Free vibrations of a beam considering different boundary conditions

Tema ovog rada su slobodne vibracije grede. U uvodu upoznali smo se sa gredom kao elementom inţenjerskih konstrukcija podvrgnutim različitim dinamičkim opterečenjima te pojavom strukturnih vibracija, u kojem smo ukratko prikazali i osnove vibracija. U radu je izraĎen matematički model za opis poprečnih vibracija Euler-Bernoullijeve grede uz četiri različita rubna uvjeta. Prikazani su detaljni izvodi Eulerove jednadţbe grede na temelju koje definiranjem rubnih uvjeta analitički dobivamo vlastite frekvencije greda. Nadalje, opisan je računalni program za konačno elementnu analizu u kojem su provedene simulacije za sva četiri rubna uvjeta uz promenu omjera duljine grede i visine poprečnog presjeka. Dan je kratak osvrt na oblike vibriranja pojedinih rubnih uvjeta i konvergenciju rezultata kod konačno elementne analize. Dobiveni analitički rezultati usporeĎeni su sa numeričkim da bi istknuli meĎusobne razlike te kako bi se pokazalo za koje omjere duljine grede i visine poprečnog presjeka moţemo upotrijebiti Euler-Bernoullijevu teoriju grede.The subject of this paper are free vibrations of beam. In the introduction we were introduced to the beam as an element of engineering constructions subjected to various dynamic loads and the phenomenon of structural vibrations, in which we quickly glossed over the basics of vibration. In the paper a mathematical model was constructed for the purpose of describing transverse vibrations of the Euler-Bernoulli beam with four different boundary conditions. Detailed derivations of the Euler beam equation are shown, which we can use to analytically obtain natural frequencies of beams by defining the boundary conditions. Furthermore, we have a description of a computer program for finite element analysis in which we carried out simulations for all four boundary conditions with alterations to the ratio of the length of the beam and the height of the cross section. There is also a brief overview of the mode shapes for certain boundary conditions and the convergence of results during the finite element analysis. The gathered analytical results were compared with the numerical values to highlight mutual differences and to show for which ratios of the length of the beam and the height of the cross section we can use the Euler-Bernoulli beam theory