## 3 Results and Discussion  
When considering the single cylinder in the presence of an inviscid fluid, the dimensionless added mass \(\alpha\) is equal to one, regardless of the vibrating frequencies. For a single cylinder in viscous fluid, the exact dimensionless added mass was calculated as [24]  
\[\alpha_{\text{ext}}=1+\frac{4\mathrm{i}K_{1}(-\mathrm{i}\sqrt{\mathrm{i} \mathrm{Re}})}{\sqrt{\mathrm{i}\mathrm{Re}}K_{0}(-\mathrm{i}\sqrt{\mathrm{i} \mathrm{Re}})}, \tag{21}\]  
where \(K_{0}\) and \(K_{1}\) are the modified Bessel functions of the third kind, and \(\mathrm{Re}=\rho\omega R_{2}/\mu\). The dimensionless added mass of a single cylinder in the fluid medium was calculated, as shown in Figure 4. The radius of the cylindrical beam R was 0.005 m. The fluid was assumed to be water, at 4 \({}^{\circ}\)C ( \(\rho_{f}\) = 1,000 kg/m\({}^{3}\) and \(\mu\) = 1.519 \(\times\) 10-3 Ns/m\({}^{2}\) ). When compared with [PERSON]'s method, the results of the exact solution (Figure 3) were in good agreement. The dimensionless added masses of the 25 cylinders were calculated using [PERSON]'s method presented in Figure 5. The gap between each cylinder was 0.001 m. The start position of the traveling wave is 0.05 m. As the number of cylinders increased, the added mass and fluid damping also increased due to more interactions between the multiple cylinders. The fluid coupling effects became significant when the structures were concentrated. Figure 6 shows the real part of the pressure that was obtained for propagating plane waves. The scattering is more significant when the frequency of the incident wave is larger. The increasing number of cylinders induced more significant scattering. Figure 7 shows the external forces from the unit pressure of the incident wave. The external forces were proportional to the input frequency of the incident wave. The forces acting on cylinders 1, 6, 11, 16, and 21 were higher than those acting on other cylinders. The external forces acting on cylinders located outside of the complex structures were influenced by the instability of the nuclear rod bundles in the fluid. The point force was applied to the fixed beam, as shown in Figure 8a. The displacement responses for the flexural vibration of a single cylindrical beam in the fluid medium were predicted, as shown in Figure 8b,c. The single cylindrical beam without water vibrated having not a time decay. Fluid damping did not occur without the presence of water, and only structural damping influenced vibration. Otherwise, the vibration magnitude of cylindrical beam considering the added mass and fluid coupling induced fast decay in comparison with only structural damping. The application of the added mass increased the total damping because of the viscous effects. The increased decay in the predicted responses was observed when the beam was placed in the water. The impact response of the nuclear rod bundles consisting of multiple cylinders was predicted, as shown in Figure 9. The resonant response of the nuclear rod bundles consisting of multiple cylinders occurred at lower frequencies. When the number of cylindrical beams \(N\) increased, the total damping of nuclear rod bundles also increased. The fluid coupling that was caused by the viscous damping of the fluid gaps between each structure was increased when the number of cylindrical beam increased. The fluid damping increased in the presence of surrounding structures due to small gaps between the vibrating structures. Figure 4: Comparison between exact solutions and calculated added masses of a single cylinder using [PERSON]’s method: (**a**) real part and (**b**) imaginary part. Figure 5: Calculated added comparison between exact solutions and calculated added masses of a single cylinder using [PERSON]’s method: (**a**) real part and (**b**) imaginary part. Figure 6: Pressure wave pattern produced by the multiple scattering of the plane wave with a wavelength of (**a**) \(k_{f}R=0.5\), (**b**) \(k_{f}R=1.0\), and (**c**) \(k_{f}R=2.0\).