# Discussion Forum

## calculating the spring constant of a rectangular cantilever

 Topics: Structural Mechanics, 3.5, Physics
RSS feed   |   Turn on email notifications   |   2 Replies   Last post: September 23, 2011 2:13pm UTC

Nico Schmidt

September 22, 2011 11:23am UTC

calculating the spring constant of a rectangular cantilever

Hello,
i'm currently simulating a cantilver used in atomic force microscopes. I'm then calculating the spring constant of a cantilever using Hooke's law and the z-displacement caused by a load.
There is a another well established method to calculate the spring constant k of rectangular cantilevers based on the Young's modulus and the geometry (see e.g. Chen, Yeh, Tai, Anal. Chem. 79, 1333, 2007):

k = ( E * w * t^3) / (4 * L^3)

E: Young's modulus E=78.7GPa
t: thickness t=0.41mm
w: width w=0.24mm
L: length L=2.94mm

Pluging the values mentioned above in the formula you get k=12807 N/m.
Modelling a cantilever of the same properties using Hooke's law and the z-displacement i get k=12669 N/m.
So both methods yield consistent numerical values.

Unfortunately when i double the width i get:
k=25614 N/m using the geometrical method
k=22190 N/m using Comsol

This is a way bigger mismatch than before, which i'm unable to explain physically.

Therefore i'm asking for your help. Since i'm started using Comsol only a few weeks ago, i might used odd parameters or forgot to take something important into account. Basically i think my inexperience with Comsol is causing this mismatch.
I'd appreciate it if you'd take a look into it.

Because the file sizes are too big to attach i uploaded them here:
rapidshare.com/files/3857110085/cantilever.rar
The .rar contains both cantilever (width 0.24mm and 0.48mm)

I'm sorry for making you download them, but if you need another hoster or if you'd like me to send them to you via email, please let me know.

Kind regards

Ivar Kjelberg

September 22, 2011 6:14pm UTC in response to Nico Schmidt

Re: calculating the spring constant of a rectangular cantilever

Hi

first to upload files, it's worth to clear the solution and the mesh, the file is then much smaller and we can always rerun it ;)

2) the difference might come from the mesh you are using, you should always check the sensitivity to the mesh, i.e. double the mesh density and confirm that the solution changes only a little (for me typically < 5%)

--
Good luck
Ivar

John

September 23, 2011 2:13pm UTC in response to Nico Schmidt

Re: calculating the spring constant of a rectangular cantilever

Also, go back and check the assumptions of your formula. I believe that the formula you're using assumes that the width of the beam is small relative to the length. A length/width ratio greater than 10 is a typical rule of thumb, which you exceed when you increase the width of your beam. COMSOL may be providing the more accurate solution here, but you may want to look up the correct formula to check.

Additionally, watch how large of a displacement that you're applying in your simulation, as the formula is only valid for small displacements.

~John

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