Airplane Performance & Dynamics Flashcards
What is the Euler - Rodriguez formula?
R = I + sinφ/φ * (φx) + (1-cosφ)/φ^2 * (φx)^2
Lateral - Directional eigendynamics: 1) Order of system 2) Real/Imaginary eigenvalues 3) Which are oscillatory? 4) Which are the most intense modes.
1) 5th order system -> 5 eigenvalues
2) 3 real eigenvalues, plus a couple of complex conjugates. Zero - root, spiral, roll-subsidiary and Dutch-Roll ( complex conjugate).
3) Only dutch-roll is oscillatory.
4) i) zero-root is associated to Δψ only and does not effect the other eigenvalues. ii) spiral is a small, either positive/negative (unstable/stable) eigenvalue which associates with Δφ, Δψ. iii) Roll-Subsidiary is a largely negative, real eigenvalue with associated with Δp, Δφ. iv) Dutch-Roll is a relatively low damped oscillatory motion, and the eigenvector is associated to significant contributions from all states. Needs to be artifically damped. Eigenvalue map of DR: β top imaginary axis, Δr positive real axis, Δφ slightly positive Im/largely negative real, Δψ and Δp.
Provide a quick definition for each of the levels and catergories proposed in the Cooper-Harper scale.
1) 3 levels of flying qualities.
I. Adequate for the mission plan
II. Adequate for the mission plan, but with increased worklaod.
III. Adequate for the mission plan, but with excessive workload.
2) 3 categories of flight phase
A. Non-terminal, requiring fast action on the controls
B. Non-terminal, accompanied with gradual action on the controls
C. Terminal Maneuvers
3) 3 classes of aircraft
1. Small/light
2. Medium weight, low-to-medium maneuveratbility
3. Large
(4. High maneuverability)
Define the hypothesis on the reference condition for linearization needed to decouple the longitudinal and lateral-directional dynamics of an A/C.
1) Roll and yaw moment = 0 (p0 = r0 = 0). Plane of symmetry does not change its orientation.
2) Lateral velocity and sideslip = 0 (v0 = β0 = 0). Plane of symmetry is not displaced laterally.
3) No rolling attitude in the plane of symmetry (φ0 =0)
Additionally
1) No side-force, Cy0 = 0;
2) No rolling moment: Lg0 = 0
3) No yawing moment: Ng0 =0
Longitudinal: 1) Order of system 2) Real/Imaginary eigenvalues 3) Which are oscillatory? 4) Which are the most intense modes.
1) 4th order system, 4 eigenvalues 2) Two couples of complej conjugate eigenvalues, Short period mode and Phygoid mode. 3) Short period has intense u, Δθ and phygoid has intense Δα, Δq
Write the equations of motion of an aircraft in generalized vector form considering a generic measuring point P. Show the generalized mass matrices, generalized state vector and generalized force vector. Show the effect of a choice of the center of gravity G as the measuring point, in simplifying the structure of the generalized equations of motion. Show the effect of aircraft symmetry with respect to a vertical body plane on the representation of the inertia tensor in a barycentric body frame (i.e. on the matrix JG).
Μp *
Show how to linearize the non-linear form of a aerodynamic force or moment.
Δfa = Δ(1/2ρU^2SCF) = ρ0 U0 ΔU SF + 1/2ρ0U0^2SΔCF
ΔMg = Δ(1/2ρU^2SCDmg) = ρ0 U0 ΔU SF + 1/2ρ0U0^2SDΔmg
Show the expression of the peturbation of a generic scalar aerodynamic coefficient Δci according to the hypothesis of linearized aerodynamics, adopted for writing the equations of dynamic equilibrium in a lienarized framework. Among the quantities appearing, highlight what are stability derivatives and control derivatives.
Δci depends on u, Δβ, Δα, Δp, Δq, Δr, Δu_dot, Δβ_dot, Δα_dot STABILITY
Δδe, Δδa, Δδr, ΔδT, CONTROL
δRe, δMa.
Which surfaces contribute to the Lift coefficient and slope? What is the expression for both?
The lift coefficient contributions are mostly from the wing and horizontal tail.
CL = CL_w + ησCL_t
CL_α = a_w + ησa_t(1-ε_α)
For a Jet: What is Endurance and Range equal to? Where are there max values? Where can they be seen on the polar diagram?
Endurance = E/C_T * ln(W) and Range = EV/C_T*ln(W)
max(Endurance) = min(Wf/dt) = max(E)
max(Range) = min(Wf/ds) = max(EV) = max(G)
max(E) –> CD = 2CD0, CL = sqrt(CD0/k)
max(EV = G) —> CD = 4/3 CD), CL = sqrt(CD0/3k)
For a Propeller driven AC: What is Endurance and Range equal to? Where are there max values? Where can they be seen on the polar diagram?
Endurance = E/V * η_p/C_p * ln(W) and Range = E * η_p/C_p * ln(W)
max(Endurance) = min(Wf/dt) = max(E/V) = max(F)
max(Range) = min(Wf/ds) = max(E)
max(F) –> CD = 4CD0, CL = sqrt(3CD0/k)
max(E) —> CD = 2CD0), CL = sqrt(CD0/k)
What is max F equal to? Which values can be found there?
max(F) –? CD = 4CD0, CL =sqrt(3CD0/k)
at max F we have: min(Pr), Minimum Descent, Fastest Climb (P), max(Endurance (P)), min(W
What is max E equal to? Which values can be found there?
max(E) –? CD = 2CD0, CL =sqrt(CD0/k)
at max E we have: minD, Best Glide, Steepest Climb (J), max(Endurance (J)), max(Range (P))
What is max G equal to? Which values can be found there?
max(G) –? CD = 4/3CD0, CL =sqrt(CD0/3k)
at max E we have: max(R (J))
What is Raymers and Roskams eqations?
Raymer: we = DWMTO^c
Roskam: logWmto = A + BlogWe
Describe short period mode:
- Higher frequency, higher damping
- Eigenvector especially intense Δq, Δα computed
- Δθ component is roughly in phase with Δα, but less intense
- u neglegible
- Short period motion is quick to die out, and difficult to maneuver
- typical period is roughly 1-2 sec
- No oscillation of airspeed
- Frequency is influenced by Cmg_a. More intensely negative Cmg_a -> Higher frequency. Less intensely negative Cmg_a -> lower frequency, possibly mode breakup
- Damping is influenced by Cmg_a, Cmg_q, Cmg_a_dot, all these depend on size of horizontal tail
Describe phygoid mode
- Lower frequency, lower damping
- Eigenvector is associated mostly to Δθ, u
- Visible and low-damped oscillation of the trajectory
_ Δα neglegible - climb angle is oscillatory
- Frequency and damping are associated to reference condition:
– Frequency: Uo+ , f-
– Damping (L/D)_0 +, damping-
Describe zero-root
It is associated with Δψ. Τhis mode does not induce a motion on any other quantity, only one component on the eigenvector. Zero frequency, always neutrally stable.
Describe roll-subsidence.
Largely negative, real eigenvalue. Eigenvector is associated to Δp, Δφ (roll rate and roll angle).
- Frequency is higher with U0
- Frequency is lower for higher Ix
- Freqeuncy is higher for high |CLG_p| damping in roll.
Describe spiral
Associated to a real eigenvalue, either positive or negative (unstable/stable) eigenvalue. Eigenvector mainly Δφ, Δψ. This is not spin.
This mode is either stable or unstable depending on:
- CLG_β < 0 (laterally stable)
-CNG_β >0 (directionally stable)
-CNG_r < 0 (damping in yaw), CLG_r >0.
Describe dutch roll
Oscillatory motion, relatively low damped. The eigenvector is associated to significant contributions from all states.
- This mode is typically stable.
- The higher CNG_β the higher the frequency.
- Damping is bound to CYβ, CNG_r.
What effects the lateral-directional dynamics eigenvalues.
CNG_β. CLG_β are highly effective in modifying the response.
-Decreasing CNG_β, the frequency of roll-sub/spiral will increase and the frequency/damping of D.R will decrease.
If CNG_b becomes small, DR may be unstable.
- If |CLG_β| is reduced, DR will increase in damping, Spiral/Roll-sub will decrease their frequency, while spiral may become unstable.
What does the pitching moment coefficient and slope depend on? Write the expressions.
The pitching moment coefficient and slopes has contributions from the wing, horizontal tail, fuselage and slender bodies such as tip tanks, pods, missiles, engine nacelles.
Cmg_w+t = Cm_AC_w + a_w(a+iw)(ξ_AC_w - ξG) + ησ(κCm_AC_t + a_t(a-e+it)(ξ_ac_t - ξγ)
Cmg_f = Cmg0 + Cmg,aa
What is the pitch up effect?
In swept-back wings, the same effect that is encountered towards the trailing edge of an aerofoil(adverse pressure gradient) is encountered towards the wing tip. The spanwise diretion pressre allon the wing increases. During a stall, the separation chordwise occurs due to an adverse pressure gradient on an airfoil. Similar effect towards the wing tip in spanwise direction. This results to separataion lines due to spanwise pressure gradient when approaching stall.
This behaviour is typical to swept-back wings and results in two effects: 1) separation of the flow is interfering with the ailerons. This results in a loss of lateral-directional control and since entering stall may result in a spin/yaw, and getting out of it required action on the rudder/aileron, the loss of aileron effectivenes may be dangerous.
2) Pitc up effect: Local loss of the lift towards the wing tip produces a positive pitching moment. The pilot does not feel an increase of required action on the elevator to enter stall. On the contrarty he/she feels a requirement when approaching stall.
What does the drag coefficient and slope depend on? How to estimate it?
Lifting surfaces: Wing, tail, vertical tail.
In general CD = CD0 + kCL^2. and CD = CDW + ησCDt + η1σ1CDVT
Non-lifting bodies, fuselage: Df is not a function of alpha, it is constant with alpha.
What does the longitudinal rotary components depend on and which are they?
CL_q and CMG_q.
They depend mainly on horizontal tail.
Δα_t = Δq*(XG - Xac_t)/U0.
ΔCL = ησa_tΔα_t = ησa_t* Δq(XG - Xac_t)/U0.
CL_q = ΔCL/Δ(qc/2U) = 2a_tησ(ξ_G - ξac_t).
Δcmg_t = ησCMg_at * Δα_t.
CMG_q < 0 always
What does the longitudinal acceleration derivative CL_alpha_dot and CMG_a_dot depend on
Mainly on the horizontal tail.
Time for the wing wake to travel the distance between the wing and horizontal tail is: Δt = (XG-Xact)/U0.
ε(t) = ε_αα(t-Δt) + ε0 and alpha(t-ΔΤ) = α(t) - a_dotΔt. Linear expansion.
CL_t = ησa_t * (α(t) + it - ε(t)).
δCl_t/δa = ησε_α(ξ_G - ξACT)
What do the longitudical control derivates depend on and how can they be calculated.
They depend only on the horizontal tail and the elevator specifially.
We add a parametrer τ = dα_t/dδe which is defined based on geometrical characteristics on the surface.
CL_t = ησa_t*(α + it - ε + τδe).
What does the lateral force Y and β slope depend on?
Typically mostly due to the vertical tail ( the fuselage does not contribute significantly whereas it produces yawing moment)
Y = Y_vt = -κq_vt S_vt a_vt *(β + γ), where γ is the sidewash effect
CY = -κη_vt σ_vt a_vt (β+γ)
CY_β = -κ η_vt σ_vy a_vt (1+γ_β)
k because of the intereference effects, due to the span of vertical tail (k = 0.8 - 1.2 typically)
η_vt (1+γ_β) because of the intereference effects of the fuselage wing etc and is a f(Α, Lc/4, σ_vt, position of AC_w, bf (fuselage diameter)
What are the contributions to yawing moment ng and its slope wrt β?
Three main contributions: The vertical tail, the wing and the fuselage.
The wing contribution is analyzed by means of strip theory. Strip theory is a simple aerodynamic model, where each 2D aerodynamic profile is acting on an isolated aerofoil. Two major features contributing to yawing moment: The wind dihedral Γ and the sweep angle Λ.
Cngβ is < 0 for Γ> 0 ανδ <0 for Γ<0.
Cng_β > 0 for Λc/4 > 0.
ng_vt = Yvt * (X_ac_t - XG)
Cng = CY(ξacvyt - ξg)
Cng_β_vt >0 for back-tailed A/C.
What does the rolling moment depend on?
Wing (Γ,Λ), Tail and Wing Mounting ( low, mid, high)
CLgβ is < 0 for Γ> 0 ανδ <0 for Γ<0.
CLg_β < 0 for Λc/4 > 0.
Wing mounting: high mounting : for β+, Lg - –> Lg_β < 0
low mounting: opposite effect.
Vertial tail: CLG = CY_vt * (-z_ac_vt)/b (usually < 0 )
What does the lateral-directional roll rate derivatives depend on.
- CY_p mostly due to the vertical tail, due to a local misalignment β due to the airspeed component pz@AC_vt. CY_p is a function of f(CYβ, geometry of the vertical tail) and sometimes wing effects if Γ,Λ significantly different than zero.
- (damping in roll) or CLG_p mostly due to wing and tail. Both are always negative, while due to the vertical tail it depends on (CYβ,t, geometry of vertical tail)
- Similar for CNG_p: Cng_p_w < 0 and Cng_p_vt >0 largely positive
What does the lateral-directional yaw rate derivatives depend on.
- CY_r depends mostly due to vertical tail and is always >0
- CLG_r depends on wing and vertical tail and both are > 0 always.
-CNG_r depends on wing and vertical tail and is <0 always.
Lateral directional acceleration derivatives:
Due to wake-up inertia effect which are hard to quantify analytically. We need a semi empirical model.
- CY_β_dot depends only on the vertical tail.
- CLG_β_dot and CNG_β_dot both functions of CY_β_dot and of geometry
Lateral directional control derivatives:
Mostly due to δr, and less due to δa.
τ2 = dβ/dδr.
Y_vt = -kq_vt a_vt * (β + γ + τ2δρ)S_vt
ng_vt = (Xac - XG)Y_vt
τ_2 < 0 by definition and therefore Cyδr_vt > 0 and CNG_δr < 0.
Rolling moment does not depend on change in δr.
Why is a longitudinal SAS needed?
Typically related to increase short-period damping.
Pilot’s bandwith up to 6.5Hz = 40 rad/s which is sufficient for controlling phugoid but not short period.
