Section 3.4 Static and Dynamic Stability, Longitudinal Pitch Dynamics

Home , Pitching moment

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

Forcing Second order Disturbance torque moment (neglected when r, p are small )

33 MAE 6530, Propulsion Systems II Simplified Pitch Axis Dynamics (2)

Neglecting Disturbance torques M  y θ = q = → M y ≈ " pitching moment" I y

M y " pitching moment coefficient" ≡ Cm = q ⋅ Aref ⋅ cref Check Units ⎛ 1 2 ⎞ Nt m2 m q = ⎜ ⋅ ρ ⋅V ⎟ q ⋅ A ⋅ c 2 ⋅ ⋅ ⎝ 2 ⎠ ref ref m  2  I y kg ⋅ m π 2 Aref = ⋅ Dref ⎛ kg − m 2 ⎞ 1 1 4 ⋅ m ⋅ m ⋅  ⎝⎜ 2 2 ⎠⎟ 2 2 c = D sec m kg ⋅ m sec "reference length" → creff ≈ Lrocmkaext 34 MAE 6530, Propulsion Systems II Simplified Pitch Axis Dynamics (3)

Pitching moment about cg q ⋅ Aref ⋅ cref Gravity acts at cg and q = ⋅C Cannot induce pitching moment I m y Depends on angle of attack q C α m 0 α

35 MAE 6530, Propulsion Systems II Simplified Pitch Axis Dynamics (4)

q ⋅ A ⋅ c θ = q = ref ref ⋅C I m y Depends on angle of attack + Control inputs

It can also be shown that … (NASA RP-1168 pp 10-22) that for angle of attack

q ⋅ Aref g(r) Fthrust sinα α = − ⋅CL + q + cos(θ − α ) − m ⋅V V m ⋅V See appendix I for derivation

36 MAE 6530, Propulsion Systems II Collected, Simplified Longitudinal Axis- Dynamics (5)

⎡ q ⋅ Aref g(r) Fthrust sinα ⎤ − ⋅CL + q + cos(θ − α ) − ⎢ m ⋅V V m ⋅V ⎥ ⎡α ⎤ ⎢ ⎥ ⎢ q ⋅ Aref ⋅ cref ⎥ ⎢q ⎥ = ⋅C ⎢ ⎥ ⎢ m ⎥ I y ⎣⎢θ ⎦⎥ ⎢ ⎥ ⎢ q ⎥ ⎢ ⎥ ⎣ ⎦

37 MAE 6530, Propulsion Systems II Linear Analysis of Longitudinal Axis-Dynamics Start with

⎡ ⎤ q ⋅ Aref ⋅CL + Tthrust α → small ⎢ − α 1 0 ⎥ ⎡ g ⋅cos(θ ) ⎤ ⎢ m ⋅V∞ ⎥ ⎢ ⎥ CL ≈ CL ⋅α ⎡ ⎤ ⎡ ⎤ V α α! ⎢ ⎥ α ⎢ ∞ ⎥ ⎢ ⎥ q ⋅ Aref ⋅cref q ⋅ Aref ⋅cref ⎢ ⎥ C ≈ C ⋅α + C ⋅q → q! = ⎢ ⋅C ⋅C 0 ⎥⋅ q + ⎢ 0 ⎥ m mα mq ⎢ ⎥ mα mq ⎢ ⎥ ⎢ I y I y ⎥ ⎢ ⎥ ⎢ ! ⎥ ⎢ ⎥ 0 q = θ! ⎣ θ ⎦ ⎢ ⎥ ⎣ θ ⎦ ⎢ ⎥ ⎢ 0 1 0 ⎥ ⎢ ⎥ q! = θ!! ⎣ ⎦ ⎣⎢ ⎦⎥

C → "stability derivative" mα C "damping derivative" mq → 38 MAE 6530, Propulsion Systems II Linear Analysis (2) q = θ! Let & Reorder States … q! = θ!! ⎡ ⎤ q ⋅ Aref ⋅CL + Tthrust ⎢ − α 1 0 ⎥ ⎡ g ⋅cos(θ ) ⎤ m ⋅V ⎢ ⎥ ⎡ ⎤ ⎢ ∞ ⎥ V α! ⎢ ⎥ ⎡ α ⎤ ⎢ ∞ ⎥ ⎢ ⎥ q ⋅ Aref ⋅cref q ⋅ Aref ⋅cref ⎢ ⎥ q! = ⎢ ⋅C ⋅C 0 ⎥⋅ q + ⎢ 0 ⎥ ⎢ ⎥ mα mq ⎢ ⎥ ⎢ I y I y ⎥ ⎢ ⎥ ⎢ ! ⎥ ⎢ ⎥ 0 ⎣ θ ⎦ ⎢ ⎥ ⎣ θ ⎦ ⎢ ⎥ ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎣ ⎦ ⎣⎢ ⎦⎥

⎡ ⎤ ⎡ ⎤ q ⋅ Aref ⋅CLα + Tthrust g ⋅cos(θ ) ⎢ − 0 1 ⎥ ⎢ ⎥ ⎡ α! ⎤ ⎢ m ⋅V∞ ⎥ ⎡ α ⎤ ⎢ V∞ ⎥ ⎢ ⎥ ⎢ ⎥ θ! = ⎢ 0 0 1 ⎥⋅ θ + ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ!! ⎥ q ⋅ Aref ⋅cref q ⋅ Aref ⋅cref ⎢ θ! ⎥ 0 ⎣ ⎦ ⎢ ⋅C 0 ⋅C ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ I mα I mq ⎥ ⎢ ⎥ ⎣ y y ⎦ ⎣ ⎦ 39 MAE 6530, Propulsion Systems II Linear Analysis (3) Write as X! = A⋅ X + G(t)

⎡ ⎤ ⎡ ⎤ q ⋅ Aref ⋅CLα + Tthrust g ⋅cos(θ ) ⎢ − 0 1 ⎥ ⎢ ⎥ ⎡ α! ⎤ ⎡ α ⎤ ⎡ α ⎤ ⎢ m ⋅V∞ ⎥ ⎢ V∞ ⎥ ⎢ ⎥ d ⎢ ⎥ ⎢ ⎥ θ! = θ → X = θ → A = ⎢ 0 0 1 ⎥ → G(t) = ⎢ 0 ⎥ ⎢ ⎥ dt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ!! ⎥ ⎢ θ! ⎥ ⎢ θ! ⎥ q ⋅ Aref ⋅cref q ⋅ Aref ⋅cref 0 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎢ ⋅C 0 ⋅C ⎥ ⎢ ⎥ ⎢ I mα I mq ⎥ ⎢ ⎥ ⎣ y y ⎦ ⎣ ⎦

Look at Eigenvalues of Linearized System

⎡ q ⋅ A ⋅C + T ⎤ ⎢ − ref Lα thrust − λ 0 1 ⎥ ⎢ m ⋅V∞ ⎥ Det A − λ ⋅ I = 0 → ⎢ 0 −λ 1 ⎥ = 0 [ ] ⎢ ⎥ q ⋅ A ⋅c q ⋅ A ⋅c ⎢ ref ref ⋅C 0 ref ref ⋅C − λ ⎥ ⎢ I mα I mq ⎥ ⎣ y y ⎦

40 MAE 6530, Propulsion Systems II Linear Analysis (4) Eigen Values ⎡ q ⋅ A ⋅C + T ⎤ ⎢ − ref Lα thrust − λ 0 1 ⎥ ⎢ m ⋅V∞ ⎥ Det A − λ ⋅ I = 0 → ⎢ 0 −λ 1 ⎥ = 0 [ ] ⎢ ⎥ q ⋅ A ⋅c q ⋅ A ⋅c ⎢ ref ref ⋅C 0 ref ref ⋅C − λ ⎥ ⎢ I mα I mq ⎥ ⎣ y y ⎦ ⎛ q ⋅ A ⋅C + T ⎞ ⎛ q ⋅ A ⋅c ⎞ ⎛ q ⋅ A ⋅c ⎞ − ref Lα thrust + λ ⋅ λ 2 − ref ref ⋅C ⋅λ + ref ref ⋅C ⋅λ = 0 ⎜ ⎟ ⎜ mq ⎟ ⎜ mα ⎟ ⎝ m ⋅V∞ ⎠ ⎝ I y ⎠ ⎝ I y ⎠

→ divide thru by − λ ⎛ q ⋅ A ⋅C + T ⎞ ⎛ q ⋅ A ⋅c ⎞ ⎛ q ⋅ A ⋅c ⎞ λ + ref Lα thrust ⋅ λ − ref ref ⋅C − ref ref ⋅C = 0 ⎜ ⎟ ⎜ mq ⎟ ⎜ mα ⎟ ⎝ m ⋅V∞ ⎠ ⎝ I y ⎠ ⎝ I y ⎠

41 MAE 6530, Propulsion Systems II Linear Analysis (4) → divide thru by − λ ⎛ q ⋅ A ⋅C + T ⎞ ⎛ q ⋅ A ⋅c ⎞ ⎛ q ⋅ A ⋅c ⎞ λ + ref Lα thrust ⋅ λ − ref ref ⋅C − ref ref ⋅C = 0 ⎜ ⎟ ⎜ mq ⎟ ⎜ mα ⎟ ⎝ m ⋅V∞ ⎠ ⎝ I y ⎠ ⎝ I y ⎠

Expand and Collect Terms to give Characteristic Equation for Linearized System

⎛ q ⋅ A ⋅C + T q ⋅ A ⋅c ⎞ ⎛ q ⋅ A ⋅c ⎞ ⎡⎛ q ⋅ A ⋅C + T ⎞ ⎤ λ 2 + ref Lα thrust − ref ref ⋅C ⋅λ − ref ref ⋅ ref Lα thrust ⋅C + C = 0 ⎜ mq ⎟ ⎜ ⎟ ⎢⎜ ⎟ mq mα ⎥ ⎝ m ⋅V∞ I y ⎠ ⎝ I y ⎠ ⎣⎝ m ⋅V∞ ⎠ ⎦

2 2 Form like :λ + 2 ⋅ς ⋅ω n ⋅λ +ω n = 0

⎛ q ⋅ A ⋅c ⎞ ⎡⎛ q ⋅ A ⋅C + T ⎞ ⎤ → ω = − ref ref ⋅ ref Lα thrust ⋅C + C n ⎜ ⎟ ⎢⎜ ⎟ mq mα ⎥ Natural Frequency ⎝ I y ⎠ ⎣⎝ m ⋅V∞ ⎠ ⎦

2 ⎛ q ⋅ A ⋅C + T q ⋅ A ⋅c ⎞ ref Lα thrust − ref ref ⋅C 2 ⎜ m V I mq ⎟ (2 ⋅ς ⋅ω n ) 1 ⎝ ⋅ ∞ y ⎠ 2 → ς = Damping ratio ω n 4 ⎛ q ⋅ A ⋅c ⎞ ⎡⎛ q ⋅ A ⋅C + T ⎞ ⎤ − ref ref ⋅ ref Lα thrust ⋅C + C ⎢ mq mα ⎥ ⎜ I ⎟ ⎜ m ⋅V ⎟ ⎝ y ⎠ ⎣⎝ ∞ ⎠ ⎦ 42 MAE 6530, Propulsion Systems II Estimating the Damping Derivatives

Xcg

Xcp

⎛ X − X ⎞ ⎜ cp cg ⎟ Small α → CN ≅ CL → CN = CL ⋅⎜ ⎟⋅q α α α ⎜ ⎟ ⎝⎜ V∞ ⎠⎟ ⎛ 2 ⎞ ⎛ 2 ⎞ X X ⎜ X X ⎟ ⎜ X X ⎟ ( cp − cg ) ( cp − cg ) ⎟ q ( cp − cg ) ⎟ cref C C C ⎜ ⎟ C ⎜ ⎟ q c C X 2 q m = − N ⋅ = − L ⋅⎜ ⎟⋅ = − L ⋅⎜ 2 ⎟⋅ ⋅ ref = − L ⋅( sm )⋅ ⋅ c α ⎜ V ⎟ c α ⎜ V c ⎟ α V ref ⎝⎜ ∞ ⎠⎟ ref ⎝⎜ ∞ ref ⎠⎟ ∞ c c ∂Cm 2 ref 2 ref δCm = ⋅q = Cm ⋅q = −CL ⋅(X sm )⋅ ⋅q → Cm = −CL ⋅(X sm )⋅ q α q α ∂q V∞ V∞

2 cref Cm = −CL ⋅(X sm )⋅ → pitch damping opposes motion ® i.e. always negative q α MAE 6530, PropulsionV∞ Systems II 43 Translational + rotational +mass Collected Longitudinal Equations of Motion

2 ⎡⎛ Vν ⎞ ⎛ q ⋅ Aref ⎞ ⎛ Fthrust sinθ ⎞ ⎤ ⎢⎜ ⎟ + ⎜ ⎟ ⋅(CL cosγ − CD sinγ ) + ⎜ ⎟ − g(r) ⎥ ⎢⎝ r ⎠ ⎝ m ⎠ ⎝ m ⎠ ⎥ Can we simplify this “mess”? ⎡V ⎤ ⎢ ⎥ r V ⋅V ⎛ q ⋅ Aref ⎞ F cosθ ⎢ ⎛ r ν ⎞ ⎛ thrust ⎞ ⎥ ⎢ ⎥ −⎜ ⎟ − ⎜ ⎟ ⋅(CL sinγ + CD sinγ ) + ⎜ ⎟ ⎢ ⎥ ⎢ ⎝ r ⎠ ⎝ m ⎠ ⎝ m ⎠ ⎥ ⎢V ⎥ ⎢ ⎥ ⎢ ν ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V ⎥ r µ ⎢r ⎥ ⎢ ⎥ g = ⎢ ⎥ ⎢ ⎥ (r) r2 ⎢ ⎥ ⎢ ⎥ Vν V ⎢ ⎥ ⎢ ⎥ tan−1 r ν r γ = = θ − α ⎢ ⎥ ⎢ ⎥ Vν ⎢ ⎥ ⎢ ⎥ x r  ⎢ ⎥ = ⎢ ⎥ →  = ⋅ν x Vν ⎢ ⎥ ⎢ ⎥ V = V 2 + V 2 ⎢ ⎥ ⎢ ⎥ r υ ⎢ ⎥ ⎢ ⎥ 1 q A g 2 ⎢α ⎥ ⎢ ⋅ ref (h) Fthrust sinα ⎥ q = ⋅ ρ(h) ⋅V − ⋅CL + q + cos(γ ) − 2 ⎢ ⎥ ⎢ m ⋅V V m ⋅V ⎥ ⎢ ⎥ ⎢ ⎥ h = r − Rearth q q ⋅ Aref ⋅ cref ⎢  ⎥ ⎢ ⋅C ⎥ ⎢ ⎥ ⎢ I m ⎥ ⎢ ⎥ ⎢ y ⎥ ⎢θ ⎥ ⎢ q ⎥ ⎢ ⎥ ⎢ ⎥ ⎣m ⎦ ⎢ ⎥ ⎢ Fthrust ⎥ ⎢ − ⎥ g ⋅ I ⎣ 0 sp ⎦ MAE 6530, Propulsion Systems II 44 Translational + rotational +mass Simplified Longitudinal Equations of Motion

⎡ q ⋅ Aref g(r) Fthrust sinα ⎤ − ⋅CL + q + cos(θ − α ) − ⎢ m ⋅V V m ⋅V ⎥ ⎡α ⎤ ⎢ ⎥ ⎢ q ⋅ Aref ⋅ cref ⎥ ⎢q ⎥ = ⋅C ⎢ ⎥ ⎢ m ⎥ I y ⎣⎢θ ⎦⎥ ⎢ ⎥ ⎢ q ⎥ ⎢ ⎥ ⎣ ⎦ Augment with longitudinal Acceleration Equation (See Appendix II)

T ⋅cosα q ⋅ A ⋅C V! = thrust − g ⋅sin(θ−α)− ref D ∞ m m

Complete with altitude and downrange ! h =V∞ ⋅sinγ =V∞ ⋅sin(θ−α)

x! =V∞ ⋅cosγ =V∞ ⋅cos(θ−α) MAE 6530, Propulsion Systems II 45 Translational + rotational +mass Simplified Longitudinal Equations of Motion Collected Body-Axis Equations ⎡ ⎤ ⎡ ⎤ V! ⎢ T ⋅cosα q ⋅ Aref ⋅CD ⎥ ⎢ ∞ ⎥ ⎢ thrust − g ⋅sin(θ−α)− ⎥ ⎢ ⎥ ⎢ m m ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g ⋅cos θ−α q ⋅ A ⋅C +T ⋅sinα ⎢ α! ⎥ ⎢ ( ) ref L thrust ⎥ ⎢ ⎥ ⎢ α! = q + − ⎥ ⎢ ⎥ ⎢ V∞ m⋅V∞ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ! ⎥ ⎢ q ⎥ ⎢ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ q! ⎢ q ⋅ Aref ⋅cref ⋅Cm ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ C C C ⎢ I L = L⎥(α)! N ⋅α ⎢ ⎥ ⎢ yy ⎥ α ⎢ ! ⎥ ⎢ h ⎥ ⎢ V sin θ−α ⎥ ⎢ ⎥ ⎢ ∞ ( ) ⎥ x! ⎢ Cm = Cm⎥ (α,q)≈ Cm ⋅α+Cmq ⋅q = ⎢ ⎥ α ⎢ V cos θ−α ⎥ ⎢ ⎥ ∞ ( ) ⎛ ⎞ ⎢ ⎥ 2 cref ⎟ ⎢ ⎥ X C C X ⎜C ⎟ q ⎢ − sm ⋅(⎥ N + D )⋅α− sm ⋅ N ⋅ ⎟⋅ ⎢ ⎥ T α ⎜ α V ⎟ ⎢ m! ⎥ ⎢ − thrust ⎥ ⎝ ∞ ⎠ ⎢ ⎥ ⎢ ⎥ ⎢ g0 ⋅ Isp ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ⎣ ⎦ MAE 6530, Propulsion Systems II 46 Model Comparison Body Axis with Pitch Dynamics Model One extra degree of freedom Local Vertical/Local Horizontal Ballistic Model ⎡ ⎤ ⎡ ⎤ V! ⎢ T ⋅cosα q ⋅ Aref ⋅CD ⎥ ⎢ ∞ ⎥ ⎢ thrust − g ⋅sin(θ−α)− ⎥ ⎢ ⎥ ⎢ m m ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ g ⋅cos θ−α q ⋅ A ⋅C +T ⋅sinα ⎢ α! ⎥ ⎢ ( ) ref L thrust ⎥ ⎢ ⎥ ⎢ α! = q + − ⎥ ⎢ ⎥ ⎢ V∞ m⋅V∞ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ! ⎥ ⎢ q ⎥ ⎢ θ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ q! ⎢ q ⋅ Aref ⋅cref ⋅Cm ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ I ⎥ ⎢ ⎥ ⎢ yy ⎥ ⎢ ! ⎥ ⎢ h ⎥ ⎢ V sin θ−α ⎥ ⎢ ⎥ ⎢ ∞ ( ) ⎥ ⎢ x! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ V∞ cos(θ−α) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ T ⎥ ⎢ m! ⎥ ⎢ − thrust ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ g0 ⋅ Isp ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ ⎦⎥ ⎣ ⎦

MAE 6530, Propulsion Systems II 47 Example Calculation

MAE 6530, Propulsion Systems II 48 Example Calculation (2)

MAE 6530, Propulsion Systems II 49 Example Calculation (2)

Fin Coefficients Barrowman …

35 2 ⎛ 4 ⎛ ⎞ ⎞ 10 ⎜ ⎝ 35⎠ ⎟ = ⎛1 + ⎞ ⎜ ⎟ 3 ⎝ ⎠ ⎜ 0.5⎟ =4.30898 (10 + 35) ⎛ 2·36.8091 2⎞ ⎜ 1 + ⎜1 + ⎛ ⎞ ⎟ ⎟ ⎝ ⎝ ⎝ 25 + 8.67923⎠ ⎠ ⎠

Helmbold … LF 2 36.80912 AR = = = 2.29885 Afin 589.387

N 2π2.29885 3 × fins = = 4.29281 2 0.5 2 + (2.298852 + 4) 2

MAE 6530, Propulsion Systems II 50 Effective CL for 4 Fin Configuration

MAE 6530, Propulsion Systems II 51 Effective CL for 4 Fin w⋅cosφ w⋅cosφ → α = = α⋅cosφ N u Configuration C = C ⋅cosφ⋅α⋅cosφ = C ⋅cos2 φ L Nα N

w⋅sinφ α = = α⋅sinφ N u

C = C ⋅sinφ⋅α⋅sinφ = C ⋅sin2 φ L Nα N

C = C = C ⋅cos2 φ +C ⋅sin2 φ +C ⋅sin2 φ +C ⋅cos2 φ L effective ∑ L i N N N N 2 2 = 2⋅CN ⋅(cos φ +sin φ) 1 → CL = ⋅ N ⋅CN → N = 4 effective 2

MAE 6530, Propulsion Systems II 52 Effective CL for 3 Fin Configuration

MAE 6530, Propulsion Systems II 53 Effective CL for

0 0 2 0 3 Fin CL = CN ⋅cos(φ +30 )⋅α⋅cos(φ +30 )= CN ⋅cos (φ +30 ) α Configuration 0 0 2 0 CL = CN ⋅cos(φ−30 )⋅α⋅cos(φ−30 )= CN ⋅cos (φ−30 ) α

2 0 2 0 2 CL = CL = CN ⋅cos (φ +30 )+CN ⋅cos (φ−30 )+CN ⋅sin φ effective ∑ i

⎛ ⎞2 2 3 1 ⎟ 3 3 1 cos2 300 cos cos300 sin sin300 ⎜ cos sin ⎟ cos2 cos sin sin2 (φ + )=( φ − φ ) =⎜ φ− φ⎟ = φ− φ φ + φ ⎝⎜ 2 2 ⎠⎟ 4 2 4 ⎛ ⎞2 2 3 1 ⎟ 3 3 1 cos2 300 cos cos300 sin sin300 ⎜ cos sin ⎟ cos2 cos sin sin2 (φ− )=( φ + φ ) =⎜ φ + φ⎟ = φ + φ φ + φ ⎝⎜ 2 2 ⎠⎟ 4 2 4

⎛ ⎞ ⎜ 2 3 2 3 1 2 3 2 3 1 2 ⎟ 3 2 2 3 CL = CN ⋅⎜sin φ + cos φ− cosφsinφ + sin φ + cos φ + cosφsinφ + sin φ⎟= CN ⋅ sin φ +cos φ = CN ∑ i ⎜ ⎟ ( ) ⎝⎜ 4 2 4 4 2 4 ⎠⎟ 2 2 N CL = CN → N = 3 QED! effective 2

MAE 6530, Propulsion Systems II 54 Example Calculation (3) Total parameter calculation C ≈ C Lα N R

C X C C m = − sm ⋅( L + D )= α α 0 − 1.22093 (4.96204 + 0.315467) =-6.44348

⎛ c ⎞ 2 ⎜ ref ⎟ Cm = −X sm ⋅CL ⋅⎜ ⎟= ( q )0 α ⎜ ⎟ ⎝⎜V∞0 ⎠⎟ (− 1.220932 (4.96204) ) 35 67.2868 100 =-0.038475

MAE 6530, Propulsion Systems II 55 Launch Simulation with Pitch Target 10,000 M Apogee altitude

2 2 I = M ⋅ L =105.87⋅2.15 = 506.373 2 ( yy )0 0 kg−m

MAE 6530, Propulsion Systems II 56 Example Calculation, Pitch Sim

MAE 6530, Propulsion Systems II 57 Example Calculation, Pitch Sim (2)

MAE 6530, Propulsion Systems II 58 Example Calculation, Pitch Sim (3)

MAE 6530, Propulsion Systems II 59 Simulation Comparisons

MAE 6530, Propulsion Systems II 60 Pitching Moment Control of Vehicle

Pitching Moment Control

q ⋅ A ⋅c q ⋅ A ⋅c M = ref ref C = ref ref C +C ⋅δ y m ( m(α,q) md ) I yy I yy

Natural Response Controlled Response Depends on α, q Depends on Actuation

MAE 6530, Propulsion Systems II 61 Launch (Rocket) Controls

MAE 6530, Propulsion Systems II 62 Decoupled Equations of Motion

• Very loose coupling between

2 longitudinal aerodynamics and pitch ⎡⎛ V ⎞ ⎛ q ⋅ Aref ⎞ ⎛ F sinθ ⎞ ⎤ ν + ⋅ C cosγ − C sinγ + thrust − g ⎢⎜ ⎟ ⎜ ⎟ ( L D ) ⎜ ⎟ (r) ⎥ dynamics and overall vehicle ⎢⎝ r ⎠ ⎝ m ⎠ ⎝ m ⎠ ⎥ ⎡V ⎤ ⎢ ⎥ r V ⋅V ⎛ q ⋅ Aref ⎞ F cosθ trajectory ⎢ ⎛ r ν ⎞ ⎛ thrust ⎞ ⎥ ⎢ ⎥ −⎜ ⎟ − ⎜ ⎟ ⋅(CL sinγ + CD sinγ ) + ⎜ ⎟ ⎢ ⎥ ⎢ ⎝ r ⎠ ⎝ m ⎠ ⎝ m ⎠ ⎥ ⎢V ⎥ ⎢ ⎥ ⎢ ν ⎥ ⎢ ⎥ • Generally pitch dynamics and ⎢ ⎥ ⎢ Outer Loop V ⎥ r vehicle trajectoryµ are controlled ⎢r ⎥ ⎢ ⎥ g = ⎢ ⎥ ⎢ ⎥ independently(r) r2 ⎢ ⎥ ⎢ ⎥ Vν V ⎢ ⎥ ⎢ ⎥ γ = tan−1 r = θ − α ν r V ⎢ ⎥ ⎢ ⎥ • Trajectoryν controlled about some ⎢ ⎥ ⎢ ⎥ x r  ⎢ ⎥ = ⎢ ⎥ →  = ⋅ν x Vν Prescribed q0(t) (outer loop) ⎢ ⎥ ⎢ ⎥ V = V 2 + V 2 ⎢ ⎥ ⎢ ⎥ r υ ⎢ ⎥ ⎢ ⎥ 1 q ⋅ A g 2 ⎢α ⎥ ⎢ ref (h) Fthrust sinα ⎥ •q Pitch= ⋅ ρ (htracking) ⋅V controlled about − ⋅CL + q + cos(γ ) − 2 ⎢ ⎥ ⎢ m ⋅V V m ⋅V ⎥ ⎢ ⎥ ⎢ ⎥ nominalh = r − Rear thq0(t) (inner loop) q q ⋅ Aref ⋅ cref ⎢  ⎥ ⎢ ⋅C ⎥ ⎢ ⎥ ⎢ I m ⎥ ⎢ ⎥ ⎢ y ⎥ ⎢θ ⎥ ⎢ q ⎥ ⎢ ⎥ ⎢ Inner Loop ⎥ ⎣m ⎦ ⎢ ⎥ ⎢ Fthrust ⎥ ⎢ − ⎥ g ⋅ I ⎣ 0 sp ⎦ MAE 6530, Propulsion Systems II 63 Example Control Strategies

MAE 6530, Propulsion Systems II 64 Questions??

MAE 6530, Propulsion Systems II 65 Appendix I: Derivation of the Longitudinal “AlphaDot” Equation

• Body Axis Coordinates, Position and Airspeed Components

x, u ⎡ u ⎤ ! ! ! ! ⎢ ⎥ V = u ⋅i + v⋅ j + w ⋅ k ≡ v ∞ ⎢ ⎥ w ⎣⎢ ⎦⎥

y, v

z, w z, w MAE 6530, Propulsion Systems II 66 Derivation of the Longitudinal “AlphaDot” Equation (2) • Body Axis Coordinates, Angular Rate Components

x ⎡ p ⎤ ! ! ! ! ⎢ ⎥ Ω = p ⋅i + q ⋅ j + r ⋅ k ≡ ⎢ q ⎥ ⎢ r ⎥ p ⎣ ⎦ ! V∞ q y

z MAE 6530, Propulsion Systems II 67 Derivation of the Longitudinal “AlphaDot” Equation (3)

• Euler Angles ⎡ p ⎤ ⎡ φ! ⎤ ⎡ 1 0 0 ⎤ ⎡ 0 ⎤ ⎡ 1 0 0 ⎤ ⎡ cos 0 sin ⎤ ⎡ 0 ⎤ ⎢ ⎥ θ − θ x ⎢ ⎥ ⎢ ⎥ ⎢ ! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ q ⎥ = ⎢ 0 ⎥ + ⎢ 0 cosφ sinφ ⎥⋅ ⎢ θ ⎥ + ⎢ 0 cosφ sinφ ⎥⋅ ⎢ 0 1 0 ⎥⋅ ⎢ 0 ⎥ ⎢ r ⎥ ⎢ 0 ⎥ ⎢ 0 −sinφ cosφ ⎥ ⎢ 0 ⎥ ⎢ 0 −sinφ cosφ ⎥ ⎢ sinθ 0 cosθ ⎥ ⎢ ψ! ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ 1 0 −sinφ ⎤ ⎡ φ! ⎤ ⎡ φ! − sinφ ⋅ψ! ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ 0 cosφ cosθ sinφ ⎥⋅ ⎢ θ! ⎥ = ⎢ cosφ ⋅θ! + cosθ sinφ ⋅ψ! ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 −sinφ cosθ cosφ ⎥ ψ! −sinφ ⋅θ! + cosθ cosφ ⋅ψ! ⎣ ⎦ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥

Earth-Fixed y Coordinates

⎡ ! ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ φ ⎥ ⎢ 1 tanθsinφ tanθcosφ ⎥ ⎢ p ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ θ! ⎥ = ⎢ 0 cosφ −sinφ ⎥ ⋅⎢ q ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ψ! ⎥ ⎢ 0 sinφ cosθ cosφ cosθ ⎥ ⎢ r ⎥ z ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎣⎢ ⎦⎥ ⎡ ⎤ ⎢ p + q⋅tanθsinφ + r ⋅tanθcosφ ⎥ ⎢ ⎥ ⎢ q⋅cosφ−r ⋅sinφ ⎥ ⎢ ⎥ ⎢ q⋅sinφ cosθ + r ⋅cosφ cosθ ⎥ ⎣⎢ ⎦⎥

MAE 6530, Propulsion Systems II 68 Derivation of the Longitudinal “AlphaDot” Equation (4)

• From Dynamics ! ! ! ! ⎡ F ⎤ ⎛ ⎞ i j k ⎢ x ⎥ u F ! d ! ! ! 1 d ⎜ ⎟ = A = (V ) + Ω ×V → ⎢ Fy ⎥ = v + p q r m dt m ⎢ ⎥ dt ⎜ ⎟ F ⎝ w ⎠ u v w ⎣⎢ z ⎦⎥ ! ! ! ⎡ ⎤ ⎡ u" ⎤ i j k Fx 1 ⎢ ⎥ ⎢ v ⎥ p q r F → ⎢ " ⎥ = − + ⎢ y ⎥ m ⎢ ⎥ ⎢ w" ⎥ u v w F ⎣ ⎦ ⎣⎢ z ⎦⎥ • Evaluating Cross Product

⎡ ⎤ ⎡ ⎤ ⎡ u! ⎤ 0 r −q ⎡ u ⎤ Fx ⎢ ⎥ 1 ⎢ ⎥ ⎢ v ⎥ r 0 p ⎢ v ⎥ F ⎢ ! ⎥ = ⎢ − ⎥⋅ ⎢ ⎥ + ⎢ y ⎥ m ⎢ ⎥ ⎢ w! ⎥ ⎢ q − p 0 ⎥ ⎢ w ⎥ F ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣⎢ z ⎦⎥

MAE 6530, Propulsion Systems II 69 Derivation of the Longitudinal “AlphaDot” Equation (5)

• Assume Axi-Symmetric Missile Profile, 2-D Flight Path

2 2 u (v, β, Fy = 0, V∞ = u + w ) ! V∞ • Translational EOM Reduce to ⎡ ⎤ Fx ⎢⎡u! = -q⋅w+Fx ⎤ ⎥ ⎢⎢u! = -q ⋅w + ⎥m ⎥ 2 2 ⎢ m ⎥ V∞ = u + w → ⎢ ⎥ ⎢ F F ⎥ w ⎢⎢ww! == q q⋅u⋅u++z ⎥z ⎥ ⎣⎢⎣⎢ m m⎦⎥ ⎦⎥ → From Kinematics w w! w w! ⋅u − u! ⋅w tanα = → 1+ tan2 α α! = − ⋅u! = u ( ) u u2 u2 2 ⎛ w ⎞ u2 + w2 u2 w! ⋅u − u! ⋅w w! ⋅u − u! ⋅w 1 tan2 1 ⎛ ⎞ ! ( + α ) = ⎜ + ⎜ ⎟ ⎟ = 2 → α = 2 2 ⋅ 2 = 2 2 ⎝ ⎝ u ⎠ ⎠ u u + w u u + w w! ⋅u − u! ⋅w w! ⋅u u! ⋅w α! = = − u2 + w2 u2 + w2 u2 + w2 MAE 6530, Propulsion Systems II 70 Derivation of the Longitudinal “AlphaDot” Equation (6)

→ From Kinematics w! ⋅u − u! ⋅w w! ⋅u u! ⋅w ! u α! = = − u2 + w2 u2 + w2 u2 + w2 V∞

→ From Dynamics

⎡⎡ FxF⎤ ⎤ ⎢u!u!== --qq⋅⋅ww++ x ⎥ ⎢⎢ m ⎥ ⎥ ⎢ m⎥ w ⎢ F ⎥ ⎢⎢w = q ⋅u + zF⎥ ⎥ ⎢⎣⎢w! = q⋅u +m z⎦⎥ ⎥ ⎣⎢ m ⎦⎥ → Substitute

⎛ Fz ⎞ ⎛ Fx ⎞ ⎛ 2 Fz 2 Fx ⎞ ⎛ 2 2 Fz Fx ⎞ q ⋅u + ⋅u -q ⋅w + ⋅w q ⋅u + ⋅u + q ⋅w − ⋅w q ⋅ u + w + ⋅u + − ⋅w ⎝⎜ m ⎠⎟ ⎝⎜ m ⎠⎟ ⎝⎜ m m ⎠⎟ ⎝⎜ ( ) m m ⎠⎟ α! = 2 − 2 = 2 = 2 V∞ V∞ V∞ V∞

→ Simplify

⎛ 2 Fz F ⎞ ⎛ 2 Fz F ⎞ q ⋅V + ⋅u + − x ⋅w q ⋅V + ⋅u + − x ⋅w ⎝⎜ ∞ m m ⎠⎟ ⎝⎜ ∞ m m ⎠⎟ ⎛ F u F w ⎞ ! q z x α = 2 = 2 = ⎜ + ⋅ − ⋅ ⎟ V∞ V∞ ⎝ m ⋅V∞ V∞ m ⋅V∞ V∞ ⎠ MAE 6530, Propulsion Systems II 71 Derivation of the Longitudinal “AlphaDot” Equation (7)

inematics x Lift ⎡ u ⎤ a ⎢ = cosα ⎥ V ⎛ F F ⎞ → ⎢ ∞ ⎥ → α! = q + z ⋅cosα − x ⋅sinα ⎢ w ⎥ ⎝⎜ m ⋅V m ⋅V ⎠⎟ ⎢ = sinα ⎥ ∞ ∞ V ⎣⎢ ∞ ⎦⎥ Drag → Resolving Forces z ⎡Fx = Tthrust − m ⋅ g ⋅sinθ − Drag ⋅cosα + Lift ⋅sinα ⎤ ⎢ ⎥ Thrust ⎣Fz = m ⋅ g ⋅cosθ − Drag ⋅sinα − Lift ⋅cosα ⎦ → Substituting

⎛ m ⋅ g ⋅cosθ − Drag ⋅sinα − Lift ⋅cosα Tthrust − m ⋅ g ⋅sinθ − Drag ⋅cosα + Lift ⋅sinα ⎞ α! = ⎜ q + ⋅cosα − ⋅sinα ⎟ ⎝ m ⋅V∞ m ⋅V∞ ⎠

MAE 6530, Propulsion Systems II 72 Derivation of the Longitudinal “AlphaDot” Equation (7)

x L ift L D C = ift , C = rag a L 1 D 1 ⋅ ρ ⋅V 2 ⋅ A ⋅ ρ ⋅V 2 ⋅ A 2 ref 2 ref

CL = lift coefficient, CD = drag coefficient

1 2 Drag ⋅ ρ ⋅V = "dynamic pressure"(q) 2

Aref = "reference area"− > typically maximum frontal area z ρ = "local air density"− > function of altitude Thrust

→ Collecting m ⋅ g ⋅cosθ ⋅cosα + m ⋅ g ⋅sinθ ⋅sinα −D ⋅sinα ⋅cosα + D ⋅cosα ⋅sinα L ⋅cos2 α + L ⋅cos2 α T ⋅sinα α! = q + + rag rag − ift ift − thrust m ⋅V∞ m ⋅V∞ m ⋅V∞ m ⋅V∞ → Simplifying

g ⋅cos(θ −α ) Lift Tthrust ⋅sinα g ⋅cos(θ −α ) q ⋅ Aref ⋅CL + Tthrust ⋅sinα α! = q + − − →→ α! = q + − V∞ m ⋅V∞ m ⋅V∞ V∞ m ⋅V∞

MAE 6530, Propulsion Systems II 73 Appendix II: Derivation of the Longitudinal “VDot” Equation Kinematics ⎛ ⎞ 1 ⎜u⋅u! + w⋅w! ⎟ V! = ⋅ 2⋅u⋅u! + 2⋅w⋅w! =⎜ ⎟= ∞ 2 2 ( ) ⎜ 2 2 ⎟ 2 u + w ⎝⎜ u + w ⎠⎟ ⎛ ⎛ F ⎞ ⎛ F ⎞⎞ ⎜ ⎜ x ⎟ ⎜ z ⎟⎟ ⎜u⋅⎜-q⋅w+ ⎟+ w⋅⎜q⋅u + ⎟⎟ ⎜ ⎜ m ⎟ ⎜ m ⎟⎟ ⎛ ⎞ ⎜ ⎝ ⎠ ⎝ ⎠⎟ ⎜ u Fx w Fz ⎟ Fx Fz ⎜ ⎟=⎜ ⋅ + ⋅ ⎟= cosα⋅ +sinα⋅ ⎜ V ⎟ ⎜V m V m ⎟ m m ⎜ ∞ ⎟ ⎝ ∞ ∞ ⎠ ⎜ ⎟ ⎝⎜ ⎠⎟ → Resolving Forces ⎡F = T −m⋅g ⋅sinθ− D ⋅cosα+ L ⋅sinα ⎤ ⎢ x thrust rag ift ⎥ ⎢ ⎥ F = m⋅g ⋅cosθ− D ⋅sinα− L ⋅cosα ⎣⎢ z rag ift ⎦⎥ Substituting

(Tthrust −m⋅g ⋅sinθ− Drag ⋅cosα+ Lift ⋅sinα) (m⋅g ⋅cosθ− Drag ⋅sinα− Lift ⋅cosα) V! = cosα⋅ +sinα⋅ ∞ m m MAE 6530, Propulsion Systems II 74 Derivation of the Longitudinal “VDot” Equation Simplifying

(Tthrust −m⋅g ⋅sinθ− Drag ⋅cosα) (m⋅g ⋅cosθ− Drag ⋅sinα) V! = cosα⋅ +sinα⋅ = ∞ m m 2 2 (Tthrust −m⋅g ⋅sinθ− Drag ⋅cos α) (m⋅g ⋅cosθ− Drag ⋅sin α) V! = cosα⋅ +sinα⋅ = ∞ m m T ⋅cosα D thrust + g ⋅(cosθsinα−sinθcosα)− rag cos2 α+sin2 α = m m ( ) T ⋅cosα q ⋅ A ⋅C q ⋅ A ⋅C T ⋅cosα thrust + g ⋅(cosθsinα−sinθcosα)− ref D = − ref D − g ⋅sin(θ−α)+ thrust m m m m

T ⋅cosα q ⋅ A ⋅C V! = thrust − g ⋅sin(θ−α)− ref D ∞ m m

MAE 6530, Propulsion Systems II 75