I have written a short paper titled:

"An Engineering Interpretation of the Complex Eigensolution of Linear Dynamic Systems".

The paper shows how the complex eigenvalues and eigenvectors interpret as physical values such as natural frequency, modal damping ratio, mode shape and mode spatial phase, and finally the modal coordinate with amplitude and phase, as explained by my good friend Claes Fredö above.

You can find the paper under my profile. Post me a request and I will share it with you.

Not sure how you solved : In controls though , a spring and a damper system displacement and frequency (low) is a say first order differential equation and the solution can be complex #s . They should be interpret based on coefficient or direction etc etc

Hi

To start. Unbounded domains have waves. Modes only exist for bounded domains, i.e. systems of finite dimension.

A mode shape is the net sum of interference from waves that form a pattern. Such interference is possible only at at a natural frequencies, i.e at frequencies where waves are able to freely propagate in the bounded domain.

The physical interpretation is

• Real values modes is that these are standing waves. Standing waves has a fix nodal (minima) position and zero net flux of energy.
• Complex modes are standing waves where energy flows from one section to the other. Complex modes has a node position that moves with time.

A condition for real valued modes is light and evenly distributed damping across the system.

For a complex mode, one part refers to the standing wave portion and the other accounts for the energy flux. Whether the former is the real or imaginary part depends on the type of mode data.

Hope this helps

Claes

I have written a short paper titled:

"An Engineering Interpretation of the Complex Eigensolution of Linear Dynamic Systems".

The paper shows how the complex eigenvalues and eigenvectors interpret as physical values such as natural frequency, modal damping ratio, mode shape and mode spatial phase, and finally the modal coordinate with amplitude and phase, as explained by my good friend Claes Fredö above.

You can find the paper under my profile. Post me a request and I will share it with you.