ME 450 Computer Project 1 FALL 2015 OUT: 11/06/15 Aircraft Altitude Control Problem DUE: 12/04/15 (2 students work as a team) A schematic diagram of the dynamics of an aircraft (in the longitudinal plane) is shown in the figure above. The sum of the lift forces applied to the aircraft wings and body is equivalent to a single lift force LW, applied at the “center of lift” CL. The center of lift does not necessarily coincide with the center of mass CG (with a positive d meaning that the center of mass is ahead of the center of lift). The mass of the aircraft is denoted by m and its moment of inertia about CG is denoted by J. The length of the aircraft and the distance between the elevator and CG are denoted by lP and lG, respectively. We assume that all angles are small enough to justify linear approximations (e.g., sin, cos1), and that the forward velocity of the aircraft remains essentially constant. The aircraft is initially cruising at a constant altitude h=h0. To affect its vertical motion, the elevator (a small surface located at the aircraft tail) is rotated by an angle E. This generates a small aerodynamic force LE on the elevator (see figure), and thus a torque about CG. This torque creates a rotation of the aircraft about CG, measured by an angle . The lift force LW applied to the wing is proportional to , i.e., LW= CZW , CZW is wing lift coefficient. Similarly, LE is proportional to the angle between the horizon and the elevator, i.e., LE= CZE (E-), CZE is elevator force coefficient. Furthermore, various types of aerodynamic force create friction torques about CG, which is proportional to ሶ, of the form ܾሶ, ܾ is lumped linear damping coefficient. Model development 1. If we only consider the translation of aircraft in the vertical plane (i.e., altitude, x(t)) and its rotation in the longitudinal plane (i.e., pitch, (t)), develop the mathematical model (includes two ODEs) that describes the equation of motion of (t) and x(t) by applying the Newton’s second law. CL CG wing LE lG d horizontal elevator α Lw lP Hint: there are three forces (or torques) involved in the model: two “spring-like” force LW= CZW and LE= CZE(E-) and a “damper-like” torque ܾሶ . When deriving the rotational EoM, find the torque generated by each of the forces. When deriving the translational EoM, because of the small angle assumption, we consider LW and LE act only along the vertical direction and are perpendicular to the long-axis of the plane (dotted line in the figure). 2. Let’s select the nominal values for the model parameters: J = 1, m = 1, b = 4, CZE = 1, CZW = 5, lG = 3, lP = 4.5, d = 0. 1) If we consider aircraft pitch angle as the system output, find the transfer function from control input elevator angle E to aircraft pitch angle , then find its poles and zeros. Comment on the stability of the system. 2) If we consider aircraft vertical position x (altitude) as the system output, Find the transfer function from control input E to aircraft vertical position x, then find its poles and zeros. Comment on the stability of the system. 3) If at time t = 0, a unit step input of the elevator angle E is applied, using initial and final value theorems, find initial and final vertical acceleration ݔሷሺ0ାሻ and ݔሷሺ∞ାሻ. Does the aircraft start in the correct direction? Why? Simulation 3. If we consider aircraft pitch angle as the output of the system, sketch the block diagram of the corresponding Input-Output model (Corresponding to the transfer function derived in question 2.1). Note that, to include the initial conditions, instead of using a single transfer function we need to use the three basic functional blocks (gain, addition/subtraction and integration). 4. Based on the block diagram in question 3, if we consider the altitude as the output of the system, sketch the block diagram of the corresponding Input-Output model (Corresponding to the transfer function derived in question 2.2). 5. Construct the block diagram obtained in question 4 in Simulink. 6. Try several initial conditions with zero input (E=0), plot the free responses of the system (for both and x). Comment on the result based on the pole locations in the complex plane, what kind of aircraft movement does the result represent? 7. Apply a unit step input of E to the system with zero initial condition, find the forced response of the system. Comment on your results. Model analysis Now let’s focus on the Input-Output model from the elevator input E to aircraft pitch angle (Corresponding to the transfer function derived in question 2.1). 8. Use both simulation in the Simulink and the analysis of the transfer function, determine the range of d (distance between center of mass CG and center of lift CL) that leads to a stable system (assume other model parameters do not change), compare your analytical and numerical simulation results. 9. Use both simulation in the Simulink and the time domain analysis of the 2nd order system, find the ranges of d that lead to underdamped, critically-damped and overdamped systems (assume other model parameters do not change), compare your analytical and numerical simulation results. 10. According to your results, comment on the following two questions: a) How does the body shape of a stable aircraft look like? b) For commercial airplanes, if the number of passengers is less than half of the total number of seats available, where should the passengers sit to gain better safety?

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