Assignment 202122

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Assignment Alessandro Crotta989893 Assignment # 1: Computation of coupled modes Class: Structural Dynamics and Aeroelasticity, Prof. Giuseppe Quaranta A.Y. 2021/22 April 21, 2022 Take the last four figures of your person code ABCD, and assemble these two numbers:DAandCB1 8 >>>< >>>: 00DA <50x1=1 50DA <99x1=.5 00CB <50x2=1 50CB <99x2= 1 (1) Consider a cantilever wing, shown in figure 1, that could be represented through a beam with the following geometric properties: Semi-span b= 11 m, Root chordc1=4.5 m, Tip chordc2=2.5 m, Sweep angle ⇤ = 34. The elastic axis is positioned at 43% of the chord. The beam is characterized by the following structural properties (with ¯ythe axis along the span of the beam) Bending sti↵nessEI(¯y) = 106 ⇣ 6.7–x25¯y ¯b ⌘ kg m2 Torsional sti↵nessGJ(¯y) = 106 ⇣ 9.38 – 6.25¯y ¯b ⌘ x1kg m2 Mass per unit spanm(¯y) = 68 –x223¯y ¯bkg/m Moment of inertia per unit spanI✓(¯y) = 160 – 51¯y ¯bkg s2 Position of the center of gravity with respect to the elastic axis (positive toward the trailing edge)xCG(¯y)=0.38 –x20.5¯y ¯bm. 1Example: 10134997!A=4 ,B=9 ,C=9 ,D=7 ,s ot h et w on u m b e r sw i l lb eDA=7 4 , CB=9 9 . 1 Figure 1: Wing planform. 1.Compute the first 4 coupled (bending-torsional) proper orthogonal fre- quencies using a Ritz-Galerkin approximation selecting the appropriate shape functionsamong a)polynomials of the typeNn(¯y)= ⇣¯y ¯b ⌘n ,w i t hn=0,...,6 for bending displacementw; b)trigonometric functions for torsionNnc(¯y) = cos ⇣ n⇡ 2 ¯y ¯b ⌘ withn= 0,...,4 andNns(¯y)= s i n ⇣ n⇡ 2 ¯y ¯b ⌘ withn=1,...,4 2.Consider the second modal shape (modes ordered starting from the lowest frequency) and compute the wing tip displacement w(b) for the mode normalized to unit modal mass. 3.Consider the second modal shape (modes ordered starting from the…