# Dynamics Modelling and Simulation for Deployment Characteristics of Mesh Reflector Antennas
## 2 Dynamic Modelling of the Truss Antenna
### Dynamic Equations of Cable Net  
In this section, the ANCF description of a flexible body is employed to predict the large deformation of cable net during deployment of the deployable antenna. With the metric that the gradient of the node is adopted as node coordinates, the ANCF mesh can precisely capture the flexible deformation as well as rigid motion [16]. Herein, a gradient deficient beam element, as shown in Figure 3, widely used in the application of the cable problem, is selected to discrete the cable net. Each element consists of two nodes with generalized coordinates expressed as  
\[\mathbf{e}=\left[\mathbf{r}_{1}^{\mathrm{T}}(0)\quad\mathbf{r}_{1,x}^{ \mathrm{T}}(0)\quad\mathbf{r}_{2}^{\mathrm{T}}(L)\quad\mathbf{r}_{2,x}^{ \mathrm{T}}(L)\right]^{\mathrm{T}} \tag{6}\]  
where \(\mathbf{r}_{i}(i=1,2)\) is the vector of the global position coordinates and \(\mathbf{r}_{i,x}\) is the vector of the gradient coordinates defined as \(\mathbf{r}_{i,x}=\partial\mathbf{r}/\partial x\). \(L\) is the length of element.  
Figure 3: Deployment of the deployable truss.  
Thus, the global position vector of an arbitrary particle on the point is defined as  
\[\mathbf{r}(x,t)=\mathbf{S}(x)\mathbf{e}(t) \tag{7}\]  
where \(\mathbf{S}\) is the shape function matrix expressed as  
\[\mathbf{S}(x)=\left[\begin{array}{cccc}S_{1}(x)\mathbf{I}_{3\times 3}&S_{2}(x) \mathbf{I}_{3\times 3}&S_{3}(x)\mathbf{I}_{3\times 3}&S_{4}(x)\mathbf{I}_{3 \times 3}\end{array}\right] \tag{8}\]  
in which \(\mathbf{I}\) is the identity matrix and the items in Equation (8) are as follows  
\[\left\{\begin{array}{ll}S_{1}=1-3\xi^{2}+2\xi^{3}\\ S_{2}=L\big{(}\xi-2\xi^{2}+\xi^{3}\big{)}\\ S_{3}=3\xi^{2}-2\xi^{3}\\ S_{4}=L\big{(}-\xi^{2}+\xi^{3}\big{)}\end{array}\right. \tag{9}\]  
where \(\xi=x/L\), \(L\) is the length of the cable element.  
Differentiating of Equation (7) with respect to time, the global velocity vector is obtained as  
\[\dot{\mathbf{r}}(x,t)=\mathbf{S}(x)\dot{\mathbf{e}}(t) \tag{10}\]  
The kinetic energy of the element is formulated as  
\[T_{e}=\frac{1}{2}\int_{0}^{L}\int_{A}\rho\dot{\mathbf{r}}^{\mathrm{T}}\dot{ \mathbf{r}}dAdx=\frac{1}{2}\dot{\mathbf{e}}^{\mathrm{T}}\mathbf{M}_{e}\dot{ \mathbf{e}} \tag{11}\]  
where \(\rho\) and \(A\) denote the density and area of the cross-section. \(\mathbf{M}_{e}\) denotes the consistent mass matrix of the cable element that can be clarified as  
\[\mathbf{M}_{e}=m_{e}\left[\begin{array}{cccc}\frac{13}{35}\mathbf{I}&\frac{ 11}{210}L\mathbf{I}&\frac{9}{20}\mathbf{I}&-\frac{13}{420}\mathbf{I}\\ &\frac{1}{105}L^{2}\mathbf{I}&\frac{13}{420}L\mathbf{I}&-\frac{1}{140}L^{2} \mathbf{I}\\ &\frac{13}{35}\mathbf{I}&-\frac{1}{210}L\mathbf{I}\\ sym&&\frac{1}{105}L^{2}\mathbf{I}\end{array}\right] \tag{12}\]  
where \(m_{e}\) is the mass of the cable element and \(\mathbf{I}\) is the identity matrix.  
The strain energy of the ANCF cable element can be obtained with the following formulation  
\[U_{e}=\frac{1}{2}\int_{0}^{L}\Big{(}E\alpha\varepsilon^{2}+EJ\kappa^{2}\Big{)}dx \tag{13}\]  
where \(E\) is the elastic modulus and \(J\) is the second moment of the cross-sectional area. The axial strain \(\varepsilon\) and curvature \(\kappa\) are defined by the following expression, respectively.  
\[\begin{array}{ll}\varepsilon=\sqrt{\mathbf{I}_{x}^{\mathrm{T}}\mathbf{r}_{ x}}-1\\ \kappa=\frac{\left|\mathbf{r}_{x}\times\mathbf{M}_{x}\right|}{\left|\mathbf{r}_{x} \right|^{3}}\end{array} \tag{14}\]  
Due to the high nonlinearity of the formula of the strain energy, numerical integration of Gaussian quadrature is adopted in this work. Note that 5 integration points are adopted to evaluate the elastic force considering a good trade-off between computational cost and numerical accuracy.  
The total kinetic energy and total strain energy of the cable net can be formulated as  
\[T_{c}=\sum_{i=1}^{n_{e}}T_{e},\quad U_{c}=\sum_{i=1}^{n_{e}}U_{e} \tag{15}\]