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Project Final Report – CUDBF April 30 th , 2009 ASEN 4028: Aerospace Senior Projects In order to guarantee that the designed control surfaces could produce the deflections necessary to trim the aircraft, analysis was done in AVL. Three cases were setup to determine if the necessary deflections were feasible. Under the worst case scenario (2 rockets on one wing during takeoff), the maximum deflection necessary from the rudder was 10 degrees, 15 degrees from the aileron and 15 degrees from the elevator. The maximum possible deflection is approximately 25 degrees. Flow separation occurs over the wing around 25 degrees of deflection. It is clear that there is enough margin for deflection for all the control surfaces before flow separation occurs. 8.1.5 Stability Analysis Stability analysis was done in AVL and in MATLAB, the code can be found in Appendix G. Non-dimensional stability derivatives were obtained from AVL for three different scenarios that mirror flight missions. The first scenario simulated a case where the centerline fuel tank is flown on the airplane for cruise conditions, the second scenario was four rockets on the airplane in cruise, and the third was the worst case scenario of two rockets on one wing tip. To get a clear understanding of the aircraft’s stability it is important to analyze stability in the longitudinal and lateral domains. The longitudinal domain refers to the stability of the aircraft about the axis of pitch. Lateral stability refers to the stability of the aircraft about the yaw and roll axis. Stability modes in the longitudinal domain are the phugoid and short period and stability modes in the lateral domain are roll subsidence, dutch roll and spiral divergence. Stability derivatives represent an incremental change in a force or moment acting on the aircraft to a corresponding change in any of the following variables: velocity, angle of attack, side slip angle, bank angle, pitch angle, pitch rate, roll rate and yaw rate. Because aircraft dynamics is fairly complex, it is necessary that the equations of motions are linearized to assess stability. The non dimensional stability derivatives outputted from AVL were dimensionalized in MATLAB to analyze dimensional stability for different modes in the longitudinal and lateral directions. Matrix methods were used to solve the set of differential equations to obtain the stability of the aircraft. The aircraft is modeled as a dynamic system where the system receives input in the form of control surface deflections implemented by the pilot. Equation 15 shows the longitudinal equations of motion. The plant that describes this system is represented by Equation 16. 0 − cos = + − sin 0 0 0 1 0 0 0 0 0 − = 0 0 0 − 1 0 0 0 0 1 Equation 15: Equations of Motion in Matrix Form for Longitudinal Stability = Equation 16: Representation of the Dynamic System for Longitudinal Stability 64
Project Final Report – CUDBF April 30 th , 2009 ASEN 4028: Aerospace Senior Projects The eigenvalues of the A matrix are the poles to the dynamic system. This analysis was performed for the three cases described previously in the longitudinal and lateral directions. The longitudinal case will be presented first. The poles plotted for the airplane with the bottle is shown in Figure 37: 0.1 0.08 Longitudinal Stability Modes for the Bottle on the Airplane Real Roots Short Period 0.06 0.04 Imaginary 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1 -7 -6 -5 -4 -3 -2 -1 0 1 Real Figure 37: Longitudinal Stability Modes for the Bottle on the Airplane The natural frequency and the period of each of the roots for the longitudinal case for the bottle are summarized in Table 7: Table 7: Longitudinal Stability for the Bottle on the Airplane Bottle on the Airplane-Longitudinal Mode Roots Time Constant (Real Roots) Natural Frequency (Complex Conjugates) Period (sec) ζ Real Root -6.04 Time constant 6.04 (1/sec) 0.17 -- Real Root -0.17 Time constant 0.17 (1/sec) 5.88 -- Short Period 0.03±0.10i ω n 0.20 (rad/sec) 31.60 0.005 65
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