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How to calculate the vibration modes of the beam?

  1. May 16, 2015 #1

    I´ve tried to calculate the vibration modes of beam in the space, without gravity or any other force.

    The forumle of the beam is this one: Cos(λL)*Cosh(λL)=1 from which I´ve calculate the roots and the resonance.

    I´ve found a programme for Matlab that calculates the vibrations modes if you enter the mass and stiffness matrix but it´s here where I´m facing some troubles. I don´t know how to build the stiffnes matrix of this beam.

    For the mass matrix I´m going to use this methog. Do you know if it´s going to work properly?


    I´ve all the dimensions, density and the young´s modulus of the material (steel).

    Thanks in advance.
  2. jcsd
  3. May 16, 2015 #2


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    Your post is a little vague.

    What kind of beam are you trying to analyze?

    What is its shape?

    How is it loaded?

    What is the beam made out of?

    Is the beam prismatic, or are there variations in its shape or construction?

    The following is a typical stiffness matrix for a beam with 6 DOF at each end node:

  4. May 16, 2015 #3
    I think that ´s exactly what I´m looking for.

    Firtly I apologize for my bad english. I might not be clear enough in my first post.

    I know all the dimensions. The beam is very simple, constant section, its a prismatic beam. I know all the dimensions, I´ll post them tomorrow.

    The beam is in the space, without any load or force on it. The material is steel and the density is 7800 kg/m3.

    Thanks for your reply, I think it will be of help.

    Do you know if the mass matrix I posted will work?

  5. May 17, 2015 #4
    The dimensions are: lenght=2 m, wide=0.05m and the thickness=0.005 m
  6. May 17, 2015 #5


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    One of the difficulties that you will encounter trying to do an eigenanalysis for a free-free beam in space is that the stiffness matrix is singular. There will be several zero eigenfrequencies (how many depending on the way you model the beam). This is not a simple problem.
  7. Sep 6, 2016 #6

    I was wondering if a tubular shaft breathing modes can be derived from bending frequency measurements?

    Kind regards,
  8. Sep 6, 2016 #7
    Many thanks for your reply. I have measured the bending frequency of a section of tubular mild steel. I was wondering if there was a method or calculation to establish breathing modes of the same tube without further testing?

    Kind regards,
  9. Sep 6, 2016 #8


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    Sensibly no is the answer .

    Just out of curiosity - are you making any distinction between breathing modes and bell modes ?
  10. Sep 6, 2016 #9


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    I assumed on first reading that you meant using entirely experimental methods .

    There are of course reasonably accurate methods for finding vibration modes analytically - are these of any interest ?
  11. Sep 6, 2016 #10
    Hello Nidum,
    Thank you for the reply. I am only familiar with the term breathing mode. I'm not sure if bell modes are similar? I was hoping to find some sort of calculator to establish breathing modes if such a calculator exists? I understand from your replies that the possibility of a direct broad spectrum "conversion" is near on impossible.

    Kind regards,
  12. Sep 6, 2016 #11
    Hello Nidum,
    Yes, I would be very interested in finding vibration modes analytically.

    Kind regards,
  13. Sep 6, 2016 #12


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    Can you tell me for what purpose ? If I understood what you are doing more clearly then I could give specific answers rather than more general ones .

    There are pure analytic solutions for some simple cases .

    More generally though the governing equations cannot be easily solved analytically and then numerical methods have to be used .

    Finite Element methods are now most commonly used for real engineering problems .
    Last edited: Sep 6, 2016
  14. Sep 6, 2016 #13
    The purpose of this is to establish the bending frequency and breathing modes of a circular tube.
  15. Sep 6, 2016 #14
    I have conducted modal impact tests and have the data for the bending frequency
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