Aeronautical engineering - structural dynamics and aeroelasticity

Assignment 2021/22

Etc

Assignment Alessandro Crotta 989893 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 numb ers: DA and CB 1 8>>>< >>>: 00  DA < 50 x1=1 50  DA < 99 x1= .5 00  CB < 50 x2=1 50  CB < 99 x2= 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 chord c1=4 .5 m, Tip chord c2=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↵ness EI (¯y) = 10 6⇣ 6.7– x25¯y ¯b ⌘ kg m 2 Torsional sti↵ness GJ (¯y) = 10 6⇣ 9.38 – 6 .25 ¯y ¯b ⌘ x1kg m 2 Mass per unit span m(¯y) = 68 – x223 ¯y ¯bkg/m Moment of inertia per unit span I✓(¯y) = 160 – 51 ¯y ¯bkg s 2 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,sothetwonumberswillbe DA =74, CB =99. 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 functions among a) polynomials of the type Nn(¯y)= ⇣¯y ¯b ⌘n,with n=0 ,..., 6 for bending displacement w; b) trigonometric functions for torsion Nnc(¯y) = cos ⇣ n⇡ 2 ¯y ¯b ⌘ with n= 0,..., 4 and Nns(¯y)= sin ⇣ n⇡ 2 ¯y ¯b ⌘ with n=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 lowest frequency) and compute the ratio between the bending rotation w0and the torsional rotation ¯✓at ¯y ¯b= 4 5. 4. Consider that the structure has modal damping equal to 2.0%. An impul- sive distributed torsional moment about the elastic axis is applied to the structure with the following distribution mt(¯y) = 10 ⇣ 1x20.5¯y ¯b ⌘ (t) N m (2) Approximating the structural dynamics using all modes with a frequency below 10 Hz, compute the time history of the root bending moment using the direct recovery approach (at least for the first 20 s). Identify the maximum bending moment reached. 5. (Optional) Are you able to show what is required to compute the bending moment using the method of mode acceleration to recover the bending mo- ment. In this case, what is the di↵erence between the maximum bending estimated using the direct recovery and this acceleration modes method? 6. (Optional) Is it possible to use the computed impulse response to compute the step response to the same input i.e., mt(¯y) = 10 ⇣ 1x20.5¯y ¯b ⌘ s(t)N m. If yes sketch how. The methods necessary to solve all steps must be written by hand on paper and a copy must be submitted together with the numeric solutions. All inte- grals could be solved using the symbolic integrator of MATLAB (or any other symbolic solver). The eigenvalues can be computed using the MATLAB eig routine (or any other numerical tool). Clarifications on the text may be asked until Friday April 22, 2022. To submit the responses and the photos of the written solution you will have to connect to a MS Form page. The answers will have to be submitted after 10 days starting from April 22, 2022 (so by 1:00 pm of May 2, 2022). The link to the MS Form page to submit the answers will be provided on April 26, 2022. DATA•persone[☐☐Ei10621786ABCD=1786Da=61CB=87i✗1=0,5'✗z=-y• b =11MSEMI-Span•C1=4,5MROOT•CZ=2,5MTIP.1= 34º •EI/Ù)= 10°6,7+5Ù[kgm2 •GJ/4-) 10°9,38 -6,25Ùkgm2 2 I •MIÙ)=68+23ÙKgn I •IO/4-)=160-51Ùkg52È ad=0,38+0,5Ùm È UN-AERODYNAMICFRAME:✗,Y4dyEACG'STRUCTURALFRAME:Ì,4-✗✗•b-=bCOSI ICOÒ4EAnùacCGLOADSoPz=L-nii+ MÒD •my=Mac+Le_(IcoÒ + MÒddl + niid =Mac+Le_(Ico + ndr ) Ò +niidIO•NECLECTINGAEDNODTNAMICFORCES:◦ Pz =-miei+ MÒD •my=- IOÒ + niid VWPb- I •Sui= |SWIÌEIW " dy + | sotto>po'DI 00BENDINGtorsioni Ù .swe= | , sèpzd + [ SotnydyÈ5II • |SÌ "EIW" dj + | . SÒÌJPO ' da = | , sè/ -miei+ mòd)dà + |SÒI - IOÒ + niid/di 00 RITZ-GALETZKINnw : ---• wlx.tl =i.eNwiIx)qui/t)=rlw91 ---MW• 0/x.tl =i.eNoi1×190il -1)=Io9,0b-11TIl|T|••• 59 | ,nw EIN-wdy-9.in + 59 [ IO GJP IoDI90+ 59 |; Nwtnnwdy9W +---O tiÀb-T•°T••- STÈ | , Nwmdniodi9.0 + STÈ | ,noIONIDI9.0 - Stò | , NÉndn-wdy-9.in =o---' / È l'T" È •• È t..-• 59in NwEI Nyudi9¥ + | , NÉnrlwdei9WNwmdIodi9.0 +...- | ..--- Iisis -+ STÒ 'T'T.. Iotonodi99 - | , NÉndn-wdy-9.in =o- IO GJP IODI99 + | ,- 1 .-I°TA Kositi= [ Sitidi = | osi/ti = siti teot°'NDICIALFunction:a/t)= | °h(t)S/t-T) DI itISTHERESPONSETOaSTEPunitcomUTEDTHROUGHTHE>< >>>: 00  DA < 50 x1=1 50  DA < 99 x1= .5 00  CB < 50 x2=1 50  CB < 99 x2= 1 (1) Consider the cantilever wing of the assignment #1. To approximate the aero- dynamic behaviour use a quasi-steady strip theory approximation, considering always that the wing flies at Mach M =0 .65 + x2·0.05. 1. Compute the flutter speed of the aircraft using as degrees of freedom all modes computed in assignment #1 with a frequency below 10 Hz. 2. Consider to fly at a dynamic pressure that is d%, where d= 15 + 2 ·x1·x2, below the flutter dynamic pressure (i.e. q= 1 d100 qF). Compute the RMS of the root bending moment, using a direct approach, considering the Von Karman PSD as input for the gust speed g= L ⇡ 1+ 8 3(1.339⌦ L)2 (1 + (1 .339⌦ L)2)11/6 (2) with⌦= ! V and L= 2500 ft = 762 m. (Hint: the problem can be solved in frequency domain. The integral required could be solved using the matlab function “trapz”, ensuring that 1Example: 10134997 ! A=4, B=9, C=9, D=7,sothetwonumberswillbe DA =74, CB =99. 1 Figure 1: Wing planform. the sampling frequency and the maximum frequency used to integrate are chosen with care). 3. Compute the response of the system to a discrete gust with the classic law “1-cosine”, using as gust intensity wg0=6 .35 m/s, and as gust length the minimal required by CS-25 standard, i.e. Lg= 50 ft. Compute the maximum root bending moment during gust using the mode acceleration approach. For all gust computations do not consider the gust penetration e↵ect. Clarifications on the text may be asked until Thursday June 3rd, 2022. To submit the responses and the photos of the written solution you will have to connect to a MS Form page. The answers will have to be submitted after 10 days starting from June 3st, 2022 (so by 1:00 pm of June 13, 2022). The link to the MS Form page to submit the answers will be provided on June 6th, 2022. 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