## 3 Numerical Modelling  
The FE models are developed using LUPOS 2023 software [43], namely a Lumped Parameter Open Source FEM code developed in Matlab, and real modal analysis is computed with MSC Nastran 2017. The prosthetic is discretised with 3D hexagonal elements (CHEXA). The characteristics of the resulting model are presented in Table 3, where the number of elements along the thickness and the width directions are also reported. The model is displayed in Figure 7. From the upper and lateral views of the image, it is clearly visible that gradual changes along the width and thickness dimensions of the real prosthetic are well discretised. Figure 6: Maxwell reciprocity test: experimental cross-inertance FRFs comparison interchanging excited (E) and measured (R) nodes and directions. Cross-inertance calculated using the modal superposition principle (Equation (2)) in black dot. Figure 7: 3D LUPOS FEM model of the foot blade prosthetic: lateral (a), top (b), and isometric (c) views. The same free-free conditions of the experimental test have been considered; thus, the model has not been constrained. Firstly, the material is considered isotropic, and its properties are reported in Table 4. The density is calculated knowing the weight and volume of the component. The weight is experimentally measured, while the volume is deduced from the prosthetic shape creation on the 3D CAD software SolidWorks 2015. The starting value of the Young modulus was 64.3 GPa, resulting from the static analysis performed by [PERSON] et al. [25] on the Cheetah Xtreme model, category 5. The numerical model has been tuned by adjusting the values of the Poisson ratio and the Young modulus to achieve a closer match with the experimental natural frequencies. The tuning procedure for the isotropic material was performed considering that the Young modulus scales equally all the natural frequencies since it is related to the stiffness. On the other hand, the Poisson ratio has effects on the relation between torsional and bending modes since it is the only parameter that affects the eigenvalues related to different mode shapes differently. Performing a parametric analysis on the Poisson ratio, it is noticed that the torsional modes are more sensitive to its variation with respect to bending ones. In detail, by varying the Poisson ratio from 0.15 to 0.45, the bending natural frequencies values slightly increase, while the torsional ones decrease by a major factor. Consequently, a Poisson ratio of 0.45 is selected to achieve the lowest natural frequency values for torsional modes and the highest for bending modes. The correspondence between numerical and experimental results is also performed by evaluating the Modal Assurance Criterion (MAC) between the simulated and experimental mode shapes (Equation (3)) [44]. It is an index that measures the correlation level between two mode shapes \(\mathbf{\Phi}_{j}\) and \(\mathbf{\Phi}_{k}\), respectively. It is defined as follows:  
\[MAC_{j,k}=\frac{\left[\mathbf{\Phi}_{j}^{T}\mathbf{\Phi}_{k}\right]^{2}}{\left[\mathbf{ \Phi}_{j}^{T}\mathbf{\Phi}_{j}\right]\left[\mathbf{\Phi}_{k}^{T}\mathbf{\Phi}_{k}\right]} \tag{3}\]  
Given its definition, the MAC index is always positive, and its range is from zero to one. When two eigenvectors are orthogonal or have very low similarity, MAC is nil, while it tends to one when the two modes are equal. Two models show the same dynamic behaviour when the MAC matrix is the identity matrix. Figure 8a represents the MAC between FEA and EMA results adopting an isotropic material for the model. The dashed line represents the iso-frequency curve. In this case, the MAC matrix is not diagonal. Indeed, besides the first mode shape, which is bending in the \(XZ\) plane, the mode order does not correspond between the numerical (FEA) and experimental results (EMA). This aspect is underlined by Table 5, where EMA and FEA (isotropic material) natural frequencies are compared, considering EMA mode ascending order.