## 3 Empirical Fatigue Damage Estimation
### Weighted Waveform Analysis
The summation of the PSDs at each frequency for all the twenty-three sensors, i.e., \(\sum_{j}PSD_{j}(f)\) (see Figure 3b) is assumed to account for the energy distribution by frequency over the entire riser and is, therefore, used for the WWA mode selection. First, the twelve separate peaks that have the highest spectral values are identified. Note that, if two peaks are closely spaced to each other in terms of frequency, the WWA method assumes that both those peaks represent one mode, and only that frequency with the larger spectral ordinate is selected. As illustrated in Figure 3b, the selected peaks include the first, third and fifth harmonics as well as the second and fourth harmonics of the input signals. Second, the associated mode numbers are determined by comparing these peak frequencies with the estimated natural frequencies of the riser. The riser is assumed to be dominated by a tensioned-string behavior and its bending stiffness is assumed to be negligible. The riser's natural frequencies are accordingly calculated using the physical properties of the riser and an assumed added mass coefficient as follows:  
\[f_{n}=\frac{n}{2L}\sqrt{\frac{T}{m+C_{a}m_{w}}}, \tag{8}\]  
where \(f_{n}\) is the \(n\)th natural frequency of the riser, \(L\) is the riser length, \(T\) is the tension in the riser, \(m\) is the mass of the riser per unit length, \(m_{w}\) is the mass of the displaced water per unit length, and \(C_{a}\) is the added mass coefficient. The added mass coefficient is one of the least understood hydrodynamic properties associated with riser VIV; its value is a function of the vibration amplitude and the current flow speed. In this study, the NDP model riser may be simplified as a tensioned string and the propagation speed of a traveling wave is given as follows:  
\[V_{wave}=\sqrt{\frac{T}{m+C_{a}m_{w}}}, \tag{9}\]  
where \(V_{wave}\) is the propagation speed of the traveling wave, which can be estimated from waterfall plots of the strain measurements. Thus, with physical properties of the NDP model riser and an estimated propagation speed of the traveling wave, the added mass coefficient, \(C_{a}\), can be calculated using Equation (9). Results show that, for the available NDP data sets, \(C_{a}\) has values of approximately zero and unity, respectively, for the uniform and sheared current data sets. For simplicity, in this study, it is assumed that \(C_{a}=0\) for the uniform current profiles, and \(C_{a}=1.0\) for the sheared current profiles. Details regarding calculation of \(C_{a}\) may be found in a related study [24]. With the WWA procedure, strain time series at locations over the entire length of the riser, including at the location of the target sensor, may be reconstructed using Equation (4). Figure 3 ccompares the energy distribution by frequency for the reconstructed CF strain PSD at sensor no. 4 (dashed red line) with the CF strain PSD computed directly from measurements (solid blue line) at the location of the target sensor; the comparison suggests that the first and higher harmonics are reasonably well represented by application of the WWA procedure. Based on the assumed mode shapes, the response over the entire length of the riser may be reconstructed after obtaining the time-varying modal weights. Figure 3d presents RMS curvature values at the locations of the twenty-four sensors based on measurements; the blue circles indicate these values at the twenty-three input sensors, and the red cross indicates the value at the target sensor. The solid line indicates the corresponding RMS curvature estimates based on the WWA procedure. The results suggest that the reconstructed curvatures reflect the presence of the first and higher harmonics and that these reconstructed curvatures match measured values reasonably well at all the sensor locations. The variability factor, representing the ratio of the estimated damage to that based directly on measurements, at the target sensor (i.e., sensor no.