large voltage will beinduced. Transducers were used to bothdetermine the resonant frequencies andto develop the 3-dimensional plots of theguitar at resonant frequencies, with thetechnicalities still to come.

Lock-In Amplifier

A lock-in amplifier is a device that isused to detect and measure small ACsignals, and it does this by “locking in”to signals of a specific frequency. Thefrequency it locks in to is the referencesignal from a power supply (in our casea pure sine wave from a functiongenerator). The lock-in we use is a dual-phase lock-in. It displays the magnitudeof the real voltage (Vreal) and theimaginary voltage (Vimag), in additionto the frequency. Vreal is the voltagethat is in phase with the reference signal,and Vimag in the voltage that is 90° outof phase with the reference signal. Inthe complex plane, Vimag (being that itis an imaginary number) represents thecomplex axis and Vreal represents thereal axis. The total voltage (Vtotal) canthus be determined by the followingequation:

2 2( ) ( )Vtotal Vreal Vimag= +The phase angle, θ, is calculated asfollows:

1tan ( / )Vimag Vrealθ −=It is easily seen that cosVreal Vtotl θ=and sinVimag Vtotl θ= .

The lock-in amplifier is extremelyvaluable to us because it can read thevoltage out of the transducer at a specificfrequency, blocking out noise or otherfrequencies we are not interested in.

HP3562A Dynamic Signal Analyzer

The HP3562A Dynamic SignalAnalyzer is a device that, among manyother things, produces frequencyspectrum plots. In our case, the signalanalyzer sources random noise, a signalconsisting of an extremely largebandwidth of frequencies, allowing us toexcite the guitar at each of its resonancefrequencies simultaneously. The graphwe look at is a cross spectrum, whichplots magnitude vs. frequency. Thecross spectrum plot is determined asfollows, where x(t) is the random noiseinto channel 1 and y(t) is the signal fromthe receive transducer into channel 2:

( ) ( ) i tspectrum x t y t e dtω∞

−

−∞

= ∫This is the Fourier transform of channelone multiplied by channel 2, where t istime and ω is angular frequency. Thepeaks determined are the resonantfrequencies.

III. Experimental Method

The data collecting process was two-fold. First, the resonant frequencieswere located using two methods: a DataAcquisition Setup (DAQ) and a whitenoise method using an HP3562ASpectrum Analyzer. Once found,another DAQ setup was used to examine

the behavior of the guitar at a particularresonance.

DAQ Setup

Figure 1 shows the DAQ setup usedfor developing plots to locate resonantfrequencies.

Fig. 1 DAQ Schematic

A C++ program written in LabWindows controls the system andproduces the plots. The programcontrols the function generator via aGeneral Purpose Interface Bus (GPIB).In the program, we set the frequencyrange from 10 to 1010 Hz. Thecomputer tells the function generator toscan this span of frequencies. Thefunction generator sends a pure sinewave into the transmit transducer,exciting the guitar into vibration. Theamplitude of the sine wave was set at.5V because it was determined that thisinput voltage resulted in clearlylocatable peaks without exceedingoutput voltage limitations of thecomputer program. The vibration ispicked up by the receive transducer. Toallow free vibration, the guitar wassuspended by rubber bands: one around

the nut, one at the beginning of the neck,and one around the end strap holder. Toimprove coupling, 50g were placed onthe transmit transducer, and 60g wereplaced on the receive transducer. Thetransducers used were from RadioShack. The induced signal is sent through anINA121 Operational Amplifier. This isa differential amplifier that reduces noise(undesired signal), especially 60Hzelectromagnetic radiation, by amplifyingthe difference between the two inputsignals. Figure 2 shows the circuitschematic.

Fig. 2 INA121 Circuit schematic

The 5.7kΩ resistor sets the gain atapproximately 9. The 1MΩ resistors inparallel with the inputs provide a pathfor the input bias current of both inputs3.The 686µF capacitors reduce the rippleof the power of the power suppliesholding the DC voltage more constant. The lock-in amplifier records thevoltages at the specific frequency, andthe computer program takes in the dataand plots of Vreal, Vimag, Vtotal, andVphase all plotted vs. frequency aredeveloped, as well as a plot of Vimag vs.Vreal. Each graph was developed foreach frequency span. To be thorough and to get a firmunderstanding of the behavior of theguitar, the receive transducer was placedat many different locations on the guitarand frequency spans were run. Thetransmit transducer was placed and kept

just in front of the bridge. All fivegraphs were developed at each location.

White Noise Method

Random noise with a peak amplitudeof 1V using the HP3562 Dynamic SignalAnalyzer was applied to the transmittransducer located in front of the bridgewith the guitar suspended with rubberbands. The random noise source outputwas sent into channel 1 and the outputfrom the receive transducer was sent intochannel 2. The cross spectrum waslooked at to determine natural modes.The final display of the cross spectrumincluded 150 averages over time, whichwas programmed into the analyzer. Itwas determined that 150 averagesproduced high resolution plots. Thesignal analyzer was also programmed toplot voltage as the amplitude rather thandecibels. Linear plots made it easier tolocate resonant peaks. Once again, thereceive transducer was moved around tovarious locations.

DAQ for 3-D Contours

Using the same DAQ setup as before,the C++ program in Lab Windows wasmodified to enable us to sit at aparticular resonance and move thereceive transducer around a coordinategrid assigned to the guitar and measureVreal, Vimag, Vtotal, and Vphase ateach location. Figure 3 is the setup used to map theguitar. The pegboard placed above theguitar has holes which were assigned aspecific (x,y) coordinate. The holeswere all spaced two inches apart fromeach other. A laser was shined downthrough each hole, illuminating thelocation on the guitar to place thereceive transducer. This enabled us to

coordinately grid the guitar withouthaving to place anything on the surfaceof it. The computer program allowed usto set the particular frequency andamplitude of the sine wave and to enterthe coordinate locations of both thetransmit (which remained stationary) andreceive transducers. For the Gibson J-45, 90Hz, 180Hz, and 786Hz weremapped out, and the sine waveamplitude was set at .1V, which wasfound to not overload the appropriatebounds of the computer program.

Fig. 3 Mapping Setup

Once the data was collected (eachlocation at a resonance had a Vreal,Vimag, Vtotal, and Vphase, the data wasfed into Lab View, which we used todevelop the 3-dimensional plots ofVreal, Vimag, and Vphase. To do this,the hard data of the voltages at each(x,y) location was organized in aMicrosoft Excel spreadsheet such that allpoints sharing the same x value were inthe same row and all and all pointssharing the same y value were put in thesame column, essentially re-creating ona spreadsheet the spatial orientation ofthe guitar. The file would then beuploaded directly into Lab View, whichwould develop the plots.

IV. Results Figure 4 is a cross spectrum plot withof the Gibson J-45 with the receivetransducer placed just below the transmittransducer.

Fig. 4 Cross spectrum plot of Gibson J-45with receive below transmit

The largest peaks are located at 90Hz,145Hz, 208Hz, and 368Hz, and smallerones are present as well. Using the DAQ setup and keeping thetransducers in the same place, a plot ofVtotal vs. frequency shows largeresonances at 90Hz, 140Hz, 185Hz,211Hz, 250Hz, 363Hz, and 427Hz. Thisgraph is Figure 7 located at the end.Also included are Figures 8 and 9-graphs of Vreal and Vimag plotted vs.frequency from the same run. At various locations on the guitar,resonances at 90Hz, 180Hz, and 786Hzwere seen quite frequently using bothmethods. We felt confident in believingthese to be resonant frequencies. Figures 5, 6, and 7 are 3-D plots ofthe Gibson J-45 at 90Hz, 180Hz, and786Hz, respectively. The neck of theguitar (not shown), which was not

measured in these graphs because thesegraphs show the front of the guitar, isparallel to the x-axis in the middle of thebody.

Fig. 5 3-D plot of Gibson J-45 Vtotal at 90Hz

Fig. 5 3-D plot of Gibson J-45 Vtotal at 180Hz

Fig. 6 3-D plot of Gibson J-45 Vtotal at786Hz

Plots of Vreal, Vimag, and Vtotalwere developed at each frequency.Figure 10 located at the end shows a plotof Vreal at 786Hz. Also, an Ibanez

RG570 electric guitar was studied lessextensively, but a resonance was locatedat 325Hz. Figure 11 at the end shows aplot of the Ibanez at 325Hz.

V. Discussion

The data obtained using the randomnoise method and DAQ setup arecompatible, although they are not 100%compatible. The reproducibility of therandom noise method is better than thereproducibility of the DAQ system.However, the DAQ system gives moreinformation, being that it produces plotsof Vreal, Vimag, Vtotal, and Vphase vs.frequency. Unfortunately, these graphsare extremely complicated and difficultto understand, but all contain relevantinformation as far as locating resonancesis concerned. A certain development inthe method would be to understandcompletely the DAQ graphs,understanding for instance why in theVreal vs. frequency graphs often peaksare negative or why at some peaksVphase is not 0°, since theoretically allof the voltage should be in phase andreal. In careful investigation into eachgraph, one could locate the trueresonances and exclude peaks that reallyare not. We are confident in saying that 90Hz,180Hz, and 786Hz are resonantfrequencies of the Gibson. However, weunderstand there are more resonances(like possibly 208Hz). The plots appearto have legitimacy to them due to thefact that more of the action is occurringnear the middle of the guitar rather thanthe edges, which one would expect. The90Hz and the 786Hz resonances havehigher amplitudes than the 180Hzresonance.

One problem with the 3-D mapping isthat the transducers are 3.7cm indiameter, which is very large.Therefore, they cover a large area on theguitar each time a measurement is taken,so to say that we are measuring an exact,finite point is incorrect. Also, possibledevelopment of the method couldinclude taking more points for eachmapping, even though this greatlyincreases both time and effort. Inaddition, the transmit transducer can bemoved around and placed at anti-nodesof various resonances, resulting inextremely high amplitudes. The amountof work that could be put in is endless! A definite improvement that could bemade is determining a relationshipbetween space and voltage. A 3-D plotconsisting of space on the vertical axisrather than voltage would provide amore accurate portrayal of the vibrationof the guitar. To do this, a calibration ofthe transducers is needed, explicitly arelationship between displacement andvoltage.

VI. Conclusions

The methods we have used do indeedprovide insight into the guitar. Resonantfrequencies can be located and plotted.However, guitar vibration is veryintricate, and the amount of time andeffort required to gain full understandingof it is limitless. Nonetheless, thetechniques we used can be enhanced anddeveloped to provide useful informationabout inherent modes of guitar bodies.

VII. Acknowledgments

I would like to thank my partner EricMoon, Dr. Steve Errede for the guidanceand opportunity, Jack Boparai for hishelp in obtaining equipment, Dr. Lee

Holloway for his help in thedevelopment of the random noisemethod, and Nicole Drummer formaking this an extremely beneficial andfun 10 weeks.

This material is based upon the worksupported in part by the NationalScience Foundation under Grant No.9987906. Any opinions, findings, andconclusions or recommendationsexpressed in this material are those ofthe author and do not necessarily reflectthe views of the National ScienceFoundation.

VIII. References

1Fletcher, Neville H and Thomas D.Rossing. The Physics of MusicalInstruments 2nd Edition. Springer Verlag,August 1998.

2Russell, Dan and Paul Pedersen.“Modal Analysis of an Electric Guitar.”http://www.kettering.edu/~drussell/guitars/coronet.html, January 29, 1999.

3Texas Instruments, Inc. “FET InputLow Power Instrumentation Amplifier.”http://www-s.ti.com/sc/ds/ina121.pdf,2002.

Fig. 7 Vtotal vs. Frequency for Gibson J- 45 with receive transducer placed below transmittransducer

Fig. 8 Vreal vs. Frequency

Fig. 9 Vimag vs. Frequency

Fig. 10 Gibson J-45 Vreal at 786 Hz

Fig. 11 Vtotal of Ibanez RG570 electric guitar at 325 Hz