## 4 System Modeling
### Water Conveying Hose Modeling  
Either Newtonian, in the same manner as the head part in the next section, or Hamiltonian approaches can be adopted to establish the motion equations of the hose. Since the deformation of the hose in our system is significant, the latter method is more intuitive. The head part is first isolated from the hose and is replaced by a vector of reaction force. Then, the modified Hamilton principle for the assembly of the hose and the water inside is written as follows [33; 34]:  
\[\int_{t_{1}}^{t_{2}}\delta Wdt+\delta\int_{t_{1}}^{t_{2}}Ldt=\int_{t_{1}}^{t_ {2}}\Big{[}m_{f}V_{h}\left(\dot{\mathbf{r}}_{L}+V_{h}\mathbf{r}^{\prime}_{L}\right) \cdot\delta\mathbf{r}_{L}\Big{]}dt \tag{4}\]  
where the right-hand terms associate with end effects due to the isolation of the head. \(\delta\) is [PERSON]'s variational operator. \(L\) is the Lagrangian including kinetic energy, bending strain energy, and gravitation. The formulations of the energies are given in the same manner in [33; 34]. The term \(\delta W\) is the virtual works done by external forces, as follows:  
\[ \begin{split}\delta W=\int_{0}^{L_{s}}\mathbf{F}_{s}^{T}\delta\mathbf{r}_{ \Delta s}ds,\ \ \mathbf{r}_{\Delta s}=\mathbf{r}_{s}-\mathbf{r}_{s0},\\ \mathbf{F}_{s}=-\mathbf{F}_{p}\delta_{(s-L_{s})}+\mathbf{F}_{r}\delta_{(Z_{s}- \sqrt{A_{h}/\pi})}+\mathbf{F}_{p_{p}}\delta_{(s)}-\mathbf{p}_{h}A_{h}\mathbf{n}_{h}\delta_{ (s-L_{s})}+\mathbf{F}_{0}\delta_{(s)}\end{split}  \tag{5}\]  
The forces mentioned in Equation (5) include the reaction force from the head \(-\mathbf{F}_{p}\); the interactive force with surroundings \(\mathbf{F}_{r}\); the force due to the pumping pressure at the hose inlet \(\mathbf{F}_{p_{p}}\) and exit pressure at the hose outlet \(-\mathbf{p}_{h}A_{h}\mathbf{n}_{h}\); the holding force due to the connection of the hose inlet to the pump outlet \(\mathbf{F}_{0}\). Additionally, the Dirac delta function \(\delta_{(\cdot)}\) indicates the part of the hose where the force applies. Substitute the Lagrangian and the kinematics in Equation (3) into Equation (4), and after a lengthy, but straightforward computation process, the governor equation of the hose conveying water is obtained as follows:  
\[(m_{f}+m_{p})\ddot{\mathbf{r}}_{s}=-2m_{f}V_{h}\dot{\mathbf{r}}^{\prime}_{s}-m_{f}V_{h }^{2}\mathbf{r}^{\prime\prime}_{s}-EI\mathbf{r}^{\prime\prime\prime\prime}_{s}+\Big{(} m_{p}+m_{f}\Big{)}\mathbf{g  
**Assumption 1**. : _The water volume contained in the head is considered undeforming and moving._  
**Assumption 2**. : _The water flow is steady. The remaining acceleration of the water will be treated as input disturbances._  
**Assumption 3**. : _External forces and torques due to wind, aerodynamic drag, and additional mechanisms are neglected in this stage of development._  
\[ \begin{array}{c}\mathbf{F}_{hb}\,=\,(-\sum_{i=1}^{4}\dot{m}_{i}\mathbf{V}_{ib}- \dot{m}_{i}\mathbf{V}_{ib}+\dot{m}_{h}\mathbf{V}_{hb})+M_{h}g_{b}+p_{h}A_{h}\overset{ \mathbf{\Lambda}}{\mathbf{n}}_{hb}+\mathbf{F}_{pb}+\mathbf{F}_{wb},\\ \tau_{hb}\,=\,(-\sum_{i=1}^{4}\mathbf{r}_{ib}\times\dot{m}_{i}\mathbf{V}_{ib}-\mathbf{r}_{ lb}\times\dot{m}_{l}\mathbf{V}_{lb})+\mathbf{\tau}_{wb}\end{array}  \tag{9}\]  
with:  
\[\dot{m}_{h}=\sum_{i=1}^{4}\dot{m}_{i}+\dot{m}_{i},\,\,\,\,\dot{m}_{j}=\rho V_{ j}A_{j},\,\,(j=h,\,\,l,\,\,i), \tag{10}\]  
\(\dot{m}_{j}\) is the mass flow rate flowing through the cross-sectional area \(A_{j}\). \(\rho\) is the water density. The water conveyed by the hose is for both generating thrust and suppressing a flame. The terms in brackets in Equation (9) are the force and torque generated by the change of the linear momentum of the water flowing through the head part. Considering the first two assumptions, the force and torque depend only on the momentum through the head inlet and the nozzles' outlet. The water inlet of the head is located at the origin of the coordinate so that no torque is created by the input water. On the other hand, \(g_{b}\) is the gravitational acceleration observed from the body-fixed frame.