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A lumped-parameter nonlinear spring-mass model which takes into account the third-order elastic stiffness constant is considered for modeling the free and forced axial vibrations of a graphene sheet with one fixed end and one free end with a mass attached. It is demonstrated through this simple model that, in free vibration, within certain initial energy level and depending upon its length and the nonlinear elastic constants, that there exist bounded periodic solutions which are non-sinusoidal, and that for each fixed energy level, there is a bifurcation point depending upon material constants, beyond which the periodic solutions disappear. The amplitude, frequency, and the corresponding wave solutions for both free and forced harmonic vibrations are calculated analytically and numerically. Energy sweep is also performed for resonance applications.

The graphene-based resonator and its application to mass sensing based on nonlinear waves have been poorly studied numerically [

where

elastic stiffness constant, and

A graphene sheet with uniform cross-section in axial vibration with fixed-free ends can be modeled by sub- stituting (1) into the standard balance of momentum equation

Here, we use

where

Equation (4), then, provides the relationship between the applied force

Our lumped parameter model is based on assuming that the density function is given by

For fixed time

Equation (7) gives the following nonlinear spring-mass equation

The corresponding autonomeous equation of (8) in which

constant in (1). Using the change of variable

where

We will show that for given initial conditions

where

We make the following observations:

1) The

2)

3) when

We prove that for any

which is an even function. Therefore it is enough to consider

Case (a):

Case (b):

Case (c):

We conclude that the bifurcation point for a given

Furthermore, for each

It is demonstrated in

Multiple scales method is often used to solve nonlinear equations with small parameters in nonlinear vibrations. Double scales are used herein to find an approximate solution of the first order for Equation (10). The solution is then compared to results obtained by numerical integration using Matlab. The new time scales are

Instead of determining the solution as a function of

sought to have the following form

Substituting (13) in (10) and identifying the term of the same power of

We will show that the solutions of Equations (14) and (15) are given by:

where

• The solution of Equation (14) has the following form:

Substituting

the following equation is obtained

To avoid unbounded solutions, we set the secular terms of the

whose solution gives

the original time using

To find

and obtain (16), where

Remark

The solution for

In this section we characterize the nonlinear spring Equation (10) by a harmonic excitation and studying the system’s nonliear responses. The equation of motion is given by:

where

We use the frequency sweep method to detect nonlinear resonnance of the nonlinear system by direct intergration. The method begins by defining a grid of frequencies around the linear resonnace and intergrate the

system at each point of frequency. The maximum displacement of the solution is then plotted against the fre- quency mesh. The curve shows a peak corresponding to the nonlinear resonnance of the system. The numerical results show the dependence of the nonlinear frequency on the magnitude of excitation and on the parameter

A simplistic nonlinear spring model is derived from the axial wave equation of a graphene sheet based on the

quadratic constitutive stress-strain equation. Using phase plane analysis, existence of periodic wave solutions

and bifurcation points depending on the parameter

method of time scales depending on

constant

parameter

The paper’s first co-author acknowledges the funding provided by the NPRP grant 08-777-1-141 from the Qatar National Research Fund (a member of Qatar Foundation) to Prof. Prabir Daripa of Texas A& M University at College Station, TX 77842, USA while working on this project.