Combined resonance of magnetically influenced nanobeam on fractional visco-Pasternak type foundation

Abstract

Nanostructures with similar shapes as macro structures such as beams, plates or shells are grown using various technological processes. Nano-scale structures mechanical behaviour is similar to macro structures with the main difference that influences of various size-effects cannot be neglected. Particulary, dynamic behaviour of nanostructures is significant for the development of new types of devices such as nanoactuators, nanosensors or resonators. Nonlinear oscillations and dissipation effects from different sources are important for the investigation of nano-scale resonators. Application of experimental and atomistic methods is limited only to specific and less complex nanostructure systems. Modified continuum based theoretical models have attracted a great attention of researchers in recent years. Here, we observe combined parametric and external resonance of a geometrically nonlinear nanobeam 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 systems, accounting size effects and dissipation effects from external medium. 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 in combined parametric and external resonance case for variable axial and transverse loads. Influences of different system parameters on frequency-amplitude response are examined on several numerical examples.