## 4 Validation of the Proposed SDoF Model
### Characteristic Response Parameters of Beam under Impact at Any Point  
The dimensionless external dynamic energy parameter is introduced in Equation (39) to convert the influence of impact location on the beam's behavior to that of the impact energy. \[\lambda=\frac{m_{0}V_{0}^{2}l_{1}}{2 BH^{3}\sigma_{0}} \tag{39}\]  
It is evident from Equation (39) that \(\lambda\) is in proportion to \(l_{1}/l\), which ranges from zero to \(1/2\). Figure 6a-d illustrates the variation trends of the maximum/final displacement and the impact duration with \(\lambda\) resulting from the SDoF and numerical models. For comparison purpose, the dimensionless maximum and final displacements, i.e., \(\delta_{m}=W_{m}/H\) and \(\delta_{f}=W_{f}/H\) obtained from the theoretical and numerical predictions, are plotted against \(\lambda\) in Figure 6a. Satisfactory agreement is observed, although the proposed SDoF model predicts a slightly larger deflection with the increase of \(\lambda\), owing to the neglect of the thinning effect of the beam with a sufficiently larger displacement, as analysed for the displacement-time history curve in Figure 5a. It is shown from Figure 6a that both the maximum and final displacements show an increasing trend with an increase of \(\lambda\). Meanwhile, the spring-back \(\Delta W=W_{m}-W_{f}\) corresponding to the gap between the \(\delta_{m}-\lambda\) and \(\delta_{f}-\lambda\) curves is seen to increase with an increase of \(\lambda\) and attains its maximum value in the central impact. To explain this phenomenon, a similar analysis to that on the slope of the unloading path can be conducted to approximate the spring-back \(\Delta W\), written as  
\[\Delta W=\frac{2E_{e}}{F_{m}}=\frac{m_{0}V_{r}^{2}}{F_{m}} \tag{40}\]  
It is revealed from the comparisons between Equations (38) and (40) that the variation tendency of \(\Delta W\) is opposite that of \(K_{u}\) with \(\lambda\). Thus, the spring-back of the beam attains its maximum value in the central impact with \(l_{1}=l_{2}\) and decreases with the increase of \(l_{1}/l\). It is evident from Figure 6b that the SDoF analysis provides reasonable agreement with the numerical predictions for the dependence of the maximum impact force on the impact location. With an increase of \(\lambda\) due to an increasing \(l_{1}/l\), the maximum impact force decreases, which can be attributed to the decrease of the beam's resistance. Specifically speaking, the dynamic loading path can be approximated as a linear model due to its characteristics, as illustrated in Figure 6b. Thus, the maximum impact force \(F_{m}\) can be approximated by  
\[F_{m}=\sqrt{2 KE_{k}} \tag{41}\]  
where \(K\) is the slope of the loading path, and \(E_{k}=m_{0}V_{0}^{2}/2\) is the impact kinetic energy of the striker. Figure 6: Dependence of characteristic parameters of an elastic–plastic beam on the impact location. COR = V\({}_{r}\)/V\({}_{0}\) is defined as the coefficient of restitution. (**a**) Variation of \(\delta_{m}\) and \(\delta_{f}\) with \(\lambda\). (**b**) Variation of \(F_{m}\) with \(\lambda\). (**c**) Variation of COR with \(\lambda\). (**d**) Variation of \(t_{m}\) and \(t_{r}\) with \(\lambda\). As pointed out in Section 4.1, there is an increase in \(K\) with a decreasing \(l_{1}/l\), which implies a greater structural resistance mainly owing to the axial force when a relatively large deformation develops in the beam. Thus, it follows from Equation (41) that a larger \(l_{1}/l\) inducing a smaller \(K\) will result in a smaller \(F_{m}\). However, it is found that in the impact cases with a relatively smaller \(\lambda\), the maximum impact force predicted by the proposed analytical model is evidently larger than the corresponding numerical prediction. This is possibly caused by the neglect of the bending wave propagation in the beam in the analytical model. In the early stage, when the beam is accelerated to obtain the same velocity of the striking mass, the bending wave propagation in the beam can result in the kinetic energy loss of the impact system.