This section studies the developed elastic contact formulation. The contact formulation suitable for the ANCF catenary systems has been developed from a sliding constraint and an elastic contact formulation [1, 2]. The assumption that comes with the sliding joint contact formulation is no pantograph–catenary separation [1, 3, 4]. The developed elastic contact formulation allows for pantograph–catenary separation, and the effect of friction on the sliding motion can be applied [5]. Two relative degrees of freedom of the translational motion between the contacting bodies are eliminated using the developed elastic contact mathematical formulation. The contact point slides in the longitudinal direction along the contact wire and the lateral direction along the panhead. The developed elastic contact force formulation is applied, and the developed elastic contact mathematical formulation is explained.

## 3.1 Contact Point and Contact Frame

The location of the contact point on the pantograph–catenary system in the case of the catenary zigzag pattern and curved track should be considered, as well as the effect of the catenary system vibration. There is lateral displacement on the panhead, and superelevation on the curved track causes vertical displacement of the contact point or its location. Therefore, the developed elastic contact formulation is used to determine the exact location of the contact point as follows:

$${{\mathbf{C}}^s}=\left[ {\begin{array}{*{20}{c}} {{{\mathbf{r}}^{cp}}^{T}{\mathbf{r}}_{x}^{c}} \\ {{{\mathbf{r}}^{cp}}^{T}{\mathbf{n}}_{1}^{c}} \\ {{{\mathbf{r}}^{cp}}^{T}{\mathbf{n}}_{2}^{c}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {C_{{e1}}^{s}} \\ {C_{{e2}}^{s}} \\ {C_{m}^{s}} \end{array}} \right]$$

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where is the absolute position vector of the catenary at the contact point defined as ANCF FE; \({{\mathbf{S}}^c}\) is the matrix of the ANCF cable element shape function; \(x,\;y\), and are spatial coordinates of the element; \({{\mathbf{r}}^p}\) is the panhead’s position vector at the contact point defined as assuming that \({{\mathbf{R}}^p}\)is the position vector of the panhead’s middle point; , where \(\bar {y}\) is the panhead’s lateral displacement; \({\mathbf{A}}\) is the transformation matrix denoted as \({\mathbf{A}}=\left[ {\begin{array}{*{20}{c}} {{{\mathbf{a}}_{\mathbf{1}}}}&{{{\mathbf{a}}_{\mathbf{2}}}}&{{{\mathbf{a}}_{\mathbf{3}}}} \end{array}} \right]\); and \({{\mathbf{r}}^{cp}}\) is a different panhead and catenary position vector denoted as . Three orthogonal vectors [6, 7] are denoted as

$${\mathbf{r}}_{x}^{c}=\left[ {\begin{array}{*{20}{c}} {{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_1}} \\ {{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}} \\ {{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_3}} \end{array}} \right],\quad {\mathbf{n}}_{1}^{c}=\left[ {\begin{array}{*{20}{c}} { - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_1}{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}} \\ {{{\left( {{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_1}} \right)}^2}+{{\left( {{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_3}} \right)}^2}} \\ { - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_3}} \end{array}} \right],\quad {\mathbf{n}}_{2}^{c}=\left[ {\begin{array}{*{20}{c}} { - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_3}} \\ 0 \\ {{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_1}} \end{array}} \right]$$

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The contact frame is defined using a single vector \({\mathbf{r}}_{x}^{c}\). This frame formulation faces the singularity problem if the vector \({\mathbf{r}}_{x}^{c}\) parallels the vector \({\left[ {\begin{array}{*{20}{c}} 0&1&0 \end{array}} \right]^T}\). Three vectors are used to define the contact frame \({\mathbf{r}}_{x}^{c}={\left[ {\begin{array}{*{20}{c}} 0&{{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}}&0 \end{array}} \right]^T},\quad {\mathbf{n}}_{1}^{c}={\left[ {\begin{array}{*{20}{c}} { - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}}&0&0 \end{array}} \right]^T}\), and \({\mathbf{n}}_{2}^{c}={\left[ {\begin{array}{*{20}{c}} 0&0&1 \end{array}} \right]^T}\).

The algebraic equation is developed to solve two sliding directions on the panhead *via* the geometric parameter. This geometric parameter is the contact point location on the flexible contact wire and the rigid panhead using the elastic contact formulation with two parameters: the flexible contact wire’s length parameter (\({s_1}\)) and the rigid panhead’s lateral displacement parameter (\({s_2}\)). Assuming \(x=x\left( {{s_1}} \right)\) and \(\bar {y}={s_2}\), the elastic contact formulation is as follows:

$${\mathbf{C}}_{e}^{s}=\left[ {\begin{array}{*{20}{c}} {C_{{e1}}^{s}} \\ {C_{{e2}}^{s}} \end{array}} \right]$$

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For the simulation time step under consideration, the generalized variables for the panhead and contact wire from the numerical integration can solve the algebraic equation \({\mathbf{C}}_{e}^{s}=0\) to obtain the flexible contact wire’s length parameter (\({s_1}\)) and the rigid panhead’s lateral displacement parameter (\({s_2}\)). Using the equation for the iterative Newton–Raphson algorithm as

$${\left( {{\mathbf{C}}_{e}^{s}} \right)_{\mathbf{s}}}\Delta {\mathbf{s}}= - {\mathbf{C}}_{e}^{s}$$

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where \({\mathbf{s}}={\left[ {\begin{array}{*{20}{c}} {{s_1}}&{{s_2}} \end{array}} \right]^T}\) and \({\left( {{\mathbf{C}}_{e}^{s}} \right)_{\mathbf{s}}}=\begin{array}{*{20}{c}} {[\partial C_{{e1}}^{s}/\partial {\mathbf{s}}}&{\partial C_{{e2}}^{s}} \end{array}/\partial {\mathbf{s}}{]^T}\), assume that the term of \({\left( {{\mathbf{C}}_{e}^{s}} \right)_s}\) is always nonzero. The Newton difference can be calculated as

$$\left[ {\begin{array}{*{20}{c}} {\Delta {s_1}} \\ {\Delta {s_2}} \end{array}} \right]= - {{\mathbf{J}}^{ - 1}}{\mathbf{C}}_{e}^{s}$$

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where the Jacobian term is \({\mathbf{J}}={\left( {{\mathbf{C}}_{e}^{s}} \right)_{\mathbf{s}}}=\begin{array}{*{20}{c}} {[\partial C_{{e1}}^{s}/\partial {\mathbf{s}}}&{\partial C_{{e2}}^{s}} \end{array}/\partial {\mathbf{s}}{]^T}=\left[ {\begin{array}{*{20}{c}} {{{\mathbf{r}}^{cp}}^{T}{\mathbf{r}}_{{xx}}^{c}+{\mathbf{r}}{{_{x}^{c}}^T}{\mathbf{r}}_{x}^{c}}&{ - {\mathbf{a}}{{_{{\mathbf{2}}}^{p}}^T}{\mathbf{r}}_{x}^{c}} \\ {{{\mathbf{r}}^{c{p^T}}}{{\mathbf{H}}_1}{\mathbf{r}}_{{xx}}^{c}+{\mathbf{r}}_{x}^{c}{\mathbf{n}}_{1}^{c}}&{ - {\mathbf{a}}{{_{{\mathbf{2}}}^{p}}^T}{\mathbf{n}}_{1}^{c}} \end{array}} \right]\) and\({{\mathbf{H}}_1}=~\frac{{\partial {\mathbf{n}}_{1}^{c}}}{{\partial {\mathbf{r}}_{{{s_1}}}^{c}}}=\left[ {\begin{array}{*{20}{c}} { - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}}&{ - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_1}}&0 \\ {2{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_1}}&0&{2{{\left( {{\mathbf{r}}_{x}^{c}} \right)}_3}} \\ 0&{ - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_3}}&{ - {{\left( {{\mathbf{r}}_{x}^{c}} \right)}_2}} \end{array}} \right]\).

## 3.2 Contact Forces

A penalty force hypothesis based on the elastic contact force formulation, which allows the pantograph–catenary separation, is used to calculate the generalized contact and friction forces on the pantograph–catenary interaction. The variables in each of the two orthogonal directions are defined as \({l_2}={\left( {{\mathbf{a}}_{2}^{c}} \right)^T}\left( { - {{\mathbf{r}}^{cp}}} \right)\) and \({l_3}={\left( {{\mathbf{a}}_{3}^{c}} \right)^T}\left( { - {{\mathbf{r}}^{cp}}} \right)\). A penalty force model is defined along the aforementioned orthogonal directions \({\mathbf{a}}_{3}^{c}\), where \(k_{k}^{{cp}}\) and \(c_{k}^{{cp}}\) are the user-defined stiffness and damping coefficients, respectively. The friction force is calculated as \({\mathbf{F}}_{t}^{{cp}}= - \mu F_{3}^{{cp}}\left( {{{\left( {{\mathbf{a}}_{1}^{c}+{\mathbf{a}}_{2}^{c}} \right)} \mathord{\left/ {\vphantom {{\left( {{\mathbf{a}}_{1}^{c}+{\mathbf{a}}_{2}^{c}} \right)} {\left\| {{\mathbf{a}}_{1}^{c}+{\mathbf{a}}_{2}^{c}} \right\|}}} \right. \kern-0pt} {\left\| {{\mathbf{a}}_{1}^{c}+{\mathbf{a}}_{2}^{c}} \right\|}}} \right)\), where \(\mu\) is the friction coefficient. The friction force along the aforementioned orthogonal directions \({\mathbf{a}}_{1}^{c}\) and \({\mathbf{a}}_{2}^{c}\) is . The virtual work principle can generate generalized forces associated with the pantograph–catenary interaction generalized coordinates, which can be expressed as

$${{\mathbf{Q}}^p}= - {\sum\nolimits_{{k=1}}^{3} {F_{k}^{{cp}}\left( {{l_k},{{\dot {l}}_k}} \right)\left( {\frac{{\partial {l_k}}}{{\partial {{\mathbf{q}}^p}}}} \right)} ^T},\quad {{\mathbf{Q}}^c}= - {\sum\nolimits_{{k=1}}^{3} {F_{k}^{{cp}}\left( {{l_k},{{\dot {l}}_k}} \right)\left( {\frac{{\partial {l_k}}}{{\partial {{\mathbf{q}}^c}}}} \right)} ^T}$$

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