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Project Final Report – CUDBF April 30 th , 2009 ASEN 4028: Aerospace Senior Projects The s-plane is a visual representation of the roots of the system to characterize the behavior of the system. Poles in the right half of the plane are unstable where as poles in the left half of the s- plane indicates stability. As seen in Figure 37 and Table 7, the real roots are located on the negative real axis. These roots are stable and have time constants of 6.04 s -1 and 0.17 s -1 . The short period is slightly unstable. The period of the short period roots were analyzed to determine if this is a concern. The period for both short period roots is 31.60 seconds. Since the period for natural instability in the short period mode is as slow as 31.60 seconds, this indicates that the pilot has 31 seconds to input control surface response to the system. As a result, this is not an issue. The analysis in the s-plane for the airplane with four rockets is shown in Figure 38: 0.15 0.1 Longitudinal Stability Modes for Four Rockets on the Airplane Real Roots Short Period 0.05 Imaginary 0 -0.05 -0.1 -0.15 -0.2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 Real Figure 38: Longitudinal Stability Modes for Four Rockets on the Airplane The natural frequency and period of each of the roots in the longitudinal case for the scenario with four rockets on the airplane are summarized in Table 8: 66
Project Final Report – CUDBF April 30 th , 2009 ASEN 4028: Aerospace Senior Projects Table 8: Longitudinal Stability for Four Rockets on the Airplane Four Rockets on the Airplane-Longitudinal Mode Roots Time Constant (Real Roots) Natural Frequency (Complex Conjugates) Period (sec) ζ Real Root -7.72 Time constant 7.72 (1/sec) 0.13 -- Real Root -0.19 Time constant 0.19 (1/sec) 5.26 -- Short Period 0.03±0.11i ω n 0.24(rad/sec) 26.23 0.005 For this flight case, there were two real, stable roots which had time constants of 7.72 s -1 and 0.19 s -1 . The short period roots were slightly unstable in this case as well. The period for the short period roots was calculated to be 26.23 seconds. Again, this is not a concern since the pilot has enough time to compensate. The poles plotted in the s-plane for the worst case scenario, i.e. the airplane with two rockets on one wing is shown in Figure 39: 0.15 0.1 Longitudinal Stability Modes for Two Rockets on the Airplane Real Roots Short Period 0.05 Imaginary 0 -0.05 -0.1 -0.15 -0.2 -12 -10 -8 -6 -4 -2 0 2 Real Figure 39: Longitudinal Stability Modes for Two Rockets on the Airplane 67
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University of Colorado Design/Build
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