Cantilever with end mass - Moment of inertia

[Dalmaril]

Hello! I have a problem and if anyone could help that would be great.

Imagine that you have a cantilever beam of length L and has an end mass m. In order to calculate the natural frequency I use the equation:

f=(1/2*pi)*SQRT(3EI/mL^3).

a) The moment of inertia in the equation is the sum of both the cantilever and the mass, correct??
b) The moment of inertia of the mass is given through Steiner theorem: I = Ic + ma^2, where Ic = (bh^3)/12 (b and h are the mass' geometric characteristics) and a is the distance from the CM of the beam, correct??
c) The problem is that when I input these parameters to the frequency equation, I am not getting Hz at the end. Is (a) and (b) correct, cause that's where I think the problem is.

Thanks in advance!

 

[TVP45]

I didn't check the formula; I only did a dimensional analysis and I get s^(-1). I'm guessing you're using ips units? If so, what units are you using for I? For m?

 

[Dalmaril]

E is in GPa , 1 Pa = 1 kg m^-1 s^-2
I is in m^4
m is in kg and L is in m.

But in the equation for the total I, there is the term of ma^2. Doesn't that have units kgm^2 or am i getting wrong somehow?

 

[TVP45]

Only a quick answer. I'm rushing off and will visit later. But, you've criss-crossed moments terms. You can't use bh^3 (an area term) together with ma^2 (a mass term). See if you can straighten it out; if not, I'll give you a hand later.

 

[Dalmaril]

Yes you are right, I saw that too :).. I need the area moment of inertia so the steiner theorem changes. I think that will do the trick. Thanks!

 

[TVP45]

Yes you are right, I saw that too :).. I need the area moment of inertia so the steiner theorem changes. I think that will do the trick. Thanks!

OK, that does indeed give you the right units. Now the big question: Have you used the correct I? Here's what you need to think about.
Is the beam considered massless?
Does the beam have significantly less mass than m?
Does the beam have a mass comparable to m?
Do you know the derivation of the formula you used?

HINT: Your formula looks like you're using an essentially massless beam and a point mass (what is the area of a point?).

I don't know what course you're in or whether you're interested in looking deeper, but here's a link you might look at (but only after you try it on your own).
http://www.vibrationdata.com/tutorials2/beam.pdf

 

[TVP45]

So, did you figure out how to do this?

 

[idagu]

hello, I think you might mistake the moment of inertia [kg*m^2] for the second moment of area [m^4].
for the calculation of stiffness k is just m^4 for interest.

 

[williamkat]

i need the formula for finding moment of inertia of rectangular beam. i have failed to know why the units are mm^4