Longitudinal Responses to Atmospheric Gusts

【飛機動力學】

附加講義

Longitudinal responses to atmospheric gusts

I. Preliminary remarks

○We are going to deal with the appearance of atmospheric gusts

⇨The two components of the gust is often referred to as horizontal gust and the vertical gust.

○What will the atmospheric gusts do to the aircraft motion?

II. Re-derive the longitudinal equations

○The problem can be resolved by re-deriving the equations of motion.

○Lets go back to the pre-treated longitudinal equations on p.33:

○To linearize this equation set, we had introduced the perturbations

--- Among these perturbations, and represent the changes in relative wind due to change in aircraft flight speed.

○With the presence of the gusts, the following modification to the perturbations of and will be necessary:

and .

--- Minus signs reflectt that gust wind changes the relative wind, which is of opposite sign of the flight speed.

⇨These changes in the perturbations of and will necessitate proper modifications to the aerodynamic terms of the linearized equations.

⇨The gravitational terms and the inertial terms of the equations, however, will not be affected by the presence of atmospheric gusts.

○Linearization of the equation set under the presence of gusts:

 For the drag equation:

1. Expansion of and that includes the gust terms:

and

--- The last term on the RHS reflects that vertical gust slants the relative wind, hence the lift vector, causing an additional drag force to appear.

--- We have also assume that and so that .

2. The resulting the drag equation.

⇨Gust effect appears as two additional terms on the RHS.

⇨These additional terms act as inputs to the longitudinal dynamics.

 For the lift equation:

1. Expansion of that includes the gust terms:

2. The resulting lift equation:

 For the pitching moment equation:

1. In previous note, we had expanded into as follows:

2. Here, the following modifications will be made:

3. The following phenomenon about the vertical gust has also been observed:

is a result of :

In such a vertical gust, the tail feels an additional (w.r.t. the wing)

---Pointing upward.

An additional change in will result:

This change in results in the following change in pitching moment:

(Because)

4. The final pitching moment equation:

⇨Note that a total of four gust terms appear in the RHS.

【Modified longitudinal equations with gust inputs】

○Time domain equation set:

※We had used: , , and .※

○Laplace domain equation set: --- Setting for simplicity.

III. Longitudinal response to vertical gusts, or downdraft

○Let's consider a constant. --- An auto-pilot can achieve this maneuver.

 Remaining equations of motion:

⇨We have substitutedfor, so that variation in vertical speed of the aircraft can be analyzed.

 Using Cramer's rule, we will have:

where

and

 For a constant vertical gust, i.e. , or where is some appropriate constant, the steady state response of will be

--- We have quoted Final value theorem for this result.

And with

we will have

;

hence, .

⇨In the end, aircraft rises or falls with same velocity as a vertical gust.

IV. Longitudinal response to horizontal gusts

○We will stick with the constant assumption. But let's go back to the equation set in terms of. The remaining equation set will be:

○For a step horizontal gust,, or , where is some constant:

 Cramer's rule will give us

 For this step gust, ; hence,

 At the steady state, we will have

⇨TheA/C is drifted with same velocity as gust..

○But we are more concerned about the aircraft response to a ramp horizontal gust, namely where is some constant:

 Such a gust, called a wind shear, is caused by strong convection of air.

 The phenomenon is called a microburst..

 Inside a microburst, where and are constants.

 We have already treated , the step portion of the gust, now we will dealt with its ramp portion, .

(A) Change of speed in a wind shear: -- Using data of Mohawk.

 For a ramp gust, ; hence,

This value of u(t) computesthe inertial flight speedof the aircraft. But we are also interested in the change in relative air speed.

⇨Change in relative air speed:.

⇨At steady state: .

 Observations:

⇨For a tail wind shear (順風),, the inertial flight speed is increasing, but the relative air speed is decreasing. Eventually, the relative air speed will decrease by an amount .

⇨Decrease in relative air speed reduces lift, causing the A/C to loss altitude.

⇨May cause crashes at take off or landing, see discussion below.

(B) Change in altitude in a wind shear:

 Rate of change in altitude:

 A is assumed; hence,

 Therefore,

and

 Using data of Mohawk as an example:

 Observations:

⇨Eventually, , indicating a A/C crash, for .

⇨This danger of crash may occur near an airport where an aircraft is in low altitude doing take off or landing.