## 4 Results and Discussion
### Viscoelastic Behavior
#### 4.2.2 Case 2  
As mentioned earlier, the load profile for case 2 is intended to more closely simulate the operation of an FESS in actual applications, such as for solar PV electric grids or EV charging. Graphs with the radial and circumferential stresses on day 1 of operation are given in Figure 6. These graphs serve as representative examples of the stress distribution for the simulated operation. While the magnitude of the radial and circumferential stresses  
Figure 5: Strength ratio (\(SR\)) data for the flywheel rotor (**a**), and peak SR in the aluminum hub and CFRP rim at various times throughout the simulated lifetime (**b**). was found to decrease in the rim and increase in the hub over time, the overall shape of the stress response at each velocity (minimum phase, \(\omega_{\text{Pmin}}\), intermediate phase, \(\omega_{\text{Pint}}\), maximum phase, \(\omega_{\text{Pmax}}\)) was found to be similar for any day of the simulated operation, so only data for the first day are provided. Comparing case 1 (Figure 3) and case 2 (Figure 6), the stress responses at \(\omega_{\text{Pmin}}\) and \(\omega_{\text{Pmax}}\) closely resemble those from case 1 at \(t=0\), \(\omega=0\) and at \(t=0\), \(\omega=\omega_{\text{limit}}\), respectively, as is expected given the similarity between angular velocities and the non-linear relationship between stress and velocity. Notably, for both cases, the circumferential stress in the hub is seen to change from initially having a positive slope (\(\omega=0\), \(\omega=\omega_{\text{Pmin}}\)) to having a negative slope at high-velocity operation (\(\omega_{\text{limit}}\), \(\omega_{\text{Pmax}}\)). Between both extremes, stresses switch from tensile to compressive with the magnitude occurring at the hub inner surface. This loading scenario resembles fatigue loading with a negative stress ratio, positive mean stress, and a comparatively high stress range. The hub design should therefore include a fatigue analysis, especially for FESS that experiences high cycle rates, i.e., numerous cycles per day. Figure 7 depicts \(SR\) data for all points along the rotor radial direction at each velocity for day 1 and day 365. Again, while magnitudes at each point are seen to vary for the different velocities, the overall shape of the \(SR\) curves at each velocity bear distinct similarities. Broadly, \(SR\) graphs exhibit similar trends as in case 1. After year 1, \(SR\) values at \(\omega_{\text{Pmax}}\) increase in the hub but decrease in the composite rim. As one would expect, \(SR\) values in Figure 7 are lower compared to data in Figure 5, due to the overall lower stress levels and the reduced time that the rotor operates at high velocity. During each phase, five key indicators are tracked throughout the simulation: (i) interface stress, (ii) peak radial tensile stress (i.e., neglecting compressive stresses, as these are found at the interface), (iii) peak circumferential stress in the CFRP rim, (iv) peak \(SR\) in the hub, and (v) peak \(SR\) in the CFRP rim. Values for each indicator recorded on day 1, 90, 180, 270, and 365 are given in Table 5. To facilitate comparisons with case 1, data from Table 5 were normalized using the day 1 value at each velocity and each location or component of interest, as depicted in Figure 8. Noting that since the peak radial tensile stress at \(\omega_{\text{Pmin}}\) is negligible throughout the simulation (see Table 5), this dataset was omitted in Figure 8.  
Figure 6: Radial (**a**) and circumferential (**b**) stress results at each velocity on day 1.