MAE331Lecture15.pdf

Transfer Functions and Frequency ResponseRobert Stengel, Aircraft Flight DynamicsMAE 331, 2014 Frequency domain view of initial condition response Response of dynamic systems to sinusoidal inputs Transfer functions Bode plots Copyright 2014 by Robert Stengel.All rights reserved.For educational use only.http://www.princeton.edu/~stengel/MAE331.htmlhttp://www.princeton.edu/~stengel/FlightDynamics.htmlReading:Flight Dynamics342-357Airplane Stability and ControlChapter 20Learning Objectives 12Fourier and Laplace Transforms3Fourier Transform of a Scalar Variable Transformation from time domain to frequency domainF !x(t )[ ] = !x( j") = !x(t )e# j"t#$$%dt, " = frequency, rad / s!x(t ) : real variable!x( j") : complex variable= a(")+ jb(")= A(")ej#(")A: amplitude! : phase anglej! : Imaginary operator, rad/s4Fourier Transform of a Scalar Variable !x(t )!x( j") = a(") + jb(")5Laplace Transform of a Scalar Variable Laplace transformation from time domain to frequency domainL !x(t )[ ] = !x(s) = !x(t )e"st0#$dts = ! + j"= Laplace (complex) operator, rad/s!x(t ) : real variable!x(s) : complex variable= a(s)+ jb(s)= A(s)ej"(s)6Laplace Transformation is a Linear Operation L !x1(t )+!x2(t )[ ] = L !x1(t )[ ]+ L !x2(t )[ ] = !x1(s)+!x2(s)L a!x(t )[ ] = aL !x(t )[ ] = a!x(s)Sum of Laplace transformsMultiplication by a constant7Laplace Transforms of Vectors and Matrices Laplace transform of a vector variableL !x(t )[ ] = !x(s) =!x1(s)!x2(s)..."#$$$%&'''Laplace transform of a matrix variableL F(t )[ ] = F(s) =f11(s) f12(s) ...f21(s) f22(s) ...... ... ...!"###$%&&&Laplace transform of a time-derivative L !! x(t )[ ] = s!x(s) " !x(0)8Laplace Transform of a Dynamic System !! x(t ) = F!x(t ) + G!u(t ) + L!w(t )System equationLaplace transform of system equations!x(s) " !x(0) = F!x(s) + G!u(s) + L!w(s)dim(!x) = (n "1)dim(!u) = (m"1)dim(!w) = (s "1)9Laplace Transform of a Dynamic System Rearrange Laplace transform of dynamic equations!x(s) "F!x(s) = !x(0)+G!u(s)+L!w(s)sI !F[ ]"x(s) = "x(0)+G"u(s)+L"w(s)!x(s) = sI "F[ ]"1!x(0)+G!u(s)+L!w(s)[ ]F to left, I.C. to rightCombine termsMultiply both sides by inverse of (sI F)10Matrix Inverse A[ ]!1= Adj A ( )A= Adj A ( )det A(n " n)(1"1)=CTdet A; C= matrix of cofactorsCofactors are signed minors of A ijth minor of A is the determinant of A with the ith row and jth column removed y = Ax; x = A!1ydim(x) = dim(y) = (n !1)dim(A) = (n ! n)ForwardInverse Numerator is a square matrix of cofactor transposesDenominator is a scalar11Matrix Inverse Examples A =a11a12a21a22!"##$%&&; A'1=a22'a21'a12a11!"##$%&&Ta11a22 ' a12a21=a22'a12'a21a11!"##$%&&a11a22 ' a12a21A =a11a12a13a21a22a23a31a32a33!"###$%&&&; A'1=a22a33 ' a23a32( ) ' a21a33 ' a23a31( ) a21a32 ' a22a31( )' a12a33 ' a13a32( ) a11a33 ' a13a31( ) ' a11a32 ' a12a31( )a12a23 ' a13a22( ) ' a11a23 ' a13a21( ) a11a22 ' a12a21( )!"####$%&&&&Ta11a22a33 + a12a23a31 + a13a21a32 ' a13a22a31 ' a12a21a33 ' a11a23a32=a22a33 ' a23a32( ) ' a12a33 ' a13a32( ) a12a23 ' a13a22( )' a21a33 ' a23a31( ) a11a33 ' a13a31( ) ' a11a23 ' a13a21( )a21a32 ' a22a31( ) ' a11a32 ' a12a31( ) a11a22 ' a12a21( )!"####$%&&&&a11a22a33 + a12a23a31 + a13a21a32 ' a13a22a31 ' a12a21a33 ' a11a23a32dim(A) = (2 ! 2)dim(A) = (3! 3)A = a; A!1= 1adim(A) = (1!1)12Matrix Inverse Examples A =1 23 4!"#$%&; A'1=4 '2'3 1!"#$%&'2='2 11.5 '0.5!"#$%&A =1 2 34 6 78 12 9!"###$%&&&; A'1='30 18 420 '15 50 4 '2!"###$%&&&10='3 1.8 0.42 '1.5 0.50 0.4 '0.2!"###$%&&&A = 5; A!1= 15 = 0.213Characteristic Matrix InversesI ! FSP " #SP(s)= s2+ c1s + c0sI ! FSP[ ]!1= Adj sI ! FSP( )sI ! FSP= CSPTs ( )"SP(s)(2 # 2)1#1 ( )Denominator is characteristic polynomial, a scalar14sI ! FSP[ ]Characteristic matrix (short-period model as example)Inverse of characteristic matrix Numerator of the Characteristic Matrix InverseAdj sI ! FSP( ) =nqq(s) n"q(s)nq"(s) n""(s)#$%%&'((Numerator is an (n x n) matrix of polynomials15For example,nqqs ( ) = k s ! z ( )(sI F)1 Distributes and Shapes the Effects of Initial ConditionssI ! FSP[ ]!1=nqq(s) n"q(s)nq"(s) n""(s)#$%%&'((s2+ c1s + c0(2 ) 2)1)1 ( )Denominator determines the modes of motionNumerator distributes each element of the initial condition to each element of the state!x(s) = Adj sI " FSP( )sI " FSP!x(0) 2 #1 ( )16Initial Condition Response in Frequency Domain Longitudinal dynamic model (time domain)!x(s) = sI "F[ ]"1!x(0) !! q(t )! ! "(t )#$%%&'((=MqM"1) LqVN*+,-./)L"VN#$%%%&'(((!q(t )!"(t )#$%%&'((,!q(0)!"(0)#$%%&'(( givenLongitudinal model (frequency domain)!q(s)!"(s)#$%%&'(( = sI ) FSP[ ])1!q(0)!"(0)#$%%&'((17Transfer Function Matrix Frequency-domain effect of all inputs on all outputs Assume control effects do not appear directly in the output: Hu = 0 Transfer function matrix H(s) = Hx sI ! F[ ]!1Gr ! n ( ) n ! n ( ) n ! m ( )= r ! m ( )181st-Order Transfer Function y s( )u s( ) = H(s) = h s ! f[ ]!1g =hgs ! f( )(n = m = r =1)Scalar transfer function (= rst-order lag) ! x t ( ) = fx t ( ) + gu t ( )y t ( ) = hx t ( )Scalar dynamic system192nd-Order Transfer Function H(s) = Hx sI ! F ( )!1s ( )G=h11h12h21h22"#$$%&''adjs ! f11( ) ! f12! f21s ! f22( )"#$$%&''dets ! f11( ) ! f12! f21s ! f22( )()**+,--g1g2"#$$%&''Second-order transfer function matrixr ! n ( ) n ! n ( ) n ! m ( )= r ! m ( ) = 2 ! 2 ( ) ! x t ( ) =! x1 t ( )! x2 t ( )!"##$%&& =f11f12f21f22!"##$%&&x1 t ( )x2 t ( )!"##$%&& +g1g2!"##$%&&u t ( )y t ( ) =y1 t ( )y2 t ( )!"##$%&& =h11h12h21h22!"##$%&&x1 t ( )x2 t ( )!"##$%&& +"Second-order dynamic system20Numerator and Denominator of 2nd-Order (sI F)1adjs ! f11( )! f12! f21s ! f22( )"#$$%&'' =s ! f22( )f12f21s ! f11( )"#$$%&''21 dets ! f11( ) ! f12! f21s ! f22( )"#$$%&'' = s ! f11( ) s ! f22( ) ! f12 f21= s2!f11 + f22( )s +f11 f22 ! f12 f21( )! s2+ 2()ns +)n2! * s ( )2nd-Order Transfer Function H(s) = Hx sI ! F ( )!1s ( )G=h11h12h21h22"#$$%&''s ! f22( ) f12f21s ! f11( )"#$$%&''s2+ 2()ns +)n2g1g2"#$$%&''22H(s) =h11 s ! f22( ) + h12 f21"#$%h11f12 + h12 s ! f11( )"#$%h21 s ! f22( ) + h22 f21"#$%h21f12 + h22 s ! f11( )"#$%"#&&&$%'''g1g2"#&&$%''s2+ 2()ns +)n2=h11 s ! f22( ) + h12 f21"#$%g1 +h11f12 + h12 s ! f11( )"#$%g2h21 s ! f22( ) + h22 f21"#$%g1 +h21f12 + h22 s ! f11( )"#$%g2"#&&&$%'''s2+ 2()ns +)n22nd-Order Transfer Function 23H(s) =h11 s ! f22( ) + h12 f21"#$%g1 +h11f12 + h12 s ! f11( )"#$%g2h21 s ! f22( ) + h22 f21"#$%g1 +h21f12 + h22 s ! f11( )"#$%g2"#&&&$%'''s2+ 2()ns +)n2 !k1 s ! z1( )k2 s ! z2( )"#$$%&''s2+ 2()ns +)n2Transfer Function Matrix for Short-Period Approximation Transfer Function Matrix (with Hx = I, Hu = 0) HSP(s) = I2 sI ! F ( )SP!1s ( )GSP =s ! Mq( )!M"! 1!LqVN#$%&'(s +L"VN( ))*++++,-....-1M/E!L/EVN)*+++,-... !! xSP t ( ) =!! q t ( )! ! " t ( )#$%%&'(( )MqM"1*LqVN+,-./0*L"VN#$%%%&'(((!q t ( )!" t ( )#$%%&'(( +M1E*L1EVN#$%%%&'(((!1 E t ( )Dynamic Equation24Transfer Function Matrix for Short-Period Approximation Transfer Function Matrix (with Hx = I, Hu = 0)HSP(s) = sI ! FLon[ ]!1GSP =s + L"VN( )M"1! LqVN#$%&'(s ! Mq( ))*++++,-....M/E!L/EVN)*+++,-...s ! Mq( ) s + L"VN( )! M"1! LqVN#$%&'(25Transfer Function Matrix for Short-Period Approximation HSP(s) =M!E s +L"VN( )#L!EM"VN$%&'()M!E 1#LqVN*+,-./# L!EVN( ) s # Mq( )$%&'()$%&&&&&'()))))s2+ #Mq +L"VN( )s # M" 1#LqVN*+,-./+ Mq L"VN$%&'()=M!Es + L"VN #L!EM"VNM!E( )$%&'()# L!EVN( ) s +VNM!EL!E1#LqVN*+,-./ # Mq$%&'()01234563$%&&&&&'()))))7SP s ( )26Transfer Function Matrix for Short-Period Approximation HSP(s) !kqn!Eq(s)k"n!E"(s)#$%%&'((s2+ 2)SP*nSPs +*nSP2 =+q(s)+!E(s)+"(s)+!E(s)#$%%%%%&'((((( dim = 2 x 1 27Scalar Transfer Functions for Short-Period Approximation !q(s)!"E(s) =M"Es + L#VN $ L"EM#VNM"E( )%&'()*s2+ $Mq + L#VN( )s $M#1$ LqVN+,-./0 + Mq L#VN%&'()*=kq s $ zq( )s2+ 21SP2nSPs + 2nSP2!"(s)!#E(s) =$ L#EVN( )s +VNM#EL#E1$ LqVN%&'()* $ Mq+,-./01234356373s2+ $Mq + L"VN( )s $M"1$ LqVN%&'()* + Mq L"VN+,-./0=k" s $ z"( )s2+ 28SP9nSPs +9nSP2Pitch Rate Transfer FunctionAngle of Attack Transfer Function28Relationship of (sI F)1 to State Transition Matrix, !(t,0)Initial condition response!x(s) = sI " F[ ]"1!x(0) =!x(t ) = " t, 0 ( ) !x(0)TimeDomain FrequencyDomain "x(s) is the Laplace transform of "x(t)!x(s) = L !x(t )[ ] = L " t, 0 ( )!x(0) #$%& = L " t, 0 ( ) #$%&!x(0)29Relationship of (sI F)1 to State Transition Matrix, (t,0)sI ! F[ ]!1= L " t, 0 ( ) #$%&=Laplace transform of the state transition matrixTherefore,30Initial Condition Response of a Single State Element (Frequency Domain) 31!x(s) = sI " F[ ]"1!x(0) !x1 s ( )!x2 s ( )!!xn s ( )"#$$$$$%&'''''=n11 s ( ) n12 s ( ) !n1n s ( )n21 s ( ) n22 s ( ) !n2n s ( )! ! ! !nn1 s ( ) nn2 s ( ) !nn2 s ( )"#$$$$$%&'''''! s ( )!x1 0 ( )!x2 0 ( )!!xn 0 ( )"#$$$$$%&'''''Initial Condition Response of a Single State Element !x2(s) = n21 s ( )! s ( ) !x1(0) + n22 s ( )! s ( ) !x2(0) +!+ n2n s ( )! s ( ) !xn(0)" p2 s ( )! s ( )Initial condition response of"x2(s)32Partial Fraction Expansion of the Initial Condition ResponseScalar response can be expressed with n parts, each containing a single mode !xi(s) = pi s ( )! s ( )=d1s " #1( ) +d2s " #2( ) +!dns " #n( )$%&'()i, i = 1, nFor each i, the coefficients ared j = s ! "j( ) pi s ( )# s ( ) s="j, j =1, n33Partial Fraction Expansion of the Initial Condition ResponseTime response is the inverse Laplace transform !xi(t ) = L"1!xi(s)[ ]= L"1d1s " #1( ) +d2s " #2( ) +!dns " #n( )$%&'()i= d1e#1t+ d2e#2t+!+ dne#nt( )i , i = 1, nEach elements time response contains every mode of the system (although some coefcients may be negligible)34Longitudinal Motions Contain Both ModesPhugoid (Long-Period) Mode Airspeed Flight Path Angle Pitch Rate Angle of Attack 35Aircraft Modes of Motion36Characteristic Polynomial of a LTI Dynamic System sI !F[ ]!1= Adj sI !F( )sI !F(n x n) Characteristic polynomial of the system is a scalar denes the systems modes of motion sI ! F = det sI ! F ( ) " #(s)= sn+ cn!1sn!1+ ... + c1s + c0!x(s) = sI " F[ ]"1!x(0) + G!u(s) + L!w(s)[ ]Inverse of characteristic matrix37Eigenvalues (or Roots) of a Dynamic System !(s) = sI " F = sn+ cn"1sn"1+ ... + c1s + c0 = 0= s " #1( ) s " #2( ) ... () s " #n( ) = 0Characteristic equation of the system ... where !i are the eigenvalues of F or the roots of the characteristic polynomial38Eigenvalues (or Roots) of a Dynamic System Eigenvalues are real or complex numbers that can be plotted in the s planes Plane !i = "i + j#i Real root !*i = "i # j$iPositive real part indicates instability Complex roots occur in conjugate pairs !i ="i39Roots of the Aircraft Dynamics Characteristic Equation!(s) = s12+ c11s11+ ... + c1s + c0 = 0= s " #1( ) s " #2( ) ... () s " #12( ) = 0 12th-order system of LTIequations 12 eigenvalues of the stability matrix, F 12 roots of the characteristic equation Characteristic equation of the systemUp to 12 modes of motion In steady, level ight, longitudinal and lateral-directional LTI perturbation models are uncoupled !(s) = s "#1( )! s "#6( )$%&'longs "#1( )! s "#6( )$%&'lat"dir = 040Lateral-Directional Modes of Motion in Steady, Level Flight!LD(s) = s " #1( ) s " #2( ) ... () s " #6( ) = 0= s " #CR( ) s " #Head( ) s " #S( ) s " #R( ) s " #DR( ) s " #*DR( )$%&' !! xLat "Dir(t ) =FLat "Dir!xLat "Dir(t ) + GLat "Dir!uLat "Dir(t ) + LLat "Dir!wLat "Dir(t )!LD(s) = s " #CR( ) s " #Head( ) s " #S( ) s " #R( ) s2+ 2$DR%nDRs +%nDR2( ) = 0Roots of the lateral-directional characteristic equation5 modes of motion (typical)CrossrangeHeadingSpiralRollDutch Roll 41Longitudinal Modes of Motion in Steady, Level Flight!Lon(s) = s " #1( ) s " #2( ) ... () s " #6( ) = 0= s " #R( ) s " #H( )s " #P( ) s " #*P( )$%&'s " #SP( ) s " #*SP( )$%&' !! xLon(t ) = FLon!xLon(t ) + GLon!uLon(t ) + LLon!wLon(t )!Lon(s) = s " #R( ) s " #H( ) s2+ 2$P%nPs +%nP2( ) s2+ 2$SP%nSPs +%nSP2( ) = 06 roots of the longitudinal characteristic equation4 modes of motion (typical)RealComplex RangeHeightPhugoidShort Period RealComplexComplexComplex 42Complex Conjugate Roots Form a Single Oscillatory Mode of Motions ! "P( ) s ! "*P( )=s ! #P + j$P( )%&'( s ! #P ! j$P( )%&'(= s2+ 2)P$nPs +$nP2( )s ! "SP( ) s ! "*SP( )=s ! #SP + j$SP( )%&'( s ! #SP ! j$SP( )%&'(= s2+ 2)SP$nSPs +$nSP2( )Short Period Roots Phugoid Roots !n: Natural frequency, rad/s ! : Damping ratio, -43Response to a Control InputNeglect initial conditionState response to control s!x(s) = F!x(s)+G!u(s)+!x(0), !x(0) ! 0!x(s) = sI "F[ ]"1G!u(s)Output response to control!y(s) = Hx!x(s) + Hu!u(s)= Hx sI " F[ ]"1G!u(s) + Hu!u(s)=Hx sI " F[ ]"1G+ Hu{ }!u(s)44Longitudinal Transfer Function Matrix With Hx = I, and assuming Elevator produces only a pitching moment Throttle affects only the rate of change of velocity Flaps produce only lift HLon(s) = HxLon sI ! FLon[ ]!1GLon=1 0 0 00 1 0 00 0 1 00 0 0 1"#$$$$%&''''nVV(s) n(V(s) nqV(s) n)V(s)nV((s) n(((s) nq((s) n)((s)nVq(s) n(q(s) nqq(s) n)q(s)nV)(s) n()(s) nq)(s) n))(s)"#$$$$$$%&''''''0 T*T00 0 L*F / VNM*E0 00 0 !L*F / VN"#$$$$$%&'''''+Lon s ( )45Longitudinal TransferFunction MatrixHLon(s) =n!EV(s) n!TV(s) n!FV(s)n!E"(s) n!T"(s) n!F"(s)n!Eq(s) n!Tq(s) n!Fq(s)n!E#(s) n!T#(s) n!F#(s)$%&&&&&&'())))))s2+ 2*P+nPs + +nP2( ) s2+ 2*SP+nSPs + +nSP2( ) There are 4 outputs and 3 inputs Douglas AD-1 Skyraider 46Longitudinal Transfer Function Matrix !V(s)!" (s)!q(s)!#(s)$%&&&&&'()))))= HLon(s)!*E(s)!*T(s)!*F(s)$%&&&'())) Input-output relationship 47Westland P.12 LysanderForssman bomber (?) 48 AEA Cygnet II, AlexanderGraham Bell, Glenn Curtiss, 1909 DEquevillery, 1908 Hargrave quadraplane (model), 1889 49 Phillips, 1907 John Septaplane, 1919 Wight Quadraplane, 1916 Phillips, 1904 Vedo Villi, 1911 Pemberton-Billings Nighthawk, 1916 50 Caproni Ca 60, 1920 Miraculously, this machine DID y the rst time in 1921- it reached a height of 60 feet, collapsed, and plummeted toward the lake just after take off, killing both pilots. Wings derived fromCa.42 bomber 51 Farman 3-engine Jabiru Heinkel 5-engine He111Z Tarrant 6-engine Tabor, 1919 Farman 4-engine Jabiru, 1923 52Scalar Transfer Function from "uj to "yiHij(s) = nij(s)!(s) = kij sq+ bq"1sq"1+ ... + b1s + b0( )sn+ cn"1sn"1+ ... + c1s + c0( )# zeros = q # poles = n Just one element of the matrix, H(s) Each numerator term is a polynomial with q zeros, whereq varies from term to term and # n 1 = kijs ! z1( )ij s ! z2( )ij... s ! zq( )ijs ! "1( ) s ! "2( )... s ! "n( )53 Denominator polynomial contains n roots Control Response of a Single State Element 54!yi s( ) = kijnij(s)!(s) !uj s( )Bode Plot(Frequency Response of a Scalar Transfer Function)55Scalar Frequency Response FunctionHij(j!) = kijj! " z1( )ijj! " z2( )ij ... j! " zq( )ijj! " #1( ) j! " #2( )... j! " #n( )Substitute: s = j" Frequency response is a complex function of input frequency, " Real and imaginary parts, or ** Amplitude ratio and phase angle ** = a(!)+ jb(!) "AR(!) ej#(!)56Short-Period Frequency Response (s = j") Expressed as Amplitude Ratio and Phase Angle Pitch-rate frequency response Angle-of-attack frequency response !q( j")!#E( j") =kqj"$ zq( )$"2+ 2%SP"nSP j"+"nSP2= ARq(") ej&q (")!"( j#)!$E( j#) =k"j#% z"( )%#2+ 2&SP#nSP j#+#nSP2= AR" (#) ej'" (#)57Bode Plot Portrays Response to Sinusoidal Control InputExpress amplitude ratio in decibelsAR(dB) =20log10AR original units ( )!"#$20 dB = factor of 10 "q( j#)"$E( j#)=kq j# % zq( )%#2+ 2&SP#nSP j# +#nSP2= ARq(#) ej'q(#)Products in original units are sums in decibels# zeros = 1 # poles = 2 58Bode Plot Portrays Response to Sinusoidal Control Input# zeros = 1 # poles = 2 59Plot AR(dB) vs. log10("input)Plot phase angle, #(deg) vs. log10("input)Asymptotes form skeleton of response amplitude ratioConstant Gain Bode Plot H( j") =1 H( j") =10 H( j") =100y t ( ) = hu t ( )Slope = 0dB / dec, Amplitude Ratio = constantPhase Angle = 060Integrator Bode Plot H( j") = 1j" H( j") = 10j"y t ( ) = hu t ( )dt0t!Slope = !20dB / decPhase Angle = !9061Differentiator Bode PlotH( j!) = j! H( j") =10 j"y t ( ) = h du t ( )dtSlope =+20dB / decPhase Angle =+9062Sign ChangeH( j!) = " hj!y t( ) = !h u t( )dt0t"H( j!) = "j!y t( ) = !hdu t( )dtSlope = !20dB / decPhase Angle =+90Slope =+20dB / decPhase Angle = !90Integral Derivative 63Multiple Integrators and DifferentiatorsH( j!) = h j!( )2y t( ) = hd2u t( )dt2H( j!) =hj!( )2y t( ) = h u t( )dt20t!0t!Slope = !40dB / decPhase Angle = !180Slope =+40dB / decPhase Angle =+180Double Integral Double Derivative 64Why Plot Vertical Lines where " = z and "n? 65AR Asymptotes change at frequencies corresponding topoles and zeros!q( j")!# E( j") =kqj" $ zq( )$"2+ 2%SP"nSP j" +"nSP2When ! = !nSP, "!nSP2+ 2#SPj!nSP2+!nSP2= j2#SP!nSP2=1j2#SP!nSP2=" j2#SP!nSP2=12#SP!nSP2e90 for positive #SPWhen ! = "zqfor negative zq( ),kqj! " zq( ) = kqzq " j "1 ( ) = "kqzqj +1 ( ) = kq zq e+45 Bode Plots of First-Order LagsHred( j!) =10j!+10( )H blue( j!) =100j!+10( )Hgreen( j!) =100j!+100( )66Bode Plot Asymptotes, Departures, and Phase Angles for First-Order Lags General shape of amplitude ratio governed by asymptotes Slope of asymptotes changes by multiples of 20 dB/dec at poles or zeros Actual AR departs from asymptotes Phase angle of a real, negative pole When " = 0, # = 0 When " = !, # =45 When # -> $, # -> 90 AR asymptotes of a real pole When " = 0, slope = 0 dB/dec When " % !, slope = 20 dB/dec 67Bode Plots of Second-Order Lags (No Zeros)Effect of Damping RatioHgreen( j!) =102j! ( )2+ 2 0.1 ( ) 10 ( ) j! ( ) +102Hblue( j!) =102j! ( )2+2 0.4 ( ) 10 ( ) j! ( ) +102Hred( j!) =102j! ( )2+2 0.707 ( ) 10 ( ) j! ( ) +10268Bode Plots of Second-Order Lags (No Zeros)Hred( j!) =102j! ( )2+ 2 0.1 ( ) 10 ( ) j! ( ) +102Effects of Gain and Natural FrequencyHgreen( j!) =103j! ( )2+2 0.1 ( ) 10 ( ) j! ( ) +102Hblue( j!) =1002j! ( )2+ 2 0.1 ( ) 100 ( ) j! ( ) +100269 AR asymptotes of a pair of complex poles When " = 0, slope = 0 dB/dec When " % "n, slope = 40 dB/dec Height of resonant peak depends on damping ratio 70Amplitude Ratio Asymptotes and Departures of Second-Order Bode Plots (No Zeros)Phase Angles of Second-Order Bode Plots (No Zeros) Phase angle of a pair of complex negative poles When " = 0, # = 0 When " = "n, # =90 When " -> $, # -> 180 Abruptness of phase shift depends on damping ratio 71MATLAB Bode Plot with asymp.mhttp://www.mathworks.com/matlabcentral/ http://www.mathworks.com/matlabcentral/leexchange/10183-bode-plot-with-asymptotes 2nd-Order Pitch Rate Frequency Response asymp.mbode.m 72Constant Gain, Integrator, and Differentiator Bode Plots Form Asymptotes for More Complex Transfer Functions+20dB/dec +40dB/dec 0dB/dec +20dB/dec 20dB/dec 73FrequencyResponse ARDepartures in theVicinity of Poles Difference between actual amplitude ratio (dB) and asymptote = departure (dB) Results for multiple rootsare additive Zero departures have opposite sign First- and Second-Order Departures from Amplitude Ratio Asymptotes74 McRuer, Ashkenas, and Graham, Aircraft Dynamics and Automatic Control, Princeton University Press, 1973First- and Second-Order Phase AnglesPhase Angle Variations in the Vicinity of Poles Results for multiple roots are additive LHP zero variations have opposite sign RHP zeros have same sign 75 McRuer, Ashkenas, and Graham, Aircraft Dynamics and Automatic ControlCurtiss Autocar, 1917 Waterman Aerobile, 1935ConsolidatedVultee 111, 1940sStout Skycar, 193176Hallock Road Wing , 1957ConvAIRCAR 116 (w/Crosley auto), 1940sTaylor AirCar, 1950s77Mitzar SkyMaster Pinto, 1970sHaynes Skyblazer, concept, 2004Lotus Elise Aerocar, concept, 200278Aeromobil, 2014Terrafugia Transition or, for the same priceTerrafugia TF-X, conceptPLUSCessna Skycatcher 162 Jaguar F Type79Next Time:Root Locus AnalysisReading:Flight Dynamics357-361, 465-467, 488-490, 509-51480Supplementary Material81Longitudinal Modes of Motion Eigenvalues determine the damping and natural frequencies of the linear systems modes of motion !ran : range mode " 0!hgt : height mode " 0#P,$nP( ) : phugoid mode#SP,$nSP( ) : short - period mode Longitudinal characteristic equation has 6 eigenvalues 4 eigenvalues normally appear as 2 complex pairs Range and height modes usually inconsequential 82Short-Period Mode AirspeedFlight Path Angle Pitch RateAngle of Attack Note change in time scale Simplied Longitudinal Modes of Motion83Lateral-Directional Modes of Motion Lateral-directional characteristic equation has 6 eigenvalues 2 eigenvalues normally appear as a complex pair Crossrange and heading modes usually inconsequential !cr : crossrange mode " 0!head : heading mode " 0!S : spiral mode!R : roll mode#DR,$nDR( ) : Dutch roll mode84Simplied Lateral Modes of MotionDutch-Roll Mode Yaw Rate Sideslip Angle 85Roll and Spiral Modes Roll RateRoll Angle Simplied Lateral Modes of Motion86Bode Plots of 1st- and 2nd-Order Lags Hred ( j") =10j" +10 ( )Hblue( j") =1002j" ( )2+ 2 0.1 ( ) 100 ( ) j" ( ) +100287Bode Plots of 3rd-Order Lags Hblue( j") =10j" +10 ( )# $ % & ' ( 1002j" ( )2+ 2 0.1 ( ) 100 ( ) j" ( ) +1002# $ % % & ' ( ( Hgreen( j") =102j" ( )2+ 2 0.1 ( ) 10 ( ) j" ( ) +102# $ % % & ' ( ( 100j" +100 ( )# $ % & ' ( 88Bode Plot of a 4th-Order System with No ZerosH( j!) =12j! ( )2+ 2 0.05 ( ) 1 ( ) j! ( ) +12"#$$%&''1002j! ( )2+ 2 0.1 ( ) 100 ( ) j! ( ) +1002"#$$%&'' Resonant peaks and large phase shifts at each natural frequency Additive AR slope shifts at each natural frequency # zeros = 0# poles = 489Left-Half-Plane Transfer Function ZeroH( j!) = j! + 10 ( )Zeros are numerator singularities H(j!) =kj! " z1( ) j! " z2( )...j! " #1( ) j! " #2( )... j! " #n( ) Single zero in left half plane Introduces a +20 dB/dec slope Produces phase lead in vicinity of zero 90Right-Half-Plane Transfer Function ZeroH( j!) = " j! "10 ( ) Single zero in right half plane Introduces a +20 dB/dec slope Produces phase lag in vicinity of zero 91Second-Order Transfer Function Zero H( j") =j" # z( ) j" # z*( )= j"( )2+ 2 0.1( ) 100( ) j"( )+1002[ ] Complex pair of zeros produces an amplitude ratio notch at its natural frequency 924th-Order Transfer Function with 2nd-Order Zero H( j") =j"( )2+ 2 0.1( ) 10( ) j"( ) +102[ ]j"( )2+ 2 0.05( ) 1( ) j"( ) +12[ ]j"( )2+ 2 0.1( ) 100( ) j"( ) +1002[ ]93Elevator-to-Normal-Velocity Frequency Response !w(s)!"E(s) = n"Ew(s)!Lon(s) #M"E s2+ 2$%ns +%n2( )Approx Ph s & z3( )s2+ 2$%ns +%n2( )Ph s2+ 2$%ns +%n2( )SP0 dB/dec +40 dB/dec 0 dB/dec 40 dB/dec 20 dB/dec (n q) = 1 Complex zeroalmost (but not quite) cancels phugoid response ShortPeriod Phugoid 94