Sketch the waveform to represent the transient response

[Connorm1]

Homework Statement

Sketch, on a set of common axes, waveforms to represent the transient
response of circuits having transfer functions with the following parameters:

a) ζ = 0.5, ω = 1×10^3 rad s^-1
b) ζ = 0.2, ω = 2×10^3 rad s^-1
c) ζ = 2, ω = 1×10^3 rad s^-1

Homework Equations

w_d=w_o√(1-ζ^2)
Time to peak overshoot = π/w_d
overshoot = e^{-(ζπ)/(√(1-ζ^2))}

The Attempt at a Solution

So with this in mind for
a)
w_d=(1*10³)√(1-0.5^2) = 866.025
Time to peak overshoot = π/w_d = 0.00363seconds
overshoot = e^{-(ζπ)/(√(1-ζ^2))} = 0.16303

b)
w_d=(2*10³)√(1-0.2^2) = 866.025
Time to peak overshoot = π/w_d = 0.00363seconds
overshoot = e^{-(ζπ)/(√(1-ζ^2))} = 0.16303

c) will have no overshoot as ζ = 2>1 (which means no oscillation).

How do i actually draw these as graphs now? is there more info i need?

do i also need to use

?

 

[rude man]

What's the input? Step? Delta? Ramp? ...
Given this you can solve your ODE but there are many places to look up under-damped, critically damped, and over-damped 2nd order systems.

 

[Connorm1]

This is what I'm unsure of, if it mentioned one of the above i could work through it via my notes... I've only encountered step change input for second order systems. Thus i understand which ones are under-damped (a/b) and which are over-damped c if this is the case. But how do i draw this? only thing i can see from the graph is y(t)=1 but how do i find x(t) & the time period...

 

[rude man]

This is what I'm unsure of, if it mentioned one of the above i could work through it via my notes... I've only encountered step change input for second order systems. Thus i understand which ones are under-damped (a/b) and which are over-damped c if this is the case.

Assume step input then.

But how do i draw this? only thing i can see from the graph is y(t)=1

Say what? The output is a constant dc voltage? And what graph? You haven't provided one for us to look at.
You have the information you need to draw the output graphs. I suggest looking up the various possible graphs depending on ζ and ω for a step input. Widely available on the web.

 

[Connorm1]

Assume step input then.Say what? The output is a constant dc voltage? And what graph? You haven't provided one for us to look at.

Apologies I think I've gotten myself confused... I think I am looking too deep into it. I'll be back when i have some form of graph! Thanks for your help @rude man

 

[rude man]

Apologies I think I've gotten myself confused... I think I am looking too deep into it. I'll be back when i have some form of graph! Thanks for your help @rude man

Good idea. Google "second order systems responses". The curves will all be there.

 

[Connorm1]

@rude man so after searching through second order system responses I have a graph that looks like this. How does it look?

 

[rude man]

That's the idea.
Make sure yu understand which waveform corresponds to your values of ω_n and ζ.

You should also understand how your ODE in your post 1 solves to those waveforms. That equation requires one or two initial conditions otherwise you get nothing from it.