# Impact of Wind on the Movement of the Load Carried by Rotary Crane
## 3 Equations of Motion for the Studied System
### Equations of Motion of the Main System  
The movement of the load as the movement of a rigid body is described by the system of two vector equations [3; 20; 21]:  
\[\left\{ \begin{array}{c}m\mathbf{a}_{C}=\mathbf{F}_{C},\\ \frac{d}{dt}\mathbf{K}_{C}=\mathbf{M}_{C},\end{array} \right.\]  
where \(m\) is the mass of the carried load, \(\mathbf{a}_{C}\) is the absolute translational acceleration vector of the load's center of mass in the global coordinate system, \(\mathbf{F}_{C}\) is the resultant vector of all the forces acting on said load, \(\mathbf{K}_{C}\) is the angular momentum vector of the load in relation to its center of mass, and \(\mathbf{M}_{C}\) is the moment vector from all the forces acting on the load in relation to the center of mass. By taking into consideration the angular velocity vector \(\boldsymbol{\omega}_{C}\) and the matrix of inertia moments \(J_{c}\), it is possible to record the vector of the load's angular momentum in relation to its center of mass:  
\[\mathbf{K}_{C}=J_{C}\boldsymbol{\omega}_{C} \tag{4}\]  
including the total angular momentum derivative of the body in the fixed system  
\[\frac{d}{dt}\mathbf{K}_{C}=\frac{\overset{\sim}{d}}{dt}\mathbf{K}_{C}+ \boldsymbol{\omega}_{C}\times\mathbf{K}_{C}, \tag{5}\]  
rotational motion (3b), using the [PERSON] summation convention, can be described as [16]:  
\[M_{j}=J_{jm}\varepsilon_{m}+\varepsilon_{jkm}\omega_{k}J_{mn}\omega_{n},\quad j =1,2,3, \tag{6}\]  
where \(\varepsilon_{jkm}\) is the Levi-Civita permutation symbol. By not including air resistances to motion, the resultant force vector \(\mathbf{F}_{C}\) constitutes the vector sum of the tension in cable \(\mathbf{S}\), load gravity force \(\mathbf{G}\), and the impact of external forces \(\mathbf{F}_{\mathbf{w}}\):  
\[\mathbf{F}_{C}=\mathbf{S}+\mathbf{G}+\mathbf{F}_{\mathbf{w}}. \tag{7}\]  
By conducting the projection of forces in the global coordinate system  
\[ \begin{array}{l}\mathbf{S}=-(S_{\xi}\mathbf{i}+S_{\eta}\mathbf{j}-S_{\zeta} \mathbf{k}),\\ \mathbf{G}=-mg\mathbf{k},\\ \mathbf{F}_{\mathbf{w}}=F_{uv}\mathbf{i}+F_{wy}\mathbf{j}+F_{wz}\mathbf{k}, \end{array}  \tag{8}\]  
and presenting the acceleration vector \(\mathbf{a}_{C}\) as the sum of the boom tip component acceleration including the center of load mass:  
\[ \begin{array}{l}a_{Cx}=a_{\Omega x}+\ddot{\xi},\\ a_{Cy}=a_{\Omega y}+\ddot{\eta},\\ a_{Cz}=a_{\Omega z}+g-\ddot{\zeta},\end{array}  \tag{9}\]  
it is possible to depict the vector Equation (3a) in the following manner:  
\[\left\{ \begin{array}{l}m\Big{(}a_{\Omega x}+\ddot{\xi}\Big{)}+S_{\zeta}+F_{ uvx}=0,\\ m\Big{(}a_{\Omega y}+\ddot{\eta}\Big{)}+S_{\eta}+F_{wy}=0,\\ m\Big{(}a_{\Omega z}+g-\ddot{\zeta}\Big{)}-S_{\zeta}+F_{wz}=0,\end{array} \right. \tag{10}\]which, after certain transformations, may be presented as [1; 20]:  
\[ \begin{array}{l}\ddot{\xi}=-a_{\Omega x}+\frac{F_{uv}}{m}+\frac{\xi}{\rho^{2}} \bigg{[}\rho\ddot{\rho}+\dot{\rho}^{2}+\xi a_{\Omega x}+\eta a_{\Omega y}-\zeta( a_{\Omega z}+g)-\left(\dot{\xi}^{2}+\dot{\eta}^{2}+\dot{\zeta}^{2}\right)\bigg{]},\\ \ddot{\eta}=-a_{\Omega y}+\frac{F_{uv}}{m}+\frac{\eta}{\rho^{2}}\bigg{[}\rho \ddot{\rho}+\dot{\rho}^{2}+\xi a_{\Omega x}+\eta a_{\Omega y}-\zeta(a_{\Omega z }+g)-\left(\dot{\xi}^{2}+\dot{\eta}^{2}+\dot{\zeta}^{2}\right)\bigg{]},\\ \ddot{\zeta}=a_{\Omega z}+g+\frac{F_{uv}}{m}+\frac{\zeta}{\rho^{2}}\bigg{[} \rho\ddot{\rho}+\dot{\rho}^{2}+\xi a_{\Omega x}+\eta a_{\Omega y}-\zeta(a_{ \Omega z}+g)-\left(\dot{\xi}^{2}+\dot{\eta}^{2}+\dot{\zeta}^{2}\right)\bigg{]}.\end{array}  \tag{11}\]  
Individual members of the boom tip acceleration (point \(\Omega\)) can be written as [1]:  
\[\begin{array}{l}a_{\Omega x}=\ \ \ \left(R_{0}\dot{\varphi}^{2}\right)\cos \varphi+\left(R_{0}\ddot{\varphi}\right)\sin\varphi+\left(\ddot{r}-r\dot{ \theta}^{2}-r\dot{\varphi}^{2}\sin^{2}\theta\right)\sin\theta\cos\  
In order to solve the initial problem of motion, the ode45 calculation procedure, based on the Runge-Kutta 4 th Order Method, was used in the Matlab environment [24; 25].