## 3 Validation of the Model  
Our previous study ([PERSON] et al., 2013) has validated the present nonlinear dynamic model with supercritical water in the static condition (\(a_{ss}^{+}=0\)), i.e., without seismic motion, against the experimental data of a supercritical water two uniformly-heated channel system ([PERSON] et al., 2013). The results indicated that the present non-linear model can predict the stability thresholds well against the experimental data, within a relative error of 10% ([PERSON] et al., 2013). To evaluate the applicability of the external force method in estimating the vibration impact that was not well validated in the past studies ([PERSON] et al., 2013; [PERSON] et al., 2013; [PERSON] et al., 2013; [PERSON] et al., 2013), the authors established a vertical vibration experimental system in Figure 2a to investigate the thermal-hydraulic characteristics of boiling two-phase flow under vertical vibrations ([PERSON] et al., 2013). The test section comprises a single-phase stand pipe (\(D_{SP}\) = 0.012 m, \(L_{SP}\) = 0.17 m), a heated section (\(D_{H}\) = 0.012 m, \(L_{H}\) =1.27 m) and an adiabatic riser (\(D_{R}\) = 0.012 m, \(L_{R}\) = 0.38 m). The additional pressure drop caused by the vertical vibration (\(\Delta P_{VIB}\)) can be estimated by the measured pressure drop (\(\Delta P(Exp. )\)) under vertical vibration subtracting the average pressure drop (\(\Delta P_{ave}\)). Thus, the non-dimensional form is:  
\[\Delta P_{VIB}^{+}(Exp.)=[\Delta P(Exp.)-\Delta P_{ave}]/\rho_{f}u_{s}^{2} \tag{16}\]  
Figure 2: The validation of external force method: (**a**) single heated channel vertical vibration experimental system ([PERSON] et al., 2013); (**b**) comparisons of the vibration-induced pressure drop using Equation (17) against the experimental values through Equation (16). In terms of external force method, the calculated value of \(\Delta P_{VIB}^{+}\) can be determined by Equation (17):  
\[\Delta P_{VIB}^{+}(Predict.)=[(\rho_{i}^{+}+\rho_{e}^{+})/2+\rho_{e}^{+}L_{R}^{+ }+\rho_{i}^{+}L_{SP}^{+}]\times a_{\text{max}}\sin(2\pi f^{+}t^{+})/Fr \tag{17}\]  
For a single heated channel vertical vibration system at atmosphere pressure, Figure 2b compares the peak magnitude of the calculated \(\Delta P_{VIB}^{+}(Predict. )\) against the peak magnitude of the experimental \(\Delta P_{VIB}^{+}(Exp. )\) among nine test cases with heat fluxes of \(q^{\prime\prime}\) = 8444, 11,029 and 13,959 W/m\({}^{2}\) and vibration frequencies of \(f\) = 0.5, 1.0 and 1.5 Hz, respectively. The results exhibit that the vibration-induced pressure drop increases as the increase in the vibration frequency, i.e., vibration acceleration. The theoretical predictions using Equation (17) can predict the experimental values based on Equation (16) within a relative error of about 20% among all the cases. It can demonstrate that external force method is appropriate to explore the impact of external vertical vibration on the rigid system. It should note that the discrepancy between the experimental data and the predicted value is smaller in the relatively small vibrations with \(f\) = 0.5 and 1.0 Hz than that in the relatively large vibration with \(f\) = 1.5 Hz. This may suggest that the external force method is more applicable for the relatively small vibration acceleration, non-resonance and insignificant resonance conditions under the vibration impacts, as in most of the cases in the present study, in which the probability of the deformation occurring in the structure parts is very limited during such vertical vibrations. In addition, at present, the experiment data of the supercritical fluid system under external vibrations is deficient in the literature. The external force method may need a further validation against the vibration experimental data of supercritical fluid system if it is available in the future. On the other hand, the actual seismic impact on a power plant is very complicated, which could be coupled with the fluctuations of coolant flow and system structures.