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Join Date: Nov 2016
Posts: 10

Natural Frequency Cantilever Beam

12/15/2016 4:47 AM

Hi all,

Many thanks in advance for your help.

My understanding of the dynamic analysis theory is: "Modal Analysis" evaluates the natural frequencies and modes, while "Frequency Response Analysis" gives the response of my system to a dynamic load. The first depends only on the system, the latter also depends on the loads.

Let's carry out a modal analysis: let's calculate a natural frequency. Our system is a cantilever beam.
The formula for natural frequency is:
ωn=√(k/m)

Now, here is a problem. What stiffness should I consider?

  1. If I consider a concentrated force on the beam extremity, the max deflection is Pl3/3EI, so I would say that the stiffness is k=F/δMax=3EI/l3
  2. If I consider a uniformly distributed load, then he max deflection is ql4/8EI and so I would consider as stiffness k=ql/δMax=8EI/l3

So I would get different natural frequencies in the two cases. Is that right?

Question n1:

Did not we say that modal analysis is independent on the load? Does it mean that the natural frequency depends on the type of load? So do I need to know what type of load I am going to apply before evaluating the natural frequency?
However, I carried on with my calculations. I assumed:

length=400mm;E=210000N/mm2; A:4mm2 I=1.33mm4; m=0.0123kg.

and I got:

  1. for the concentrated load at extremity Tn=2π/ωn=2π/√(k/m)=2π/√(3EI/l3/m)=0.19s
  2. for the distributed load Tn=2π/ωn=2π/√(k/m)=2π/√(8EI/l3/m)=0.12s

Then I used a FEA software, input the same values for l, E,A, etc. and it gave me the results in the picture.
As you can see the results are different from mine, so:
Question n2&3:

  1. Why FEA gives different results from my calculations? Did I make a mistake in the way I interpret the theory?
  2. The FEA software did not ask for any loads when calculating these modal frequencies. But previously, did not we say that we need to know the type of load to estimate the natural frequency?

Thank you very much for your great help

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#1

Re: Natural frequence cantilever Beam

12/15/2016 4:59 AM

"Did not we say that modal analysis is independent on the load? Does it mean that the natural frequency depends on the type of load?"

If we said it's independent on the load, it means that is does not depend on the load.

(Would've been better phrased "independent of the load").

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Posts: 10
#2
In reply to #1

Re: Natural frequence cantilever Beam

12/15/2016 5:20 AM

"If we said it's independent on the load, it means that is does not depend on the load."

Sorry.

Of course I did not express myself clearly.
I meant:
From theory, we know that the natural frequency should not depend on the load, but, on the other hand, when I actually try to calculate it, I see that the natural frequency formula depends on the stiffness k, which depends on the type of load that we are applying. I mean: k is equal to k=F/δMax=3EI/l3 for concentrated load on the extremity and k=ql/δMax=8EI/l3 for uniformly distributed load

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#5
In reply to #1

Re: Natural frequence cantilever Beam

12/16/2016 4:40 PM

I cannot name the tune, but if it had two natural frequencies, it would have an interesting beat! It might end up with a rogue amplitude of oscillation at some point in time?

I think OP needs to study the k function over the length of the cantilever, as integral dk/dl. This includes the moment of each point, including the beam and the load added to that point, does it not?

In a linear approximation, the moment will be defined at the total length of the cantilever if the load is a large number times the mass of the cantilever structure itself. That is because the moment contributions are very small until the point of attachment of load.

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#3

Re: Natural Frequency Cantilever Beam

12/15/2016 3:09 PM

It's not my area of expertise, but the way I see it, the force causing the deflection is the inertia of the mass of the cantilever, so I would say that uniformly distributed load would be the correct stiffness.

As to why the Finite Element Analysis does not agree, I have no idea. Maybe this will help:

https://sites.google.com/site/lucidanalysis/examples/nm_cantilever

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#4

Re: Natural Frequency Cantilever Beam

12/15/2016 3:33 PM

Guys!

I have just realized my mistake! I will write it here, in case somebody is interested or has something to add!

The equation is valid for a 1dof system, so I had to consider the equivalent 1dof mass-spring system! What is it?
It is a system made up of 1 point with an equivalent stiffness and an equivalent mass. The equivalent stiffness must be considered considering the concentrated equivalent force divided by the deflection of the point where this force is applied. So, the stiffness is well calculated. But I have to consider also an equivalent mass! What I was doing with my previous equation was assuming that the equivalent mass was equal to the total mass of the system as if my system was equivalent to one point at its extremity with all the mass! This is wrong!

Rayleigh gives us info about how to do this! And in a cantilever beam I have to assume only 23% of the mass!

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