Vibration Analysis Of The Carbon Nanotubes By A Hamiltonian-based Method

Nanobeams have become one of the most promising new materials for nanotechnology due to their novel electronic and mechanical properties.They are being extensively utilized as key structural components in nano-/micro-electromechanical systems(NEMS/MEMS),e.g.,nanosensors,nanoactuators.Consequently,understanding the dynamic behaviors of the nanobeams is of great importance to improve the design of the NEMS/MEMS.However,the existing method is not suitable for the beam-like structure in the nanoscale.The classical continuum mechanics was found inapplicable to nanoscale structures.The atomistic simulation and experimental study are very time consuming and expensive for the complex or large-sized nanobeam system.Therefore,there is a great demand for an effective method to the dynamic analysis of the nanoscale beam-like structures.To this end,a new Hamiltonian-based approach is presented for finding exact solutions for transverse free and forced vibrations of single nanobeam and double-nanobeam-systems resting on a foundation or embedded in an elastic medium.The continuum model is established within the frameworks of the symplectic mathematics and Eringen's nonlocal Euler-Bernoulli and Timoshenko beam beams.In the Hamiltonian system,the dual variables of the displacement and angel of rotation are the generalized shear force and bending moment,respectively.The higher-order governing partial differential equation in the classical Lagrangian system is simplified as a set of lower-order ordinary different equations by introducing an unknown vector composed of fundamental variables and their dual variables.The vibration of single nanobeam or double-nanobeam-system is finally reduced to a symplectic eigenproblem.The symplectic eigenfunctions are obtained by the method of separation of variables.Exact frequency equations,vibration modes are obtained by using symplectic eigenfunctions and end conditions.Analytical solutions for steady-state forced vibration of the nanobeams are derived by the relations of wave propagation in the nanobeams.Numerical results verify the accuracy and efficiency of the present method.A systematic parametric study on the small size effect,the coefficient of the foundation,the boundary condition and the material property is provided also.The comprehensive results could serve as benchmark results for verifying numerically obtained solutions.