Analytical solutions for thermal vibration of nanobeams with elastic boundary conditions

A nonlocal Euler beam model with second-order gradient of stress taken into consideration is used to study the thermal vibration of nanobeams with elastic boundary. An analytical solution is proposed to investigate the free vibration of nonlocal Euler beams subjected to axial thermal stress. The effects of the nonlocal parameter, thermal stress and stiffness of boundary constraint on the vibration behaviors of nanobeams are revealed. The results show that natural frequencies including the thermal stress are lower than those without the thermal stress when temperature rises. The boundary-constrained springs have significant effects on the vibration of nanobeams. In addition, numerical simulations also indicate the importance of small-scale effect on the vibration of nanobeams.