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INFLUENCE OF THE MATERIAL CONSTANTS ON THE LOW FREQUENCY MODES OF A FREE GUITAR PLATE
809134—JSV 194/4 (issue)—MS 1433
Journal of Sound and Vibration (1996) 194(4), 640–644
INFLUENCE OF THE MATERIAL CONSTANTS ON THE LOW FREQUENCY MODES OF A FREE GUITAR PLATE A. E Departamento de Fı´ sica, Universidad Publica de Navarra, Campus de Arrosadia s/n, 31006 Pamplona, Navarra, Spain (Received 24 October 1994, and in final form 27 October 1995)
1. The sound production by traditional (that is, not electronic) musical instruments is, from a physical point of view, a classical mechanics subject. In this sense, its physical bases are well established and no fundamental phenomena are expected. Nevertheless, musical instruments are, in general, highly complex vibrating systems— although their components could be simple oscillators with many parameters having an influence on the final sound quality, in some cases in a way that is not well understood. Recently, the availability of advanced experimental equipment and powerful computers has made possible important advances in the analysis of these complex systems, providing physical explanation of the craftsmen’s procedures. As in the construction of a good quality instrument the main work is to shape the different pieces by semi-empirical methods and to put them together, a theoretical prediction of the final changes due to hypothetical modifications would represent a fundamental contribution. In the computer analysis of an instrument, or a part of it, two different aspects can be considered: on the one hand, the geometric aspect and the surrounding constraints, and on the other hand the material properties. In this work, attention is focused in the latter problem. This is because of the observed high sensitivity of the eigenfrequency values to the elastic parameters’ variations and the wide dispersion in the measured material data. Indeed, to determine elastic constants is not an easy task. Therefore, the work reported here is concerned with establishing the influence (and so the necessary degree of accuracy) of the different material parameter variations, in the case of a guitar plate. 2. Because there are no significant differences in geometry among classical guitar plates, the characteristics of the wood should determine their behaviour. However, wood is a very general material in the sense that there are many varieties. Therefore, it is necessary to specify which kind of wood is supposed to be used. In good quality guitars, spruce is the most common choice for the plate. In order to study the spruce plate’s behaviour in a reasonable range of variability, the material characteristics of a group of seven samples as obtained by Bucur [1] by means of ultrasonic techniques have been considered. In this reference it can be observed that the parameters differ significantly from one sample to another, the most important variations being those of the Young’s moduli and density, due to the different spruce subspecies, geographic origins and/or times of ageing by natural drying. As far as the author knows, this collection could be considered a significant one for a guitar maker. 640 0022–460X/96/290640 + 05 $18.00/0
7 1996 Academic Press Limited
641
To obtain the dynamic behaviour in the lowest frequency range, the finite element method [2] (ABAQUS software in a Convex computer) has been used. The plate was divided into 1088 order-two quadrangular shell elements jointed by 3444 nodes. The element size (and therefore the quantity of elements) was chosen by studying the convergence of the results for a rectangular plate of dimensions similar to that of the guitar (maximum height 492 mm, maximum width 372 mm, thickness 3 mm). The anisotropic characteristics were considered by putting the longitudinal direction (that of the grain) on the plate plane and parallel to the symmetry axis, as is customary in most guitars. The plate was assumed to have free-edge boundary conditions. To determine the sensitivity of the normal modes and frequencies to different parameters, the following ‘‘mean’’ values were used: r = 400 kg/m3; EL = 100, ET = 2, GLR = 8·5, GLT = 8·0, GTR = 0·4 (×108 N/m2 ); sLT = 0·49. Each parameter was varied individually, covering the experimental data and calculations were repeated. 3. RESULTS
In Figure 1, the nodal lines for the lowest six normal modes and the associated frequencies are indicated, as obtained for the ‘‘mean’’ value parameters. The variation of the parameters caused the following effects. 3.1. A wide variation, from 0·395 to 0·8, has been taken into account. The normal modes do not change perceptibly, and the eigenfrequencies are similar to the reference values, slightly lower for smaller s values, and vice versa (0·03% for the fundamental mode and 1% for the sixth one). 3.2. It should be reasonable to believe that the density decreases as the age of drying increases. Unfortunately, no data about initial values are available for establishing a time-dependence law. Indeed, it seems clear that there are a lot of additional factors (related to geographic origin, age of the tree, etc.) which have some influence on this, as for the rest of material parameters.
Figure 1. Nodal lines and eigenfrequencies of the six lowest normal modes as obtained from the finite element analysis with use of the ‘‘mean’’ parameter values.
642
Figure 2. The dependence of the eigenfrequencies on the longitudinal Young’s modulus. As can be seen, interchange in the mode order is present due to the large influence of this modulus on the frequencies. For example, the seventh mode appears below the sixty one incidentally (the numeration is that for the ‘‘mean’’ values case, marked with an arrow).
Density values differ approximately 10% from the mean value (except for two cases). In any case, density variations do not change the normal modes’ patterns, and frequency values and density are related: when the density increases, the eigenfrequencies decrease, according to the inverse of the square root of the density, as can be expected, and fi /fj ratios remain unchanged. 3.3. ’ These are the most characteristic anisotropic parameters in wood. Unfortunately, as can be seen in reference [1], they have strong deviations from one subspecies to another, and even for different samples belonging to the same subspecies. The normal modes’ evolution has been studied with EL varying from 70 to 150, ET from 1 to 4 (×108 N/m2 in both cases), and the results are presented in Figures 2 and 3. As can be seen, the order of the normal modes does not remain the same for all the values, due
Figure 3. As Figure 2, but with variation of the transversal Young’s modulus.
643
Figure 4. The dependence of the eigenfrequencies on the GLR shear modulus.
to the strong dependence of the frequency values for some of them. The numeration is that of the reference case (EL = 100, ET = 2). Modes 1, 2 and 3 are nearly EL -independent, whereas modes 4, 5 and 6 rise when EL increases. On the other hand, ET variations have a visible effect on all the frequency values, increasing them when ET goes up. These changes are more evident in the odd modes, mainly in the third one. 3.4. GLR and GLT are nearly equal, GLR being slightly higher than GLT , and GTR is much smaller than the others, close to a 1:20 relation. From sample to sample, variations of around 15% can be observed in each shear modulus. GLT and GTR variations, in the samples’ range, do not produce any visible change in the nodal lines, and the frequencies are only slightly affected, by 1% for the highest modes at the most. On the contrary, GLR variations produce visible variations (around 10%) in the eigenfrequency values, as can be seen in Figure 4, although vibration patterns do not change significantly. This is the only parameter affecting mode 2, and this fact seems clear, upon taking into account the torsional character of this mode. In modes 5 and 6 (with a mixture of torsional and flexural characteristics) the influence of this parameter is as important as those of EL and ET . 4. CONCLUSIONS
The vibrational behaviour of a guitar plate as a function of the mechanical parameters of wood has been studied for a meaningful set of samples by means of the finite element method. It has been found that the most influential parameters for the normal frequencies and modes are Young’s moduli. Density has been found to be a parameter that alters the frequency values, although it does not change either the relative frequency relations or the pattern of the modes. The shear moduli do not affect the results significantly, except in the GLR case. Finally, the Poisson ratio has hardly any influence on the results, over a very wide range of variations. From these facts, it seems clear that precise determinations of the Young’s moduli and density, mainly, and the GLR value for each sample are necessary if a quantitative study
644
of vibrational behaviour is desired. The rest of parameters could be taken as the ‘‘mean’’ values without causing significant deviations in the results. 1. V. B 1987 Journal of the Catgut Acoustical Society 47, 42–48. Varieties of resonance wood and their elastic constants. 2. O. C. Z 1980 The Finite Element Method. New York: McGraw-Hill.
Journal of Sound and Vibration (1996) 194(4), 640–644
INFLUENCE OF THE MATERIAL CONSTANTS ON THE LOW FREQUENCY MODES OF A FREE GUITAR PLATE A. E Departamento de Fı´ sica, Universidad Publica de Navarra, Campus de Arrosadia s/n, 31006 Pamplona, Navarra, Spain (Received 24 October 1994, and in final form 27 October 1995)
1. The sound production by traditional (that is, not electronic) musical instruments is, from a physical point of view, a classical mechanics subject. In this sense, its physical bases are well established and no fundamental phenomena are expected. Nevertheless, musical instruments are, in general, highly complex vibrating systems— although their components could be simple oscillators with many parameters having an influence on the final sound quality, in some cases in a way that is not well understood. Recently, the availability of advanced experimental equipment and powerful computers has made possible important advances in the analysis of these complex systems, providing physical explanation of the craftsmen’s procedures. As in the construction of a good quality instrument the main work is to shape the different pieces by semi-empirical methods and to put them together, a theoretical prediction of the final changes due to hypothetical modifications would represent a fundamental contribution. In the computer analysis of an instrument, or a part of it, two different aspects can be considered: on the one hand, the geometric aspect and the surrounding constraints, and on the other hand the material properties. In this work, attention is focused in the latter problem. This is because of the observed high sensitivity of the eigenfrequency values to the elastic parameters’ variations and the wide dispersion in the measured material data. Indeed, to determine elastic constants is not an easy task. Therefore, the work reported here is concerned with establishing the influence (and so the necessary degree of accuracy) of the different material parameter variations, in the case of a guitar plate. 2. Because there are no significant differences in geometry among classical guitar plates, the characteristics of the wood should determine their behaviour. However, wood is a very general material in the sense that there are many varieties. Therefore, it is necessary to specify which kind of wood is supposed to be used. In good quality guitars, spruce is the most common choice for the plate. In order to study the spruce plate’s behaviour in a reasonable range of variability, the material characteristics of a group of seven samples as obtained by Bucur [1] by means of ultrasonic techniques have been considered. In this reference it can be observed that the parameters differ significantly from one sample to another, the most important variations being those of the Young’s moduli and density, due to the different spruce subspecies, geographic origins and/or times of ageing by natural drying. As far as the author knows, this collection could be considered a significant one for a guitar maker. 640 0022–460X/96/290640 + 05 $18.00/0
7 1996 Academic Press Limited
641
To obtain the dynamic behaviour in the lowest frequency range, the finite element method [2] (ABAQUS software in a Convex computer) has been used. The plate was divided into 1088 order-two quadrangular shell elements jointed by 3444 nodes. The element size (and therefore the quantity of elements) was chosen by studying the convergence of the results for a rectangular plate of dimensions similar to that of the guitar (maximum height 492 mm, maximum width 372 mm, thickness 3 mm). The anisotropic characteristics were considered by putting the longitudinal direction (that of the grain) on the plate plane and parallel to the symmetry axis, as is customary in most guitars. The plate was assumed to have free-edge boundary conditions. To determine the sensitivity of the normal modes and frequencies to different parameters, the following ‘‘mean’’ values were used: r = 400 kg/m3; EL = 100, ET = 2, GLR = 8·5, GLT = 8·0, GTR = 0·4 (×108 N/m2 ); sLT = 0·49. Each parameter was varied individually, covering the experimental data and calculations were repeated. 3. RESULTS
In Figure 1, the nodal lines for the lowest six normal modes and the associated frequencies are indicated, as obtained for the ‘‘mean’’ value parameters. The variation of the parameters caused the following effects. 3.1. A wide variation, from 0·395 to 0·8, has been taken into account. The normal modes do not change perceptibly, and the eigenfrequencies are similar to the reference values, slightly lower for smaller s values, and vice versa (0·03% for the fundamental mode and 1% for the sixth one). 3.2. It should be reasonable to believe that the density decreases as the age of drying increases. Unfortunately, no data about initial values are available for establishing a time-dependence law. Indeed, it seems clear that there are a lot of additional factors (related to geographic origin, age of the tree, etc.) which have some influence on this, as for the rest of material parameters.
Figure 1. Nodal lines and eigenfrequencies of the six lowest normal modes as obtained from the finite element analysis with use of the ‘‘mean’’ parameter values.
642
Figure 2. The dependence of the eigenfrequencies on the longitudinal Young’s modulus. As can be seen, interchange in the mode order is present due to the large influence of this modulus on the frequencies. For example, the seventh mode appears below the sixty one incidentally (the numeration is that for the ‘‘mean’’ values case, marked with an arrow).
Density values differ approximately 10% from the mean value (except for two cases). In any case, density variations do not change the normal modes’ patterns, and frequency values and density are related: when the density increases, the eigenfrequencies decrease, according to the inverse of the square root of the density, as can be expected, and fi /fj ratios remain unchanged. 3.3. ’ These are the most characteristic anisotropic parameters in wood. Unfortunately, as can be seen in reference [1], they have strong deviations from one subspecies to another, and even for different samples belonging to the same subspecies. The normal modes’ evolution has been studied with EL varying from 70 to 150, ET from 1 to 4 (×108 N/m2 in both cases), and the results are presented in Figures 2 and 3. As can be seen, the order of the normal modes does not remain the same for all the values, due
Figure 3. As Figure 2, but with variation of the transversal Young’s modulus.
643
Figure 4. The dependence of the eigenfrequencies on the GLR shear modulus.
to the strong dependence of the frequency values for some of them. The numeration is that of the reference case (EL = 100, ET = 2). Modes 1, 2 and 3 are nearly EL -independent, whereas modes 4, 5 and 6 rise when EL increases. On the other hand, ET variations have a visible effect on all the frequency values, increasing them when ET goes up. These changes are more evident in the odd modes, mainly in the third one. 3.4. GLR and GLT are nearly equal, GLR being slightly higher than GLT , and GTR is much smaller than the others, close to a 1:20 relation. From sample to sample, variations of around 15% can be observed in each shear modulus. GLT and GTR variations, in the samples’ range, do not produce any visible change in the nodal lines, and the frequencies are only slightly affected, by 1% for the highest modes at the most. On the contrary, GLR variations produce visible variations (around 10%) in the eigenfrequency values, as can be seen in Figure 4, although vibration patterns do not change significantly. This is the only parameter affecting mode 2, and this fact seems clear, upon taking into account the torsional character of this mode. In modes 5 and 6 (with a mixture of torsional and flexural characteristics) the influence of this parameter is as important as those of EL and ET . 4. CONCLUSIONS
The vibrational behaviour of a guitar plate as a function of the mechanical parameters of wood has been studied for a meaningful set of samples by means of the finite element method. It has been found that the most influential parameters for the normal frequencies and modes are Young’s moduli. Density has been found to be a parameter that alters the frequency values, although it does not change either the relative frequency relations or the pattern of the modes. The shear moduli do not affect the results significantly, except in the GLR case. Finally, the Poisson ratio has hardly any influence on the results, over a very wide range of variations. From these facts, it seems clear that precise determinations of the Young’s moduli and density, mainly, and the GLR value for each sample are necessary if a quantitative study
644
of vibrational behaviour is desired. The rest of parameters could be taken as the ‘‘mean’’ values without causing significant deviations in the results. 1. V. B 1987 Journal of the Catgut Acoustical Society 47, 42–48. Varieties of resonance wood and their elastic constants. 2. O. C. Z 1980 The Finite Element Method. New York: McGraw-Hill.