The constitutive nonlinear equation of the system was derived using the Energy method and Hamilton's principle. A lumped mass model was developed to study the essential dynamics of the graphene cantilever. A key focus was on investigating the dynamic pull-in conditions of the system under both constant and harmonic excitation.
In addition, we investigated the effect of the excitation frequency on the dynamic response of the graphene cantilever beam under harmonic loading. The equilibrium position is reached when the restoring force of the beam matches the electrostatic force [32]. This study contributed valuable insights into the dynamic behavior of the model under a constant load scenario.
In addition, the drag phenomenon of the same lumped mass model but excited by harmonic loading was investigated in two separate research works: [12] and [20]. 30] delved into the study of the nonlinear equation of the graphene beam and the existence of several natural frequencies of the system. The interaction of the attractive electrostatic force due to the potential difference and the nonlinear restoring force of the beam is expected to produce high-frequency oscillations.
This section focuses on the derivation of the constitutive nonlinear equation for a cantilever beam made of graphene using Hamilton's principle, an essential concept in variational mechanics.
Constitutive stress-strain equation for graphene
This characterization of the stress-strain behavior of the graph will be used in the upcoming modeling section.
Model equation for Euler-Bernoulli beam made of graphene
By integrating the stress-strain equation (2.1) with respect to strain, we obtain the strain energy density, which represents the energy stored in the material per unit volume. This energy density is a measure of the potential energy stored in the beam due to deformation. By integrating this quantity over the entire volume of the beam, we can determine the total potential energy of the system.
Then the total potential energy 𝐸𝑝𝑜𝑡 can be expressed as 𝐸𝑝𝑜𝑡 = 1. 2.8) Inserting the axial stress in Eq. The kinetic energy of a beam can be calculated from the mass distribution along the length of the beam and the velocity of its individual mass elements. The work 𝑊 done by the external force on the cantilever beam at the free end can be written as.
Hamilton’s Principle
The boundary term in time vanishes in Eq. 2.22) due to the boundary conditions imposed on the virtual displacement. To express the equation above Eq. 2.23) in terms of displacement variation 𝛿𝑤 various integration by parts should be implemented as shown below. 2.26). By definition, the variation 𝛿𝑤 and 𝛿𝑢 is arbitrary, therefore each group term must be zero to fit the equation Eq. 2.26), which leads to the following equation of motion and boundary conditions:
Analytic solution
This static response of the system is associated with the concave form of the deflection function 𝑤(𝑥). Note that as 𝐷 approaches zero, the analytical solution in (3.14) coincides with the classical deflection equation for a cantilever beam under a point load, where the beam is assumed to be a linear elastic material and that the deflection is small compared to the length of the beam. It also assumes that the load is applied perpendicular to the longitudinal axis of the beam and that the beam has a uniform cross-sectional area.
Lumped mass model
Therefore, to study the essential dynamics of the graphene beam undergoing a point force at the free tip, we use one-mode Galerkin approach.
Dimensionless single-degree-of-freedom model
Constant voltage
We then focus on the phase diagram, which plays a crucial role in understanding the dynamics of the system. The periodic solutions of Eq. 5.1) correspond to closed curves or loops in the phase diagram, also called limit cycles. Of particular interest is the curve, which separates the regions of the system with different dynamics, known as separatix.
If the initial conditions of the system lie within the separatix, the solution is periodic; otherwise it is not periodic [32]. To determine the range of positive parameter values of 𝛼 and 𝜆 that lead to periodic solutions, we need to analyze the separation, which occurs when the horizontal axis on the right-hand side of Eq. Consequently, the system exhibits an oscillatory or periodic solution ifℎ𝛼,𝜆(𝑦*)≤0for some positive parametric values of 𝛼and 𝜆.
Another crucial parameter in MEMS devices is the retraction time, which represents the time it takes for the system to collapse. The integration of this expression over the interval [0,1] corresponds to the distance that the tip of the beam must travel to reach the fixed electrode, leading to the occurrence of the retraction phenomenon. Using a similar approach, we can determine the oscillation period 𝑇 for our system by subtracting 𝑑𝑡 from Eq.
Time-dependent voltage
Then, using the second derivative test, we can find that 𝑓𝛼(𝑦) has a local maximum at the least critical point 𝑦1 = 2𝛼−3+.
Constant voltage
Time-dependent voltage
6-5 show that when the excitation frequency Ω is chosen to be close to the natural frequency ω𝑛, the vibration amplitude of the graphene cantilever beam increases significantly compared to its behavior under constant stress conditions. Letting the excitation frequency be exactly the same as the natural frequency results in a significant increase in the vibration amplitude of the graphene cantilever beam. However, this increment is accompanied by the occurrence of a traction instability scenario in which the free tip of the beam collapses into the fixed electrode, see Fig. 3b.
In this thesis, a comprehensive analysis of the static and dynamic behavior of a graphene cantilever beam subjected to electrostatic actuation at its free tip was performed. First, an analytical solution to the nonlinear static problem was derived and its consistency with the classical linear solution was demonstrated in the limit where the second-order elastic stiffness constant approaches zero. For the case of constant voltage excitation, the system exhibited periodic solutions when the values of the parameters 𝛼 and 𝜆 lay below the separatix curve, as illustrated in Fig.
The dependence of the deflection amplitude, frequency and pull-in time on the excitation parameter𝜆 for a fixed value of𝛼 was demonstrated in this chapter. In addition, it was observed that the maximum deflection amplitude occurred just below the separatix value for the given 𝛼. Furthermore, simulations of the cantilever beam under harmonic load excitation showed that choosing an excitation frequency near the resonance frequency of the beam could lead to structural collapse, even though the parametric values were below the pull-in conditions.
Appendix
2] Alexander A Balandin, Suchismita Ghosh, Wenzhong Bao, Irene Calizo, De-Salegne Teweldebrhan, Feng Miao, and Chun Ning Lau. Nonlinear vibrational behavior of graphene resonators and their applications in sensitive mass detection. Nanoscale Research Papers. 6] Suchismita Ghosh, Irene Calizo, Desalegne Teweldebrhan, Evghenii P Pokatilov, Denis L Nika, Alexander A Balandin, Wenzhong Bao, Feng Miao, and C Ning Lau.
Non-local mass nanosensor model based on the damped vibrations of a single-layer graphene sheet influenced by an in-plane magnetic field. On the application of Sturm's theorem to the analysis of dynamic pull-in for a graphene-based mems model. Analysis of the mass model for the cantilever beam subjected to Grob's swelling pressure.
![Figure 2-1: A schematic of a deflected beam under a point load at the free tip According to the Euler-Bernoulli beam theory, the cross section of the beam re-mains plane and perpendicular to the beam’s centerline [32]](https://thumb-ap.123doks.com/thumbv2/azdokorg/10963436.0/14.918.248.673.434.731/figure-schematic-deflected-according-bernoulli-section-perpendicular-centerline.webp)
