## 3 Modelling and Control
### Approaching Flow Stage (AFS)
\[l_{1}=\frac{L}{\pi}\text{cot}\,\text{h}\bigg{(}\frac{\pi A}{L}\bigg{)}, \tag{10}\]  
\[L_{1}=-\bigg{[}l_{1}+\frac{A}{2}-\bigg{(}\frac{2L}{\pi}\text{tan}\,\text{h} \bigg{(}\frac{\pi A}{L}\bigg{)}\bigg{)}\bigg{]}. \tag{11}\]  
Considering an input signal of \(\theta(t)=\phi\sin(\alpha t)\) to excite the vessel motion, the Equation (9) solution yields,  
\[\ddot{\theta}_{1}+\frac{g}{l_{1}}\theta_{1}=\bigg{[}\bigg{(}\bigg{(}\frac{A}{ 2}-L_{1}-l_{1}\bigg{)}\frac{\alpha^{2}\phi}{l_{1}}\bigg{)}+\frac{g\phi}{l_{1}} \bigg{]}\sin(\alpha t). \tag{12}\]  
The characteristic polynomial of the differential Equation (12) is in form of: \(\text{s}^{2}+\omega^{2}\). \(\text{s}\) is the laplace variable, and \(\omega^{2}=\frac{g}{l_{1}}\) is the corresponding frequency equation for a simple pendulum. Figure 4.0  
As [PERSON] states in [29], the slosh time response solution of (12), expressed in Equation (13), will let us compare analytically, the angle elevation of \(\theta_{1}\) against \(\theta\). On the right side of Figure 6, \(\theta_{1}(t)\) is plotted for values in the range of \(\pi<\alpha<2\pi\). The dark plot is the input signal \(\theta(t)=\phi\sin(\alpha t)\). It is observed here that the free surface angle \(\theta_{1}(t)\) tends to increase the elevation between angles as \(\alpha\) increases and until \(\alpha\) reaches the resonant frequency \(\Big{(}\frac{g}{l_{1}}\Big{)}^{1/2}\). It is clear now that this elevation will increase a non-linear and unwanted behavior of the slosh. On the left side of Figure 6, a valid range for \(\alpha\) is used for several plots of \(\theta_{1}(t)\). Notice that the elevation here is marginal, keeping \(\alpha\ll\frac{g}{l_{1}}\). This will retain a linear model of the slosh, and it will make both angles of rotation, \(\theta_{1}\) and \(\theta\), change alike. The results in Section 4, illustrate the equivalence of this pendulum model of the slosh with respect to the time responses in Figure 4.1  
As concluded earlier in the graphs of Figure 6, a valid range of \(\alpha\) can be obtained in an input elevation signal of \(\theta\) so the elevation of \(\theta_{1}\) retains linearity, and ASF with SIM provides a back elevation of the slosh with equivalent angle value of \(\theta\). Figure 7 shows the proposed timeline response of ASF with SIM for the elevation of the angles. In Figure 7, \(t_{0}\) is the period of time where the mentioned input signal of \(\theta(t)\) is applied to vessel 1 allowing the free surface angle \(\theta_{1}\) to turn at the same rotational velocity as \(\theta\) and equally reach angle \(\phi\). This time period corresponds to transition 1 in Figure 4.2 In the next period of time, \(t_{1}\), \(\theta_{1}\) returns to zero, while \(\theta\) remains at that initial conditional angle \(\phi\). As soon as \(\theta_{1}\) crosses  
Figure 4.3 \(\theta_{1}\) transient time response for an excitation of \(\theta(t)=\theta_{0}\sin(\alpha t)\). Low frequency values of \(\alpha\) to the (**left**), and high frequencies values of \(\alpha\) to the (**right**). Figure 4.4 Proposed timeline of slosh induction method (SIM) pouring. AFS ranges from the initial tilting to the end of the control action. CFS initiates at the end of AFS and continues till the end of pouring. of time, \(t_{2}\), initiates with the control action of the slosh. The goal of this control is to turn the surface elevation to zero and to avoid pure undamped behavior of the slosh. Recall that an oscillating response of the slosh will be presented if no control action is taken in the pendulum model. The optimal control action would lead \(\theta_{1}\) to zero as soon as possible so the particle outflow velocity become constant at fixed \(h_{q}\), allowing it to migrate from AFS to CFS with a stable pouring. At the beginning of \(t_{3}\), where the slosh finds free exit to flow out of the vessel, AFS will end and CFS will begin. \[