1 Introduction

Devices of nano-electromechanical systems (NEMSs) have been thoroughly investigated for their potential applications in building high sensitive sensors^[1], probes^[2], filters^[3], and resonators^[4-6]. However, due to the nano-size of the nano-devices and the nano-distances between the driving electrostatic polar plates, nonlinear phenomena, e.g., nonlinear geometry and nonlinear electrostatic forces, exist widely in nano-devices, which will seriously affect the working function and stability of the nano-devices. There exist many nonlinear phenomena in nano-devices than in conventionally large size devices. One important reason for such nonlinear effects is the change in the capacitance of the electrostatic driving devices with nano-structures. Under the nonlinear system of nanobeams, electrostatic coupling shows a relatively abundant action of parametric and external excitation coupling, e.g., nonlinear dynamic characteristics, which implies that the response of the nonlinear system consists of a complex alternate evolution process of periodic motion, quasi periodic motion, and chaotic motion along with changes in the system parameters. It is also prone to the additional nonlinear phenomena such as spring hardening or softening, sudden jump, hysteresis, pull-in instability, and frequency shift^[7-11]. Since the resonance of the nonlinear vibration of nanobeams performs many complex features, it is essential to study the resonant control of the nonlinear vibration of the nanobeams. Therefore, an investigation on the effects of the nonlinear factors on the control parameters and the method of the general calculation of the control parameters will be carried out.

The stability of the nonlinear systems and several types of nonlinearities are highly sensitive to the physical parameters, initial amplitude, and excitation frequency^{[5-6, 12]}. It is important to identify the bifurcation points and bifurcation parameters to design and control the systems under parametric excitations^[13]. Ke^[14] investigated a double-sided electro mechanically driven nano-tube resonator with the consideration of the van der Waals force. The nonlinear response of electrostatically actuated micro-resonators of the cantilever beam at the near-half natural frequency was analyzed with the consideration of the nonlinearities of electrostatic and Casimir forces. Caruntu and Knecht^[15] analyzed the nonlinear response near-half natural frequency of an electrostatically actuated NEMS cantilever.

The electrostatic driving method was usually used for the resonant excitations in resonators. However, many problems might occur when the electrostatic structure was used for the control^[16]. Since the synchronization control with the same frequency of driving forces produced by a parallel plate capacitor controller may bring superposition of vibration modes and the different frequencies of the excitation and control forces may cause mixed-frequency phenomena, the effective control performances are difficult to obtain. The magnetic force control, which requires a strong magnetic field so as to control the nanobeam vibration with the Lorentz force, suffers from two drawbacks. One is that it is contrary to the concepts of the nano-scale design for nano-devices to design large volume equipments to produce strong magnetic field. The other is that the magnetic force control usually requires a layer of metal film covering on nanobeams to create a loop current and generate the magnetic control force. The plating of the metal film on the nanobeams involves a complicated technology, which increases the manufacturing costs of nano-resonators and the nanobeam weight. It is a difficult work to find a suitable controller to control the nanobeam vibration.

The feedback control of vibration systems needs the electrical signals containing the vibration information of nanobeams, which is not easy to be obtained because of the nano-scale vibration amplitude. Kim et al.^[17] found that the graphene film had the characteristics that the resistance changed with the deformation of the nanobeams. The graphene film has several unique advantages, such as ultra-thin thickness, ultra-light weight, less effect on the nanobeam vibration, and definite changes in the resistance with deformation, and can be used as the sensors for nano-devices^[17-19]. The graphene film can be applied on the surface of nanobeams to serve as a sensor to the extract strain signals of the nanobeam vibration as the control information to provide input to the controller and then to finally achieve effective control performances for the nonlinear vibration of nanobeams.

Although the research and applications of NEMSs have expanded in recent years, NEMS devices and systems are designed mostly by trial and error. The prototype of proposed NEMSs has not been tested and modified until the desired performance is achieved. These slender nanobeams are excited via electrostatic forces. Nanobeam resonators display distinctive nonlinear characteristics due to the relatively small dimensions, which arise from a number of sources, including large deflections (geometric nonlinearities), electrostatic actuation, and Casimir effects. The electrostatic actuation can create a variety of nonlinear parametric resonances depending on the system parameters, excitation frequency, and excitation voltage^[20-22]. The control of the nonlinear vibration of NEMS devices and the achievement of the steady state vibration are important tasks in NEMS designs.

Pull-in instability, as an inherently nonlinear and crucial effect, continues to become important for the design of electrostatic MEMSs and NEMS devices. Controlling the pull-in instability and further enhancing the performance of MEMSs and NEMS devices with electrostatic actuation and sensing have been a hot research topic in recent years^[23]. The pull-in instability of a cantilever nano-actuator model incorporating the effects of the surface, the fringing field, and the Casimir attraction force has been widely investigated^[24]. An analytical model for calculating the pull-in voltage of a stepped cantilever-type radio frequency micro electro-mechanical system switch was developed based on the Euler-Bernoulli beam and a modified couple stress theory, and was validated by comparison with the finite element results^[25]. Since the pull-in instability control is universal, it is a hot research topic. The axial control forces produced by the parallel capacitive controller can give a way to control the pull-in of the nanobeam and enhance the stability of the resonator vibration. However, the axial control of the pull-in instability is a limit in the research contents of the paper, and is going to be conducted in the future research.

There are two purposes to present this work. First, the graphene sensor is used in the nonlinear vibration control of nanobeams, which deforms with the vibrations of the nanobeams, and produces a voltage signal. The vibration voltage signal is the input to the closed-loop control vibration controller, which is used as a control signal to control the nonlinear vibrations of the nanobeams. Second, an axial parallel capacitor controller is designed to control the nonlinear vibrations of the nanobeams. The nonlinearity of the electrostatic force is considered, and the nonlinear response of the electrostatically actuated nanobeams is analyzed at the near half-natural frequency. The electrostatic force and viscous damping are included in the model. The method of multiple scales is used directly in solving the partial-differential equation of motion and boundary conditions of the resonator.

2 Control system model

A flexible nanobeam suspended over a grounded substrate is considered (see Fig. 1). The beam is electrostatically actuated by applying AC voltage between the beam and the underlying plate. This paper investigates the case in which there is no DC polarizing voltage and the case with only AC voltage. The electrostatic force is considered. The length of the beam is relatively large compared with its width and thickness.

A voltage series circuit made up by the graphene resistance r and the divider resistance R is shown in Fig. 1. A control voltage U is applied on the circuits of the graphene resistance and the divided resistance. The graphene resistance changes with the nanobeam deformation^[17]. The relationship between the resistance change and the nanobeam deformation can be written as follows:

(1)

where w is the nanobeam deflection, and t' is the time. The coordinates x₁ and x₂ are the boundary values of the graphene sensor. ρ_d is the equivalent resistance coefficient of graphene, h is a half of the nanobeam thickness, and A_d is the axial cross section area of the graphene film.

The control voltage of the graphene sensor vibration can be approximated as follows:

(2)

where g_d and g_f are the linear and nonlinear feedback gains of the control voltages. The control force of the parallel plate capacitor, which is vertical to the axial direction of the controller, can be written as follows:

(3)

where

and a and b are the length and the width of the nanobeam axial capacitor controller, respectively. d₀ is the distance between the two poles of the capacitor controller. ε₀=8.854× 10^-12 C²⋅N^-1⋅m^-2 is the permittivity of vacuum.

Assume that the lateral displacement w and the axial displacement u are actuated by the electric excitation loads. Then, the strain can be written as follows:

(4)

where (⋅)' represents . The equation of motion is derived from the Hamilton as follows:

(5)

where K, Π, and W are the potential energy, the kinetic energy, and the external work, respectively. They are defined by

(6)

(7)

(8)

where x and l are the longitudinal coordinate and the beam length, respectively, the dots represent , b^* is the coefficient of the viscous damping per unit length, m is the nanobeam mass per unit length, E is Young's modulus, and I is the cross sectional moment of inertia.

The first-order fringing field correction of the electrostatic force per unit nanobeam length is^[26]

(9)

where s is the nanobeam width, g is the distance between the two poles of the capacitor excitatory, and V is the applied voltage.

In order to simplify the calculation, the axial displacement is ignored here, i.e., u=u'==0. Therefore, the Euler-Bernoulli hypothesis can be held. Considering the nonlinear vibration of the system, the partial-differential equation of motion and the boundary conditions can be written as follows:

(10)

where A and ρ are the cross-sectional area and the material density, respectively.

The boundary conditions can be expressed as follows:

(11)

The following nondimensional variables are now introduced:

(12)

where u, z, and t are the nondimensional displacement of the nanobeams, the nondimensional longitudinal coordinate, and the nondimensional time, respectively. The nondimensional motion equation and the boundary conditions are

(13)

(14)

where

Considering δ, f, and b to be small parameters, the excitation force and damping are taken as weak terms. Expanding the right-hand side of Eq. (13) around u=0, retaining the terms up to the third power of u, and setting all these terms to a slow scale by multiplying them, we have

(15)

3 Direct approach with method of multiple scales

A first-order expansion of the nondimensionl displacement u can be written as follows^[27]:

(16)

where T₀=t is the fast time scale, and T₁=εt is the slow time scale. Substituting Eq. (16) and the time derivatives into Eq. (15) and equating the terms of the same order of ε, we have

(17)

(18)

(19)

where The first-order approximate solution to Eq. (17) can be written as follows:

(20)

The nondimensional voltage is^[15]

(21)

where Ω^* is the nondimensional excitation frequency, and

(22)

4 Resonance and stability near half-natural frequency

In this section, the excitation frequency resonance at the near-half natural frequency is investigated. The excitation frequency can be written as follows:

(23)

where σ is the detuning parameter. The square of the voltage V is given by^[15]

(24)

Substitute Eqs. (20) and (24) into Eq. (18). Then, the secular terms are collected, and equal zero. One can notice that there are additional secular terms at the near-half natural frequency. With the solvability condition and stating that the right-hand side is orthogonal to every solution of the homogenous problem, we have

(25)

where

Expressing A_k in the polar form, we have

(26)

Separating the real and imaginary parts of the equation of the secular terms, the amplitude a_k and the phase γ_k of the response are governed by the following polar form of modulation equations:

(27)

(28)

where

When the nonlinear term v_k is positive, the nonlinear vibration system is a hard spring. When the nonlinear term v_k is smaller than 0, the nonlinear vibration system behaves as a soft spring. When the nonlinear term v_k equals 0, the nonlinear vibration system behaves as a linear spring. The critical control voltage, which can eliminate the nonlinear terms, can be written as follows:

(29)

The critical control voltage is a function of the actuated voltage, the length and width of the nanobeam, the gap between the nanobeams, the actuated plate, and the parameter of the axial controller.

The steady state solutions to Eq. (15) for the superharmonic resonance response correspond to the fixed points of Eqs. (27) and (28), which can be obtained by setting D₁a_k=D₁γ_k=0, i.e.,

(30)

(31)

With Eqs. (30) and (31), we obtain the frequency-response equation as follows:

(32)

The response amplitude is a function of the external detuning and the excitation amplitude.

Let E_k=a_k². Equation (32) can be rewritten as follows:

(33)

where

The derivative of Eq. (33) with respect to σ can be obtained. Let Then, at the peak of the resonance, we have

(34)

Substituting this equation into Eq. (31), we can find that cos γ_k=0. Assume that Then, we have

(35)

The peak amplitude of the superharmonic resonance a_kmax can be written as follows:

(36)

5 Saddle node bifurcation control

When p₀≠0, let (i=1, 2, 3, 4, and 5). Equation (33) can be rewritten as follows:

(37)

Let , and substitute it into Eq. (37). Then, we have

(38)

where

Equation (38) can be seen as a three-parameter universal unfolding

of the bud g(x, α)=x⁵-α^[28-29].

Case A β₁=0 (see Fig. 2)

(i) Bifurcation point set

(ii) Hysteresis set

(iii) Double limit point set

Case B β₂=0 (see Fig. 3)

(i) Bifurcation point set

(ii) Hysteresis set

(iii) Double limit point set

Case C β₃=0 (see Fig. 4)

(i) Bifurcation point set

(ii) Hysteresis set

(iii) Double limit point set

When p₀=0, Eq. (33) can be rewritten as follows:

(39)

The discussion of the case is the same as the analysis of the above case.

6 Numerical analysis

The nondemensional approximate solutions of the vibration system can be written as follows:

(40)

Substituting the approximate solutions (40) into Eq. (15) and using the Galerkin discretization method, we have

(41)

where

Let x₁=q, and x₂=. Then, we can obtain the state-space equation as follows:

(42)

(43)

The state equations (42) and (43) are a nonlinear, parametric, and excited, and time-varying model. The control performance is simulated in the following part.

7 Case study

The behavior of an electrostatically actuated clamped-clamped supported nanobeam is investigated. The nanobeam is modelled as an Euler-Bernoulli beam. The divided resistance r equals R/20. Tables 1-3 give the physical characteristics of a nanobeam, a graphene, and a control capacitor, respectively.

Figure 5 gives the frequency-amplitude curves for different control feedback gains g_f when the excited voltage is fixed and the linear feedback gain g_d equals 100. There are multi-solution phenomena when the nonlinear control feedback gains g_f are set to be 150 and 300. The vibration system is unstable. However, there are no multi-solution phenomena when g_f=226. There are no obvious changes for the peak amplitude when the control voltage varies. The point of the peak amplitude shifts to the left when the nonlinear control feedback gain g_f is set to be 150. However, the point of the peak amplitude shifts to the right when the nonlinear control feedback gain g_f is set to be 300. There are softening and hardening spring characteristics for the vibration system when g_f=150 and g_f=300.

Figure 6 depicts the frequency-amplitude curves for different actuated voltages without control. The amplitude of the steady-state oscillation increases when the excitation voltage increases. There are multi-solution phenomena for V₀=1.8 V. The vibration system is unstable. While there are no multi-solution phenomena when V₀=1.6 V or 1.4 V. The peak amplitude increases with the increase in the actuated voltage system, and the vibration system tends to be unstable. It shows that, the actuated voltage shifts the resonance to low frequency, and can introduce a resonance amplitude dependency, a jump, and instabilities.

Figure 7 shows the frequency-amplitude curves for different control feedback gains g_d when the excited voltage is fixed and the nonlinear feedback gain g_f equals 226. The vibration system behaves as a linear spring when the control nonlinear feedback gain g_f equals 226. The nonlinear vibration phenomenon can be avoided by selecting a suitable control voltage. The variations of the stiffness have almost no or few effects on the pick amplitude. However, bending the backbone curves has a significant effect. Hence, there must be a suitable stiffness to avoid the jump.

Figure 8 shows the curves of the peak amplitude of the superharmonic resonance for the nonlinear vibration system varying with the actuated voltage for different gap distances between the nanobeam and the actuated plate. The amplitude increases with the increase in the actuated voltage. For a fixed value of the actuated voltage, the amplitude grows with the increase in the gap distance between the nanobeam and the actuated plate.

The curves of the critical control voltage of the superharmonic resonance for the nonlinear vibration system varying with the actuated voltage for different gap distances between the nanobeam and the actuated plate are shown in Fig. 9. The control voltage increases with the increase in the actuated voltage. With a fixed value of the actuated voltage, the control voltage ascends when the gap distance between the nanobeam and the actuated plate decreases.

Figure 10 shows the curves of the control voltage of the superharmonic resonance for the nonlinear vibration system varying with the actuated voltage for different lengths of the axial controller. The control voltage increases with the increase in the actuated voltage. For a fixed value of the actuated voltage, the control voltage increases with the decrease in the length of the axial controller.

Figure 11 shows the curves of the control voltage of the superharmonic resonance for the nonlinear vibration system varying with the actuated voltage for different gap distances between the nanobeam and the control actuated plate. The control voltage increases with the increase in the actuated voltage. For a fixed value of the actuated voltage, the control voltage increases with the increase in the gap distance between the nanobeam and the control actuated plate.

Figure 12 shows the curves of the product of the eigenvalues of the superharmonic resonance for the nonlinear vibration system varying with the linear feedback gains for different nonlinear feedback gains. The eigenvalue product of the superharmonic resonance increases with the increase in the linear feedback gain for fixed values of the nonlinear feedback gains. For a fixed value of the linear feedback gain, the eigenvalue product of the superharmonic resonance increases with the increase in the nonlinear feedback gain.

Figure 13 gives the control performance of the time response for the uniform nanobeam on the steady-state amplitude for Mode 1 without controllers. The numerical solutions are carried out with the Matlab function ODE45, which uses an explicit Runge-Kutta algorithm. Figure 14 gives the numerical result of the control performance of the nonlinear vibration system with the axial capacitor controller. In the numerical calculation, the control damping μ_e and the tune coefficient σe are approximative replacements of μ and σ, respectively. The frequency of the excitation force is 11.15. The feedback gains of the displacement and velocity are g_d=100, and g_f=226, respectively. We can find that the vibration amplitude is mitigated evidently. The result in Fig. 13 is quite comparable and in agreement with that presented by Caruntu and Knecht^[15].

8 Conclusions

The nonlinear response control of the electrostatically actuated nanobeam at the near-half natural frequency is studied by considering the electrostatic force nonlinearity. The axial capacitor controller is used to control the nonlinear phenomena of the nano-remonstrator modelled as an Euler-Bernoulli thin beam. The gap between the nanobeam and the substrate is determined by the magnitude of the nonlinear terms actuated by the electrostatic force. The peak amplitude decreases with the increase in the gap, and increases with the increase in the actuated voltage. The peak amplitude of the superharmonic resonance is found to have a relationship with the damping, external detuning, and excitation amplitude.

The sufficient conditions for ensuring the system stability are stated in the paper. The saddle-node bifurcation is studied, and the stable and unstable ranges of the saddle bifurcation of the clamped-clamped supported nanobeam are reported. It is possible to avoid the nonlinear vibration phenomena by selecting a suitable control voltage. The relationship between the eigenvalue product of the superharmonic resonance and the nonlinear feedback gains is given.

The amplitude increases with the increase in the actuated voltage. The control voltage of the superharmonic resonance for the nonlinear vibration system varies with the actuated voltage for different gaps between the nanobeam and the control actuated plate. The control voltage increases with the increase in the actuated voltage. The vibration amplitude of the nonlinear vibration system is mitigated evidently with the displacement and velocity controllers.