Nonlinear free and forced vibrations of graphene nanoplatelet reinforced microbeams with geometrical imperfection

Nonlinear free/forced vibration of a functionally graded graphene nanoplatelet (GNP) reinforced microbeam having geometrical imperfection which is rested on a non-linear elastic substrate have been studied in the present research. Graphene Platelets have been uniformly and non-uniformly scattered in the cross section area of the microbeam. Non-uniform distribution of GNPs is considered to be linear or non-linear type. Geometric imperfection is considered similar to the first vibration mode of microbeam. Size effects due to micro-rotations are captured in this study by means of modified couple stress elasticity. In the case of forced vibration, a uniform harmonic load is exerted to the top surface of microbeam. Harmonic balance method has been implemented to solve the non-linear governing equation of microbeam having quadratic and cubic nonlinearities. In this regard, frequency-amplitude curves are obtained and their trends are studied by changing of GNP amount and distribution, geometric imperfection, forced amplitude and hardening foundation.