13.2.3: Aircraft motion exercise

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In order to obtain the 4D trajectory, the flight will be divided into three phases: climb, cruise, and descent.

1. The climb phase will be assumed to be a symmetric flight into the vertical plane non considering any wind and assuming the heading angle to be zero. Therefore, the ODE system to be used is as follows:

\[m \dot{V} = T - D - m \cdot g \cdot \sin \gamma, \nonumber\]

\[m V \dot{\gamma} = L - m \cdot g \cdot \cos \gamma,\nonumber\]

\[\dot{x}_e = V \cdot \cos \gamma,\nonumber\]

\[\dot{h}_e = V \cdot \sin \gamma,\nonumber\]

\[\dot{m} = -T \cdot \eta.\nonumber\]

where according to BADA, \(\eta = ( \tfrac{C_{f1}}{1000*60} ) \cdot (1 + \tfrac{V}{C_{f2}} )\), with \(V\) in knots.

In order to integrate the system, one should:

1.1 Set the initial conditions for all the state variables, initial and final time. This conditions must be selected according to typical values of aircraft performance.

1.2 Set the control variables \((T(t), C_L (t))\),⁷ for instance to the following values:
* \(T = 0.8 \cdot T_{\max}\), where \(T_{\max} = C_{tc1} \cdot (1 - \tfrac{h_e}{C_{tc2}} + C_{tc3} \cdot h_e^2)\), \(h_e\) in feet,
* \(C_L = C_{L_{opt}}\).

1.3 Use a suitable numerical method to solve the resulting system.

2. The cruise phase will be assumed to be a symmetric flight into the horizontal plane not considering any wind. Therefore, the ODE system to be used is as follows:

\[m \dot{V} = T - D,\nonumber\]

\[m V \dot{\chi} = L \sin \mu, \nonumber\]

\[\dot{x}_e = V \cos \chi,\nonumber\]

\[\dot{y}_e = V \sin \chi, \nonumber\]

\[\dot{m} = -T \eta,\nonumber\]

being \(L = \tfrac{mg}{\cos \mu}\).

In order to solve the system, one should:

2.1 Set the initial conditions for all the state variables, initial and final time. Initial conditions and initial time will coincide with the final conditions of the previous phase. Set the final time according to typical values of aircraft performance.

2.2 Set the control variables \((T(t), \mu (t))\) to the following values:
* \(T = 0.5 * T_{\max}\),
* \(\mu = 0\).

2.3 Use a suitable numerical method to solve the resulting system.

3. The landing phase will be assumed to be gliding performance not considering any wind. Therefore, the ODE system to be used is as follows:

\[m \dot{V} = -D - mg \sin \gamma, \nonumber\]

\[m V \dot{\gamma} = L - mg \cos \gamma, \nonumber\]

\[\dot{x}_e = V \cos \gamma,\nonumber\]

\[\dot{h}_e = V \sin \gamma, \nonumber\]

3.1 Set the initial conditions for all the state variables, initial and final time. Initial conditions and initial time will coincide with the final conditions of the previous phase. Set the final time according to typical values of aircraft performance.

3.2 Set the control variable \((C_L(t))\) to the following value:
* \(C_L = C_{L_{opt}}.\)

3.3 Use a suitable numerical method to solve the resulting system.

7. Notice that \(C_L\) acts as control.