That's good to know. You can identify the "Gain" in this problem as being the equivalent of Gc(s) = 0.78, but that misses the crucial point the the "Gain" K can be made to show up in the denominator of the closed loop function explicitly. Now for any value of gain including 0.78 you can see what happens to the roots, and you can see how ω_sub_0 and ζ change as the gain changes. To be more specific If we consider the closed loop function as:

G(s) = 10 * K / (14.99*(s^2) + 13.33*s + (1+10*K))
​

Now we have an expression that is much more useful in examining questions about all sorts of things. We still need to transform into standard form and substitute the correct value of K.

 

That's good to know. You can identify the "Gain" in this problem as being the equivalent of Gc(s) = 0.78, but that misses the crucial point the the "Gain" K can be made to show up in the denominator of the closed loop function explicitly. Now for any value of gain including 0.78 you can see what happens to the roots, and you can see how ω_sub_0 and ζ change as the gain changes. To be more specific If we consider the closed loop function as:

G(s) = 10 * K / (14.99*(s^2) + 13.33*s + (1+10*K))​

Now we have an expression that is much more useful in examining questions about all sorts of things. We still need to transform into standard form and substitute the correct value of K.

But it seems like the meaning of K was pretty well established. He had a standard form for a second order (low-pass) response given as:

\(
G(s) = \frac{K \omega^2}{s^2+2 \zeta \omega s + \omega^2}
\)

So K is the DC gain of the overall system.

He just needed to match that up to the G(s) that he was starting from.

 

He did show that form at the end of the first post, but I was looking at the problem from a slightly different perspective. He'll get to the problem I was thinking of soon enough.

 

Hello guys.So i have this transfer function G(s)= 7.8 / ( 14.99*(s^2) + 13.33*s+8.8 ) and i need to find the gain K and the damping ratio ζ .What should i do? When i had an one order tranfer function i compared it with the the classic type k/(τs+1) . Should i do the same now ? I mean , i have to compare it with κ*(ω^2) / (s^2) + 2*ζ*ω*s+(ω^2) ???

Hi,

I think i see what happened here. You wrote:
κ*(ω^2) / (s^2) + 2*ζ*ω*s+(ω^2)

when you should have written:
κ*(ω^2) / [(s^2) + 2*ζ*ω*s+(ω^2) ]

because that second part should all be in the denominator. That will make it easy to match up with:
7.8 / ( 14.99*(s^2) + 13.33*s+8.8 )

because we know three things to start with:
w^2=8.8/14.99
2*ζ*ω=13.33/14.99
K*w^2=7.8/14.99

From inspection you can see that all we did was divide top and bottom by 14.99 to get s^2 alone, then match up the individual parts to the canonical form so we could see what all the parts equate to.

 

so after some calculations
K = 0.88
ζ = 0.34
but i have one more question. i need to find the time constant(im not sure if its called like that ) τ .
can i find it through ω ?

 

Oops...Never mind your K was correct, but

I get ζ = 0.580 as follows
2*ζ*ω = 13.33 / 14.99 ≈ .889
ζ = .889 / (2*ω) ≈ 0.444 / ω = 0.444 / 0.7662 ≈ 0.580
which is underdamped.

Solve for the roots. Look at the real part of the complex conjugate pair and that will give you the exponential decay of the oscillation of the underdamped system.

 

so after some calculations
K = 0.88
ζ = 0.34
but i have one more question. i need to find the time constant(im not sure if its called like that ) τ .
can i find it through ω ?

What does the time constant mean for an underdamped second-order response?

 

I suppose it means something akin to the exponential decay of a first order system. In the sense that after some number of time constants, the oscillations are bounded to a small(?) region about the steady state value.

 

I suppose it means something akin to the exponential decay of a first order system. In the sense that after some number of time constants, the oscillations are bounded to a small(?) region about the steady state value.

But doesn't the damping factor provide that information?

 

I think it probably does because you can relate ζ to the real part of the complex conjugate roots.

 

well i dont know , i guess i have to ask my teacher about that
btw u were right about the price of ζ .

 

Hi,

I think i see what happened here. You wrote:
κ*(ω^2) / (s^2) + 2*ζ*ω*s+(ω^2)

when you should have written:
κ*(ω^2) / [(s^2) + 2*ζ*ω*s+(ω^2) ]

because that second part should all be in the denominator.

Using tex, it's so easy to see that κ*(ω^2) / (s^2) + 2*ζ*ω*s+(ω^2) is

\(
\frac{K \omega^2}{s^2}+2 \zeta \omega s + \omega^2
\)

and κ*(ω^2) / [(s^2) + 2*ζ*ω*s+(ω^2) ] is

\(
\frac{K \omega^2}{s^2+2 \zeta \omega s + \omega^2}
\)

 

Hi,

I think i see what happened here. You wrote:
κ*(ω^2) / (s^2) + 2*ζ*ω*s+(ω^2)

when you should have written:
κ*(ω^2) / [(s^2) + 2*ζ*ω*s+(ω^2) ]

I chose not to harp on that (and it was HARD to hold myself back) because the TS seemed to understand what he meant and was struggling with other things. And, frankly, after not harping on it in my first response I promptly forgot about it altogether.

 

so after some calculations
K = 0.88
ζ = 0.34
but i have one more question. i need to find the time constant(im not sure if its called like that ) τ .
can i find it through ω ?

Hi,

I think you should check your basic math here.
Let me repeat the three little equations:

w^2=8.8/14.99
2*ζ*ω=13.33/14.99
K*w^2=7.8/14.99

so we get:
w^2=0.58705803869246164109406270847231
2*ζ*ω=0.88925950633755837224816544362909
K*w^2=0.52034689793195463642428285523682

and so taking the square root of w^2 we get:
w=0.76619712782838180063278043383521

and since we have the second equation we can calculate the damping factor now:
2*ζ*ω=0.88925950633755837224816544362909
ζ = 0.88925950633755837224816544362909/(2*ω)
ζ =0.88925950633755837224816544362909/(2*0.76619712782838180063278043383521) = 0.58030725647456417059289563539906

The time constant can be found from the real part(s) of the solution of the denominator set equal to zero, or for underdamped cases just:
T=1/ζ*ω

so the exponential factor would be:
e^(-ζ*ω*t)

It is very common to call the denominator of the exponential part the "time constant" in the same manner that we do for an RC circuit such as:
e^(-t/RC)

where the time constant here is RC (which of course is R*C).
It is usually represented by the Greek letter tau: τ
so we have:
e^(-t/τ)

and that is actually very common, and if two time constants then:
e^(-t/τ1)+e^(-t/τ2)

I've also seen it represented as in:
e^(-t*τ)

so here τ is the inverse of the previous, so for the resistor capacitor version it would be:
τ=1/RC

 

Using tex, it's so easy to see that κ*(ω^2) / (s^2) + 2*ζ*ω*s+(ω^2) is

\(
\frac{K \omega^2}{s^2}+2 \zeta \omega s + \omega^2
\)

and κ*(ω^2) / [(s^2) + 2*ζ*ω*s+(ω^2) ] is

\(
\frac{K \omega^2}{s^2+2 \zeta \omega s + \omega^2}
\)

Hi,

Yeah i'll have to play around with that a little sometime.