Both Paths Satisfy the Dynamic Equations

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1 Liearized Equatios ad Modes of Motio Robert Stegel, Aircraft Flight Dyamics MAE 331, 008 Liearizatio of oliear dyamic models Nomial flight path Perturbatios about the omial flight path Modes of motio Nomial ad Actual Flight Paths Nomial or referece trajectory, cotrol, ad disturbace histories x N t, u N t, w N t for t i [t o,t f ] Actual path, perturbed by Iitial coditio variatio Cotrol variatio Disturbace variatio x : aircrafts dyamic state u : aircrafts cotrol iput w : aircraft s disturbace iput xt, ut, wt for t i [t o,t f ] Copyright 008 by Robert Stegel. All rights reserved. For educatioal use oly. Actual dyamics ca be expressed as the sum of the omial dyamics plus perturbatio effects Both Paths Satisfy the Dyamic Equatios x N t = f[x N t,u N t,w N t,t] x t = f[xt,ut,wt,t] x N t = f[x N t,u N t,w N t,t] Aircraft equatios of motio r I = H I B v B = L I B B v B = 1 m F v B B B B = I 1 B M B B I B B x t = x N t + x t = f[x N t + xt,u N t + ut,w N t + wt,t] f[x N t,u N t,w N t,t] + f f f xt + ut + x u w wt where the partial-derivative matrices are evaluated alog the omial path How Is System Respose Calculated? Liear ad oliear, time-varyig ad time-ivariat dyamic models Numerical itegratio of ordiary differetial equatios i the time domai Liear, time-ivariat dyamic models State trasitio i the time domai Trasformatio from the time domai to the frequecy domai, solutio via trasfer fuctios, ad iverse trasformatio back to the time domai

2 Numerical Itegratio of Ordiary Differetial Equatios Give Iitial coditio, cotrol, ad disturbace histories xt 0 ut, wt for t i [t o,t f ] Path or trajectory is approximated by executig a umerical algorithm Noliear equatios of motio x t = f[xt,ut,wt,t], xt 0 give Liear, time-ivariat equatios of motio x t = Fxt + Gut + Lwt, xt o give Rectagular Euler Itegratio xt k = xt k1 + xt k1,t k Two Algorithms for Itegratig Ordiary Differetial Equatios [ ] t, t = t k t k1 xt k1 + f xt k1,ut k1,wt k1 Trapezoidal modified Euler Itegratio ode3 xt k xt k1 + 1 [ x 1 + x ] where [ ] t [ ] t x 1 = f xt k1,ut k1,wt k1 x = f xt k1 + x 1,ut k,wt k Perturbatios from the Nomial Coditios Liearized Equatio Approximates Perturbatio Dyamics Solve for the omial path ad the perturbatio path separately Differece betwee omial ad actual paths: xt o = xt o x N t o xt = xt x N t x t = x t x N t x : aircraft s dyamic state perturbatio u : aircraft s cotrol iput perturbatio Differece betwee omial ad actual iputs: ut = ut u N t w : aircraft s disturbace iput perturbatio wt = wt w N t x N t = f[x N t,u N t,w N t,t] x t f f f xt + ut + x u w wt = Ftxt + Gtut + Ltwt where the Jacobia matrices of the liear model are evaluated alog the omial trajectory Ft = f x x= x N t u= u N t w= w N t ; Gt = f u x= x N t u= u N t w= w N t dimx = 1 dimu = m 1 dimw = s 1 ; Lt = f w x= x N t u= u N t w= w N t

3 Sesitivity to state perturbatios f 1x1 f 1x f 1x Ft = f = x1 x x x x= x N t u= u N t w= w N t f f x1 x Lt = f w x= x N u= u t N t w= w N t Jacobia Matrices Express the Solutio Sesitivity to Small Perturbatios f x x= x N t u= u N t w= w N t Sesitivity to disturbace perturbatios f 1w1 f 1w f 1ws f = w1 w ws f f w1 w x t = Ftxt + Gtut + Ltwt f ws x= x N t u= u N t w= w N t Sesitivity to cotrol perturbatios Gt = f u x= x N t u= u N t w= w N t dimx = 1 dimu = m 1 dimw = s 1 f 1u1 f 1u f 1um = u1 u um f f u1 u dimf = dimg = m diml = s f um x= x N t u= u N t w= w N t How Do We Calculate the Partial Derivatives? Ft = f x x= x N t u= u N t w= w N t Gt = f u x= x N t u= u N t w= w N t Lt = f w x= x N t u= u N t w= w N t Aalytically Symbolic maipulatio of aalytical models of F, G, ad L Numerically First differeces i fx,u,w Noliear equatio Liearizatio Example dimx = 31 dimu = 1 dimw =11 x 1 x + dw 1 x = x = a x 3 x x 3 c x c 1 x 1 + x + a 1 x 3 x 1 + b 1 u 1 + b u + b 3 x 1 u 1 Aalytical calculatio of the Jacobia matrices Ft = f = a x x= x N t 1 u= u N t w= w N t c 1 + b 3 u 1N Gt = f u x= x N t u= u N t w= w N t 0 0 = b 1 b b 3 0 Why are F, G, ad L fuctios of time? = f 1 f f 3 a a + a 1 c 1 3c Lt = f w x= x N u= u t N t w= w N t d = 0 0 Numerical Estimatio of the Jacobia Matrices x 1 + x 1 x 1 * x 1 x 1 x 1 x 1 x 1 x f x 1 * f x + x L 1 f L 1 * f x + x L 1 f x L 1 * f x L 1 L x x x x x + x x + x L x 1 x x x 1 + x 1 x 1 * x 1 x 1 x 1 x f x * f x + x Ft L f L * f x + x L L x x x x L L x 1 x L L L L x 1 + x 1 x 1 * x 1 x 1 x 1 x f x * f x L f x L * f L L x x x + x x + x L L x 1 x x=x N t u=u N t w=w N t

4 Separate Solutios for Nomial Path ad Perturbatios from the Path Origial oliear equatio describes omial dyamics x N = x N x N t = f[x N t,u N t,w N t,t] x N + dw 1N = a x N c x 3 3N + c 1 + x N + a 1 + b 1 u 1N + b u N + b 3 u 1N Liear, time-varyig equatio describes perturbatio dyamics x t = Ftxt + Gtut + Ltwt ; xt o give x x = a 1 x 3 c 1 + b 3 u 1N x 1 t o x t o give x 3 t o a a + a 1 c 1 3c, x N give x d u x + b 1 b u x 3 b 3 0 w 1 ; 0 Rate of chage of Traslatioal Velocity u = X /m gsi + rv qw = f 1 v = Y /m + gsi cos ru + pw = f w = Z /m + gcos cos + qu pv = f 3 Rigid-Body Equatios of Motio Scalar Notatio Rate of chage of Traslatioal Positio x I = cos cos u + cos si + si si cos v + si si + cos si cos w = f 4 y I = cos si u + cos cos + si si si v + si cos + cos si si w = f 5 z I = si u + si cos v + cos cos w = f 6 Rate of chage of Agular Velocity I xy ad I yz = 0 p = I zz L + I xz N { I xz I yy I xx I zz p + [ I xz + I zz I zz I yy ]r}q = f 7 I xxi zz I xz [ ] I yy = f 8 = f 9 q = M I xx I zz pr I xz p r r = I xz L + I xx N { I xz I yy I xx I zz r + [ I xz + I xx I xx I yy ]p}q I xxi zz I xz Rate of chage of Agular Positio = p + qsi + rcos ta = f 10 = qcos rsi = f 11 = qsi + rcos sec = f 1 Aircraft has mirror symmetry State Vector x 1 u x x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 1 v w x y z = p q r * First six elemets of the state are logitudial variables Secod six elemets of the state are lateraldirectioal variables x 1 x x 3 x 4 x 5 x 6 x 7 x 8 x 9 Rearrage the State Vector State Vector x 10 x 11 x 1 ew u w x z q = x Lo = x LatDir v y p r * + Logitudial Equatios of Motio Dyamics of positio, velocity, agle, ad agular rate i the vertical plae u = X /m gsi + rv qw w = Z /m + gcos cos + qu pv x I = cos cos u + cos si + si si cos v + si si + cos si cos = x 1 = f 1 = x = f w = x 3 = f 3 z I = siu + si cos v + cos cos w = x 4 = f 4 q = M I xx I zz pr I xz p r x 5 = f 5 [ ] I yy = = qcos rsi = x 6 = f 6

5 Lateral-Directioal Equatios of Motio Dyamics of positio, velocity, agle, ad agular rate out of the vertical plae v = Y /m + gsi cos ru + pw y I = cos si u + cos cos + si si si v + si cos + cos si si p = I zz L + I xz N { I xz I yy I xx I zz p + [ I + I I I xz ]r}q I xxi zz zz yy zz I xz r = I xz L + I xx N I xz I yy I xx I zz r + I + I I I { [ xz ]p}q xx xx yy I xxi zz I xz = x 7 = f 7 w = x 8 = f 8 = = x 9 = f 9 x 10 = f 10 = p + qsi + rcos ta = x 11 = f 11 = qsi + rcos sec = x 1 = f 1 Sesitivity to Small Perturbatios i the Motio 1 x 1 stability matrix for the etire system f 1x1 f 1x f 1x f 1u f 1w f 1 Ft = x1 x x = u w f f x1 x f f x 1 f 1 f 1 u w 1 x 1 stability matrix has four 6 x 6 blocks that distiguish logitudial ad lateral-directioal effects Effects of logitudial perturbatios o logitudial motio F = F Lo Lo F LatDir LatDir F Lo Effects of lateral-directioal perturbatios o logitudial motio F LatDir Effects of logitudial perturbatios o lateral-directioal motio Effects of lateral-directioal perturbatios o lateral-directioal motio Sesitivity to Small Cotrol Iput Perturbatios Sesitivity to Small Disturbace Iput Perturbatios Cotrol iput vector Et Tt Ft ut = At Rt SFt Elevator, deg or rad Throttle, Flaps, deg or rad Aileros, deg or rad Rudder, deg or rad Side Force Paels, deg or rad Cotrol iput perturbatios Et Tt Ft ut = At Rt SFt Disturbace iput vector Disturbace iput perturbatio u w t w w t q wt = w t v w t p w t r w t Axial wid, m /s Normal wid, m /s Pitchig wid shear, deg/s or rad /s Lateral wid, m /s Rollig wid shear, deg /s or rad /s Yawig wid shear, deg /s or rad /s u w t w w t q wt = w t v w t p w t r w t 1 x 6 cotrol effect matrix has four 6 x 3 blocks that distiguish logitudial ad lateral-directioal effects Effects of logitudial cotrols o logitudial motio G = G Lo Lo G LatDir LatDir G Lo Effects of lateral-directioal cotrols o logitudial motio G LatDir 1 x 6 disturbace effect matrix has four 6 x 3 blocks that distiguish logitudial ad lateral-directioal effects Effects of logitudial disturbaces o logitudial motio L = L Lo Lo L LatDir LatDir L Lo Effects of lateral-directioal disturbaces o logitudial motio L LatDir Effects of logitudial cotrols o lateral-directioal motio Effects of lateral-directioal cotrols o lateral-directioal motio Effects of logitudial disturbaces o lateral-directioal motio Effects of lateral-directioal disturbaces o lateral-directioal motio

6 Restrict the Nomial Flight Path to the Vertical Plae Restrict the Nomial Flight Path to Steady, Level Flight With mirror symmetry ad o lateral-directioal motio u N = X /m gsi N q N w N w N = Z /m + gcos N + q N u N x I N = cos N u N + si N w N z I N = si N u N + cos N w N q N = M I yy N = q N Nomial lateral-directioal equatios are of o cocer i determiig the omial flight path x LatDirN = 0 x LatDirN = 0 Lateral-directioal perturbatios eed ot be zero x LatDirN 0 x LatDirN 0 Nomial State Vector x 1 u N x w N x 3 x N x 4 z N x 5 q N x 6 = x Lo = N x 7 x LatDir 0 N x 8 0 x x 10 x 11 x 1 N Specify omial airspeed V N ad altitude h N = z N Calculate coditios for trimmed equilibrium flight See Flight Dyamics for a solutio method 0 = X /m gsi N q N w N 0 = Z /m + gcos N + q N u N V N = cos N u N + si N w N 0 = si N u N + cos N w N 0 = M I yy 0 = q N Trimmed State Vector is costat u w x z q Trim u Trim w Trim V = N t z N 0 Trim Decouplig of Small Perturbatios i Steady, Symmetric, Level Flight Assume the airplae is symmetric ad its omial path is steady, level flight Small logitudial ad lateral-directioal perturbatios are ucoupled from each other 1 x 1 system is block diagoal costat F = F Lo 0 G = G Lo 0 L = L Lo 0 0 F LatDir 0 G LatDir 0 L LatDir Decoupled Logitudial ad Lateral- Directioal Equatios Describe Small Perturbatios from Steady, Level Flight Two 6 x 6 systems x Lo t = F Lo x Lo t + G Lo u Lo t + L Lo w Lo t x LatDir t = F LatDir x LatDir t + G LatDir u LatDir t + L LatDir w LatDir t x 1 x x 3 x Lo = x 4 x 5 x 6 Lo u w x = z q x 1 x x 3 x LatDir = x 4 x 5 x 6 LatDir v y p = r * +

7 Fourier Trasform of a Scalar Variable Fourier trasform [ ] = x j = xte jt F xt xt : x j : real variable complex variable Iverse Fourier trasform dt, = frequecy, rad /s = a + jb j = Ae A : amplitude : phase agle F -1 [x j ] = xt = 1 x je jt d xt x j = a + jb Laplace Trasform of a Scalar Variable Laplace trasform [ ] = xs = xte st L xt Iverse Laplace trasform L -1 [xs] = xt = 1 dt 0 j + s = + j = Laplace operator, rad /s xse st ds Laplace trasformatio is a liear operatio L [x 1 t + x t] = L [x 1 t] + L [x t] = x 1 s + x s L[ axt ] = al [xt] = axs Laplace Trasforms of Vectors ad Matrices Laplace trasform of a vector variable x 1 s L [xt] = xs = x s Laplace trasform of a matrix variable [ ] = As = L At a 11 s a 1 s s s Laplace trasform of a time-derivative L [ x t] = sxs x0 Laplace Trasform of a Dyamic System System equatio x t = F xt + Gut + Lwt Laplace trasform of system equatio sxs x0 = F xs + G us + Lws Rearrage Laplace trasform of dyamic equatio sxs F xs = x0 + G us + Lws [ si F]xs = x0 + G us + Lws xs = [ si F] 1 [x0 + Gus + Lws ] dimx = 1 dimu = m 1 dimw = s 1

8 Matrix Iverse Matrix Iverse y = Ax; x = A 1 y; dimx = dimy = 1; dima = dima = 11 A = a; A 1 = 1 a dima = T a 1 A = a 11 a 1 a ; A 1 = 1 a 11 a = 1 a 11 a 11 a 1 a 11 a 1 [ A] 1 = Adj A A = CT det A ; = Adj A det A 11 C = matrix of cofactors Cofactors are siged miors of A ij th mior of A is the determiat of A with the i th row ad j th colum removed a 31 a 31 dima = 3 3 a 11 a 1 a 13 a 1 a 13 a 11 a 13 a 31 a 11 a 1 a 31 A = ; A 1 = a 1 a 13 a 11 a 13 a 11 a 1 a a a 1 a 31 + a 13 a 13 a 31 a 1 a 11 a 1 a 13 a 1 a 13 a 31 a 11 a 13 a 31 a 11 a 13 a 31 a 11 a 1 a 31 a 11 a 1 = a 11 + a 1 a 31 + a 13 a 13 a 31 a 1 a 11 T Characteristic Polyomial of a Dyamic System xs = [ si F] 1 [x0 + Gus + Lws ] [ si F] 1 = Adj si F si F x Characteristic polyomial of the system is a scalar si F = det si F s = s + a 1 s a 1 s + a 0 Eigevalues or Roots of a Dyamic System Characteristic equatio of the system s = s + a 1 s a 1 s + a 0 = 0 s s = 0 = s 1 where! i are the eigevalues of F or the roots of the characteristic polyomial Eigevalues are complex umbers that ca be plotted i the s plae Complex cojugate i = i + j i * i = i j i s Plae

9 si F Lo = s rage Eigevalues of Logitudial Modes = det si F Lo Lo s = s 1 s s 6 s height s phugoid s * phugoid s short period s * short period s + P P s + P s + SP SP s + SP = 0 = s ra s hgt x Lo t = F Lo x Lo t + G Lo u Lo t + L Lo w Lo t Eigevalues determie the dampig ad atural frequecies of the liear system!s modes of motio Logitudial characteristic equatio has 6 eigevalues 4 eigevalues ormally appear as complex pairs ra : rage mode 0 hgt : height mode 0 P, P : phugoid mode SP, SP : short - period mode si F LD = s crossrage Eigevalues of Lateral- Directioal Modes x LatDir t = F LatDir x LatDir t + G LatDir u LatDir t + L LatDir w LatDir t = det si F LD LD s = s 1 s s 6 = 0 s Dutch roll s headig s spiral s roll s * Dutch roll = 0 = s cr s head s S s R s + DR DR s + DR Lateral-directioal characteristic equatio has 6 eigevalues eigevalues ormally appear as a complex pair cr : crossrage mode 0 head : headig mode 0 S : spiral mode R : roll mode DR, DR : Dutch roll mode Next Time: Logitudial Dyamics