## 4 Validation of the Proposed SDoF Model
### Characteristic Response Parameters of Beam under Impact at Any Point
For the case with a smaller \(\lambda\), the bending wave propagation is somewhat significant due to the boundary effect. Nevertheless, generally speaking, compared to the numerical results, the current analytical model with a relative error within 5.4% gives a satisfactory prediction of the maximum impact force. It is observed from Figure 6c that the numerical and analytical results generally show the same variation trends of the rebound velocity with \(\lambda\), despite a slight difference, especially for a smaller \(\lambda\). It is notable that \(V_{r}\) increases with an increasing \(\lambda\) induced by an increasing \(l_{1}/l\), which implies that the more symmetrical load allows for more elastic energy absorbed by the beam. It can be explained as follows. With the aid of Equation (39), a larger \(l_{1}/l\) with a specific kinetic energy can be equivalent to a larger kinetic energy in a central impact case. Besides, it has been demonstrated that a larger impact energy results in a larger elastic strain energy, i.e., a larger rebound velocity [19]. Thus, the rebound velocity attains its maximum value in the central impact corresponding to the largest \(\lambda\) amongst the examined cases with different impact locations. Furthermore, the variations of loading and unloading durations, namely \(t_{m}\) and \(t_{r}\), respectively, with \(\lambda\) are plotted in Figure 6d, which shows that \(t_{m}\) and \(t_{r}\) increase with an increasing \(\lambda\). This characteristic can be explained from the point of view of momentum. At \(t=t_{m}\), the impact force reaches its maximum value, and the velocity of the striker decreases to zero. Thus, it follows from the principle of momentum that the initial momentum \(l_{0}\) and the residual momentum \(l_{f}\) of the impact system can be expressed as  
\[I_{0}=m_{0}V_{0}=\int_{0}^{t_{m}}F(t)dt\;\;I_{f}=m_{0}V_{r}=\int_{t_{m}}^{t_{f }}F_{u}(t)dt \tag{42}\]  
For simplicity, two linear functions are adopted to model the variations of impact force with time during loading and unloading. Thus, the duration during loading, \(t_{m}\), and the duration during unloading, \(t_{r}=t_{f}-t_{m}\), can be approximated from Equation (42), written as  
\[t_{m}=\frac{2m_{0}V_{0}}{F_{m}}\;\;t_{r}=\frac{2m_{0}V_{r}}{F_{m}} \tag{43}\]  
Now, combined with the variation tendencies of \(F_{m}\) and \(V_{r}\) with \(\lambda\) shown in Figure 6b,c, respectively, it is revealed from Equation (43) that \(t_{m}\) and \(t_{r}\) both increase with an increase of \(\lambda\), which results in an increase of the impact duration \(t_{f}\). Thus, it is revealed from Equation (43) that \(t_{m}\) and \(t_{r}\) both increase with an increase of \(\lambda\), which results in an increase of the impact duration \(t_{f}\). In order to quantitatively assess the accuracy of the analytical model, it is necessary to analyse the ratios of numerical to analytical predictions in terms of the characteristic response parameters, i.e., \(W_{m}\), \(W_{f}\), \(F_{m}\), \(COR\), \(t_{m}\) and \(t_{r}\). Table 1 presents the corresponding mean values and the standard deviations of the ratios obtained for the four different values of \(\lambda\). It is notable that for all the analysed parameters, the mean values are quite close to \(1\), while the standard deviations approach zero. This demonstrates that the proposed analytical model, although with some simplifying assumptions, can offer a satisfactory approximation of the dynamic response of a beam on condition of a heavy mass with a low impact velocity, at least within the range of parameters examined (the largest value of \(\lambda\) is 4.9). Nevertheless, it should be mentioned that the necking phenomenon can become significant with further increase in the structural deflection due to a larger \(\lambda\). Thus, one should be extremely cautious and careful when applying this model to the impact cases with comparatively larger initial kinetic energy.