Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method

In the present work, damped free and forced vibrations of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field due to surface effects surrounded on a non-linear viscoelastic medium using nonlocal Euler-Bernoulli beam theory multiple time scales method are investigated. Lorentz force equation is used obtain vertical applied voltage system. The as well combinational foundation considered, finally, dynamic equilibrium equations used, motion extracted. following, Galerkin methods analytical solutions extracted vibrational responses relevant coefficients discovered for two both simple support (S-S) clamped (C-C) boundary conditions. , different parameters such non-local parameter electric-magnetic fields, effect hardness-linear damping nonlinear considered foundation, field, base modes forms considering effects, etc. will be studied. As one results this study, presence has increased curvature deviation right stiffening effect. other words, factor increase Also, it predictable that load affect position moves away from center nanotube toward supports, amplitude response decreases significantly, relative reduction greater C-C condition than S-S condition. It also important note location no rate curve peak, degree nonlinearity