Engineering Finding the natural frequency of transfer function (2s) / (3s^2+5s+2)

AI Thread Summary
The discussion centers on determining the natural frequency of a transfer function, specifically addressing the challenges posed by the presence of an 's' in the numerator of the function (2s) / (3s^2+5s+2). Participants clarify that in such cases, the 's' can be factored out, allowing the analysis to proceed similarly to a standard second-order system. Concerns are raised about the implications of this approach on the system's oscillatory behavior, particularly regarding damping and the absence of natural frequency when damping is high. The conversation also touches on the importance of verifying results through Laplace transforms and partial fraction decomposition. Overall, the participants work through the complexities of the problem while sharing insights on control system analysis.
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Homework Statement
See attached PDF
Relevant Equations
(ω_n)^2 / [s^2 + 2ζ(ω_n) + (ω_n)^2]
In the context of control systems, if I have a vibratory second-order system, (ω_n)^2 / [s^2 + 2ζ(ω_n) + (ω_n)^2], I know how to get the natural frequency ω_n. So, if I have something like 2 / (3s^2+5s+2), I know how to get the natural frequency ω_n.

However, if I instead have something like (2s) / (3s^2+5s+2), I'm not sure what to do. Do I just ignore the s on the numerator and proceed like if it wasn't there or what?

Any input would be greatly appreciated!

Edit:
P.S.
For what it's worth, I'm doing this as part of a larger problem, so if I made a mistake and it's impossible for a vibratory second-order system to have an s on the numerator, just let me know. The problem is question 3 from the PDF document
 

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First,G(s)=y2/u1 it is not equal G1(s) multiplied by G2(s) since y1 in G1(s) is calculated when is no load after y1.In case there is a load [L2 series with r2] then y2/u1=2s/(3s^2+6s+2) and not 2s/(3s^2+5s+2).
Second:
However, since 6^2-4*3*2=12>0 sqrt(+12) is real and then no oscillation is expected [the damping factor is more than 1 and no ω]
 
If approximations don't work you can do the Laplace transform to verify. The equation you provided you can do do a partial fraction and use a table.
 
Thanks for your input, Babadag, but I still don't fully understand how to compute the connection of the two "sub-circuits". I also don't understand why multiplying the two transfer functions isn't good enough. Could you please elaborate on those?

Having said that, I figured out how to do the problem (as you can see the final answer is what you said), but I'm not sure if that's how the problem statement intended I do it. For what it's worth, here is what I did.:

Equation 1:
u_1 = (1 Ω)(I_A (t)) + (1 H) d [I_A (t) - I_B (t)] /dt
U_1 (s) = (1 Ω)(I_A (s)) + (1 H) s [I_A (s) - I_B (s)]
Removing the units for visual clarity,
U_1 (s) = I_A (s) + s I_A (s) - s I_B (s)

Equation 2:
0 = (3 H)(d/dt I_b (t)) + (2 Ω)(I_b (t)) + (1 H)[ d (I_b (t) - I_a (t) /dt]
0 = (3 H)(s I_B (s)) + (2 Ω)(I_B (s)) + (1 H)[ s (I_B (s) - s I_A (s)]
Removing the units for visual clarity,
0 = 3 s I_B (s) + 2 I_B (s) + s I_B (s) - s I_A (s)
0 = 4s I_B(s) + 2 I_B (s) - s I_A (s)
0 = (4s + 2) I_B (s) - s I_A (s)
I_A (s) = (4s + 2)/s I_B (s)

Equation 3:
y_2 (t) = (2 Ω)( I_b (t) )
Y_2 (s) = (2 Ω) I_b (s)
Removing the units for visual clarity,
Y_2 (s) = 2 I_B (s)

Then, we mix equations 1 and 2 to get the following, Equation 4,
Y_1 (s) / U_1 (s) = [2 I_B (s)] / [I_A (s) + s I_A (s) - s I_B (s)]

Then, mixing Equation 2 and Equation 4, we get
Y_1 (s) / U_1 (s) = [2 I_B (s)] / [{(4s + 2)/s I_B (s)} + s {(4s + 2)/s I_B (s)} - s I_B (s)]
Y_1 (s) / U_1 (s) = [2 I_B (s)] / [{(4s + 2)/s I_B (s)} + {(4s + 2) I_B (s)} - s I_B (s)]
Y_1 (s) / U_1 (s) = [2 I_B (s)] / [(4s + 2)/s I_B (s) + (4s + 2) I_B (s) - s I_B (s)]
Y_1 (s) / U_1 (s) = 2 / [(4s + 2)/s + (4s + 2) - s]
Y_1 (s) / U_1 (s) = 2s / [s(4s + 2)/s + s(4s + 2) - s^2]
Y_1 (s) / U_1 (s) = 2s / [4s + 2 + 4s^2 + 2s - s^2]
Y_1 (s) / U_1 (s) = 2s / [6s + 2 + 3s^2]
Y_1 (s) / U_1 (s) = 2s / [3s^2 + 6s + 2]

And, Joshy, thanks for the partial fractions and table tip; I later realized that while doing another problem.

What do you mean about an approximation, though?

I'm not too familiar with this stuff, but I've only ever seen stuff about a linear approximation, but that was in the time domain, not frequency domain. Were you suggesting the possibility of using a linear approximation in the frequency domain and then getting the inverse Laplace transform of that?

P.S.
Sorry for the late response!
 

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