I saw that problem with spring connected circularly and found that for one of the normal mode frequency is zero, and could not explain why.

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    $\begingroup$ That sounds like a uniform translation. $\endgroup$ – garyp Jan 7 '16 at 19:01
  • $\begingroup$ @garyp: your answer is right. I explained it a bit more. $\endgroup$ – Kevin Kostlan Jan 7 '16 at 19:45
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    $\begingroup$ Hint: Solve for the modes, not just the frequencies. Then it will be completely obvious. $\endgroup$ – DanielSank Jan 7 '16 at 19:48

The zero frequency mode corresponds to a merry-go-round of masses rotating in the same direction. No springs are being stretched/compressed so there is nothing (apart from friction of course) to stop and reverse the motion of the masses.

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