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

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

A STRUCTURED APPROACH TO USING A RECTANGULAR BRACE TO DESIGN A SOUNDBOARD SECTION FOR A DESIRED NATURAL FREQUENCY Patrick Dumond Natalie Baddour pdumo057@uottawa.ca nbaddour@uottawa.ca Department of Mechanical Engineering University of Ottawa 161 Louis Pasteur, CBY A205 Ottawa, Canada K1N 6N5

the fact that these methods are very labor intensive, rendering them cost-prohibitive, as well as difficult to implement into a structured manufacturing process. For these 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-anderror 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 specifically with finding matrices from a set of given eigenvalues [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 illposed, meaning there generally exists many solutions [7].

In design, the existence of many solutions is potentially beneficial, giving the designer options. However, physical 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 recently proposed using the generalized Cayley-Hamilton

ABSTRACT The manufacture of acoustically consistent wooden musical 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 considered in this paper. The theory of inverse eigenvalue problems seeks to address the problem by creating representative system matrices directly from the desired natural frequencies of the system. The goal of this paper is to demonstrate how the generalized Cayley-Hamilton theorem 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 system at a desired value is then calculated with the proposed 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 soundboards of wooden musical instruments.

1. INTRODUCTION Many aspects of the manufacture of wooden musical instruments have been addressed and rendered consistent.

However, acoustical consistency still remains unattainable in most situations [1]. This is a consequence of the fact that the material of choice for many musical instruments is wood, a natural material that exhibits high variability in its material properties. By definition, wood has inconsistent properties because its growth is directly related to the highly variable climate of its environment.

Luthiers have been compensating for these inconsistent material properties in guitars by means of various methods 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 Copyright: Š 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, distribution, and reproduction in any medium, provided the original author and source are credited.

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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 stiffness 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.

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 constraints thus becomes an exercise of the forward modeling 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 natural 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 frequencies via design of the braces.

Known force R

Excellent Good Mediocre Discard

2. MODEL Figure 2. Quality control measurement of soundboard plate used in industry

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.

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 generalized 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(KλM), then substituting (M-1K) for λ in the polynomial gives the zero matrix [18], [19]. Thus, by building the model in the forward sense, and by leaving relevant design 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 fundamental 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 associated frequency [21].

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. z

Ly

y

brace

x

0 hp 0

hc plate x1 x2 Lx

Figure 1. Orthotropic plate reinforced with a rectangular brace.

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

The forward model is discretized using the assumed shape method, similar to the model used in [5]. The assumed 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 equations of motion, from which the mass and stiffness matrices are extracted [16]. The system is assumed simply

2.4 Cayley-Hamilton Algorithm Using the facts that the wood’s mechanical properties vary based on its radial stiffness, and that the brace thickness controls the brace-plate system’s fundamental frequency, the forward model is created using the assumed

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

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 implemented into the solution:

shape method while leaving these two parameters as variables. 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 function 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 rectangular plate such that

0.013 ≤ hb ≤ 0.016 m 1× 107 ≤ b1 ≤ 9 ×108 rad s 1× 107 ≤ b2 ≤ 9 × 108 rad s

(4)

1× 107 ≤ b3 ≤ 9 × 108 rad s mx

w( x, y, t ) =

my

 nx π x   n y π y   qn n (t ) (1)  sin  Lx   L y  x y

∑ ∑ sin 

nx =1 n y = 2

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.

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 supported 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 manufacture 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 fundamental frequency, the Cayley-Hamilton theorem is used. To do so, the characteristic polynomial is created using the desired frequency,

p ( λ ) = ( λ − a ) ⋅ ( λ − b1 ) ⋅ ( λ − b2 ) ⋅ ( λ − b3 )

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 widespread use in the industry. The mechanical properties of Sitka spruce are obtained from [17] and are given in Table 1.

Material properties Density – µ (kg/m3) Young’s modulus – ER (MPa) Young’s modulus – EL (MPa) Shear modulus – GLR (MPa) Poisson’s ratio – νLR Poisson’s ratio – νRL

(2)

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

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 CayleyHamilton equation can be written by substituting (M-1K) for λ into equation (2).

p ( K , M ) = c4 (M −1 K ) 4 + c3 (M −1 K )3 +c3 ( M −1 K )2 + c1 ( M −1 K ) + c0 I = 0

Values 403.2 850 ER / 0.078 EL × 0.064 0.372 νLR × ER / EL

Since wood is an orthotropic material, the ‘R’ and ‘L’ subscripts used in Table 1 refer to the radial and longitudinal property directions of wood, respectively. The wood used in making instrument soundboards is generally quartersawn. Therefore the tangential property direction can be neglected. Since wood properties are highly variable, the values presented in Table 1 represent a statistical average and are hence used as a benchmark for further analysis.

(3)

3.2 Model Dimensions The dimensions used for the model throughout the analysis of the brace-plate system are shown in Table 2.

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 solutions, according to Bézout’s theorem [22]. From the set of all possible solutions, complex solutions can be immediately 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

Dimensions Length – Lx (m) Length – Ly (m) Length – Lb (m) Reference – x1 (m) Reference – x2 (m) Thickness – hp (m) Thickness – hb (m) Thickness – hc (m)

Values 0.24 0.18 0.012 Lx / 2 – Lb / 2 x1 + Lb 0.003 0.012 hp + hb

Table 2. Dimensions of brace-plate model.

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

Clearly, adjusting the thickness of the brace also has an effect on the other natural frequencies. These can be seen in Table 5.

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.

Young’s modulus ER (MPa) 750 800 813 850 900 950

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. Therefore, 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 provided 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.

Brace thickness hc (m) 0.01576 0.01536 0.01527 0.01500 0.01466 0.01435

b1 (Hz) 774 782 784 790 798 806

b2 (Hz) 1360 1363 1364 1366 1370 1374

b3 (Hz) 2653 2650 2650 2648 2645 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 solution in certain cases. An example is shown in Table 6.

Young’s modulus ER (MPa) 750

Brace thickness hc (m) 0.01359

a (Hz) 687

b1 (Hz) 570

b2 (Hz) 1149

b3 (Hz) 2185

mx

my

Natural frequency (Hz)

1

1

687

Table 6. Alternate brace thickness solution satisfying the physical constraints.

These results, along with their importance are discussed below.

2

1

790

2

2

1366

1

2

2648

Modeshapes

4. DISCUSSION From these results, it is evident that designing a braceplate system starting with a desired fundamental frequency is possible. Table 4 clearly shows that by adjusting the thickness of the brace by small increments (10-5 m, machine limit), it is possible to compensate for the variation in the radial stiffness of the plate (ER) so that the fundamental 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 fundamental 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 fundamental 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 frequencies is found in [5]. This involves the use of scalloped 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 simultaneously. Furthermore, by increasing the number of variable parameters such as brace position or number of braces, many more system frequencies could be controlled.

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 previous 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) 750 800 813 850 900 950

Brace thickness hc (m) 0.01576 0.01536 0.01527 0.01500 0.01466 0.01435

Fundamental Frequency a (Hz) 687 687 687 687 687 687

Table 4. Results of the inverse model analysis.

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

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.

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 braceplate 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 thickness 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 fundamental 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 associated with a certain modeshape is absolutely critical.

In situations requiring the specification of a fundamental frequency, it is clear that to achieve consistency in this desired frequency, the thickness of the brace must increase when the stiffness of the plate decreases and viceversa. The method demonstrated in this paper provides for a precise way in which these values can be found.

[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.

5. CONCLUSION

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

Kluwer Academic Publishers, 2004.

In this paper, we demonstrate a direct method for adjusting the natural frequencies of various systems including musical instruments. The method also directly demonstrates the ability to compensate for variations in the radial stiffness of wooden plates, thereby making it possible to create brace-plate systems having a consistent fundamental 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 potential 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.

[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.

Acknowledgments The authors would like to acknowledge the generous support provided by the Natural Sciences and Engineering Research Council of Canada.

[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.

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[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.

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.

[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.

[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.

[18] A. W. Knapp, “Basic Algebra,” in Basic Algebra, Boston: Birkhäuser, p. 219.

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

Mineola, New York: Dover Publications, 1959.

[19] F. R. Chang and H. C. Chen, “The generalized Cayley-Hamilton theorem for standard pencils,”

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