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Since microplates are extensively used in MEMS devices such as microbumps, micromirrors, and microphones, this work aims to study nonlinear vibration of an electrically actuated microplate whose four edges are clamped. Based on the modified couple stress theory (MCST) and strain equivalent assumption, size effect and damage are taken into consideration in the present model. The dynamic governing partial differential equations of the microplate system were obtained using Hamilton’s principle and solved using the harmonic balance method after they are transformed into ordinary differential equation with regard to time. Size effect and damage effect on nonlinear free vibration of the microplate under DC voltage are discussed using frequency-response curve. In the forced vibration analysis, the frequency-response curves were also employed for the purpose of highlighting the influence of different physical parameters such as external excitation, damping coefficient, material length scale parameter, and damage variable when the system is under AC voltage. The results presented in this study may be helpful and useful for the dynamic stability of a electrically actuated microplate system.

Since a microelectromechanical system (MEMS) received a considerable amount of attention in recent years, numerous works related to nonlinear responses and characteristics because of intrinsic existence of nonlinearity of these microdevices have been carried out in recent years. Using the multiple scale method, Younis and Nayfeh [

However, size-dependent behavior of microstructures has been experimentally validated [

In the meantime, during manufacturing and operating process damage, that is, fatigue damage [

Consider a movable microplate with mass density

Electrically actuated microplate.

With the introduction of only one extra material parameter, material length scale parameter, Yang et al. [

Kirchhoff’s displacement components

Substituting (

Substituting (

The stress-strain relations can be given with respect to Poisson’s ratio

With Cauchy’s stress and couple stress expressed above, bending moments

Substituting (

With Hamilton’s principle, we have

Substituting stresses and strains into (

The virtual kinetic energy is given as

The virtual work done by electric force and damping force can be expressed as

By substituting (

Boundary conditions are as follows:

The following dimensionless parameters are introduced:

With application of Taylor series expansion into (

Due to the fact that a fully clamped rectangular microplate is extensively used in MEMS applications [

First, note that, in the boundary conditions in (

Submitting (

For convenience, (

After eliminating

In this section, the nonlinear size-dependent vibration analysis of the microplate is studied under the influence of size effect and damage effect. Free vibration and forced vibration analysis of the microplate under DC voltage and AC voltage are discussed in detail accordingly. The frequency-response curves are presented for various system parameters such as material length scale parameter, damage variable, damping ratio, and external AC voltage. Note that geometric and physical parameters of the microplate are listed in Table

Geometric and material properties of the microplate.

Property | Value |
---|---|

Length, | |

Width, | |

Thickness, | |

Initial gap, | |

Poisson’s ratio, | 0.06 |

Young’s modulus, | |

0.2 | |

Dielectric constant, |

For static analysis, (

The harmonic balance method is feasible for finding solutions of both weakly and strongly nonlinear problems. A general solution of (

Submitting (

The variation of amplitude-frequency response of the microplate with various values of damage variable is presented in Figure

Damage effect on frequency-response curves of the microplate.

Influence of size effect on frequency-response curves is given in Figure

Size effect on frequency-response curves of the microplate.

As for dynamic analysis, (

Similarly, the harmonic balance method is employed for solving (

Substituting (

Using

The frequency-response curves of the system with damage variable

The frequency-response curves of the system for different amplitude of the AC voltage

Figure

The frequency-response curves of the system for different damping coefficients

Figure

The frequency-response curves of the system for different damage variable

Figure

The frequency-response curves of the system for different nondimensional material length scale parameters

The nonlinear size-dependent vibration of a microplate with damage was explored by employing MCST and the strain equivalent assumption in this research. The nonlinear governing partial differential equations were transformed into nonlinear ordinary differential equations via Galerkin’s scheme and further solved numerically by the harmonic balance method. Numerical results indicate that damage effect and size dependency both have obvious influences on the static and dynamic behaviors of the microplate system. It was concluded that, on one hand, the hardening-type nonlinear behavior of the microplate system enhances when it encounters damage; on the other hand, the system exhibits a weaker nonlinear behavior, greater nondimensional frequency, and lower amplitude of the microplate midpoint as size effect gets obvious.

The Matlab simulation and control program data used to support the findings of this study are available from the corresponding author.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to acknowledge with great gratitude for the supports of the National Science Foundation of China (grant nos. 51778551 and 11272270).