The propagation of forced vibrations in coupled plates of guitars.

Curtu, Ioan ; Stanciu, Mariana ; Savin, Adriana 等

1. INTRODUCTION

Dynamical behaviour of guitar is linked by the kinetic energy of the strings.

The transfer of the vibration energy of the strings is done through the solid system ([s.sub.1]--string)--solid ([s.sub.2]--bridge)--solid ([s.sub.3]--sound board)--fluid ([f.sub.1]--the air from the cavity)--solid ([s.sub.4]--back)--solid ([s.sub.5]--side)--fluid ([f.sub.1]--the air from the cavity and the sound hole). The speed emission of acoustic waves through solid (string-bridge-plate-box) is higher than the one through air, therefore the mechanical structure of the instrument influences the acoustic quality (Curtu, 2008).

In this paper we analyzed which are the areas of plates with maximum and minimum amplitudes, how the vibrations are propagates in wooden structures and the radiation fields of plate.

2. LITERATURE OVERVIEW

Description of acoustical and structural guitar modes is far extensive but each reference depicts certain approaches in this field. Much of the researches in musical acoustic carried out at the University of Wales in Cardiff where the finite element analysis were used by Roberts (1986), Wright (1996). Recently studies were made up by Paul M. Shaheen (2004), Becache, Chaigne, Derveaux and Joly (2004), Bader (2005), Serghei Vladimirovici (2006), Elejabarrieta, Ezcurra and Santamaria (2007). Compared to the work of mentioned references, we modeled the classical guitar bodies according with the Romanian typical bracing patterns and with materials properties of Romanian resonance spruce.

3. DYNAMICAL BEHAVIOR ANALYSIS

3.1 The modelled structure

First, there was geometrically modeled a classical guitar's body incorporating the 3 top-braces plate taking into account the dimensions used in the musical instruments factory S.C. Hora S.A. Reghin, Romania. For dynamical analysis with FEM there were used shell type elements with 4 nodes (QUAD4), being very helpful the Patran Nastran 2004 package. The structures were clamped in the area of neck and body jointing. The program was used for the parameters: thicknesses h=2.5 mm, modulus of elasticity E=14000MPa, density [rho] = 450 kg/[m.sup.3], Poisson's coefficient [upsilon] = 0.36, damping factor [delta]=0.02 and shear modulus G=5000 MPa. The values of these parameters were taken from the specialized literature in this domain.

3.2 The load case and frequency response

It was applied a unitary force (F=1 N) in the node P1 placed on the bridge area of the soundboard. The force varied in the frequency range between 0 to 1000 Hz. The dynamic response was analyzed in the nodes placed on the top plate, on the back and on the sides.

3.3. Results and discussion

A series of quantitative and qualitative aspects regarding to the dynamical behaviour of acoustic box are presented in Fig. 1, 2, 3, 4, 5, and 6.

The amplitudes decrease in the modes further from the force's point of application on the cross axis of the plate, which, on the one hand, assumes that the vibration is transmitted longitudinally, being favored by the elastically characteristics and on the other hand it is necessary the assurance of a width big enough for the vibrations to radiate on a useful optimum surface (Fig. 1). With increasing the distance between the point of application force and a certain point of structures, the nodes of plate resonate at a higher frequency. Compare to the frequency response of the top, the vibration amplitude and the resonance frequency in the back is different. First, it can be noticed that the point P1 on the back which is corresponding with the point of application force on the top, has the highest amplitude at the frequency 280 Hz (Fig. 2). Secondly, the variation curves of amplitude versus frequency for the points of the back plate are similarly as shape (Fig. 2). Third, emplacement of the internal struts, the thickness and the pattern increase the mass and the stiffness of structures which conduct to the lowest frequency and modification of the longitudinal flexural modes.

Fig. 3 shows the comparison between the amplitudes of five points placed on the soundboard, back and sides. It can be noticed that the top plate vibrate in a large band of frequency. The common frequency for all parts of acoustic box is around 220 Hz, value obtained by Wright (1996) too.

[FIGURE 1 OMITTED]

The coupled plates in the acoustic box vibrate at the lower frequencies compared to the free plates (Fig. 4). The normal frequency has close value with the resonance frequency which means that the plate under the forced vibrations resonates in the same time with fundamental frequency. The eigenfrequencies occurring in the assembled instrument do not correspond to the eigenfrequency of the separate parts of the guitar, aspect noticed by Bader (2005) too. The study on vibration behaviour of the guitar box demonstrated the coupling between top and back plates through the air (Fig. 5).

[FIGURE 2 OMITTED]

[FIGURE 5 OMITTED]

Due to the initial boundary conditions, first the top and the plate is moving in phase; then they move out of phase to alternately increase and decrease the cavity volume as shown in Fig. 5 and finally the travelling of vibration in the plates is repeated at a higher frequency and with more nodal and anti-nodal areas.

4. CONCLUSION

The frequency response of the ligno-cellulose plates with different structure and material characteristics offers useful information about the way in which the frequency of the excitation force overlaps with the plate's normal frequency. The modal shapes show the elastically behavior of the plate, respectively the areas in which are formed the node and the amplitude loops for different vibration modes. One of the most important characteristics of the stringed instruments is the possibility of resonating at excitation frequency of the strings. On this line, the amplification box made up of the ligno-cellulose plates takes over this function, being the reason of the necessity of analyzing the frequency response of the plates with FEM. The resonating domain of the plates takes shape between the bridge (the force's application point) and the sound hole. From the analysis of the dynamical behavior of the plates it can be noticed that there are a series of factors which influence the frequency response and the propagation of waves in plates: the plates' structure, the material's features, the magnitude and the application point of the force. The dynamical analysis of the plates aimed just the structural analysis, the acoustical one being supposed to be researched in the future.

This study is a part of a wider research whose purpose is optimizing structures in order to improve the acoustic performance of classical guitar. The future work consists of experimental research on the frequency characteristics of the plate's structures.

ACKNOWLEDGEMENT

This paper is supported by Romanian Ministry of Education and Research under Research of Excellence Program--Contract PNII 71-016/2007. This work was accomplished under the technical support of Transilvania University of Brasov, Romania.

5. REFERENCES

Bader, R. (2005). Computational Mechanics of the Classical Guitar, Springer-Verlag Berlin Heidelberg N.Y., ISBN 3540-25136-7, in Netherland

Becache, E., Chaigne, A., Derveaux, G., Joly, P. (2005). Numerical simulation of a guitar. In: Computers and Structures, 83,. 107-126.

Curtu, I., Stanciu, Mariana, Itu, C., Grimberg, R., (2008) Numerical Modeling of the Acoustic Plates as Constituents of Stringed Instruments, in Proceedings of the 6th International Conference of DAAAM Baltic Industrial Engineering, 24-26th April 2008, ISBN 978-9985-59-783-5, p. 53-58, Tallinn, Estonia.

Haines, D. (2000). The essential mechanical properties of wood prepared for musical instruments. In: Catgut Acoustic Society Journal, 4(2), pp. 20-32.

Rossing, T., Fletcher, N. (2004). Principle of Vibration and Sound--second edition. Springer Science, New York, ISBN 0-387-40556-9.

Vladimirovici, S. (2004). The Computational Method of The Musical Instruments Body Parts. PhD Thesis, The State Thechnical University of Marii, Russian Federation, in Russian.

Wright, H., (1996). The Acoustics and psychoacoustics of the guitar. PhD Thesis, University of Wales, 1996.

Fig. 3. Comparison between amplitudes and resonance
frequency of different point placed on the top, back and ribs

Amplitude [mm]

Measured
point Top Back Sides

P1 220 280 200
P2 620 280 220
P3 300 220 220
P4 220 220 200
P5 220 220 620

Note: Table made from bar graph.

Fig. 4. Comparison of guitar body response with and without
coupling to the fundamental mode

Frequency [Hz]

Structures Free vibration Forced vibration

plate with 3 braces 262 260
top of box 210 200
back of box 210 220

Note: Table made from bar graph.