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

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In summary, on page 548 of the book Automatic Control Systems, Benjamin C. Kuo discusses the analysis of a transfer function with a maximum value, Mr, for |H(jw)| when w ranges from 0 to infinity. He states that if the damping ratio, ζ, is negative, the system is unstable and the value of Mr loses meaning. However, the function |H(jw)| remains the same regardless of ζ being positive or negative. Mr is also independent of ζ being negative or positive. The response of a system with negative damping to a sinusoidal input would have the same amplitude and angle, but negative. For a small impulse, such as thermal noise, the system would have unusual characteristics as it
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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?
 
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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.
 
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What is its response to a small impulse, e.g., thermal noise?
 
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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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Related to When ζ is negative, Mr ceases to have any meaning? (Book Benjamin Kuo)

1. What does the value of ζ being negative mean in relation to the book "Mr" by Benjamin Kuo?

When ζ is negative, it means that the damping ratio of the system described in the book is over-damped. This means that the system will eventually return to its equilibrium position but will do so slowly and without any oscillations.

2. How does the negative value of ζ affect the meaning of "Mr" in the book by Benjamin Kuo?

The negative value of ζ changes the behavior of the system described in the book. This means that the system will behave differently and the meaning of "Mr" may not be applicable in this scenario.

3. Is a negative value of ζ always present in the book "Mr" by Benjamin Kuo?

No, a negative value of ζ is not always present in the book. The value of ζ can vary depending on the specific system being described in the book. In some cases, the damping ratio may be positive, resulting in a different behavior of the system.

4. How does the concept of damping ratio relate to the overall message of the book "Mr" by Benjamin Kuo?

The concept of damping ratio is used to describe the behavior of a system in response to external forces. In the book "Mr", this concept may be used to illustrate the effects of different factors on the characters and their actions.

5. Can you provide an example of when a negative value of ζ would be relevant in the context of the book "Mr" by Benjamin Kuo?

One example could be when a character in the book is faced with a challenging situation and their emotional response is over-damped, meaning they may not show immediate or intense reactions but will eventually come to terms with the situation in a calm and controlled manner.

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