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When ζ is negative, Mr ceases to have any meaning? (Book Benjamin Kuo)

  1. Oct 20, 2014 #1
    In the book Automatic Control Systems, Benjamin C.Kuo, 7th edition, on page 548, he says:
    https://imagizer.imageshack.us/v2/720x154q90/540/ML8zmu.jpg [Broken]
    He is doing an analysis of the following transfer function:
    http://imagizer.imageshack.us/a/img911/4839/XPKxoO.gif [Broken]
    Mr is the maximum value that | H (jw) | can reach with w ranging from 0 to infinity.

    He says that if ζ is negative, the system is unstable and the value of Mr ceases to have any meaning.
    I disagree with that. Whereas if ζ is negative, it is clear that this puts the two complex poles to the right side of the real axis, but the function | H (jw) | is exactly the same in the cases of ζ be positive or negative and it is not by fact a transfer function having its poles right that Mr ceases to have meaning.
    Mr will be:
    http://imageshack.com/a/img538/9372/tFTOaZ.gif [Broken] (as discussed on page 546)
    independent of ζ to be negative or positive.
    Ie, it is not because the system is unstable to a step in the time that their analysis in the frequency domain loses meaning.
    I would like to be corrected if my view is wrong. Thank you.
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Oct 20, 2014 #2

    billy_joule

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    If a system could exist with a negative damping ratio, what would it do when disturbed? Would there be a resonant peak for this system?
     
  4. Oct 20, 2014 #3
    For a sinusoidal input, the amplitude is the same as it would if ζ were positive, so the maximum is the same. And the angle is the same, but negative.
    I think that's it.
     
  5. Oct 23, 2014 #4

    NascentOxygen

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    Staff: Mentor

    What is its response to a small impulse, e.g., thermal noise?
     
  6. Oct 23, 2014 #5

    billy_joule

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    Nope..
    Damping, like friction, is an energy loss.

    If a system with negative friction or negative damping could exist it would have very unusual characteristics...
    It would defy energy conservation and would make a fine perpetual motion machine.
     
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