Investigation of influence of pre-stresses on Viola da Gamba sound using FEM.

Transcription

1 Investigation of influence of pre-stresses on Viola da Gamba sound using FEM. Tomasz Jan Wilczyński Faculty of Mechanical Engineering and Robotics, AGH University of Science and Technology, Krakow, Poland. Roman Filipek Faculty of Mechanical Engineering and Robotics,AGH University of Science and Technology, Krakow, Poland. Paweł Wilczyński Faculty of Mathematics and Information Science, Warsaw University of Technology, Warsaw, Poland. 1. Introduction 1 Summary This study relates to one of the most popular string instruments in the Renaissance and Baroque eras. Based on a scheme by Nicola Bertrand from the year 1720, a three-dimensional geometrical model of a viola da gamba has been created. The authors focused on modeling the exact shape of the convexity of the top plate in order to obtain the highest reliability of the results. A proper selection of a finite number of element types has guaranteed a consistent combination of individual elements of the instrument. The main objective of the study was to achieve the effect of inducing pre-stresses on a variation of proportions between modal frequencies. The survey methodology was based on a modification of the structure which influences the pre-stresses. Next, a numerical sequence experiment involving modal analysis was carried out in order to investigate the aforementioned aspect. The first modification concerns the placement and the length of the instrument's soundpost, connecting the top plate with the back plate. It is worth noting that changes in the angle of inclination of the soundpost were also taken into account. Another aspect concerns the placement of the bridge, and the influence of the tension of 7 gut strings on the bridge, and the effects of the bridge pressure on the top plate. The material used in this case was wood, specifically spruce and sycamore, and an average humidity of 12% was assumed. The whole study was carried out with the use of the Finite Element Method (FEM). Viola da gamba is a one of the famous instruments in the Renaissance and Baroque eras. Its shape is similar to the double bass. There are 5 main types of viola da gamba in existence. They are as follows: soprano, tenor, treble bass, bass, and double bass. They were used in viol consort ensembles. In this article, an investigation of the bass viola was made. The general rule of playing viola da gamba is similar to playing the cello, but the main difference relates to the strings. Nowadays, the strings of bowed string instruments the made of steel, but for violas the main material is catgut. It causes many difficulties with sound propagation, esspecialy in the case of multiple bows being used while using the same amount of force. Furthermore, another important dissimilarity between present day and ancient bowed string instruments is the number of the strings, 4 and 6-7 respectively. Essentially, the researched instrument has been tuned to 440 [Hz], which is a different tuning than in the Baroque era, where the general tuning value, based on the A1 sound, was 415 Hz and 460 Hz [2]. In general, the main objective of the study was to obtain the effect of pre-stresses on a variation of proportions between modal frequencies. The survey methodology was based on a modification of the structure which influences the pre-stresses. A numerical sequence experiment involving modal analysis was carried out in order to investigate the forementioned aspect. The methods used to achieve the pre-stress effect were a specific

2 Figure1. The scheme of Nicolas Bertrand's viola da gamba bass. placement of the soundpost, the force load effect, specific boundary conditions, and a strictly selected FEM data set. Moreover, in order to obtain the highest reliability of results, a convexity of the upper soundboard as well as specific wood materials were used. This paper includes two methods of investigation, one for an instrument with pre-stresses and one without. The next step is the comparison. Most of the presented results contain dependences and relations between the input parameters and their outcomes. The most relevant fact is that there is very little knowledge about the viola da gamba based on an the available articles, books, etc. [1]. This is why the authors of this paper wanted to achieve new results in order to fill the gaps in the literature. 2. Parameters and data set. The construction of the viola da gamba is similar to the cello, but with some differences. An exploded view of the cello is presented in [1] p The aforementioned differences relate to additional reinforcement plates over a lower soundboard (which is not convexed, as opposed to the cello), and a sliding steel spike. The scheme of the researched instrument is presented in Figure 1. Based on that scheme, a numerical model has been created, though excluding the corner blocks, as well as the edging strip of wood attached to both soundboards. These parts have been added to the model due to the advice of instrument builders. The material used in this case was wood, specifically spruce and sycamore, and an average humidity of 12% was assumed. The upper soundboard, soundpost and the bass bar are made of spruce, and the rest of the elements are made of sycamore. All of the mechanical properties of wood, such as: - Poisson s ratios for various species µ, - modulus of elasiticity E, - modulus of rigidity G, - density ρ, were based on [3, 6]. Despite of the anisotropic nature of wood, the numerical instrument has been modeled using orthotropic and isotropic wood materials. The most important parts, in order to achieve results of a higher reliability, have orthotropic properties

3 (for instance, the soundboards, bass-bar, reinforcement plates, bottom block), and the rest of body has isotropic properties. The orthotropic coordinate system of the principal axes, used for the aforementioned parts of the instrument, is [N]. The whole layout of reaction forces is presented in Figure 3, and the calculated values are: N = 513 [N], T2 = 1163 [N]. Moreover, the values of the listed angles are: φ1=10.4º, φ2=15.3º. Table 1. String parameters. Name of sounds A D G c e a d Relative frequencies [Hz] shown in Figure 2. Figure 2. Three principal axes of wood with respect to grain direction and growth rings. In order to calculate strings' tension on bridge, string equation 1 has been resolved. 2 u $, (1) x (x,t) = 1 2 c 2 u 2 t (x,t) 2 where: u(x, t) - vertical displacement of the string from the x axis at position x and time t c - speed of sound within a material The boundary conditions are: $ u(0,t) = 0, $ u(l,t) = 0. Based on [4], the tensile strength of the string has been calculated from equation 2. $ T = ρ'4π 2 f 2, (2) where: $ ρ' = m - linear density of the string, l In order to make a proper estimation, some of parameters need to be measured. All of the measured and calculated data are listed in Table 1, and the distribution of reaction forces is presented in Figure 3. Figure 3. Distribution of reaction forces over the instrument. Taking into account the convexity proportion of the bridge, an average tensile strength is T1 = 1141 Length of string [mm] Mass of string [mg] Calculated tensile strength [N] Processing and mathematical model In this research, modal analysis was performed. It takes into account the initial pre-stress and deformation of the structure introduced by the impact of static forces on the viola da gamba and their placement. The model was built using the finite element method (FEM), with ANSYS package calculation. When building the model, both shell and solid elements were applied. The former were used for bouts and reinforcement plates spread over the lower soundboard. These are 8 nodes of the Shell281 element. For modeling additional reinforcements, a 20-node Solid186 element was applied. SHELL281 is suitable for analyzing thin to moderately-thick shell structures. The element has eight nodes with six degrees of freedom at each node: translations in the x, y, and z axes, and rotations about the x, y, and z-axes [5]. SOLID186 is a higher order 3-D 20-node solid element that exhibits quadratic displacement behavior. The element is defined by 20 nodes having three degrees of freedom per node: translations in the nodal x, y, and z directions. The SOLID186 homogeneous structural solid is well suited for modeling irregular meshes [5].

4 The data flow starts in three-dimensional geometry model. A very important fact is the upper soundboard average thickness, which is equal to 3,5 * 10-3 [m]. The next step is setting up the materials for each of the parts. Furthermore, after the meshing operation (presented in Figure 4), the tension on the bridge and the upper soundboard effect has been set. The total number of elements is The aspect ratio ranges Modal analysis based on a prior, non-linear, preloaded status has been performed by using the linear perturbation analysis procedure. The effect on the structure from the previous static analysis was included. from 4 to 5, depending on the calculations of each experiment. Figure 4. Mesh over the instrument. A different kind of pressure is generated by the soundpost, because of the effect of lengthening through the lower soundboard. All loadings and displacement are presented in Figure 5. Figure 5. Loadings and displacement. After the loadings, the static structural and prestress analysis was made. The last step was performing the modal analysis. The whole linear perturbation analysis is shown in Figure 6. Furter explanation of linear perturbation analysis is available in [5]. Figure 6. Linear perturbation analysis [5]. The tangent matrix Kti can be used in linear perturbation analysis in order to obtain the effect of preload, as the linear stiffness matrix does not give the correct solution without preloading. Kti is the global tangent matrix which can be symbolically segregated into other matrices, as follows: t $ K i = K M i + K N C i + K i (3) Ki M = the part of the tangent stiffness due to the properties of the material. Ki N = the stress stiffening matrix introduced by the non-zero stresses of the structure. Ki C = the total stiffness matrix contributed from the contact elements of the model. The equation of motion for an undamped structure with no time-varying forces, displacements, or pressures applied, is $ M u + K t i u = 0 (4) For a linear system, free vibrations will be harmonic:

5 $ u = φ j cos(ω i t), φj = eigenvector representing the mode shape of the ith natural frequency, ωi =ith natural circular frequency (radians per unit of time),t = time. Equation 4 becomes $ ( ω 2 j + K t i )φ j = 0 (5) The program calculates the eigensolution 2 eigenvalues $ λ j = ω i using the following equation $ K (6) t i φ j = λ j Mφ j {φj} = eigenvector, [M] = structural mass matrix. The experiment has been designed with the following input parameter changes in mind: soundpost placement, diameter value, and length of the soundpost. A very important aspect is using contacts. The overall number of contacts is 75, with 7 of them being set manually. Each contact is bounded with asymmetric behaviour. Using the contacts between the bridge and the upper soundboard, a pinball region equal to 4*10-3 [m] was used. Some of the most relevant contacts are shown in Figure 7. Figure 7. Selected contacts. 4. Results The first comparison of results of the model with and without pre-stresses can be seen in Table 2. Frequency number Without pre-stress Mode frequencies [Hz] With prestress 1 100,79 85, ,5 96, ,14 116, ,64 129, ,11 136, ,73 171, ,08 177, ,46 191, ,56 193, ,86 218, ,44 237, ,97 254, ,64 262, ,68 273, ,21 294, ,22 300, ,66 303, ,92 307, ,98 313, ,32 321, ,54 339, ,33 343, ,46 351, ,43 369, ,51 378, ,68 390, ,78 405, ,56 419, ,76 423, ,5 429,86

6 Table 2. Mode frequencies. These are calculated for the same placement and diameter of the soundpost, but for a different length. This difference is equal to 3*10-3. ue to the fact that the beginning of the coordinate system for x and y is in the same place as the remote force in Figure 5, the soundpost's placement and diameter is as follows: - x = 0,046 [m], - y = 0,297 [m], - d = 0,009 [m]. Further results are presented in Table 3, where mode figures for three selected modes can be seen. Table 3. Modes with or without pre - stresses. modes without pre-stress This comparison showes big differences between each mode of the model, with and without prestresses. The most visible differences are for modes 3 and 10. Additional results relate to the experiment made in the study. The range of changes in placement, diameter, and length of the soundpost is as follows: - diameter = 0,0081-0,0099 [m], - x = 0,036-0,056 [m], - y = 0,286-0,308 [m], - length = real length + (0 to 3 [mm]). modes with pre-stress

7 Table 4. Outcomes after an experiment. mode 1 mode 5 In the first figure in table 4 mode 1 is presented with x and y placement of soundpost relation. The main reasult is, that frequencies decresing when y values going down. But, for decreasing x values, frequencies increasing. It causes, a more rigit instrument structure while soundpost is closer to middle of instrument. In mode 5 case, when y values going down the frequencies also decresing.

8 But, for x values, situation is trhrough the opposite way. Furthermore, for relation between length of soundpost to y placement, freqiencies going up for longer soundpost. According to soundpost decimeter. Generally frequencies going up for bigger diameter in both cases. Unfortunately, due to a large set of results, only the most powerful and fruitful results have been included in this article. It is obvious that the results in their entirety correspond non-linearly with each other. 5. Conclusions The main conclusions are: - all frequencies decrease for the pre-stresses. - the modes, in the vast majority, are different between models with and without pre-stresses. - the proportionality of mode frequencies in both cases is mostly different. - in the majority of cases, a lower value placement of y gives lower mode frequencies. - in many cases, a middle range placement of x gives higher mode frequencies. - generally, with the soundpost length ranging [mm], mode frequencies grew. Moreover, the proposed numerical model gives more reliable results because of its very good reproduction of the upper soundboard convexity, as well as proper model properties, such as: - low aspect ratio (good meshing) - all symetric and bouded contacts - good boundary conditions. This results in a much accurate and faster computation. [2] H. W. Myers: Renaissance Viol Tunings: a Reconsideration. Journal of The Viola da Gamba Society of America, Vol. 44 (2007-8) [3] United States Department of Agriculture Forest Service: Wood Handbook - Wood as an Engineerng Material. Forest Products Laboratory, FPL-GTR-190, Madison, [4] L. E. Kinsler, A. R. Frey, A. B. Coppens, J. V. Sanders: Fundamentals of Acoustics, 4th Edition. John Wiley $ Sons. New York, [5] Ansys Inc.: ANSYS Inc. PDF Documentation for Release last accessed [6] V. Bucur: Acoustics of Wood. T. E. Timell, R. Wimmer (eds.). Springer, Berlin, Acknowledgement The authors would like to appreciate Mr. Andrzej Pancerz for his useful advice over the course of the research. References [1] T. D. Roosing: The Science of String Instruments. Springer, New York, 2010.