[財會考試]附加講義

1、.【飛機動力學】附加講義Longitudinal responses to atmospheric gustsLongitudinal responses to atmospheric gustsI. 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 atmospher

2、ic gusts do to the aircraft motion? II. Re-derive the longitudinal equations The problem can be resolved by re-deriving the equations of motion. Lets go back to the pre-treated longitudinal equations on p.33: To linearize this equation set, we had introduced the perturbations - Among these perturbat

3、ions, 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 f

4、light 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

5、 equation set under the presence of gusts: l 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 an

6、d 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. l For the lift equation: 1. Expansion of that includes the gust terms: . 2. The resulting lift equation: l For the pitching mom

7、ent 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 upw

8、ard. 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 us

9、ed: , , and . Laplace domain equation set: - Setting for simplicity. III. Longitudinal response to vertical gusts, or downdraft Lets consider a constant . - An auto-pilot can achieve this maneuver. l Remaining equations of motion: We have substituted for , so that variation in vertical speed of the

10、aircraft can be analyzed. l Using Cramers rule, we will have: , where and . l 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

11、 rises or falls with same velocity as a vertical gust.IV. Longitudinal response to horizontal gustsWe will stick with the constant assumption. But lets 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:l Cramers r

12、ule will give us l For this step gust, ; hence, l At the steady state, we will have. The A/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: l Such a gust, called a wind shear, is caused by strong

13、convection of air. l The phenomenon is called a microburst. l Inside a microburst, where and are constants. l 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. l For a ramp gust, ; hence, l T

14、his value of u(t) computes the inertial flight speed of the aircraft. But we are also interested in the change in relative air speed. Change in relative air speed: . At steady state: . l Observations: For a tail wind shear (順風), , the inertial flight speed is increasing, but the relative air speed i

15、s 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: l Rate of change in altitude: . l A is assumed;