# Transient Responses of Repeated Transverse Impacts on Beams
## 5 Results and Discussion
### Parameter Determinations  
For the eigenfunction method, the convergence study on the number of truncated modes is conducted for the 91 st repeated impact. Table 4 lists the main impact behaviors of the 91 st repeated impact by combining the eigenfunction method and KK model [43]. It shows that good convergence can be achieved when truncation number reaches 20. In the following studies, the mode number is chosen as 20 for the analysis. In this paper, the first sub-impact is focused if the sub-impacts occur [30; 46].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Sphere (Gr 15)** & **Simply Supported Beam (Q345)** \\ \hline \(R_{s}\) & 35.000 (mm) & \(L\) & 780.0 (mm) \\  & & \(h\) & 27.80 (mm) \\  & & \(b\) & 60.00 (mm) \\ \(\rho_{s}\) & 7800 (kg/m\({}^{3}\)) & \(\rho_{b}\) & 7800 (kg/m\({}^{3}\)) \\ \(m_{s}\) & 1.400 (kg) & \(m_{b}\) & 10.929 (kg) \\ \(E_{s}\) & 208,000 (MPa) & \(E_{b}\) & 210,000 (MPa) \\ \(\sigma_{Ys}\) & 2555 (MPa) & \(\sigma_{Yb}\) & 345 (MPa) \\ \(v_{s}\) & 0.3 & \(v_{b}\) & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material properties and geometry.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(H\) & **Number of** & \(H\) & **Number of** & \(H\) & **Number of** & \(H\) & **Number of** \\ (mm) & **Impacts** & **(mm)** & **Impacts** & **(mm)** & **Impacts** & **(mm)** & **Impacts** \\ \hline
50 & 4 & 130 & 7 & 210 & 3 & 290 & 3 \\
60 & 3 & 140 & 2 & 220 & 8 & 300 & 2 \\
70 & 3 & 150 & 2 & 230 & 2 & 310 & 1 \\
80 & 3 & 160 & 2 & 240 & 5 & 320 & 1 \\
90 & 3 & 170 & 2 & 250 & 2 & 330 & 3 \\
100 & 2 & 18 For the spring-mass method shown in Figure 5, \(k_{1}\) is the nonlinear local contact stiffness, \(k_{2}\) is the static stiffness of the beam at the impact location and \(m_{e}\) is the equivalent mass of the beam at the impact location. It is a simplified model to simulate the repeated impact responses, in which the bending waves travel from the impact location to the ends of the beam and back many times during the impact, i.e., the beam behaves in a quasi-static manner and the wave propagation effect is neglected [4].  
Assuming that the static load P is applied at the center of the beam, the deflection at the center of the simply supported beam is:  
\[w_{b}\ =\ PL^{3}/48E_{b}I \tag{26}\]  
Hence, the static stiffness of the center of the beam can be calculated as:  
\[k_{2}\ =\ P/w_{b}\ =\ 48E_{b}I/L^{3} \tag{27}\]  
The equivalent mass \(m_{e}\) is also related to the impact location, and is defined by the circular frequency of the spring-mass model \(\omega\ =\ \sqrt{k_{2}/m_{e}}\). Assuming that \(\omega\) is equal to the fundamental nature circular frequency of the simply supported beam \(\omega_{1}\), the equivalent mass \(m_{e}\) can be obtained as  
\[m_{e}\ =\ 48\rho_{b}AL/\pi^{4} \tag{28}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Nodes** & \(\mathbf{t_{c}}\) **(ms)** & \(\mathbf{V_{r}}\) **(m/s)** & \(\mathbf{\delta_{r}}\) **(mm)** & **COR** \\ \hline
10 & 0.11000 & 0.11902 & 0.23238 & 0.36159 \\
20 & 0.11200 & 0.10761 & 0.22679 & 0.39040 \\
50 & 0.11200 & 0.10746 & 0.22681 & 0.39034 \\
100 & 0.11200 & 0.10745 & 0.22682 & 0.39036 \\
200 & 0.11200 & 0.10745 & 0.22682 & 0.39036 \\ \hline \end{tabular}
\end{table}
Table 4: Convergence of the truncated mode number.  
Figure 5: Two degree-of-freedom spring-mass model.  
The equations of motion of the spring-mass can be expressed as:  
\[\left\{\begin{array}{l}m_{s}\tilde{w}_{s}-m_{s}g~{}=~{}-F(t)\\ m_{e}\tilde{w}_{b}+k_{2}m_{b}~{}=~{}F(t)\end{array}\right. \tag{29}\]  
where the indentation \(\delta~{}=~{}w_{s}-w_{b}\). Equation (28) can be solved by using Runge-Kutta integration method.  
To ensure the applicability of eigenfunction method and spring-mass method, the time interval \(\Delta\tau\) should be small enough. For the following studies, \(\Delta\tau~{}=~{}10^{-3}\) ms.