Experimental Research Regarding the Dynamic Behaviour of the Classical Guitar IOAN CURTU, MARIANA D. STANCIU Department of Strength of Materials and Mechanical Vibrations Transilvania University of Braşov, Adress: B-dul Eroilor nr. 29, Braşov 500036 ROMANIA e-mail: curtui@unitbv.ro, mariana.stanciu@unitbv.ro, RAIMOND GRIMBERG, ADRIANA SAVIN National Institute of Research& Development for Technical Physics of Iasi Address: B-dul D. Mangeron nr. 47, Iaşi, 700050, ROMANIA e-mail: grimberg@phys-iasi.ro, asavin@phys-iasi.ro Abstract: This paper presents the results of some experimental work regarding the dynamic behaviour of the classical guitar manufactured by SC Hora SA Reghin Romania. From the different types of guitars made by the Romanian factory, the guitar with 5 radial braces was the one analyzed. To establish the influence of the neck on the guitar stiffness and its dynamic response, the acoustic box (without the neck and the bridge) and the guitar body (with the neck) were investigated. Chladni figures, frequency responses, power spectrum density and dynamic mass were collected and compared in both cases. This research is useful to improve the acoustic quality of a guitar from the point of view of construction optimization. Key-Words: Chladni figures, frequency response, guitar body, stiffness, power spectral density 1 Introduction The acoustic quality of a guitar depends on the physical, mechanical (elastic) and dynamic characteristics of the wood species used in the construction of the different components of the guitar, namely: the geometry and the sizes of the elements, the density, the moisture content, the Young s modulus, the stiffness modulus, the shear resistance, the bending strength, the sound velocity in wood, the acoustic impedance, the acoustic radiation, the quality factor [3], [5], [9], [11]. Between the component parts of guitars (the neck, the plates from the acoustic box, the bridge, the strings) there exists a power connection which assures the static, the dynamic and the acoustic equilibrium of the musical instrument as it shown in Fig. 1. Fig. 1. The mechanical structure of the guitar [4] During the guitar s manufacturing, each added element can change the stiffness and, implicitly, the dynamic behaviour of the instrument [10], [12]. These structural modifications can improve or not the guitar s tone. 2 Previous work Many theoretical and experimental studies on different types of guitars were investigated from numerous points of view: physical, psychoacoustical, mechanical, mathematical, musical, and the problems are still open. Due to the complexity of the phenomena of stringed instruments, each researcher dealt with the aspects which he considered relevant to him. Some of them, like Elejabarrieta, Ezcurra and Santamaria (2004, 2007), performed the modal analysis and the vibration behaviour on the classical guitar in its different construction stages using FEM [7], [8]. Others Bernard Richardson, Howard Wright with some colleagues from Cardiff University and Ra Inta from University of New South Wales tried to establish the influence of the individual components, monitoring the quantitative and qualitative contributions [10], [20], [13], [14]. Paul M. Shaheen (2004) focused on the ISSN: 1790-5095 53 ISBN: 978-960-474-061-1
sensitivity analyses of the folk guitar simulating the building process [15]. Serghei Vladimirovici (2006) researched the influence of the Russian strutting system on the dynamic behaviour of the plates and managed to improve the quality of the sound of the guitar according to the design and the dimensions of the fan bracing [19]. For Becache, Chaigne, Derveaux and Joly (2004) the simulation coupling between the inner fluid and the structure represented one of the theoretical studies [2]. Stanciu, Curtu, Itu (2008) focused on the modelling of the different types of classical guitar manufactured in their country and on analysing the dynamic behaviour of the plates as free structures and as parts of the guitar body [6], [16], [17], [18]. Bader (2005) brought into focus the physical sounds of the instruments from a musicological point of view. He approached the difference between continual and discrete mechanics, elaborating an interesting theory about impedance of second order to solve the problem of overtones [1]. 3 Experimental Analyses The present paper deals with the scientific study of music. The work aims to analyze the influence of the guitar neck on the frequency response of the acoustic box. 1-frequency generator 2-amplifier 3-vibration mass 4-force transducer 5-sample/structure 6 -accelerometers 7 spring support 8 conditioner 9 DAQ 10 personal computer to displaying and processing data Fig.3. The block diagram representation The signal capture and display were achieved through the program developed in LabVIEW, and the data obtained under a numerical form were processed using Microcal Origin [12]. To determine the Chladni figures, the top plate of the guitar body was covered with a thin uniform layer of sand having a 100-150 grit size Case 1 Case 2 Fig. 2. The investigated structures Firstly, the guitar body without its neck was investigated (Fig. 2. and, secondly, the entire guitar (Fig. 2.. In both cases the bridge was neglected. Under analysis was the classical guitar with 5 radial braces on the top plate. The method used was performed under standardized conditions. An artificial simulation of a string vibration was created by means of the B&K 8210 mini-shaker. The experimental stand was built according to the scheme in Fig. 2 and for each sample. The captured signals were amplified and filtered through a conditioner (8), then were sent to the acquisition device (DAQ) (9) which was connected to the computer (10). c) Fig. 4. The accelerometers layout 3 Results and Discussion 3.1. Chladni patterns The Chladni figures in both cases of the structures were obtained for each exciting frequency as shown in Table 1. It can be noticed that nodal lines formed on the top plate are different from the ones formed on the back plate, regardless of the analyzed structure. This is due to several factors: the plate material (the top plate made of spruce, the back plate made of maple), the strutting system (the soundboard with 5 radial braces and the bottom plate with 3 transversal ribs), the presence of the sound hole on the top versus the full back plate, the contact between the ISSN: 1790-5095 54 ISBN: 978-960-474-061-1
excitation source and the soundboard. Comparing the simple acoustic box with the guitar body, it can be remarked that the Chladni figures of the top plates differ, in spite of the similar conditions of the test (boundary conditions, materials, value of the periodical force, frequency). The neck attached to the guitar body, in the second case, is the only different element. The increased stiffness of the structure due to the presence of the neck leads to the increase of the frequency that produced the same modal shape (from 110 Hz, in first case, to 246 Hz, in the second case). The presence of the neck produced a different dynamic behaviour of the back plate. Table 1. Chladni figures of the simple acoustic box and of the guitar body - selection 110 329 413 C A S E 1 compared to the dynamic behaviour of the guitar body without its neck. Fig. 5. The vibrations plot of the top plate of the guitar Fig. 6 a and b display the time histories of the soundboard ( and of the back plate ( for both of the analyzed cases. Under the same input conditions, the structures responded differently. The amplitudes of the guitar plates are higher than the magnitude of the acoustic box. In both cases the nonlinearities are not present. C A S E 2 3.2. The Fourier analyses The main results of Fourier analyses are represented in Figs. 5-12. Fig. 5 shows the time evolution of the guitar soundboard captured at different points: A 1, A 2, A 3 A 7. For the present plot, the lower amplitude was recorded by the A 3 and A 7 accelerometers placed near the impedance head (A 3 ) and on the fret (A 7 ). The highest amplitudes of the soundboard were obtained in the area between the bridge and the sound hole. The same fact was noticed using the finite elements method in our previous studies [6], [16]. As symmetrical standpoints, the measurements from the A 2 and A 4 points revealed the same values of amplitudes of symmetrical areas, but opposite-phase movements. The modifications of the system dynamic behaviour due to the presence of the neck were Fig. 6 Comparison between the vibration output of the considered structures In Fig. 7 a and b the frequency responses of the top plates at 196 Hz, for cases 1 and 2, are illustrated. In ISSN: 1790-5095 55 ISBN: 978-960-474-061-1
case of the acoustic box without its neck, overtones appear. Contrary, with the increasing of the structure rigidity, the guitar body resonates at excitation frequency, amplifying that frequency. The capacity of the top plate to filter and amplify the desired frequency represents one of the criteria for the acoustic quality of a guitar. In accordance with some luthiers, the harmonics are given by the back plate and by the coupling between the solid parts and the inner fluid of the guitar body. In Fig. 7 c, the frequency response of the points around the sound hole was analyzed. Numerous overtones could be noticed. The internal fluid vibrates, produces and absorbs all resonance frequencies and harmonics. The effect of this phenomenon is noticed mainly around the sound hole. Another analysis of the frequency response is the power spectral density displayed in Fig. 8 a, b, c and d. c) c) Fig. 7 The frequency spectrum d) Fig. 8 The power spectral density ISSN: 1790-5095 56 ISBN: 978-960-474-061-1
Comparing the plot of energy dissipation in cases of different stiffness (Fig. 8 a free plate with 5 radial braces, b acoustic box of the guitar with 5 fan bars on the top plate, c guitar body with neck), a various dynamic behaviour of the structures can be remarked due to numerous factors. One of them is the increasing of rigidity as a result of added elements. In the case 2 - the guitar, the nonlinearities of energy distribution lack or are minimal. The back plate (Fig.8. d) has a similarly curve of energy dissipation like soundboard, with mention that it vibrates in opposite phase (Fig. 8 d). 4 Comparison with FEM Results In previous research, the dynamic behaviour of the plates and of certain structures made of plates was analyzed using the finite element method (FEM). structures, boundary conditions, strutting system). The numerical results regarding the vibrations of the classical guitar neck are presented in Fig. 9. Due to the difficulty of measuring the vibration modes of the neck, modal analyses were performed, using the FEM. As a result of the experimental and numerical investigations, it could be noticed that, due to its dimensions, its shape and its stiffness, the neck of the guitar vibrates at low frequencies (under 80 Hz). From a musical point of view, the neck influences the acoustics of the sounds at low and medium frequencies. From a mechanical point of view, the guitar, as a complex structure, is exposed, during vibrations, to complex stimuli (bending, traction, torsion). These aspects could be emphasized mainly by the finite elements method, usage being made of the field of tensions, deformations and displacements. 5 Conclusion The paper wanted to establish the way in which the increase of the general stiffness of the guitar (due to the presence of the neck) influences the dynamic behaviour of the acoustic box. Under analysis were only particular guitar structures / plates and for this, the results cannot be generalized. The contributions of this study aim at finding the manufacturing and technological solutions for the improvement in the acoustics of the musical instrument made by SC Hora SA Reghin. [21]. Being a complex structure, both as shape, materials and from the point of view of the mechanical phenomena, by making use of the experimental method, the simplification of the real conditions was sought, which lead to obtaining some particular results. The experimental results widely validate the theoretical ones. The combination of the two methods allowed the outlining of several more phenomena specific to vibrations. At present, the acoustic quality of a guitar in its real functioning conditions is under investigation. Fig. 9 Modal shapes of the guitar obtained with FEM Similar conditions were simulated (materials, density, Young modulus, thickness, types of 6 Acknowledgments This work was accomplished under the following grants: PNII 71-0161/2007 project manager: Prof. Dr. Grimberg Raimond, INCDFT Iasi, scientific responsible P3 Prof. Dr. Eng. Curtu Ioan, Transilvania University of Brasov and TD cod 182, no. 222/2007, project responsible: Ph.D. Eng. Stanciu Mariana Domnica. ISSN: 1790-5095 57 ISBN: 978-960-474-061-1
We are also grateful to the Technical Staff of S.C. HORA S.A. Reghin Romania for the logistic support. References: [1] Bader, R., Computational Mechanics of the Classical Guitar,Springer-Verlag Berlin Heidelberg N.Y., ISBN 3-540-25136-7, in Netherland, 2005. [2] Bécache, E., Chaigne, A., Derveaux, G., Joly, P., Numerical simulation of a guitar. Computers and Structures, Vol 83, 2005, pp. 107 126 [3] Bucur, V., Acoustic of Wood. Springer-Verlag Berlin Heidelberg New York, 2006 [4] Curtu, I., Stanciu, Mariana, Itu, C., Grimberg, R., Numerical Modeling of the Acoustic Plates as Constituents of Stringed Instruments, in Proc. of the 6 th International Conference of DAAAM Baltic Industrial Engineering, ISBN 978-9985-59-783-5, Tallinn, Estonia.24-26 th April 2008, p. 53-58. [5] Curtu, I., Stanciu, M. D., Grimberg, R., Correlations between the Plate s Vibrations from Guitar s Structure and Physical, Mechanical and Ellastically Characteristics of the Composite Materials, Proceeding of the 9 th WSEAS Int. Conf. on Acoustic & Music: Theory and Applications (AMTA 08), ISBN 978-960- 6766-74-9Bucharest Romania 24-26 June 2008, pp. 55-60. [6] Curtu I., Stanciu M. D., Savin A., The propagation of forced vibrations in coupled plates of guitars, Proceeding of the 19 th International DAAAM Symposium "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions" ISSN 1726-9679, Trnava, Slovacia 22-25 Octombrie 2008, pp. 345-346. [7] Ezcurra, Amaya 1, Elejabarrieta, M Jesús Santamaría, C., Internal Fluid Influence on the Dynamic Behavior of the Resonance box of the Guitar, http://www.sea-acustica.es [8] Elejabarrieta María Jesús, Ezcurra Amaia; Santamaría Carlos, Resonance Box of the Guitar Structure-Fluid Interaction, Proceeding of 19th International Congress on Acoustics Madrid, 2-7 september 2007, http://www.seaacustica.es [9] Haines, D., The essential mechanical properties of wood prepared for musical instruments. Catgut Acoustic Society Journal, 2000, Vol 4(2):20-32. [10] Inta Ra, The acoustics of the steel string guitar, PhD Thesis, The University of New South Wales, Australia, September, 2007. [11] Rossing, T., Fletcher, N., Principle of Vibration and Sound second edition. Springer Science, New York, 2004. [12] Rosca, I. C., Vibratii mecanice, Ed. Infomarket Brasov, ISBN 973-8204-24-0, 2002. [13] Russell, D., Modal analysis of an Acoustic Folk Guitar, PhD., Applied Physics, Kettering University, 1998. [14] Russel, D. and Paul Pedersen, Modal Analyses of an Electric Guitar, Kettering University January 29, 1999 http://www.kettering.edu/~drussell/guitars, [15] Shaheen, P. M., Sensitivity Analysis of the Natural Frequency and Modal Effective Weight of Mode (0,0) of the Top and Back Plate of an Acoustic Steel-String Guitar using FEM, PhD Thesis, 2004. [16] Stanciu, M. D., Curtu, I., Itu, C., Influence of strengthening bars of guitar s plates on the normal modes of vibrations using, Proc. of the 19 th International DAAAM Symposium "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions" ISSN 1726-9679, Trnava, Slovacia 22-25 Octombrie 2008, pp.1295-1296. [17] Stanciu M. D., Cretu N. C., Rosca I. C., Curtu I., Experimental Research Regarding the Underdamped Free Vibration of Lignocellulose Plates from Guitar's Body, ProLigno ISSN 1841-4737., vol. 4, Nr. 4-2008, pp. 41-54. [18] Stanciu, Mariana, Curtu, I.., Itu, C., Grimberg, R., Dynamical Analysis with Finite Element Method of the Acoustic Plates as Constituents of the Guitar, ProLigno, Vol. 4, No. 1, March 2008, pp. 41-52. [19] Vladimirovici, S. Calculation Method for the Component Elements of Guitar, PhD Thesis. Technical State Institute Marii, 2004. [20] Wright, H., The Acoustics and psychoacoustics of the guitar. PhD Thesis. University of Wales, College of Cardiff. 1996. [21] *** Technical Documentation of S.C. Hora S.A. Reghin Romania. ISSN: 1790-5095 58 ISBN: 978-960-474-061-1