Spacecraft and Aircraft Dynamics Matthew M. Peet Illinois Institute of Technology Lecture 11: Longitudinal Dynamics
Aircraft Dynamics Lecture 11 In this Lecture we will cover: Longitudinal Dynamics: Finding dimensional coefficients from non-dimensional coefficients Eigenvalue Analysis Approximate modal behavior short period mode phugoid mode M. Peet Lecture 11: 2 / 25
Review: Longitudinal Dynamics Combined Terms u+ θgcosθ 0 = X u u+x w w +X δe δ e +X δt δ T ẇ + θgsinθ 0 u 0 θ = Z u u+z w w +Zẇ ẇ +Z q θ +Z δe δ e +Z δt δ T θ = M u u+m w w +Mẇ ẇ +M q θ +M δe δ e +M δt δ T M. Peet Lecture 11: 3 / 25
Force Coefficients Force/Moment Coefficients can be found in Table 3.5 of Nelson M. Peet Lecture 11: 4 / 25
Nondimensional Force Coefficients Nondimensional Force/Moment Coefficients can be found in Table 3.3 of Nelson M. Peet Lecture 11: 5 / 25
State-Space From the homework, we have a state-space representation of form u ẇ θ = [ A ] u w θ +[ B ][ ] δ e δ T q q Where we get A and B from X u, Z u, etc. Recall: Eigenvalues of A define stability of ẋ = Ax. A is 4 4, so A has 4 eigenvalues. Stable if eigenvalues all have negative real part. M. Peet Lecture 11: 6 / 25
Natural Motion We mentioned that A is Stable if eigenvalues all have negative real part. Now we say more: Eigenvalues have the form λ = λ R ±λ I ı If we have a pair of complex eigenvalues, then we have two more concepts: 1. Natural Frequency: ω n = λ 2 R +λ2 I 2. Damping Ratio: d = λ R ω n M. Peet Lecture 11: 7 / 25
Natural Frequency Natural frequency is how fast the the motion oscillates. Closely related is the Definition 1. The Period is the time take to complete one oscillation τ = 2π ω n M. Peet Lecture 11: 8 / 25
Damping Ratio Damping ratio is how much amplitude decays per oscillation. Even if d is large, may decay slowly is ω n is small Closely related is Definition 2. The Half-Life is the time taken for the amplitude to decay by half. γ =.693 λ R M. Peet Lecture 11: 9 / 25
State-Space Example: Uncontrolled Motion C172: V 0 = 132kt, 5000ft. u.0442 18.7 0 32.2 u ẇ q =.0013 2.18.97 0 w.0024 23.8 6.08 0 q θ 0 0 1 0 θ M. Peet Lecture 11: 10 / 25
State-Space Example: Uncontrolled Motion Using the Matlab command [u,v] = eigs(a), we find the eigenvalues as Phugoid (Long-Period) Mode and Eigenvectors Short-Period Mode and Eigenvectors λ 1,2 =.0209±.18ı.1717.2826 ν 1,2 =.0748.9131 ±.1685 0 ı.1038.1103 λ 3,4 = 4.13±4.39ı 1 0 ν 3,4 =.0002.001 ±.0000001.0000011 ı.0008.0055 Notice that this is hard to interpret. Lets scale u and q by equilibrium values. M. Peet Lecture 11: 11 / 25
State-Space Example: Uncontrolled Motion After scaling the state by the equilibrium values, we find the eigenvalues unchanged (Why?) as Phugoid (Long-Period) Mode λ 1,2 =.0209±.18ı but clearer Eigenvectors.629.0213 ν 1,2 =.0218.0016 ±.0007.0001 ı.138.765 Natural Frequency: ω n =.181rad/s Damping Ratio: d =.115 Period: τ = 34.7s Half-Life: γ = 33.16s Motion dominated by variables u and θ. M. Peet Lecture 11: 12 / 25
Modal Illustration M. Peet Lecture 11: 13 / 25
State-Space Example: Long Period Mode M. Peet Lecture 11: 14 / 25
State-Space Example: Uncontrolled Motion Using the Matlab command [u,v] = eigs(a), we find the eigenvalues as Short-Period Mode λ 3,4 = 4.13±4.39ı and Eigenvectors.0049.004 ν 3,4 =.655.396 ±.409.495 ı.006.0423 Natural Frequency: ω n = 6.03rad/s Damping Ratio: d =.685 Period: τ = 1.04s Half-Life: γ =.167s Motion dominated by variables w and q. M. Peet Lecture 11: 15 / 25
Modal Illustration M. Peet Lecture 11: 16 / 25
State-Space Example: Short Period Mode M. Peet Lecture 11: 17 / 25
State-Space Modal Approximations Now that we know that longitudinal dynamics have two modes: Short Period Mode Phugoid Mode (Long-Period Mode) Short Period Mode: fix u = 0 and w = 0. study variation in θ and q. Similar to Static Longitudinal Stability Long Period Mode: fix q = 0 and w = 0. study variation in θ and u. Now we develop some simplified expressions to study these modes. M. Peet Lecture 11: 18 / 25
Short Period Approximation For the short period mode, we have the following dynamics: [ ẇ ] [ ] Z α [ u = u 0 1 w ] [ ] Z δe 0 q M α + M u0 + u 0 u αzα 0 M q +M α q M δe + M α+z δe u 0 [ ] θ = A sp +B q sp δ e δ e To understand stability, we need the eigenvalues of A sp. Eigenvalues are solutions of det(λi A sp ) = 0. Thus we want to solve det(λi A sp ) = λ 2 (M q + Z α u 0 +M α )λ+( Z αm q u 0 M α ) = 0 We use the quadratic formula (Lecture 1): λ 3,4 = 1 2 (M q +M α + Z α )± 1 (M q +M α + Z α ) u 0 2 u 2 Z α 4(m q M α ) 0 u 0 M. Peet Lecture 11: 19 / 25
Short Period Approximation Frequency and Damping Ratio λ 3,4 = 1 2 (M q +M α + Z α )± 1 (M q +M α + Z α ) u 0 2 u 2 Z α 4(m q M α ) 0 u 0 This leads to the Approximation Equations: Natural Frequency: Damping Ratio: d sp = 1 2 ω sp = M q Z α u 0 M q +M α + Zα u 0 ω sp M. Peet Lecture 11: 20 / 25
Short Period Mode Approximations Example: C127 Approximate Natural Frequency: 481 431 ω sp = +27.7 = 6.10rad/s 219 True Natural Frequency: ω sp = 6.03rad/s Approximate Damping Ratio: True Damping Ratio: So, generally good agreement. d sp = 4.32 2.20 1.81 2 6.10 d sp =.685rad/s =.683rad/s M. Peet Lecture 11: 21 / 25
Long Period Approximation Long period motion considers only motion in u and θ. [ u θ] [ ][ Xu g u = Zu u 0 0 θ] This time, we must solve the simple expression Using the quadratic formula, we get det(λi A) = λ 2 X u λ Z u u 0 g = 0 X u ± λ 1,2 = X 2 u +4Zu u 0 g 2 M. Peet Lecture 11: 22 / 25
Long Period Approximation X u ± Xu 2 +4Zu u 0 g λ 1,2 = 2 This leads to the Approximation Equations: Natural Frequency: Damping Ratio: ω lp = Z ug u 0 d lp = X u 2ω lp M. Peet Lecture 11: 23 / 25
Long Period Mode Approximations Example: C127 Approximate Natural Frequency: ω lp =.181rad/s True Natural Frequency: ω lp =.208rad/s Approximate Damping Ratio: d lp =.115rad/s True Damping Ratio: d lp =.106rad/s So, generally good, but not as good. Why? M. Peet Lecture 11: 24 / 25
Conclusion In this lecture, we covered: How to find and interpret the eigenvalues and eigenvectors of a state-space matrix Natural Frequency Damping Ratio How to identify Long Period Eigenvlaues/Motion Short Period Eigenvalues/Motion Modal Approximations Phugoid and Short-Period Modes Formulas for natural frequency Formulae for damping ratio M. Peet Lecture 11: 25 / 25