Steady state vibration analysis and mitigation of single-walled carbon nanotubes based on nonlocal Timoshenko beam theory

A direct analytical approach is used to suppress the steady state vibrations of a nanotube resting on Winkler foundation. Lumped masses are only used to suppress vibration by imposing points of zero displacements. The formulation of the problem is based on the constitutive equations of Eringen’s nonlocal theory of elasticity to accommodate the small length scale effect. In addition, the Timoshenko beam model is used to allow for the effects of transverse shear deformation and rotary inertia. An exact solution is obtained using the Green’s functions. The proposed method is efficient, convenient and exact. In addition, it is free from numerical inaccuracy and exhibits less computational time compared to other methods. Effects of nonlocal parameter and stiffness of Winkler foundation on non-dimensional vibration frequencies is investigated as well. Different boundary conditions at tube ends are considered.