1 Characterization of the Non - Damped Modal Response of a Portuguese Guitar Including its Twelve Strings Jorge A. S. Luis Dept. Engenharia Mecânica, Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal Abstract The Portuguese guitar is a pear-shaped twelve-string musical instrument (six pairs of strings). This thesis is dedicated to the study of the analysis of the un-dampened vibrational response of a Portuguese guitar, including its twelve strings. This study is motivated by the interest in predicting, in the project stage, which geometrical characteristics (dimensions) allow for a certain amount of modal characteristics, in this case, focusing on the three first frequencies associated with the modes (0,0), (0,1) and (0,2) in the guitar sound board. Besides the body, part of the arm is included in the simulation, as well as the twelve strings tuned in the standard Lisbon Portuguese guitar tuning (B, A, E, B, A, D). As far as the author knowledge is concerned, there is no study which such characteristics in the current literature, connecting CAD (computer assisted drawing) with a un-dampened modal analysis with pre-tension in the cables. The strings were tuned to the centesimal and the frequencies of modes (0,0), (0,1) and (0,2) adjusted to values found in the literature. Keywords: Portuguese guitar, structure, strings, vibration, acoustics. 1. Introduction The goal of this thesis is the development of prediction methodologies for the structural and acoustic response of a Portuguese guitar, through the analysis of the un-dampened modal vibrational response, including its twelve strings, so as to predict, in the project stage, which geometrical characteristics mainly dimensions, allow for a certain amount of modal characteristics, and doing so, furthering the contemporary knowledge of these instruments, so that the results produced may be used by the industries that build these instruments. Guitar-makers like Ervin Somogyi [1], whose California produced guitars can reach 31000 dollars, and Gerald Sheppard [2], compare hand-made guitars to industrially produced guitars in their articles, where they stress the quality and uniqueness of hand built guitars, while recognizing the quality of some industrially produced guitars as well as the unique characteristics of some of these guitars. With the introduction of new materials and the further investigation of previously existing ones, it is possible, in my opinion, to produce very high quality industriously produced guitars. In [3], “The sound of a concert guitar will be clean every string and frets" adding that the goal of state-of-the-art technology nowadays is to replace the subjective quality assessments usually associated with the concert guitar, with simulations and experimental science, in order to increase quality and lower costs. It follows that the introduction of models with the ability to produce an effective simulation, within the possibilities of computer simulation, using the Finite Element Method (FEM), for instance, would be invaluable for music instruments industry. It would also be an excellent contribution to applied science.
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Characterization of the Non - Damped Modal Response of a Portuguese Guitar Including its Twelve Strings
Jorge A. S. Luis
Dept. Engenharia Mecânica, Instituto Superior Técnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Abstract
The Portuguese guitar is a pear-shaped twelve-string musical instrument (six pairs of strings).
This thesis is dedicated to the study of the analysis of the un-dampened vibrational response of a
Portuguese guitar, including its twelve strings. This study is motivated by the interest in predicting, in
the project stage, which geometrical characteristics (dimensions) allow for a certain amount of modal
characteristics, in this case, focusing on the three first frequencies associated with the modes (0,0),
(0,1) and (0,2) in the guitar sound board. Besides the body, part of the arm is included in the
simulation, as well as the twelve strings tuned in the standard Lisbon Portuguese guitar tuning (B, A,
E, B, A, D). As far as the author knowledge is concerned, there is no study which such characteristics
in the current literature, connecting CAD (computer assisted drawing) with a un-dampened modal
analysis with pre-tension in the cables. The strings were tuned to the centesimal and the frequencies
of modes (0,0), (0,1) and (0,2) adjusted to values found in the literature.
3. Dynamic analysis According to the theory of elasticity, the dynamic behaviour of a linear elastic solid, for small deformations, is (in Cauchy’s form) [8]
𝜎!",!!𝑓!!𝜌!𝑢! (1)
where 𝜎!" is the stress tensor, 𝑓! is the sum of the force vectors acting on the body, 𝜌! is the density of the solid, 𝑢! is the displacement vector and I, j = x, y, z.
The weak form can be obtained by the residual method whose function of choice is based on the Galerkin method. In this way, the approximate solution by finite methods in terms of nodal displacements can be written as:
𝑴𝒖 + 𝑲𝒖 = 𝒇 (2)
where 𝑢 is the nodal displacement, M is the global mass matrix, K is the global stiffness matrix and 𝒇
is the force vector. These matrices and force vector can be obtained by FE (Finite Element)
assembling of the following elements:
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𝒎𝒆 = 𝜌𝑵!𝑵𝑑𝑉! (3)
𝒌𝒆 = 𝑩!𝑪𝑩𝑑𝑉! (4)
𝒇𝒆 = 𝑵!𝑭𝒗𝑑𝑉! + 𝑵!𝑭𝒔𝑑𝑆! (5)
were N is the matrix of the element’s shap function, B is the matrix of the extensions – nodal displacements, C is the constitutive law matrix, 𝑭𝒗 is the vector of the volume forces and 𝑭𝒔 is the vector of the surface nodal forces. Equation 2 restrained to static analysis gives
𝑲𝒖 = 𝒇 (6)
Considering thermal expansion due to temperature change and despised (for being too small) the variation of the elastic constants can apply the principle of superposition.
𝜎!" = 𝐶 !"#$ 𝜀!" − 𝛼!"∆𝑇 (7)
The tensions can be obtained through Hooke’s law. Rewriting (2) in the frequency domain for free vibration conditions we get
𝑲 + 𝒌𝒈𝒆𝒐𝒎 + 𝝀𝑴 𝒖 = 𝟎, (8)
where 𝜆 is a diagonal matrix of the frequencies squared and u is the nodal displacements vector
matrix (one per column) of the corresponding vibration modes and 𝒌𝒈𝒆𝒐𝒎 is the geometric matrix.
4. Results for the model without strings The problem analysed in this section corresponds to the determination of the first frequencies
of the guitar without the strings, with the boundary conditions of guitar free in space, as was
considered in the experimental model of [4]. Table 3 presents the final dimensions (the ones which
better approximate the modes and frequencies obtained by [4]), attained after several Finite Elements
(FE) analyses.
Table 3 – Dimensions and brace distances of the analysed guitar model.
(lxh mm) Top Back Distance to the exact point Between braces Gluing Belt