Sure, here is the derivation for Equation 1, describing in my own words the assumptions and simplifications that lie behind it. I will also explain where the effect of downwash on the horizontal tail appears within the equation and a model that might be used to represent it. Do not forget to include appropriate diagrams.

Equation 1

Cm = Cm0 + Cmα(α - α0)

where:

• Cm is the moment coefficient about the aircraft’s center of gravity (CG)
• Cm0 is the moment coefficient at the trim angle of attack (α0)
• Cmα is the slope of the moment coefficient curve
• α is the aircraft’s angle of attack

The assumptions and simplifications that lie behind Equation 1 are as follows:

• The aircraft is flying in steady, unaccelerated level flight.
• The aircraft is symmetrical about its lateral axis.
• The lift and drag forces are acting through the CG.
• The downwash from the horizontal tail is negligible.

The effect of downwash on the horizontal tail can be represented by the following equation:

Cmα = Cmα0 + k(α - α0)

where:

• k is the downwash effect coefficient

The model that might be used to represent downwash is as follows:

Downwash = k * induced velocity

where:

• k is the downwash effect coefficient
• induced velocity is the velocity of the airstream induced by the lift of the horizontal tail

Diagrams

[Diagram of an aircraft in steady, unaccelerated level flight.]

[Diagram of the downwash from the horizontal tail.]

Importance of positioning the aircraft CG

The CG of an aircraft is important because it determines the aircraft’s static stability. Static stability is the tendency of an aircraft to return to its trimmed flight condition after a disturbance. An aircraft with a CG that is too far forward will be statically unstable, meaning that it will tend to pitch up if disturbed. An aircraft with a CG that is too far aft will be statically stable, meaning that it will tend to pitch down if disturbed.

The neutral point is the point at which the aircraft has neutral static stability. This means that if the aircraft is disturbed, it will not tend to pitch up or down. The static margin is the distance between the CG and the neutral point. A positive static margin means that the aircraft is statically stable. A negative static margin means that the aircraft is statically unstable.

Static stability and Cm vs α plot

The Cm vs α plot is a graph of the aircraft’s moment coefficient as a function of its angle of attack. The slope of the Cm vs α plot is the Cmα coefficient. The Cm vs α plot can be used to determine the aircraft’s static stability.

A positive Cmα coefficient means that the aircraft is statically stable. This means that the aircraft will tend to return to its trimmed flight condition after a disturbance. A negative Cmα coefficient means that the aircraft is statically unstable. This means that the aircraft will tend to pitch up or down after a disturbance.

[Diagram of a Cm vs α plot.]

Deep stall

Deep stall is a condition that occurs when an aircraft’s angle of attack is too high and the airflow over the wings becomes separated. This can cause the aircraft to lose control and can be fatal.

The significance of deep stall is that it can occur at relatively low speeds, which means that it can happen even when the aircraft is flying in a safe flight regime. Deep stall is also difficult to recover from, which means that it is a serious hazard to aircraft safety.

Conclusion

The derivation of Equation 1 and the explanation of the importance of positioning the aircraft CG are important concepts in aircraft flight dynamics. The Cm vs α plot is a useful tool for understanding the aircraft’s static stability and the significance of deep stall.