Second Order Eigenvalues

[PhilipFry]

I'm new here to mathcad and I'm trying to find the eigenvalues of a second order spring/mass system without damping. I have a 6 by 6 matrix and I would prefer not to solve it symbolically because the solution would be too long. Is there a command to solve this system, or do I need to refresh my diffeq skills.

 

[GregLocock]

Six by six? wow.

Oh well, assuming that's right as we all know the eigenvalues L are the solution to the equation

determinant(A-L.I)=0

You need to know the oh so cryptic command

L:=eigenvals(A)

Works for me!

Now I've got to figure out how a 2DOF system can have 6 frequencies...







Cheers

Greg Locock

 

[PhilipFry]

Wait, I thought the eigenvals command was for a first order diffeq. In other words, I thought the system solved Cs+k=0, instead of ms^2+cs+kx=0. So, it seems to me like my answer using Mathcad would be in terms of s^2 instead of s. Am I off-base here?

As far as why I have a 6x6 matrix:
Simple, I do not have kinematic contraints holding me to y=0 or x=0. Also, I have a degree of freedom when it comes to rotation, so since I have two bodies, each with 3 DOF...

 

[GregLocock]

m.w^2.x+c.w.x+k.x=0

doesn't need s^2

Cheers

Greg Locock

 

[PhilipFry]

Good deal! Thanks for the help.

 

[Diquan]

Here is another way:

You have your matrix of masses: M
And your stiffness matrix: K (already condensated)

From the book: Dynamics of Structures, 2nd edition from Anil Chopra (if you have it good!), chapter 10.15 page 440.

You can solve it just like this:

http://img71.exs.cx/img71/7093/mathcad.jpg

As you can see the example is wrong because the first mode is the 3rd row of "omega" and 3rd column of "phi" (1st mode) is negative, but this is just an example, you get the idea.