Can a brace be used to control the frequencies of a plate?
 Patrick Dumond^{1}Email author and
 Natalie Baddour^{1}
DOI: 10.1186/219318012558
© Dumond and Baddour; licensee Springer. 2013
Received: 26 March 2013
Accepted: 7 October 2013
Published: 24 October 2013
Abstract
Although many improvements in the manufacturing of guitars have been made recently, one aspect that has often been overlooked is that of the acoustical consistency of the final manufactured product. The aim of this paper is to create a better understanding of the effect of a brace on the frequencies of vibration of the bracesoundboard system. This paper seeks to shed light on why a luthier ‘tunes’ braces when a guitar soundboard is handmanufactured. A simple analytical model of a rectangular brace and soundboard is derived from first principles using Kirchhoff plate theory in order to develop insight into the effect of the soundboard’s stiffness and brace thickness on the frequencies of the combined system. Natural frequencies and modeshapes of the combined system are calculated via the assumed shape method. Results show that by adjusting the thickness of the brace in order to compensate for the stiffness of the plate, one of the natural frequencies of the combined system can be adjusted to meet a desired value. However, simultaneously adjusting several natural frequencies cannot be done with a rectangular brace. Therefore modifications to the shape of the brace are explored.
Keywords
Frequency matching Assumed shape method Braceplate system Musical instrument Plate vibration Orthotropic material Tuning Designforfrequency Spectrum controlIntroduction
The scientific study of guitars and other stringed musical instruments has been around for over a half century and many improvements to their production manufacturing have also been made (Richardson 1990; Chaigne 1999; French 2008a). In spite of this, manufactured instruments often do not sound as good as instruments built by hand by experienced luthiers. With precision tooling, production manufactured instruments can be built to strict dimensional tolerances and yet, acoustical consistency of the final product is still not ensured – meaning that two instruments emerging from the same production line will be dimensionally identical but acoustically different (French 2008b). There are two principal reasons for the lack of acoustical consistency. The first is that wood is a natural material, with natural variations so that soundboards that are dimensionally identical may be acoustically quite different. The other reason is that the tuning process used by the experienced luthiers to handbuild and tune instruments is based on years of experience and tradition but scientifically is not well understood. This tuning process is what luthiers use to make up for the natural variation in the acoustical properties of wood and since this tuning process is not well understood analytically, it is impossible to replicate autonomously. The goal of this paper is thus to develop a better analytical understanding of the tuning process of a guitar soundboard during its manufacture. The ultimate end goal is that this understanding can eventually be used to develop an automated tuning process to be used during guitar manufacture.
On the other hand, for productionmanufactured instruments, current practice in the industry is to test the acoustic quality of a soundboard by measuring its stiffness across the grain during the manufacturing process (French 2008a). The deflection across the grain of the soundboard without braces is measured under a known load. Based on a certain set of deflection ranges, the soundboards are judged to be of higher or lower acoustical quality and are therefore used in different product lines. This test gives an idea as to the soundboard’s stiffness and thus the resulting frequency range of the final product. Based on the soundboard’s stiffness test, a brace from a collection of predimensioned braces is chosen for that soundboard. A certain quality control is obtained since the manufacturer is aware of the range of soundboard stiffness for which their dimensioned braces produce decent instruments.
The fact that dimensional changes to the braces have a substantial impact on the frequency spectrum of the guitar is known but the extent of this effect is still unknown. However, research into the acoustics of musical instruments has begun to explain the fundamental interaction between various components of an instrument (Caldersmith 1995) and also the effect on the resulting sound field. Loosely generalized, research into guitars and other instruments has focused on their radiated sound fields or on their modal properties. Over the years, several experimental and computational techniques have been applied to the study of guitars, and in particular soundboards. One area of investigation has been the visualization of guitar box and soundboard resonances via holography and laser interferometry (Jansson 1971; Stetson 1981; Firth 1977; Richardson 2010; Jovicic 1977).
The radiation fields of the guitar have also been studied numerically and experimentally (Brooke & Richardson 1993;a; Brooke & Richardson 1991;b; Lai & Burgess 1990; Hill et al. 2004). Recent work has demonstrated that a relatively small number of measured parameters are required to predict the sounds radiated by a guitar (Richardson 2010; Hill et al. 2004; Richardson 2005). Interestingly, Brooke and Richardson conclude that there are no simple relationships between the modal properties of instruments and their estimated “quality” (Brooke & Richardson 1993a). At first glance this is surprising, although this is likely the scientific equivalent to the musical statement that just because an instrument is in tune does not ensure it is a “quality” instrument. On the other hand, most musicians will reject out of tune instruments as unplayable therefore one might infer that “good” modal properties are necessary but not sufficient to ensure a decent instrument. However, there does not seem to be a consensus on what constitutes “good” modal properties.
Elejabarrieta, Ezcurra, and Santamaría performed an extensive set of experimental and numerical studies on a guitar through its entire construction process by a master luthier. They experimentally tested the soundboard at each construction step as modifications were made by the luthier, in order to understand the effect of his modifications on the vibration properties of the soundboard (Elejabarrieta et al. 2000). This was followed by a finite element analysis of the same data (Elejabarrieta et al. 2001). As the guitar continued to be built by the same luthier, they continued with their work by analyzing the resonance box and its modes of vibrations, along with the effect of the soundhole on the acoustic modes of the box (Elejabarrieta et al. 2002a). Their next step was an experimental and finite element analysis of the coupling of the vibration modes of the structural (soundboard and backplate) and acoustic (the box as a Helmholtz resonator) modes once the box of the guitar had been assembled (Elejabarrieta et al. 2002b). Finally, they investigated the fluidair interaction of the guitar box in which the interior gas was changed both experimentally and numerically (Ezcurra et al. 2005). This set of work contributed greatly to the understanding of how each component of the guitar contributes to the final frequencies of vibration but did not offer any suggestions as to how an instrument designer might modify a given component to achieve a specific acoustical objective. Boullosa experimentally measured the radiation efficiency and frequency content of the vibrations of a classical guitar (Boullosa 2002; Boullosa et al. 1999) but offers little in the way of insight as to which components of the guitar or their modifications contribute to either. Torres and Boullosa also studied the effect of the bridge on the vibrations of the soundboard both with finite elements and with laser vibrometry (Torres & Boullosa 2009). Chaigne and various collaborators focused their research on the timedomain modelling of the guitar (Chaigne 1999; Bécache et al. 2005; Derveaux et al. 2003) with the intent to better understand the vibroacoustical behavior via physical and numerical modeling. Their intent was for this to be used as a tool for the estimation of quantities that are hard to measure experimentally as for example the estimation of the relative structural losses and radiation losses in the sounds generated by the guitar.
In other (nonmusical) disciplines, stiffened plates have been previously studied by various methods (Fox & Sigillito 1980; Barrette et al. 2000; Peng et al. 2006; Hong et al. 2006). However these studies have focused on the structural properties of such systems rather than the acoustical properties of the interaction between the beam and plate elements.
Despite the work that has been done to understand the mechanics of the instruments, little has been done to improve the acoustical consistency of manufactured guitars in large part because the tuning process that is used for handbuilt instruments is not well understood and thus cannot easily be replicated. The goal of this paper is to begin to develop a better understanding of this tuning process via a simple analytical model to represent the vibrations of an instrument soundboard and a supporting/tuning brace. A simple model is sufficient to answer the question at hand, therefore a simple model that can yield the most physical insight is chosen for the analysis. The question is: can we change the dimensions of the brace to make up for changes in stiffness of the soundboard so that the combined bracesoundboard system has desired frequencies of vibration? This is the essence of what luthiers do when they handtune an instrument during its construction; they make small changes to the structural properties in order to produce desired changes in acoustical properties. An idea similar to this has been considered analytically for a xylophone (OrdunaBustamante 1991), where the effect of an undercut on the bar on its tuning was modelled and analyzed. However, to the best of the authors’ knowledge, this type of analysis has not been considered for a guitar.
Current research shows that it is the first few modes of the coupled system that are necessary in order to tune the soundboard during manufacturing (Hutchins & Voskuil 1993). In this work, the assumed shape method is used to analyze the continuous system, with the modes of the soundboard without the brace used as the solution building blocks. The assumed shape method is a global element method detailed in (Meirovitch 1996a). It has been shown to be particularly usefully in the modal analysis of stiffened and orthotropic plates (Xu et al. 2010; McIntyre & Woodhouse 1988). The assumed shape method has specifically been chosen as the analysis tool because it can solve the mass and stiffness matrices analytically; thereby each matrix entry is a function of all physical parameters. This helps give vibration insight about the model’s natural frequencies and corresponding modes (Bisplinghoff et al. 1996).
A brief outline of the paper is as follows. Section 2 presents the analytical model, specifically chosen based on the pertinent background information provided in Section 1. Section 3 demonstrates the results of the research and Section 4 is a discussion of the results. The effects of scalloped braces are dealt with in Section 5. Finally, conclusions are given in Section 6.
Analytical model of the platebrace system
A. Modeling assumptions
The soundboard is modeled as a thin rectangular Kirchhoff plate and the brace is modeled as a thicker section of the same plate. A simple rectangular geometry is assumed in order to enable the closedform solution of a simple plate (without the brace) to be used as the trial functions for the assumed shape method. Further, since the solution of a rectangular plate is known in closed form, this will enable a direct comparison and enable the understanding of the effect of the brace on the vibration properties of the plate.
The Kirchhoff plate theory assumes small transverse deflections and neglects transverse normal and shear stresses, as well as rotary inertia. Although this is an accurate assumption for the plate, due to the brace’s thicknesstowidth aspect ratio, it may imply a certain error in that region of the soundboard. Also, because of the method in which the brace thickness is added to that of the plate in the kinetic and strain energy expressions, it was necessary to change the direction of the grain of the plate, in this region only, to match that of the brace. This is reasonable since the plate is thin and the properties of the brace dominate in this region. The plate is also assumed to be simply supported all around, although in reality it is somewhere between simply supported and clamped (Meirovitch 1996b). It has been assumed that the system is conservative in nature, which allows damping to be neglected. Although there is a certain amount of damping found in wood, its effects on the lower natural frequencies is thought to be minimal and has been neglected. This is justified because the tuning process (adding and adjusting the dimensions of the braces) has a greater effect on the lower frequencies than on the higher frequencies (Hutchins & Voskuil 1993).
The subscripts represent the direction of the plane in which the material properties act. Therefore, E _{ x } is the Young’s modulus along the xaxis, E _{ y } along the yaxis and ν _{ xy } andv _{ yx } the major Poisson’s ratios along the xaxis and yaxis respectively. We begin with the expressions for strain and kinetic energies for an orthotropic plate and then consider modifications to these when the brace is added to the plate.
B. Strain energy for an orthotropic plate
where h is the thickness of the plate.
C. Kinetic energy for an orthotropic plate
where the dot represents the time derivative, ρ is the mass per unit area of the plate such that ρ = μ ⋅ h, μ is the material density and h is the plate’s thickness.
D. Strain energy for a plate modified with an attached brace
To account for the modification of the plate by adding a brace to it, as seen in Figure 3, the strain and kinetic energies are modified to account for additional thickness between x _{1} and x _{2}.
where the subscripts p and c denote the plate alone and combined plateandbrace system.
E. Kinetic energy for a plate modified with an attached brace
F. The assumed shape method
The assumed shape method is chosen because it allows us to use the flatplate modeshapes as the fundamental building blocks of the solution, thereby permitting observation of how the addition of the brace affects those fundamental modeshapes. This method also permits greater flexibility in analyzing the effects of changes in brace dimensions since it enables the creation of an analytical solution from which numerical solutions can be quickly obtained for various thicknesses of the brace. The equations of motion are derived using a computer algebra system (Maple). This yields mass and stiffness matrices where each matrix entry is a function of all physical parameters (dimensions, density, stiffnesses, etc.). The effect of any parameter on the system’s eigenvalues can then easily be examined without having to reestablish the entire system model. Using the assumed shape method, it has also been found that only two additional odd or even trial functions more than the one of interest are required for convergence (depending on whether it is itself odd or even). The finite element method was also considered. While the FE method offers significant advantages over the other approximate methods, namely its ability to model complex systems and boundaries and a high numerical accuracy, its disadvantage for the purpose of this work is its inability to make use of the known mode shapes of the system without the brace. Furthermore, the FE method also requires a large number of degrees of freedom in order for the solution to converge to accurate results. Contrary to the nature of the global functions approach, the finite element method uses local functions which extend over small subdomains of the system (Meirovitch 1996b), thus comparison of the global behaviour to the exact solution of a simple plate problem cannot be directly incorporated into this solution approach.
The ${\varphi}_{{n}_{x}{n}_{y}}$ are the chosen discrete spatial trial functions and ${q}_{{n}_{x}{n}_{y}}\left(t\right)$ are the generalized (timedependent) coordinates. Also, m _{ x } and n _{ x } represent the mode number and trial function number in the x direction respectively and m _{ y } and n _{ y }represent the same in the y direction. Next, the trial functions are chosen so as to satisfy the geometric boundary conditions and be complete in order to ensure convergence of the solution (Meirovitch 2001). No other considerations of the boundary conditions need to be taken into account. A simply supported plate implies boundary conditions such that the transverse displacement w of the perimeter of the plate is zero (Meirovitch 1996b).
which is then used in the strain and kinetic energy equations of the modified plate.
Here, ω is the system’s natural frequency, ϕ the phase shift and $\overrightarrow{\mathbf{A}}$ is a magnitude vector of dimension(m _{ x } ⋅ m _{ y }) × 1. Then replacing the assumed harmonic solution into the equation of motion, Eq. (14) an eigenvalue problem is obtained, from which the natural frequencies and modeshapes are found.
Results
The purpose of this analysis is to verify if it is possible to alter the dimensions of the brace so as to obtain a desired set of natural frequencies from the coupled system, knowing their respective properties before assembly. Both symbolic and numerical computational tools are used.
A. Material properties
Material properties for Sitka spruce as an orthotropic material (Forest Products Laboratory (US) 1999 )
Material Properties  Values 

Density – μ (kg/m^{3})  403.2 
Young’s modulus – E _{ R } (MPa)  850 
Young’s modulus – E _{ L } (MPa)  E _{ R } / 0.078 
Shear modulus – G _{ LR } (MPa)  E _{ L } × 0.064 
Poisson’s ratio – ν _{ LR }  0.372 
Poisson’s ratio – ν _{ RL }  ν _{ LR } × E _{ R } /E _{ L } 
B. Plate and brace dimensions
Test specimen dimensions
Dimensions  Values 

Length – L _{ x } (m)  0.24 
Length – L _{ y } (m)  0.18 
Length – L _{ b } (m)  0.012 
Reference – x _{ 1 } (m)  L _{ x } / 2 – L _{ b } / 2 
Reference – x _{ 2 } (m)  x _{ 1 } + L _{ b } 
Thickness – h _{ p } (m)  0.003 
Thickness – h _{ b } (m)  0.012 
Thickness – h _{ c } (m)  h _{ p } + h _{ b } 
C. Frequencies and modeshapes
Results for the assumed shape with 10x10 trial functions (orthotropic)
m _{ x }  m _{ y }  Assumed shape method  

Natural Frequency (Hz)  Modeshape  
1  1  590 

2  1  703 

2  2  930 

1  2  1015 

2  3  1185 

1  3  1248 

3  1  1273 

2  4  1551 

1  4  1598 

2  5  2051 

The dip in the center of the xaxis for the assumed shape method modeshapes is the location of the brace, which is visible on the modeshapes. The brace stiffens this area and limits the amount of displacement that can occur. Since the Kirchhoff model is used, shear and rotary inertia are neglected, which may induce some inaccuracies near the brace as the aspect ratio is larger there.
D. Effect of the brace on the natural frequencies of the combined system
Comparison of the orthotropic brace, plate and combined system natural frequencies
m _{ x }  m _{ y }  Brace natural frequencies  Plate natural frequencies  Combined system natural frequencies  % Increase by adding 

(Hz)  (Hz)  (Hz)  the brace  
1  1  873  166  590  256% 
2  1  873  530  703  33% 
2  2  3491  662  930  40% 
1  2  3491  330  1015  207% 
2  3  7855  923  1185  28% 
1  3  7855  629  1248  98% 
3  1  873  1146  1273  11% 
2  4  13965  1321  1551  17% 
1  4  13965  1054  1598  52% 
2  5  21820  1853  2051  11% 
1  5  21820  1604  2090  30% 
These results show intuitive trends which help verify the model and help in the understanding of the effect that adding a brace has on the coupled system. A detailed discussion about the relationship between these results is given in Section 4.
E. Tuning
In order to verify the feasibility of tuning braces to a plate having a predetermined crossgrain stiffness, it is necessary to look at effects of a change in both the Young’s modulus in the radial direction E _{ R } and of the brace thickness h _{ b }, on the modified orthotropic plate of Figure 3. Although the lowest five natural frequencies carry importance, only two will be observed during the variation in structural properties. This is because frequencies that have a mode of vibration which contains a node at the location of the brace are not as affected by the brace as those which have a mode which passes through it. Therefore the two frequencies observed during this analysis are the first and fourth natural frequencies of the orthotropic platebrace system. The second and third modeshapes have a node at the location of the brace and are not as affected by the brace, contrary to the first and fourth modeshapes which don’t. This can be observed in Table 4, where the first and fourth modes use only one trial function along the xaxis such that ω _{1}:m _{ x } = 1, m _{ y } = 1 and ω _{4}:m _{ x } = 1, m _{ y } = 2.
In the same way as the previous case, it can be seen from Figure 6 that when h _{ b } increases so do the 1st and 4th natural frequencies of the combined system.
The system is compensated so that the 1st natural frequency is held constant
Young’s modulus  Brace thickness  1st natural frequency  % change of ω _{ 1 }from  4th natural frequency  % Change of ω _{ 4 }from 

E _{ R }(MPa)  h _{ b }(m)  ω _{ 1 }(Hz)  E _{ R } = 850  ω _{ 4 }(Hz)  E _{ R } = 850 
600  0.0150  590  0.0%  884  13% 
650  0.0142  590  0.0%  913  10% 
700  0.0136  592  0.3%  941  7% 
750  0.0130  591  0.2%  967  5% 
800  0.0125  592  0.3%  992  2% 
850  0.0120  590  0%  1015  0% 
900  0.0116  591  0.2%  1038  2% 
950  0.0112  590  0.0%  1059  4% 
1000  0.0109  592  0.3%  1080  6% 
1050  0.0105  589  0.2%  1098  8% 
1100  0.0102  589  0.2%  1117  10% 
The system is compensated so that the 4th natural frequency is held constant
Young’s modulus  Brace thickness  1st natural frequency  % change of ω _{ 1 }from  4th natural frequency  % change of ω _{ 4 }from 

E _{ R }(MPa)  h _{ b }(m)  ω _{ 1 }(Hz)  E _{ R } = 850  ω _{ 4 }(Hz)  E _{ R } = 850 
600  0.0909  934  58%  1015  0.0% 
650  0.0683  927  57%  1015  0.0% 
700  0.0387  907  54%  1015  0.0% 
750  0.0194  781  32%  1015  0.0% 
800  0.0144  661  12%  1015  0.0% 
850  0.0120  590  0%  1015  0% 
900  0.0104  541  8%  1015  0.0% 
950  0.0093  508  14%  1015  0.0% 
1000  0.0084  482  18%  1014  0.1% 
1050  0.0077  462  22%  1014  0.1% 
1100  0.0071  446  24%  1014  0.1% 
Once again, forcing the fourth natural frequency to be more or less constant causes the first natural frequency to vary considerably from its value at E _{r} = 850MPa. These results are further discussed in the next section.
Discussion
Based on the results presented in Section 3, the analysis has demonstrated clear trends in the behaviour of a soundboard having a brace across its width. Specific points are discussed herein.
A. Material properties
The first thing to note during this analysis is the use of the statistical average values of spruce’s material properties. It is obvious that these material properties vary on a specimen by specimen basis. However the assumption was made that there is a relationship between the radial stiffness E _{ R } and the other properties. While this is definitely alluded to by the (Forest Products Laboratory (US) 1999), it is unclear how much variation is actually present in these relationships. Based on years of luthier experience in using the crossgrain stiffness as a measure of soundboard quality, it would appear that the relationship between this stiffness and other properties is more consistent than the properties themselves. It would, however, be quite interesting to further investigate this phenomenon, as this is has been found to be a great way of modeling the material properties of wood.
Clear, quartersawn, musical instrument spruce has been shown to have remarkably consistent microscopic properties in spring and summer growth. Furthermore, spruce displays an abrupt transition period which means that microscale properties are generally in line with macroscale mechanical properties (Kahle & Woodhouse 1994). Therefore, by avoiding visual imperfections as is currently being done in industry, it is reasonable to assume that bulk material properties taken on a specimenbyspecimen basis is an adequate measure of overall system performance. However, it would be interesting to investigate how localized changes in material properties would affect system performance, especially at the location of the brace. Modifications to the method, by increasing the number of zones of interest in the kinetic and strain energy equations for example, could take into account microscale variations in the material specimen. On the other hand, more rigorous testing of material properties would be necessary to populate the input information required for the analysis. On the macroscale, CNC machinery could be reprogrammed to shape braces as required by the material property measurements of the plate.
To increase the accuracy of the model by including frequencydependent damping properties, an analytical approach such as that described in (McIntyre & Woodhouse 1988) could be used to modify the mathematical model in order to take damping into account.
B. Dimensions
The dimensions used on the test specimen consisting of the simply supported rectangular plate and brace across the width, are based on typical dimensions of those used on guitar soundboards. The plate itself having only one brace is typical of the area on a soundboard around the lower bout where is positioned one of the legs of the typical xbrace pattern. This leads to a model which produces a set of the lowest frequencies within the acoustical range sought by a typical musical instrument (e.g. A0C8 or 27.5 − 4186.01Hz).
C. Effect of the brace on plate modes
When results are compared between the plate alone and the plate with the brace as seen in Table 4, it is clear that the addition of the brace affects modes for which the location of the brace is not a node, such as m _{ x } = 1, more than a mode having a node at the location of the brace, such as m _{ x } = 2. This is also clear from the percentage increase in the natural frequencies for the m _{ x } = 1 mode, which is much higher than for the m _{ x } = 2 modes. It is also impossible to avoid a slight increase in the natural frequencies for the m _{ x } = 2 modes or all other even modes because their nodes are along a line and the brace does in fact have a finite width. This finite brace width causes local stiffening to occur around it, affecting the curvature of the even modes and thereby also increasing their frequencies.
This table also demonstrates that for modes directly affected by the brace, the natural frequencies in fact fall somewhere between those of the brace and plate alone. These results are those which are expected from basic vibration theory. For the modes not directly affected by the brace, a simple increase in the original plate’s natural frequency is observed due to the forced changes in the mode’s curvature.
D. Tuning
The effects of variations in the crossgrain stiffness of the plate, measured as E _{ R }, are observed in Figure 5. It is clear that when the stiffness across the grain is reduced so too are the natural frequencies, as expected. Similarly, the effects of the brace’s thickness, measured as h _{ b }, are considered in Figure 6. As the brace’s thickness increases, so does its natural frequencies since its stiffness increases at a larger rate than its mass.
Luthiers use this phenomenon in order to adjust the braces to a given soundboard, changing the thickness of the brace in certain sections to compensate for changes in the stiffness of the soundboard. This adjustment process is tested analytically whereby the brace is adjusted inversely to the plate’s crossgrain stiffness in order to hold the first natural frequency of the combined system constant. These results can be seen in Table 5. While the first natural frequency variation falls well below the 1% human hearing threshold for sound variation (Chaigne 1999), the variation in the fourth natural frequency lies above it.
A second attempt was made to tune the fourth natural frequency, modifying the properties of the brace so that the fourth natural frequency remained constant despite variations in the stiffness of the soundboard. As can be seen in Table 6, a wider adjustment span is required for the brace in order to achieve consistency in the fourth natural frequency. This time, the frequency variation of the fourth natural frequency lies well below the 1% threshold. However, the variation in the first natural frequency is wider than the first attempt.
These results indicate that it is possible to produce an acoustically consistent set of braceplate assemblies that have at least one desired natural frequency. Conversely, it does also indicate that a rectangular brace is not suitable for adjusting multiple frequencies.
After obtaining these results, it has become clear that adjustments to the shape of the brace itself are required. This has led back to the debate on whether or not scalloped braces have an acoustical role in producing more consistent instruments. This being an interesting topic on its own, further investigation of the scalloped brace is found in Section 5.
E. Sources of error
Evidently, improved accuracy in the calculation of the natural frequencies of the modified plate could be obtained by simply incorporating shear deformation and rotary inertia into the plate model. Nevertheless, other assumptions were also made which have an impact on the preciseness of the calculated values.
The first assumption was that the mass of air which would normally surround the soundboard of a musical instrument has been neglected. Including the mass of air surrounding the soundboard would in fact decrease the natural frequencies because the mass of air acts to increase the total inertia of the soundboard (Leissa 1993).
To simplify the model, the assumption was also made that the soundboard is simply supported when in fact it is probably somewhere between simply supported and clamped (Meirovitch 1996b). Since clamped edges prevent rotation at the edge, local stiffening occurs. This leads to an increase in the natural frequencies.
Damping was also neglected during the analysis, which allowed for a much simpler model. Although this assumption is justified due to the fact that a musical instrument is designed to sustain rather than to absorb vibration, it is the damping or decay time of specific partial frequencies, because of wood’s distinct properties, which help give a wooden instrument its tone (Chaigne 1999). Therefore, to improve the acoustical preciseness of the model, damping would need to be included in the analysis.
Finally, in order to create the orthotropic system, the direction of the grain at the location of the brace was changed for the plate. This would in fact slightly modify the stiffness properties of the system in this location, thereby influencing the natural frequencies.
Analysis of a scalloped brace
Based on the results of Section 3, it is clear that a rectangular brace can be modified to control one of the lower frequencies, but alone is unable to control multiple frequencies at once. Based on these previous results, insight was obtained as to what needs to be done in order to tune at least two of the lower modes, therefore a preliminary analysis was performed based on the hypothesis that modifying the shape of the brace itself has the ability to control more than one of the system’s natural frequency.
A. Modeling of the scalloped brace
where h _{ bo } is the height of the brace at its ends and center. This h _{ b } is then substituted into the kinetic and strain energy equations as used for the modified plate model in the assumed shape method.
B. Results
Comparison of brace geometry on a simply supported orthotropic modified plate
m _{ x }  m _{ y }  System natural frequencies with rectangular brace  System natural frequencies with scalloped brace  % change in frequencies 

(Hz)  (Hz)  
1  1  592  621  4.9% 
2  1  711  743  4.5% 
2  2  950  1003  5.6% 
1  2  1063  1053  −0.9% 
2  3  1211  1279  5.6% 
C. Discussion
Although the natural frequencies have a clear increase in value throughout for the scalloped brace compared to the rectangular brace, there is a marked difference for the 1 × 2 mode, where there is in fact a reduction in the natural frequencies of the system. The original thickness of the scalloped brace is equal to that of the rectangular brace, which adds additional material to the system, and thus increases the natural frequencies. Since the peaks of the scalloped brace occur at the maximum displacement locations of the 4th mode of vibration in the direction of the brace (1 × 2 mode), it increases the inertia at these locations by increasing the mass locally. This is the reason for the reduction in the natural frequency observed for the 4th modeshape. These peaks also minimize the effect of extra mass on the other modes of vibration, because their maximum displacement is found to be either at the center of the brace where the brace’s thickness goes unchanged or the brace is in fact at a location of one of their nodes.
These discoveries lead to the theory that what a luthier is in fact doing when scalloping a brace, is adjusting two or more modes at once, or at least controlling which modes are affected the most by the bracing since not all modes are equally affected. It is evident that the exact shape chosen based on the polynomial of Eq. (17) may not be the optimal solution. Further investigation into the scallop shape itself is necessary to further grasp the magnitude of its effect on the frequency spectrum of the soundboard. Research on understanding the effects of a scalloped shape brace on the natural frequencies of a braceplate system is ongoing, but shows great promise in a field that deserves to be explored. Only preliminary results have been discussed herein.
Conclusions
In this paper, the effect on natural frequencies of adding a brace to a soundboard was modeled and analyzed in order to better understand how luthiers tune a musical instrument. The assumed shape method was used in the analysis and the insight gained by using this approach was tremendous.
First, the analysis has shown that the acoustic properties of a soundboard can be modified by adjusting the thickness of a brace. In fact, it has shown that specific natural frequencies can be controlled. However, the rectangular brace used for most of the analysis has been found to have the ability to control only one frequency at a time.
Scalloped braces have been shown to be a solution for which multiple natural frequencies of a soundboard can be adjusted. Further investigation on this subject should be explored. However, it does indeed help to clarify the purpose of using scalloped braces, as has been done for hundreds of years, and also gives hope that it is possible for a wooden musical instrument manufacturing process to include acoustical consistency.
Nomenclature
A: Magnitude vector
D: Stiffness function
E: Young’s modulus
G: Shear modulus
h: Thickness
k: Stiffness
K: Stiffness matrix
K _{ s }: Shear correction factor
m: Mass
M: Mass matrix
L: Longitudinal axis parallel to the wood grain
L _{ x }: Length along the x axis
L _{ y }: Length along the y axis
m: Mode number
n: Trial function number
q: Generalized coordinates
Q: Generalized nonconservative forces
R: Radial axis normal to the growth rings and perpendicular to the wood grain
S: Stiffness components
t: Time
T: Kinetic energy of the system
T: Tangential axis tangent to the growth rings and perpendicular to the wood grain
u: Displacement along the x axis
U: Strain energy of the system
v: Displacement along the y axis
V: Potential energy of the system
w: Displacement along the z axis
w _{ o }: Displacement along the z axis of the plate’s neutral plane
W: Work
x: Axis direction and position
y: Axis direction and position
z: Axis direction and position
γ: Shear strain
ϵ: Normal strain
μ: Density
ν: Poisson’s ratio
ρ: Mass per length or mass per area
σ: Normal stress
τ: Shear stress
φ: Discrete spatial trial functions
ω: Natural frequency
Declarations
Acknowledgements
Special thanks go out to Dr. Frank Vigneron for help with certain aspects of this study, Dr. Michel Labrosse for help in setting up the finite element model used to verify our approach and finally Robert Godin of Godin Guitars for giving important insight into the guitar manufacturing industry.
Authors’ Affiliations
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