# Form Finding of Asymmetric Cable Network Reflector with Flexible Ring Truss
## 2 Form-Finding Method for Asymmetric Cable Network Reflector  
### Equilibrium Equation  
The asymmetric cable network reflector is designed using a combination of the force density method, a genetic algorithm, and an iterative technique. The boundary nodes of the ring truss are called \"fixed nodes\" and nodes on the cable network are called \"free nodes\". Assuming that the structure of the cable net has \(m\) members, \(n\) free nodes, \(n_{f}\) fixed nodes, and \(n_{s}=n+n_{f}\), its topology can be expressed using an incidence matrix \(C_{s}(\in R^{m\times n_{s}})\)[27]. Supposing that the member \(k\) connects nodes \(i\) and \(j\) (\(i<j\)), the \(i\)th and \(j\)th elements of the \(k\)th row of \(C_{s}\) are set to \(1\) and \(-1\), respectively. This can be represented as:  
\[C_{S}(k,g)= \begin{cases}+1&\text{for }g=i\\ -1&\text{for }g=j\\ 0&\text{otherwise}\end{cases}  \tag{3}\]  
The columns of the matrix \(C_{s}\) can be rearranged such that the free nodes precede the fixed nodes in the numbering sequence. Hence, \(C_{s}\) can be divided into two parts as:  
\[C_{s}=[C\,C_{f}] \tag{4}\]  
where \(C\in R^{m\times n}\) and \(C_{f}\in R^{m\times n_{f}}\) describe the connectivity of the members to the free and fixed nodes, respectively. As an example, for a simple two-dimensional truss shown in Figure 4a, which consists of seven members and eight nodes, including two free nodes (\(n=2\)) and six fixed nodes (\(n_{f}=6\)), the connectivity matrix \(C_{s}\) is given in Figure 4b. The force density method is typically used to control both the geometry and the pre-stress distribution [28]. Using this method, we can transform a set of non-linear equilibrium equations of the nodes into a set of linear equations, which will be explained in the following. Figure 3: Asymmetric paraboloid-shaped net model. If \(R\) members of the cable net structure meet at the node \(i\), then the equilibrium equations of node \(i\) can be derived as:  
\[ \begin{split}\sum_{1}^{R}\frac{x_{k}-x_{i}}{l_{i,k}}s_{i,k}+p_{xi}=0 \\ \sum_{1}^{R}\frac{y_{k}-y_{i}}{l_{i,k}}s_{i,k}+p_{yi}=0\\ \sum_{1}^{R}\frac{z_{k}-z_{i}}{l_{i,k}}s_{i,k}+p_{zi}=0\\ \end{split}  \tag{5}\]  
Further, define  
\[ \begin{split} u&=C_{s}\cdot x_{s}=C\cdot x+C_{f} \cdot x_{f}\\ v&=C_{s}\cdot y_{s}=C\cdot y+C_{f}\cdot y_{f}\\ w&=C_{s}\cdot z_{s}=C\cdot z+C_{f}\cdot z_{f}\\ \end{split}  \tag{6}\]  
where \(s_{i,k}\) and \(l_{i,k}\) denote the tension force and length of the \(k\)th member, respectively. The symbols \(p_{xi}\), \(p_{yi}\), and \(p_{zi}\) denote the column vectors of the external forces of the nodes. The set of equations in Equation (5) can be linearized for each member by introducing the force density as:  
\[q_{j}=\frac{s_{i}}{l_{j}} \tag{7}\]  
The force density values are the target values that need to be calculated during the form-finding process. For a general structure with \(w\) members and \(j\) nodes, the equilibrium equations in the \(x\)-direction are written as:  
\[C_{s}^{T}\cdot Q\cdot C_{s}\cdot x_{s}=p_{x} \tag{8}\]  
In a similar way, the equilibrium equations in the \(y\)- and \(z\)-directions are expressed as:  
\[C_{s}^{T}\cdot Q\cdot C_{s}\cdot y_{s}=p_{y} \tag{9}\]  
\[C_{s}^{T}\cdot Q\cdot C_{s}\cdot z_{s}=p_{z} \tag{10}\]  
Since the external forces and weight of the whole structure are ignored during the form-finding process, the equilibrium equations in the \(x\)-coordinate are changed to  
Figure 4: A two-dimensional cable net. \[\left[ \begin{array}{c}C^{T}\\ C^{T}_{f}\end{array} \right]\cdot\left[Q\right]\cdot\left[C\;C_{f}\right]\cdot \left\{ \begin{array}{c}x\\ x_{f}\end{array} \right\}=\left\{ \begin{array}{c}0\\ p_{xf}\end{array} \right\} \tag{11}\]  
which can be expressed as:  
\[\left\{ \begin{array}{c}C^{T}\cdot Q\cdot C\cdot x+C^{T}\cdot Q\cdot C_{f} \cdot C_{f}\cdot x_{f}=0\\ C^{T}_{f}\cdot Q\cdot C\cdot x+C^{T}_{f}\cdot Q\cdot C_{f}\cdot x_{f}=p_{xf} \end{array} \right.