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
