Free vibration analysis of an aircraft wing

Aeroelastika je znanost koja se bavi elastičnim svojstvima strukture izložene protoku okolnog fluida. Zrakoplovna elastična konstrukcija krila se može opisati i ispitati modalnom analizom. Rezultati modalne analize su vlastite frekvencije i oblici vibriranja. Vlastita frekvencija je karakteristično svojstvo konstrukcije kod koje bi vanjskom pobudom ili rezonancijom došlo do vibracija najvećih mogućih amplituda. Takve vibracije su štetne, a ponekad dovode do destrukcije. Iz tog razloga se vlastite frekvencije trebaju poznavati. U radu je opisana teorija slobodnih vibracija prema teoriji tankih greda (Euler - Bernoulli teorija). Euler- Bernoulli teorija tankih greda se koristi samo za tanke grede i za osciliranje u samo jednom stupnju slobode. Izrađen je računarski model prema matemtičkoj fomulaciji teorije u progrmaskom jeziku Python. Korišten je NACA4415 aerodinamički profil krila s najdužom tetivom 1000 mm, duljina krila odnosno konzolne grede je 5000 mm, a materijal krila/konzole je Al6061. Napravljena je teorijska i analiza metodom konačnih elemenata konzolne grede i punog modela NACA4415 krila. Izrađeni su modeli krila s konstrukcijsim elementima koji se nalaze u pravom krilu. Krila su izrađena od uzdužnica kružno-prstenastog poprečnog presjeka, tri različita tipa rebra (rebro vrha, normalno rebro i rebro za tank goriva) i od jednodijelne oplate koja okružuje prethodno opisani kostur krila. Modeli se razlikuju po debljini jednodijelne oplate, ali su im jendake vanjske mjere. Debljine stijenka krila “s” su 8 mm, 10 mm, 12 mm, 14 mm i 16 mm. Napravljena je analiza statičke čvrstoće krila s najmanjom debljinom stijenke (krilo s8) i modalna analiza svih modela. Prikazani su i uspoređeni rezultati oblika vibracija vlastitih frekvencija i vrijednosti vlastitih frekvencija. Rezultati pokazuju da se greda s odgovarajućom teorijom može koristiti umjesto kompliciranog modela, te da Euler - Bernoulli teorija daje približna rješenja numeričkoj metodi za niže frekvencije i za modele kojima su debljine bar 10 manje od dužine. Analize utjecaja debljine stijenke na karakteristike slobodnih vibracija upućuje na to da, pri malim debljinama stijenke, čak duplo deblja stijenka ne smanjuje drastično veličine vlastitih frekvencija. Drugim rječima, s povećanjem debljine stijenke oplate ne može se drastično smanjiti vrijednost vlastitih frekvencija.Aeroelasticity is the science that deals with the elastic properties of a structure exposed to the flow of the surrounding fluid. The aircraft elastic wing structure can be described and examined by modal analysis. The results of modal analysis are natural frequencies and mode shapes. The natural frequency is a characteristic property of the structure in which vibrations of the largest possible amplitudes would occur by external excitation or by resonance. Such vibrations are harmful and sometimes lead to destruction. For this reason, natural frequencies need to be known. The paper describes the theory of free vibrations according to the theory of thin beams (Euler - Bernoulli theory). Euler-Bernoulli theory of thin beams is used solely for thin beams and for oscillation in only one degree of freedom. A computer model based on the mathematical formulation of the theory in the Python programming language was developed. The NACA4415 aerodynamic wing profile with the chord 1000 mm, with the length of the wing (cantilever beam) of 5000 mm, and the material of the wing/console Al6061. A theoretical and finite element method analysis of the cantilever beam and of the NACA4415 flying wing model was performed. Wing models with construction elements which can be found in the real aircraft wing were made in CATIA. The wings are made of stingers with circular-annular cross-sections, three different types of ribs (tip rib, normal rib and fuel tank rib) and a one-piece skin surrounding the previously described wing skeleton. The models differ in the thickness of the one-piece skin, but their external dimensions are the same. The wing wall thicknesses “s” are 8 mm, 10 mm, 12 mm, 14 mm and 16 mm. An analysis of the static strength of the wing with the smallest wall thickness (wing s8) and a modal analysis of all aforementioned models were made. The results of natural frequency vibration shapes and natural frequency values are presented and compared. The results show that the beam with right theory can be used instead of the complicated model, and that the Euler - Bernoulli theory provides approximate solutions to the numerical method for lower frequencies and for models with thicknesses at least 10 less than the length. Analysis of the influence of skin thickness on the characteristics of free vibrations suggests that, at small wall thicknesses, even twice as thick wall does not drastically reduce the magnitude of natural frequencies. In other words, with an increase in the wall thickness of the skin, the value of the natural frequencies cannot be drastically reduced