In what follows, the equations of simplified 2 and 3 DOF models for low frequency vibro-acoustic response of the resonant chamber (guitar body) will be presented, trying to keep clarity and conciseness as possible. The meaning of each index adopted in the development of the equations below is shown in
Table 1 (for an explanation of guitar components, refer to
Figure 1 below).
Table 2 presents the parameters appearing in the equations of the 4 DOF model
[1]; it should be noted that those appearing in the equations for 2 and 3 DOF models
[2][3][4] are also covered in this table, but some table entries do not apply for the simpler models. Damping coefficients are presented separately on the last line because these parameters are to be defined through experimental modal analysis.
About the damping parameters in
Table 2, it is important to emphasize at this point that these coefficients alone would imply, for instance, a purely viscous, or viscoelastic/hysteretic dissipation behaviour, which is clearly a non-physical assumption considering the predicted forms of energy loss in the modelling of vibro-acoustic coupled systems; i.e., the energy balance in this cases should include, at least, some sort of viscous, viscoelastic and acoustic radiation forms of damping
[5][6]. Indeed, for those parameters related to energy dissipation, that equivalent modelling approach mentioned earlier is almost always adopted—with damping ratios, loss factors or another dissipation quantifier accounting alone for all forms of energy loss.
1.1.1. Simplified 2 Degrees-of-Freedom Model
Now, suppose the tensions in guitar strings transmit a force of magnitude
f through the bridge (see
Figure 1), that can be represented as acting pointwise, up and down, in the centroid of top plate. Thus, the equations of motion representing the dynamic coupling with the air column DOF displayed in
Figure 2 can be written as
Figure 2. Schematic of the simple 2 DOF model: (a) Cut plane view of a guitar resonant chamber 3D CAD model, constructed with shells, planes and lines for subsequent FE analysis. (b) Representation as a simple 2 DOF model, considering only the motion of top plate and air column.
Considering the adiabatic, linearized form, of the equations of acoustics, the pressure variation
Δp may be expressed in terms of the total volume
V inside the resonant chamber, the sound velocity
c and mass density
ρ of the surrounding air, and of the (idealized) chamber volume increment
ΔV=Atxt+Aaxa [7], as
Thus, substituting Equation (
2) into (
1) and writing it in matrix form, i.e.,
Mx¨+Kx=f, yields
Quantities appearing on the left-hand side of Equation (
3) could be viewed separately into two classes: those described in
Table 2 depend on geometric and/or material properties of guitar components, and thus shall be treated as adjustable parameters of the equivalent system, while the remaining ones (speed of sound and mass density) are physical properties of the surrounding air, and so it is more reasonable to consider them as system-independent constants. On the other hand, the driving force
f would be truly impossible to obtain, both experimentally or analytically, because this excitation motion is rather imposed by an effectively distributed conjugate on the interfacing area between the top plate and the bridge; i.e., its much higher stiffness impels it to rotate when forced to and fro by the tension of guitar strings, acting some distance above the plate. Theoretically, once the plate is adequately approximated (to some level of accuracy) as a clamped or hinged plate
[1] an equivalent pivot distance could be considered, via static equilibrium of moments, in order to compare the driving force over the top plate with the tension in guitar strings. This approach has not been explored up to this moment, however, to the authors’ knowledge. Nevertheless, analytical values for modal parameters can be readily extracted from this matrix equation, and in this way it has been extensively used for model comparison
[8][2][9], structural modifications prediction based on eigenvalues sensitivity
[9][4] or frequency domain model updating
[10].
Unlike the mentioned references, the previous equations were written in undamped form. While this choice was made primarily to simplify the presentation, it has a more profound justification stemming from the authors’ previous works
[11] that is: given the levels of uncertainty in these simplified models, the use of more complex, non-proportional damping formulations would not cause any perceivable alteration in measurable outputs. Thus, since a proportionally damped analytical system produces the same eigenproperties as its undamped version, and taking into consideration the prominence of modal parameters over the forced response for the envisaged applications, damping factors can be confidently considered here as fine tuning adjustments. Among the myriad of available methods to extract damping coefficients (along with mode shapes and natural frequencies) from measured responses, Subspace-Stochastic Identification (SSI) has proven to be a very convenient, output-only, alternative
[11].
1.1.2. Simplified 3 and 4 Degrees-of-Freedom Models
From the definitions presented above for 2 DOF, modelling extensions can be readily devised.
Figure 3 shows schematic representations for two of these possible expansions, resulting in 3 and 4 DOF models, respectively. The most obvious approach then is to consider the flexibility of back plate in a symmetrical manner as was made for the top plate. This in turn adds up three more free parameters: the mass, the stiffness and one more modal damping coefficient, and the system can now vibrate with three linearly independent modes. Thus, following
[4], the resultant displacements vector and the corresponding mass and stiffness matrices for the 3 DOF model are given as
Figure 3. Schematic of 3 and 4 DOF simplified models; (a) 3 DOF: allowing for motions on top and back plates, and air column through the hole. (b) 4 DOF: basically allowing for the same motions of the 3 DOF model, plus one for the entire structure (free condition).
The next modeling extension, considering mass and motion for the ribs, is a little bit trickier and thus needs deeper attention. First of all, the motion allowed for the ’ribs’ in fact means that the whole guitar is free to move. Thus, this is not exactly about freeing a motion in the same sense as it was done for the back plate, but rather allowing a fully free boundary condition. As the governing equations of the 4 DOF model in
[1] are not explicitly presented in matrix form, the resulting stiffness matrix for this model was derived, and is written below as