Abstract
Phase plane analysis of the nonlinear spring-mass equation arising in modeling vibrations of a lumped mass attached to a graphene sheet with a fixed end is presented. The nonlinear lumped-mass model takes into account the nonlinear behavior of the graphene by including the third-order elastic stiffness constant and the nonlinear electrostatic force. Standard pull-in voltages are computed. Graphic phase diagrams are used to demonstrate the conclusions. The nonlinear wave forms and the associated resonance frequencies are computed and presented graphically to demonstrate the effects of the nonlinear stiffness constant comparing with the corresponding linear model. The existence of periodic solutions of the model is proved analytically for physically admissible periodic solutions, and conditions for bifurcation points on a parameter associated with the third-order elastic stiffness constant are determined..
Highlights
The importance of graphene can be highlighted in the title “The Arms Race for Graphene is Officially On” of an article published in the Wall Stress Daily on May 21, 2014 [20]
A headphone equipped with a graphene membrane has become a real life application of graphene [16, 23], the mathematical model for the membrane reported in the work [23] considered graphene as a linear elastic material and a linear lumped parameter model based on Hooke’s law was used to estimate the frequencies of vibrations
We demonstrate that the second order material constant D is an important factor in modeling the patterns of graphene in vibrations and stability of electrostatic pull-in devices just as reported in [9] for mechanically operated devices [22]
Summary
The importance of graphene can be highlighted in the title “The Arms Race for Graphene is Officially On” of an article published in the Wall Stress Daily on May 21, 2014 [20]. It is assumed that a voltage is applied to the system creating an electrostatic force to pull the plate towards the substrate This is commonly considered as the simplest model for stability study of electrostatic pull-in devices. We demonstrate that the second order material constant D is an important factor in modeling the patterns of graphene in vibrations and stability of electrostatic pull-in devices just as reported in [9] for mechanically operated devices [22].
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