Cantilever with end mass - Moment of inertia

In summary,The moment of inertia of a rectangular beam is given by: I = Ic + ma^2, where Ic is the area moment of inertia and ma is the mass moment of inertia.
  • #1
Dalmaril
3
0
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!
 
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  • #2
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?
 
  • #3
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?
 
  • #4
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.
 
  • #5
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!
 
  • #6
Dalmaril said:
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
 
  • #7
So, did you figure out how to do this?
 
  • #8
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.
 
  • #9
i need the formula for finding moment of inertia of rectangular beam. i have failed to know why the units are mm^4
 

1. What is a cantilever with end mass?

A cantilever with end mass is a type of structure where one end is fixed and the other end has a concentrated mass attached to it. This structure is commonly used in engineering and architecture as it allows for longer spans and can support heavy loads.

2. What is the purpose of a cantilever with end mass?

The purpose of a cantilever with end mass is to distribute the load of a structure across a larger area, allowing for longer spans and greater stability. It also helps to reduce bending and stress on the fixed end of the structure.

3. How is the moment of inertia calculated for a cantilever with end mass?

The moment of inertia for a cantilever with end mass is calculated by taking into account the mass of the end load and the length of the cantilever. The formula for moment of inertia is I = mL^2, where m is the mass and L is the length of the cantilever.

4. What factors affect the moment of inertia for a cantilever with end mass?

The moment of inertia for a cantilever with end mass can be affected by several factors, including the mass of the end load, the length of the cantilever, and the material properties of the structure. The distribution of the mass along the length of the cantilever can also affect the moment of inertia.

5. How does the moment of inertia impact the stability of a cantilever with end mass?

The moment of inertia plays a crucial role in the stability of a cantilever with end mass. A higher moment of inertia means a greater resistance to bending and a more stable structure. This is why engineers and architects carefully calculate and consider the moment of inertia when designing cantilever structures.

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