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

1. Oct 20, 2014

### xorg

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. Oct 20, 2014

### billy_joule

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?

3. Oct 20, 2014

### xorg

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.

4. Oct 23, 2014

### Staff: Mentor

What is its response to a small impulse, e.g., thermal noise?

5. Oct 23, 2014

### billy_joule

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.