Applied Acoustics 73 (2012) 1168–1173

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Applied Acoustics

journal homepage: www.elsevier .com/locate /apacoust

Effects of using scalloped shape braces on the natural frequenciesof a brace-soundboard system

Patrick Dumond ⇑, Natalie BaddourDepartment of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, CBY A205, Ottawa, Canada K1N 6N5

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 September 2011Received in revised form 16 March 2012Accepted 30 May 2012Available online 23 June 2012

Keywords:Frequency matchingScalloped braceBrace-plate systemPlate vibrationOrthotropic properties

0003-682X/$ - see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.apacoust.2012.05.015

⇑ Corresponding author.E-mail addresses: pdumo057@uottawa.ca (P. Dum

(N. Baddour).

Many prominent musical instrument makers shape their braces into a scalloped profile. Although reasonsfor this are not well known scientifically, many of these instrument makers attest that scalloped bracescan produce superior sounding wooden musical instruments in certain situations. The aim of this paper isto determine a possible reason behind scalloped shaped braces. A simple analytical model consisting of asoundboard section and a scalloped brace is analyzed in order to see the effects that changes in the shapeof the brace have on the frequency spectrum of the brace-soundboard system. The results are used to ver-ify the feasibility of adjusting the brace thickness in order to compensate for soundboards having differ-ent stiffness in the direction perpendicular to the wood grain. It is shown that scalloping the brace allowsan instrument maker to independently control the value of two natural frequencies of a combined brace-soundboard system. This is done by adjusting the brace’s base thickness in order to modify the 1st naturalfrequency and by adjusting the scalloped peak heights to modify the 3rd natural frequency, both of whichare considered along the length of the brace. By scalloping their braces, and thus controlling the value ofcertain natural frequencies, musical instrument makers can improve the acoustic consistency of theirinstruments.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Like most musical instruments, stringed instruments are de-signed and built to produce sound by allowing their body to vibratewhen excited by some outside force. In the case of stringed musicalinstruments, a set of strings are usually plucked, strummed ortapped by the musician in order to set these in motion. The vibrat-ing motion of the strings is then transmitted to the body of theinstrument through the soundboard which in turn displaces theair around it, creating pressure waves perceived by the listenerto be the characteristic tones of a particular instrument. Certainstringed musical instruments such as the flat top guitar, the man-dolin or the piano use braces on the underside of their soundboardso that it remains thin enough to be within the desired natural fre-quency range while still maintaining the structural integrity re-quired to withstand the large string tension. Although the bracesare structural in nature, their shape and size greatly affect the fre-quency spectrum of the soundboard system. The braces modify thefrequency spectrum of the soundboard by locally modifying itsmass and stiffness.

The physics of musical instruments have been thoroughly stud-ied over the last half century, with most research focusing on under-

All rights reserved.

ond), nbaddour@uottawa.ca

standing the key elements of sound production and radiation as canbe seen in various books published over the past few decades [1–3].With the growth in popularity and capabilities of numerical simula-tion, many researchers have focused on trying to reproduce parts orwhole instruments as accurately as possible [4–8]. However, due tothe variable nature of wood, it is often necessary to adjust a numer-ical model’s parameters based on those of actual material speci-mens used in instrument construction. As such, other studies haveproduced numerical simulations which parallel the constructionof an instrument so as to be as accurate as possible while verifyingthe effects of various construction stages on the frequency spectrumof the instrument [9,10]. These studies in particular have helpedverify the effects of the addition or removal of material on an instru-ment’s spectrum. These results have also allowed researchers tocreate construction methods and instruments of better acousticquality [11,12]. More recently, instrument makers have also at-tempted to share and explain their empirical expertise gained overyears of construction and fine tuning experience [13–15]. However,their explanations are not always scientific in nature. It is clear thata certain gap still exists between the scientific understanding ofmusical instrument construction and the nature of hand-builtinstruments.

One such unexplored example is the reason why many promi-nent instrument makers shape the braces on their instrument’ssoundboard. While most instrument makers disagree on thereasons behind the shaping of braces, many will agree that it can

Fig. 1. Shape of a scalloped brace.

P. Dumond, N. Baddour / Applied Acoustics 73 (2012) 1168–1173 1169

improve an instrument’s sound. Known as scalloped braces, theyhave a shape similar to those of Fig. 1.

This paper seeks to justify the use of scalloped braces by deter-mining why, in certain situations, they produce a superior sound-ing musical instrument. Based on previous research, it wasthought that it may be possible to independently control the valueof two natural frequencies using a scalloped-shaped brace [16].The goal of this paper is to verify this hypothesis.

Current research shows that it is the first few natural frequen-cies of the soundboard that determine its acoustic quality [11].Therefore, we focus our attention on the first few eigenvalues ofthe combined brace-soundboard system and our solution approachwill exploit this. Previous research also suggests that the requiredfrequency accuracy for a musical instrument must fall within a 1%margin due to the ear’s sensitivity to pitch [17], although it is un-clear if this required frequency margin applies to the soundboardalone. Common practice in the industry is to use the stiffness per-pendicular to the wood grain as a measure of the acoustical qualityof a soundboard [18]. In this paper, we determine if it is possible tocompensate for the variance in the stiffness perpendicular to thegrain (Young’s modulus in the wood’s radial direction) by adjustingthe thickness at various points along the brace.

2. Analytical model

A simple model is created in order to investigate whether or notthe value of two natural frequencies can be controlled using a scal-loped brace. The model of Fig. 2 represents a section of musicalinstrument soundboard usually supported by a single brace.

The assumed shape method is the method chosen for investi-gating the effects of changes in the thickness of the brace. Themethod itself is well explained in [19]. This method is chosen be-cause it allows us to use the flat-plate modeshapes as the funda-mental building blocks of the solution, thereby permittingobservation of how the addition of the scalloped brace affects thosefundamental modeshapes. This method also permits greater flexi-bility in analyzing the effects of the scalloping since it enables thecreation of an analytical solution from which numerical solutionscan be quickly obtained for various parameter modificationsincluding changes in the thickness of the scalloped brace. Themethod uses 3 � 3 trial functions during the analysis. The equa-tions of motion are derived using a computer algebra system (Ma-ple). This yields mass and stiffness matrices, where each matrixentry is a function of all physical parameters (dimensions, density,stiffness, etc.). The effect of any parameter on the system’s eigen-values can then be easily examined without having to re-establishthe entire system model.

Ly

x

yz

Lx

0

0hp hc

x1 x2

brace

plate

Fig. 2. Orthotropic plate reinforced with a scalloped brace.

2.1. Modeling assumptions

The soundboard is modeled as a thin rectangular plate and thebrace is modeled as a thicker section of the same plate. A simplerectangular geometry is assumed in order to enable the closed-form solution of a simple plate (without the brace) to be used asthe trial functions for the assumed shape method. The plate isassumed thin so that linear Kirchhoff plate theory can be used.The Kirchhoff assumptions state that when compared to the plate’sthickness, deflections are small. Kirchhoff plate theory also ne-glects transverse normal and shear stresses, as well as rotary iner-tia. Although this is an accurate assumption for the plate, due tothe brace’s thickness-to-width aspect ratio, it may imply a certainerror in that region of the soundboard. Also, because of the methodin which the brace thickness is added to that of the plate in the ki-netic and strain energy expressions, it was necessary to change thedirection of the grain of the plate, in this region only, to match thatof the brace. This is a reasonable assumption since the plate is thinin comparison to the brace and there is a solid link of wood gluewhich bonds them together. Since the stiffness of the brace is muchgreater than that of the plate in this region, it is the brace whichdominates the stiffness properties. The plate is also assumed tobe simply supported all around, although in reality it is somewherebetween simply supported and clamped [3]. It has been assumedthat the system is conservative in nature, which allows dampingto be neglected. Although there is a certain amount of dampingfound in wood, its effects on the lower natural frequencies isthought to be minimal and has been neglected. This is justified be-cause the lower frequencies have a larger effect on the sound-board’s perceived pitch than do the higher frequencies [11].

2.2. Kinetic and strain energies

Because the brace is modeled as a thicker section of the platebetween x1 and x2, the standard kinetic and strain energies mustbe modified accordingly by breaking them down into three distinctsections. For an orthotropic plate, the kinetic and strain energiescan be found in [20].

With the addition of the brace, the kinetic energy for the sound-board and brace system becomes

T ¼12

Z x1

0

Z Ly

0_w2qp dydxþ 1

2

Z x2

x1

Z Ly

0_w2qc dy dx

þ 12

Z Lx

x2

Z Ly

0_w2qp dydx ð1Þ

where Lx and Ly are the dimensions of the plate in the x and y direc-tions respectively, the dot above the transverse displacement vari-able w represents the time derivative, q is the mass per unit areaof the plate such that

qp ¼ l � hp and qc ¼ l � hc; ð2Þ

l is the material density and hp and hc are the thickness of the plateand combined brace-plate sections respectively.

The strain energy is also modified by the addition of the braceand becomes

U ¼12

Z x1

0

Z Ly

0½Dxpw2

xx þ 2Dxypwxxwyy þDypw2yy þ 4Dkpw2

xy�dydx

þ 12

Z x2

x1

Z Ly

0½Dxcw2

xx þ 2Dxycwxxwyy þDycw2yy þ 4Dkcw2

xy�dydx

þ 12

Z Lx

x2

Z Ly

0½Dxpw2

xx þ 2Dxypwxxwyy þDypw2yy þ 4Dkpw2

xy�dydx ð3Þ

Table 1Material properties for Sitka spruce as an orthotropic material.

Material properties Values

Density – l (kg/m3) 403.2Young’s modulus – ER (MPa) 850Young’s modulus – EL (MPa) ER/0.078Shear modulus – GLR (MPa) EL � 0.064Poisson’s ratio – mLR 0.372Poisson’s ratio – mRL mLR � ER/EL

Table 2Model dimensions.

Dimensions Values

Length – Lx (m) 0.24Length – Ly (m) 0.18Length – Lb (m) 0.012Reference – x1 (m) Lx/2–Lb/2Reference – x2 (m) x1 + Lb

Thickness – hp (m) 0.003Thickness – hbo (m) 0.012Thickness – hc (m) hp + hb

1170 P. Dumond, N. Baddour / Applied Acoustics 73 (2012) 1168–1173

where the subscripts on w refer to partial derivatives in the givendirection, as per standard notation, the stiffnesses D are section-specific because of the change in thickness h from x1 to x2:

Dxp ¼Sxxh3

p

12Dyp ¼

Syyh3p

12

Dxyp ¼Sxyh3

p

12Dkp ¼

Gxyh3p

12

ð4Þ

and

Dxc ¼Sxxh3

c

12Dyc ¼

Syyh3c

12

Dxyc ¼Sxyh3

c

12Dkc ¼

Gxyh3c

12

ð5Þ

where G is the shear modulus and the S are stiffness componentsthat are defined as

Sxx ¼Ex

1� mxymyx

Syy ¼Ey

1� mxymyx

Sxy ¼ Syx ¼myxEx

1� mxymyx¼ mxyEy

1� mxymyx

ð6Þ

Here, the subscripts represent the direction of the plane inwhich the material properties act. Therefore, Ex is the Young’smodulus along the x-axis, Ey along the y-axis and mxy and vyx arethe major Poisson’s ratios along the x-axis and y-axis respectively.

2.3. Scalloped brace shape

To accommodate the scalloped shape brace in the energy equa-tions, the variable thickness of the brace hb such that hc = hp + hb

must be taken into account. In order to model the scalloped shape,a second-order piece-wise polynomial function is chosen. Thispolynomial function puts the peaks of the scallops at 1=4 and 3=4 ofthe brace. The function is given by

hb ¼

k � y2 þ hbo for y < Ly

4

k � y� Ly

2

� �2þ hbo for Ly

4 6 y 6 3Ly

4

k � ðy� LyÞ2 þ hbo for y > 3Ly

4

8>>><>>>:

ð7Þ

where hbo is the height of the brace at its ends and center and k isthe scallop peak height adjustment factor which is a real valuewhose range is the subject of investigation. For the purpose ofdimensioning, the width of the brace is identified as Lb.

3. Results

3.1. Material properties

The material chosen for analysis of the soundboard is Sitkaspruce due to its common usage in the industry. Material proper-ties for Sitka spruce are obtained from the US Department ofAgriculture, Forest Products Laboratory [21]. Since properties be-tween specimens of wood have a high degree of variability, theproperties obtained from the Forest Products Laboratory are anaverage of specimen samplings. The naturally occurring propertiesof wood cause it to act as an orthotropic material. Material proper-ties of Sitka spruce are seen in Table 1. The subscripts ‘R’ and ‘L’ re-fer to the radial and longitudinal property directions of woodrespectively. These property directions are adjusted accordinglyfor both the plate and the brace.

3.2. Soundboard dimensions

A typical section of a soundboard structurally reinforced by asingle brace is modeled using the same dimensions throughoutthe analysis. The pertinent dimensions are given in Table 2. In or-der to avoid confusion, subscript ‘p’ stands for plate, ‘b’ for braceand ‘c’ for combined plate and brace.

3.3. Benchmark values

To set a benchmark for further investigation, a set of values forthe natural frequencies are computed and will be used for compar-ison purposes. This benchmark is based on a Young’s modulus of850 MPa in the soundboard’s radial direction, a brace base heightof 0.012 m and a scallop peak height adjustment factor of 1. Thefirst five natural frequencies and their modeshapes are found inTable 3.

In Table 3, mx and my represent the mode numbers in the x andy directions respectively. Also, the dip in the center of x-axis of themodeshapes represents the location of the brace, which locallystiffens the area and limits the maximum amplitude possible.

3.4. Effects of the soundboard stiffness and brace thickness

We consider the effects of a change in both the Young’s modu-lus in the radial direction ER and of the brace base thickness hbo, onthe scalloped brace-soundboard system. Although several of thelowest natural frequencies carry importance, only two will be ob-served during the variation in structural properties. This is becausefrequencies that have a mode of vibration which contain a node atthe location of the brace are not as affected by the brace as thosewhich have a mode which passes through it. Therefore the two fre-quencies observed during this analysis are the 1st and 3rd naturalfrequencies of the system. The 2nd and 5th modeshapes have anode at the location of the brace and are not as affected by thebrace, contrary to the 1st and 3rd modeshapes which do not, asseen from the figures in Table 3. Although the 4th natural fre-quency does not contain a node at the location of the brace, thebrace’s current scalloped design of Eq. (7) is not well suited to ad-just this frequency and will therefore not be held constant. This isbecause the scalloped shape’s peaks do not correspond with themaximum amplitudes of the fourth modeshape.

Table 3First five natural frequencies of the benchmark soundboard.

mx my Natural frequency (Hz) Modeshape

1 1 619

2 1 820

1 2 1131

1 3 1346

2 2 1400

Table 4Natural frequency variation with changes in ER.

Young’s modulus ER

(MPa)1St natural frequency x1

(Hz)3Rd natural frequencyx3 (Hz)

750 582 1062800 601 1097850 619 1131900 637 1164950 655 1196

Table 5Natural frequency variation with changes in hbo (k = 1).

Brace thickness hbo

(m)1st natural frequency x1

(Hz)3rd natural frequency x3

(Hz)

0.0110 579 10850.0115 599 11080.0120 619 11310.0125 640 11550.0130 661 1180

Table 6Compensation for variations in soundboard stiffness.

Young’smodulus ER

(MPa)

Brace basethicknesshbo (m)

Peak heightadjustmentfactor k

1St naturalfrequency x1

(Hz)

3Rd naturalfrequency x3

(Hz)

750 0.0123 2.0 619 1128800 0.0121 1.6 620 1133850 0.0120 1.0 619 1131900 0.0118 0.6 618 1134950 0.0120 �0.2 620 1134

y

x

A

A

Fig. 3. Location of cross-section for modeshape comparison.

P. Dumond, N. Baddour / Applied Acoustics 73 (2012) 1168–1173 1171

In order to see what effects changes in the soundboard’s stiff-ness and brace’s base thickness have on the overall naturalfrequencies of the system, each of soundboard stiffness and bracebase thickness is varied from its benchmark value. When one isvaried, the other is held constant at the benchmark value. As canbeen seen in Tables 4 and 5, the natural frequencies increase whenboth the soundboard’s stiffness and scalloped brace base thicknessincrease respectively.

3.5. Previous results

It was shown in [16] that by adjusting the thickness of a rectan-gular brace, one can control the value of the fundamental fre-quency of the brace-soundboard system. The rectangular braceachieves this by limiting the maximum amplitude of the domeshaped fundamental frequency. Reducing the stiffness of the braceincreases the vibration amplitude of the fundamental modeshape.The value of other natural frequencies, which have maximum

amplitudes along the length of the brace, can also be controlled.However, a rectangular brace alone does not offer enough freedomto control the value of more than one frequency independently.

3.6. Controlling the value of two natural frequencies

In order to control the value of two natural frequencies inde-pendently, a modification to the shape of the brace is made. Theresulting scalloped shape brace increases the freedom with whichfrequencies can be controlled. The 1st natural frequency is ad-justed by changing the base thickness of the brace, hbo. The 3rd nat-ural frequency is adjusted by changing the scallop peak heights ofthe brace by modifying the peak height adjustment factor, k. Thenatural frequencies are adjusted to within 1% of those calculatedat the benchmark of ER = 850 MPa. Table 6 shows these natural fre-quencies which have been made consistent by compensating forvariations in the soundboard’s cross-grain stiffness.

3.7. Effects of the brace on the modeshapes

By adjusting the shape of the brace, a musical instrument makercan control the value of two natural frequencies in the soundboardsystem. However, by changing the shape of the brace, the mode-shapes are also affected, Figs. 3 and 4.

To demonstrate the effect of the scalloped brace on each mode-shape, a cross section of the three lowest modeshapes is taken andanalyzed. Fig. 3 shows the location in the plate, where the cross-section is taken, while Fig. 4 demonstrates the effect on that(cross-section of) modeshapes. Fig. 4, part (a) shows a cross-sec-tion of modeshape 1 � 1 for the plate without a brace, with a rect-angular brace as in [16] and with the scalloped brace. The 3D shapeof the modeshape is shown in the top right corner of the figure, andshows that the 1 � 1 modeshape is a dome. It can be observed thatadding a brace flattens out the dome shape but changing the rect-angular brace to a scalloped brace does not have a significant effecton the average shape for this particular mode. Looking at the sec-ond (2 � 1) modeshape, shown in part (b) of Fig. 4, it is clear thatthe brace has very little effect on the modeshape due to its locationalong the nodal line of that modeshape. Finally, looking at the 1 � 2modeshape in Fig. 4c, it is clear that adding peaks on the braceat the location of the soundboard’s maximum amplitude ofvibration significantly affects the modeshape and therefore alsoits frequency.

xy

xy

xy

x

x

x

a

b

c

Fig. 4. Section ‘‘A–A’’ comparison of mx �my modeshapes: a. 1 � 1, b. 2 � 1and c. 1 � 2.

Brace

1st mode of vibration

2nd mode of vibration

Fig. 5. Scalloped brace with affected frequencies.

Fig. 6. Effect of a negative peak height adjustment factor.

1172 P. Dumond, N. Baddour / Applied Acoustics 73 (2012) 1168–1173

4. Discussion

Based on the results, it becomes clear that it is possible to adjusttwo natural frequencies at once by modifying the shape of only onebrace. This is made evident by Fig. 5. This is not possible with a sin-gle rectangular brace [16]. Therefore, the scalloping of the brace al-lows for greater control over two values in the frequency spectrumof the system.

Fig. 5 demonstrates that the maximum amplitude of the 1stnatural frequency occurs at the center of the brace, where the basethickness of the scalloped brace is the predominant factor in deter-mining the magnitude of this natural frequency. Conversely, themaximum amplitude of the 3rd natural frequency occurs at the

location of the scalloped shape’s peaks. Therefore the height ofthese peaks becomes the predominant factor in determining themagnitude of this natural frequency. Also interesting is the factthat if the soundboard stiffness is high enough an inverted scal-loped brace may be required as seen by the negative peak heightadjustment factor required to keep the 3rd natural frequency con-stant for a soundboard having a stiffness of ER = 950 MPa in Table 6.This negative peak height adjustment factor would cause the braceto look like the one in Fig. 6.

It is then evident that what a musical instrument maker is doingwhen shaping his braces, is in fact empirically controlling the valueof multiple natural frequencies so as to optimize the acoustic qual-ity of his instrument. Without a scientific understanding of theprocess, the methods used are those developed from years of expe-rience. Although a great understanding of the scientific principlesbehind the construction of musical instruments is not a prerequi-site to producing great sounding instruments, tradition and historyhave repeatedly proven that a good ear and loads of experience goa long way. However, these attributes are not always available tomass-manufacturers at a reasonable cost.

5. Conclusions

In this paper, the effects and reasoning behind using scallopedbraces to acoustically improve a brace-soundboard system weremodeled and analyzed in order to better understand how musicalinstrument makers control the sound of their musical instruments.The assumed shape method was used in the analysis and the in-sight gained by using this approach was tremendous.

This study has demonstrated that it is possible to modify a scal-loped shaped brace in order to control the values of two distinctnatural frequencies in a wooden brace-soundboard system. Thisis done by adjusting the base thickness and scallop peak heightsof the brace in order to compensate for the 1st and 3rd natural fre-quencies respectively, since the maximum amplitudes of each ofthese frequencies occurs at these locations.

It is therefore likely that when musical instrument makersshape their braces, they are in fact controlling the value of multiplenatural frequencies in order to improve the overall acoustic qualityof their instruments. This helps clarify the reasons behind why somany instrument makers have been shaping their braces.

Acknowledgments

Special thanks go out to Dr. Frank Vigneron for help with certainaspects of this study and Robert Godin of Godin Guitars for givingimportant insight into the guitar manufacturing industry.

P. Dumond, N. Baddour / Applied Acoustics 73 (2012) 1168–1173 1173

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