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Section 3.4 Static and Dynamic Stability, Longitudinal Pitch Dynamics

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Section 3.4 Static and Dynamic Stability, Longitudinal Pitch Dynamics Section 3.4 Static and Dynamic Stability, Longitudinal Pitch Dynamics MAE 6530, Propulsion Systems II 1 Static Vs. Dynamic Stability Static Stability Dynamic Stability Dynamic Stability MAE 6530, Propulsion Systems II 2 Static Versus Dynamic Stability (2) Static Instability Statically Stable, Dynamically Unstable Static and Dynamically Stable MAE 6530, Propulsion Systems II 3 Airframe Static Stability 4 MAE 6530, Propulsion Systems II Airframe Dynamic Stability MAE 6530, Propulsion Systems II 5 Center of Pressure •Aerodynamic Lift, Drag, and Pitching Moment Can be though of acting at a single point … the Center of Pressure (CP) of the vehicle •Sometimes (and not quite correctly) referred to as the Aerodynamic Center (AC) •For our purposes applied to an axisymmetric rocket Pitching Moment configuration, AC and CP are synonymous MAE 6530, Propulsion Systems II 6 Center of Pressure MAE 6530, Propulsion Systems II 7 Flight Vehicle Static Stability • If center of gravity (cg) is forward of the Cp, vehicle responds to a disturbance by producing aerodynamic moment that returns Angle of attack of vehicle towards angle that existed prior to the disturbance. (static stability) • If cg is behind the center of pressure, vehicle will respond to a disturbance by producing an aerodynamic moment that continues to drive angle of attack further away from starting position. (static instability) MAE 6530, Propulsion Systems II 8 Static Stability, Rocket Flight Example • During flight small wind gusts or thrust offsets cause the rocket to "wobble” … change attitude • Rocket rotates about center of gravity (cg) • Lift and drag both act through center of pressure (Cp) • When cp is behind cg, aerodynamic forces provide a “restoring force” … rocket is said to be “statically stable” • When Cp ahead of cg, aerodynamic forces provide a “destabilizing force” … rocket is said to be “unstable” • Condition for a statically for a stable rocket is that center of pressure must be located behind longitudinal center of gravity. MAE 6530, Propulsion Systems II 9 Static Stability, Rocket Flight Example (2) MAE10 6530, Propulsion Systems II Static Stability, Rocket Flight Example (3) MAE11 6530, Propulsion Systems II Weather Vane Analogy of Static Stability MAE12 6530, Propulsion Systems II Static Margin and Pitching Moment • Static margin used to characterize static stability and controllability of aircraft and missiles. • For aircraft systems … Static margin defined as non- dimensional distance between center of gravity (cg) and aerodynamic center (ac) of the aircraft. • For missile systems … Static margin defined as non- dimensional distance between center of gravity (cg) and the center of pressure (Cp). • Static Stability requires that the pitching moment Cm about the rotation point, become negative as we increase C : L 13 MAE 6530, Propulsion Systems II Pitching Moment Analysis α Cm0 à pitching moment about Cp Cm(a) à pitching moment about cg α cref = “chord” reference length C m 0 Sum Moments abut cg ⎛ ⎞ X cg − XCp ⎟ C C ⎜ ⎟ C cos C sin m (α)= m +⎜ ⎟⋅( L ⋅ α+ D ⋅ α) 0 ⎜ c ⎟ ⎝ creegf ⎠ ⎛ ⎞ ⎜ XCp − X cg ⎟ → Define ... X =⎜ ⎟ → Cm (α)= Cm −Xsm ⋅(CL ⋅cosα+CD ⋅sinα) sm ⎜ ⎟ 0 ⎝⎜ creg ⎠⎟ cref → Linearize Pitching Moment Equation ∂Cm ... Cm (α)= Cm + ⋅α 0 ∂α 14 MAE 6530, Propulsion Systems II Pitching Moment Analysis (2) ∂C ⎛∂C ∂C ⎞ → Define ... C = m = −X ⋅⎜ L ⋅cos −C ⋅sin + D ⋅sin +C ⋅sin ⎟= mα sm ⎜ α L α α CDD ⋅cosαα⎟ ∂α ⎝⎜ ∂α ∂α ⎠⎟ ⎡ ⎛∂C ⎞ ⎛ ∂C ⎞ ⎤ −X ⋅⎢ ⎜ L +C ⎟⋅cosα−⎜C − D ⎟⋅sinα ⎥ sm ⎢ ⎜ D ⎟ ⎜ L ⎟ ⎥ ⎣⎢ ⎝ ∂α ⎠ ⎝ ∂α ⎠ ⎦⎥ ∂C∂∂CC ⎛⎛⎛∂⎛⎛⎛C∂∂CC ⎞⎞⎞ ⎛⎛ ⎛ ∂C∂C⎞⎞ ⎞ ⎞⎞ ⎞ →→Sm→SamlSlam lla lalp ap praoppxrpiormxoiaxmitmiaoatnitoi o.n.n. .. m =mm =−=X−X⋅⎜⎜⋅⋅⎜⎜⎜⎜ L L+L++CCC⎟⎟⋅⋅⎟⋅−−−⎜⎜CC⎜C−−− DD⎟⎟D⋅⋅⎟⋅22⎟⎟2 ⎟ α αα sm ssmm⎜⎜⎜⎜⎜⎜ DD⎟D⎟α⎟αα⎜⎜ ⎜LL L ⎟⎟ α⎟αα⎟⎟ ⎟ ∂α∂∂αα ⎝⎜⎝⎜⎝⎜⎝⎜⎝⎜∂⎝⎜α∂∂αα ⎠⎟⎠⎟⎠⎟ ⎝⎜⎝⎜ ⎝⎜ ∂αα∂α⎠⎟⎠⎟ ⎠⎟ ⎠⎟⎠⎟ ⎠⎟ ⎛⎛⎛ ⎞⎞ ⎛ ⎞ ⎞ 2 ∂Cm ∂∂CCLL ⎟⎟ ∂CD ⎟ 22⎟ →→SmNaelgl lαec at pαpr otexrimma .t..i oCn ...α = C=−XX ⋅⋅⎜⎜⎜ ++CC ⎟⎟⋅⋅αα−⎜C − ⎟⋅α ⎟ m ( ) m0 ssmm ⎜ DD⎟⎟ L ⎟ ⎟ ∂α ⎝⎜⎝⎜⎝⎜∂∂αα ⎠⎟⎠⎟ ⎝⎜ ∂α ⎠⎟ ⎠⎟ ∂C ⎛∂C ⎞ m ⎜ L ⎟ → Cm = = −Xsm ⋅⎜ +CD ⎟ α ∂α ⎝⎜ ∂α ⎠⎟ 15 MAE 6530, Propulsion Systems II Pitching Moment Analysis (3) Linear Airfoil Theory “Stall point” ∂C ∂C L C C L “Linear region” CL = CL + ⋅α ≡ L + L + ⋅α 0 ∂α 0 0 ∂α C 2 C = C + L D D0 π⋅ε⋅ Ar ⎡ ⎤ ⎢ ∂CL ⎥ ⎢ ⎥ → ∂α > 0 ⎢ ⎥ CL0 ⎢ C ⎥ ⎣⎢ D ⎦⎥ ∂C m → "C ".....Ideally... ∂α m α ∂C ⎡ L ⎤ static static ⎛ ∂CL ⎞ ⎢ ⎥ Cm = −Xsm ⎜ + CD ⎟ → ∂α > 0 → Xsm > 0... → Cm < 0... α ⎝ ∂α ⎠ ⎢ ⎥ stability α stability C ⎣ D ⎦ 16 MAE 6530, Propulsion Systems II Pitching Moment Analysis (4) For a Rocket Static margin is the distance between the CG and the CP; divided by body tube diameter. ∂C m → "C ".....Ideally... ∂α m α ∂C ⎡ L ⎤ static static ⎛ ∂CL ⎞ ⎢ ⎥ Cm = −Xsm ⎜ + CD ⎟ → ∂α > 0 → Xsm > 0... → Cm < 0... α ⎝ ∂α ⎠ ⎢ ⎥ stability α stability ⎣ CD ⎦ 17 17 MAE 6530, Propulsion Systems II Pitching Moment Analysis (6) “Pike” stable unstable stable • Even Xsm > 0 (static stability) rockets become unstable at higher angles of attack C • “Strong stability” region limited to mα very low angle of attack range MAE 6530, Propulsion Systems II α,deg. 18 Pitching Moment Analysis (5) MAE 6530, Propulsion Systems II 19 Achieving Static Stability 20 20 MAE 6530, Propulsion Systems II Static Margin = Degree of Static Stability 21 21 MAE 6530, Propulsion Systems II How Much Static Stability? 22 22 MAE 6530, Propulsion Systems II How Much Static Stability? (2) 23 23 MAE 6530, Propulsion Systems II Calculating the Static Margin •Key to calculating static margin is estimate of location of longitudinal center of pressure at low angles of attack •Barrowman equations provide simple, accurate technique for Axi-symmetric rockets •cg is measured as the longitudinal balance point of the rocket. •As a rule of thumb, Cp distance should be aft of the cg by at least one rocket diameter. -- "One Caliber stability”. MAE 6530, Propulsion Systems II 24 Calculating the Static Margin (2) N -- total normal FORCE n -- sectional normal pressure differential … + transitions + boat tail MAE 6530, Propulsion Systems II 25 Calculating the Static Margin (3) N + N + N + N +... = nose tail trns boat = … + q ⋅ Aref C +C +C +C +... Nnose Ntail Ntrns Nboat … + transitions + boat tail C +C +C +C +... ⋅ ( Nα Nα Nα Nα ) α nose tail trns boat of Nose, Fin &Transition Geometry MAE 6530, Propulsion Systems II 26 Calculating the Static Margin (4) maximum body diameter max max MAE 6530, Propulsion Systems II 27 Calculating the Static Margin (3) XN = center of pressure location for nose section XT = center of pressure location for transitions a (CNa)N = normal force derivative for nose (CNa)T = normal force derivative for transition a max max MAE 6530, Propulsion Systems II 28 Calculating the Static Margin (4) XF = center of pressure location for fin groups (CNa)F = Normal force derivative for fin group max a Static Margin (Xsm) = (Xcp – Xcg)/dmax a a a a (CNa)R = total normal force derivative X – measured aft from a a nose of vehicle Xcp a a Small α → C ≅ C Nα Lα MAE 6530, Propulsion Systems II 29 Example: Stable Static Margin Vehicle 20 cm Ogive max 10 cm Nose 10 cm 8 cm max 5 cm 30 cm 10 cm 5 cm 10 cm 12 cm 4 cm 16 cm 50 cm 3 30 MAE 6530, Propulsion Systems II Example: Unstable Static Margin Calculation 12.5 cm Constant diameter tube Ogive 5.54 cm max (No transition section) Nose 5.54 cm 5.54 cm max 0 cm 12.5 cm 10 cm 0 cm 5.25 cm 6.5 cm 2.77 cm 9 cm 27 cm 3 31 MAE 6530, Propulsion Systems II Rotational Dynamics of a Rigid Body. 32 MAE 6530, Propulsion Systems II p q r Simplified Pitch Axis Dynamics General Rotational Dynamics ⎡ . ⎤ p ⎛ 2 2 ⎞ ⎡ I I I ⎤ ⎢ ⎥ q ⋅r I y − Iz + q − r I yz + p ⋅q Ixz − r ⋅ p Ixy ⎡ ⎤ x − xy − xz ( ) ( ) ( ) ( ) M x ⎢ ⎥ ⎢ . ⎥ ⎜ ⎟ ⎢ ⎥ ⎢ −I I −I ⎥⋅ ⎢ q ⎥ = ⎜ r ⋅ p I − I + r 2 − p2 I + q ⋅r I − p ⋅q I ⎟ + ⎢ M ⎥ xy y yz ⎜ ( z x ) ( ) xz ( xy ) ( yz ) ⎟ y ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ I I I ⎥ ⎢ ⎥ ⎜ 2 2 ⎟ M − xz − yz z r ⎜ p ⋅q I − I + p − q I + r ⋅ p I − q ⋅r I ⎟ ⎢ z ⎥ ⎣ ⎦ ⎢ ⎥ ⎝ ( x y ) ( ) xy ( yz ) ( xz ) ⎠ ⎣ ⎦ ⎣ ⎦ . .. M I I y ( z − x ) → q = θ = + r ⋅ p Neglect Cross Products of Inertia I y I y Forcing Second order Disturbance torque moment (neglected when r, p are small ) 33 MAE 6530, Propulsion Systems II Collected Equations ⎡ ⎤ ⎡ ⎤ I I q r I q2 r 2 I p q I p r ⎡ ⎤ I −I −I ⎡ ⎤ ⎢ ( yy − zz )⋅ ⋅ + yz ⋅( − )+ xz ⋅ ⋅ − xy ⋅ ⋅ ⎥ M ⎢ xx xy xz ⎥ ⎢ p! ⎥ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ −Ixy I yy −I yz ⋅⎢ q! ⎥ = ⎢ (Izz − Ixx )⋅ p⋅r + Ixz ⋅(r − p )+ Ixy ⋅q⋅r − I yz ⋅ p⋅q ⎥ + Ly ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −I −I I ⎥ ⎢ r! ⎥ ⎢ 2 2 ⎥ N xz yz zz ⎣⎢ ⎦⎥ ⎢ (Ixx − I yy )⋅ p⋅q + Ixy ⋅( p −q )+ I yz ⋅ p⋅r − Ixz ⋅q⋅r ⎥ ⎢ z ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Principal Axes ⎡ ⎤ ⎡ ' ⎤ ⎢ Ixx −Ixy −Ixz ⎥ ⎢ I 0 0 ⎥ ⎢ ⎥ ⎢ xx ⎥ I ≈ Real,symmetrix → Find Axis Where → ⎢ −I I −I ⎥ =U T ⋅Γ⋅U → Γ = ⎢ 0 I ' 0 ⎥ ⎢ xy yy yz ⎥ ⎢ yy ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −I −I I ⎥ ⎢ 0 0 I ' ⎥ ⎣ xz yz zz ⎦ ⎣ zz ⎦ Euler 's Equations → Ixy = Ixz = I yz = 0 → Simplify ⎡ ⎛ ⎞ ⎤ ⎢ I yy − Izz ⎟ M ⎥ p! =⎜ ⎟⋅q⋅r + x ⎢ ⎜ ⎟ ⎥ ⎡ ⎤ ⎢ ⎝⎜ Ixx ⎠⎟ Ixx ⎥ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ I ⋅ p! ⎢ (I yy − Izz )⋅q⋅r ⎥ M ⎢ xx ⎥ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎛ ⎞ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ I − I ⎟ Ly ⎥ ⎢ I q! ⎥ ⎢ I I p r ⎥ ⎢ L ⎥ q! ⎜ zz xx ⎟ p r yy ⋅ = ⎢ ( zz − xx )⋅ ⋅ ⎥ + y → ⎢ =⎜ ⎟⋅ ⋅ + ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎜ I ⎟ I ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎝ yy ⎠ yy I ⋅r! ⎢ I − I ⋅ p⋅q ⎥ N ⎢ ⎥ ⎣⎢ zz ⎦⎥ ⎢ ( xx yy ) ⎥ ⎣⎢ z ⎦⎥ ⎢ ⎛ ⎞ ⎥ ⎣ ⎦ ⎢ Ixx − I yy ⎟ N ⎥ r! =⎜ ⎟⋅ p⋅q + z ⎢ ⎜ ⎟ ⎥ ⎢ ⎝⎜ Izz ⎠⎟ Izz ⎥ ⎣⎢ ⎦⎥ Collected Euler Equations ⎡ ⎤ ⎛ I I ⎞ ⎢ ⎜ yy − zz ⎟ ⎥ ⎢ ⎜ ⎟⋅q⋅r ⎥ ⎡ M ⎤ ⎜ ⎟ ⎢ x ⎥ ⎢ ⎝⎜ Ixx ⎠⎟ ⎥ ⎢ ⎥ ⎢ I ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ xx ⎥ ⎢ p! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎛ I − I ⎞ ⎥ ⎢ ⎥ ⎜ zz xx ⎟ ⎢ M ⎥ ⎢ ⎥ ⎢ ⎜ ⎟⋅ p⋅r ⎥ y ⎢ q! ⎥ ⎢ ⎜ I ⎟ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎝ yy ⎠ ⎥ ⎢ I ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ yy ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ r! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎛ I − I ⎞ ⎢ ⎥ ⎢ ⎜ xx yy ⎟ ⎥ ⎢ N ⎥ ⎢ ⎥ ⎢ ⎜ ⎟⋅ p⋅q ⎥ z ⎢ ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ ⎥ = ⎝⎜ Izz ⎠⎟ + ⎢ I ⎥ + disturbance torques ⎢ ! ⎥ ⎢ ⎥ zz ⎢ φ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ p tan sin q tan cos r ⎢ ⎥ ⎢ ! ⎥ ⎢ + θ⋅ φ⋅ + θ⋅ φ⋅ ⎥ ⎢ 0 ⎥ ⎢ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ cosφ⋅q−sinφ⋅r ⎥ ⎢ 0 ⎥ ⎢ ψ! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ⎢ sinφ cosφ ⎥ ⎢ 0 ⎥ ⎢ ⋅q + ⋅r ⎥ ⎢ ⎥ ⎢ cosθ cosθ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥ Simplified Pitch Axis Dynamics General Rotational Dynamics ⎡ .
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