Lecture 8 : Dynamic Stability

Transcription

1 Lecture 8 : Dynamic Stability Or what happens to small disturbances about a trim condition

2 1.0 : Dynamic Stability Static stability refers to the tendency of the aircraft to counter a disturbance. Dynamic stability is concerned with how fast the aircraft returns to trim condition after a disturbance. Note that

3 1.1 Quantifying dynamic stability The 6 DOF flight dynamics is a set of 9 ODEs of the form x = f (x, u ) e.g. x = { u, v, w, p, q, r,,, } T are the states or {,,V T, p, q, r,,, } T and u = { a, e, r, } T are the controls

4 The dynamic stability analysis procedure is as follows: Step 1. Find the trim point x o, u o such that dx/dt = 0 or solve f (x, u ) = 0 Step 2. Consider a small disturbance about the trim condition, i.e. set x = x o + x, u = u o + u Expanding the ODEs about the trim condition d( x o + x)/dt = f(x o, u o ) + [ f/ x ] x + [ f/ u ] u +.

5 For the trim point dx o /dt = 0, f (x o, u o ) = 0 Ignore higher order terms, the disturbance is governed by a linear system of ODEs of the form: x = [ A ] x + [ B ] u Step 3. Check the asymptotic stability of this linear system i.e. How fast will the disturbance decrease to zero?

6 1.2 Basic trim condition steady, symmetric, level flight The basic trim condition is characterized by steady flight - no rotational rates p, q, r = 0 symmetric flight no sideslip, r = 0 wings level no banking, a = 0 In this case the trim equations simplify to :

7 i = T L D V T W vertical forces horizontal forces pitching moment

8 Written in terms of aerodynamic coefficients, e.g.... ½ (h) V 2 S C L (, e) + T(h, V, ) sin = W ½ (h) V 2 S C D (, e) - T(h, V, ) cos - = 0 C m (, e) = 0 Any observations?

9 Question 1: How many equations and how many unknowns are there? Answer : Question 2 : Which "unknowns" need to be specified? Answer :

10 Question : What happens if we specify h and then solve for corresponding, V, e? 60 Sample F5 1g flight envelope kft Mach

11 Figure 1.1: Typical aerodynamic & thrust characteristics Note the parabolic nature of C D

12 2.0 Longitudinal and lateral dynamic stability models For a "moderate range of,, p, q, r, the behaviour of the aerodynamics allow the separation of the EOM's into 2 decoupled sets Longitudinal states & controls x = {, V T, q, } or {w, u, q, } u = { e, } i.e. the forward, up-down translation and pitching motion. Lateral states & control x = {, p, r, } or {v, p, r, } u = { a, r} i.e. the sideward, rolling and yawing motion. This allows us to work with a smaller linear system when determining the stability of a trim point.

13 2.1 Math digression Why stability is related to eigenvalues 1. To solve the linear system x = [ A ] x, we look for solutions of the form x = v e t where v and are unknown. 2. This implies that v e t = [ A ] v e t or [ A - I ] v = 0 3. For non trivial solutions we require det A - I = 0 4. This is the eigenvalue problem. Hence are given by the eigenvalues of A and v are the corresponding eigenvectors

14 2.3 Longitudinal linear (small disturbance) model x = [ A ] x + [ B ] u ( s omitted) x = {, u/v To, q, } T u = { e} Z Z u 1 + Z q 0 X [A] = X u X q -g/v To [B] = m m u m q Z X m 0

15 2.4 Lateral linear (small disturbance) model x = [ A ] x + [ B ] u ( s omitted) x = {, p, r, } T u = { a, r} T Y Y p Y r - 1 g/v To l [A] = l p l r 0 [B] = n n p n r Y a l a n a 0 0 Y r l r n r

16 The subscripted terms in the linear models are the control stability derivatives. They are divided into a) static derivatives eg : Z Z u Z b) dynamic derivatives eg : Z q c) rate derivatives eg : Z This is the starting point for the dynamic stability analysis

17 3.0 Dynamic stability analysis Stability derivatives for aircraft trimmed at V To = 660 ft/s (201 m/s) Longitudinal stability derivatives Z = Z u = Z q = 0 X = X u = X q = 0 m = m u = 0 m q = Lateral stability derivatives Y = Y p = 0 Y r = 0 l = l p = l r = n = n p = n r =

18 Longitudinal stability derivatives Z = Z u = Z q = 0 X = X u = X q = 0 m = m u = 0 m q = A = Z Z u 1 + Z q 0 X X u X q -g/v To m m u m q =

19 Form the characteristic equation A - I = 0 A - I = The characteristic equation is a quartic polynomial = 0 With the roots (eigenvalues) = j j

20 Plotting the eigenvalues in the complex plane short period mode = j x j x phugoid mode = j x

21 The eigenvector for each is obtained by solving Av = v For the short period v = j j j j For the phugoid v = j j j j

22 3.1 Interpreting the longitudinal eigenvalues & eigenvectors Recall the disturbances are x = {, u/v To, q, } T In terms of the eigenvalues and vectors, x = C i v e t. The contribution from the short period mode takes the form: e t { cos ( t) or sin ( t) }

23 Now interpret the behaviour of the phugoid mode e t { cos( t) or sin( t) } Which states are strongly affected by the phugoid mode?

24 Summary : Longitudinal modes 1. The longitudinal eigenvalues typically consists of two well separated complex pairs called the short period mode (stable) and the phugoid mode (marginally stable or unstable). 2. The short period mode is oscillatory and relatively well damped. It affects disturbances in and q. 3. The phugoid mode is oscillatory and takes a long time to decay or grow. It affects disturbances in u and.

25 3.2 FAR Dynamic stability, paragraphs (a) & (d) (a) Any short period oscillation not including combined lateraldirectional oscillations occurring between the stalling speed and the maximum allowable speed appropriate to the configuration of the airplane must be heavily damped with the primary controls -- (1) Free; and (2) In a fixed position (d). Any long-period oscillation of flight path, phugoid oscillation, that results must not be so unstable as to increase the pilot's workload or otherwise endanger the airplane.

26 Lateral stability derivatives Y = Y p = 0 Y r = 0 l = l p = l r = n = n p = n r = [A] = Y Y p Y r - 1 g/v To l l p l r 0 n n p n r =

27 Similarly for the lateral eigenvalues and vectors The eigenvalues are : , j The lateral eigenvalues typically consists of a stable real eigenvalue (roll), a stable complex pair (Dutch roll) and a marginally stable or unstable real eigenvalue (spiral) Homework : Find the eigenvectors

28 3.3 Interpreting the lateral eigenvalues & eigenvectors roll mode e t x = {, p, r, } T Dutch roll mode spiral mode e t {cos( t) or sin( t)} e t

29 Summary : Lateral modes 1. The lateral eigenvalues typically consists of a stable real eigenvalue (roll), a stable complex pair (Dutch roll) and a marginally stable or unstable real eigenvalue (spiral). 2. The roll mode affects p and. 3. The Dutch roll is a coupled oscillatory roll-yaw motion (p and r). 4. The spiral mode affects mainly and hence r.

30 3.4 FAR Part 23. Section Dynamic stability, paragraph (b) (b) Any combined lateral-directional oscillations ("Dutch roll") occurring between the stalling speed and the maximum allowable speed appropriate to the configuration of the airplane must be damped to 1/10 amplitude in 7 cycles with the primary controls -- (1) Free; and (2) In a fixed position. Question : What s the constraint on the eigenvalue?

31 3.5 A comparison with FAR Part Dynamic Stability (a) Any short period oscillation, not including combined lateraldirectional oscillations, occurring between 1.13 VSR and maximum allowable speed appropriate to the configuration of the airplane must be heavily damped with the primary controls (1) Free; and (2) In a fixed position. (b) Any combined lateral-directional oscillations ("Dutch roll") occurring between 1.2 VS and maximum allowable speed appropriate to the configuration of the airplane must be positively damped with controls free, and must be controllable with normal use of the primary controls without requiring exceptional pilot skill.