Aerospace Engineering - Airplane performance and dynamics

Full exam

AIRPLANE PERFORMANCE AND DYNAMICS A.A. 2025/26 – 1st EXAM TERM – JANUARY 16th , 2026 Teacher: C.E.D. Riboldi General Rules•The delivery must beclearly markedwith the SURNAME, NAME and STUDENT PERSONALCODE at the top of the sheet, inthis orderand all written incapital letters. •Answers must be written using a pen, not a pencil. •Graphs and drawings must be readable and complete. •THE MAXIMUM NUMBER OF PAGES FOR ALL ANSWERS IS FOUR Questions1. Consider the following generalized expression for the non-linear dynamics and rotational kinematicsof an aircraft in flight, where pointGis the center of gravity, andBrepresents a body reference: (MB G˙ wG B+wG B×MB GwG B=τG B ˙ eB 321=SB−1 321ωB B/I Elaborating on that expression: a. Define explicitly the content ofMB Gat matrix level, and that ofwG Bat scalar level, for a generic aircraft. b. Define a typical split ofτG Bin terms of aerodynamic/propulsive and gravity forces and mo- ments, highlighting (in implicit form) the most general dependence on state and control vari- ables of these quantities, when assuming a writing in a body reference. Define the content of a typical control array in the process. c. Introduce the hypothesis of static equilibrium, and based on the outcome of pointb.showwhat is its effect on the equation for dynamic equilibrium in body components. Clearly state and show whether from this writing any state scalar variables can be defined constant. d. Apply the hypothesis of static equilibrium to the kinematic equations in body components,and show whether any components of the rotational rateωI B/Iin an inertial frameIcan be defined constant from this writing. (Note: recall where needed thatSB−1 321R|B T I→B= cosψcosθ sinψcosθ 0 −sinψcosψ0 tanθcosψtanθsinψ1 ). e. Define the scalar body components of the rotational rateωB B/I, and by recalling that R|I B I→B= cosθcosψsinϕsinθcosψ−cosϕsinψcosϕsinθcosψ+ sinϕsinψ cosθsinψsinϕsinθsinψ+ cosϕcosψcosϕsinθsinψ−sinϕcosψ−sinθsinϕcosθcosϕcosθ , proof from previous points that the derivative˙ ψis constant in static equilibrium. f. Summarize from the previous points the characteristics of the most generic solution for staticequilibrium, in terms of the components of˙ eB 321,v G/Iandω B/I, and the corresponding identity of that type of tra jectory. g. Directly answer (Y/N): is a non-symmetric flight condition compatible with the generic solu-tion of static equilibrium described at point f.? 1 AIRPLANE PERFORMANCE AND DYNAMICS A.A. 2025/26 – 1st EXAM TERM – JANUARY 16th , 2026 Teacher: C.E.D. Riboldi 2. Consider an aircraft on a steady, straight, horizontal flight tra jectory. With the help of accuratequalitative sketches and analytic definitions: a. Explain the effect on longitudinal static stability of a standard puller propeller configuration. b. Explain the effect on longitudinal static stability of a pusher propeller configuration.c. Show the typical contribution of a low-mounted engine to static equilibrium around the pitchaxis. d. Furthermore, answer this directly (Y/N): can an aircraft with a puller propeller be staticallystable in the longitudinal plane? 3. Consider the model for the pitch moment coefficientCmGwith respect to the center of gravity of an aircraft built in a standard back-tailed configuration, as in the following equation: CmG=C m ACW+aW α(ξACW −ξ G) +ησat (α(1−ϵα)−ϵ 0+it +τ δe)(ξ ACt −ξ G) +cf mG0+cf mGαα Assume the following linear relationship for describing the behavior of the hinge moment coefficient of the elevator: Ch,e=C h,eαα+C h,eδeδ e According to the nomenclature just introduced: a. Define the meaning of each parameter appearing in the definition ofCmG. b. Define and provide an explicit expression of the floating angle of the elevator.c. Provide an expression of the derivativeCmGαin a stick-fixed condition, and state the sign it should take for stick-fixed static stability. Clearly highlight the tail contribution within the expression, comment about its sign and state whether it is typically stabilizing or not for a back-tailed aircraft. d. Making use of the expression of the floating angle of the elevator, work out an expressionof the derivativeCmGαin a stick-free condition, such to highlight the relationship with the corresponding stick-fixed value. e. State whether the level of static stability is increased or reduced in a stick-free conditioncompared to a stick-fixed condition, and justify your statement by accurately showing the typical sign taken by the parameters of the expression found at point d. 4. Consider the preliminary weight sizing problem for a general aviation aircraft.a. Introduce a typical weight break-down for an aircraft in that category, and write it in a formsuitable for starting an algorithm for the solution of the weight sizing problem. b. Introduce the expression of a typical regression of historical data for empty weight vs. take-offweight, according to the form proposed by Raymer. c. Introduce the concept of fuel fraction for the generic leg of a mission, then show how fuelfractions combine, and show where the result of this combination enters the algorithm for solving the weight sizing problem. d. Show on a suitable plot the intersection representing the solution of the weight sizing problemin terms of empty weight and take-off weight, clearly stating the identities of the intersecting curves. 2