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The influence of damping on the dynamical behavior of the electrostatic parallel-plate and torsional actuators with the van der Waals (vdW) or Casimir force (torque) is presented. The values of the pull-in parameters and the number of the equilibrium points do not change whether there is damping or not. The ability of equilibrium points is varied with the appearance of damping. One equilibrium point is an unstable saddle with a different damping coefficient, the other equilibrium point is a stable node when the damping coefficient is greater than some critical value, and otherwise it is a stable focus. Then there are two heteroclinic orbits passing from the unstable saddle point to the stable node or focus.

Inherent instability such as in pull-in phenomenon and stiction exists in both microelectromechanical (MEM) and nanoelectromechanical (NEM) actuators. Such instability is due to some kind of surface force, i.e. electrostatic, van der Waals (vdW), Casimir and capillary forces. Although vdW and Casimir forces can be neglected when designing a MEM actuator, they play important roles at nanoscales [

A typical MEM (NEM) parallel-plate (or torsional) actuator is made up of two conducting electrodes, one is typically fixed and the other, which is controlled by an equivalent mechanical spring, is movable (or rotary) [

Using a one-dimensional (1D) model, the pull-in parameters have been analytically obtained by many researchers when electrostatic [

In this paper, the influence of damping on the dynamical behavior of the electrostatic parallel-plate and torsional actuators with the vdW or Casimir force (torque) is presented, and the results are compared with those in Refs. [

In this paper, we will use a 1D lumped model (1DLM) to discuss the influence of damping on the dynamical behavior of the parallel-plate and torsional models with electrostatic, vdW and Casimir forces. Then, as in previously published papers [

For the parallel-plate model with damping, the system can be simplified to a 1DOF as shown in

The system is a typical mass-spring-damping one. For the different intermolecular forces, the applied forces on the same model are the electrostatic, vdW, or Casimir forces. Then the equation of motion is

The electrostatic force _{elec} (neglecting the fringing force) acting between the planes with potential difference _{vdW} and the retarded Casimir force, respectively, are
_{0} is the permittivity of vacuum within the gap, ^{2}^{2} is the Hamaker constant which lies in the range (0.4–4)10^{−19} J, ^{−34} Js, ^{8}ms^{-1}.

Introducing dimensionless variables: _{0}^{2}/2^{3}, ^{vdW} =^{4}, ^{c}= π^{2}^{5}, and characteristic time

According to the definition of these parameters, physically meaningful solutions exist in the region 0 < ^{vdW} denotes the order of magnitude of ratio between the vdW and elastic forces, ^{C} denotes the order of magnitude of ratio between the Casimir and elastic forces.

For the torsional structure, the simplified 1DLM is shown in

The 1DOF is the torsional angle, ^{2} /3 is the rotational inertia of the upper rotational beam when the mass is uniformly distributed, k_{θ}_{θ}

Introducing dimensionless variables: _{max}, _{θ}, _{Θ}_{θ}_{θ}T_{θ}_{Θ}_{0}^{3}^{2}/2_{θ}g^{3},

According to the definition of these parameters, physically meaningful solutions exist in the region 0 < _{Θ}

In the following sections, the four dimensionless

In this section, we discuss the stability of the stationary equilibrium of the above equations, then we should set zero the velocity and acceleration for each model.

In this part, we just consider the electrostatic force and the corresponding equation from

Setting, d^{2}Δ/d^{2}= 0, dΔ/d

According to the critical condition

For this case, using the same procedure as Case I and using

The variation of the pull-in parameters _{PI} and _{PI} with ^{vdW} are plotted in

Similarly, by using the

The variations of the pull-in parameters _{PI} and _{PI} with ^{C} are plotted in _{*}” corresponds to

At the point “°”, it implies that there is no vdW or Casimir force on the structure. The results are consistent with those in

The governing equation for the system just subjected to the electrostatic force is:

To get the pull-in parameters, we also set d^{2}^{2}=0, d

Using

The variations of the pull-in parameters _{PI} and _{θ}_{PI} with
_{*}” corresponds to

Similarly by

The variations of the pull-in parameters _{PI} and _{ΘPI}_{*}” corresponds to

At the point “°”, it implies that there is no vdW or Casimir torque on the structure. These results are consistent with those in

In this section, we just discuss the dynamical behavior of the parallel-plate model with the electrostatic and vdW forces.

To discuss the dynamical behavior of

The stationary solutions of this system can be obtained by setting zero of the right-hand side of ^{vdW}, we solve ^{vdW}. We plot the variation of ^{vdW}, the solution is shown in _{0}^{2}/2^{3} is positive, then the solution is physical meaningful when the solution curves are on the right of

In order to check the stability of the equilibrium points, we need the Jacobian matrix of

We first discuss the stability of the equilibrium points with the given parameters _{1},0)and (_{2},0) satisfying the inequality _{1}< _{*}<_{2}.

Firstly, we consider the equilibrium point stability of the special state that there is no electrostatic force on the upper movable beam. Then substituting
_{1} < _{*} into

Its corresponding eigenvalues are

Here, we discuss the property of the eigenvalues when the damping coefficient is positive. Because _{1}< _{*}, then
_{1,2} are all real, and they all are absolutely negative. This means the equilibrium point (_{1},0) is a stable node. According to the property of node, this point is an equilibrium point at first. At this position, the elastic force is equal to the vdW force, and the parallel-plate actuator keeps balance state. When we add a small perturbation on the upper movable beam, the perturbation will die out at the stable node. When
_{1,2} are a pair of complex conjugates, and the real parts of them are absolutely negative. This means the equilibrium point (_{1},0) is a stable focus. According to the property of focus, this point is also an equilibrium point at first. When we add a small perturbation on the upper movable beam, then the trajectory close to the equilibrium position resembles a spiral. Above all, at the point of (_{1},0), the real parts of the eigenvalues are negative, this equilibrium point (_{1},0) is always stable. Subsequently, we take
_{2} > _{*} into _{2},0) is a saddle point. At equilibrium position, if we add a small perturbation on it, the trajectory of the upper movable beam will leave the equilibrium position because one of the eigenvalues is positive. We then call this equilibrium state unstable.

Secondly, applying the same method to discuss the stability of the two solutions with any different given ^{vdW}, we plot the bifurcation diagram as

According to the properties of the stable node, stable focus, and saddle point, there exist two heteroclinic orbits which depart from the unstable saddle point and be end at the stable point. In order to see the movement process of the equilibrium points, we draw the phase portraits with ^{vdW} equal to 0.03, 0.07 and 0.09, respectively. These phase portraits are shown in

In ^{vdW}. By observing ^{vdW}, one is the stable node (marked by “°”), and the other is the unstable saddle point (marked by “×”). There are two heteroclinic orbits between the unstable saddle point and the stable node. We note that the heteroclinic orbit is convergent to the stable node from the unstable saddle point with exponent. In Figures ^{vdW}. By observing Figures ^{vdW}, one is the stable focus (marked by “°”), and the other is the unstable saddle point (marked by “×”). There are two heteroclinic orbits between the unstable saddle point and the stable node. We note that the heteroclinic orbit is convergent to the stable focus from the unstable saddle point spirally, which is different from the stable node because of the difference of their eigenvalues.

From these four figures, we also note that the stable point which is node or focus, and unstable saddle point move to the point “_{*}” from opposite direction with ^{vdW} is increasing. These two points turn into the pull-in point
^{vdW}.

According to the eigenvalue _{1}, λ_{2} is positive when the damping coefficient is negative. At this time, the system is unstable, which should be avoided in engineering applications.

Until now, the dynamical behavior of

The influence of damping on the dynamical behavior of the electrostatic parallel-plate and torsional actuators with the vdW or Casimir force (torque) is presented. First, we studied the variation of two pull-in parameters with another parameter with different surface forces (torques), we get two special points for each case shown in Figures _{*}” illustrates the actuator will lose its stability even though there is no applied voltage. With the appearance the vdW or Casimir force (torque), the pull-in parameters are all decreasing. From Figures

Secondly, we studied the stability of equilibrium points. One equilibrium point is an unstable saddle with different damping coefficient, the other is a stable node when damping coefficient is greater than some critical value, and otherwise it is a stable focus. Then there are two heteroclinic orbits passing from the unstable saddle point to the stable node or focus. Compared with the results in [

As a matter of fact, there are numerous possible sources of dissipation and damping in NEM actuators, which may broadly be classified as either intrinsic or extrinsic. Extrinsic dissipation or damping, such as gas (squeeze film) friction, clamping loss and surface loss, results from interaction of the actuator microstructure with the environment; whereas intrinsic dissipation or damping, such as thermoelastic relaxation, phonon-phonon and phonon-electron interaction, results from properties of the resonating material. The dissipation and damping mechanisms in NEM actuators are quite complicated [

WHL was supported by the National Natural Science Foundation of China (Grant No. 10602062) and YPZ was supported by the National Basic Research Program of China (973 Program, Grant No. 2007CB310500) and National High-tech R&D Program of China (863 Program, Grant No. 2007AA04Z348).

1DLM for the parallel-plate actuator.

1DLM for the torsional actuator.

Comparison between vdW and Casimir forces with variation of the pull-in displacement _{PI} with parameter

Comparison between vdW and Casimir forces with variation of the pull-in parameter _{PI} with parameter

Comparison between vdW and Casimir torques with variation of the pull-in angle _{PI} with parameter _{Θ}

Comparison between vdW and Casimir torques with variation of the pull-in parameter _{Θ}_{PI} with parameter _{Θ}

Variation of equilibrium points with parameter ^{vdW} with vdW force.

Bifurcation diagram: variation of equilibrium points with parameter ^{vdW} for different

Heteroclinic orbits with ^{vdW} 0.03,0.07,0.09, respectively when

Heteroclinic orbits with ^{vdW} = 0.03 when

Heteroclinic orbits with ^{vdW} = 0.07 when

Heteroclinic orbits with ^{vdW} = 0.09 when