MATLAB FEM for natural frequencies and mode shapes of a beam

In summary, the conversation discusses modeling a beam with 3 elements and finding the first 6 natural frequencies and corresponding mode shapes. The Homework Equations section presents a DET formula and shape functions for the beam. The Attempt at a Solution section mentions using eigenvectors and eigenvalues to find the natural frequencies, but expresses uncertainty about determining the mode shapes. The conversation also mentions a preferred method involving continuity and beam equations. The discussion concludes with a suggestion to use a maximum of 3 natural frequencies for the beam and offers assistance with deriving the method if needed.
  • #1
psuaero
4
0

Homework Statement



Model the above beam with 3 elements(image provided in attachment).
Calculate and list the first 6 natural frequencies
Plot the mode shapes corresponding to each of the natural frequencies

Homework Equations


DET([tex]M^{-1}[/tex]*K + [tex] \omega[/tex]^2)=0

shape functions?
H1=1-[tex]\frac{3x^{2}}{l^{2}}[/tex]+[tex]\frac{2x^{3}}{l^{3}}[/tex]

H2=x-[tex]\frac{2x}{l}[/tex]+[tex]\frac{x^{3}}{l^{2}}[/tex]

H3=[tex]\frac{3x^{2}}{l^{2}}[/tex]-[tex]\frac{2x^{3}}{l^{3}}[/tex]

H4=-[tex]\frac{x^{2}}{l}[/tex]+[tex]\frac{x^{3}}{l^{2}}[/tex]

The Attempt at a Solution


My code Assembles the reduced global mass matrix(M) and reduced global stiffness matrix(K)
each of which is 6x6. I found the natural frequencies using eigenvectors and eigenvalues:

[v,d]=eig(M^-1*K). where v contains the eigenvectors and d has eigenvalues

My problem is in finding the mode shapes. I'm not sure where to begin, here is my guess:

realizing that the middle element shares a node with both end elements I can reduce the eigenvectors from 6x1 to 4x1. should I multiply the shape functions and mode shapes and sum them to get the equation for the mode shape. for example:

w=H(1)*v(1,1)+H(2)*v(2,1)+H(3)*v(3,1)+H(4)*v(4,1)

I just want to make sure that my mode shapes are correct since I can't find them in the notes. Also when I did the above the mode shape wasn't what I expected. for the first mode i would expect a parabolic shape.
 

Attachments

  • beam.jpg
    beam.jpg
    4.3 KB · Views: 697
Physics news on Phys.org
  • #2
Hello, I don't favour the approach you are using to determine the natural frequencies of beams. Perhaps you ae forced to use that method.

My perfered method involves a series of continuity and beam equations that must be satisified during a natural frequency. It is similar to a Holzer method for determining torsional resonances.

I worked on the method for a while so it could be used on a variety of beam constraints, comparison with experimental and other theoretical methods was very good.

I can scan in some of my notebook scribbles when deriving the method if you would like. If you are stuck with using eigenvectors etc, it won't be of any use.

Generally speaking, the maximum number of natural frequencies a beam can have is equal to the degree of freedom for the beam. If you are told to use 3 elements for the beam, it can only have 3 distinct natural frequencies. Sure it will have the same frequencies in the orthagonal plane but that's not very exciting.

Good luck
 

1. What is MATLAB FEM and how does it work?

MATLAB FEM (Finite Element Method) is a numerical analysis technique used to solve complex engineering and scientific problems. It involves dividing a structure into smaller, simpler elements, and using mathematical equations to determine the behavior of each element. These equations are then combined to solve for the overall behavior of the structure. MATLAB FEM uses the MATLAB programming language to implement the Finite Element Method.

2. How can MATLAB FEM be used to calculate natural frequencies and mode shapes of a beam?

MATLAB FEM can be used to calculate the natural frequencies and mode shapes of a beam by creating a 3D model of the beam, defining the material properties and boundary conditions, and then solving for the eigenvalues and eigenvectors of the model. The eigenvalues represent the natural frequencies of the beam, and the eigenvectors represent the mode shapes.

3. What are natural frequencies and mode shapes?

Natural frequencies are the frequencies at which a structure or system will naturally vibrate without any external force applied. Mode shapes are the patterns of vibration that occur at each natural frequency. In the case of a beam, mode shapes represent the different ways in which the beam can bend or deform when vibrating at a specific natural frequency.

4. Can MATLAB FEM be used to analyze different types of beams?

Yes, MATLAB FEM can be used to analyze different types of beams, including simple beams, cantilever beams, and continuous beams. The properties and boundary conditions of the beam may need to be adjusted accordingly, but the overall process for calculating natural frequencies and mode shapes remains the same.

5. Are there any limitations to using MATLAB FEM for natural frequencies and mode shapes of a beam?

Like any numerical analysis method, MATLAB FEM has its limitations. It is important to ensure that the model accurately represents the real-life beam and that the boundary conditions and material properties are correctly defined. Additionally, MATLAB FEM may not be suitable for highly complex or non-linear problems. It is always best to consult with an experienced engineer or scientist when using MATLAB FEM for critical applications.

Similar threads

  • MATLAB, Maple, Mathematica, LaTeX
Replies
32
Views
2K
  • Mechanical Engineering
Replies
2
Views
818
  • Advanced Physics Homework Help
Replies
1
Views
894
  • Engineering and Comp Sci Homework Help
Replies
1
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
31
Views
1K
  • Mechanical Engineering
Replies
3
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
6
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
1
Views
914
Replies
2
Views
950
  • Engineering and Comp Sci Homework Help
Replies
26
Views
2K
Back
Top