# How to calculate the vibration modes of the beam?

• O_o_O_o_O
In summary: The problem is that I do not know how to construct the stiffness matrix.I would be very interested in finding a way to do this without further testing.Thank you for your time.In summary, Nidum is trying to calculate the vibration modes of a beam in space, but is having difficulty because the stiffness matrix is singular. He has measured the bending frequency of a section of steel tube, but does not know how to construct the stiffness matrix.
O_o_O_o_O
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?

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

Your post is a little vague.

What kind of beam are you trying to analyze?

What is its shape?

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:

SteamKing said:
Your post is a little vague.

What kind of beam are you trying to analyze?

What is its shape?

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:

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.

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 ?

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.

Hello Nidum,
Yes, I would be very interested in finding vibration modes analytically.

Kind regards,
Jim.

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:
The purpose of this is to establish the bending frequency and breathing modes of a circular tube.

I have conducted modal impact tests and have the data for the bending frequency

## 1. How do I determine the natural frequency of a beam?

The natural frequency of a beam can be determined by solving the equation of motion for the beam using its material properties, geometry, and boundary conditions. This can be done analytically or through numerical methods, such as finite element analysis.

## 2. What is the formula for calculating the vibration modes of a beam?

The formula for calculating the vibration modes of a beam is given by the solution to the eigenvalue problem of the beam's equation of motion. This solution will provide the natural frequencies and corresponding mode shapes of the beam.

## 3. How do I choose the appropriate boundary conditions for calculating vibration modes?

The appropriate boundary conditions for calculating vibration modes depend on the specific characteristics of the beam, such as its supports and applied loads. These conditions can be determined through knowledge of the beam's physical properties and an understanding of the expected behavior under different loading scenarios.

## 4. Can the vibration modes of a beam be affected by its material properties?

Yes, the material properties of a beam, such as its density, stiffness, and damping, can affect its natural frequencies and mode shapes. A change in material properties can result in a shift in the beam's natural frequencies and potentially alter its mode shapes.

## 5. Are there any simplifications or assumptions made when calculating the vibration modes of a beam?

Yes, there are often simplifications and assumptions made when calculating the vibration modes of a beam. These can include assumptions about the beam's geometry, material properties, and loading conditions. It is important to carefully consider these assumptions and their potential impact on the accuracy of the results.

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