Uniqueness of eigenvectors and reliability

[Ronankeating]

Dear All,

In general eigenvalue problem solutions we obtain the eigenvalues along with eigenvectors. Eigenvalues are unique for each individual problem but eigenvectors are not, since the case is like that how we can rely that solution based on the eigenvector is correct. Because if solution is X(eigenvectors) then 10*X, 20*X, 30*X, etc..will also conform with (K-w²*M)*X=0 eigenvalue problem. And sometimes we use those eigenvectors to find exact solution e.g. K*X = F and how reliable can be those solution even if the eigenvector is normalised?

Regards,

 

[MabAsakura]

I think the eigenvalues are what makes the solution reliable because eigenvalues are particular while eigenvectors are general.

 

[Ronankeating]

I think the eigenvalues are what makes the solution reliable

No doubt about that,

But as I show in my previous post, sometimes we go directly for the solution originating from eigenvectors e.g. linear elasticity problems. Suppose that you solve an engineering problem K*x = F can you say that K*10*x = F is also solution, it really doesn't look very reasonable?

 

[MabAsakura]

Hrm. In that case, wouldn't you have to set up boundary conditions? I.E. set reasonable limits as to which answers "make sense" and which do not. From a pure mathematics POV, yeah, the answers you choose would be arbitrary answers because there will be other, perfectly good answers that you could've picked but chose not to. But in engineering there are such things as a ludicrous answer.

 

[boneh3ad]

Your constant K would be adjusted accordingly. Or your set of constants, if the problem had multiple constants.

 

[Ronankeating]

Your constant K would be adjusted accordingly. Or your set of constants, if the problem had multiple constants.

Can you elaborate that a bit more? How the K will be adjusted?

That K is global stiffness matrix which depends on element cross sectional mechanical properties and X is global displacement/rotations matrix. After matrix multiplication two of them the result is F global load vector. The glitch here is that, in my case X are eigenvectors obtained from eigenevalue solution and since X and 10*X are both solution it becomes something like 5kN and 50kN are both solution which is not coherent.

Regards,

 

[boneh3ad]

Well any set of eigenvectors (or eigenfuctions if you are in function space) and their scalar multiples all equally satisfy the criteria of being eigenvectors for the particular system. However, your system likely has boundary conditions, which should determine the correct scaling for the eigenvectors so that you get the ones that should fit properly into your Hooke's Law equation.

Full disclosures, it has been a long time since I have done any elasticity problem. I am going off memory here.

 

[Ronankeating]

Dear All,

Thank you in advance does anybody can shed more detailed light on that one.

Regards,

 

[windsim]

for F=kX ,is the linear static formulation of the system.
the K matrix is generally singular, you must add some boundary conditions to eliminate the singularity,thus the function has an unique solutioin.

for eigenvalue problem , it's the scope of dynamics.the eigenvalue is related to the system natural frequency, eigenvector is the vibration mode of the correspoding frequency. component of the eigenvector is the relative vibrating amplitude of each node.
once the initial conditons and the boundary conditions are specified,the vibrating amplitude of each node is unique, i think u can reference books on mechanical vibration.