Alternative Expressions for the Velocity Vector Velocity restricted to the vertical plane. Longitudinal Equations of Motion

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1 Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flig Dynamics MAE 33, 008 Separate solutions for nominal and perturbation flig paths Assume that nominal path is steady and in the vertical plane Therefore, perturbation model is linear and time-invariant LTI x t = 0 = f[x t,u t,w t,t] x Lon t = F Lon x Lon t G Lon u Lon t L Lon w Lon t Copyrig 008 by Robert Stengel All rigs reserved For educational use only tp://wwwprincetonedu/~stengel/mae33ml tp://wwwprincetonedu/~stengel/fligdynamicsml onlinear Dynamic Equations u = X /m gsin qw w = Z /m gcos qu x I = cos u sin w z I = sinu cos w q = M /I yy = q onlinear Dynamic Equations, neglecting range and altitude u = f = X /m gsin qw w = f = Z /m gcos qu = M /I yy = f 4 = q Longitudinal Equations of Motion Symmetric aircraft Motions in the vertical plane Flat earth State Vector, 6 components x x x 3 x 4 x 5 x 6 x 4 u Axial Velocity w Vertical Velocity x Range = x Lon6 = = z Altitude q Pitch Rate Pitch Angle State Vector, 4 components x u Axial Velocity x w = x x 3 Lon4 = = Vertical Velocity q Pitch Rate Pitch Angle Velocity vector can be expressed in Cartesian components Inertial frame v x, v z Body frame u, w Polar coordinates V, Referenced to inertial axes Referenced to body axes V, Alternative Expressions for the Velocity Vector Velocity restricted to the vertical plane v x = V cos V sin v z = u = V cos w V sin V v = x v z sin v z /V V u w = sin w /V Replace Cartesian body components of velocity by polar inertial components Replace X and Z by T, D, and L u = f = X /m gsin qw w = f = Z /m gcos qu = M /I yy = f 4 = q x u Axial Velocity x w = = Vertical Velocity x 3 q Pitch Rate x 4 Pitch Angle V = f = m T cos i D mgsin = f = mv T sin i L mgcos = M /I yy = f 4 = q x V Velocity x = = Flig Path Angle x 3 q Pitch Rate x 4 Pitch Angle i = Incidence angle of the thrust vector with respect to the centerline Hybrid Longitudinal Equations of Motion Replace pitch angle by angle of attack = V = f = m T cos i D mgsin = f = mv T sin i L mgcos = M /I yy = f 4 = = q mv T sin i L mgcos x V Velocity x = = Flig Path Angle x 3 q Pitch Rate x 4 Angle of Attack

2 Why Transform Equations and State Vector? ominal Equations of Motion Phugoid long-period mode is primarily described by velocity and flig path angle Short-period mode is primarily described by pitch rate and angle of attack Hybrid linearized equations allow the two modes to be studied separately Effects of phugoid perturbations on phugoid motion F Lon = F Ph SP F Ph Effects of phugoid perturbations on short-period motion F SP Ph F SP = F small Ph F Ph 0 small F SP 0 F SP Effects of short-period perturbations on phugoid motion Effects of short-period perturbations on short-period motion x t = 0 = f[x t,u t,w t,t] V = 0 = f = m T cos i [ D mgsin ] = 0 = f = mv T sin i q = 0 = f 3 = M I yy [ L mgcos ] = 0 = f 4 = q mv T sin i x V Velocity x = = Flig Path Angle x 3 q Pitch Rate x 4 Angle of Attack [ L mgcos ] E Elevator Setting u = T = Thrust Setting F Flap Setting dimx = 4 dimu = 3 w = V wind Parallel Wind = wind Perpendicular Wind dimw = Perturbation Equations of Motion on-dimensional Stability Derivative otation x Lon t = F Lon x Lon t G Lon u Lon t L Lon w Lon t, f f f f V q V t f f f f t V q = - q f 3 f 3 f 3 f 3 t V q f 4 f 4 f 4 f 4 V q / x=x t u=u t w=w t 0, f V t 3E f t - 3E qt f 3 t 3E f 4 3E / f 3T f 3T f 3 3T f 4 3T f 3F f 3F f 3 3F f 4 3F x=x t u=u t w=w t 0, f f V wind wind 3Et f f V 3Tt wind wind - 3Ft f 3 f 3 V wind wind f 4 f 4 V wind wind / dimx = 4 dimu = 3 dimw = x=x t u=u t w=w t V wind wind 0 C TV C DV C LV C mv C T V C D V C L V C m V C Dq C Lq C mq C D q C L q C m q C D C L C m C D C L C m

3 Velocity Dynamics onlinear equation V = f = T cos D mgsin m V = m C T V cos C DV = m C T cos V First rows of sensitivity matrices F, G, and L V, = m mgcos, q = m C D q = gcos, S V = m C T sin C D V S C T cos C D V S S S C V D S mgsin E = m C V D E S T = m C V T T cos, S F = m C V D F S = V wind V Thrust incidence angle neglected wind = Flig Path Angle Dynamics onlinear equation = f = mv [ T sin L mgcos ] = mv C T sin V S C V L S mgcos Second rows of sensitivity matrices F, G, and L f V = C TV sin C LV V mv S C T sin C L V S C T sin C L V S mgcos, mv f, = [ mgsin, ] = gsin, V mv f q = V C Lq mv S f = C T cos C L V mv S f E = V C LE mv S f T = V C TT sin mv S f F = V C LF mv S f = f V wind V f wind = f Pitch Rate Dynamics onlinear equation Angle of Attack Dynamics onlinear equation = M = C m V Sc I yy Third rows of sensitivity matrices F, G, and L V = V C mv I yy = 0 q = V C mq I yy Sc = V C m I yy Sc I yy Sc C m V Sc C m may include thrust as well as aerodynamic effects E = V C me I yy T = F = Sc V C mt Sc I yy V C mf Sc I yy = V wind V wind = = f 4 = = q mv Fourth rows of sensitivity matrices F, G, and L V = f V = f q = f q = f [ T sin L mgcos ] E = f E T = f T F = f F = f V wind V wind = f

4 Lockheed YF-A Velocity-Dependent Derivative Definitions Pitch-Moment Coefficient Sensitivity to Angle of Attack Mach number effects are a principal source of velocity dependence C D M C D M = C D V /a C DV C LV C mv = a C D V C D V = C a D M C L V = C a L M C m V = C a m M C D M 0 a = Speed of Sound M = Mach number C D M > 0 C D M < 0 For small angle of attack and no control deflection M B = C m q Sc C m o C mq q C m q Sc c x cm x cpwing x cm x cp l = C Lwing C L = C wing l Lwing C L c c c c x C m C net h cm h cpnet C Lnet h cm h cpnet = C cm x cpnet Lnet = C mwing C m referenced to wing area, S Horizontal Tail Lift Sensitivity to Angle of Attack C L = V tail aircraft V V Tail = Airspeed at vertical tail scrubbing lowers V Tail, propeller slipstream increases V Tail S, elas C S L! = Wing downwash angle at the tail elas = Aeroelastic correction factor Pitch-Rate Derivative Definitions Pitch rate derivatives are usually expressed in terms of a normalized pitch rate q ˆ = qc V C m ˆ q = C m q ˆ = C m qc = V C c mq V C mq M B = C m q Sc C m o C mq q C m q Sc = C m q = c C m V q ˆ often tabulated used in pitch-rate equation M q = C m q V c Sc = C mq ˆ V V V Sc = C Sc m q ˆ 4

5 Angle of Attack Distribution Due to Pitch Rate Horizontal Tail Lift Due to Pitch Rate Aircraft pitching at a constant rate, q rad/s, produces a normal velocity distribution along x w = qx Corresponding angle of attack distribution = w = qx V Angle of attack perturbation at tail center of pressure = ql V V l = horizontal tail distance from cm Incremental tail lift due to pitch rate, referenced to tail area, S L = C L V S Incremental tail lift coefficient due to pitch rate, referenced to aircraft reference wing area, S S C L = C aircraft L = C L - 0 S,- aircraft / 0 = C L Lift coefficient sensitivity to pitch rate referenced to wing area aircraft q C L C Lq = C L aircraft l V aircraft ql V Differential pitch moment due to pitch rate M q C mq = C mq V Sc = C Lq - = C L / /, = C L l V aircraft l V Moment Coefficient Sensitivity to Pitch Rate of the Horizontal Tail l V V Sc 0 l c V Sc Coefficient derivative with respect to pitch rate l = C L l c c c Coefficient derivative with respect to normalized pitch rate is insensitive to velocity C mq ˆ = C m C = m ˆ q qc = C L l q ˆ = qc c V V V Wing Lift and Moment Coefficient Sensitivity to Pitch Rate Straig-wing incompressible flow estimate Etkin C Lq ˆ = C Lwing h cm 075 wing C mq ˆ = C Lwing h cm 05 wing Straig-wing supersonic flow estimate Etkin C Lq ˆ = C Lwing h cm 05 wing C mq ˆ = wing 3 M C L wing h cm 05 Triangular-wing estimate Bryson, ielsen C Lq ˆ wing = 3 C L wing C m ˆ q wing = 3AR

6 Dimensional Stability-Derivative otation Dimensional stability derivatives portray acceleration sensitivities to state perturbations Redefine force and moment symbols for simplicity D m D V = m C T V cos C DV V f = C T cos C L mv V = V C m I yy Sc M L m L S C T cos C D V S, D V S L V M I yy M Thrust and lift effects are combined and represented by one symbol Thrust and drag effects are combined and represented by one symbol Dimensional Stability-Derivative otation Stability derivatives portray acceleration sensitivities to state perturbations V D V f V L V V V L V V = gcos f = g sin V q D q f q L q V D V M V = 0 q M q M = g sin V q L q V f L V L V F Lon = F Ph SP F Ph F SP Ph F SP Linear, Time Invariant Stability Matrix Effects of phugoid perturbations on phugoid motion D V gcos D q D L V g L V sin q L V V V = M V 0 M q M L V V g sin L q - V V 0 L, / V Effects of phugoid perturbations on short-period motion Effects of short-period perturbations on phugoid motion Effects of short-period perturbations on shortperiod motion Control- and Disturbance- Effect Matrices Control-effect derivatives portray acceleration sensitivities to control input perturbations D E T T D F L G Lon = E /V L T /V L F /V M E M T M F L E /V L T /V L F /V Disturbance-effect derivatives portray acceleration sensitivities to disturbance input perturbations D Vwind D wind L Vwind /V L wind /V L Lon = M Vwind M wind L Vwind /V L wind /V

7 Second-Order Models of Longitudinal Motion Initial-Condition Responses of Business Jet at Two Time Scales Assume off-diagonal blocks are negligible Approximate LTI Phugoid Equation F Lon = F Ph ~ 0 ~ 0 F SP Same four 4 th -order responses viewed over different periods of time 0-00 sec Reveals Phugoid Mode 0-6 sec Reveals Short-Period Mode x Ph = V D V gcos L V g V V sin V Approximate LTI Short-Period Equation x SP = q M q M, L / q - V L q 0 V T,T L,T V M E L E V D,T V L V V V wind M E L V wind Comparison of Fourth- and Second-Order Model Responses Fourth Order Full and approximate linear models Phugoid Time Scale Short-Period Time Scale Comparison of Fourthand Second-Order Models and Eigenvalues Fourth-Order Model F = G = Eigenvalue Freq rad/s e-03 4e-0j 678E-0 4E e-03-4e-0j 678E-0 4E e00 83e00j 4E-0 30E e00-83e00j 4E-0 30E00 Phugoid Approximation F = G = Eigenvalue Freq rad/s e-03 36e-0j 678E-0 37E e-03-36e-0j 678E-0 37E-0 Short-Period Approximation F = G = Eigenvalue Freq rad/s Second Order e00 83e00j 4E-0 30E e00-83e00j 4E-0 30E00 Approximations are very close to 4 th -order values because natural frequencies are widely separated

8 Approximate Phugoid Roots Approximate Phugoid Equation = 0 Characteristic polynomial x Ph = V D V g V L V 0 V T,T L,T V,T Effect of Airspeed and L/D on Approximate Phugoid atural Frequency and n g V 387 /V m /s T = / n 045V sec L /D eglecting compressibility effects si F Ph atural frequency and damping ratio n = gl V /V = D V glv /V = det si F Ph s = s D V s gl V /V = s n s n eglecting compressibility effects n g V 387/V m /s T = / n 045V sec L /D Velocity atural Frequency Period L/D m/s rad/s sec Approximate Phugoid Response to a 0 Thrust Increase What is the steady-state response? Approximate Short- Period Roots Approximate Short-Period Equation L q = 0 x SP = q M q M L q V Characteristic polynomial M,E L,E V,E s = s L L M q s M M q V = s, n s, n atural frequency and damping ratio L L n = M M M q V q = V L M M q V V

9 Effects of Airspeed, Altitude, Mass, and Moment of Inertia on Approximate atural Frequency and of a Figer Aircraft Approximate Short-Period Response to a 0-Rad Pitch Control Step Input Airspeed variation at constant altitude Airspeed Dynamic Pressure Angle of Attack atural Frequency Period m/s P deg rad/s sec Mass variation at constant altitude Mass Variation atural Frequency Period rad/s sec Pitch Rate, rad/s Angle of Attack, rad Altitude variation with constant dynamic pressure Moment of inertia variation at constant altitude Airspeed Altitude atural Frequency Period m/s m rad/s sec Moment of Inertia Variation atural Frequency Period rad/s sec Approximate Short-Period Response to a 0-Rad Pitch Control Step Input ormal load factor at the center of mass n z = V g q = V ormal Load Factor, g!s at cm Aft Pitch Control L L E E g V V ormal Load Factor, g!s at cm Forward Pitch Control ext Time: Lateral-Directional Dynamics What are the similarities and differences?