# Flap and Wing Dynamics for a Light Sport Aircraft Analysis Using a Topological Model
## 1 Introduction
### Incidence Matrix  
By attaching to each bar element a vector, with the end in one joint and the top in the other, and taking for each branch the direction of travel given by the direction of the vector thus constructed, the oriented graph from Figure 4 is obtained. Figure 4: (**a**) Planar four-bar mechanism; (**b**) vector contour; (**c**) associated graph; and (**d**) the tree of the associated graph. Figure 3: A straight lever with multiple joints is replaced with triangles. Let \(r+1\) be the number of nodes and \(s\) the number of branches of the obtained graph. The complete incidence matrix \([A]=\left[\varepsilon_{ij}\right]\) of the associated graph is defined as a matrix having the dimensions \((r+1)\times s\) (\(s>r\)) where  
* \(\varepsilon_{ij}=1\) if branch \(j\) is incident at node \(i\) and comes out of the node;
* \(\varepsilon_{ij}=-1\) if branch \(j\) is incident at node \(i\) and enters the node;
* \(\varepsilon_{ij}=0\) if branch \(j\) is not incident at node \(i\). These operations offer us a matrix \([A]\) that looks like the following:  
\[ \begin{array}{ccccc}&\mathit{nodes}\backslash\mathit{branches}\\ &1&2&3&\ldots&s\\ &1&\left[\begin{array}{ccccc}\varepsilon_{11}&\varepsilon_{12}&\varepsilon_ {13}&\ldots&\varepsilon_{1s}\\ \varepsilon_{21}&\varepsilon_{22}&\varepsilon_{23}&\ldots&\varepsilon_{2s}\\ \vdots&&&&\vdots\\ \vdots&&&&\vdots\\ \varepsilon_{r+1,1}&\varepsilon_{r+1,2}&\varepsilon_{r+1,3}&\ldots&\varepsilon _{r+1,s}\end{array}\right]\\ \end{array}  \tag{1}\]  
where  
\[\varepsilon_{ij}\in\{-1,\ 0,\ 1\};\ i=1,2,\ldots,r+1;\ j=1,2,\ldots,s. \tag{2}\]  
A few examples will clarify the associated graph and the incidence matrix problem below. **Example 1**. : _The associated graph of the four-bar linkage (Figure 3(a)) is shown in Figure 3(c). Figure 3(b) shows the vector contour that defines the single loop of the mechanism. The extended incidence matrix, if using the notation in Figure 3(c), is as follows:_  
\[ \begin{array}{ccccc}\mathit{nodes}\backslash\mathit{branches}\\ &1&2&3&4\\ &1&0&0&-1\\ &2&3\\ &4&0&-1&1\end{array} \]  
The matrix of the fundamental loops (in our case, only one loop) is  
\[ \begin{array}{ccccc}\mathit{cycle}\backslash\mathit{branches}\\ &1&2&3&4\\ &\left[B_{f}\right]=1\left[\begin{array}{ccccc}1&1&1&1&1\end{array}\right] \end{array} \]  
**Example 2**. : _A slightly more complex mechanism is shown in Figure 5. Three independent loops are presented in Figure 6. The associated graph and tree are attached to this mechanism (Figure 7)._  
The extended incidence matrix is as follows:  
\[ \begin{array}{ccccc}\mathit{nodes}\backslash\mathit{branches}\\ &1&2&3&4&5&6&7&8&9\\ &1&0&0&-1&0&0&0&0\\ &2&\left[\begin{array}{ccccc}1&0&0&-1&0&0&0&0\\ -1&1&0&0&-1&0&0&0\\ 0&-1&1&0&1&0&0&0&0\\ 0&0&-1&1&0&0&0&-1\\ &0&0&0&0&-1&1&0&0\\ &6&0&0&0&0&-1&1&0\\ &0&0&0&0&0&0&-1&1\end{array}\right]\\ \\ \end{array} \]The matrix of the fundamental loops is  
Figure 5: A sketch of the mechanism. Figure 6: Three vector contours of the mechanism. Figure 7: (**a**) Associated graph. (**b**) A tree of the graph. In Graph Theory, the complete matrix of fundamental loops is defined as the matrix \(B_{a}=[b_{ij}]\), where  
\[ \begin{array}{ll}\blacksquare&b_{ij}=1\text{ if the branch }j\text{ is part of the loop }i\text{ having a positive orientation;}\\ \blacksquare&b_{ij}=-1\text{ if side }j\text{ is part of the loop }i\text{ having a negative orientation;}\\ \blacksquare&b_{ij}=0\text{ if branch }j\text{ is not part of loop }i.\end{array}  \tag{8}\]  
Here, \(\eta_{ij}\in\{-1,0,1\}\); \(i=1,2,\ldots,n_{b};j=1,2,\ldots,s\); and \(n_{b}\) represents the total number of graph loops. According to Graph Theory [1; 2; 3], the rank of the matrix \([B]_{a}\) is \(s-r\). This matrix contains redundant information (some rows, describing a loop, can be obtained as a linear combination of other rows). The matrix \([B]_{f}\), with the rank \(s-r\), considering \(s-r\) linear independent rows, is the complete matrix \([B]_{a}\).