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## Prove ( Or Disprove) that Lower EigenValues always Have More Dominant Corresponding EigenVectors in Multiple DOF System

Assuming that I have a multiple DOF system of second order differential equations of this form with n degree of freedom:

where C=αM+βK.

For this kind of system we can compute the eigenvalue( frequency) and eigenvector( normal modes).

We know that for computation purpose, only a few lowest eigenvalue modes need to be taken into account because higher modes rarely impact the total response.

My question is, is it possible to prove ( or disprove) that, under any sort of initial condition, that the lower the eigenvalue is, the more dominant the mode is in contributing to the total response of the system? Or, is it that although generally the higher the eigenvalue is the smaller the mode contribution, but nonetheless there are some exceptional modes that don't follow this pattern?

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