## 3 Results  
### Numerical Analysis Using FEM  
Several design modifications were analysed by numerical modelling using the finite element method in terms of the shape of the hole, its size and the number of holes. In the present paper, the most advantageous variant of the design modification of the existing  
Figure 3: Measuring stand for the realization of dynamic analysis. Figure 2: View of the torque measurement carried out on the flange that is part of the pivot joint. component is presented, which decreased the overall mass of the manipulator and did not require any design modifications to other components. When defining the boundary conditions, it was taken into account that the considered flange is loaded only by torque. SolidWorks 2018 software was used for the numerical analysis. The finite element model without structural modifications was composed of 16,674 nodes and 8311 volume elements with quadratic approximation. The finite element model with structural modifications was composed of 260,099 nodes and 157,783 volume elements with quadratic approximation. Figures 4 and 5 show the equivalent (von Mises) stress field and the displacement field for the original and modified shape of the flange under consideration at a maximum load of 200 Nm. From the figures, it is clear that the above design modification increased the stress levels around the concentrators, i.e., in the areas between the oval holes chosen as the application locations of the strain gauge sensors. As already mentioned, the presented methodology is based on comparing the principal strains registered by the strain gauge at the location of its application. In Figure 6, for illustration, the values of the principal strains at a maximum torque load of 200 Nm are presented. At the location of the applied strain gauge, the principal strains were numerically computed as \(\varepsilon_{1}\) = 446 \(\mu\varepsilon\) and \(\varepsilon_{3}\) = \(-452\)\(\mu\varepsilon\). Figure 4: Fields obtained using maximum torque of 200 Nm for the original shape of the flange. (**a**) Equivalent (von Mises) stress; (**b**) total displacement. Figure 5: Fields obtained using maximum torque of 200 Nm for the modified shape of the flange. (**a**) Equivalent (von Mises) stress; (**b**) total displacement. Table 1 shows the results of the above numerical FEA corresponding to the minimum (28 Nm) and maximum (200 Nm) operational loads of the flange with structural modification. Figure 7a shows the equivalent (von Mises) stress field obtained at the location of the applied strain gauge. It is obvious that the values of the equivalent stresses (about 43 MPa) can be considered constant in the area of the strain gauge measuring grids. The following section presents the procedure for validating numerical modelling results by experimental measurements using the strain gauge method. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Torque** & \(\sigma_{\text{eqv,max}}\)** & \(\sigma_{\text{eqv,SG}}\)** & \(\tau_{\text{SG}}\)*** & \(2\varepsilon_{\text{45}}\) & \(\text{Ures}_{\text{max}}\) \\
**(Nm)** & **(MPa)** & **(MPa)** & **(MPa)** & **(\(\mu\varepsilon\))** & **(mm)** \\ \hline
28 & 7.02 & 6.12 & 3.51 & 127 & 0.003 \\
200 & 49.98 & 43.36 & 25.03 & 898 & 0.024 \\ \hline \hline \end{tabular}  
* \(\sigma_{\text{eqv,max}}\) is the maximum value of the equivalent (von Mises) stress at the edge of the hole; ** \(\tau_{\text{eqv,SG}}\) is the value of equivalent (von Mises) stress read in the strain gauge application area; *** \(\tau_{\text{SG}}\) is the value of shear stress read in the strain gauge application area.  
\end{table} 
Table 1: Values of equivalent and shear stresses, strains and maximum total displacements for the flange under consideration at the minimum and maximum operating loads. Figure 6: Fields of principal strains obtained using maximum torque of 200 Nm for the modified shape of the flange. (**a**) \(\varepsilon_{1}\); (**b**) \(\varepsilon_{3}\). Figure 7: Location of the applied strain gauge.