Aerospace Engineering 3521: Flight Dynamics. Prof. Eric Feron Homework 6 due October 20, 2014

Transcription

1 Aerospace Engineering 3521: Flight Dynamics Prof. Eric Feron Homework 6 due October 20,

2 Problem 1: Lateral-directional stability of Navion With the help of Chapter 2 of Nelson s textbook, we established conditions for lateral-directional stability of an aircraft by looking at two distinct motions: First, we looked at aircraft weathercocking motion and the yaw moment generated by a given side slip angle β. Then we looked at the roll moment arising from a given sideslip angle β. In this problem, we are interested in comparing the complete lateraldirectional equations of motion of the aircraft with the following two constrained motions: First, pure weathercocking motion; the aircraft is free to yaw about its center of gravity, but its center of gravity is fixed and the aircraft cannot roll. Second, pure sideslip motion: The aircraft is free to roll and move sideways, but its yaw angle remains fixed. For all numerical computations, we use the Navion model available in Nelson s textbook, your course notes, and you past homeworks. 1. From the general linearized lateral-directional equations of motion of the aircraft, extract the pure weathercocking equations of motion. Begin by writing the constraints induced on all states by pure weathercocking. Use the beginning of Nelson s Chapter 4 to guide your work. 2. Is the pure weathercocking equation of motion stable? Compute all required eigenvalues and eigenvectors and simulate the weathercocking motion of the Navion assuming the aircraft is initially oriented 20 degrees away from the airflow direction (think of a wind tunnel experiment). 3. From the linearized lateral-directional equations of motion of the aircraft, extract the pure sideslip equations of motion of the aircraft. Begin by writing the constraints forced on all states due to the requirement for pure sideslip motion. 4. Is the pure sideslip motion of the Navion stable? Compute all required eigenvalues and eigenvectors. Simulate the motion of the Navion assuming the Navion is originally banked at 20 degrees to the right and its lateral speed is originally zero. 5. Consider now the full lateral directional equations of motion of the Navion. Do they describe a stable system? Discuss the merits of the static stability analysis of aircraft as described in chapter 2 in the light of the full equations of motion. 2

3 Problem 2: Aircraft rotation at landing We are considering the Navion again. We are interested in studying the behavior of the aircraft near landing. We assume that the Navion is originally flying straight and level (at 176 ft/sec, zero pitch, and zero AOA) and all control variables are at zero. We assume the landing is performed using elevator only (even though this is a coarse assumption, since engine power and flaps are also used in practice, especially to slow down the aircraft to 61 knots IAS ( News/1999/June/1/Navion)). In what follows, we discuss a landing procedure where the Navion landing speed is, in fact, much higher than the 61 knots (IAS) found in the literature. 1. Compute the elevator setting δeapp for the Navion to a 4 degree approach glide slope on final approach. Is the Navion pitched up, or down? What is the speed of the Navion? Its angle of attack? 2. At landing, the Navion needs to be pitched about 10 (ten) degrees further up from straight and level flight to protect the nose-wheel from hard touch-down and prevent propeller contact with the ground. Compute the elevator value δe rot for the aircraft to be trimmed at 12 (twelve) degrees pitch angle. At that trim pitch angle, what are the other variables like (AOA, speed, elevator setting). 3. Assume the aircraft is on approac Compute the trajectory (longitudinal, vertical position and speed, pitch angle) followed by the Navion as it transitions from its final approach glide slope to a 12 degree pitch angle. Assume the elevator input is a step from δeapp to δe rot. Show drawings of the Navion (position, attitude) at different locations during its transition. 4. Based on the simulation above, suggest a landing procedure for the Navion (neglect the ground effect of the runway). Discuss in detail. Are there any particular issues to be concerned about by this landing procedure, other than speed, which we know is too high)? 3

4 Problem 3: Landing with heavy cross-wind Landing in the presence of heavy cross-wind is a demanding exercise, for which there exist various procedures, including crabbing and side-slipping. The two procedures are shown in Fig. 1. Foar all practical applications, we assume that the Navion flies at 176 ft/sec, and the cross-wind comes from the right and blows at 25 ft/sec (15 knots - this value is the max recommended value for a runway with medium to poor breaking action). 1. Consider the crabbing final approach configuration and assume the runway must point exactly north. Compute the heading the aircraft must have for its ground track to be exactly aligned with the runway. 2. A hard landing on the runway while crabbing is usually a bad idea (except for the 747, which is to aviation what a chevy pick-up is to farming): Lateral stress is exercised on the landing gear, and as the aircraft quasi-instantly aligns with the runway, the passengers do not like it. Following a reasoning similar to that discussed in Problem 2, design a de-crabbing procedure aimed at aligning the aircraft heading with the runway. From Quiz 1, discuss the possibility of aligning the aircraft with a last-second pair of kicks in the rudder (together with the proper pair of kicks in the ailerons to keep the aircraft wings level). Figure out the timing and amplitude of the rudder and aileron kicks to get the desired result: The aircraft is aligned with the runway, but its airspeed has not changed too much yet. Show a numerical simulation of the aircraft position and attitude (φ, ψ, lateral speed relative to runway) as a function of time. Give an estimate of the time interval available for the aircraft to land decently after the de-crabbing maneuver is completed (that is, the appropriate aileron and rudder kicks have been applied). 3. Consider the forward slip final approach configuration. Compute the sideslip angle the aircraft for its ground track to be aligned exactly with the runway. What aileron and rudder settings do you have to apply in order to result in the desired forward slip? What is the additional vertical velocity generated by the forward slip configuration, and how can you remedy it (think longitudinal controls). 4. Landing the aircraft while forward slipping may cause damage to the landing gear, as the aircraft touches down with only one side of the 4

5 Figure 1: Navion landing in crabbing (left) or forward slipping (right) configuration for landing 5

6 landing gear (the right side in our case), and may over-stress the system. Moreover, since the aircraft is banked, the wingtips and/or engines may strike the ground, which is not a good idea. Discuss the possibility of leveling the wings of the aircraft without changing the heading just before landing via appropriate (pairs of) kicks in the rudder and ailerons. What are the appropriate kicks (timing, amplitude) that will level the aircraft. Show a numerical simulation of the aircraft position and attitude (φ, ψ, lateral speed relative to runway). Give an estimate of the time interval available for the aircraft to land decently after the leveling maneuver is completed. Please look at landing to get more amazing facts about cross-wind landings. As much as the 747 is an aviator s pick-up, the B-52 (same manufacturer) is his Honda Prelude (check Prelude in detail to understand why I say that). 6