natural frequency from eigen analysis

[civil2013]

Hi, i am trying to find the natural period of a vertical cantilever beam which is fixed at bottom and free at other end., i worked out the global M & K matrices and i have the eqn in the form [M]-w^2[K] = 0, the M & K are not diagonal matrices, but square symetric matrices of rank 6. i understand that w2 is the cyclic frequency, how do i solve this characteristic polynomial the matrix way.. thanks in advance,

 

[KootK]

What you're looking for are the values of w that satisfy:

DET[[M]-w^2[K]] = 0

For the generalized matrix solution to eigenvalue problems, Google will be your friend. It's pretty intense. For small systems only, it can be done by hand.

If your problem is just a uniform cantilever, there are much easier methods to find the natural frequency. Rayleigh's method is one such method.


I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[PittEng88]

I agree with KootK, Rayleigh's method would be a lot easier. A 6x6 eigenvalue problem is no small task!

 

[civil2013]

thanks guy, i have found the soln myself... the matrix was converted to diagonal form, and then the soln was easy..could you guys give me a link/note on rayeligh and any comparison with eigen analysis.. appreaciate your help... thanks.

 

[PittEng88]

Check out Dynamics of Structures by Anil Chopra. It has some pretty good examples of both eigenvalue problems and Rayeligh's method.

 

[civil2013]

hi,

i still face some problmes... my eqn to solve is of the form K-w2M=0 which is a generalised eigen eqn.. the matrix is of 6x6 size... whats the easy way to get the eigen pairs.. any handnotes on how to convert this to standard eigen eqn format ie; A-lamdaI=0 ?? appreciate your assistance.. thanks.

 

[civil2013]

hi, i found the soln myself.. just by multiplying throughout with m^-1, i converted the generalised eqn to stnd eigen format and then it was easier... i am using mathcad to get the eigen vectors, when i calculate the eigen vectors using the command - genvecs(Kr, Mr) - where the Kr, Mr are the stiffness and mass matrices, i get a sect of eigen vectors which is not matching when i am using the command eigenvecs(A) where A= Kr.Mr^-1.. screenshot attached..appreaciate help....also, i want to know how i could get the mode shape of vibration from the eigen vectors.... thanks

 

[PittEng88]

If I'm not mistaken, I think you might have the equation backwards. It should be {[K] - ω[sup]2[/sup][M]}[X] = 0

 

[civil2013]

hi PittEng88, what made u think that i have put the eqn backwards... i meant (K-w2M)x as u have mentioned, did u notice any place where i mentioned other ways? appreciate ur help.. thanks.

 

[PittEng88]

Sorry I meant in your original post you have: [M]-w^2[K] = 0. I can't really tell from your screenshot what is going on though, without seeing the whole problem. I was just merely pointing out that if you used the equation that you had in your original post, it would not work.

 

[civil2013]

hi PIttEng88,

i am attaching the mathcad(PTC) file along with the pdf copy too.i have given my explanations on the same, kindly review and get back to me. thanks.

https://drive.google.com/folderview?id=0B9gSrTCMo9R3d3lkeWJnMkZjNzA&usp=sharing

 

[PittEng88]

I will take a look at it when I get a chance.

 

[civil2013]

hi,
basically I used two methods to solve the [K]-w2[M]=0 eqn.

method 1: I retained the generalized eqn, then used the command genvals(K,M) in Mathcad to get the Eigen values and genvecs(K,M) to get the Eigen vectors

method 2: I multiplied through out with [M^-1] and so the eqn is now of the form [M^-1][K]-w2=0 which is standard Eigen format. assume [A]=[M^-1][K].... I calculated the Eigen value in Mathcad using eigenvals(A) and eigenvecs(A)... Eigen value remained same as that of method 1, but Eigen vector was not the same, how do I get the vectors as obtained in method1, also, how to derive the modeshape from the Eigen vectors..

thanks,

 

[PittEng88]

So, after looking at what you did I do have a few comments/suggestions/.

1.) You said that you are looking to analyze a cantilever column, which you split into two elements (3 nodes). When you do this you should only have a 4x4 stiffness matrix due to the boundary condition of the fixed support. Considering the other two nodes, each will have two DOF, a lateral displacement and a rotation.

2.) I also noticed in the stiffness matrix that you had the axial stiffness of the cantilever column included as well. Due to the direction of the motion of the cantilever, this is not necessary and will over complicate your analysis. Unless the motion you are looking to analyze is along the the column longitudinal axis.

2.) The mass matrix should be a diagonal matrix. Considering the column to be split into two elements, you should have two lumped masses.

3.) I would suggest getting a book on structural dynamics, to help explain things further.

 

[civil2013]

hi,i am putting my thoughts/reply below in blue

1.) You said that you are looking to analyze a cantilever column, which you split into two elements (3 nodes). When you do this you should only have a 4x4 stiffness matrix due to the boundary condition of the fixed support. Considering the other two nodes, each will have two DOF, a lateral displacement and a rotation. The whole structure is divided into 2 beam elements (actually frame) with 3 DOFs at each node (2 lateral and 1 rotation)- considering one single plane.so for element, its a 6x6 matrix, I combined the element matrices to get 9x9 matrix for the global stiffness matrix, then chopped out rows/columns from 1 to 3 as my initial node has no translations/rotations. I am pretty sure about the global [K] matrix as I modeled the same cantilever in sacs software, applied a load, found the deflection, and this is matching the deflection when I worked it out using disp=[K[sup]_1[/sup]]*{F}.. it was matching exactly the software output.

2.) I also noticed in the stiffness matrix that you had the axial stiffness of the cantilever column included as well. Due to the direction of the motion of the cantilever, this is not necessary and will over complicate your analysis. Unless the motion you are looking to analyze is along the the column longitudinal axis. I was considering axial load on the top when I checked the deflection, so included axial stiffness also, as said in point 1, the dof for frame was considered, not that of a beam.

2.) The mass matrix should be a diagonal matrix. Considering the column to be split into two elements, you should have two lumped masses. I was using consistent mass approach, and I verified the natural timeperiod with a commericial software, results are matching exact for mode 1. I compared the freq of cantilever for mode 1 in software & on Mathcad, its matching. what I need is how to arrive at mode shape from here.because I have converted the generalized eqn to std Eigen format, I understand that the Eigen values will remain same, but not Eigen vectors, I need to know how to get the Eigen vectors for the real case and then derive the modeshapes.

3.) I would suggest getting a book on structural dynamics, to help explain things further.

 

[civil2013]

hi kootK, PittEng88, any update/suggestions on the thread.
thanks,

 

[jrbaus]

Its been too long since I took a look at dynamics from first principles (due for a refresher), but I can remember solving eigenvalues using Sturm's method. I have attached my algorithm from my university days which is implemented in Matlab. I can email m file if it is more helpful. I believe this implementation came from "Matrix Analysis and Structural Dynamics" by Franklin Y. Cheng.

Sorry if I have misunderstood your question - I thought I would lob it out there anyway.... Hopefully I haven't completely missed the mark.

Regards Jake