Advanced Problems of Lateral- Directional Dynamics!

Transcription

1 Advanced Poblems of Lateal- Diectional Dynamics! Robet Stengel, Aicaft Flight Dynamics! MAE 331, 216 Leaning Objectives 4 th -ode dynamics! Steady-state esponse to contol! Tansfe functions! Fequency esponse! Root locus analysis of paamete vaiations Residualization Flight Dynamics! ! Copyight 216 by Robet Stengel. All ights eseved. Fo educational use only Stability-Axis Lateal-Diectional Equations With idealized aileon and udde effects i.e., N!A = L!R =!!t!"t!!!pt!!t N N " N p Y *1 " g = V N V N L L " L p 1!t!"t!pt!t + ~ N + R L + A ~!+ A!+ R "!x 1!x 2!x 3!x 4 "! "! = =!p! Yaw Rate Petubation Sideslip Angle Petubation Roll Rate Petubation Roll Angle Petubation "!u 1!u 2 = "! A! R " = Aileon Petubation Rudde Petubation 2

2 Lateal-Diectional Chaacteistic Equation s = s 4 + L p + N + Y " VN s 3 * Y + N " L N p + L " Y p VN + N " -, VN + L p +. + L " N p g V N * + Y " VN L N p L p N +, + g V N L " N L N " = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a = -./ s Typically factos into eal spial and oll oots and an oscillatoy pai of Dutch oll oots / s2 DR s = s " S s " R s n s + n 3 Business Jet Example of Lateal-Diectional Chaacteistic Equation s = s ".883 s s s Slightly unstable Spial Stable Roll Lightly damped Dutch oll Dutch oll Roll Spial Dutch oll 4

3 Initial yaw ate 4 th -Ode Initial-Condition Responses of Business Jet Initial sideslip angle Initial oll ate Initial oll angle Initial oll angle and ate have little effect on yaw ate and sideslip angle esponses Initial yaw ate and sideslip angle have lage effect on oll ate and oll angle esponses 5 Appoximate Roll and Spial Modes Roll ate is damped by L p Roll angle is a pue integal of oll ate!!p!!" = L p 1!p!" + L A! A Chaacteistic polynomial has eal oots! RS s = s s " L p S = Neutal stability R = L p Geneally < 6

4 !!!"! = Appoximate Dutch Roll Mode N N " * Y -, 1/ + V N. Y " V N!!" + N R Y R V N!R Chaacteistic polynomial, natual fequency, and damping atio! DR s = s 2 " N + Y * VN s + N 1" Y +, VN ndr = N 1" Y VN + N Y VN + N Y VN + N Y VN 1 DR = " N + Y VN 2 N 1" Y VN -./ With negligible side-foce sensitivity to yaw ate, Y Y! ndr = N " + N " VN DR = N + Y " Y VN * 2 N " + N " VN 7 Effects of Vaiation in Pimay Stability Deivatives! 8

5 N " Effect on 4th - Ode Roots Goup!s tems multiplied by N " to fom numeato Denominato fomed fom emaining tems of!s Root Locus Gain = Diectional Stability N " > Dutch Roll s = ds + N " ns = k ns ds = 1 = N " s z 1 s 1 s 2 s z 2 s n s + n Zeo Zeo Roll Spial Dutch Roll N " < Positive N!! Inceases Dutch oll natual fequency! Damping atio deceases but emains stable! Spial mode dawn towad oigin! Roll mode unchanged Negative N " destabilizes Dutch oll mode 9 N Effect on 4 th - Ode Roots Root Locus Gain = Yaw Damping Dutch Roll N < N > s = ds + N ns = k ns ds = "1 = N s " z 1 s " 1 s µ n s + n s " 2 s n s + n Negative N! Inceases Dutch oll damping! Daws spial and oll modes togethe Positive N destabilizes Dutch oll mode Zeo Roll Zeo Spial Zeo Dutch Roll 1

6 L p Effect on 4 th - Ode Roots Root Locus Gain = Roll Damping k ns ds = "1 = s = ds + L p ns = L p s s µ n s + n s " 1 s " 2 s n s + n 2 Negative L p! Roll mode time constant! Daws spial mode towad oigin Positive L p destabilizes oll mode L p : negligible effect on Dutch oll mode L p can become positive at high angle of attack Dutch Roll Zeo L p < L p > Roll Zeo Spial Dutch Roll Zeo 11 Coupling Stability Deivatives! 12

7 Dihedal L " Effect on 4th -Ode Roots Root Locus Gain = Dihedal Effect ns = g VN N p s z 1 s = ds + L " g VN N p k ns L " ds = 1 = s S s R s ndr s + ndr s = s s ".883 s s Bizjet Example Dutch Roll L " < L " > Roll Zeo Spial Dutch Roll Negative L "! Stabilizes spial and oll modes but...! Destabilizes Dutch oll mode Positive L " does the opposite 13 Stabilizing Lateal- Diectional Motions Sola Impulse Povide sufficient L " to stabilize the spial mode Povide sufficient N to damp the Dutch oll mode Oiginal Root Locus Inceased N How can L " and N be adjusted atificially, i.e., by closed-loop contol? 14

8 Roll Acceleation Due to Yaw Rate, L 2 "V L! C N l Sb 2I xx b = C lˆ 2V N "V N 2 2I xx Sb = C "V N lˆ Sb2 4I xx Root Locus Gain = Roll Due to Yaw Rate L < L > s = ds + L N p ns = kns ds = "1 = L N p s " z 1 s " 1 s " 2 s " z 2 s n s + n 15 Yaw Acceleation Due to Roll Rate, N p 2 "V N p! C N np Sb 2I zz b = C n ˆp 2V N 2 "V N 2I zz Sb = C "V N n ˆp Sb2 4I xx N p > N p < Root Locus Gain = Yaw due to Roll Rate s = ds + N p ns = s n s + n kns ds = "1 = N p s s " z 1 s " 1 s " 2 16

9 Oscillatoy Roll-Spial Mode! RSes = s " S s " R o s n s + n RS The chaacteistic equation factos into eal o complex oots Real oots ae oll mode and spial mode when L! N < L N! Complex oots poduce oll-spial oscillation o lateal phugoid mode when N p L! N > L N! L! + L Y! /V N 2 and g V N L! N " L N! < 1 17 Roll-Spial Oscillation of the M2-F2 Lifting Body Test Vehicle 18

10 4 th -Ode Fequency Response! 19 Yaw Rate and Sideslip Angle Fequency Responses of Business Jet 4 th -Ode Response to Aileon and Rudde 2 nd -Ode Response to Rudde! j"! A j"! j"! R j"! j"! R j"!" j! A j!" j! R j!" j! R j Yawing esponse to aileon is not negligible Yaw ate esponse is pooly chaacteized by the 2 nd -ode model below the Dutch oll natual fequency Sideslip angle esponse is adequately chaacteized by the 2 nd -ode model 2

11 Roll Rate and Roll Angle Fequency Responses of Business Jet 4 th -Ode Response to Aileon and Rudde 2 nd -Ode Response to Aileon!p j"! A j"!p j"! R j"!p j"! A j"!" j! A j!" j! R j!" j! A j Roll esponse to udde is not negligible Roll ate esponse is maginally well chaacteized by the 2 nd -ode model Roll angle esponse is pooly chaacteized at low fequency by the 2 nd -ode model 21! j"! A j" Fequency and Step Responses to Aileon Input!" j! A j!p j"! A j"!" j! A j!v t!y t! t!p t!" t!" t Yaw/sideslip sensitivity in the vicinity of the Dutch oll natual fequency Roll ate esponse is elatively benign Ratio of oll angle to sideslip esponse is impotant to the pilot 22

12 Fequency and Step Responses to Rudde Input! j"! R j"!" j! R j!p j"! R j"!" j! R j!v t!y t! t!p t!" t!" t Yaw esponse vaiability nea and below the Dutch oll natual fequency Significant oll ate esponse nea the Dutch oll natual fequency Lightly damped yaw/sideslip esponse would be had to contol pecisely 23 Next:! Lateal-Diectional Flying Qualities! Leaning Objectives LCDP " /" d /! Eigenvectos Pilot-Vehicle Inteactions PIO 24

13 Supplemental Mateial! 25 Review Questions!!! What is a flying qualities thumbpint?!!! What is the best Coope-Hape pilot ating?!!! What is MIL-F-8785C?!!! What ae the Aiplane Types?!!! What ae the Flight Phases?!!! What ae the Levels of Pefomance?!!! What is the Time to Double?!!! What is the Contol Anticipation Paamete?!!! What is the C* Citeion?!!! What is the Gibson Dopback Citeion fo Pitch Angle Contol?! 26

14 Bizjet Fouth- and Second-Ode Models and Eigenvalues Fouth-Ode Model F = G = Eigenvalue Damping Feq. ad/s Unstable e e+j 8.32E E e e+j 8.32E E+ Dutch Roll Appoximation F = G = Eigenvalue Damping Feq. ad/s e e+j 9.55E E e e+j 9.55E E+ Roll-Spial Appoximation F = G = Eigenvalue Damping Feq. ad/s nd -ode-model eigenvalues ae close to those of the 4 th -ode model Eigenvalue magnitudes of Dutch oll and oll oots ae simila 27 Steady-State Response!!x S = "F "1 G!u S 28

15 Equilibium Response of 2 nd -Ode Dutch Roll Model Equilibium esponse to constant udde! SS!" SS Y " N " V N = 1 N * Y " -, N + N " / + V N. N R!R SS Y N R * V! S = " N!R Y S N + N * V N N R! S = "!R Y S N + N * V N Steady yaw ate and sideslip angle ae not zeo 29 Equilibium Response of 2 nd - Ode Roll-Spial Model Equilibium state with constant aileon!p SS!" SS = L p 1 1 L * A!* A SS but! " L p 1 1! = " 1 L p taken alone!p S = "L p "1 L A!A S!p S = " L A L p!t S = " t L A L p! A S! A S dt Steady oll ate popotional to aileon Roll angle, integal of oll ate, continually inceases 3

16 Equilibium Response of 4 th -Ode Model Equilibium state with constant aileon and udde deflection! S!" S!p S! S N N " N p Y *1 " g = * V N V N L L " L p 1 *1 ~ N + R L + A ~!+ A S!+ R S! S!" S!p S! S g L V * A N " + g L N V " N * R N g g L V * A N L N V N * R N, Y N " + N " / V -. N 1 L, Y * A + L " + L " / -. V N 1 N * R = g L V " N + L N " N!* A S!* R S 31 32

17 NTSB Simulation of Ameican Flight 587 Flight simulation deived fom digital flight data ecode DFDR tape 33 Reduction of Model Ode by Residualization! 34

18 Appoximate Low-Ode Response Dynamic model ode can be educed when! One mode is stable and well-damped, and it and is faste than the othe! The two modes ae coupled "!!x fast!!x slow " fast = F fast F slow slow F fast F slow "!x fast!x slow + " G fast G slow!u Expess as 2 sepaate equations!!x f = F f!x f + F s f!x s + G f!u!!x s = F f s!x f + F s!x s + G s!u 35 Appoximation fo Fast-Mode Response Assume that fast mode eaches steady state vey quickly compaed to slow-mode esponse!!x f " " F f!x f + F s f!x s + G f!u!!x s = F f s!x f + F s!x s + G s!u Steady-state solution fo!x fast! F f "x f + F s f "x s + G f "u F f "x f = F s f "x s G f "u!x f = "F "1 f F f s!x s + G f!u 36

19 Adjust Slow-Mode Equation fo Fast-Mode Steady State Substitute quasi-steady!x fast in diffeential equation fo!x slow s!!x s = "F f F "1 f F f s!x s + G f!u = F s " F s f F "1 f f F s + F s!x s + G s!u!x s + G s " F s f F "1"1 f G f Residualized diffeential equation fo!x slow!!x s = F s!x s + G s!u!u whee F s = F s! F s f F!1 f " f F s G s = " G s! F s f F!1 f G f 37 Model of the Residualized Roll-Spial Mode Yawing motion is assumed to be instantaneous compaed to olling motions Residualized oll/spial equation!!p!!" + Y N p L *. g + L V *, - " L p N / L V N * L * N N + Y N * + N *. + Y, - V N / N * + N *., - V N / 1 = f 11 f 12 1!p!" +!p!" + 38

20 Roots of the 2 nd -Ode Residualized Roll-Spial Mode si! F RS " = s 1 1 "! f 11 f 12 1 = RSes " * L = s 2! L p! N + L Y /V N - p, + N + N Y /V / s + g * L N! L N - N. V, N + N + N Y /V / N. s! R o s n s + 2 n = s! S RS = Fo the business jet model! RSes = s s ".18 = = s ".98 s +1.1 = s " S s " R Slightly unstable spial mode Simila to 4 th -ode oll-spial esults s = s ".883 s s s