Free vibration of a nanogrid based on Eringen’s stress gradient model

Abstract

This article deals with the free vibration analysis of a two-member nanogrid. First, the governing differential equations of flexural and torsional vibrations are found based on nonlocal elasticity theory. Second, the problem under study is formulated, considering the interaction of bending and torsion by direct method. It contains four partial differential equations and twelve boundary and compatibility conditions. Third, the problem is derived by using the variational method. Fourth, the problem at hand is analytically solved and natural frequencies of the structure are obtained. Finally, a finite element solution is developed for the problem at hand. A two-node six-degree-of-freedom nanogrid element is introduced. Comparison of both exact and numerical results shows an excellent agreement between findings via both methods. Findings can be served as benchmarks for future researchers.

Communicated by Chandrika Prakash Vyasarayani

Keywords:

• Free vibration
• nanogrid
• Eringen’s differential nonlocal elasticity law
• bending-torsion coupling
• exact solution
• finite element solution