a 2024 06 14 solution

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STRUCTURAL DYNAMICS AND AEROELASTICITY Prof. Giuseppe Quaranta June 14, 2024 Exercise 1: Static aeroelasticity Consider the structure shown in the figure. A flexi- ble straight wing with span b and chord c, clamped at one end and connected to the ground with a linear spring of characteristics k at the leading edge of the wing tip. The elastic characteristics of the wing, that could be represented as a beam, are EJ for bending and GJ for torsion, and the elastic axis is at distance d from the leading edge. The distance between the elastic axis and the aerodynamic center is e. The wing is immersed in a uniform incompressible flow of asymptotic speed V , and the strip theory with lift slope coefficient CLα could be used to model the aerodynamic behavior. Use the Ritz-Galerkin for- mulation using a single, well-chosen, shape function for each unknown field. 1.a) Compute the dynamic pressure of divergence. 1.b) Is it possible to choose the spring constant k so that there will be no divergence dynamic pressure? If yes/no explain why and how. 1.c) Provide a qualitative description of how the dynamic pressure of divergence will change if the linear spring attachment point is moved from the leading edge to the trailing edge. Exercise 2: Dynamic stability Consider the symmetric structure shown in the picture, composed of two flexible clamped wings, with span L and chord c, that can be modeled as a beam with uniform bending stiff- ness EJ and a torsional stiffness GJ to be con- sidered infinite. The mass per unit surface of the beam is equal m. The central lifting surface must be considered rigid, with the same cord c and span b and the same mass per unit surface m. This rigid surface is connected to the two flexible wings so that it can rotate at 50% of the chord. Two torsional…