Combined resonance of a nonlocal nanobeam on fractional Pasternak-type viscoelastic foundation

Abstract

In this communication, we observe the interaction of fundamental parametric resonances with subharmonic resonances of order one-half of a geometrically nonlinear nonlocal nanobeam model resting on a fractional Pasternak-type viscoelastic foundation. Euler-Bernoulli beam theory and nonlinear von Karman strain-displacement relation 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 multiplescales method to obtain an approximated analytical solution for the frequency-amplitude response. Combined parametric resonance from axial load and subharmonic resonance under external excitation are examined for different parameters of the model. Validation of the multiple scales solution against numerical solution in the phase plane and Poincare map will be provided.