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Fluid–structure coupling in the guitar box: numerical and experimental comparative study
Applied Acoustics 66 (2005) 411–425 www.elsevier.com/locate/apacoust
Fluid–structure coupling in the guitar box: numerical and experimental comparative study A. Ezcurra a
a,*
, M.J. Elejabarrieta b, C. Santamarı´a
c
Departamento de Fı´sica, Universidad Pu´blica de Navarra, Campus Arrosadı´a, 31006 Pamplona, Spain b Departamento de Meca´nica, Mondragon Unibertsitatea, 20500 Mondragon, Guipuzcoa, Spain c Departamento de Fı´sica Aplicada II, Universidad del Paı´s Vasco, Apdo.644-48080 Bilbao, Spain Received 1 June 2003; received in revised form 1 May 2004; accepted 1 July 2004 Available online 29 September 2004
Abstract Resonance boxes are common to many musical instruments and determine the radiated sound to a great extent. The behaviour of the structure and the air inside must be understood as a whole, the complexity of which is increased by the presence of sound holes. In this work, we present a comparative study of the guitar box in which the interior gas is changed both experimentally and numerically. Modal patterns, natural frequencies and quality factors are determined when the box is full of helium, air and krypton, respectively. This allows us to characterise the soundboard–back plate coupling via the cavity fluid, stressing the role of the structural and acoustic uncoupled modes. This could help guitar makers, allowing them to modify the final modes by means of structural modifications. Moreover the methodology, together with the developed finite element model, proves to be valid for studying the dynamic fluid–structure coupling in any arbitrary mechanical system, including cavities connected to the surrounding air. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Guitar; Vibrational properties; Resonance box
*
Corresponding author. Tel.: +34 948169580; fax: +34 948169565. E-mail address: [email protected] (A. Ezcurra).
0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.07.010
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1. Introduction Musical instruments are complex vibrating systems (both structurally and in their interaction with the surrounding air), which radiate sound waves [1]. In the case of string instruments, the strings radiate only a small amount of sound directly, but they excite the bridge and the top plate, which in turn transfer energy to the air cavity, ribs, and back plate. Sound is radiated efficiently by the vibrating plates and through the sound hole. In this way resonance boxes amplify and filter string vibrations and accordingly determine the final sound [1–3]. Although the construction is obviously centred in the material structure of the box, the term ‘‘resonance box’’ itself makes the importance of the cavity clear. In most string instruments, the wooden structure and the air contained inside form a whole whose complexity is increased by the presence of sound holes, and therefore their dynamic behaviour cannot be understood individually. However, despite its essential role, it must be noted that very little research has been done on the influence on string instruments of the fluid contained within the cavity [4–7]. In previous studies, we have investigated the vibrational dynamics of the guitar box by means of experimental measurements and numerical calculation [8]. The aim of the present paper is to examine in detail the way in which structural modes couple with fluid modes in the low-frequency range (up to 400 Hz). At the same time, we show that the fluid-structure coupling including the role of the sound hole can be successfully modelled numerically. A comparative study of the vibrational behaviour of the guitar box full of different fluids (krypton and helium in addition to air) is presented. On the one hand, the experimental response of the guitar box was determined by modal analysis [9]; on the other hand finite element analysis [10] was applied to calculate the dynamic response. These two techniques were chosen because they are quantitative and complementary, allowing their results to be compared. Consequently the numerical results are discussed in relation to the experimental measurements. In this way, the influence of fluid loading on the vibration patterns, natural frequencies and frequency response of the instrument, together with the participation factors [10] of the uncoupled modes (wooden structure and contained fluid) on the coupled modes, can be analysed.
2. Experimental 2.1. Modal analysis A skilled luthier built a good quality guitar box whose dynamic behaviour was studied step by step by means of modal analysis. The experimental device along with the guitar box description has been published elsewhere [8,11]. In this case the modal analysis technique has been applied to the resonance box when filled with helium, krypton and air (this last case was previously described separately [8]) in order to compare their respective effects on the dynamical behaviour. Among the available approaches, the measurement of the frequency response function (FRF) was chosen.
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The structure was set into vibration by means of a transducer hammer acting on successive points, and the response was recorded at one fixed point by an accelerometer. The frequencies studied were in the 0–800 Hz range (1 Hz resolution), and the estimation of the modal parameters was carried out in more reduced bands. Both the excitation and the response measurement were made perpendicular to the plates, the most significant direction regarding the vibration and acoustic radiation of the final instrument. The response point was located at the position of the bridge, next to the bass string, and 230 excitation points were distributed over the soundboard (1 1 5) and the back plate (1 1 5). The number and location of the selected measurement points was chosen to enable measurement of all the modes expected to occur in the frequency range studied. The estimation of the modal parameters was global, since each modal analysis was made up of 230 response functions. Therefore, the reported natural frequencies, the vibration modes and the quality factors do not depend on the particular response point. As for the boundary conditions, the ribs were fixed by means of polyurethane foam to a metallic mould, in order to prevent the soundboard and back plate perimeters from moving in the analysed frequency band. The box cavity was successively filled with HeliumN50 and KryptonN45 (ISO9002) and, in each case, was kept topped up (to compensate for diffusion) by a continuous flow controlled by a pressure governor. The external medium was air in both cases. Furthermore, in the case of krypton (denser than air), the box was placed with the soundboard (and therefore the sound hole) facing upwards. On the contrary, the box was turned 180° in the case of helium, so that the sound hole faced downwards. In order to guarantee that the cavity was really full, preliminary measurements were made, and the frequency value of the first natural mode was considered to be a good indicator. Thus, first of all this frequency was determined in both cases (helium and krypton) and its stability was continuously controlled during the measurements. 2.2. The numerical model The numerical model used to model the resonance box is based on the finite element method, and is described in previous articles [8,12]. In summary, a guitar box model (whose geometrical design and material parameters were similar to the real one) was assembled from its individual components. The vibrational dynamic was studied at every constructional step, determining the evolution of the modal parameters. The structural modes corresponding to the box (the empty wooden structure) with the ribs fixed can be found in [8] and are shown schematically in Fig. 1. The enclosed fluid was studied separately using a procedure [13] that allows the role of the sound hole to be included. This procedure develops a preliminary model (the box immersed in a much larger air cavity) to calculate the length correction [14] to the thickness of the opening. This correction together with the pressure value on its top includes all the effects due to the sound hole and to the external medium. The final model describes the proposed experimental situation (that is, where the external medium is air and the cavity is full of air, krypton or helium) by substituting the external medium for the length correction. The gases are characterised by means
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Fig. 1. Scheme of the uncoupled modes corresponding to the wooden structure. Table 1 Mechanical parameters (density, bulk modulus and sound velocity) of the gases in standard atmospheric conditions
Helium Air Krypton
q (kg/m3)
B (kPa)
c (m/s)
0.166 1.29 3.47
168 142 177
1040 342 229
of their mechanical parameters in standard atmospheric conditions summarized in Table 1. Using the procedures described above, the uncoupled vibration modes corresponding to the wooden structure and to the three contained fluids were obtained. On these uncoupled modes, the modal coupling method was successively applied to calculate the vibration patterns and natural frequencies of the coupled modes due to the fluid–structure interaction. As with the normal mode measurements, the numerical results include the participation factors of the uncoupled modes in the coupled modes. Consequently, this numerical study contributes to the analysis of the experimental data in addition to the prediction of the vibrational behaviour.
3. Results and discussion 3.1. Internal fluid modes by the finite element method Fig. 2(a) shows the fundamental (A0) modes when the guitar box was filled with helium, air and krypton, calculated using the preliminary model (that is, the box immersed in a much larger air cavity). The diagrams show the pressure distribution both inside and outside the box. Fig. 2(b) shows the resulting meshes in which the length correction absorbs the effects of the radiation and of the external medium. The length correction is 8 mm for krypton, 15 mm for air and 56 mm for helium. Finally, Fig. 2(c) shows the pressure distributions and the frequency values for the
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Fig. 2. Acoustic models: (a) A0 modes corresponding to the preliminary model. Pressure distribution inside and outside the box can be seen. (b) Definitive meshes including the length corrections. (c) A0 modes corresponding to the definitive models.
A0 mode calculated using the model. The frequencies and patterns depend on the gas, with the frequency values appearing to be closely related to density. All the three pressure distributions illustrated in Fig. 2 show a variation in pressure throughout the cavity, indicating that the A0 mode is not a pure Helmholtz mode. This is more evident in the case of krypton whereas helium is closer to the ideal even pressure distribution. The absolute pressure values corresponding to the different gases are not comparable, in this mode and in the rest, because the modes have been independently normalised. However, for a specific gas, the pressures at different positions within the resonance box can be compared. Thus pressure
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Fig. 3. A1, . . ., A5 acoustic modes corresponding to the inside fluid.
differences between the upper and lower part of the box are 25% for helium, 60% for air and 80% for krypton. Fig. 2(a) shows the pressure variation over the sound hole. This variation is especially large for helium and very small for krypton, while air exhibits a medium variation. The vibrational patterns corresponding to the upper modes A1, A2, etc., are similar for the three cases, and are shown in Fig. 3 together with the natural frequencies. Therefore, it is the size and shape of the cavity that determines the stationary wave patterns, with the type of fluid having no influence on them (although it does determine the frequency values). As might be expected, the highest frequencies correspond to helium, the lightest gas. The frequencies of the acoustic modes of air and krypton, on the other hand, are in the same range as those of the structural modes (see Fig. 1); this fact will make the low-frequency coupling easy. The coupling for helium will be possible only through the A0 mode since the remaining acoustic frequencies are over 1 kHz. 3.2. Coupled modes 3.2.1. Finite element method As was demonstrated in a previous paper [8], the air contained in the box provides a coupling between the soundboard and back plate, giving rise to the box modes. Moreover it acts as an added mass because the natural frequencies decrease in comparison with those of the structural modes. Furthermore, the interaction between structural and acoustical modes depends on the vibrational character of the structural modes. For example, some structural modes are unable to couple and therefore appear only in one part, the soundboard or the back plate, as in the case of the transversal flexural modes. The three gases were chosen because of their different densities. The aim was to study their influence on the final behaviour of the instrument. On the one hand the results will allow us to validate the whole process, and on the other hand the comparative study will stress the nature of the coupling. Figs. 4(a), 5(a) and 6(a)show the calculated vibration patterns and natural frequencies of the guitar box when filled with helium, air and krypton, respectively.
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Meanwhile, Fig. 7 shows the uncoupled modes (see Figs. 1–3) that form each coupled mode in terms of the participation factors. The coupled modes are indexed by the capital letters TB (referring to the top–back coupling) followed by the mode number, starting at the lowest frequency. As can be seen, the patterns present the same vibrational character and appear in the same order in the three cases. As forms of the vibrational patterns have been previously analysed [8] only the main differences among them will be discussed. The two lowest modes TB(1) and TB(2) present the same aspect regardless of the type of gas, with a single antinodal zone at the position of the bridge. In TB(1) the soundboard and back plate vibrate with opposite phase whereas they vibrate inphase in TB(2). The main difference between the three gases is the amplitude ratio between top and back plates. Thus, in TB(1) the maximum amplitude in the soundboard is 2.7 times greater than in the back plate when the box is full of krypton, 3 times greater in the case of air, and 14 times when the gas is helium. It follows that the greater the density of the gas, the more similar the vibration amplitudes of the front and back plates. In terms of the fluid, the A0 mode is dominant when the box is vibrating in the TB(1) mode. In terms of the structure, the fundamental modes of the soundboard – (1,1)s – and of the back – (1,1)b – are dominant in all cases, the participation factor of the soundboard being slightly greater for air and krypton, and
Fig. 4. (a) Calculated and (b) experimental coupled modes corresponding to the box full of helium.
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Fig. 5. (a) Calculated and (b) experimental coupled modes corresponding to the box full of air.
very much greater for helium (so in this case the back remains nearly motionless). The frequency of the TB(1) mode is very sensitive to the contained fluid, the denser the gas the lower the frequency (in good agreement with the added mass effect of the fluid on the structure). On the other hand, since this mode is conditioned by the A0 resonance, their frequencies must correlate in some way, but the results indicate that they are not purely proportional. In short, TB(1) depends on the contained fluid, both in its frequency values (it varies 47% between krypton and helium) and in the amplitude relation between soundboard and back plate. The vibration amplitudes of the soundboard and back plate are similar in TB(2) for air and krypton, although in the latter case the antinodal region in the back plate is much smaller. The behaviour is clearly different for helium, in which the vibration amplitude of the back plate is twice that of the top plate. The modal participation makes this fact clear: (1,1)s and (1,1)b participation factors are similar for air and krypton whereas the second factor is three times greater than the first for helium. Regarding the acoustic modes, the increase of the A1 participation factor is remarkable in the case of krypton. Finally the TB(2) frequency is not very sensitive to fluid, the biggest difference being 6%. As can be seen from the figures, in the TB(3) mode, the vibrational pattern in the soundboard is similar for all three gases. Meanwhile, the back plate is nearly
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Fig. 6. (a) Calculated and (b) experimental coupled modes corresponding to the box full of krypton.
motionless in all cases: the fluid is not able to couple the soundboard and back plate. Acoustic modes A2 and A3 contribute to this pattern in similar proportions for all three gases. In the structural modes the (2,2)b contribution in the krypton case is surprising, although its interaction with (2,1)s is not detected in the vibration pattern. Helium presents the highest TB(3) mode frequency (equal to (2,1)s frequency), being some 10% higher than the value for krypton. Mode TB(4) presents the biggest differences among the three cases. In the case of helium, it did not appear at all. In the case of krypton and air, the zone close to the sound hole (both in the soundboard and in the back plate) vibrates, noticeably in the case of krypton, with the air case quite being similar to mode TB(1). The TB(4) mode is created by structural modes (1,1)s and (1,1)b, mainly interacting with the A0 air mode. As the participation factors are similar to those of the TB(1) mode the same considerations on the influence of the soundboard and the geometry of the cavity can be made in this case. Given this behaviour we can infer that this mode forms the lowfrequency triplet together with TB(1) and TB(2) [1,5]. The calculated TB(4) is more complex than the theoretically predicted mode as the structural modes (1,2)b, (1,2)s and (1,3)b are present too. This mode could not be experimentally determined, probably due to the proximity of TB(3).
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Fig. 7. Participation factors of the structural and acoustic modes on the coupled modes corresponding to the box full of helium, air and krypton, respectively.
Mode TB(5) takes different forms depending on the gas, although it maintains its longitudinal flexion character. The mode is particularly different in the case of krypton. It presents a single antinodal zone, around the sound hole, both in the
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soundboard and in the back plate, moving in phase opposition. With air and helium, the soundboard and back plate present two antinodal zones, that of the sound hole having the greatest amplitude. Three acoustic modes contribute to TB(5): A0, A1 and A4. The participation factors for air are nearly the same as those for krypton, A0 making a slightly greater contribution in this last case. The same structural modes contribute to the TB(5) mode for all three fluids, but to varying degrees. Thus the (1,1)s and (1,1)b participation factors are slightly greater than the (1,2)s and (1,2)b participation factors for krypton. The difference increases, but in an opposite sense, for air. Using helium increases the (1,2)b participation up to three times that of (1,2)s, the fundamental mode contribution being nearly non-existent. With regard to frequencies, this mode presents a tendency similar to the preceding ones, being 11% higher for helium than for krypton. Mode TB(6) also has a longitudinal flexion character, but now the top and back plates vibrate in phase. In all cases the soundboard presents a greater vibration amplitude, the maximum being positioned around the sound hole. The amplitude relationship between the upper and lower part of the soundboard is similar for the three gases. However, the back plate appears different depending on the gas. Thus it appears nearly motionless for helium, this being clear from the participation factors of 70% and 27% for the (1,2)s and (1,2)b modes, respectively. Krypton and air present nearly equal participation factors for these two structural modes, and also for the front and back plate amplitudes. The participation factors of the acoustic modes are nearly equal for the three gases, except for the presence of mode A5 for krypton; this is due to its proximity to TB(6) frequency value. The frequency of mode TB(6) is equal for the three gases. Mode TB(7) presents a transverse flexural character in the back plate, the soundboard remaining almost motionless, except in the case of krypton when the soundboard vibrates slightly. This fact is clear from the participation factors, with the (2,1)s mode making a 25% contribution for krypton, 7% for air and below 1% for helium. The acoustic mode participation shows a common dominance of mode A2, which presents a longitudinal nodal line like (2,1)b and (2,1)s. With helium, unlike the rest, there is a notable participation of the A0 mode. The frequency of mode TB(7) differs up to 5% between helium and krypton. Mode TB(8) is quite similar for air and krypton, both in shape and in the participation factor of the structural modes. However, the acoustic mode participation is different, with a relevant contribution of A5 for krypton, which increases its TB(8) mode frequency to 3% more than the corresponding TB(8) frequency with air in the cavity. The appearance of the mode with helium is very different: the nearness of A0 frequency to the structural mode frequencies leads to a strong coupling between soundboard and back plate. To sum up the calculated coupled modes, the fluid modes determine the patterns and frequencies. Whenever the soundboard interacts with the back plate the acoustic mode A0 appears; therefore the frequency and pressure distribution of A0 determine its coupling with the structural modes, and the final modes of the box. Furthermore the frequency and shape of the stationary waves in the cavity, A1, . . ., A5, influence the structural – acoustic couplings.
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As set out below, the experimental results will show that the numerical simulation is a suitable and useful method to understand and describe the dynamic coupling between the wooden structure and the fluid contained within it. 3.2.2. Experimental results Figs. 4(b), 5(b) and 6(b) show the measured vibration patterns, natural frequencies and quality factors of the eight lowest modes of the guitar box with the ribs fixed, when filled with helium, air and krypton, respectively. The vibrational amplitude of the modes cannot be compared with one another, since each pattern was normalised to unit modal mass. The vibration modes obtained have a normal character, since their MPC (Modal Phase Co-linearity) [9] is between 91% and 99% for krypton and air, and between 83% and 99% for helium. This index indicates that the imaginary part of the pattern is practically negligible compared with the real part, and so the relative phase for the 230 degrees of freedom was 0° or 180°. Consequently, the modes can be considered as normal, with proportional viscous damping. When comparing these results with the calculated modes (Figs. 4(a), 5(a) and 6(a)) it can be seen that the experimental and calculated modes are similar and appear in the same order. Furthermore the upper zone of the resonance box remains motionless, regardless of the type of gas, in the analysed frequency range. The following paragraphs are devoted to the analysis of the modes presented, accounting for the modal participation factors. Although the pattern of mode TB(1) does not depend on the gas, the vibration amplitudes differ. In the case of helium the amplitude of vibration is less in the back plate than it is in the soundboard, whereas it is similar in the case of air and krypton. This is in good agreement with the finite element results. Regarding mode TB(2) the vibration amplitude is higher on the soundboard than on the back plate for air and krypton, whilst the opposite is true for helium. This is the same tendency predicted by the calculation, although here the differences are smaller. Mode TB(3) presents the same vibrational pattern for the soundboard in the three studied cases (the back plate being motionless). As can be seen, the numerically predicted and experimentally measured patterns coincide in this mode. This is also the case in mode TB(7), which like TB(3) has a transverse flexural character. In mode TB(7), krypton, unlike the rest, provokes vibration in the soundboard; this behaviour was predicted by the calculations. As for modes TB(5) and TB(6) the vibration patterns coincide at first glance. However the agreement is not so good when it comes to comparing the vibration amplitudes of the top and back plates. Experimentally the upper and the lower parts of the soundboard (and of the back plate) vibrate with the same amplitude when the box is full of helium; yet in the results of the numerical model, the lower zone of the soundboard presents much weaker vibration amplitude. The agreement is much better for air and krypton. Finally the experimental and calculated TB(8) mode patterns are similar for air and krypton. On the other hand, the transverse flexural character of this mode is not experimentally detected in the soundboard in the case of helium.
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Regarding the frequency values, there are differences between the numerical predictions and the experimental measurements. The experimental measurements show that the predicted tendencies are accurate, in that the order of appearance of the coupled modes is the same using both methods. Furthermore the experimental measurements confirm that the denser the gas the smaller the coupled frequencies. This effect is more pronounced at lower frequencies. The maximum deviation between the measured and predicted frequency values appears in mode TB(3) (12% for air and krypton and 14% for helium). Since the effect of the fluid in this mode is minimal, we must conclude that the model underestimates the stiffness of the wooden box. Taking into account the high number of variables, the results indicate that the numerical method is a valid approximation to describe the vibrational dynamics of the fluid contained in the resonance box and the fluid–structure coupling. If this were not the case, the different mechanical properties of the three gases would have greatly increased the deviations. On the contrary, the deviations are similar and in the same sense. It can be concluded therefore that they are due to the estimated elastic parameters of the woods used and/or to the FEM method, which tends to underestimate the eigenfrequencies [10]. The quality factors (see Figs. 4(b), 5(b) and 6(b)) are very sensitive to the particular gas. Although no direct link could be deduced, it is likely that this parameter is connected with the modal participation factors of the acoustic modes, especially of mode A0. When the A0 participation factor is the same for different gases, the density determines the quality factor. This is certainly true in the case of mode TB(1), where the highest quality factor corresponds to helium. When the A0 mode does not contribute to the coupled mode – that is, its participation factor is zero – the quality factors are similar for the three gases. A good example of this is the TB(3) mode. On the other hand, the higher the participation factor of mode A0, the lower the quality factor in the coupled mode (see the rest of the coupled modes). As can be seen, the influence of mode A0 on the coupled modes is considerable, and it is worthy of a specific report.
4. Conclusions Finite element and modal analysis methods have been applied to a guitar box filled with three gases in order to study in detail its vibrational behaviour with the ribs fixed, emphasising the fluid–structure coupling and the influence of the type of fluid. The results allow us to draw some conclusions about the fluid–structure coupling, the dynamic behaviour of the guitar box and the appropriate methodologies to approach complex vibrating systems: Concerning the cavity modes. The six lowest acoustic modes of the guitar box present the same pattern, independently of the type of fluid, for the three studied gases. The pressure distribution inside the box is the same for the three gases in all the modes except the Helmholtz resonance.
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In the Helmholtz resonance, the pressure field is nearly uniform in the case of helium, and presents a strong gradient pointing to the upper part of the cavity in the case of krypton (air exhibiting an intermediate behaviour). So, the modal A0 pattern depends on fluid density; the denser the gas, the bigger the difference between A0 mode and the pure Helmholtz resonator. Concerning the dynamic behaviour of the resonance box. The type of fluid determines the modal patterns and frequencies of the whole resonance box. Since the A0 mode is present at all modes which exhibit soundboard–back plate interaction via the fluid, its frequency and pressure pattern determine the coupling with the structural modes. In this sense the Helmholtz mode determines the coupled modes to a great extent. The frequencies and patterns corresponding to A1, . . ., A5 modes determine their coupling with the structural modes. The experimental measurements show that the quality factor is very sensitive to the type of fluid contained in the box. Concerning the methodology. Combining modal analysis and the finite element method is useful in analysing different structural designs and their effect on the dynamic response of the resonance box. Finally, the developed finite element model proves to be a useful tool for studying the dynamic fluid–structure coupling in any arbitrary mechanical system, including cavities connected to the surrounding air.
References [1] Fletcher NH, Rossing TD. The physics of musical instruments. New York: Springer; 1991. [2] Jansson EV. Acoustical properties of complex cavities and measurements of resonance properties of violin-shaped and guitar-shaped cavities. Acustica 1977;37(4):211–21. [3] Caldersmith G. Plate fundamental coupling and its musical importance. Catgut Acoust Soc 1981;36:21–7. [4] Christensen O, Vistisen BB. Simple model for low-frequency guitar function. J Acoust Soc Am 1980;68(3):758–66. [5] Christensen O. Qualitative models for low frequency guitar function. J Guitar Acoust 1982;6:10–25. [6] Richardson BE, Roberts GW. The adjustment of mode frequencies in guitars: a study by means of holographic interferometry and finite element analysis. In: SMACÕ83, vol. II. Publ Roy Swed Acad Music 1985;46(2):285–302. [7] Bissinger G, Hutchins CM. Further evidence for coupling between plate and enclosed air vibrations in string instruments. Catgut Acoust Soc Newslett 1983;40:18–9. [8] Elejabarrieta MJ, Ezcurra A, Santamarı´a C. Coupled modes of the resonance box of the guitar. J Acoust Soc Am 2002;111(5):2283–92. [9] Ewins DJ. Modal testing: theory and practice. England: Research Studies Press Ltd.; 1984. [10] Zienkiewicz OC, Taylor RL. The finite element method. London: McGraw Hill Book Company; 1967. [11] Elejabarrieta MJ, Ezcurra A, Santamarı´a C. Evolution of the vibrational behaviour of a guitar soundboard along successive construction phases by means of the modal analysis technique. J Acoust Soc Am 2000;108(1):369–78.
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[12] Elejabarrieta MJ, Ezcurra A, Santamarı´a C. Vibrational behaviour of the guitar soundboard analysed by the finite element method. Acustica – Acta Acust 2001;87:128–36. [13] Elejabarrieta MJ, Ezcurra A, Santamarı´a C. Air cavity modes in the resonance box of the guitar: the effect of the sound hole. J Sound Vib 2002;252(3):584–90. [14] Kinsler LE, Frey AR, Coppens AB, Sanders JV. Fundamentals of acoustics. New York: Wiley; 1988.
Fluid–structure coupling in the guitar box: numerical and experimental comparative study A. Ezcurra a
a,*
, M.J. Elejabarrieta b, C. Santamarı´a
c
Departamento de Fı´sica, Universidad Pu´blica de Navarra, Campus Arrosadı´a, 31006 Pamplona, Spain b Departamento de Meca´nica, Mondragon Unibertsitatea, 20500 Mondragon, Guipuzcoa, Spain c Departamento de Fı´sica Aplicada II, Universidad del Paı´s Vasco, Apdo.644-48080 Bilbao, Spain Received 1 June 2003; received in revised form 1 May 2004; accepted 1 July 2004 Available online 29 September 2004
Abstract Resonance boxes are common to many musical instruments and determine the radiated sound to a great extent. The behaviour of the structure and the air inside must be understood as a whole, the complexity of which is increased by the presence of sound holes. In this work, we present a comparative study of the guitar box in which the interior gas is changed both experimentally and numerically. Modal patterns, natural frequencies and quality factors are determined when the box is full of helium, air and krypton, respectively. This allows us to characterise the soundboard–back plate coupling via the cavity fluid, stressing the role of the structural and acoustic uncoupled modes. This could help guitar makers, allowing them to modify the final modes by means of structural modifications. Moreover the methodology, together with the developed finite element model, proves to be valid for studying the dynamic fluid–structure coupling in any arbitrary mechanical system, including cavities connected to the surrounding air. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Guitar; Vibrational properties; Resonance box
*
Corresponding author. Tel.: +34 948169580; fax: +34 948169565. E-mail address: [email protected] (A. Ezcurra).
0003-682X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2004.07.010
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1. Introduction Musical instruments are complex vibrating systems (both structurally and in their interaction with the surrounding air), which radiate sound waves [1]. In the case of string instruments, the strings radiate only a small amount of sound directly, but they excite the bridge and the top plate, which in turn transfer energy to the air cavity, ribs, and back plate. Sound is radiated efficiently by the vibrating plates and through the sound hole. In this way resonance boxes amplify and filter string vibrations and accordingly determine the final sound [1–3]. Although the construction is obviously centred in the material structure of the box, the term ‘‘resonance box’’ itself makes the importance of the cavity clear. In most string instruments, the wooden structure and the air contained inside form a whole whose complexity is increased by the presence of sound holes, and therefore their dynamic behaviour cannot be understood individually. However, despite its essential role, it must be noted that very little research has been done on the influence on string instruments of the fluid contained within the cavity [4–7]. In previous studies, we have investigated the vibrational dynamics of the guitar box by means of experimental measurements and numerical calculation [8]. The aim of the present paper is to examine in detail the way in which structural modes couple with fluid modes in the low-frequency range (up to 400 Hz). At the same time, we show that the fluid-structure coupling including the role of the sound hole can be successfully modelled numerically. A comparative study of the vibrational behaviour of the guitar box full of different fluids (krypton and helium in addition to air) is presented. On the one hand, the experimental response of the guitar box was determined by modal analysis [9]; on the other hand finite element analysis [10] was applied to calculate the dynamic response. These two techniques were chosen because they are quantitative and complementary, allowing their results to be compared. Consequently the numerical results are discussed in relation to the experimental measurements. In this way, the influence of fluid loading on the vibration patterns, natural frequencies and frequency response of the instrument, together with the participation factors [10] of the uncoupled modes (wooden structure and contained fluid) on the coupled modes, can be analysed.
2. Experimental 2.1. Modal analysis A skilled luthier built a good quality guitar box whose dynamic behaviour was studied step by step by means of modal analysis. The experimental device along with the guitar box description has been published elsewhere [8,11]. In this case the modal analysis technique has been applied to the resonance box when filled with helium, krypton and air (this last case was previously described separately [8]) in order to compare their respective effects on the dynamical behaviour. Among the available approaches, the measurement of the frequency response function (FRF) was chosen.
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The structure was set into vibration by means of a transducer hammer acting on successive points, and the response was recorded at one fixed point by an accelerometer. The frequencies studied were in the 0–800 Hz range (1 Hz resolution), and the estimation of the modal parameters was carried out in more reduced bands. Both the excitation and the response measurement were made perpendicular to the plates, the most significant direction regarding the vibration and acoustic radiation of the final instrument. The response point was located at the position of the bridge, next to the bass string, and 230 excitation points were distributed over the soundboard (1 1 5) and the back plate (1 1 5). The number and location of the selected measurement points was chosen to enable measurement of all the modes expected to occur in the frequency range studied. The estimation of the modal parameters was global, since each modal analysis was made up of 230 response functions. Therefore, the reported natural frequencies, the vibration modes and the quality factors do not depend on the particular response point. As for the boundary conditions, the ribs were fixed by means of polyurethane foam to a metallic mould, in order to prevent the soundboard and back plate perimeters from moving in the analysed frequency band. The box cavity was successively filled with HeliumN50 and KryptonN45 (ISO9002) and, in each case, was kept topped up (to compensate for diffusion) by a continuous flow controlled by a pressure governor. The external medium was air in both cases. Furthermore, in the case of krypton (denser than air), the box was placed with the soundboard (and therefore the sound hole) facing upwards. On the contrary, the box was turned 180° in the case of helium, so that the sound hole faced downwards. In order to guarantee that the cavity was really full, preliminary measurements were made, and the frequency value of the first natural mode was considered to be a good indicator. Thus, first of all this frequency was determined in both cases (helium and krypton) and its stability was continuously controlled during the measurements. 2.2. The numerical model The numerical model used to model the resonance box is based on the finite element method, and is described in previous articles [8,12]. In summary, a guitar box model (whose geometrical design and material parameters were similar to the real one) was assembled from its individual components. The vibrational dynamic was studied at every constructional step, determining the evolution of the modal parameters. The structural modes corresponding to the box (the empty wooden structure) with the ribs fixed can be found in [8] and are shown schematically in Fig. 1. The enclosed fluid was studied separately using a procedure [13] that allows the role of the sound hole to be included. This procedure develops a preliminary model (the box immersed in a much larger air cavity) to calculate the length correction [14] to the thickness of the opening. This correction together with the pressure value on its top includes all the effects due to the sound hole and to the external medium. The final model describes the proposed experimental situation (that is, where the external medium is air and the cavity is full of air, krypton or helium) by substituting the external medium for the length correction. The gases are characterised by means
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Fig. 1. Scheme of the uncoupled modes corresponding to the wooden structure. Table 1 Mechanical parameters (density, bulk modulus and sound velocity) of the gases in standard atmospheric conditions
Helium Air Krypton
q (kg/m3)
B (kPa)
c (m/s)
0.166 1.29 3.47
168 142 177
1040 342 229
of their mechanical parameters in standard atmospheric conditions summarized in Table 1. Using the procedures described above, the uncoupled vibration modes corresponding to the wooden structure and to the three contained fluids were obtained. On these uncoupled modes, the modal coupling method was successively applied to calculate the vibration patterns and natural frequencies of the coupled modes due to the fluid–structure interaction. As with the normal mode measurements, the numerical results include the participation factors of the uncoupled modes in the coupled modes. Consequently, this numerical study contributes to the analysis of the experimental data in addition to the prediction of the vibrational behaviour.
3. Results and discussion 3.1. Internal fluid modes by the finite element method Fig. 2(a) shows the fundamental (A0) modes when the guitar box was filled with helium, air and krypton, calculated using the preliminary model (that is, the box immersed in a much larger air cavity). The diagrams show the pressure distribution both inside and outside the box. Fig. 2(b) shows the resulting meshes in which the length correction absorbs the effects of the radiation and of the external medium. The length correction is 8 mm for krypton, 15 mm for air and 56 mm for helium. Finally, Fig. 2(c) shows the pressure distributions and the frequency values for the
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Fig. 2. Acoustic models: (a) A0 modes corresponding to the preliminary model. Pressure distribution inside and outside the box can be seen. (b) Definitive meshes including the length corrections. (c) A0 modes corresponding to the definitive models.
A0 mode calculated using the model. The frequencies and patterns depend on the gas, with the frequency values appearing to be closely related to density. All the three pressure distributions illustrated in Fig. 2 show a variation in pressure throughout the cavity, indicating that the A0 mode is not a pure Helmholtz mode. This is more evident in the case of krypton whereas helium is closer to the ideal even pressure distribution. The absolute pressure values corresponding to the different gases are not comparable, in this mode and in the rest, because the modes have been independently normalised. However, for a specific gas, the pressures at different positions within the resonance box can be compared. Thus pressure
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Fig. 3. A1, . . ., A5 acoustic modes corresponding to the inside fluid.
differences between the upper and lower part of the box are 25% for helium, 60% for air and 80% for krypton. Fig. 2(a) shows the pressure variation over the sound hole. This variation is especially large for helium and very small for krypton, while air exhibits a medium variation. The vibrational patterns corresponding to the upper modes A1, A2, etc., are similar for the three cases, and are shown in Fig. 3 together with the natural frequencies. Therefore, it is the size and shape of the cavity that determines the stationary wave patterns, with the type of fluid having no influence on them (although it does determine the frequency values). As might be expected, the highest frequencies correspond to helium, the lightest gas. The frequencies of the acoustic modes of air and krypton, on the other hand, are in the same range as those of the structural modes (see Fig. 1); this fact will make the low-frequency coupling easy. The coupling for helium will be possible only through the A0 mode since the remaining acoustic frequencies are over 1 kHz. 3.2. Coupled modes 3.2.1. Finite element method As was demonstrated in a previous paper [8], the air contained in the box provides a coupling between the soundboard and back plate, giving rise to the box modes. Moreover it acts as an added mass because the natural frequencies decrease in comparison with those of the structural modes. Furthermore, the interaction between structural and acoustical modes depends on the vibrational character of the structural modes. For example, some structural modes are unable to couple and therefore appear only in one part, the soundboard or the back plate, as in the case of the transversal flexural modes. The three gases were chosen because of their different densities. The aim was to study their influence on the final behaviour of the instrument. On the one hand the results will allow us to validate the whole process, and on the other hand the comparative study will stress the nature of the coupling. Figs. 4(a), 5(a) and 6(a)show the calculated vibration patterns and natural frequencies of the guitar box when filled with helium, air and krypton, respectively.
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Meanwhile, Fig. 7 shows the uncoupled modes (see Figs. 1–3) that form each coupled mode in terms of the participation factors. The coupled modes are indexed by the capital letters TB (referring to the top–back coupling) followed by the mode number, starting at the lowest frequency. As can be seen, the patterns present the same vibrational character and appear in the same order in the three cases. As forms of the vibrational patterns have been previously analysed [8] only the main differences among them will be discussed. The two lowest modes TB(1) and TB(2) present the same aspect regardless of the type of gas, with a single antinodal zone at the position of the bridge. In TB(1) the soundboard and back plate vibrate with opposite phase whereas they vibrate inphase in TB(2). The main difference between the three gases is the amplitude ratio between top and back plates. Thus, in TB(1) the maximum amplitude in the soundboard is 2.7 times greater than in the back plate when the box is full of krypton, 3 times greater in the case of air, and 14 times when the gas is helium. It follows that the greater the density of the gas, the more similar the vibration amplitudes of the front and back plates. In terms of the fluid, the A0 mode is dominant when the box is vibrating in the TB(1) mode. In terms of the structure, the fundamental modes of the soundboard – (1,1)s – and of the back – (1,1)b – are dominant in all cases, the participation factor of the soundboard being slightly greater for air and krypton, and
Fig. 4. (a) Calculated and (b) experimental coupled modes corresponding to the box full of helium.
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Fig. 5. (a) Calculated and (b) experimental coupled modes corresponding to the box full of air.
very much greater for helium (so in this case the back remains nearly motionless). The frequency of the TB(1) mode is very sensitive to the contained fluid, the denser the gas the lower the frequency (in good agreement with the added mass effect of the fluid on the structure). On the other hand, since this mode is conditioned by the A0 resonance, their frequencies must correlate in some way, but the results indicate that they are not purely proportional. In short, TB(1) depends on the contained fluid, both in its frequency values (it varies 47% between krypton and helium) and in the amplitude relation between soundboard and back plate. The vibration amplitudes of the soundboard and back plate are similar in TB(2) for air and krypton, although in the latter case the antinodal region in the back plate is much smaller. The behaviour is clearly different for helium, in which the vibration amplitude of the back plate is twice that of the top plate. The modal participation makes this fact clear: (1,1)s and (1,1)b participation factors are similar for air and krypton whereas the second factor is three times greater than the first for helium. Regarding the acoustic modes, the increase of the A1 participation factor is remarkable in the case of krypton. Finally the TB(2) frequency is not very sensitive to fluid, the biggest difference being 6%. As can be seen from the figures, in the TB(3) mode, the vibrational pattern in the soundboard is similar for all three gases. Meanwhile, the back plate is nearly
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Fig. 6. (a) Calculated and (b) experimental coupled modes corresponding to the box full of krypton.
motionless in all cases: the fluid is not able to couple the soundboard and back plate. Acoustic modes A2 and A3 contribute to this pattern in similar proportions for all three gases. In the structural modes the (2,2)b contribution in the krypton case is surprising, although its interaction with (2,1)s is not detected in the vibration pattern. Helium presents the highest TB(3) mode frequency (equal to (2,1)s frequency), being some 10% higher than the value for krypton. Mode TB(4) presents the biggest differences among the three cases. In the case of helium, it did not appear at all. In the case of krypton and air, the zone close to the sound hole (both in the soundboard and in the back plate) vibrates, noticeably in the case of krypton, with the air case quite being similar to mode TB(1). The TB(4) mode is created by structural modes (1,1)s and (1,1)b, mainly interacting with the A0 air mode. As the participation factors are similar to those of the TB(1) mode the same considerations on the influence of the soundboard and the geometry of the cavity can be made in this case. Given this behaviour we can infer that this mode forms the lowfrequency triplet together with TB(1) and TB(2) [1,5]. The calculated TB(4) is more complex than the theoretically predicted mode as the structural modes (1,2)b, (1,2)s and (1,3)b are present too. This mode could not be experimentally determined, probably due to the proximity of TB(3).
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Fig. 7. Participation factors of the structural and acoustic modes on the coupled modes corresponding to the box full of helium, air and krypton, respectively.
Mode TB(5) takes different forms depending on the gas, although it maintains its longitudinal flexion character. The mode is particularly different in the case of krypton. It presents a single antinodal zone, around the sound hole, both in the
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soundboard and in the back plate, moving in phase opposition. With air and helium, the soundboard and back plate present two antinodal zones, that of the sound hole having the greatest amplitude. Three acoustic modes contribute to TB(5): A0, A1 and A4. The participation factors for air are nearly the same as those for krypton, A0 making a slightly greater contribution in this last case. The same structural modes contribute to the TB(5) mode for all three fluids, but to varying degrees. Thus the (1,1)s and (1,1)b participation factors are slightly greater than the (1,2)s and (1,2)b participation factors for krypton. The difference increases, but in an opposite sense, for air. Using helium increases the (1,2)b participation up to three times that of (1,2)s, the fundamental mode contribution being nearly non-existent. With regard to frequencies, this mode presents a tendency similar to the preceding ones, being 11% higher for helium than for krypton. Mode TB(6) also has a longitudinal flexion character, but now the top and back plates vibrate in phase. In all cases the soundboard presents a greater vibration amplitude, the maximum being positioned around the sound hole. The amplitude relationship between the upper and lower part of the soundboard is similar for the three gases. However, the back plate appears different depending on the gas. Thus it appears nearly motionless for helium, this being clear from the participation factors of 70% and 27% for the (1,2)s and (1,2)b modes, respectively. Krypton and air present nearly equal participation factors for these two structural modes, and also for the front and back plate amplitudes. The participation factors of the acoustic modes are nearly equal for the three gases, except for the presence of mode A5 for krypton; this is due to its proximity to TB(6) frequency value. The frequency of mode TB(6) is equal for the three gases. Mode TB(7) presents a transverse flexural character in the back plate, the soundboard remaining almost motionless, except in the case of krypton when the soundboard vibrates slightly. This fact is clear from the participation factors, with the (2,1)s mode making a 25% contribution for krypton, 7% for air and below 1% for helium. The acoustic mode participation shows a common dominance of mode A2, which presents a longitudinal nodal line like (2,1)b and (2,1)s. With helium, unlike the rest, there is a notable participation of the A0 mode. The frequency of mode TB(7) differs up to 5% between helium and krypton. Mode TB(8) is quite similar for air and krypton, both in shape and in the participation factor of the structural modes. However, the acoustic mode participation is different, with a relevant contribution of A5 for krypton, which increases its TB(8) mode frequency to 3% more than the corresponding TB(8) frequency with air in the cavity. The appearance of the mode with helium is very different: the nearness of A0 frequency to the structural mode frequencies leads to a strong coupling between soundboard and back plate. To sum up the calculated coupled modes, the fluid modes determine the patterns and frequencies. Whenever the soundboard interacts with the back plate the acoustic mode A0 appears; therefore the frequency and pressure distribution of A0 determine its coupling with the structural modes, and the final modes of the box. Furthermore the frequency and shape of the stationary waves in the cavity, A1, . . ., A5, influence the structural – acoustic couplings.
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As set out below, the experimental results will show that the numerical simulation is a suitable and useful method to understand and describe the dynamic coupling between the wooden structure and the fluid contained within it. 3.2.2. Experimental results Figs. 4(b), 5(b) and 6(b) show the measured vibration patterns, natural frequencies and quality factors of the eight lowest modes of the guitar box with the ribs fixed, when filled with helium, air and krypton, respectively. The vibrational amplitude of the modes cannot be compared with one another, since each pattern was normalised to unit modal mass. The vibration modes obtained have a normal character, since their MPC (Modal Phase Co-linearity) [9] is between 91% and 99% for krypton and air, and between 83% and 99% for helium. This index indicates that the imaginary part of the pattern is practically negligible compared with the real part, and so the relative phase for the 230 degrees of freedom was 0° or 180°. Consequently, the modes can be considered as normal, with proportional viscous damping. When comparing these results with the calculated modes (Figs. 4(a), 5(a) and 6(a)) it can be seen that the experimental and calculated modes are similar and appear in the same order. Furthermore the upper zone of the resonance box remains motionless, regardless of the type of gas, in the analysed frequency range. The following paragraphs are devoted to the analysis of the modes presented, accounting for the modal participation factors. Although the pattern of mode TB(1) does not depend on the gas, the vibration amplitudes differ. In the case of helium the amplitude of vibration is less in the back plate than it is in the soundboard, whereas it is similar in the case of air and krypton. This is in good agreement with the finite element results. Regarding mode TB(2) the vibration amplitude is higher on the soundboard than on the back plate for air and krypton, whilst the opposite is true for helium. This is the same tendency predicted by the calculation, although here the differences are smaller. Mode TB(3) presents the same vibrational pattern for the soundboard in the three studied cases (the back plate being motionless). As can be seen, the numerically predicted and experimentally measured patterns coincide in this mode. This is also the case in mode TB(7), which like TB(3) has a transverse flexural character. In mode TB(7), krypton, unlike the rest, provokes vibration in the soundboard; this behaviour was predicted by the calculations. As for modes TB(5) and TB(6) the vibration patterns coincide at first glance. However the agreement is not so good when it comes to comparing the vibration amplitudes of the top and back plates. Experimentally the upper and the lower parts of the soundboard (and of the back plate) vibrate with the same amplitude when the box is full of helium; yet in the results of the numerical model, the lower zone of the soundboard presents much weaker vibration amplitude. The agreement is much better for air and krypton. Finally the experimental and calculated TB(8) mode patterns are similar for air and krypton. On the other hand, the transverse flexural character of this mode is not experimentally detected in the soundboard in the case of helium.
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Regarding the frequency values, there are differences between the numerical predictions and the experimental measurements. The experimental measurements show that the predicted tendencies are accurate, in that the order of appearance of the coupled modes is the same using both methods. Furthermore the experimental measurements confirm that the denser the gas the smaller the coupled frequencies. This effect is more pronounced at lower frequencies. The maximum deviation between the measured and predicted frequency values appears in mode TB(3) (12% for air and krypton and 14% for helium). Since the effect of the fluid in this mode is minimal, we must conclude that the model underestimates the stiffness of the wooden box. Taking into account the high number of variables, the results indicate that the numerical method is a valid approximation to describe the vibrational dynamics of the fluid contained in the resonance box and the fluid–structure coupling. If this were not the case, the different mechanical properties of the three gases would have greatly increased the deviations. On the contrary, the deviations are similar and in the same sense. It can be concluded therefore that they are due to the estimated elastic parameters of the woods used and/or to the FEM method, which tends to underestimate the eigenfrequencies [10]. The quality factors (see Figs. 4(b), 5(b) and 6(b)) are very sensitive to the particular gas. Although no direct link could be deduced, it is likely that this parameter is connected with the modal participation factors of the acoustic modes, especially of mode A0. When the A0 participation factor is the same for different gases, the density determines the quality factor. This is certainly true in the case of mode TB(1), where the highest quality factor corresponds to helium. When the A0 mode does not contribute to the coupled mode – that is, its participation factor is zero – the quality factors are similar for the three gases. A good example of this is the TB(3) mode. On the other hand, the higher the participation factor of mode A0, the lower the quality factor in the coupled mode (see the rest of the coupled modes). As can be seen, the influence of mode A0 on the coupled modes is considerable, and it is worthy of a specific report.
4. Conclusions Finite element and modal analysis methods have been applied to a guitar box filled with three gases in order to study in detail its vibrational behaviour with the ribs fixed, emphasising the fluid–structure coupling and the influence of the type of fluid. The results allow us to draw some conclusions about the fluid–structure coupling, the dynamic behaviour of the guitar box and the appropriate methodologies to approach complex vibrating systems: Concerning the cavity modes. The six lowest acoustic modes of the guitar box present the same pattern, independently of the type of fluid, for the three studied gases. The pressure distribution inside the box is the same for the three gases in all the modes except the Helmholtz resonance.
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In the Helmholtz resonance, the pressure field is nearly uniform in the case of helium, and presents a strong gradient pointing to the upper part of the cavity in the case of krypton (air exhibiting an intermediate behaviour). So, the modal A0 pattern depends on fluid density; the denser the gas, the bigger the difference between A0 mode and the pure Helmholtz resonator. Concerning the dynamic behaviour of the resonance box. The type of fluid determines the modal patterns and frequencies of the whole resonance box. Since the A0 mode is present at all modes which exhibit soundboard–back plate interaction via the fluid, its frequency and pressure pattern determine the coupling with the structural modes. In this sense the Helmholtz mode determines the coupled modes to a great extent. The frequencies and patterns corresponding to A1, . . ., A5 modes determine their coupling with the structural modes. The experimental measurements show that the quality factor is very sensitive to the type of fluid contained in the box. Concerning the methodology. Combining modal analysis and the finite element method is useful in analysing different structural designs and their effect on the dynamic response of the resonance box. Finally, the developed finite element model proves to be a useful tool for studying the dynamic fluid–structure coupling in any arbitrary mechanical system, including cavities connected to the surrounding air.
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