## 2 Experimental Modal Analysis
### Experimental Modal Analysis Results  
The Experimental Modal Analysis (EMA) has been carried out to evaluate the modal properties of the prosthesis, i.e., the natural frequencies (\(f_{r}\)), damping ratios (\(\zeta_{r}\)), and mode shapes (\(\mathbf{\Phi}_{r}\)), which are identified in the frequency range from 0 Hz to 750 Hz using PolyMAX algorithm [42], and 6 mode shapes are obtained (Table 2). Figure 3: Reference geometry for experimental test. Figure 2: Details of the implemented hardware: referring to accelerometer nomenclature of Figure 3, tri-axial node 303 (**a**), tri-axial nodes 207 and 311 (**b**), tri-axial node 216 and mono-axial node 314 (**c**). The reference system considered in the description of the mode shapes is the same described in Section 2.1, as illustrated in the plots of the experimental mode shapes (Figure 4). As clearly evident from the table, the principal mode shapes of the prosthetic are the bending in the \(XZ\) plane and the torsion around \(Z\) combined with bending in the \(XY\) plane. This is due to the prosthesis geometry since the thickness of the cross-section is less than the width in \(Y\) direction. Furthermore, it is reasonable to have as first mode the bending in the vertical plane since the limb prosthetic must deform along the \(Z\) direction in order to promote the storing and restoring energy function, helping athletes to perform the running gait. In order to check the reliability of the results, the experimental (Exp) and the synthesised auto-inertance FRFs are compared, as shown in Figure 5. The synthesised ones are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode n.** & \(f_{r}\) **[Hz]** & \(\zeta_{r}\) **[\%]** & **Mode Type** \\ \hline
1 & 86.33 & 0.38 & 1 st bending \(XZ\) \\
2 & 226.49 & 1.06 & Torsion \(RZ\) /1 st bending \(XY\) \\
3 & 275.25 & 0.24 & 2 nd bending \(XZ\) \\
4 & 371.77 & 0.41 & Torsion \(RZ\) /2 nd bending \(XY\) \\
5 & 603.34 & 0.48 & 3 rd bending \(XZ\) \\
6 & 735.58 & 1.95 & Torsion \(RZ\) /3 rd bending \(XY\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Natural frequencies (\(f_{r}\)) and damping ratios (\(\zeta_{r}\)) identified in the frequency range from 0 Hz to 750 Hz. Figure 4: Experimental mode shapes plotted using LUPOS software. obtained using the following two methods: using TestLab (Syn), where the identified modal set and both lower and upper residuals (\([LR]\) and \([UR]\)) are considered, Equation (1) [42], and with the modal superposition (MS) principle, Equation (2), where no residuals are taken into account. In Equations (1) and (2), \(A_{j,k}(\omega)\) is the inertance FRF computed for each \(\omega\) between the two DoFs \(j\) and \(k\), while \(\omega_{r}\) and \(\zeta_{r}\) are the natural frequency and the damping ratio of the \(r\)-th mode shape \(\mathbf{\Phi}_{k}\). For the auto-inertances, the two DoFs are the same: \(k=j\). For sake of clarity, the excited (E) and measured (R) nodes with their directions are mentioned in the title of each graph. \[A_{j,k}(\omega)=\frac{\bar{x}_{j}(\omega)}{F_{k}(\omega)}=-\omega^{2}\frac{x_{ j}(\omega)}{F_{k}(\omega)}=\sum_{r=1}^{n}\frac{-\omega^{2}\ \Phi_{j,r}\Phi_{k,r}}{\omega_{r}^{2}+2\,\mathrm{i}\,\zeta_{r}\,\omega_{r}\, \omega-\omega^{2}}+[LR]-\omega^{2}[UR] \tag{1}\]  
\[A_{j,k}(\omega)=\frac{\bar{x}_{j}(\omega)}{F_{k}(\omega)}=-\omega^{2}\frac{x_ {j}(\omega)}{F_{k}(\omega)}=\sum_{r=1}^{n}\frac{-\omega^{2}\ \[PERSON]_{j,r}\[PERSON]_{k,r}}{\omega_{r}^{2}+2\,\mathrm{i}\,\zeta_{r}\,\omega_{r}\, \omega-\omega^{2}} \tag{2}\]  
As clearly visible in Figure 5, really good agreement between the auto-inertances is obtained for nodes 207 and 303 in the frequency range between 0 Hz and 400 Hz. Furthermore, the synthesised auto-inertance values resulting from TestLab are closer to the experimental outcomes. Indeed, they are more reliable with respect to the modal superposition calculus since, in the first computation model, both lower and upper residuals are considered. The linearity of the component is then verified by analysing the Maxwell reciprocity.