Damping ratio and Maximum overshoot relation

[zoom1]

There is a certain equation relating both Mp (max. overshoot) and damping ration. Which is;

Mp = e^{(-ζ*pi)/(1-ζ2)1/2}

What I get from that equation is for every system a certain damping ratio will result the system in a certain amount of max. overshoot.

That sounds ridiculous, because every system must have its own inner dynamics, right ?

Can someone please explain me that phenomena ?

 

[milesyoung]

What I get from that equation is for every system a certain damping ratio will result the system in a certain amount of max. overshoot.

That equation only holds for second order systems (the definition of damping ratio relates to a second order ODE). If the dominant dynamics in your system is of second order, then you can use it to some extent.

I'm not sure what you mean by inner dynamics. If you know the damping ratio of a system, then you have characterized one aspect of its dynamic response (which includes max. percent overshoot).

 

[zoom1]

That equation only holds for second order systems (the definition of damping ratio relates to a second order ODE). If the dominant dynamics in your system is of second order, then you can use it to some extent.

I'm not sure what you mean by inner dynamics. If you know the damping ratio of a system, then you have characterized one aspect of its dynamic response (which includes max. percent overshoot).

Ok, you're right regarding this equation to belong to 2nd order systems.
From your bottom line what I conclude is, for every 2nd order system, a certain amount of damping leads a certain amount of max overshoot, which will be exactly the same for all the 2nd order systems.
Well, at that point another question popped-up; The max. overshoot is the max. oscillation in a signal, however every signal is not bounded to reach that max. value with the same damping ration, am I correct ? So, at that point systems (2nd order systems) can vary with the same damping ratio ?

 

[milesyoung]

That equation gives you the max. percent overshoot of the steady state value of the system step response. It will be the same for any other system with the same damping ratio. It is not a bound on the peak value during the first period of oscillation - it's the actual peak value (but only with a step function as the input).

Edit:
The natural response of any second order system is completely characterized by its damping ratio and natural frequency of oscillation. If they're both the same for any two systems, then they have the same natural response.

 

[Runei]

Exactly, I concur. The max overshoot is related to when the system is excited with a step input :)

 

[AlephZero]

That sounds ridiculous, because every system must have its own inner dynamics, right ?

You are right. The formula only applies for a system which can be modeled with just one degree of freedom. In other words, it only applies to systems where the "inner dynamics" don't have any significant effect on the amount of overshoot.

Of course this is a useful model in many situations - but not in EVERY situation!

 

[milesyoung]

What does "inner dynamics" mean?

The OP had some misconceptions with regards to the time response of second order systems. Leading him on with this probably won't do him/her any favors.

 

[AlephZero]

What does "inner dynamics" mean?

I took it to mean a MDOF system as opposed to a SDOF.

I think your use of "second order system" is also misleading, because MDOF systems (at least, mechanical ones) are also modeled by second order ODEs, but the OP's formula for overshoot doesn't necessarily apply to them.