Eigen period?

I am currently studying the heave response of a semi submersible and keep coming across Eigen period?

I have scoured the internet for a few hours now to find simply what they are but had no luck at all.

Can anyone direct me to a decent website or have the time to briefly and simply explain this.

Thank you in advance.

 

You've probably already done this but in case you haven't, I found lots of mention of "eigen period" after googling this: Eigen period in the study of heave stability for semi submersibles

 

It is the natural frequency and its mode shape.

 

The eigen period is the inverse of the frequency of natural oscillations of a body. The name comes from the fact that it is a solution of the eigenvalue problem relative to the equation of the free-oscillatory motion of the body:
x' + A x = 0
The associated eigenvalue lambda is the solution of the equation:
det |A - lambda I| = 0 .
Once you calculate the value of lambda, the eigen period t is given by:
t = 2 pi lambda
and the natural frequency is:
f = 1/t.

Cheers

P.S.
Ad Hoc was faster...

 

D's maths is much better than mine...I cant even multiple 2 x2 these days!!

Or put another way, in words.

Since you're in the UK you will no doubt have heard of the Millennium Bridge across the river Thames?....when it was opened in 2000, I think it (hence its name), it started to sway and vibrate within just days of its opening. It was subsequently closed down for repairs!!

The bridge like all structures has a natural period of vibration and has various modal shapes associated with it. The people walking across the bridge their walking become synchronised with each other (unsure if this was because it started to sway or other way around).

Some of the natural modes of vibration of the bridge coincided with the frequency or period of the sideway movement of walking of those people on the bridge. Thus the natural frequency and its mode shape is important and must be reviewed against all external forcing functions and their periods of motion.

 

Funny that, the Millenium Bridge. After the Tacoma narrows etc and the 'break step when marching' across bridges it was surprising that it 'wobbled' so much. My own inclination for a solution was to 'tune' the tensions to different frequency or pitch to avoid a coincident point when people crossed it. You deliberately use oscillation to get sound from stringed instruments and get scale length halving to same note an octave apart, plus harmonic periods.

Not sure what the eventual solution was but probably along those lines. If i remember right, it dampened some of the pitch frequencies to stop them reaching that coincident point.