Correlations between the Plates Vibrations from the Guitar s Structure and the Physical, Mechanical and Elastically Characteristics of the Composite Materials IOAN CURTU MARIANA DOMNICA STANCIU Department of Strength of Materials and Vibrations University Transilvania of Brasov Address: B-dul Eroilor nr. 29, Brasov, 500036, ROMANIA. RAIMOND GRIMBERG National Institute of Research& Development for Technical Physics of Iasi Address: B-dul D. Mangeron nr. 47, Iaşi, 700050, ROMANIA Abstract: Due to the acoustic, aesthetic and processing properties (workability, finishing, joints) the wood and the ligno-cellulose composites are the most valued materials for the musical instruments construction. The guitar is made up of a complex structure, formed by a vertical wall in a curve shape (technologically named sides ) and two faces made up of ligno-cellulose plates, so that it should be formed an acoustic box which amplifies the vibrations of strings. The acoustic quality is determined by the contribution of each element (shape, dimensions, the nature of the used material, the characteristics of the excitation forces etc). This paper focuses on dynamical behaviour of every element and the whole structure. Firstly, the plates were considered simple, with different inner ribs, with various thicknesses and variable elasticity. Secondly, the plates were modelled as an assembly forming the body of guitar. It had been analyzed the structural behaviour of the plates under free and forced vibrations and also the acoustic box. The modelling with FEM permitted the emphasizing of some aspects, such as: the plate s and the box s normal modes of vibrations, the magnitude of the vibrations amplitudes, reaching the resonance frequency at different values from the acoustic domain of the structure etc. The results obtained had been compared to the ones obtained by other researchers with analytical, numerical and experimental methods. The obtained data refer to the Romanian guitars produced at S. C. Hora S. A.Reghin. Key-Words: finite element method (FEM), normal modes, cyclic stresses, resonance wood, modal shape 1 About the wood s acoustic qualities The acoustic quality of a mechanical structure is intrinsically linked to the material s properties. The acoustical characteristics of the wooden material from the plates structures are influenced on the one hand by the elasticity of the material along and perpendicular to the fibres and on the other hand depend on the internal friction phenomena, caused by the energy of vibrations [13], [14], [15]). The wood s behaviour under the acoustic waves depends on the one hand on the sound energy which makes contact with it and on the other hand it depends on the quality of the wooden material, respectively on the macro and microscopic structure of the wood, the wood s humidity, the temperature of the wood, the elastically properties, and it also depends on the direction of wooden structure in accordance with the sound source (longitudinal, transversal, radial, tangential, complex) [2], [4], [7], [9], [12]. The vibrations produced in the wood the internal frictions. These modify the sound energy forming the resonance phenomenon and also produce heat, due to the intermolecular energetic exchanges. The most important acoustic property of the wood is the reception capacity of the sounds with similar or identical frequency with the frequency of its membranes [7]. Depending on the period of external frequency of excitation, the ISBN: 978-960-6766-74-9 55 ISSN 1790-5095
resonance appears when the pulsation of the forced vibration goes near the normal frequency vibration of the wood [5] [17]. 2 The structures modelling with FEM This paper s purpose is to establish the ways in which the structural and the material s characteristics influence the plates resonance frequency as independent structures and also as incorporated structures in the amplification box. For the analysis of the dynamical behaviour of the plates and of the ligno-cellulose acoustic box, with different structures, thicknesses, densities and modulus of elasticity it was used the finite element method (FEM). First, there were geometrically modelled 4 versions of acoustic plates: the simple plate, the cross braces plate, the 3 top-braces plate and the 5 top-braces plate (Fig. 1), taking into account the dimensions used in the musical instruments factory S.C. Hora S.A. Reghin, Romania [3], [4], [21]. aimed the clamped-edge plates. The program was run for different parameters: thicknesses (h=1.5, 2, 2.5, 3 and 3.5 mm), modulus of elasticity (E= 10000, 12000 and 14000 MPa), density (ρ= 350, 400, 450 and 500 kg/m 3 ), Poisson s number remaining constant (υ =0.36) and shear modulus (G=5000 MPa). The values of these parameters were taken from the specialized literature [2], [5]. 3 The results of the plates modelling As a result of the analysis with FEM numerous outcomes were obtained: modal shapes of the normal modes of vibrations and from these are presented the first 5 modes in Fig.3, values of the frequencies f 1...10 and values of the amplitudes for the first 10 modes, which were analyzed and emphasized through figures and charts. f=191.65 Hz f=210.22 Hz f=262.1 Hz f=264.17 Hz Fig. 3 The modal shapes for mode (0,0), for E=14000 MPa, h=2.5 mm, and ρ= 450 kg/m 3 Fig. 1 The versions of the modelled plates Then it was modelled the guitar s body incorporating the versions of the plates presented above (Fig. 2). It could be noticed that the normal frequencies for the mode (0,0) have higher values as the plate s stiffening system is more complex [18]. The acoustic field is between the sound hole and the bridge, varying in magnitude and intensity, depending on the number and the position of the top braces in the analyzed cases (Fig.3) [11], [16]. Table 1 Comparison between the results obtained by different authors [15] Bécache 2005 Stanciu 2008 Wright 1996 Vladimirovici 2004 Fig. 2 The modelling of the guitar s body in different constructive versions For dynamical analysis with FEM there were used shell type elements with 4 nodes (QUAD4) using Patran Nastran 2004 package. The limit conditions ISBN: 978-960-6766-74-9 56 ISSN 1790-5095
Comparing the obtained results to the ones from the specialized literature (Table 1), it can be noticed that there are a lot of similarities regarding the modal shapes [1], [15], [19], [20]. In terms of the frequencies values, the differences that appear are due either to the numerical method that was used, or to the set properties of the material, or to the type of the analyzed guitar (plate). Frequency [Hz] 300 200 100 0 126 117 111 105 ρ=350 kg/m3 ρ=400 kg/m3 ρ=450 kg/m3 ρ=500 kg/m3 1.5 2.0 2.5 3.0 3.5 Thickness of plates [mm] 277 259 244 232 Fig. 4 The frequencies variation for the first normal mode of vibration according to the plate s thickness The fundamental frequencies values and the values of the higher harmonics are influenced by the density, modulus of elasticity and thickness (Fig. 4, 5 and 6). The normal frequencies increase along with the increasing of the plates thickness (Fig.4). Fig. 6 The variation of the fundamental frequencies with the increasing of the modulus of elasticity 4 The results of the acoustic box modelling The resonance wood used in the building of the Romanian classical guitars has the following properties: reduced density, the sound s high velocity in wood, high acoustic radiation, high modulus of elasticity, low logarithmic decrement (internal friction) endwise, relatively low values of the other elastic constants. [2], [7], [21]. For establishing the frequencies and the vibration modes of the plates from the guitar s body there were modelled 4 types of acoustic boxes (Fig. 2), the program runs for the following parameters: E=10000, 12000 and 14000 MPa, ρ=400, 450 and 500 Kg/m 3, h=2.5 mm, ν=0.36 and G=5000 MPa. Plate Top plate Back plate 1 (0,0)) (0,0) (0,0) Fig.5. The variation of the fundamental frequency a the harmonics for the plate with 5 top braces The lighter is the wood (has low densities 350 400kg/m 3 ), the higher is the capacity of vibration under high frequencies (Fig. 5). Along with the increasing of the plates material density, the frequencies decrease with almost 16 %, the differences being obvious between the higher harmonics. The increasing of the modulus of elasticity of the plate s material leads to the increasing of the fundamental frequency, regardless of the plate s structure (Fig. 6). 2 3 4 (2,0) (0,0) (0,0) (0,2) (0,0) (2,0) (3,0) (0,0) (2,0) Fig. 7 Comparisons between the first four normal modes of vibrations ISBN: 978-960-6766-74-9 57 ISSN 1790-5095
The modal shapes for the plates analyzed as individual structures and also as subassemblies of the sound box differentiated as the stiffening system is more complex (0, 1, 3 or 5 ribs). The face and the back of the guitar s body have modal shapes and different vibration frequencies, as it can be seen in the Fig. 7 and Fig. 8. As an effect of the bracing of the sound box s faces with the sides, it appears the standing-waves through the overlapping of the direct waves over the reflected ones [6], [10]. These ones generate the higher harmonics (Fig. 8). 5 6 7 8 Plate Top plate Back plate (2,3) (0,2) (0,2) (2,2) (2,0) (2,0) (4,0) (0,2) (0,2) Mod 1 2 3 4 7 Fig.9 The moving in phase and opposite phase of the plates from the structure of the simple box From the Fig. 10 it can be noticed that the first frequency that the two plates go into opposite phase for different structures of the box is the proper one for mode 3 and 4. This phenomenon, named breathing mode appears at almost low frequencies for the boxes with a great number of stiffening elements [10]. The increasing of the number of stiffening elements in the box s structure doesn t influence so much the instrument s acoustics. Their presence has an increasing role of the structure s reliability, so that the most common bodies of the classical guitar should contain plates with cross braces and with 3 radial braces. As opposed to the plates, along with the increasing of the elasticity of the acoustic box s faces, the normal frequency of oscillation decrease (Fig. 11). 3 448 Hz 3 563 Hz 2 289 Hz 2 290 Hz (5,3) (2,0) (3,0) Fig.8 Comparison between the normal modes of vibrations of the plate with 3 top braces and of the box with the same type of plate An interesting phenomenon regarding the structural analysis of the plates from the guitar s body is represented by the vibrations in phase and opposite phase of the face and the back. The phenomenon of standing-wave propagation in the individual and assembled plates begins with the face and back plate in phase, the face plate having higher amplitudes. Then it is produced the alteration of phase of the plates vibration, these ones reaching the point of opposite phase (Fig. 9). This process is restarted for the higher modes (1,1), (0,2), (2,0), (2,2). From the composition of these structural and aero-acoustic processes it is produced a variety of the sound quality and of the frequencies ranges which the acoustic box is able to amplify. Fig. 10 Comparisons between the ways of producing the opposite phase moving of the constituents of the box Comparing the dynamical behaviour of the individual plates to the ones of the acoustic boxes, it can be seen that the plates assembly into the structure of the amplification box goes to the decreasing of the normal frequencies distances and also to a linear variation of the harmonics (Fig. 12). ISBN: 978-960-6766-74-9 58 ISSN 1790-5095
Fig. 11 The variation of the normal frequencies depending on the structure of the box This has a practical importance because it could be noticed that in terms of acoustics and musical, the low and medium frequencies (between 80-380 Hz) are responsible for the sound s radiation through the structure of the guitar. Fig. 12 The influence of the plates assembly in the acoustic box upon its normal frequencies The influence of the modulus of elasticity E upon the normal frequencies is similar for the plates and for the resonance boxes, as it can be seen in the Fig.8 (the frequencies increase along with the decreasing of the modulus of elasticity). The plates structure and the way in which these are assembled or not in the guitar s body correlated with the material s elasticity produce natural frequencies with different values: lower with 10-15 % in the case of acoustic boxes than the free plates (Fig. 13). 5 Conclusion In this paper it was done a structural analysis of the plates and of the guitar s body seeking to establish the correlations between the free vibrations and the mechanical, elastically and physical characteristics of the analyzed structures. The modal shapes for different cases of structures revealed the elastically behaviour of the plate, respectively the areas in which are produced the nodes and the antinodes for different modes of vibrations. Fig. 13 The influence of the modulus of elasticity upon the fundamental frequencies of the plates and the acoustic boxes One of the most important characteristics of the stringed instruments is the possibility of resonating at the excitation frequency of the strings. In this respect the amplification box made up of lignocellulose plates takes over this function, therefore it was necessary the analysis of the normal modes of vibration, for the plates as individual structures and also as subassemblies of the acoustic box. The acoustic field of the structures of plates and boxes is outlined between the bridge (the application point of the force) and the acoustic hole. From the analysis of the dynamical behaviour of the plates it could be noticed that there is a series of factors that influence the normal modes of vibration of the structures from ligno-cellulose plates: the plates structure (shape, thicknesses, dimensions, back braces), the material (elasticity, density, damping factor of wood), the plates assembly in the structure of the acoustic box. Generally the fundamental frequency has similar values regardless of the plates structure (the layout of the back braces); the plates with the density 350 kg/m 3 have higher frequencies than the ones with the density 500 kg/m 3, the difference between the extreme frequencies at the first mode of vibration is 35 Hz and at the mode 10 has the value of 187 Hz. This it reflects in the acoustic quality of the musical instruments. The increasing of the density leads to higher differences between the frequencies of the higher harmonics. The plates with low frequencies have the sound power higher, a phenomenon obtained through the usage of the plates with high density (450-550 kg/m 3 ). Knowing the normal modes of vibration and the frequencies corresponding to them represents a theoretical and practical importance in terms of the resonance phenomenon of the plate, of the way in which materials with different properties can be ISBN: 978-960-6766-74-9 59 ISSN 1790-5095
valued in the acoustic field and not only. The acoustic quality of the plates influences the acoustic performances of the guitars. The experimental investigations regarding to the dynamical and acoustical characteristics of the plates structures will emphasize the concordance between the results of the numerical modelling and the real ones. The researches will continue with the experimental investigations of the acoustical and structural behaviour of the integrated plates in the sound box of the classical guitar. References [1] Bécache, E., Chaigne, A., Derveaux, G., Joly, P., Numerical simulation of a guitar. Computers and Structures Vol 83, 2005, pp. 107 126. [2] Bucur, V., Acoustic of Wood. Springer-Verlag Berlin Heidelberg New York, 2006. [3] Cotta, N., Proiectarea si tehnologia fabricarii produselor industriale din lemn (Design and Technology of Industrial Wood Products). Editura Didactica şi Pedagogicǎ, Bucureşti, România, 1983. [4] Curtu, I., Stanciu, M., BABA, M., Aspects regarding to the influence of the anisotropy on the statically modeling of the guitar, Proceedings of the 2nd International Conference Computational mechanics and virtual engineering Brasov, Romania, 2007. [6] Gordon, W., Towards a physical model of the guitar, PhD Thesis, University of Wales, Cardiff, Wales, 1991. [7] Haines, D., The essential mechanical properties of wood prepared for musical instruments. Catgut Acoustic Society Journal, Vol 4(2):20-32, 2000. [8] Ra Inta, The acoustics of the steel string guitar, PhD Thesis, The University of New South Wales, Australia, September, 2007. [9] Rossing, T., Fletcher, N., Principle of Vibration and Sound second edition. Springer Science, New York, 2004. [10]Russell, D., Modal analysis of an Acoustic Folk Guitar., PhD., Applied Physics, Kettering University, 1998. [11]Shaheen, P. M., Sensitivity Analysis of the Natural Frequency and Modal Effective Weight of (0,0) of the Top and Back Plate of an Acoustic Steel-String Guitar using FEM, PhD Thesis, 2004. [12]Shigeru Yoshikawa (2007) Acoustical classification of wood for string instruments, The Journal of the Acoustical Society of America, Vol. 122, nr.1, pp: 574-580, 2007. [13]Stanciu, M., Curtu, I. Aspects concerning the mechanical structures on the classical and acoustic guitars, Proceedings of ICWSE 2007, pp. 438 445. [14]Stanciu, M., Curtu, I. Researches concerning the acoustic properties of tone wood used on musical instruments structures. In: Procc. of 11 th International Research / Expert Conference Trends in the Development of Machnery and Associated Technology 2007, 807-810. [15]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. [16]Torres Torres, J., A., Boullosa, R. Ruiz, (2006), Obtención de modulos de elasticidad para simular tapa real de guitarra mediannte MEF, Originalmente publicado en el XXI Congreso de la Sociedad Mexicana de Instrumentación, 2006 Ensenada B. C. Mexico. [17]Urma, Dem.: Acoustic and Music, Scientific and Encyclopedic Printing House, Bucharest, 1983. [18]Vernet, D., Influence of the guitar bracing using Finite element method, Technical Report The University of New South Wales and Ecole Normale Superiore, august 2001. [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. Acknowledgement 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, University Transilvania Brasov, TD cod 182, no. 222/2007, project responsible: Ph.D. Eng. Stanciu Mariana Domnica. Also we are grateful to the Technical Staff of S.C. HORA S.A. Reghin Romania for the logistic support. ISBN: 978-960-6766-74-9 60 ISSN 1790-5095