## 2 Parametric Analysis of a Steady Motion of Underwater Glider in a Vertical Plane
Write, alternatively,  
\[\frac{R_{B}-R_{W}}{R_{y}}=\sqrt{1+\varepsilon^{2}(\alpha)}, \tag{9}\]  
or  
\[R_{y}=\frac{R_{B}-R_{W}}{\sqrt{1+\varepsilon^{2}(\alpha)}}. \tag{10}\]  
On the other hand, it follows from the moment equation that  
\[ \begin{array}{l}(R_{B}-R_{W})x_{1B}\cos\beta+R_{W}[(x_{1B}-x_{1W})\cos\beta \\ -z_{1W}\sin\beta]-x_{1Y}R_{y}=0,\end{array}  \tag{11}\]  
Expressing \(R_{y}\) in (11) with Equation (10), one can find  
\[\frac{R_{B}-R_{W}}{R_{W}}=\frac{\Delta R_{B}}{R_{W}}=\frac{\Delta V}{V_{0}}= \eta=\frac{y_{1W}\sin\beta-(x_{1B}-x_{1W})\cos\beta}{x_{1B}\cos\beta-x_{1p} \cos\alpha/\sqrt{1+\varepsilon^{2}(\alpha)}}, \tag{12}\]  
where \(\eta\) represents the relative differential buoyancy, with \(\eta>0\) for the ascent and \(\eta<0\) for the descent. It is adopted above that the excess of buoyancy \(R_{B}-R_{W}=\Delta R_{B}=\rho g\Delta V=\rho gV_{0}\eta\) is obtained by means of _buoyancy engine_ (BE), which realizes a variation of buoyancy in the fore part of the UG for the same weight of the vehicle and that in the state of equilibrium, \(R_{W}=\rho gV_{0}\), where \(V_{0}\) is the volumetric displacement of the UG in equilibrium mode on the surface. With the Equation (10), one can derive an expression for the speed of the UG on the trajectory. Writing lift and drag through corresponding force coefficients  
\[R_{y}=C_{y}\frac{\rho L_{0}^{2}}{2}S_{wing},R_{x}=C_{x}\frac{\rho L_{0}^{2}}{ 2}S_{wing}, \tag{13}\]  
where \(S_{wing}\) indicates the reference area; taking into account that the inverse hydrodynamic quality can be written as \(\varepsilon(\alpha)=R_{x}/R_{y}=C_{x}/C_{y}\), we come to the following expression for the speed of UG:  
\[U_{0}=\sqrt{\frac{2 gV_{0}\eta}{S_{wing}\sqrt{C_{y}^{2}+C_{x}^{2}}}}, \tag{14}\]  
A ratio \(\eta V_{0}/S_{wing}=\Delta V_{0}/S_{wing}\) in (14) is associated with the _volumetric loading_ upon the wing system. Its magnitude is significantly smaller for the UG of the _flying wing_ (FW) type than for the UG configured as a _winged body of revolution_ (BRW), which, similarly to the case of airplanes, results in a decrease of the speed on a trajectory when passing from fuselage configuration to flying wing configuration. Note that the drag coefficient entering (14) can be represented as a sum of the induced drag and viscous drag coefficients  
\[C_{x}=C_{xi}+C_{x0}. \tag{15}\]  
If the pitch angle equals \(\beta=90^{\circ}(\alpha=0)\), the UG will execute a vertical ascent (descent). Therewith \(C_{y}=0\) and \(C_{xi}=0\), and the speed of vertical motion equals  
\[U_{y|_{C_{y}=0}}=\sqrt{\frac{2 gV_{0}\eta}{S_{wing}C_{x0}}} \tag{16}\]  
Compose ratio of the speed on trajectory and the speed of vertical displacement at \(\alpha=0\)  
\[\frac{U_{0}}{U_{0|}C_{y}=0}=\sqrt{\frac{C_{x0}}{\sqrt{C_{y}^{2}+C_{x}^{2}}}}, \tag{17}\]  
or representing the induced drag coefficient in (15) in Prandtl format as  
\[C_{xi}=k_{i}\frac{C_{y}^{2}}{\pi\lambda}, \tag{18}\]  
where \(k_{i}=C_{x}^{\alpha\alpha}/C_{y}^{\alpha 2}\), and \(\lambda\) is the wing aspect ratio; rewrite (17) as  
\[\frac{U_{0}}{U_{0|}C_{y}=0}=\sqrt{\frac{C_{x0}}{\sqrt{C_{y}^{2}+\left(k_{i} \frac{C_{y}^{2}}{\pi\lambda}+C_{x0}\right)^{2}}}}, \tag{19}\]  
The dependence of (19) as a function of lift coefficient \(C_{y}\) for different magnitudes \(C_{x0}\) is shown in Figure 3, from which one can draw an important qualitative conclusion: for an UG with wings, its speed on the trajectory is less than the speed of vertical ascent/descent. Let us consider the relationship between the angle of trajectory and the angle of attack of the vehicle in more detail. Figure 3: Speed on trajectory of the UG, related to the speed of vertical descent/ascent, plotted versus the lift coefficient.