## 2 Materials and Methods
### Deriving New Quantities from Full-Field Receptances
#### 2.2.3 Estimating the Dynamic Stresses  
By exploiting the _complex-valued_ FRF formulation of the _Stress FRF tensor_\(\mathbf{H_{\sigma}}(x,y,\omega)\) as in Equation (5), (B.2), dynamic _Stresses_\(\mathbf{\Sigma}(x,y,\omega)\in\mathbb{C}\) are easily computed as:  
\[\mathbf{\Sigma}(x,y,\omega)=\mathbf{H_{\sigma}}(x,y,\omega)F(\omega)\in\mathbb{C}, \tag{8}\]  
with components as expressed in [N/m\({}^{2}\)], but where a \(F(\omega)\in\mathbb{C}\) force is used, as in the general modelling of real-life excitations proposed in Section 2.3, under the linearity assumption between structural responses and excitations, grounding Equation (1).  
There are two equivalent means--the diagonalisation base being the same--of obtaining the detailed maps of the dynamic _Principal Stresses_\(\mathbf{\Sigma}_{P}(x,y,\omega)\)16\(\mathbb{C}\): (i) by the  
Figure 5: Examples of SLDV-based _von Mises equivalent stress FRF maps_ from optical techniques in Equation (7) from shaker 2: at 121 Hz in (**a**), at 127 Hz in (**b**), at 246 Hz in (**c**), at 250 Hz in (**d**), at 279 Hz in (**e**), at 284 Hz in (**f**).  
diagonalisation of the dynamic _Stresses_\(\mathbf{\Sigma}(x,y,\omega)\in\mathbb{C}\) in Equation (8); (ii) by exploiting the FRF-based formulation of the _Principal Stress FRF maps_\(\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)\) with a proper \(F(\omega)\in\mathbb{C}\) excitation. All this means:  
\[\begin{split}\mathbf{\Sigma}_{p}(x,y,\omega)&=diag[\mathbf{ \Sigma}_{p_{1}}\mathbf{\Sigma}_{p_{2}}\mathbf{\Sigma}_{p_{3}}](x,y,\omega)\\ &=\mathbf{H}_{\mathbf{P}\sigma}(x,y,\omega)F(\omega)=diag[\mathbf{H}_{p_{1} \sigma},\mathbf{H}_{p_{2}\sigma},\mathbf{H}_{p_{3}\sigma}](x,y,\omega)F(\omega)\in \mathbb{C}.\end{split} \tag{9}\]  
Figure 6: Examples of SLDV-based _von Mises equivalent stress FRF maps_ from optical techniques in Equation (7) from shaker 2: at 333 Hz in (g), at 336 Hz in (h), at 496 Hz in (i), at 752 Hz in (l), at 755 Hz in (m), and at 991 Hz in (n).  
Figure 7: Examples of _von Mises equivalent stress FRF graphs_ from SLDV optical technique in Equation (7) in 20–1023 Hz range: from shaker 1 in (a) and from shaker 2 in (b).  
As above in the FRF formulation of Equation (7), the _Principal Stresses_\(\mathbf{\Sigma}_{p}(x,y,\omega)\) coming from Equation (9), version (i), can be re-arranged--according to the functional \(\mathcal{F}_{vM}()\) defined in Equation (C.2) for general multi-axes stress definition--in the _von Mises Equivalent Stress_\(\mathbf{\Sigma}_{vM}(x,y,\omega)\in\mathbb{C}\):  
\[\mathbf{\Sigma}_{vM}(x,y,\omega)=\mathcal{F}_{vM}(\mathbf{\Sigma}_{p}(x,y,\omega))\in \mathbb{C}. \tag{10}\]  
But here working with the _Principal Stresses_\(\mathbf{\Sigma}_{p}(x,y,\omega)\) coming from Equation (9), version (ii), is preferred in obtaining the _von Mises Equivalent Stress_\(\mathbf{\Sigma}_{vM}(x,y,\omega)\), because the FRF-based approach evaluates \(\mathbf{H}_{e_{vM}}(x,y,\omega)\) only once, while varying the excitation:  
\[\mathbf{\Sigma}_{vM}(x,y,\omega)=\mathbf{H}_{e_{vM}}(x,y,\omega)F(\omega)\in\mathbb{C}. \tag{11}\]  
Albeit both Equations (10), (11) give the same result, Equation (11) is preferred in this approach, or whenever force/input shaping optimisation targets are sought, for the multiple excitations used afterwards in Section 3.