Dynamic analysis of nano scaled continuous and discrete structures based on nonlocal finite element formulation (NL–FEM)

In this master of science thesis, free vibration analysis of continuous and discrete structures is performed by using nonlocal elasticity. Axial rod and beam as continuous systems and truss and frame as discrete systems are modelled at the nano–scale. In the first of examined topics in the study, the nonlocal vibration of simple axial, Love–Bishop and torsional rods embedded in the elastic media is investigated. The nonlocal transverse vibration of Euler–Bernoulli and Timoshenko beams lying on Winkler–Pasternak elastic media and under thermal environment is studied in the second problem. For this type structures, firstly, the obtained governing equations via Hamilton's Principle are solved with the help of separation of variables that is analytical method. Then, finite element formulation that takes into consideration nonlocality and other environmental effects is presented. In the third and fourth problem types, size–dependent vibration of truss and frame is explained, respectively. With in the scope of numerical results, the nondimensional frequencies of truss and frames with different geometries as well as rods and beams with general and attached boundary condions are calculated for different parameters such as element sizes, mode number, atomic parameter, elastic media/foundation parameter, temperature, attachment ratio and discrete system geometric parameters and a detailed discussions of these numerical results is carried out. The obtained results are compared by analytical and the developed finite element methods. The fact that finite elements give sufficient accuracy results to the analytical values show the usability of the method in the nonlocal free vibration problems. Finally, attained conclusions from all numerical results and discussions are outlined. Fundamentally, with this study, it is aimed to understand the mechanical behaviors of different types of nano–scaled structures by presenting the finite element results of some analyses which cannot be analytically performed.