I presume, that the real issue is that this problem must be solved
self-consistently - ie. the electrostatic force changes the cantilever
distance, which again changes the electrostatic force. Of course, the
problem is harder still, in that one end is anchored and one is free,
giving a varying distance between the cantilevers going down the length
of them.

My best guess is to either assume, that the cantilevers are static, and
then calculate the electrostatic force, which should be fairly easy. Of
course, this is only valid if the cantilevers really don't bend that
much compared to the total distance between them (delta(D)/D << 1). A
more complicated version of the same is to find the shape of the
cantilever as a function of force applied homogeneously. This should
also be possible if the geometry is not to strange - see textbooks on
AFM cantilever calculations or at
http://www.ntmdt.com/SPM-Techniques/index.html. This assumes that the
distance modulation can be described by looking only at one of the
cantilevers bending - a quick fix for 2 cantilevers could be multiplying
the force by 2, which is of course not correct, but will probably put
you in the ballpark.

Alternatively, making a finite-element simulation (for example in
Comsol), should be quite easy, but of course won't supply you with any
nice analytical expression. To be honest, I think this is the only way
to go, if delta(D)/D is anywhere close to 1.

Hope it helps,

// Morten Aaroe
Ph.D-student
Department of Physics
Technical University of Denmark

Andrew O'Grady wrote:
> Hi everyone,
>
> I need to model the electrostatic bending of two cantilever beams towards
> each other, and am running into some difficulty. I wonder if anyone has had
> to solve this problem before. I am looking for either an approximate
> analytical model, or a paper which might describe this scenario used in an
> actual MEMS device. This is similar to a comb drive device except that the
> fingers are not very stiff, and they start to bend towards each other.
>
> The model consists of two parallel fingers around 100um long, 1 um thick and
> 2um wide, with a gap of 2um in between them. One cantilever beam is anchored
> on the left side, and the other beam is anchored on the right side. When I
> apply a voltage between the two cantilevers, I can see the tip of one beam
> start to be pulled towards the anchor of the other beam.
>
>