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free-free beam natural frequency
Posted Dec 9, 2009, 10:07 PM EST MEMS & Nanotechnology, MEMS & Nanotechnology 6 Replies
I first conducted a simulation to calculate the natural frequency of the first mode of a beam of 4m length, 40cm height and 70cm in width. the theoretical calculation gives me f1 = 128 Hz but Comsol gives me 0 Hz
do you have an idea about the problem?
thank you in advance
a "0" for a structural mode means a (or several) DoF that is not blocked, if you have 6 of them at about "0" Hz, its typically that you are looking at the first 6 (x,y,z, Rx,Ry,Rz) free-free modes, and your 128 Hz will show up as mode no 7 (increase the number of mods in the solve tab to see this)
This was assuming a 3D model, if you are in 2D its 3 modes x,y,Rz that would be at "0".
If you have only 1 or 2 modes at "0" Hz then typically you have not constrained enough DoF to block one face.
I did a quick check in 3D default values (steel) I get 125.65, 206.30, Hz for mode "7 & 8". But if I fixee one of the small faces I get 20.49, 35.17 and 123.2, 161.72 Hz for the first 4 modes respectively in Rz and Ry . First, is the meshing fine enough ? by doing an overkill going from 1'794 to 273'873 Dof the frequencies changes to respectively 125.39 and 206.14 in free-free and 20.41, 35.05, 122.26, 156.20 Hz in clamped- free mode, so default values are about OK
Furthermore, one must be carefull to observe the mode shapes, in free-free and in clamped-free they differ, the two first in clamped-free cannot show up in free-free because of symemtry reasons. In fact a free free mode beam behaves as a symmetric L/2 length beam hence there is a factor 4 in frequency difference for this mode between clamped-free and free-free for this specific mode, try to set a symmetry boundary codition on the small edge instead of fixed, and compare to the free-free values.
Analytically in a simplified way the first eigenmodes are typically for clamped-free f=gamma/2/pi*sqrt(K/M) with k=3*E*I/L^3 with L the beam length, E the young modulus, M the total mass (mostly expressed as m*L=M the mass per length or M=rho*h*b*L with rho the density) I=h*b^3/12 the inertia with h width, b thickness (in the normal direction of rotation) amd gamma is a (mass) partcipation factor depending on the way the beam is clamped (sqrt(gamma)=(1.875, 4.694, 7.855) for the first clamped-free modes, while sqrt(gamma)=(4.730, 7.8532) for the first free-free modes (note: these are for the same given flexure mode, while the 2 cases from COMSOL are the first modes in two perpendicuzlar directions). This gives typically for E=200[GPa] and rho= 7850[kg/m^3] the value of steel: f1 = gamma*b/4/pi/L^2*sqrt(E/rho) = 5.8[Hz]*1.875^2 = 20.38[Hz] in clamped-free and 5.8[Hz]*4.73^2=129.71[Hz] in free-free
All above in simplified non-shear theory, ignoring "nu" which is an acceptable first approcimation.
Furthermore as the heigt to width ratio is only 0.7/0.4=1.75 we should see a mode difference of this ratio (sqrt(h*b^3/b*h^3)=b/h = 1.75), indeed 35.17/20.49=1.716 and 206.3/125.65=1.64, good enough
Well as you have not precised your boundary conditions, neither the material, I might be wrong with some of the numerical values, but the rest I believe is more or less OK, no? There is quite a lot to say about mode shapes...
For the theor, have a look at "Formulas for Natural Frequency and mode shape" R.D. Blevin, Kruger, 1979, reprint 2001, ISBN 1-57524-184-6
Hope this helps
you wrote: a "0" for a structural mode means a (or several) DoF that is not blocked, if you have 6 of them at about "0" Hz, its typically that you are looking at the first 6 (x,y,z, Rx,Ry,Rz) free-free modes, and your 128 Hz will show up as mode no 7 (increase the number of mods in the solve tab to see this)
that mean the first natural frequency occurs at the 7 mode in the case of free-free beam.
thank you very much
sorry for the late reply, but you are perfectly right, in a 3D representation of a free-free-structure it is the 7th mode that is the first interesting (non-trivial one), as the first 6 represent the free motion: 3 displacements and 3 rotations in a 3D space.
For you second question about stress concetration, I need a little more info.
Are you studying stress concetrnation at a "sharp edge" ?
in which case FEM will tend to give you "infinity" as you have a singularity at the edge. These singularities on "sharp edge effects" are typical for structural or even ACDC electric or magnetic fields.
one must "soften" the edges, to a reasonnable "engineering" level (note that: if you put a radius, the FEM meshing will simplify it to a polygonal shamfer).
If you need some litterature, I can only advise to study through:
"Peterson's Stress Concetration Factors" by W.D. Pilkey, Wiley&Sons, 1997.
For Eigenmodes take a look at:
"Formulas for natural frequencies and mode shapes" R.D. Blevins Krieger, 2001
These are classical approaches, but are the best exercices you might get to fit to FEM. It would be a nice project that someone went systematically through these two books and ported the examples to a FEM tool like COMSOL and reported the differences
Have fun Comsoling
Single Edge Crack: the KI parametre converge just for 1000 elements
but the maximum value of Von Mises stress increase all time and do not converge
thank you in advance
unfortunately i do only have paper books (yet ;), but if you try on Wiki or on the web I'm sure you can find the relevant analytical formulas
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