How to calculate the vibration modes of the beam?

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Discussion Overview

The discussion revolves around calculating the vibration modes of beams, specifically a prismatic beam in space without external forces. Participants explore methods for constructing stiffness and mass matrices, as well as the challenges associated with eigenanalysis for free-free beams. Additionally, there is a side discussion on deriving breathing modes from bending frequency measurements of tubular structures.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a formula for beam vibration modes and seeks assistance in building the stiffness matrix for a prismatic beam made of steel.
  • Several participants request clarification on the beam's characteristics, including shape, loading conditions, and material properties.
  • A participant notes that eigenanalysis for a free-free beam is complicated due to the singularity of the stiffness matrix, leading to zero eigenfrequencies.
  • Another participant inquires about deriving breathing modes from bending frequency measurements of a tubular shaft, expressing interest in analytical methods for vibration mode calculations.
  • There is a distinction made between breathing modes and bell modes, with some participants questioning the relationship between the two.
  • One participant mentions the use of finite element methods for solving governing equations in more complex cases.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and interest in the methods for calculating vibration modes, with no consensus on the best approach or the feasibility of deriving breathing modes analytically. The discussion remains unresolved regarding the specifics of the stiffness matrix and the relationship between breathing and bell modes.

Contextual Notes

Limitations include the lack of detailed assumptions regarding the beam's loading conditions and the potential complexity of the stiffness matrix for free-free beams. The discussion also highlights the challenges in establishing breathing modes without further testing.

Who May Find This Useful

Readers interested in vibration analysis, structural engineering, and modal testing may find this discussion relevant, particularly those working with beams and tubular structures in theoretical or practical applications.

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Hi,

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?

SLGHkgO.png


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

Thanks in advance.
 
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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:

eIInC.png
 
SteamKing said:
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:

eIInC.png

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?

Greetings
 
The dimensions are: lenght=2 m, wide=0.05m and the thickness=0.005 m
 
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.
 
Hello,

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

Kind regards,
Jim.
 
Hello,
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,
Jim.
 
Sensibly no is the answer .

Just out of curiosity - are you making any distinction between breathing modes and bell modes ?
 
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 ?
 
  • #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,
Jim.
 
  • #11
Hello Nidum,
Yes, I would be very interested in finding vibration modes analytically.

Kind regards,
Jim.
 
  • #12
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:
  • #13
The purpose of this is to establish the bending frequency and breathing modes of a circular tube.
 
  • #14
I have conducted modal impact tests and have the data for the bending frequency
 

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