Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams

Abstract

An analytical formulation combined with a fractional-order time derivative damping model has been developed to conduct a comprehensive study on the large amplitude free and forced vibration response of carbon nanotubes (CNTs)/fiber/polymer laminated multiscale composite beams. The Caputo fractional derivative of order α is employed to incorporate the viscoelastic material having nonlinear behavior. The governing equations of CNTs/fiber/polymer composite (CNTFPC) beams are coupled second order nonlinear partial FDEs (fractional differential equations) which are derived based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity. Halpin–Tsai equations and fiber micromechanics are used in hierarchy to predict the bulk material properties of the multiscale nanocomposite. The carbon nanotubes are assumed to be uniformly distributed and randomly oriented through the epoxy resin matrix. Discretized by the Galerkin approximation, the perturbation method of multiple time scales is employed to obtain the nonlinear natural frequencies, amplitude–frequency equation and time history of the beams with hinged–hinged boundary conditions. The effects of the Caputo fractional derivative order, beam geometry, volume fraction of fibers and weight percentage of SWCNTs and MWCNTs on the nonlinear oscillation of the CNTFPC beams are investigated through a detailed parametric study. It is found that nonlinear natural frequencies, amplitude–frequency relationship and time history are characterized by viscoelastic damping coefficient which are connected with the natural frequency by the exponential relationship with a negative fractional exponent.