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EE/Aroe/Meche/Math 575 HW3 EE/AroE/MechE/Math 575 HW3 Problem 1 Modi¯ed F-8 Aircraft Longitudinal Dynamics. 1. Introduction. The F-8 is an old fashioned aircraft that has been used by NASA as part of their digital ”fly by wire" research program. We have modi¯ed the equations of motion by including a large ”flap- eron" on the wings so as to obtain two control variables in the longitudinal dynamics of the plane. This flaperon does not exist in the F-8 aircraft. However, such surfaces exist in other recent aircraft and provide some additional flexibility for precision maneuvers. 2. Modeling In Figure 1 we show an aircraft with a Cartesian coordinate system ¯xed to its center of grav- ity (cg). This coordinate system, which is called "stability axes" coordinate frame, has the x-axis pointed toward the nose of the aircraft, the y-axis out the right side, and the z-axis pointed down. We shall assume that the aircraft is flying in the vertical plane with its wing level (i.e. banking or turning) so that we can study its motion in the vertical plane, i.e. its longitudinal dynamics. The important variables that characterize the aircraft in this motion are the horizontal velocity of the plane denoted by u, the pitch angle θ which is the angle of the x-axis with respect to the dθ horizontal, the pitch rate is the rate of change of the pitch angle, q = dt , and the angle of attack ®, which is the angle of the nose with respect to the velocity vector of the aircraft. Holding the wings at the angle of attack ® with respect to the incoming wind, which necessitate a tail, is what provide the lift force needed to fly, i.e., to just balance the force of gravity, but it also produces drag; both lift and drag are approximately proportional to the angle of attack for small ®. The flight path angle, γ, de¯ned by γ = θ ¡ ® is the angle between the aircraft velocity vector and the horizontal. As its nature implies, γ de- scribes the trajectory of the aircraft in the vertical plane. The longitudinal motion of the aircraft is controlled by two hinged aerodynamics control surfaces, as shown in Figure 1: the elevator is located on the horizontal tail, and we shall use the elevator angle ±e(t) as control variable; the flaperons are located on the wing, and we shall use the flaperon angle ±f (t) as another control variable. Most people are familiar with elevators; the flaperons are just like the ailerons except that they move in the same direction. Deflection of either of these two surfaces downward causes the airflow to be deflected downward; this produces a force that induces a nose-down moment about the cg. As the aircraft rotates, it changes its angle of attack which results in additional forces and moments (we shall show the equation below). The longitudinal motion of the aircraft is also influenced by the thrust generated by the engines. However, in this problem we shall ¯x the thrust to be a constant and we shall not use it as a dy- 1 namic control variable. (Actually, the dynamic coordination of the thrust, elevator, and flaperons becomes important and signi¯cant when the aircraft is in its landing con¯guration). In general, the di®erential equations that model the aircraft lonfgitudinal motions are nonlinear. These nonlinear di®erential equations can be linearized at a particular equilibrium (steady-state) flight condition which is characterized by constant airspeed, altitude, cg location, "trimmed" angle of attack ®, "trimmed" pitch angle θ (so that γ = 0), and "trimmed" elevator angle ±e to maintain zero pitch rate. One can then obtain a system of linear time invariant di®erential equations that describe the deviation of the relevant quantities from their constant equilibrium (trimmed) values. 3. Linearized F-8 Dynamics. The following linear di®erential equations model the longitudinal motions of the F-8 aircraft about the following equilibrium flight conditions: Altitude: 20; 000 ft = 6095 meters Speed: Mach 0:9 ¡ 281:58 m=s = 916:6 ft=sec Dynamic Pressure: 550 lbs=ft2 = 26; 429 N=m2 Trim Pitch Angle 2:25 deg. Trim Angle of Attack 2:25 deg. Trim Elevator Angle ¡2:65 deg The following four variables represent perturbations form the equilibrium values: q(t) = pitch rate rad=sec u(t) = perturbation from horizontal velocity ft=sec ®(t) = perturbed angle of attack from trim (rad) θ(t) = perturbed ptich angle from trim (rad) The two control variables are: ±e(t) = elevator deflection from trim (rad) ±f (t) = flaperon deflection (rad) Neglecting the actuator dynamics the linearized state equations are: q_(t) = ¡0:8q(t) ¡0:0006u(t) ¡12:0®(t) ¡19:0±e(t) ¡2:5±f (t) u_(t) = ¡0:014u(t) ¡16:64®(t) ¡32:2θ(t) ¡0:66±e(t) ¡0:5±f (t) ®_ (t) = q(t) ¡0:0001u(t) ¡1:5®(t) ¡0:16±e(t) ¡0:6±f (t) θ_(t) = q(t) The solution of the di®erential equations above provides the dynamic response of all the state variables. From those we can calculate, if desired, the flight path angle using Eq. (1). Another quantity of interest is the normal acceleration, nz(t), which is the acceleration of the cg along the z-axis and whose units are in g0s (1g = 32:2ft=sec2), as follows nz(t) = 42®(t) + 4:4±e(t) + 6:5±f (t) 2 4. Problem Statement For the time being you are asked to simulate the above di®erential equations so as to get a feeling for the F-8 dynamics. Note: there is no need to turn in all transients with your homework. Just explain what you have observed, and turn in only those transients that you have found to be interesting. 4.1 Change of variable Rewrite all the equations so that all the angles are expressed in degrees rather than in radiants, and the pitch rate is in deg/sec. leave velocities in ft=sec and the norma acceleration is g0s. De¯ne the state vector as x0 = [ q(t) u(t) ®(t) θ(t)] and the control vector as 0 u = [ ±e(t) ±f (t)] and write the equations in the standard state-space form. x_ (t) = Ax(t) + Bu(t) 4.2 Unforced transient responses Set the controls to zero and study the unforced transient response for the initial conditions ®(0) = 1deg: all he other zero. Repeat for θ(0) = 1deg:, all other zero. You should plot all state variables, flight path angle, and normal acceleration vs. time. You should see two types of oscillations one fast and one slow, so adjust the integration step accordingly. The slow oscillation is due to the so called phugoid mode and the fast oscillation is due to the so called short-period mode. Describe the motion of the aircraft, and try to see if you can ¯gure out the energy explanation associated with these motions. Compute the eigenvalues and the eigenvectors of the A matrix and identify them with the phugoid and the short-period modes. 4.3 Forced Transient Response Set all initial states to zero. Apply only one unit (1 degree) step to the elevator and plot the aircraft transients. Repeat for a unit step (1 degree) to the flaperons. Explain what the aircraft is doing during the transients and at steady-state. 4.4 Steady-State Analysis 3 Since we have two independent control variables, we should be able to control two aircraft variables independently. Suppose that all the initial states are zero. Remark: In all statements that follows, the notions of "up", "down", "straight", etc. neglect the fact that the trim variables are nonzero. Find what should be the steady-state elevator and the flaperon angles so that as t ! 1 θ(t) !¡1; γ(t) ! 0 which means that the aircraft is flying straight with nose pointing down! Apply the required eleva- tor and flaperon angles and observe the resulting transients in a transient simulation of the aircraft dynamics starting from zero initial conditions. Find what should be the steady state elevator and flaperon angles so that as t ! 1 θ(t) ! 0; γ ! +1 which means that the aircraft nose points along the horizontal, while it is climbing. Note that conventional aircrafts with only elevator control cannot fly like this! Such flight be- comes possible because we have added the flaperon. This is what multivariable control is all about! However, there are limitations on the the flaperon e®ectiveness because of physical limits ont he control surfaces (about 25 degrees). Thus, we cannot obtain arbitrarily large angular deviations from trim (at any rate for a large deviations the linearized model will lose its validity). Also since the open loop responses are oscillatory we can use feedback to make such transitions "better". We will do that in the future. 4.5 Singular Value Analysis Compute the transfer function matrix between the control vector u and the y(t) = [ θ γ ]0. Plot the singular values as function of frequency. Can you identify the phugoid and the short- period modes from the plots? Comment on the di±culty of inverting the system at di®erent frequencies. Identify the sinusoidal inputs which leads to the maximum ampli¯cation at the output. Repeat for the inputs which have the least ampli¯cation (this should give you an idea of the di±cult direction to control). 4 x θ α γ horizontal v : velocity δf δe z y Figure 1: Definition of variables for longitudinal dynamics 5.
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