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Static as well as dynamic analyses have been performed on clamped-clamped carbon nanotube (CNT) resonator. The nonlinear CNT model is investigated with a novel discretization technique: a differential quadrature method (DQM) to discretize the spatial variables and a finite difference method (FDM) for limit-cycle solutions. Parametric study is performed by varying the electric load, as well as the initial curvature (due to fabrication). It is found that the pull-in voltage decreases nonlinearly with initial curvature and linearly with residual stresses. The eigenvalue problem is also solved to obtain the bending natural frequencies of the system as function of the DC voltage as well as the initial curvature of the CNT. Frequency-response curves near some selected resonant frequencies are plotted to better understand the nanotubes' dynamic behavior. Different linear and nonlinear phenomena are depicted such as dynamic pull-in, hardening, and softening behavior and veering of the odd modes. We have found that even when exciting the CNT near its first natural frequency, the vibration mode located at the veering process significantly alters the CNT's motion and hence may decrease its overall quality factor.

Since their discovery [

Static and dynamic analyses of CNTs rise difficult issues in solving the highly nonlinear governing equation. In this case the nonlinearity is both of the geometric type (mid-plane stretching) and of the loading type (electric load) [

Previous studies have focused on mechanical as well as electric response of CNTs exploring their interesting characteristics such as Young’s modulus and pull-in properties. Experimental investigations of the vibration of electrostatically actuated CNTs have been conducted to characterize their effective Young modulus (Poncharal et al. [

Using the nonlocal Timoshenko model, the effect of temperature change on mode shapes is investigated [

Many researchers have fabricated and tested systems based on CNTs such as torsional electromechanical systems [

The objective of the paper is to solve both static and dynamic nonlinear problems using novel discretization technique: differential quadrature method and finite difference method. These methods have been applied successfully to MEMS electrostatic actuators in order to investigate their nonlinear dynamics [

A hollow cylindrical Euler-Bernoulli beam of length

Schematic of the CNT resonator.

The differential quadrature method (DQM) is used to solve the space dependent partial differential equation by transforming it into ordinary differential equations describing the motion of the CNT with respect to time at

In this section, we solve the algebraic system obtained by dropping the time dependent terms in (

Variation of the normalized static deflection with DC voltage for different numbers of grid points.

Residual stresses have thermal origins due to fabrication techniques of the CNT [

In Figure

Variation of the pull-in voltage with the residual stresses at different initial curvatures.

We investigate next the effect of the initial curvature into the static response of the CNT. The initial curvature

The results in Figure

Variation of the normalized static deflection with DC voltage for different initial curvatures

Variation of the pull-in voltage with the initial curvatures

Assuming that superposition principle applies to the CNT deflection, we split it into static part due to DC voltage and dynamic part due to AC voltage assumed to be harmonic, that is,

At a given applied DC voltage, we investigate the effect of variation of initial curvature on natural frequencies described by the parameter

Variation of the 9 first natural frequencies with

We can see clearly that even frequencies are insensitive to the variation of curvature, whereas the odd frequencies vary with slack. One can see that odd frequencies do not intersect, but they diverge in a manner called frequency or mode veering [

The same phenomenon of mode veering is observed in Figure

Variation of the 9 first natural frequencies with DC voltage at

For initial curvature

To further clarify the veering process with the DC voltage depicted in Figure

Mode shaped of the CNT without the initial curvature for

In this section, we present the dynamic behavior of the CNT by generating their frequency-response curves for different sets of parameters. For this, we have to solve the dynamic equation (

To construct the limit-cycle solutions and the frequency-response curves of the CNT, we seek periodic orbits of the system with period

The frequency-response curves are obtained by plotting the variation of the maximum deflection of the CNT at its midpoint as function of the excitation frequency. This excitation is obtained by a DC load superimposed to an AC harmonic load around one of the natural frequencies of the CNT.

In Figures

Frequency-response curves near the 1st and 9th natural frequencies when

Frequency-response curves of

We study modes veering phenomenon by investigating the dynamic behavior of the CNT before and after veering. According to Figure

Concerning the first veering between

Frequency response near the first and third natural frequencies when first mode is excited before veering at (a)

Frequency response near the third and fifth natural frequencies when third mode is excited before veering at (a)

In this paper, the nonlinear dynamics of a CNT have been investigated using a novel discretization technique: FDM and DQM. In the static part, CNT responses to DC electric load were calculated solving the static equation using DQM. We have studied the static CNT behavior and how it is affected by residual stress as well as initial curvature. The natural frequencies of the CNT have been calculated for different DC voltages and initial curvatures. This study revealed the existence of the phenomenon of modes veering as both DC voltage and initial curvature are varied.

Dynamic analysis was also investigated for the CNT response to harmonic electric load near the natural frequencies. Frequency-response curves have been plotted and validated under different excitation conditions.

Veering of the modes has been analyzed by comparing the dynamic behavior of the CNT before and after veering. Veering of the modes with initial curvature as parameter is found to exchange dynamic properties between modes. In fact, the maximum deflection at resonant frequency of lower mode before veering is higher than that after veering, and maximum deflection at resonant frequency of higher mode increases considerably during veering of the modes. This proves that energy of the system is transferred from one mode to the other during veering.