[mems-talk] Electrostatic bending of two cantilevers

[Morten Aarøe]

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.
>
>