Nickolay,

To find the spring constant of a thin plate, treat it as an elastic
cantilever beam under flexure. For some force, F, applied at the end of the
beam, calculate the curvature of the beam as a function of position along
its length. Integrate the curvature to find the slope of the cantilever,
and then integrate the slope to find the deflection, d.

You will now have an equation of the form:

y = (1/k) F

where (1/k) is some expression dependent on the geometry of the beam. k is
the spring constant of your plate.

An example follows. For more complex examples, or for other ways of solving
this sort of problem, please see Riley, Sturges, and Morris, "Mechanics of
Materials", or your favorite mechanics text.

Good luck with your work!

-Craig McGray
 6211 Sudikoff Laboratory
 Dartmouth College
 Hanover, NH 03755



-- EXAMPLE --------------------

Consider a rectangular beam with constant length, width, and thickness: L,
w, t. The beam is fixed at x=0, where both its deflection and slope are 0.
A force is applied at the end of the beam, where x=L. The bending moment at
any point along the beam is a function of the applied force and the moment
arm of the force:

M = F*x

The curvature of the beam, given the above bending moment, will depend on
the Young's modulus of the beam material, Y, and the second moment of
cross-sectional area of the beam. For our rectangular beam, the 2nd moment
of cross-sectional area is:

I = (1/12) w (t^3)

So we have:

curvature:   ddy  =  M/(Y*I)  =  12*F*x/(Y*w*(t^3))

Integrating this with respect to x gives the slope:

slope:   dy  =  6*F*(x^2)/(Y*w*(t^3)) + C1        where C1 is a constant

Integrating again gives:

deflection:   y  =  2*F*(x^3)/(Y*w*(t^3)) + C1*x + C2     where C2 is a
constant

Since we know the cantilever beam is fixed at x=0, we have the following
boundary conditions:

dy  =  0      at x=0
y  =  0     at x=0

This allows us to solve for C1 and C2. Happily, in this example they are
both 0. We now have the deflection as a function of both x and F:

y  =  2*F*(x^3)/(Y*w*(t^3))

However, you are most likely only interested in the spring constant at the
tip of the cantilever where the force is being applied. So we can
substitute L for x, yielding:

y  =  2*F*(L^3)/(Y*w*(t^3))

So, for the rectangular cantilever described above, the spring constant is:

k = Y*w*(t^3)/(2*(L^2))




Nickolay Lavrik wrote:

> Does anyone know the equation (or the reference where it can be found)
> that can be used to estimate a spring constant of a thin plate (such as
> microcantilever) based on the plate geometry (thickness, width, length)
> and Young's modulus of the material ?
>
> You help will be much appreciated.
>
> --
> Nickolay Lavrik
> University of Tennessee
> Department of Chemistry
> 420 Buehler Hall
>
> Ph.(865) 974-6174
>