## 2 Materials and Methods
### Spectral Methods for Cumulative Damage and Fatigue Life Assessment in Brief  
The _equivalent range of stress cycles_\(S_{eq}\) is targeted by any spectral method for fatigue life estimation, because \(S_{eq}\) represents the cumulative damage inferred by the known excitation spectrum on the specific component, due to its involved structural dynamics. The _experiment-based Stress FRF maps_\(H_{\sigma}(x,y,\omega)\) in Equation (5) and the selected _complex-valued spectrum_ of \(F(\omega)\) offer unprecedented abilities in mapping, in each location \((x,y)\) of the sensed surface, the _equivalent range of stress cycles_\(S_{eq}(x,y)\), by means of a full-field extension of the fatigue spectral methods. Many _spectral methods_ (e.g., [46; 47; 48; 49; 50; 88; 89; 90; 88]) are based on punctual evaluations, in each location \((x,y)\) of the spatial domain, of \(m_{k}(x,y)=\int_{0}^{\infty}f^{k}PSD_{\Sigma_{eq}}(x,y,\omega)d\omega\), the _k-th order moments_ of the frequency by \(PSD_{\Sigma_{eq}}(x,y,\omega)\), the _Power Spectral Density_ (PSD) of _von Mises Equivalent Stress_, from which we can obtain other parameters, such as the _effective frequency_\(F_{\text{{zeros}}}(x,y)=F_{\text{{zc}}}(x,y)=\sqrt{m_{2}(x,y)/m_{0}(x,y)}\), the _expected number of peaks per unit time_\(F_{\text{{peaks}}}(x,y)=F_{p}(x,y)=\sqrt{m_{4}(x,y)/m_{2}(x,y)}\), and the _irregularity factor_\(\gamma(x,y)=\gamma_{2}(x,y)=F_{\text{{zc}}}(x,y)/F_{p}(x,y)=m_{2}(x,y)/ \sqrt{m_{0}(x,y)m_{4}(x,y)}\). The \(PSD_{\Sigma_{eq}}(x,y,\omega)\) map can be computed, from the statements for the _von Mises Equivalent Stress_\(\Sigma_{vM}(x,y,\omega)\in\mathbb{C}\) in Section 2.2.3 and Equation (11), as:  
\[PSD_{\Sigma_{eq}}(x,y,\omega)=2\Sigma_{vM}(x,y,\omega)\Sigma_{vM}^{*}(x,y, \omega)/\Delta\omega\in\mathbb{R}, \tag{13}\]  
where \(()^{*}\) is the complex conjugate operator, \(\Delta\omega\) is the frequency domain spectral spacing, and the multiplier 2 is for counting only the positive frequencies in a symmetric domain. As anticipated above, when adopting varying _complex-valued coloured noises_ for \(F(\omega)\), the here preferred formulation for evaluating the _PSD_ maps of _von Mises Equivalent Stress_ is that in Equation (11), exploiting itself the _von Mises Equivalent Stress_\(FR_{\sigma_{vM}}(x,y,\omega)\in\mathbb{C}\) of Equation (7). For the sake of exemplifications, the latter \(H_{\sigma_{vM}}(x,y,\omega)\) were shown, from both shakers, in the selected maps of the tiles--at single frequencies--in Figures 3,4-5,6, and in the whole frequency domain graphs of Figure 7, in the single dof 550. Note that any other17 spectral method, than what here simulated, can exploit the proposed experiment-based full-field approach. Specifically, all the spectral methods taking advantages from true _complex-valued amplitude and phase_ relations--without the limiting multi-axes _von Mises Stress_ equivalence--are expected to reach new achievements by means of the _experiment-based full-field Stress_\(FR_{\sigma}(x,y,\omega)\) in Equation (5), by the related _principal FRF components_\(H_{\sigma_{v}}(x,y,\omega)\) in Equation (6), or by the dynamic _Stresses_\(\Sigma(x,y,\omega)\) of Equation (8) with their _principal components_\(\Sigma_{\mathbf{P}}(x,y,\omega)\) in Equation (9). Once a specific fatigue spectral method18 is adopted to evaluate \(S_{eq}(x,y)\), it is possible to calculate the _Time-to-Failure spatial distribution_\(T_{failure}(x,y)\) ([s] or [h]), or its reciprocal _Frequency-to-Failure spatial distribution_\(F_{failure}(x,y)\) ([1/s] or [1/h]), evaluated across all the dofs \((x,y)\) of the maps, function of \(S_{eq}(x,y)\), of \(F_{p}(x,y)\) and of the \(K_{r}\) fatigue strength coefficient and \(b\) exponent19, as:  
Footnote 18: Here detailed in Equation (D.2) of Appendix D for the [PERSON]’s semi-empirical, but widely spread, approach [46]. Footnote 19: Here \(K_{r}\)=4.42e+43 Pa\({}^{b}\), \(b\)=4.81 for the 7075-T6 aluminium alloy in [58].