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Project Final Report – CUDBF April 30 th , 2009 ASEN 4028: Aerospace Senior Projects The natural frequency and period of each of the roots for the longitudinal case for two rockets on one wing are summarized in Table 9: Table 9: Longitudinal Stability for Two Rockets on the Airplane Two Rockets on the Airplane-Longitudinal Mode Roots Time Constant (Real Roots) Natural Frequency (Complex Conjugates) Period (sec) ζ Real Root -10.50 Time constant 10.50 (1/sec) 0.10 -- Real Root -0.19 Time constant 0.19 (1/sec) 5.26 -- Short Period 0.04±0.13i ω n 0.24 (rad/sec) 26.23 0.005 As seen in the preceding figure, there are two real, stable roots. The time constants for these roots are 10.50 s -1 and 0.19 s -1 . The short period is unstable in this scenario as well, but, it has a period of 26.23 seconds which is enough time for the pilot to input control. For lateral stability, the modes concerned are spiral divergence, dutch roll and roll subsidence. In aircraft design, there exists a tradeoff between spiral divergence and dutch roll stability. The equations of motion for lateral stability analysis are shown in Equation 17 and the plant that models the dynamic system is represented in Equation 18: cos ( − ) = 0 0 1 0 0 0 0 0 0 0 1 0 − = ⁄ 0 0 1 0 0 − ⁄ 0 1 Equation 17: Equations of Motion in Matrix Form for Lateral Stability = Equation 18: Representation of the Dynamic System for Lateral Stability The visual representation of the poles in the s plane for the three cases described previously are shown in Figure 40, Figure 41, and Figure 42. 68
Project Final Report – CUDBF April 30 th , 2009 ASEN 4028: Aerospace Senior Projects Lateral Stability Modes for the Bottle on the Airplane 0.1 Roll Subsidence Spiral Divergence Dutch Roll 0.05 Imaginary 0 -0.05 -0.1 -0.15 -0.2 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 Real Figure 40: Lateral Stability Modes for the Bottle on the Airplane The natural frequency and period of each of the roots for the lateral case for the bottle on the airplane are summarized in Table 10: Table 10: Lateral Stability for the Bottle on the Airplane Bottle on the Airplane-Lateral Mode Roots Time Constant (Real Roots) Natural Frequency (Complex Conjugates) Period (sec) ζ Roll Subsidence -0.71 Time constant 0.71 (1/sec) 1.41 -- Spiral Divergence 0.0001 Time constant -0.0001 (1/sec) 10,000 -- Dutch Roll 0.00±0.11i ω n 0.11 (rad/sec) 58.68 -- For the case with the airplane carrying the bottle, the airplane was stable in roll subsidence, neutrally stable in spiral divergence and barely unstable in dutch roll. The time constant for the roll subsidence mode was determined to be 0.71 s -1 , and the time constant for spiral divergence was -0.0001 s -1 and the period for dutch roll was determined to be 58.68 sec and 10,000 seconds for the spiral divergence. This is clearly not a concern for stability. 69
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