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A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method

In this study, size‐dependent thermo‐mechanical vibration analysis of nanobeams is examined. Size‐dependent dynamic equations are obtained by implementing Hamilton's principle based on Timoshenko beam theory and then combined with stress equation of nonlocal elasticity theory. The separation of variables total method and finite element formulation is utilized to solve the eigenvalue problem. Local and nonlocal stiffness and mass matrices are firstly derived by using a weighted residual method for the finite element analysis. The accuracy of the finite element solution is demonstrated by comparisons with the earlier studies. Then, nondimensional frequencies of nanobeams with different boundary conditions based on a nonlocal finite element method are presented for vibration analysis that cannot be analytically solved under different parameters. It is aimed to emphasize the importance of the nonlocal finite element method in the size‐dependent vibration behavior of nanobeams which form different components of nano‐electro‐mechanical systems.