## 3 Assessment of Flow-Induced Vibration of Fuel Assembly
### Strain Along Fuel Pin Length and Vibration Statistics  
Next, we assessed the strain along the fuel pin length and evaluated the vibration statistics. In Figure 7, we show the trend of the strain amplitude along the central fuel pin. Recall that near the inlet, the fuel pin is supported, whilst the other extremity is free to move. Therefore, we observed the highest strain amplitude near the inlet and a decreasing trend in the amplitude towards the outlet. In the figure, we again show the rms strain amplitude determined with the 21 FBGs on the central fuel pin in the direction perpendicular to the support rails, as well as parallel with the support rails. We found that the trend of the vibration amplitude along the fuel pin in the direction parallel to the support rails essentially resembles the second bending mode of a classical cantilevered beam whilst in the perpendicular direction the trend follows the third bending mode of such a cantilevered beam. The slightly different trend of the two directions of motion can be the result of the different support conditions of the fuel pin at the fuel assembly inlet. A modal analysis could confirm if these trends indeed correspond to the respective bending modes. These trends are not significantly influenced by the flow velocity, except for the amplitude as illustrated in the previous section. In order to estimate the displacements associated with the measured strains we approximated the fuel pins as cantilevered beams, and applied discretized classical beam theory to formulate strain-data-dependent displacement equations of the fuel pins (according to the study described in [32]). From these equations, and taking only bending moments into account, we estimated the mean displacement to be 33.8 \(\upmu\)m in the direction parallel with the support rails, and 20.9 \(\upmu\)m in the perpendicular direction. These values are in line with those previously determined in [29].  
Mean values of the vibration amplitudes are useful but they do not describe the entire vibration signal. Moreover, absolute values of the vibration amplitude at a certain time are not meaningful since the fuel pin excitation is due to forces from turbulent flow, which is a random phenomenon. Given this random nature of the turbulent forces and the central limit theorem, we expect the average dynamic strain of the fuel pin to adhere to a normal or Gaussian distribution. To evaluate the entire vibration, we have to determine the distribution and nature of the vibration. In Figure 8, we show a histogram of the average measured strains on the fuel pins in the direction parallel with the support rails (i.e., averaged over 40 FBGs), and at nominal MYRRHA flow speed (i.e., 1.72 m/s). It is clear from the distribution that excursions exceeding on average 2 \(\upmu\epsilon\) very rarely occur. More specifically, 99.9% of the measured average strains are below 2 \(\upmu\epsilon\). The spread or standard deviation of the distribution corresponds to what we determined for the mean strain amplitude in Figure 6. A very similar histogram can be constructed for the direction perpendicular to the support rails. Finally, to evaluate the Gaussian nature of the distribution, we determined the kurtosis, or the so-called fourth central moment, which should  
Figure 6: The measured vibration amplitude (with standard deviations) of the fuel assembly (averaged over all fuel pins) for the different evaluated flow velocities.  
be equal to 3 for a Gaussian distribution. We calculated that the kurtosis of the histogram in Figure 8 equals 3.03, thus illustrating the Gaussian nature of the distribution. These results provide information on the location of the highest strain on the fuel pin, as well as on the frequency of excursions with a higher amplitude.