# 2. Experiment. 3. Numerical model 2 J. ALEJANDRO TORRES, J. LUIS VILLARREAL AND R. RAMÍREZ

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2 2 J. ALEJANDRO TORRES, J. LUIS VILLARREAL AND R. RAMÍREZ FIGURE 1. Analogy for the lowest mode shapes in a guitar top plate, through a soap membrane vibrating in its first (top) and second (bottom) modes. In the present paper, two tools were developed to visualize features related with bending and acoustic waves properties. One consisted in an experiment involving a soap membrane to show vibrational behaviors, and the other one in software to scientific visualization of data of discrete models of two guitar top plates. 2. Experiment The objective of the experiment described below is naked eye visualization of the most prominent deflections of a vibrating structure, called mode shapes [9]. It can be achieved in a classroom. The experiment only requires soap solution to wash dishes; and two rods jointed by thread, forming a rectangular frame. Two pencils for this latter task can be useful, but bigger is better. The rods together are submerged and extracted from the solution. A soap membrane is created once the frame is quickly extended. Comparing with a guitar top plate, the membrane is very flexible and it has very low mass encouraging two characteristics: mode shapes at very low frequencies (even less than one cycle per second) and visible deflections of the surface (in the order of centimeters). Both characteristics are desirable due the objective of the experiment that it is being proposed. Hand shaking slowly the membrane, the first mode shape can be easily distinguished. All the membrane surface moves in phase and in the same direction (Fig. 1 top). If the shaking frequency is increased, other higher mode shapes can be detected. For the second mode shape, a nodal line (a line without deflections) crossing the membrane is detected (Fig. 1 bottom). A video performing this experiment is available in Ref. 10. Although the effect of air in the film is, by far, larger than for guitar top plates, explained features results be analogous. Comparing the explained modes of the membrane with the lowest mode shapes of a mounted top plate (Fig. 1 right). 3. Numerical model In order to highlight differences in behaviors of top plates with structural modifications, a top plate with two kinds of FIGURE 2. Steps for data calculation and visualization. fan bracing were used in this work: a symmetrical and other one with asymmetrical design. Methodology is briefly schematized in four steps (Fig. 2), which the last step was bolded to remark the work developed in the present paper. In fact, the whole calculated data can be obtained using batch files completely included in Ref. 11. The first step refers to simulated modal analyses performed using commercial finite element (FE) software. The outline of the top plate was restricted to zero displacements (similar to the condition mounted in the soundbox). This simulation has shown good accuracy comparing with experimental measurements, as it was previously reported in Ref. 12. In the second step, it is indicated that sound radiation of the top plate was calculated from mode shapes obtained in the first step. Calibrated numerical methods based on Fourier Acoustics were used for this purpose. Details of the complete algorithm can be found in Ref. 13. Sound radiated by the mode shape of the highest frequency was calculated to show features of the data related with acoustic waves. For the third step, a Matlab function was written to array the data to be visualized. Input files of this function must be individual text data in a row. Output files are arrayed in the standard format of the Visualization Toolkit format (vtk files) [14]. This open-source format allows multidimensional data handling; through structured and unstructured points, nodal conectivity and fields, among many others specifications. In the four step, scientific visualization tools were applied exploring the previously calculated vibroacoustic data of the guitar top plate. Procedures related with this last step are detailed in Subsec Scientific visualization software Scientific visualization is concerned with exploring very large data sets and information graphically to gain a quick understanding. The difference with presentation graphics is

4 4 J. ALEJANDRO TORRES, J. LUIS VILLARREAL AND R. RAMI REZ F IGURE 3. Scientific visualization software displaying two diferrent top plate designs (left windows) and some fully turnables mode shapes (center and right windows). of points having the same phase [18]) crossing air, internal inspection of the medium is necessary. This detection is a hard task, because visualizing the whole data is not possible, at least for a single display. An analogy of this complication can be done figuring a brick of Neapolitan ice cream. Flavors in the ice cream take the place of sound pressure values: vanilla flavor would correspond to no pressure fluctuation, chocolate to compress, and strawberry to expansion. You can see at the outside of the brick, but you can not locate where more chocolate, vanilla or strawberry inside are. It would be necessary slicing the brick in a determined zone to inspection it. Slicing is one of several applied techniques to visualize inside of this kind of brick data. The quick update of a model for dynamic visualizations is also difficult and often beyond of the capabilities for current standard software. A time domain animation used to require froze the model in sequential images to obtain a movie, creating a file incapable of being three-dimensionally rotated. Interaction only with frozen models is inefficient and not adequated in exploration. If it is necessary to explore another perspective in the model, you would have record another movie, generating an undesirable delay of the analysis process. The time domain performance of the developed software (allowing real time interaction of dynamic data) facilitates a better understanding of wave propagation behaviors, helping to study multidimensional information. In point of fact, exploration of the dynamical and kinematical performance in the acoustic data exhibit sufficient accuracy and a faithful display for relevant propagation characteristics, by means of combining use of software and techniques proposed in the present work. Figure 4 shows a software screen with sound radiation by a mode shape at 738 Hz. At left is located the toolbar. At its top is located the controls of the dynamic animation of the data. Below these controls, the colorscale bar is plotted for compression and expansion values of the air pressure. At the center of the toolbar, it is a medium-size window with the edges of a brick sliced by three orthogonal planes. This medium-size window is the control for exploring the data in the big windows (identified using a and b). Bottom of the toolbar have a small window which brings the whole data to the selected big window to exploration. Sound radiation was calculated on an air brick of 1.05 m over the top plate. Explorations detailed in the next paragraph can be done handling interactively a window, but in the present paper two windows (Fig. 4(a) and Fig. 4(b) are plotted due that frozen images will be explained). Spherical wavefronts. In Fig. 4(a) a slice is in contact with the structural source. The co-ordinate system indicated in the top plate with the symmetrical design without deflections shown in Fig. 3 has been preserved. Hard contrast between colored pressures can be basically interpreted as phase change (expansion to compression and vice versa). Spherical wavefronts of the acoustic waves are distinguished on the pressure data on orthogonal slices to the source plane. The location of the slicing planes allows seeing the whole air in contact with the top plate. Directivity. In Fig. 4(b) the brick data has been turned. Orthogonal slices were moved to the co-rodinate system origin. Sound radiation is preferentially generated for x and z postitive values, as it was marked. In the side of more sound radiation (x and z positive), wavefronts almost reach the source plane; unlike the less sound radiation side (x negative and z

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