Combined sub-harmonic resonances of nanobeam on fractional visco-Pasternak type foundation

Abstract

In this work, we observe combined parametric and external sub-harmonic resonances of order one-third of a geometrically nonlinear nonlocal nanobeam model resting on a fractional visco- Pasternak type foundation. Euler-Bernoulli beam theory, nonlinear strain-displacement relation and nonlocal elasticity constitutive equation are employed to obtained fractional order governing equation for the transverse vibration of a system. Under the assumption of small fractional damping, we used the perturbation multiple-scales method to obtain an approximated analytical solution for the frequency-amplitude response for variable axial and transverse external loads. Several numerical examples are given to show the effects of different parameters on frequencyamplitude response.