Characterization of the Non - Damped Modal Response of a Portuguese Guitar Including its Twelve Strings

Transcription

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, 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 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. 1

2 In this work, a modal characterization of the structure with and without strings is reviewed, and the mode tuning is explored, considering the presence of strings. As far as the author knowledge there is no publication on FEM in this field that includes the strings. 1.1 Revision of the literature One of the first studies of the modal response of a Portuguese guitar was actually made by [4]. The authors obtained the modal characterization of the guitar soundboard in fully mounted Lisbon and Coimbra guitars. They compared and described the frequency response curves for several specimens, as well as some significant vibrational modes. In the experimental setup the guitar was suspended by rubber bands on a rigid structure. The authors made a detailed modal identification in one of the guitars. They defined a grid of 114 possible impact points, in the sound board as well as the arm, so that they could identify possible coupled movements. The authors reached the following conclusions: 1) frequencies bellow 200 Hz are not significant for the sound radiation, 2) frequencies between 121 Hz and 160 Hz are due to cavity resonances (Helmholtz), and 3) between 250 and 450 Hz, there is, at least one resonance or resonance group that is responsible for a significant part of the radiated spectrum (see figure 1). The most important is the monopole mode (0,0) (shown in figure 1a), which is the mode that radiates the sound more efficiently, as opposed to the longitudinal dipole (0,1) (see figure 1b), where the adjacent antinodes move in anti-phase, eliminating the air movement inside the box. The tripole longitudinal mode (0,2) (see figure 1c) shows up at 635 Hz. Others appear above these, but are not mentioned here. a) b) c) Figure 1 - Modal forms of three resonances of one of the guitars in study: a) f!,! = 275 Hz ; b) f!,! = 360 Hz; c) f!,! = 6355 Hz (fonte: [4]). In [5], the author modulates the Portuguese guitar, using MEF, for the first time. The modal analysis of the guitar produces results that are compared with the experimental results in [4]. The dimensions of the guitar and of the harmonic braces were supplied beforehand. From these dimensions the author performed the modal analysis and obtained the results, which are the first 20 modes of vibration of the guitar, amongst which she compared the frequencies closest to the ones obtained by [4]. In [6], the authors use a Portuguese guitar model that is based on the reference geometry [5] to study the influence of the inclusion of the guitar strings and its respective modal response. The 2

3 twelve steel strings are stretched in a way that their natural resonances correspond to the frequencies of the Portuguese guitar notes. For that, they use pre-tensions related to temperature variations in the strings and so obtain the frequencies corresponding to the right tones of the Portuguese guitar. 2. Methodology The strategy adopted in this study uses a CAD program to model the guitar s geometry according to the dimensions, which were taken from [7], of Álvaro Merceano da Silveira s Portuguese guitar, including harmonic braces, gluing belt (linings) and the tail block, that constitute the grid and arm with thickness variation, see figures 2 and 3. Simplifications: neither the part of the arm that supports the fan (string holding mechanism), nor the volute (wood decorative piece) are considered in the adopted geometry, as these elements were ignored to the modal behavior of the guitar and at this step the box is modeled with shell elements. Due to the difficulties in finding the properties of some woods, the arm is modeled with mahogany instead of Brazilian cedar, and the parts that should be modeled with spruce are modeled with SitkaSpruce. The side of the box opposed to the arm is projected from a circumference centered in the exact point i.e. 205mm of radius instead of being bulging with 165mm of distance from the exact point of the nut, see figure 2. The strings are all considered bulk (actually some are spiral shaped) and modeled by beam elements. Instead of a continuum string, two are considered, one from nut (opposite side of de arm) to bridge and another from the bridge to the nut (near head of guitar). a) b) c) d) Figure 2 a) Álvaro Merceano da Silveira s guitar (source [17]), b) CAD model, c) front view of the model d) side view of the model. In the Portuguese guitar, the soundboard, the back and the sides are glued through a gluing belt that acts as reinforcement as well as being useful in the fixing of the braces. Also in the model used, a grid is used that unites all the braces and the tail block, figure 3. a) b) c) Figure 3 Gluing grid with the braces a) Front view, b) side view c) perspective. 3

4 Afterwards the geometry is exported to the finite elements program (EF) and the modal analysis is performed with the goal of obtaining the natural frequencies and the vibrational modes of the guitar. The obtained frequencies and modal forms for the three first modes, namely modes (0,0), (0,1), (0,2) and respective natural frequencies are compared with the frequencies and the three first vibration modes of the guitar tested experimentally by [4], see figure 1. This process is reproduced interactively, varying, in each iteration, at least one of the following dimensions: gluing belt, harmonic braces of the back or harmonic braces of the soundboard or all at once (three pairs of braces) or the distances between them and the distance to the exact point (position of the bridge) while checking the effect on the modal forms and frequencies. When the deviations relative to the experimental values of [4] are considered acceptable, as far as this work in concerned, the bridge is positioned (protuberance of 17 mm of height by 4 mm of length, in mahogany) on the soundboard, followed by the installation of just one string, and finally the installation of all strings. In the case of the strings the method used is based on the one proposed in [6]: insertion of a pre-tension in the strings by introducing a reference temperature and the variation of the temperature of each string, depending on the desired frequency, followed by a static analysis with pre-tensions and modal analysis of the complete model (with strings). Thereby it is possible to create a tension in each string, so that its natural vibration frequency in the model s complete structure corresponds to the string s natural vibration frequency in the tuned guitar. An iterative process is also used in the case of the installation of the strings, where the results obtained with the model are compared with the experimental results of [4]. The installation of the bridge and the strings produces a downward dislocation of the frequencies and change the vibration modes. In this case, the variation of the distances and dimensions of the braces is not enough to adjust the results of the EF analysis to the values of [4], so, the geometry of the guitar is corrected (by replacing the 205mm radius in the half circumference opposite the arm by an elyptical arch that makes the guitar bulging with 165mm of the exact point to the nut (opposite side of the arm), see figure 4. With this new geometry and with all the components (bridge and strings), a new iterative adjustment is made on the dimensions and distances of the braces, so that it is possible to obtain the frequencies and modes of vibration adjusted to the values of [4]. To speed up the convergence of the iterative process, several thicknesses for the soundboard are tried so as to obtain an estimate of the thickness of the braces. a) b) c) a) Figure 4 Modified CAD model. a) perspective of the guitar with strings, b) front-view, c) sideview. 4

5 The final guitar geometry, external dimensions included, are thus determined by this iterative process. Special attention was given to work [5], which is the source of the material data, see table 1, with the exception of the material used in the strings. In this work, the author begins with pre-determined dimensions, that she keeps constant, and performs the modal analysis, comparing the frequencies obtained with the frequencies in [4], not considering whether these modal frequencies correspond, except in the case of the first one. In this case it becomes relatively easy to find all the coincident frequencies with some modal form. Table 1- Characteristics of the materials (source [5]) Wood Sitka- Spruce Indian Rosewood Density (Kg/m 3 ) E 1 (10 8 E 2 (10 8 E 3 (10 8 G 23 (10 8 G 13 (10 8 G 12 (10 8 ν 12 ν 13 ν Ebony Mahogany In [6], the author placed the strings in the model created and analysed by [5], with previously defined dimensions and the same vibration modes and frequencies found by [5], which he kept fixed, having afterwards adjusted the strings to the nearest hundredth. In this case the two last modes do not correspond to the modes obtained by [4]. Nothing is said regarding the effect of the string placement on the guitar body. The material used to model the strings was a stainless steel AISI INOX 302, already employed by [6], whose mechanical properties are presented in Table 2. Table 2 Stainless steel (AISI INOX 302) properties [6]. Density [Kg/m^3] 7900 Young's modulus [GPa] 193 Thermal Expansion [1/oC] 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: 5

6 m e = ρn! NdV! (3) k e = B! CBdV! (4) f e = N! F v dv! + N! F s ds! (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, F v is the vector of the volume forces and F s 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 + k geom + 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 k geom 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 Brace 1 12x15 12x15 Brace 2 8x16 8x16 Brace 3 8x18 8x Mesh Convergence Between braces 52 Gluing Belt Table 4 presents the results obtained with four meshes, corresponding to the size parameters of the finite elements which vary from 4,49x10!! m to 8,98x10!! m. 6x12 Table 4 Frequencies obtained with different meshes. Mesh/ number of elements Modes 1 st Mesh 2 nd Mesh 3 rd Mesh 4 th Mesh Frequency (Hz) 1 306,59 291,27 283,93 283, ,73 376,13 371,02 369, ,95 552,7 553,85 549,6 Figure 5 presents three meshes, 1 st, 2 nd and 4 th of Table 4, respectively. 6

7 a) b) c) Figure 5 - a) 1 st mesh; b) 2 nd mesh; c) 4 th mesh. In Table 4 we see that the variation between the third and the fourth meshes is not significant anymore, and, therefore, it can be validated as a fairly accurate approximation for the objectives of this work. It is true that the number of elements increases from mesh to mesh, which also corresponds to an increase in computation time; however, since the computation time is not too high (130,8s of CPU time), mesh 4 was chosen for this analysis. 4.2 Modal analysis Now we present the three vibration modes of the guitar, obtained through FE analysis, which better approximate the modes and frequencies obtained by [4]. a) b) c) Figure 6 - Modal forms and frequencies. a) Mode (0,0); b) Mode (0,1); Mode (0,2) Table 5 presents the results and the deviations of the frequencies obtained in the FE analysis, in comparison with the results obtained by [4]. 4.3 Guitar with strings Table 5 Frequencies of modes 1,2 and 3, experimental and obtained through FE. Mode Shapes Frequency (Hz) Experimental FE Analysis Deviation (%) ,04 2, ,62 2, ,6 13,45 At this point the objective is to place and tune the strings of the FE model. The guitar is assumed to be free in space as in the experimental model of [4]. FE analyses are performed with two different models using the mesh with the characteristics indicated in section 4.1. The modal analysis is first performed in the CAD model, presented in the previous section, and afterwards in the modified CAD model (see Figure 4). 7

8 4.4 Mesh When the strings are placed and tuned, the CPU time increases drastically. It is important to note that the strings are tuned iteratively and, therefore, the software has to be run several times in order to tune each string and the whole set. Based on the results presented in Table 6, a mesh with elements was chosen. Table 6 Different meshes (with strings) Mesh/ number of elements Modes 1 st Mesh 2 nd Mesh 3 rd Mesh 4 th Mesh Frequency (Hz) 1 283,78 268,83 266,89 267, ,81 380,76 378,57 377, ,34 511,06 507,67 506,1 Details of the strings and bridge are represented in Figure 7. a) b) Figure 7 Detail of the strings. a) Whole set of strings; b) Strings and bridge. 4.5 Modal analysis of the initial model with bridge and of the initial model with one string Successive FE analysis of the model were undertaken: with bridge and without strings, with one string, and with the whole set of strings, using in the model the dimensions of Table 3. The results are presented in Table 7. Table 7 Frequencies of modes 1,2 and 3, experimental and obtained through FE analysis (initial Mode Shapes model). Frequency (Hz) Initial model Experimental With bridge and one With bridge With all strings string Deviation Deviation FE FE FE (%) (%) 1º ,03 227,96 17,45 193,05 29,8 2º ,7 14,81 268,58 25,39 307,74 14,5 3º ,27 29,25 442,07 30,38 426,56 32,8 Deviation (%) In the modified model, the following FE analysis were carried out: with bridge and without strings, with a 5 mm thick top and with the whole set of strings. As a result of an iterative approximation to the results of [4], the braces 2 (see table 3) of the top and back were changed to 8x18 mm instead of 16x18 mm. In the case of the last column, which corresponds to the guitar with the whole set of strings, the dimensions of braces 2 and 3 (see table 3 and figure 4) of the top and back were changed to 8x25 mm. The obtained results are shown in Table 8. 8

9 Table 8 Frequencies of modes 1,2 and 3, experimental and obtained through FE analysis (modified model) Frequency (Hz) Modified Model With bridge and 5 mm of With bridge (without Mode thickness of the top (without strings) Shapes Experimental strings) With all strings EF Deviation Deviation EF Deviation (%) EF (%) (%) ,29 3,17 276,14 0,41 268,34 2, ,4 2,33 366,12 1,67 380,76 5, ,23 17,9 629,04 0,94 511,06 19,5 Figure 7a) shows string 11 in the fundamental form and the respective tuning frequency rounded to the nearest hundredth. Figure 7b) shows the 3 rd mode of vibration with all the strings attached. The fundamental frequencies of the remaining strings and the two remaining modes of vibration were found in a similar way, see table 9. a) b) Figure 7 a) Frequency and mode shape of the eleventh string. b) All strings on guitar and third modal shape of vibration Table 9 String diameters for the Portuguese guitar and experimental frequencies (source [6]). Frequencies obtained through FEM String Diameter[mm] Frequency values [Hz] FEM Frequency values [Hz] - from [9] B (b4) 0,24 493, ,880 A (a4) 0,25 440, ,000 E (e4) 0,32 329, ,630 B bordão (b3) 0,50 246, ,940 A bordão (a3) 0,64 220, ,000 D (d4) 0,44 293, ,660 D bordão (d3) 0,79 146, , Results and conclusions. For the initial guitar model without the strings, the modes (0,0), (0,1) and (0,2) were obtained in the correct form. The frequencies estimated through FE analysis had deviations of 2,84%; 2,6% and 13,4%, respectively, when compared with the experimental values of [4]. From the dimensions of the braces and the relative distances between them and between the braces and the exact point (where the bridge is placed, see Figure 2 c), only one of the dimensions was changed: the middle braces (close to the hole, Figure 2 c) with 18x8 mm (height x thickness) changed to 18x16 mm. 9

10 In the case of the guitar (modified model) with strings, the correct modes were also obtained and the deviations in the frequencies were of 2,4%, 5,45% e 19,5%, respectively, when compared to the experimental values of [4]. Regarding the geometry, two pairs of central braces were changed, from 18x8 mm to 25x8; furthermore, the guitar was redesigned in order to better approximate the reference model from [7], thus improving the results. The strings were tuned to the nearest hundredth through the application of a pre-tension to each string. The effect of the bridge placement on the two previous cases was also verified. These results support the hypothesis that the behaviour of the guitar can be reasonably well predicted during the design stages, demonstrating that it is possible to quantify the effects caused by changes in the guitar structure, bridge and string placement and string tuning. Acknowledgments to supervisor I would like to thank my supervisor, Professor Miguel de Matos Neves, for the support and advice provided during the writing of this thesis, as well as for the availability to make our schedules compatible for the orientation meetings. References [1] Ervin Somogyi, Some thoughts on the difference between handmade and factory made guitars, [2] Gerald Sheppard,The Question of Handcrafted Versus, Mass-Produced Instruments, 2015, ( [3] Torres T. Jesús Alejandro, Análise modal de la tapa armónica de una guitarra clássica mediante la aplicación de Ansys, Instituto Tecnológico de Querétaro. p ( 2). [4] O. Inácio, F. Santiago and P. Caldeira Cabral, The Portuguese Guitar Acoustics: Part 1 Vibroacoustic Measurements, paper ID: 187 p (2) [5] Raquel Vaz, Modelo de elementos finitos da Guitarra Portuguesa, Universidade Técnica de Lisboa, Instituto Superior Técnico, Mestrado em Engenharia Aeroespacial 2011/12 [6] João P. B. Lourenço and Miguel M. Neves Modal Analysis of a Portuguese Guitar using a 3D Finite Elemente Model with string tension Acoustics Conference 2013 and Sound and Music Computing Conference, pp.1-8, [7] [8] Claus Jurgen Bathe, Finite Element Procedures, Prentice Hall, New Jersey, [9] José Maria Campos dos Santos e Guilherme Orelli Paiva, Vibroacoustic Numerical Analisys of a Brazilian Guitar Resonance Box, 2013 ESSS Conference & Ansys Users Meeting, Atibaia, Brasil, Abril