# Flap and Wing Dynamics for a Light Sport Aircraft Analysis Using a Topological Model
## 3 Kinematic Analysis of the Control Mechanism  
We will perform a partition of \(\left[B\right]_{f}\), \(\left[B\right]_{f}=\left[\left[B\right]_{fu}\vdots\left[B\right]_{fk}\right]\) made after a rearrangement of the columns of \(\left[B\right]_{f}\) so that the matrix \(\left[B\right]_{fu}\) corresponds to the elements of the first class and \(\left[B\right]_{fk}\) to the elements of the second class. \[ \begin{array}{l}\mathit{cycle}\backslash\mathit{branch}\\ 1\\ 2\\ 3\\ 4\\ 5\\ 6\\ \end{array} \begin{array}{l}3&4&5&6&7&8&10&11&12&13&14&15&1&2&9\\ 1&  
where \[[\backslash L_{n}\backslash]=[\backslash\quad l_{3}\quad l_{4}\quad l_{5}\quad \ldots\quad\ldots\quad l_{13}\quad l_{14}\quad l_{15}\quad\backslash]\] (16) and \[[\backslash L_{c}\backslash]=[\backslash\quad l_{1}\quad l_{2}\quad l_{9}\quad\backslash]\] (17)
* \([\backslash C\backslash]\), the matrix with the cosine of the angles made by the bars with the horizontal axis, partitioned in \([\backslash C_{n}\backslash]\) (unknown motion) and \([\backslash C_{c}\backslash]\) (known motion) \[[\backslash C\backslash]= \begin{bmatrix}[\backslash C_{n}\backslash]&\vdots&0\\ \ldots&\vdots&\ldots\\ 0&\vdots&[\backslash C_{c}\backslash]\end{bmatrix} \] (18) where \[[C_{n}]=[\backslash\quad\cos\theta_{3}\quad\cos\theta_{4}\quad\cos\theta_{5} \quad\ldots\quad\ldots\quad\cos\theta_{13}\quad\cos\theta_{14}\quad\cos\theta_ {15}\quad\backslash]\] (19) and \[[C_{c}]=[\backslash\quad\cos\theta_{1}\quad\cos\theta_{2}\quad\cos\theta_{9} \quad\backslash]\] (20)
* \([S]\), the matrix with the sine of the angles made by the bars with the horizontal axis, partitioned in \([\backslash S_{n}\backslash]\) (unknown motion) and \([\backslash S_{c}\backslash]\) (known motion) \[[\backslash S\backslash]= \begin{bmatrix}[\backslash S_{n}\backslash]&\vdots&0\\ \ldots&\vdots&\ldots\\ 0&\vdots&[\backslash S_{c}\backslash]\end{bmatrix} \] (21) where \[[S_{n}]=[\backslash\quad\sin\theta_{3}\quad\sin\theta_{4}\quad\sin\theta_{5} \quad\ldots\quad\ldots\quad\sin\theta_{13}\quad\sin\theta_{14}\quad\sin\theta_ {15}\quad\backslash]\] (22) and \[[S_{c}]=[\backslash\quad\sin\theta_{1}\quad\sin\theta_{2}\quad\sin\theta_{9} \quad\backslash]\] (23) We define the column vectors of size \(s\) as \[\{\omega\}= \begin{cases}\omega_{n}\\ \omega_{4}\\ \vdots\\ \vdots\\ \vdots\\ \omega_{15}\\ \cdots\\ \omega_{1}\\ \omega_{2}\\ \omega_{9}\end{cases} \text{; }\left\{\omega^{2}\right\}= \begin{cases}\omega_{n}^{2} \\ \omega_{n}^{2}\\ \cdots\\ \omega_{c}^{2}\end{cases} = \begin{cases}\omega_{n}^{2}\\ \omega_{4}^{2}\\ \vdots\\ \vdots\\ \omega_{15}^{2}\\ \cdots\\ \omega_{2}^{2}\\ \omega_{9}^{2}\end{cases} \text{; }\left\{\varepsilon\right\}= \begin{cases} \varepsilon_{n}\\ \cdots\\ \varepsilon_{c}\end{cases} = \begin{cases}\varepsilon_{3}\\ \varepsilon_{4}\\ \vdots\\ \vdots\\ \varepsilon_{15}\\ \cdots\\ \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{9}\end{cases} \] (24) In subsequent calculus, it must be considered that \(\omega_{1}=\omega_{9}=0\); \(\varepsilon_{1}=\varepsilon_{9}=0\); with the elements \(1\) and \(9\) being fixed elements. In these formulas, we have denoted \(l_{i}\)--the length of the numbered element with \(i\); \(\theta_{i}\)--the angle made by the vector \(\overline{l}_{i}\) with the axis \(Ox\) of the global coordinate system \(xOy\); \(\overline{\omega}_{i}\)--the angular velocity of the element numbered with \(i\); \(\overline{\varepsilon}_{i}\)--the angular acceleration of the element \(i\). By applying this theory, it is possible to obtain the unknown velocities and acceleration vectors as a function of the known vectors of the velocities and accelerations, with the relations  
\[\{\omega_{n}\}=-[P_{n}]^{-1}[P_{c}]\{\omega_{c}\}, \tag{25}\]  
and  
\[\{\epsilon_{n}\}=[P_{n}]^{-1}\Big{(}-[P_{c}]\{\epsilon_{c}\}+[Q_{n}]\Big{\{} \omega_{n}^{2}\Big{\}}+[Q_{c}]\Big{\{}\omega_{c}^{2}\Big{\}}\Big{)}. \tag{26}\]  
where  
\[