Using a Brace to Design a Soundboard Section for a Desired Natural Frequency

A STRUCTURED APPROACH TO USING A REC-

TANGULAR BRACE TO DESIGN A SOUNDBOARD

SECTION FOR A DESIRED NATURAL FREQUENCY

Patrick Dumond Natalie Baddour

[email protected] [email protected]

Department of Mechanical Engineering

University of Ottawa

161 Louis Pasteur, CBY A205

Ottawa, Canada K1N 6N5

ABSTRACT

The manufacture of acoustically consistent wooden musi-

cal instruments remains economically demanding and can

lead to a great deal of material waste. To address this, the

problem of design-for-frequency of braced plates is con-

sidered in this paper. The theory of inverse eigenvalue

problems seeks to address the problem by creating repre-

sentative system matrices directly from the desired natu-

ral frequencies of the system. The goal of this paper is to

demonstrate how the generalized Cayley-Hamilton theo-

rem can be used to find the system matrices. In particular,

a simple rectangular brace-plate system is analyzed. The

radial stiffness of the plate is varied in order to model

variations typically found in wood which is quartersawn.

The corresponding thickness of the brace required to keep

the fundamental natural frequency of the brace-plate sys-

tem at a desired value is then calculated with the pro-

posed method. It is shown that the method works well for

such a system and demonstrates the potential of using this

technique for more complex systems, including sound-

boards of wooden musical instruments.

1. INTRODUCTION

Many aspects of the manufacture of wooden musical in-

struments have been addressed and rendered consistent.

However, acoustical consistency still remains unattaina-

ble in most situations [1]. This is a consequence of the

fact that the material of choice for many musical instru-

ments is wood, a natural material that exhibits high varia-

bility in its material properties. By definition, wood has

inconsistent properties because its growth is directly re-

lated to the highly variable climate of its environment.

Luthiers have been compensating for these inconsistent

material properties in guitars by means of various meth-

ods for years. The most prominent method currently in

use is to adjust the shape of the soundboard’s braces in

order to attain a more consistent frequency spectrum from

this part of the instrument [2]. These methods have had

varying degrees of success and mostly depend on the skill

and experience of the luthier. Worsening the situation is

the fact that these methods are very labor intensive, ren-

dering them cost-prohibitive, as well as difficult to im-

plement into a structured manufacturing process. For the-

se reasons, most manufacturers only use material that has

mechanical properties which fit within their set criteria.

Such an approach leads to much waste [3].

Like most design problems, a design is first created

from experience and then iteratively refined in order to

achieve the desired parameters. This is especially true of

systems in which certain eigenvalues are desired [4]. For

guitar soundboards, luthiers begin with a certain design,

remove material from the braces in small increments and

then check the system’s natural frequencies (eigenvalues)

until a desired solution converges. It has been shown in

previous work that it is indeed possible to alter certain

frequencies of a soundboard system by simply adjusting

the shape of the braces [5]. While effective, this trial-and-

error method is not optimal. A better approach would be

to design/ construct the system directly from the desired

natural frequencies (eigenvalues).

In order to achieve this, we turn to the field of study

known as inverse eigenvalue problems, which deals spe-

cifically with finding matrices from a set of given eigen-

values [6], [7]. A rather young area, inverse eigenvalue

problems use knowledge of matrix algebra and numerical

methods to create matrices that yield a desired frequency

spectrum (set of eigenvalues) or a partial spectrum. It is

well known that inverse eigenvalue problems are ill-

posed, meaning there generally exists many solutions [7].

In design, the existence of many solutions is potential-

ly beneficial, giving the designer options. However, phys-

ical constraints do need to be applied in order to make a

system physically realizable. Most methods for inverse

eigenvalue problems involve the use of well-developed

matrix theory for matrices with a specific structure (e.g.

Jacobi and band matrices) and then apply appropriate

numerical algorithms to solve for the unknown matrices

from the known desired eigenvalues [8], [9], [10], [11],

[12], [13], [14]. The structure of the matrices generally

implies various physical constraints. However, there exist

very few methods that can solve for matrices having a

more general unstructured form. The goal of this paper is

to demonstrate the use of a technique that has been re-

cently proposed using the generalized Cayley-Hamilton Copyright: © 2013 Patrick Dumond and Natalie Baddour. This is an

open-access article distributed under the terms of the Creative Commons

Attribution License 3.0 Unported, which permits unrestricted use, distri-

bution, and reproduction in any medium, provided the original author

and source are credited.

Proceedings of the Stockholm Music Acoustics Conference 2013, SMAC 2013, Stockholm, Sweden

613

theorem [15]. In particular, this method is interesting

since it allows the use of any matrix structure. Therefore,

a suitable matrix structure can be determined by other

means, for instance from modeling the forward dynamics

of the problem. Implementing particular desired con-

straints thus becomes an exercise of the forward model-

ing process.

In this paper, we apply the Cayley-Hamilton technique

to a simple rectangular brace-plate system in order to

design the combined brace-plate system to a desired natu-

ral frequency. In doing so, we demonstrate that a braced

plate can be designed directly from knowledge of the

desired fundamental frequency. This approach is novel

because it would allow the construction of wooden

soundboards having a consistent set of natural frequen-

cies via design of the braces.

2. MODEL

2.1 Problem Statement

Given a desired fundamental frequency, construct a

brace-plate system as described by a mass matrix M and a

stiffness matrix K.

All mechanical properties of the system are a function

of the radial stiffness ER of the wooden specimen, which

is assumed known and given (and which tends to vary

from specimen to specimen). All dimensional (geometric)

properties of the brace-plate system are assumed to be

specified and fixed except for the thickness of the brace

hc, the design variable for which we must solve.

2.2 Forward Model

The model is based on a typical section of a guitar

soundboard, where a single brace is used to structurally

reinforce the weaker plate direction. The model is shown

in Figure 1.

Figure 1. Orthotropic plate reinforced with a rectangu-

lar brace.

The forward model is discretized using the assumed

shape method, similar to the model used in [5]. The as-

sumed shape method is an energy method which uses

global plate elements within the kinetic and strain energy

plate equations in order to determine the system’s equa-

tions of motion, from which the mass and stiffness matri-

ces are extracted [16]. The system is assumed simply

supported, conservative and the material properties are

assumed orthotropic.

The forward model is created assuming Sitka spruce

and all material properties are related to the radial stiff-

ness of the wood specimen (i.e. the Young’s Modulus in

the radial direction, ER) as indicated in [17]. The use of

ER is chosen because quality control practice observed in

industry use the stiffness across the grain of the wooden

soundboards, measured as in Figure 2.

Figure 2. Quality control measurement of soundboard

plate used in industry

It is assumed that an exact measurement of ER could be

obtained in a similar fashion as that described above and

which could be used in the calculations.

2.3 Inverse Model

The goal of this paper is to reconstruct the brace-plate

system from a desired fundamental frequency. The gen-

eralized Cayley-Hamilton theorem inverse eigenvalue

method is used [15]. The generalized Cayley-Hamilton

theorem states that if p(λ) is the characteristic polynomial

of the generalized eigenvalue problem (K,M), where K

and M are square matrices obtained from p(λ)=det(K-

λM), then substituting (M-1

K) for λ in the polynomial

gives the zero matrix [18], [19]. Thus, by building the

model in the forward sense, and by leaving relevant de-

sign parameters as variable, it is possible to design the

brace-plate model for the fundamental frequency.

It is shown in [20] that in order to adjust the funda-

mental frequency of the brace-plate system to a desired

value, it is necessary to adjust the thickness of the brace.

A cross section of the fundamental modeshape is shown

in Figure 3. It is clear that the brace affects the maximum

amplitude of this modeshape, thus also affecting the as-

sociated frequency [21].

Figure 3. Cross section of the brace-plate system's fun-

damental modeshape

2.4 Cayley-Hamilton Algorithm

Using the facts that the wood’s mechanical properties

vary based on its radial stiffness, and that the brace thick-

ness controls the brace-plate system’s fundamental fre-

quency, the forward model is created using the assumed

Excellent

Good

Mediocre

Discard

Known force

R

0

hp

Ly

x

y z

Lx

0

hc

x1 x2

brace

plate


614

shape method while leaving these two parameters as vari-

ables. Thus, the mass matrix M is a function of hc, the

height of the brace-plate system at the (assumed fixed)

location of the brace, and the stiffness matrix K is a func-

tion of hc and also ER, the plate’s radial stiffness. Here,

we use 2 × 2 trial functions in the assumed shape method.

Hence, 4th

order square matrices are created. The trial

functions used are those of the simply supported rectan-

gular plate such that

1 2

( , , ) sin sin ( )

yx

x y

x y

mmyx

n n

x yn n

n yn xw x y t q t

L L

ππ

= =

= ∑∑ (1)

where m are the modal numbers, q the time function and

w is the displacement variable normal to the plate. The

displacement variable w is then used directly in creating

the kinetic and strain energy equations of the simply sup-

ported rectangular plate. These equations are broken into

three sections as shown in Figure 1 in order to take into

account the brace. This procedure is well described in [5].

It is assumed that the ER of the wood selected for manu-

facture is measured during the manufacturing process and

used as input information into the stiffness matrix. This

leaves hc as the only unknown parameter, appearing in

both the mass and stiffness matrices.

In order to solve these matrices from the desired fun-

damental frequency, the Cayley-Hamilton theorem is

used. To do so, the characteristic polynomial is created

using the desired frequency,

( ) ( ) ( ) ( ) ( )1 2 3p a b b bλ λ λ λ λ= − ⋅ − ⋅ − ⋅ − (2)

where a is the desired frequency and b1-b3 are unknown

values which need to be found. Since we have assumed 2

× 2 trial functions so that the mass and stiffness matrices

are both 4 × 4, the characteristic polynomial must be

fourth order, as shown in equation (2). Subsequently, p(λ)

is expanded so that the polynomials coefficients can be

found. Once the polynomial is created, the Cayley-

Hamilton equation can be written by substituting (M-1

K)

for λ into equation (2).

( )

1 4 1 3

4 3

1 2 1

3 1 0

, ( ) ( )

( ) ( ) 0

p K M c M K c M K

c M K c M K c I

− −

− −

= +

+ + + =

(3)

where cn are the coefficients of λ in p(λ) determined via

equation (2). As stated in [15], equation (3) produces

sixteen equations, of which only four are independent.

Solving the equations on the main diagonal for the four

unknowns (hb, b1, b2, b3) produces 44 = 256 possible solu-

tions, according to Bézout’s theorem [22]. From the set

of all possible solutions, complex solutions can be imme-

diately eliminated as not being physically meaningful.

Clearly, further constraints must be added to the solution

in order to get a solution which fits within the desired

physical limits. These physical limits are based on the

maximum and minimum brace dimensions which are

required to compensate for the range of plate stiffnesses

used during the analysis, as well as the range of natural

frequencies which can be obtained using these system

dimensions. Thus, the following constraints are imple-

mented into the solution:

7 8

1

7 8

2

7 8

3

0.013 0.016 m

1 10 9 10 rad s

1 10 9 10 rad s

1 10 9 10 rad s

bh

b

b

b

≤ ≤

× ≤ ≤ ×

× ≤ ≤ ×

× ≤ ≤ ×

(4)

Solving the four equations obtained from equation (3)

within the constraints provided by (4) yields a physically

realistic solution which satisfies the desired fundamental

frequency, as well as the system’s parameters.

3. RESULTS

3.1 Material Properties

The material used for the brace-plate system during the

analysis is assumed to be Sitka spruce, due to its wide-

spread use in the industry. The mechanical properties of

Sitka spruce are obtained from [17] and are given in Ta-

ble 1.

Material properties Values

Density – µ (kg/m3) 403.2

Young’s modulus – ER (MPa) 850

Young’s modulus – EL (MPa) ER / 0.078

Shear modulus – GLR (MPa) EL × 0.064

Poisson’s ratio – νLR 0.372

Poisson’s ratio – νRL νLR × ER / EL

Table 1. Material properties of Sitka spruce [17].

Since wood is an orthotropic material, the ‘R’ and ‘L’

subscripts used in Table 1 refer to the radial and longitu-

dinal property directions of wood, respectively. The

wood used in making instrument soundboards is general-

ly quartersawn. Therefore the tangential property direc-

tion can be neglected. Since wood properties are highly

variable, the values presented in Table 1 represent a sta-

tistical average and are hence used as a benchmark for

further analysis.

3.2 Model Dimensions

The dimensions used for the model throughout the analy-

sis of the brace-plate system are shown in Table 2.

Dimensions Values

Length – Lx (m) 0.24

Length – Ly (m) 0.18

Length – Lb (m) 0.012

Reference – x1 (m) Lx / 2 – Lb / 2

Reference – x2 (m) x1 + Lb

Thickness – hp (m) 0.003

Thickness – hb (m) 0.012

Thickness – hc (m) hp + hb

Table 2. Dimensions of brace-plate model.


615

These dimensions refer to those shown in Figure 1, where

‘p’ refers to the plate’s dimensions, ‘b’ refers to the

brace’s dimensions and ‘c’ refers to the dimensions of the

combined system. These dimensions are chosen based on

the dimensions of a typical guitar soundboard section

reinforced by a single brace.

3.3 Benchmark Values

In order to demonstrate the working values of the model,

a benchmark is set using the statistical averages for the

properties of Sitka spruce as defined in Table 1. There-

fore, a plate with a radial stiffness of ER = 850 MPa to

which a brace is attached with a combined brace-plate

thickness of hc = 0.015 m is investigated. Using these

values and the forward model, the eigenvalue problem

can be solved to find a fundamental natural frequency of

687 Hz for the brace-plate system. In order to see the

effect on the overall system, the four frequencies pro-

vided using 2 × 2 trial functions are shown in Table 3. In

this case mx and my represent the mode numbers in the x

and y directions respectively.

mx my Natural frequency

(Hz) Modeshapes

1 1 687

2 1 790

2 2 1366

1 2 2648

Table 3. System's natural frequencies at benchmark

values.

3.4 Analysis

After determining the benchmark values, an analysis is

performed using the inverse method described in the pre-

vious section. As the plate’s radial stiffness varies, the

thickness of the brace-plate section is calculated such that

the fundamental frequency of the brace-plate system is

kept consistent at 687 Hz. The results of the computations

can be found in Table 4.

Young’s

modulus

ER (MPa)

Brace

thickness

hc (m)

Fundamental

Frequency

a (Hz)

750 0.01576 687

800 0.01536 687

813 0.01527 687

850 0.01500 687

900 0.01466 687

950 0.01435 687

Table 4. Results of the inverse model analysis.

Clearly, adjusting the thickness of the brace also has an

effect on the other natural frequencies. These can be seen

in Table 5.

Young’s

modulus

ER (MPa)

Brace

thickness

hc (m)

b1

(Hz)

b2

(Hz)

b3

(Hz)

750 0.01576 774 1360 2653

800 0.01536 782 1363 2650

813 0.01527 784 1364 2650

850 0.01500 790 1366 2648

900 0.01466 798 1370 2645

950 0.01435 806 1374 2642

Table 5. Calculated frequencies of the inverse model

analysis.

Interestingly, the constraints indicated in equation (4),

although physically strict, allow for more than one solu-

tion in certain cases. An example is shown in Table 6.

Young’s

modulus

ER (MPa)

Brace

thickness

hc (m)

a

(Hz)

b1

(Hz)

b2

(Hz)

b3

(Hz)

750 0.01359 687 570 1149 2185

Table 6. Alternate brace thickness solution satisfying

the physical constraints.

These results, along with their importance are discussed

below.

4. DISCUSSION

From these results, it is evident that designing a brace-

plate system starting with a desired fundamental frequen-

cy is possible. Table 4 clearly shows that by adjusting the

thickness of the brace by small increments (10-5

m, ma-

chine limit), it is possible to compensate for the variation

in the radial stiffness of the plate (ER) so that the funda-

mental frequency of the combined system is equal to that

of the benchmark value of 687 Hz. The results obtained

using the Cayley-Hamilton theorem algorithm match

those values obtained using the forward model exactly

when compared to the benchmark values in Table 3.

In modifying the thickness of the brace, the fundamen-

tal frequency is not the only frequency which is modified.

Table 5 shows that frequencies b1 to b3 are affected, with

a varying degree of magnitude. To be precise, while fun-

damental frequencies a remain consistent, frequencies b1

to b3 vary by 2%, 0.6% and 0.2% respectively based on

the variation of ER. Therefore, it is important to ensure

that there is a good understanding of what the brace can

control. A detailed discussion of varying the shape of the

brace in order to simultaneously control two natural fre-

quencies is found in [5]. This involves the use of scal-

loped shaped braces such as the one shown in Figure 4.

Using such a brace, one can control both the fundamental

frequency, as well as one of its higher partials simultane-

ously. Furthermore, by increasing the number of variable

parameters such as brace position or number of braces,

many more system frequencies could be controlled.


616

Figure 4. Scalloped shaped brace

Moreover, it was a surprise to find that within the strict

physical constraints of (4), there is more than one brace-

plate system (solution) that satisfies the Cayley-Hamilton

theorem of equation (3). From Table 6 it can be seen that

an alternate solution to the system exists, different from

the one presented in Table 4, for a plate having a radial

stiffness of 750 MPa. In this case, by reducing the thick-

ness of the brace, it is still possible to achieve a system

having the desired frequency of 687Hz. However, the

desired frequency is no longer the fundamental frequency

but rather becomes the second frequency and the funda-

mental has been replaced with a fundamental frequency

of 570 Hz. It is important to keep this phenomenon in

mind while designing a system. This is especially true if

the order in the spectrum of a certain frequency associat-

ed with a certain modeshape is absolutely critical.

In situations requiring the specification of a fundamen-

tal frequency, it is clear that to achieve consistency in this

desired frequency, the thickness of the brace must in-

crease when the stiffness of the plate decreases and vice-

versa. The method demonstrated in this paper provides

for a precise way in which these values can be found.

5. CONCLUSION

In this paper, we demonstrate a direct method for adjust-

ing the natural frequencies of various systems including

musical instruments. The method also directly demon-

strates the ability to compensate for variations in the radi-

al stiffness of wooden plates, thereby making it possible

to create brace-plate systems having a consistent funda-

mental frequency. This structured approach represents a

significant improvement over heuristic methods currently

in use. Although only a simple model was presented in

this paper, the concept of design-for-frequency using the

Cayley-Hamilton method was demonstrated as a proof of

concept for future work in the field of musical acoustics.

It is clear that much work needs to be done in order to

apply this method of design to actual musical instrument

soundboards. However, this technique holds great poten-

tial for creating system matrices of complex systems from

a set of desired frequencies. In doing so, it promises to

greatly benefit the advancement of the manufacturing of

acoustically consistent wooden musical instrument

soundboards.

Acknowledgments

The authors would like to acknowledge the generous

support provided by the Natural Sciences and Engineer-

ing Research Council of Canada.

6. REFERENCES

[1] R. M. French, “Engineering the Guitar: Theory and

Practice,” in Engineering the Guitar: Theory and

Practice, 1st ed., New York: Springer, 2008, pp.

159–208.

[2] R. H. Siminoff, The Luthier’s Handbook: A Guide to

Building Great Tone in Acoustic Stringed

Instruments. Milwaukee, WI: Hal Leonard, 2002.

[3] M. French, R. Handy, and M. J. Jackson,

“Manufacturing sustainability and life cycle

management in the production of acoustic guitars,”

International Journal of Computational Materials

Science and Surface Engineering, vol. 2, no. 1, pp.

41–53, Jan. 2009.

[4] A. Schoofs, F. V. Asperen, P. Maas, and A. Lehr, “I.

Computation of Bell Profiles Using Structural

Optimization,” Music Perception: An

Interdisciplinary Journal, vol. 4, no. 3, pp. 245–254,

Apr. 1987.

[5] P. Dumond and N. Baddour, “Effects of using

scalloped shape braces on the natural frequencies of

a brace-soundboard system,” Applied Acoustics, vol.

73, no. 11, pp. 1168–1173, Nov. 2012.

[6] M. T. Chu and G. H. Golub, Inverse Eigenvalue

Problems: Theory, Algorithms, and Applications.

Oxford University Press, USA, 2005.

[7] G. M. L. Gladwell, Inverse problems in vibration.

Kluwer Academic Publishers, 2004.

[8] D. Boley and G. H. Golub, “A survey of matrix

inverse eigenvalue problems,” Inverse Problems,

vol. 3, no. 4, p. 595, 1987.

[9] M. T. Chu and J. L. Watterson, “On a Multivariate

Eigenvalue Problem: I. Algebraic Theory and a

Power Method,” SIAM J. Sci. Comput., vol. 14, pp.

1089–1106, 1993.

[10] R. Erra and B. Philippe, “On some structured inverse

eigenvalue problems,” Numerical Algorithms, vol.

15, no. 1, pp. 15–35, 1997.

[11] F. W. Biegler-König, “Construction of band matrices

from spectral data,” Linear Algebra and its

Applications, vol. 40, pp. 79–87, Oct. 1981.

[12] G. H. Golub and R. R. Underwood, “The block

Lanczos method for computing eigenvalues,” in

Mathematical Software III, J.R. Rice., New York:

Springer, 1977.

[13] F. W. Biegler-König, “A Newton iteration process

for inverse eigenvalue problems,” Numerische

Mathematik, vol. 37, no. 3, pp. 349–354, 1981.

[14] M. T. Chu, “A Fast Recursive Algorithm for

Constructing Matrices with Prescribed Eigenvalues

and Singular Values,” SIAM Journal on Numerical

Analysis, vol. 37, no. 3, pp. 1004–1020, Jan. 2000.


617

[15] P. Dumond and N. Baddour, “A Structured

Approach to Design-for-Frequency Problems Using

the Cayley-Hamilton Theorem,” Computers &

Structures, vol. Submitted: CAS-D-13–00301, Apr.

2013.

[16] L. Meirovitch, “Principles and Techniques of

Vibrations,” in Principles and Techniques of

Vibrations, Upper Saddle River, NJ: Prentice Hall,

1996, pp. 542–543.

[17] Forest Products Laboratory (US), “Wood Handbook,

Wood as an Engineering Material,” Madison, WI:

U.S. Department of Agriculture, Forest Service,

1999, pp. 4.1–13.

[18] A. W. Knapp, “Basic Algebra,” in Basic Algebra,

Boston: Birkhäuser, p. 219.

[19] F. R. Chang and H. C. Chen, “The generalized

Cayley-Hamilton theorem for standard pencils,”

Systems & Control Letters, vol. 18, no. 3, pp. 179–

182, Mar. 1992.

[20] P. Dumond and N. Baddour, “Toward improving the

manufactured consistency of wooden musical

instruments through frequency matching,” in

Transactions of the North American Manufacturing

Research Institution of SME, 2010, vol. 38, pp. 245–

252.

[21] P. Dumond and N. Baddour, “Effects of a Scalloped

and Rectangular Brace on the Modeshapes of a

Brace-Plate System,” International Journal of

Mechanical Engineering and Mechatronics, vol. 1,

no. 1, pp. 1–8, 2012.

[22] J. L. Coolidge, Treatise on Algebraic Plane Curves.

Mineola, New York: Dover Publications, 1959.


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Using a Brace to Design a Soundboard Section for a Desired Natural Frequency

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