Complete course materials for AIRPLANE PERFORMANCE AND DYNAMICS in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Politecnico di Milano Scuola di ingegneria industriale e dell’informazione Corso di laurea magistrale in ingegneria aeronautica Airplane performance and dynamics notes Realizzazione e svolgimento a cura di: Parma Giuseppe Russo Alessandro Docenti di riferimento: Cacciola Stefano
Complete course materials for AIRPLANE PERFORMANCE AND DYNAMICS in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Politecnico di Milano Scuola di ingegneria industriale e dell’informazione Corso di laurea magistrale in ingegneria aeronautica Airplane performance and dynamics notes Realizzazione e svolgimento a cura di: Parma Giuseppe Russo Alessandro Docenti di riferimento: Cacciola Stefano
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Politecnico di Milano Scuola di ingegneria industriale e dell’informazione Corso di laurea magistrale in ingegneria aeronautica Airplane performance and dynamics notes Realizzazione e svolgimento a cura di: Parma Giuseppe Russo Alessandro Docenti di riferimento: Cacciola Stefano Filippoli Giorgio Anno Accademico 2025-2026
Chapter 2. Equilibrium and Static Stability For LIFT: L L L= (α − α0 ) + (δE − δE0 ) α δE L L HE /α 1 = (α − α0 ) + − (α − α0 ) + (HE − HE0 ) α δE HE /δE HE /δE L HE /α L L/δE = − · (α − α0 ) + (HE − HE0 ) (2.9) α HE /δE δE HE /δE For PITCHING MOMENT: MP MP MP = (α − α0 ) + (δE − δE0 ) α δE MP MP HE /α 1 = (α − α0 ) + − (α − α0 ) + (HE − HE0 ) α δE HE /δE HE /δE MP HE /α MP MP /δE = − · (α − α0 ) + (HE − HE0 ) (2.10) α HE /δE δE HE /δE So by imposing a force to the stick from a physical point of view we end up to generate a specific lift and a specific pitching moment. From a mathematical point of view, we have exactly the same model structure as for the Position Ctrl model. Therefore, for the Force Ctrl Model, we can rewrite the previ- ous equation as follows: ′ ′ L = L (α − α0 ) + L (HE − HE0 ) α HE ′ (2.11) MP M′P MP = (α − α0 ) + (HE − HE0 ) α HE Then we will apply the equilibrium: ( L = W cos(γ) MG = 0 Because from a mathematical point of view there isn’t any change in the shape between this model and the position control model, except that there are different variable, without computing anything we know that the solution remain the same (obviously the structure of the solution is intended). So we can obtain the important points that interest us. 40
Chapter 4. Kinematics (evolution of Euler angles) 4.2.1 Simple application of the theorem Referring to Fig. 4.3, consider a point P moving at constant pace u0 along axis i1 of an inertial triad I. Another triad N rotates at a constant angular velocity ω around i3 . Figure 4.3: Example of an application of the moving axes theorem. In this configuration, the velocity of point P , VP , in I results to be u0 VPI = 0 . (4.10) 0 Whereas its derivative, again viewed in I, is 0 I dVP = 0 . (4.11) dt 0 Moreover, the angular velocity vector of N with respect to I is 0 N ω N /I = 0 . (4.12) ω At first, try to compute the time derivative of VP in N without using the moving axes theorem. The rotation tensor associated to the motion of N with respect to I is cos(ωt) − sin(ωt) 0 RI→N = sin(ωt) cos(ωt) 0 . (4.13) 0 0 1 80
Chapter 10. Stability & control derivatives 10.2 Estimation of stability & control derivatives To this end we have different methods: 1. Analytical methods: fast, easy to interpret, but excessively approximated models, that can be accepted for a preliminary design, but not for the final model. 2. Empirical/semi-empirical methods: we are considering data coming from experi- mentation, so very realistic data. A very relevant database is DATCOM, where are gathered many experiments for the last 50 years. The only drawback is the lack of informations for very unusual aircrafts. 3. Numerical methods: in general here we have "vortex lattice" and CFD methods. It is impossible to use these tools at the initial part of the design because you need to have the shape of the aircraft and moreover, the process that takes to resolve these numerical methods is typically very long, so it’s good for final refinements. 4. Testing: wind tunnel and flight testing In this course we’ll focus on analytical methods, because we want to have interpret-ability of the result an understand what is the link between the geometrical data and the aero- dynamics derivatives; if we can have this link we have the link between the geometrical features and the modes of the system. Now let’s begin with the different analytical methodologies used to estimate the deriva- tives, starting with the longitudinal ones. 119
Chapter 12. Conceptual design and preliminary weight estimation The graph now has to be read in a different way: • Maximize wing loading in order to design a lighter and cheaper wing; • maximize power loading to have a lighter and cheaper engine 12.3 Final results Returning back to our blank paper, we are now able to list all the pieces of information that we have: What can we do now with these data? WT O Maximum takeoff weight From weight estimation WE Empty weight From weight estimation WF Fuel weight From weight estimation WP L Payload From weight estimation CLmax Maximum CL (clean conf.) From SMP CLmax LA Maximum CL (LA conf.) From SMP CLmax T O Maximum CL (TO conf.) From SMP λ Aspect ratio Guess value WT O Wing loading Design SMP choice S (T /W )T O Thrust-to-weight ratio Design SMP choice (W/P )T O Power loading Design SMP choice S Wing area From SMP + Weight est. T, P Available thrust, power From SMP + Weight est. NEN G Number of engines From SMP + Weight est. • From thrust/power we can select our engine; • with wing area and the coefficients we can start to design the wing. • and so on... 159
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