Guitar Top-Plate Modelling Using Finite Element Method Techniques 4398 (D- 2)

Derek O'Gorman and Dermot J. FurlongTrinity CoUegeDublin 2, Ireland

Presented at ^uD,the 101stConvention1996 November 8-11Los Angeles, California

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AN AUDIO ENGINEERING SOCIETY PREPRINT

GUITAR TOP-PLATE MODELLING

USING FINITE ELEMENT METHOD TECHNIQUES

Derek O'Gorman and Dermot J. Furlong,

Electronics and Electrical Engineering Department,

Trinity College,

Dublin 2,

Ireland.

ABSTRACT

Results of finite element method modeling of guitar top plates are presented and compared withChladni vibration patterns of real top plates. Use of thefinite element model for the interactiveinvestigation of the effects of bracing pattern, plate thickness and material property variations isoutlined with reference to the application of such models as effective design tools.

0. INTRODUCTION.

Modern day attempts at understanding the functioning of guitars have extended the traditionalcraft approach into the realms of science and engineering. Methods of describing the guitar

system have included the use of mathematical and system models in efforts to simplify and

comprehend the fundamental properties that govern the instruments behaviour [4,5]. Muchresearch has been concentrated solely on the top plate and its effect on the tonal quality of the

instrument[3,6]. Such research emphasizes the importance of the bracing pattern of the plate

relative to the size, shape, thickness and materials used in its construction. The lack of consensus

among luthiers as to the ideal 'play off' between these parameters is further evidence as to the needfor improved analysis methods. This paper presents the results of research into computer aided

simulation of the top plate. The simulation approach is considered as a" melting pot" for both the

craft and scientific methods of understanding and analysing the properties of the guitar.

1. MODES, FREQUENCY MATCHING AND COUPLING.

In order to appreciate the significance of the top plate modal patterns to be discussed in subsequentsections, a brief introduction to strings, plates and resonance phenomena is necessary.

Modes of vibration are the vibrational patterns that any object may vibrate with. The vibration

medes of the guitar are complex and arise due to the interactive energy transfer between the string

and the top plate. Before considering this interactive process, the two important factors that

influence the degree to which one vibrating system can influence another must be considered. These

are the frequency match between the driving frequencies and the vibration modes of the driven

object and the cohpling between the two systems[l]. If the driver frequency is operating at any of

the modal frequencies of the driven object the individual mode corresponding to that frequency will

be excited. If, however, the driver operates at frequency values that fall between any of the modesof the object then each of the modes are excited to a certain degree. For a complex excitation the

frequencies of vibration do not change but due to the match between frequencies of the driver and

the resonant frequencies of the modes of the driven object, the amplitude of the vibrations will

change. Thus, a high amplitude harmonic in the driving frequency that does not match a resonantfrequency of any vibration mode, is likely to be diminished. The opposite of this is also tme: aweak harmonic in the driving vibration may be enhanced if it matches one of the modal resonant

frequencies.

Coupling refers to the method or means by which one system is "connected" to another. Systemscan either be loosely or tightly coupled. In a loosely coupled system the driver only brings about

large amplitude vibrations at the resonant frequency of the driven object. In a tightly coupled

system the driver and object are connected by a stiff link. This results in the pattern of the driverdominating over that of the driven object. The amplitude of vibration is nearly identical to that of

the vibrator amplitude and remains so for all frequencies. The quality factor, Q, is the term used to

describe the degree of coupling. Systems with a high Q are loosely coupled and vice versa.

2, THE STRING AND TOP PLATE INTERACTION

In the context of guitar function, the plucking of the string results in a complicated vibrationalpattern that consists of a fundamental and many harmonics. This complicated pattern is imposed

on the top-plate, a sound body vibrator with its own complex vibration patterns. The complexdriving vibration can be considered as being made up of many sinusoidal vibrations. Each of these

can excite every vibrational mode of the plate at varying amplitudes. The overall sound bodyvibration becomes the sum of all the effects from each harmonic. The growth and decay of each

harmonic of the driving vibration may be different and in turn each vibratory mode of the sound

body has its own growth and decay rates due to the coupling effects. Thus, the attack and decaytiming of any vibration mode will be due to the joint action of the two time functions. The

interactivity between the top plate and the driving string also results in the pattern of the drivenvibration modes feeding back to the driving string which further affects the vibration modes of the

string. This action/reaction process continues until the vibrational energy dies away.

3. UNDISCIPLINED NOTES (WOLF TONES)

In the case where a plucked string's fundamental frequency closely matches a particularly strong

resonant frequency of the top plate a phenomenon that is analagous to the wolf tone effect,common in bowed instruments, occurs. In such a case the energy from the vibrating string is

rapidly transferred to the plate. The rapid absorbtion of all the strings energy, due to initially

intense plate vibrations, encourages the note to be very loud. At this instance the plate, inaccordance with the interactive string/body process, attempts to drive the string. However, when

the plate is subject to very strong resonant modes it has a limited ability to act as an efficient driverfor the string. This results in little energy being redirected back to the string, the overall effect

being a damping of the string vibration which shortens the sustain and decay rates of the note[I,2].

4. THE QUALITIES OF A GOOD GUITAR TONE

When considering the tonal qualities of a guitar, it is necessary to realise that sound is a subjective

phenomenon. The individual preference of one aspect of musical sounds over another is proof ofthis in itself. However, the subjective nature of that which sounds optimal varies only

slightly when considering the tone of a guitar. Most luthiers and researchers alike agree that there

are certain qualities desirable in all acoustic guitars. The need for an instrument that is both loudand evocative in its musicality is paramount. To this end many strive to produce a powerful

instrument, with a brilliant tone, improved tonal sustain, an evocative treble and a tree bass

fundamental [3]. When considering these facets with respect to the guitar top plate, the

effectiveness of the plate vibrations must be considered. For a quality bass fundamental the largerthe vibrational area of the top plate, the more convincing the bass will tend to be. As notes

progress to the higher frequencies the areas of vibration should be smaller, with the effect of treble

side vibration regions playing an important role in producing a quality note while still receivingsupport from relevant vibrational regions throughout the plate. Ideally amplitudes of vibration

should not be too strong, particularly in the vibrational area most related to the frequency of thenote, with a proportionate distribution of the strings energy throughout the plate. The frequenciesthat such vibrations occur at are determined by the thickness of the plate, the materials used and

most importantly by the bracing pattern.

5. FEM MODELING OF THE GUITAR TOP PLATE

OUTLINE OF MODELING PROCEDURE

At this juncture consideration is given to the procedure undertaken for simulating and analysingguitar top plates. The aim was to produce an accurate finite element model of a real top plate

using an existing FEM software package (in this instance ANSYS) and to then solve the modes ofvibration of the simulated plate. Validation was undertaken using standard Chladni vibration

patterns on the real plate. Comparisons between the simulated and real model vibration patternswere then undertaken to determine the accuracy of the simulated model and its results. This

validation becomes the departure point from which future varied designs could be modeled.

DESIGN CRITERIA

The initial top plate design presented is that used on classical guitars manufactured by George

Lowden in 1994 (Figure 1). The soundboard is made of cedar and the spruce struts are shaped and

tapered in accordance with the stiffening effect required of any particular region of the plate. Thecross (or "X") bracing, a feature normally associated with steel string guitars, serves a somewhat

similar purpose as on steel string guitars insofar as it provides stmctoral integrity. However, theextension of the cross brace into the upper bout region through apertures in the harmonic bar, is to

encourage the whole soundboard to vibrate more efficienfiy in an effort to improve bass response.The soundboard is also tapered from the bass side to the treble side (2.55mm to 2.3mm).

This particular design produced quality guitars with good response, yet most were slightly flawed.

There was a problem with certain undisciplined notes on the high 'E' string, specifically 'F' and'B' flat (130' and 18thfret respectively). Usually these notes were loud and very short, a description

that closely resembles the wolf tones normally associated with bowed instruments. There were

other problem notes (e,g. zero fret 'G' string) but due to the fact that the string length was longerand the note's frequency was lower, these problem notes were not so noticeable. In order to deal

with this problem, Lowden redesigned the treble side of the plate to that of the 1995 design (Figure2). This modification dealt very effectively with the undisciplined notes, with all high notes (12 _

fret upwards) on the high 'E' string exhibiting a very marked improvement in their sustain.

The scope of the criteria involved here is twofold in terms of attempting to produce an accuratefinite element model of a guitar top plate. In the first instance, there is a particular top plate at hand

to model and validate, that of the 1994 design. Furthering this, there is a basis to work from in

terms of comparing any alterations. However, such design alterations have already been consideredand implemented by Lowden resulting in the 1995 plate design which dealt quite effectively with

the undisciplined notes. The logical step therefore is to extend the procedure to a second finite

element model (the 1995 plate) in order to provide a basis for comparing the effects of designalteration while simultaneously analysing the possible reasons why such alterations dealt so

effectively with the undisciplined notes. Thus, there is not only the scope to simulate and validate a

given top plate but also the possibility of exploring a specific problem.

1994 MODEL AND VALIDATION

The generated finite element model of the 1994 top plate is shown in Figures 3(a) and3(b). Inorder to ensure accurate geometric corroboration between the struts of the simulated model and

those of the real plate, both were left rectangular in section and profile. This was in order to reduce

errors that may have been introduced while attempting to accurately model the intricate shaping ofthe struts. The taper in the soundboard, however, was modeled accurately. The material properties

of cedar and spruce were associated with the soundboard and struts, respectively. Graindirectionality was also incorporated into the material properties by ensuring the respective Youug's

modulii were correctly set in both the transverse and axial directions. The model presented does notinclude the bridge, as the real plates used for validating the model by means of Chladni patterns are

also without a bridge. The solid model mesh consists solely of 3-dimensional solid brick and ·tetrahedral elements. The natural modes of vibration were solved under conditions resembling those

ora plate in a guitar body (all boundary nodes had all degrees of freedom set to zero in order to

effectively 'clamp' the plate steady).

The first simulated modal pattern for the plate is shown in Figure 4. This is typical of the first

mode of vibration for any plate, with only one large region of displacement. It can be noticed thatthe extension of the struts into the upper bout proves quite effective in encouraging a significant

area of the board to vibrate. The second modal pattern is shown in Figure 5 with its complimentarymode shown in Figure 6. It can be noted that both these modes also encourage the majority of the

surface area of the plate to vibrate, but with two distinct regions.

The model was validated by the traditional method of Chladni vibration patterns. A steal ring rigwas constructed in order to clamp the plate firmly at the edges. Fine sand was spread across the

surface of the plate and the modes of vibration were agitated by two methods. In the first instance a

Ling Dynamic Systems model 200 vibrator was used to excite the plate at various points around

the surface. The second method used to excite the modes of the plate involved suspending aloudspeaker at a nominal distance directly over the plates surface. In both cases a sine wave

generator was used to drive the system. The induced plate vibrations resulted in the migration of

the sand particles from regions of displacement (antinodes) to the regions of zero displacement(nodes).

The combined results from these two tests proved the model to be more than just acceptably

accurate and provided further information about the nature of the top plates coupling and

frequency match characteristics. In the instance of using the vibrator (tight coupling) to drive theplate, the first three modal patterns are reproduced quite accurately. This can be seen from photo

comparative results in Photos 2,4 and 6, The position of the cross indicates the point at which the

plate was excited. Different positions of the vibrator were experimented with in all instances and

those presented are considered to give optimum results.

By comparison, the modes agitated by the speakers (effectively air resonated and loosely coupled)are somewhat less accurate in the case of some of the lower frequencies. The first mode ofvibration is shown in Photo 1 and proves quite an accurate match to that of the simulated model.

The second (air excited) mode in Photo 3 appears to have a reduced vibrational region whereas the

third mode (Photo 5) appears to correlate more accurately with the simulated result. Thisfluctuation between air and vibrator excited patterns continues for some higher frequency modes,

yet the air vibration mode patterns become more correlated with the simulated patterns at the

highest frequencies. Some of the air excited patterns proved more accurate than others, but the lackof this accuracy in the latter was not judged to be too severe. This was the opposite of the

attempts to excite higher frequency modes using the vibrator, with the modal frequency patternsabove 724Hz being difficult to reproduce. The tenth and twelfth air excited modal pattems (Photos7 and 8) are shown to demonstrate the increased correlation between these and the simulated

modal patterns (Figures 7 and 8).

The two sets of validation methods presented were deemed to prove that the finite element model

was a more than reasonable representation of the top plate. The results indicate that the iow

frequency modes are more tightly coupled (lower Q) than the higher frequency modes due to thefact that the low frequency modes appear to be dependent on a vibrator that is more forceful

and as such dominates the extent to which these modes are excited. The dependency of the higher

resonant frequencies of the plate appears to .be on the degree to which the frequencies of the drivermatches these resonant frequencies as opposed to the extent to which they can be driven.This is important when one considers that any musical note is rich in harmonics that have a low

amplitude. In order that such harmonics are enhanced for any guitar note, high Q resonance is

important for such weaker harmonics that closely match these resonant frequencies. This qualitymust be considered paramount if the notes produced on any guitar are to be rich and melodic withthe degree of frequency matching determining the extent to which any of the higher modes areexcited and also as to which modes are excited first.

COMPARATIVE RESULTS FOR BOTH MODELS

The fact that the top plate design for the 1995 classical guitars produced by Lowden proved soeffective in dealing with the undisciplined notes, provides a prompt to model this plate design and

apply the FE model to comparative tests. The FE model of the 1995 plate is shown in Figure 9

with the notable change of the strutting pattern on the treble side of the guitar. The first three

modal patterns of the model are also shown in Figures 10 to 12. Comparing these three patternsalone to those of their 1994 counterparts provides some insights into the effect of the new bracing

pattern on the quality of the instrument.

The first modal pattern is identical to that of the 1994 model pattern with the exception that the

frequency is lower by approximately 4Hz. This could be considered as approximation error on thepart of the FE simulation package. However, the second modal pattern has a considerably reduced

modal frequency and the area of vibration is also reduced and tempered. The third modal vibration

pattern has a higher modal frequency with virtually no lower bout vibration pattern. The fact thatboth these modes have altered resonance frequencies indicates that it is not unreasonable to assume

that the resonant frequency of the first mode has indeed been slightly lowered, albeit by an

insignificant amount. The analysis of the next two lower modes however leads to a reasonable

argument as to the existence of low frequency undisciplined notes.

Recall the discussion on the 1994 validation test in which the second vibration mode was only

accurately excited by the vibrator whereas the third mode was both air (high Q) and vibrator

(low Q) excited. Consider further that a low frequency undisciplined note (the example given beingthat of zero fret 'G') has a harmonic that could be (undesirably) somewhat frequency matched with

the resonant frequency of the third mode. When the note is played, the partially excited largevibrational area of the second mode, coupled with the undesired resonance of the third mode due to

the matched harmonic, effectively combined to produce the undisciplined note. This theory is

suggested due to the fact that in the 1995 model, the third modal pattern has virtually no lowerbout vibration that could act as a contribution towards the support of a note. The indication is that

the 1995 model ensures that the high Q resonant mode has been effectively damped and shifted to ahigher frequency to reduce interference effects and that the low Q modal pattern dominates with

more controlled vibrational regions.

The higher frequency notes that were undisciplined were those of 'F' and 'B' fiat on the high 'E'

string (approx. 698Hz and 932Hz respectively). In order to compensate for the fact that the strutson the model are not shaped, as would have been the real scenario, a slightly extended frequency

range, relative to the fundamental frequency of the respective real undisciplined notes, is to beconsidered for both the 1994 and 1995 FE models. Due to the fact that the struts on the FE model

effectively increase the stiffening of the plate, an ideal range would be that of a number of modalfrequencies higher than the note's fundamental (stiffer plates result in higher resonance modes)[3]

and the nearest lower mode. In the first instance, that of the 'F' note, these are the sixth (6t°),

seventh (7_h),eighth (8th), and ninth (9 th)modal patterns for both the 1994 and 1995 models. Thesepatterns occur at 602Hz, 703Hz, 724Hz and 782Hz for the 1994 model (Figures 13 to 16). The

most notable feature of these modal patterns is that there is a significant amount of vibrationaround the treble region in all cases. The two modes most active on the treble side are those

occurring at 703Hz (7a modal pattern) and 782Hz (9thmodal pattern) with the point of maximumdisplacement in the 7* modal pattern case being in the treble region itself. Comparing the same setof patterns (6 to 9) in the 1995 model there are marked changes. The frequencies are only altered

slightly in the case of the 6th(608Hz), 7th(697Hz) and 8th(732Hz) resonances whereas the 9 th

resonance has a considerable decrease in the frequency (763Hz) (Figures 17 to 20). In all cases

however, the vibrational area in the altered region appears to be consistently more controlled withreduced variations across the four resonance range and a considerable spatial tempering of the

treble region for the 7thmodal pattern alone.

The phenomenon of frequency matching and a high Q factor now provides a basis for

understanding the nature of the undisciplined high notes in the 1994 model and the improvement of

the 1995 model in this respect. In the first instance the fundamental of the note would have closelymatched the 7* modal pattern resonant frequency of the plate (which would have been slightly

lower than the simulated modal frequency due to reduced stiffness resulting from brace shaping).

The close frequency match would result in this particular resonance pattern being strongly excited.Noting that the particular undisciplined note occurs on the treble string, the regions of vibration on

the treble side play an important role in supporting this note with further support from the rest of

the plate's vibration. Comparing now the 1995, 7* medal vibration pattern, it can be seen that the

resonant frequency is only slightly lowered which would imply that the frequency matching wouldstill be quite strong for the note. However, if the treble regions are examined in both cases, it can

be seen that the point of maximum displacement for the mede in the 1994 case occurs in the treble

region, but that this maximum displacement point is shifted to the bass side, upper bout in the 1995

case. This would indicate the importance of the location and amplitude of displacement ofresonating regions relative to the frequency of any note. Relative to this discussion on treble notes,

the pattern and location of all the vibrating regions on the treble side (and particularly in the treble

region) are paramount in providing the correct support for the note. In the case of the 1994 designit appears that the maximally excited treble region would rapidly absorb most of the strings energy

resulting in the undisciplined note. In the 1995 case, the shifting of the maximally excited region to

the base side upper bout resulted in the more controlled absorption of the strings energy in thetreble region, with correct combined support from all other vibarational regions, leading to a note

with proportionate loudness and sustain. A similar explanation applies to the undisciplined noteoccurring at 'B' flat but is not entered into here as the principles are the same as those already

given for the 'F' note.

CONCLUSION

The presented finite element models of the two real top plates are considered to have proved the

validity of this modeling technique for analysing the behaviour of the guitar top plate. It is perhaps

fortunate that there was the scope to model both a given top plate that had contributed to slightflaws (when used in fully constructed guitars) and the top plate that corrected the problem. The

modeling of the plates in "out of body" conditions appears to have contributed useful insights intothe behaviour of both plates and the comparative results provided a basis for understanding both

the existence and elimination of the undisciplined notes.

REFERENCES

[1] S. Handel, "Listening: An Introduction to the Perception of Auditory Events", BradfordBooks, The MIT Press, 1993.

[2] C. Taylor; "Exploring Music: The Science and Technology of Tones", IOP Publishing Ltd.,1992.

[3] Michael and Nicholas Kasha, "Applied Mechanics and the Modem String Instrument. ClassicalGuitar", The Journal of Guitar Acoustics, Issue 6, September 1982.

[4] Ian Firth, "Guitars: Steady State and Transient Response", Journal of Guitar Acoustics,Issue 6, September 1982.

[5] Erik Jansson, "Acoustics for the Guitar Maker", Function, Construction and Quality of theGuitar, Royal Swedish Academy of Music, 1983.

[6] Thomas Rossing, "Plate Vibrations and Applications to Guitars", Journal of Guitar Acoustics,Issue 6, September 1982.

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