### A. Modeling assumptions

To analytically investigate the brace/soundboard interaction, we analyze the natural frequencies and modeshapes of a rectangular plate model with an attached cross-brace, as shown in Figure 3.

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 closed-form 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 thickness-to-width 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 orthotropic properties of wood are modeled, therefore its longitudinal and radial properties are of interest, labelled *L* and *R* respectively. The only material properties that need to be considered independently in these directions are Young’s modulus, *E* and Poisson’s ratio, *ν*. For an orthotropic plate the stress–strain relationships are given by (Riley et al. 2006)

\begin{array}{l}{\sigma}_{x}={S}_{\mathit{xx}}{\u03f5}_{x}+{S}_{\mathit{xy}}{\u03f5}_{y}\\ {\sigma}_{y}={S}_{\mathit{yx}}{\u03f5}_{x}+{S}_{\mathit{yy}}{\u03f5}_{y}\\ {\tau}_{\mathit{xy}}=G\phantom{\rule{0.12em}{0ex}}{\gamma}_{\mathit{xy}}\end{array}

(1)

where the *S* are stiffness components are

\begin{array}{l}{S}_{\mathit{xx}}=\frac{{E}_{x}}{1-{\nu}_{\mathit{xy}}{\nu}_{\mathit{yx}}}\\ {S}_{\mathit{yy}}=\frac{{E}_{y}}{1-{\nu}_{\mathit{xy}}{\nu}_{\mathit{yx}}}\\ {S}_{\mathit{xy}}={S}_{\mathit{yx}}=\frac{{\nu}_{\mathit{yx}}{E}_{x}}{1-{\nu}_{\mathit{xy}}{\nu}_{\mathit{yx}}}=\frac{{\nu}_{\mathit{xy}}{E}_{y}}{1-{\nu}_{\mathit{xy}}{\nu}_{\mathit{yx}}}\end{array}

(2)

The subscripts represent the direction of the plane in which the material properties act. Therefore, *E*
_{
x
} is the Young’s modulus along the *x*-axis, *E*
_{
y
} along the *y*-axis and *ν*
_{
xy
} and*v*
_{
yx
} the major Poisson’s ratios along the *x*-axis and *y*-axis 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

Using the stress–strain relationships of Eq. (1), the strain energy for an orthotropic plate is given by (Timoshenko & Kreiger 1964).

U=\frac{1}{2}{\displaystyle \underset{0}{\overset{{L}_{x}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}\left[{D}_{x}{w}_{\mathit{xx}}^{2}+2{D}_{\mathit{xy}}{w}_{\mathit{xx}}{w}_{\mathit{yy}}+{D}_{y}{w}_{\mathit{yy}}^{2}+4{D}_{k}{w}_{\mathit{xy}}^{2}\right]}}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x

(3)

where *L*
_{
x
}and *L*
_{
y
}are the dimensions of the plate in the *x* and *y* directions, *w* = *w*(*x*, *y*)is the transverse displacement and the subscripts on *w* refer to partial derivatives in the given direction. The plate’s stiffnesses *D* are given by

\begin{array}{l}{D}_{x}=\frac{{S}_{\mathit{xx}}{h}^{3}}{12}{D}_{y}=\frac{{S}_{\mathit{yy}}{h}^{3}}{12}\\ {D}_{\mathit{xy}}=\frac{{S}_{\mathit{xy}}{h}^{3}}{12}{D}_{k}=\frac{{G}_{\mathit{xy}}{h}^{3}}{12}\end{array}

(4)

where *h* is the thickness of the plate.

### C. Kinetic energy for an orthotropic plate

The orthotropic properties of the plate only affect its stiffness and not its density, so its kinetic energy is the same as for an isotropic plate

T=\frac{1}{2}{\displaystyle \underset{0}{\overset{{L}_{x}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}{\dot{w}}^{2}}}\rho \phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x

(5)

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}.

From the expression for strain energy, Eq. (3), the only term affected by the change in thickness between *x*
_{1} and *x*
_{2} is the stiffness *D*. Therefore, the integral of Eq. (3) is split into three separate parts so that the strain energy becomes

\begin{array}{l}U=\frac{1}{2}{\displaystyle \underset{0}{\overset{{x}_{1}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}\left[{D}_{\mathit{xp}}{w}_{\mathit{xx}}^{2}+2{D}_{\mathit{xyp}}{w}_{\mathit{xx}}{w}_{\mathit{yy}}+{D}_{\mathit{yp}}{w}_{\mathit{yy}}^{2}+4{D}_{\mathit{kp}}{w}_{\mathit{xy}}^{2}\right]}}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x\\ \phantom{\rule{1.5em}{0ex}}+\frac{1}{2}{\displaystyle \underset{{x}_{1}}{\overset{{x}_{2}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}\left[{D}_{\mathit{xc}}{w}_{\mathit{xx}}^{2}+2{D}_{\mathit{xyc}}{w}_{\mathit{xx}}{w}_{\mathit{yy}}+{D}_{\mathit{yc}}{w}_{\mathit{yy}}^{2}+4{D}_{\mathit{kc}}{w}_{\mathit{xy}}^{2}\right]}}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x\\ \phantom{\rule{1.5em}{0ex}}+\frac{1}{2}{\displaystyle \underset{{x}_{2}}{\overset{{L}_{x}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}\left[{D}_{\mathit{xp}}{w}_{\mathit{xx}}^{2}+2{D}_{\mathit{xyp}}{w}_{\mathit{xx}}{w}_{\mathit{yy}}+{D}_{\mathit{yp}}{w}_{\mathit{yy}}^{2}+4{D}_{\mathit{kp}}{w}_{\mathit{xy}}^{2}\right]}}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x\end{array}

(6)

The stiffnesses *D* are now section-specific because of the change in thickness *h* from *x*
_{1} to *x*
_{2}:

{D}_{\mathit{xp}}=\frac{{S}_{\mathit{xx}}{h}_{p}^{3}}{12}{D}_{\mathit{yp}}=\frac{{S}_{\mathit{yy}}{h}_{p}^{3}}{12}{D}_{\mathit{xy}}{p}_{}=\frac{{S}_{\mathit{xy}}{h}_{p}^{3}}{12}{D}_{\mathit{kp}}=\frac{{G}_{\mathit{xy}}{h}_{p}^{3}}{12}

(7)

and

{D}_{\mathit{xc}}=\frac{{S}_{\mathit{xx}}{h}_{c}^{3}}{12}{D}_{\mathit{yc}}=\frac{{S}_{\mathit{yy}}{h}_{c}^{3}}{12}{D}_{\mathit{xy}}{c}_{}=\frac{{S}_{\mathit{xy}}{h}_{c}^{3}}{12}{D}_{\mathit{kc}}=\frac{{G}_{\mathit{xy}}{h}_{c}^{3}}{12}

(8)

where the subscripts *p* and *c* denote the plate alone and combined plate-and-brace system.

### E. Kinetic energy for a plate modified with an attached brace

Similar to the method used to modify the strain energy term, the kinetic energy can also be written to take into account the change in thickness from *x*
_{1} to *x*
_{2}:

\begin{array}{l}T=\frac{1}{2}{\displaystyle \underset{0}{\overset{{x}_{1}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}{\dot{w}}^{2}}}{\rho}_{p}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x+\frac{1}{2}{\displaystyle \underset{{x}_{1}}{\overset{{x}_{2}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}{\dot{w}}^{2}}}{\rho}_{c}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x\\ \phantom{\rule{1.5em}{0ex}}+\frac{1}{2}{\displaystyle \underset{{x}_{2}}{\overset{{L}_{x}}{\int}}{\displaystyle \underset{0}{\overset{{L}_{y}}{\int}}{\dot{w}}^{2}}}{\rho}_{p}\phantom{\rule{0.12em}{0ex}}\mathrm{d}y\mathrm{d}x\end{array}

(9)

where the density per unit area *ρ* is now calculated as:

{\rho}_{p}=\mu \cdot {h}_{p}\phantom{\rule{1.2em}{0ex}}\mathrm{and}{\rho}_{c}=\mu \cdot {h}_{c}

(10)

### F. The assumed shape method

The assumed shape method is chosen because it allows us to use the flat-plate 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 re-establish 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 first step of the assumed shape method is to approximate the transverse displacement *w*(*x*, *y*, *t*) as (Meirovitch 2001)

w\left(x,y,t\right)={\displaystyle \sum _{{n}_{x}=1}^{{m}_{x}}{\displaystyle \sum _{{n}_{y}=2}^{{m}_{y}}{\varphi}_{{n}_{x}{n}_{y}}\left(x,y\right)\cdot}}\phantom{\rule{0.12em}{0ex}}{q}_{{n}_{x}{n}_{y}}\left(t\right)

(11)

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 (time-dependent) 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).

Here, the modeshapes of the simply supported rectangular plate (without the brace) are known and these will be used as the trial functions in Eq. (11), so that

{\varphi}_{{n}_{x}{n}_{y}}=sin\left({n}_{x}\cdot \pi \cdot \frac{x}{{L}_{x}}\right)\cdot sin\left({n}_{y}\cdot \pi \cdot \frac{y}{{L}_{y}}\right)

(12)

Applying the trial functions of Eq. (12) to Eq. (11), gives a discrete series

w\left(x,y,t\right)={\displaystyle \sum _{{n}_{x}=1}^{{m}_{x}}{\displaystyle \sum _{{n}_{y}=2}^{{m}_{y}}sin\left({n}_{x}\cdot \pi \cdot \frac{x}{{L}_{x}}\right)\cdot sin\left({n}_{y}\cdot \pi \cdot \frac{y}{{L}_{y}}\right)\cdot}}\phantom{\rule{0.12em}{0ex}}{q}_{{n}_{x}{n}_{y}}\left(t\right)

(13)

which is then used in the strain and kinetic energy equations of the modified plate.

Once the strain and kinetic energies have been assembled, Lagrange’s equations are used to find the equations of motion which can then be written in matrix form as

\mathbf{M}\overrightarrow{\ddot{\mathbf{q}}}+\mathbf{K}\overrightarrow{\mathbf{q}}=\overrightarrow{\mathbf{0}}

(14)

where *M* is the mass matrix and *K* is the stiffness matrix given. Additionally, \overrightarrow{\mathbf{q}} is the generalized coordinate vector

\overrightarrow{\mathbf{q}}={\left[\begin{array}{cccc}\hfill {q}_{11}\hfill & \hfill {q}_{12}\hfill & \hfill {q}_{21}\hfill & \hfill \begin{array}{cc}\hfill {q}_{22}\hfill & \hfill \dots \hfill \end{array}\hfill \end{array}\right]}^{T}

(15)

Letting the generalized coordinate system have a harmonic solution as in (Meirovitch 1996b), then

\overrightarrow{\mathbf{q}}=\overrightarrow{\mathbf{A}}cos\left(\mathit{\omega t}+\varphi \right)

(16)

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.