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

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Discussion Overview

The discussion revolves around the implications of a negative damping ratio (ζ) in control systems, specifically referencing Benjamin Kuo's analysis in "Automatic Control Systems." Participants explore the meaning of the maximum value of the transfer function's magnitude (Mr) in the context of system stability and frequency response.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant argues that Mr retains meaning even when ζ is negative, suggesting that the transfer function's behavior remains unchanged regardless of the sign of ζ.
  • Another participant questions the behavior of a system with a negative damping ratio when disturbed, specifically regarding the presence of a resonant peak.
  • It is noted that for sinusoidal inputs, the amplitude and phase angle of the system's response would be similar to those with positive ζ, despite the negative damping ratio.
  • A later reply emphasizes that negative damping could imply unusual characteristics, potentially violating energy conservation principles and suggesting perpetual motion scenarios.

Areas of Agreement / Disagreement

Participants express differing views on the implications of a negative damping ratio, with no consensus reached on whether Mr ceases to have meaning or retains its relevance in the analysis of system behavior.

Contextual Notes

The discussion involves assumptions about the nature of damping and stability, as well as the definitions of transfer functions and their properties in the frequency domain. The implications of negative damping on energy conservation are also raised but remain unresolved.

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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
He is doing an analysis of the following transfer function:
http://imagizer.imageshack.us/a/img911/4839/XPKxoO.gif
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 (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.
 
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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?
 
billy_joule said:
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?
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.
 
What is its response to a small impulse, e.g., thermal noise?
 
xorg said:
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.

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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