# cantilever beam model

I'm trying to figure out a simple way of determining the first natural
frequency of a cantilever beam. I figured it would be just like a
mass-spring system where the frequency is equal to the square root of
spring constant over mass...but how do I determine the equivalent mass
and spring constant of a cantilever beam?
Thanks,
Nikyu
You know, I'm sure, that the most accurate way to solve this problem is to work with the PDE that describes the beam deflection Y as a function of distance X and time T.
You say you want a simple method, based on a mass-spring type of model. Some people call this a "lumped-parameter" model -- if you Google that term together with cantilever, beam, and vibrate I bet you'll find your answer.
To find the spring constant, use the equation for (static) bending of a beam with a force applied at the end, and your spring constant is just the force divided by the deflection.
To get the mass, you lump a certain fraction (I forgot how much, sorry) of the beam mass at the free end.
get a copy of Mark's Mechaincal Engineering Handbook
check out
my recollection is that Meffective = 1/3 mass total but my memory could be faulty

has the answer embeded in the equations but I'm too lazy to back calculate it.
cheers Bob
And for that you use the tip stiffness?
That's a fine wiki article, a model of accuracy, brevity and wit, worth every penny. It would be nice if someone could check it, actually.
Another approach is by considering the upper and lower bounds.
An upper bound solution would be to take all the mass and the stiffness half way along the beam.
A lower bound solution would be to use all the mass, and the tip stiffness.
So you know the frequency must fall into that band.
sqrt(8*k/m)>F/2/pi>sqrt(k/m)
Bob's suggestion of sqrt(3*k/m) would fall nicely in the middle of that band.
A better approximation might be to say (after thinking about the likely mode shape) that the stiffness 3/4 of the way along, and 1/2 the mass, is probably pretty close.
F/2/pi ~ sqrt(k*(4/3)^3/(m*.5)) = sqrt((128/27)*k/m) ~ sqrt(4.3*k/m)
and so on and so forth
Another way of calculating it would be to build an FEA model.
SO the OP has at least 3 good options (PDE, Rayleigh Ritz, FEA, guessing).
In practice R-R is terrific for real conditions, such as strangely shaped beams, and springy foundations.
Cheers
Greg Locock
The simplest way is to look up the correct formula. For beams with a variety of B.C., the natural frequency is omega=2 pi f=a sqrt(EI/(mL^4)), where m=mass per unit length. For a cantilever, a=3.52.
Been a LONG time since I checked this but I'm pretty sure that
using 1/3 of the total mass AND the tip stiffness (3EI) / (L^3)
gives a very good approximation, so good that FEA is hardly worth the effort
cheers Bob
Sorry, I only took a 5 sec look at the WIki article...............it was just something to get him thinking about energy methods

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