# Pantograph-Catenary Interaction Prediction Model Based on SCSA-RBF Network
## 3 Pantograph-Catenary Interaction Prediction Model
### Consideration of a Piecewise Linear Panhead Spring  
Three parameters, \(k_{31}\), \(k_{32}\), and \(S_{0}\), will be involved if the panhead spring exhibits piecewise linear stiffness. When the spring force \(S(t)<S_{0}\), the spring stiffness is \(k_{31}\); when \(S(t)\geq S_{0}\), the spring stiffness is \(k_{32}\). Similarly, the ranges of the three decision variables were determined according to engineering practice. Then, the decision vector was extended to \(\mathbf{Y}=\{v,\,m_{3},\,m_{2},\,m_{1},\,k_{31},\,k_{32},\,S_{0},\,k_{2},\,c_{1 }\}^{\text{T}}\).  
It was assumed that the dynamic characteristics of the nonlinear panhead spring are equivalent to that of a linear spring over the entire analysis period, and the equivalent stiffness \(k_{3}\) could be expressed as  
\[k_{3}=\eta k_{31}+(1-\eta)k_{32} \tag{4}\]  
where \(\eta\) represents the percentage of the sum of time when the spring stiffness is expressed as \(k_{31}\) in the total time. The following steps were taken to make \(\mathbf{Y}\) equivalent to \(\mathbf{X}\): (1) Take \(\mathbf{X}^{0}=\{v,\,m_{3},\,m_{2},\,m_{1},\,k_{3}{}^{0},\,k_{2},\,c_{1}\}^{ \text{T}}\), that is, \(k_{3}{}^{0}=k_{31}\) and \(r=0\). (2) Based on the pantograph-catenary interaction prediction model in Section 3.1, the predicted value of the vertical displacement of panhead and the second lumped mass, denoted as \(\widehat{z}_{3}\left(t_{j}\right)\) and \(\widehat{z}_{2}\left(t_{j}\right)\), and the predicted value of the spring force \(\widehat{S}\left(t_{j}\right)=k_{3}{}^{r}\left(\widehat{z}_{2}\left(t_{j} \right)-\widehat{z}_{3}\left(t_{j}\right)\right)\) were obtained. Count the number of time sampling points  
Figure 2: Pantograph–catenary interaction prediction model.  
with \(\widehat{S}\left(t_{j}\right)<S_{0}\) in the whole analysis period and divide it by the total number of time sampling points to obtain \(\eta\), set \(k_{3}{}^{\prime}{}^{+1}=\eta k_{31}+(1-\eta)k_{32}\). (3) If \(\left|k_{3}{}^{\prime}{}^{+1}-k_{3}{}^{\prime}{}^{\prime}\right|<1\) N/m, \(\mathbf{X}=\mathbf{X}^{\prime}\), and stop iteration; otherwise, set \(\tau=\tau+1\), \(\mathbf{X}^{\prime}=\left[\
u\), \(m_{3}\), \(m_{2}\), \(m_{1}\), \(k_{3}{}^{\prime}{}^{\prime}\), \(k_{2}\), \(c_{1}{}^{\dagger}\), and repeat step (2).  
For the equivalent decision vector \(\mathbf{X}\), the same method in Section 3.1 was used to obtain the boundary conditions \(\widehat{\widetilde{z}}_{3}\left(t_{j}\right)\), \(\widehat{z}_{3}\left(t_{j}\right)\), and \(\widehat{z}_{3}\left(t_{j}\right)\). Combined with the initial conditions, the dynamic equation (1) of the three-mass model of the pantograph was solved to obtain the model-predicted value of the pantograph-catenary contact force \(\widehat{F}_{c}(t)\). In the process of solving the dynamic equation, at each time integration point, if \(S(t)<S_{0}\), replace \(k_{3}\) in Equation (1) with \(k_{31}\); if \(S(t)>S_{0}\), replace \(k_{3}\) with \(k_{32}\).  
Hereinafter, the pantograph-catenary interaction prediction model was extended to the case of a piecewise linear panhead spring. Furthermore, the Latin hypercube sampling method was used to draw set D in a nine-dimensional sample space of \(\mathbf{Y}\). Set D was considered as the test set to evaluate the generalization ability of the pantograph-catenary interaction prediction model in the case of a piecewise linear panhead spring.