20 06 2022 text

Page 1

STRUCTURAL DYNAMICS AND AEROELASTICITY Prof. G. Quaranta, June 20, 2022 Exercise 1: Static aeroelasticity Consider the cantilever wing mounted in a Wind Tunnel (WT) shown in the figure. The wing has a span L, a chord c, bending stiffness EI, torsional stiffness GJ and mass per unit length m, all constant along the span. With the symbol e it is indicated the distance between the elastic axis and the aerodynamic centre, while d indicates the distance between the EA and the centre of gravity. At the wing tip there is a rigid aerodynamic body, of span b, and with same inertia properties per unit span of the flexible wing, and same aerodynamic prop- erties. This surface is controlled by an ideal ac- tuator that allow to impose to this surface a ro- tation angle β around the elastic axis between it and the flexible wing. The wing aerofoils are all symmetric and the wing is immersed in an incompressible flow. Using the strip theory and considering also the effect of gravity forces: 1.a) Determine the divergence dynamic pressure. 1.b) Supposing to fly at a lower dynamic pressure than point 1.a), at an angle of arttack α0. Identify the angle of rotation of the tip surface β necessary to obtain for the flexible model the same root shear force obtained at the same flight condition by the rigid wing with β = 0. 1.c) Plot a qualitative diagram of the shear along the span and assess if the maximum bending load is reached by the rigid wing or by the flexible wing un case 1.b) Use the Ritz-Galerkin approximation with a sigle, well chosen shape function. Exercise 2: Stability analysis Consider the structure shown in the picture. A cantilever beam of length L with bending stiffness EI, tor- sional stiffness GJ, m mass per unit length and Jp polar moment of in- ertia per unit length is…