Primary and subharmonic resonance features of a graphene nanoplatelet strengthened beam in a thermal ambient directly attacking nonlinear integro-partial differential governing equations

Abstract

The nonlinear aspects of advanced structures may be utilized for design purposes and making use of advantages and avoiding disadvantages of such structures. On the other hand, developing information from secondary resonance examinations of advanced structures is worthwhile for engineering objectives such as mass detection, energy harvesting and damage detection of systems likewise structures. Moreover, a precise solution methodology discards any possibility of qualitative errors and incorrect prejudice. Accordingly, for the first time, the nonlinear primary resonance alongside the subharmonic resonance of an advanced beam structure strengthened by means of graphene nanoplatelets in a thermal ambient is addressed in details employing the direct method of multiple scales. With the aim of defeating to the disadvantageous influences of the discretization procedure on the nonlinear dynamics of the graphene nanoplatelet strengthened beam, the nonlinear integro-partial differential equation of motion in accompanied by the in-homogeneous boundary conditions are attacked directly resorting to the method of multiple scales. It is shown that how an FGX distribution pattern of graphene nanoplatelets for a beam subjected to a subharmonic resonance excitation relative to an FGO distribution pattern effectively extends the range of excitation frequency that leads to a bistable solution. This deduction is very proficient for harvesting energy scopes. A similar finding is observed for a CC beam with respect to a CS beam. It is seen that when an FGX beam is exposed to a subharmonic resonance in a wider range of force amplitude experiences a non-trivial equilibrium state relative to an FGO beam. However, the strengthening of the beam with graphene nanoplatelets decreases the lower bound of the excitation frequency required for the activation of subharmonic resonance more significant for an FGX distribution pattern. Moreover, the primary resonance frequency response shows different trend prior to and after the situation of the external force frequency to the first natural frequency ratio equal to one for an FGX and an FGO graphene nanoplatelet strengthened beams. The presented results open a new view of the future of engineering designs for various engineering scopes such as energy harvesting, mass detection and damage detection purposes employing advanced structures.