# Laplace transform question

#### ham3388

Joined Jul 3, 2012
97
Hi dear friends.....

could some one please help me , its alredy done just want to know wheather im correct or no

my regards

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#### mlog

Joined Feb 11, 2012
276
I would say you're mostly correct. You have correctly found the natural frequencies and damping ratios. However, you can pretty much solve this problem by inspection. Let's look at it.

(1), (2), & (3) all have the same natural frequency, √2 rad/s. (4) is twice the frequency. That will be important later.

Ask yourself, which of this waveforms look like they're close to critical damping? I would say B. So which of the transfer functions have a damping ratio closes to unity? That would be (1). So I agree with you on (1)≈B.

So you have A, C, & D remaining to match with (2), (3), & (4). Which of the remaining waveforms has the shortest peak time, which would be consistent with the higher natural frequency? It looks like D to me. So I would match (4)≈D, and I believe this disagrees with your answer. If you calculate t_p for each transfer function, you'll find only (4) has a t_p < 2 seconds. So the math supports the eye test here.

A word of caution here: not only does the natural frequency affect t_p, but so does the damping ratio. It should be clear that a higher natural frequency and a lower damping ratio gives a shorter t_p. (4) has the highest natural frequency and the lowest damping ratio, which makes it a perfect candidate for the shortest peak time, t_p.

What's left? A & C, and (2) & (3). Between A & C, which has the greatest overshoot? Between (2) & (3), which has the smallest damping ratio, which would be consistent with the greatest overshoot?

#### ham3388

Joined Jul 3, 2012
97
Thanks mlog....

So I get

1~~B
2~~C
3~~A
4~~D

Cheers