## 3 Results
### Eigenfrequency Study  
Figure 4 shows the relative variation in the first ten eigenfrequencies for the (a) free and (b) fixed feet boundary conditions. In particular, the variation in the eigenfrequencies is normalized with respect to the Normal Model.  
In free boundary conditions, the modal frequencies that change the most when varying the shape of the legs are associated with modes 1, 4, 6, 7, 8. The mode shapes associated with these modes present a displacement concentrated in the legs (Figures A2 and A3). In particular, the mode shapes of modes 6 and 7 exhibit the most different mode shapes if we compare the Normal Model to Model 30.  
In fixed boundary conditions, modes 3, 4, 5, and 6 are the most affected in terms of frequency by the change in the shape of the legs. The mode shapes associated with modes 3, 5, and 6 that occur in plane \(xy\) are also reported in other studies [9,15] as the most relevant.  
A summary of the eigenfrequencies of the first 10 modes under different boundary conditions is given in Tables 2 and 3. Figure 5 allows one to visualize the same eigenfrequencies and to compare them for each boundary condition. It is apparent that the boundary conditions have the largest impact on the natural frequencies. If the feet are fixed, the first frequency decreases by almost 1000 Hz with respect to the free boundary conditions. Something similar happens when using springs for the boundary conditions and there are many more modes in the mid-frequency range (1-4 kHz) than in the free boundary case. This dependence on the boundary conditions shows that it is not possible to directly obtain insights into how the bridge behaves in the cello from its free body behavior. Accurately describing the boundary conditions of a real instrument goes far beyond the scope of this article and should be tackled in future research.  
Figure 4: (**a**) Changes in natural frequencies \(f_{n}\) in free boundary conditions. (**b**) Variation in natural frequencies \(f_{n}\) in fixed boundary condition. Frequencies from different models are shown as a function of the mode number \(n\); \(f_{0}^{n}\) is the frequency of the \(n\)th mode for the Normal Model. (**c**) Mode shapes and corresponding natural frequencies for \(n=3,\ldots,6\) in constrained boundary conditions, \(|d|\) being the norm of displacement, shown on the Normal Model.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Mode Number** & **(a) Free** & **(b) Fixed** & **(c) Spring** & **(d) Spring+** \\ \hline
1 & 1227 & 333 & 160 & 498\({}^{m}\) \\
2 & 1311 & 1580 & 309\({}^{m}\) & 861 \\
3 & 1331 & 2316 & 968 & 1597 \\
4 & 2337 & 2918 & 1090 & 2231 \\
5 & 3306 & 3224 & 1507 & 2364 \\
6 & 3753 & 3734 & 2149 & 3076 \\
7 & 4298 & 4003 & 2535 & 3083 \\
8 & 4768 & 5369 & 3087 & 3250 \\
9 & 5347 & 6504 & 3246 & 4082 \\
10 & 5889 & 7252 & 3592 & 4887 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Values of the first 10 natural frequencies (in \(Hz\)) of the Normal Model in different boundary conditions. Case (c) only includes springs at the feet, while case (d) both springs at the feet and at the upper arch. Modes where the bridge moves rigidly are denoted with \({}^{m}\). All mode shapes are available in Figures A2, A4, A6, and A8.  
Figure 5: Modal frequencies as a function of the mode number, from Tables 2 and 3. Note that in this case, eigenfrequencies are rearranged in order to take care of mode switches. Each plot compares the values obtained for the Normal model and Model 30 in a specific boundary condition. (**a**) Free boundary conditions. (**b**) Fixed boundary conditions. (**c**) Boundary condition with springs at the feet only. (**d**) Boundary condition with both springs at the feet and at the upper arch.