# Transient Responses of Repeated Transverse Impacts on Beams
## 3 Solution of Transient Impact Responses by Eigenfunction Expansion Method
### Solving Procedure for Repeated Transverse Impact on Beams  
The sphere-beam impact system is shown in Figure 2. A sphere of radius R and mass \(m_{s}\) impacts transversely at the center of the beam. During impact, the contact force \(F(t)\), i.e., the interaction force between the sphere and the beam, is generated in the local contact area due to the local indentations between the beam and the sphere. It's worth noting that the local indentation is small in size compared with the dimensions of the contact bodies. Hence, the plane-section assumption of Euler-Bernoulli beam is still valid. By neglecting the influence of local contact plastic deformation, the overall beam response is assumed elastic. Therefore, the eigenfunction expansion method can be applied to solve the dynamic beam deflection \(w_{b}(x,t)\) resulted from impacts.  
The motion equation of the sphere is  
\[m_{s}\ddot{w}_{s}(t)\ =\ m_{s}g-F(t) \tag{18}\]  
where \(w_{s}(t)\) is the displacement of the sphere, and \(g\) is the gravity acceleration.  
The local indentation at the impact location is calculated as,  
\[\delta(t)\ =\ w_{s}(t)-w_{b}(L/2,t) \tag{19}\]  
To solve the repeated impacts, the indentation in Equation (19) needs to be solved step by step. Dividing the time from 0 to \(t\) into small time interval \(\Delta\tau\) and assuming the contact force \(F(t)\) as constant for each \(\Delta\tau\), the indentation \(\delta(t)\) could be solved [1].  
During the time interval \(((N-1)\Delta\tau,\ N\Delta\tau)\), Equation (19) becomes a discretized equation,  
\[\delta(t)\ =\ \frac{1}{2}\bigg{(}g-\frac{F_{N}}{m_{s}}\bigg{)}(\Delta\tau)^{2 }+\dot{w}_{s(N-1)}\Delta\tau+w_{s(N-1)}-w_{b}(L/2,t) \tag{20}\]  
where \(\dot{w}_{s(N-1)}\),\(w_{s(N-1)}\) are the centroid velocity and centroid displacement of the sphere when \(t\ =\ (N-1)\Delta\tau\), \(F_{N}\) is the contact force at \(t\ =\ N\Delta\tau\). To ensure the accuracy, \(\Delta\tau\) should be small enough.  
Therefore, Equation (20) could be written as the function of \(F_{N}\),  
\[\delta(t)\ =\ f(F_{N}) \tag{21}\]  
Furthermore, at \(t\ =\ N\Delta\tau\), the contact force can be obtained once the local indentation \(\delta(t)\) is known, that is,  
\[F_{N}\ =\ G(\delta(t)) \tag{22}\]  
Substituting Equation (21) into Equation (22), the following equation is obtained  
\[F_{N}\ =\ G(f(F_{N})) \tag{23}\]  
Figure 2: Sphere-beam impact system.  
The \(F_{N}\) in the nonlinear Equation (23) could be solved by bisection method, and then \(\delta(t)\) could be solved from Equation (21) once \(F_{N}\) is known. The dynamic deflection \(w_{b}(x,t)\) can be obtained by using Equation (6). The acceleration, the velocity and the displacement of the sphere at time \(t~{}=~{}N\Delta\tau\) could be solved by Equation (18), Equation (24) and Equation (18), respectively.  
\[\dot{w}_{SN}~{}=~{}\ddot{w}_{SN}\Delta\tau+\ddot{w}_{s(N-1)} \tag{24}\]  
\[w_{SN}~{}=~{}\frac{1}{2}\ddot{w}_{SN}\Delta\tau^{2}+\dot{w}_{s(N-1)}\Delta\tau +w_{S(N-1)} \tag{25}\]  
For the repeated impact problem in this paper, the impact velocity of the \(i\)-th impact is marked as \(V_{0i}\). The beam is assumed to be stationary but deformed at the local contact area when subsequent impacts initiate.