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Derive the moment equation around the centre of gravity of this King Air 200, as a function of the total lift , the tail lift , the centre of gravity distance , the tail arm and the aerodynamic centre moment .

Tips:

1. Explicitly indicate multiplication with a * symbol, indicate division with a / symbol, and use an underscore to indicate subscripts (so is written as "l_cg").

2. Use the sign convention "Pitch up = positive".

3. Take a positive moment in the aerodynamic centre.

4. Use a capital 'H' for all subscripts denoting the horizontal tail.

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2. Non-dimensionalise the equation just derived and give the result below. You are allowed to use the quantities , , , , , , and .

Tips:

1. This problem segment does not support a double subscript, so write as "C_mac" and as "C_LH".

2. Don't introduce the tail volume as the quantity yet.

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3. Derive this equation with respect to the angle of attack and give your result for . You are allowed to use the quantities , , (the tail volume), (the wing lift coefficient gradient) and (the tail lift coefficient gradient).

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4. What is the requirement on the position of the centre of gravity? You are allowed to use the quantities , , (the tail volume), (the wing lift coefficient gradient) and (the tail lift coefficient gradient).

1 answer

To derive the moment equation around the center of gravity (CG) of the King Air 200, we will consider the forces and moments acting on the aircraft:

Step 1: Moment Equation Derivation

  1. Definitions and Variables:

    • Let \( L_{total} \) be the total lift generated by the wings.
    • Let \( L_H \) be the lift generated by the horizontal tail.
    • Let \( x_{CG} \) be the distance of the center of gravity from the aerodynamic center (AC).
    • Let \( x_H \) be the tail arm (the distance from the AC to the horizontal tail).
    • Let \( M_{AC} \) be the moment about the aerodynamic center.

  2. Moment Equation:
    Taking moments about the aerodynamic center and considering the sign convention where "Pitch up = positive", the equation can be written as:

    \[ M_{CG} = M_{AC} - L_{total} * x_{CG} + L_H * x_H \]

    Adjusting for the moment about the CG this takes the form: \[ M_{CG} = M_{AC} - L_{total} * (x_{CG}) + L_H * (x_H) \] Here, we are assuming that the distance \( x_{CG} \) is measured from the aerodynamic center towards the CG.

Step 2: Non-dimensionalisation

Next, we need to non-dimensionalise the equation in terms of relevant quantities. For non-dimensionalisation, we will use \( C_{mac} \) (mean aerodynamic chord) and \( C_{LH} \) (lift coefficient):

The non-dimensional moment around CG can be expressed as: \[ M_{CG} / (q * S_{ref} * C_{mac}) = M_{AC} / (q * S_{ref} * C_{mac}) - C_L * (x_{CG} / C_{mac}) + C_{LH} * (x_H / C_{mac}) \]

Step 3: Derivative with respect to Angle of Attack

Now we take the derivative of the moment equation with respect to the angle of attack (\( \alpha \)) to yield an expression for \( \frac{dM_{CG}}{d\alpha} \).

Using the given variables and including the corresponding lift coefficients gradients:

  1. The lift coefficient of the wings is \( C_L \), and its gradient \( \frac{dC_L}{d\alpha} \) can be denoted as \( C_{L_{wing}}' \).
  2. For the horizontal tail, the lift coefficient and its gradient can be denoted as \( C_{LH} \) and \( C_{L_H}' \) respectively.

The relationship becomes: \[ \frac{dM_{CG}}{d\alpha} = \left( \frac{dM_{AC}}{d\alpha} - C_{L_{wing}}' * L_{total} * x_{CG} + C_{L_H}' * L_{H} * x_H \right) \]

Step 4: Requirement on the Position of the Centre of Gravity

For steady level flight, we want moments to act such that pitching moments are neutral or controlled. This is often defined by the requirement that moments about the CG are balanced.

To achieve this, we can set:

\[ M_{CG} \approx 0 \]

Thus, the condition for the position of the center of gravity becomes: \[ L_{total} * x_{CG} - L_H * x_H \leq M_{AC} \]

In terms of tail volume \( V_H \), we describe: \[ V_H = (x_H / C_{mac}) * (L_H / (q * S_{ref})) \]

This implies that we need the center of gravity to be located such that the positive moment induced by the tail lift overcomes the moment due to the total lift acting at the CG. In symbols: \[ x_{CG} \leq \frac{M_{AC} + L_H * x_H}{L_{total}} \]

This outlines the requirement for the CG position concerning dynamics and stability.