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Effects of van der Waals Force and Thermal Stresses on Pull-in Instability of Clamped Rectangular Microplates

View Article: PubMed Central - PubMed

ABSTRACT

We study the influence of von Kármán nonlinearity, van der Waals force, and thermal stresses on pull-in instability and small vibrations of electrostatically actuated microplates. We use the Galerkin method to develop a tractable reduced-order model for electrostatically actuated clamped rectangular microplates in the presence of van der Waals forces and thermal stresses. More specifically, we reduce the governing two-dimensional nonlinear transient boundary-value problem to a single nonlinear ordinary differential equation. For the static problem, the pull-in voltage and the pull-in displacement are determined by solving a pair of nonlinear algebraic equations. The fundamental vibration frequency corresponding to a deflected configuration of the microplate is determined by solving a linear algebraic equation. The proposed reduced-order model allows for accurately estimating the combined effects of van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflection profile with an extremely limited computational effort.

No MeSH data available.


Sketch of an electrostatically actuated rectangular microplate.
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f1-sensors-08-01048: Sketch of an electrostatically actuated rectangular microplate.

Mentions: We consider a rectangular plate-like body of longer side ℓ and shorter side φℓ, φ ∈]0,1], occupying the three-dimensional region Ω × (−h/2, h/2) as depicted in Figure 1. The mid-surface Ω is described by rectangular Cartesian coordinates (x1, x2) aligned with the longer and with the shorter sides, respectively. We assume that the initial gap g0 between the two conductors and the thickness h of the deformable plate are much smaller than the characteristic length ℓ. In the proposed model, we assume that g0 and h can be of the same order of magnitude. Therefore, the maximum displacement that the device can undergo is of the order of the plate thickness h, but it is much smaller than the characteristic length ℓ, since h/ℓ ≪ 1. This implies that strains in the deformable electrode are small. Under these assumptions, we use the von Kármán plate theory to account for large deflections and small strains, see for example [41]. Neglecting the effect of the rotatory inertia, the von Kármán plate equations in Cartesian coordinates are [41](1a)ϱhω¨+D∂2∂xk∂xk(∂2ω∂xj∂xj)−h∂∂xj(σjk∂ω∂xk)−Fe−Fs=0,(1b)ϱu¨i−∂σij∂xj=0,i=1,2.


Effects of van der Waals Force and Thermal Stresses on Pull-in Instability of Clamped Rectangular Microplates
Sketch of an electrostatically actuated rectangular microplate.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3927533&req=5

f1-sensors-08-01048: Sketch of an electrostatically actuated rectangular microplate.
Mentions: We consider a rectangular plate-like body of longer side ℓ and shorter side φℓ, φ ∈]0,1], occupying the three-dimensional region Ω × (−h/2, h/2) as depicted in Figure 1. The mid-surface Ω is described by rectangular Cartesian coordinates (x1, x2) aligned with the longer and with the shorter sides, respectively. We assume that the initial gap g0 between the two conductors and the thickness h of the deformable plate are much smaller than the characteristic length ℓ. In the proposed model, we assume that g0 and h can be of the same order of magnitude. Therefore, the maximum displacement that the device can undergo is of the order of the plate thickness h, but it is much smaller than the characteristic length ℓ, since h/ℓ ≪ 1. This implies that strains in the deformable electrode are small. Under these assumptions, we use the von Kármán plate theory to account for large deflections and small strains, see for example [41]. Neglecting the effect of the rotatory inertia, the von Kármán plate equations in Cartesian coordinates are [41](1a)ϱhω¨+D∂2∂xk∂xk(∂2ω∂xj∂xj)−h∂∂xj(σjk∂ω∂xk)−Fe−Fs=0,(1b)ϱu¨i−∂σij∂xj=0,i=1,2.

View Article: PubMed Central - PubMed

ABSTRACT

We study the influence of von Kármán nonlinearity, van der Waals force, and thermal stresses on pull-in instability and small vibrations of electrostatically actuated microplates. We use the Galerkin method to develop a tractable reduced-order model for electrostatically actuated clamped rectangular microplates in the presence of van der Waals forces and thermal stresses. More specifically, we reduce the governing two-dimensional nonlinear transient boundary-value problem to a single nonlinear ordinary differential equation. For the static problem, the pull-in voltage and the pull-in displacement are determined by solving a pair of nonlinear algebraic equations. The fundamental vibration frequency corresponding to a deflected configuration of the microplate is determined by solving a linear algebraic equation. The proposed reduced-order model allows for accurately estimating the combined effects of van der Waals force and thermal stresses on the pull-in voltage and the pull-in deflection profile with an extremely limited computational effort.

No MeSH data available.