I'm having really tough time to figure out how to inverse Laplace tranformations. I've got an example, apparently it's a walk in the park, but I just don't get it.

How do you go from this:

To this:

Is it just practise that gets you out of this easily?

 

Of course it is practise and a bit of tricks!

One way to do it is to brake the initial expression into smaller bits, which you can find the inverse Laplace from the table, by using partial fractions.

 

I'm having really tough time to figure out how to inverse Laplace tranformations. I've got an example, apparently it's a walk in the park, but I just don't get it.

How do you go from this:

To this:

Is it just practise that gets you out of this easily?

You can get to that result by means of doing partial fractions.

 

OK, I'll bite. Can either of you show us how to do this by partial fractions?

 

I usually skip any preliminary simplifications unless their blatently obvious because in general it takes as long to perform algebraic manipulations like completing the square and factoring or whatever else as it does to just go ahead and do the fractional expansion with what you have (IMO), whatever works for you though. I just distributed the 10 back in and used the quadratic to get the complex conjugate factors then solved pretty easy for the K's. Hope this helps if you are still confused or even more confused I'll try to clear it up if you ask, and hopefully I didn't work this too fast and overlook something.

-10s+60020____________ = K1 + K2____________ + K3__________
s(s+2000+6000i)(s+2000-6000i) s (s+2000+6000i) (s+2000-6000i)

solving: K1=0.0015
K2=0.0013<55.3
K3=0.0013<-55.3

0.0015 + 0.0013<55.3___ + 0.0013<-55.3__
S (s+2000+6000i) ( s+2000-6000i)

L-1 = 0.0015 + (0.0026e-2000t cos(6000t-55)) u(t)

For some reason I can't get the text to align right it should be:
(K1/s) + (K2/(s+2000+6000i)) + (K3/(s+2000-6000i))
the same thing in the other fraction the top doesn't line up very well I apoligize.

 

I think following 2 sites give good information about partial fraction
http://www.purplemath.com/modules/partfrac3.htm
http://en.wikipedia.org/wiki/Partial_fraction

In Laplace transformation (L{..}), aim of partial fraction is to brake the equation into set of known units. Like in your example:
L{Sin at} = s/(s^2+a^2)
L{Cos at} = a/(s^2+a^2)
and,
L{e^-bt.f(t)} = F(s+b) so during partial fraction you try to obtain a component like
(s+b)/{(s+b)^2+(a)^2}, and likewise.....