Application of Differential Quadrature method in free vibration analysis of nanobeams based on various nonlocal theories

Conducting experiments with nanoscale size is quite difficult to handle. In this regard, mathematical modeling and their solutions play a vital role in the field of nanotechnology. In the present work, the governing differential equations are converted into a single unknown function and then Differential Quadrature (DQ) method is employed to investigate free vibration of nanobeams based on different beam theories like Euler–Bernoulli, Timoshenko, Reddy and Levinson in conjunction with nonlocal elasticity theory for the first time. Application of DQ method in the governing differential equation converts the problem to a generalized eigenvalue problem and the solution of this gives the frequency parameters. It is observed that frequency parameters are overpredicted in Euler–Bernoulli beam theory than other types of beam theories. It is also concluded that the effect of nonlocal parameter is more in case of higher vibration modes. Present results are compared with other available results and are found to be in good agreement.