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Laplace help please!

northy16

6

Removed

(edited 9 years ago)

Reply 1

TeeEm

19

Original post by northy16

Hi,

Have a maths problem and was wondering if you could help?

I have a function: 1/s * [1/(sT+1)]^2

where T is a constant and I need to get the inverse laplace transform of it but I'm not sure how to do it.

I was advised to get it into the form 1/s(s+a)^2 then get the inverse laplace to give something along the lines of: 1/a^2 (1-e^-at-ate^-at).

I realise I could also do partial fraction expansion, but I haven't done much of this stuff for a while and just can't remember what to do.

Anyway long story short if anyone can provide any useful links or videos that show how to do similiar examples it'd be greatly appreciated. If anyone wants to solve it for me even better!

look at this link

http://madasmaths.com/archive/maths_booklets/advanced_topics/laplace_transforms.pdf

all questions have solutions

Reply 2

ztibor

10

Original post by northy16

Hi,

Have a maths problem and was wondering if you could help?

I have a function: 1/s * [1/(sT+1)]^2

where T is a constant and I need to get the inverse laplace transform of it but I'm not sure how to do it.

I was advised to get it into the form 1/s(s+a)^2 then get the inverse laplace to give something along the lines of: 1/a^2 (1-e^-at-ate^-at).

I realise I could also do partial fraction expansion, but I haven't done much of this stuff for a while and just can't remember what to do.

Anyway long story short if anyone can provide any useful links or videos that show how to do similiar examples it'd be greatly appreciated. If anyone wants to solve it for me even better!

You have to use partial fraction

\displaystyle \frac{1}{s\cdot (sT+1)^2}=\frac{A}{s}+\frac{B}{sT+1}+\frac{C}{(sT+1)^2}

Summarizing the fractions the numerator will be:

\displaystyle A(sT+1)^2+Bs(sT+1)+Cs=s^2(AT^2+BT)+s(2AT+B+C)+A

this equal with 1, the original numerator, so

\displaystyle AT^2+BT=0

\displaystyle 2AT+B+C=0

\displaystyle A=1

Solving simultaneously and assuming T=/=0

A=1 B=C=-T

So with inverse Laplace

\displaystyle f(t)=L^{-1}\left (\frac{1}{s}\right )-L^{-1}\left (\frac{1}{s+\frac{1}{T}}\right )-\frac{1}{T}L^{-1}\left (\frac{1}{\left (s+\frac{1}{T}\right )^2}\right )

So we get

\displaystyle f(t)=1-e^{-\frac{1}{T}}-\frac{1}{T}\cdot t\cdot e^{-\frac{1}{T}}

using that

L(t^k)=\frac{k!}{s^{k+1}}

and when

\displaystyle L^{-1} (s) =f(t)

then

\displaystyle L^{-1} (s+c) =e^{-ct}\cdot f(t)

from the shifting theorem

(edited 9 years ago)

Reply 3

ztibor

10

Original post by northy16

Thanks for the help. However I have one last question. In the equation: 1 - e^-1/T - 1/T * t * e^-1/T how do you get rid of the little t? As in say i wanted to put T as equal to 25x10-3.

Sorry if thats a stupid question, but not done Laplace for years.

EDIT: also shouldn't it be:

\displaystyle f(t)=1-e^{-\frac{1}{T}t}-\frac{1}{T}\cdot t\cdot e^{-\frac{1}{T}t}

Yes Of Course

EDIT2: also this link has the laplace transform table: http://www.dartmouth.edu/~sullivan/22files/New%20Laplace%20Transform%20Table.pdf

if you look at number 33. its in the same form as how we had it, but there is a 1/a^2 there. How come the answer you gave doesnt have the 1/a^2.

The book sais:

\displaystyle L^{-1} \left (\frac{1}{s\cdot (s+a)^2}\right )=\frac{1}{a^2}\left (1-e^{-at}-a\cdot t\cdot e^{-at}\right )

THe question was

\displaystyle L^{-1} \left (\frac{1}{s\cdot (sT+1)^2}\right )

Using the textbook 'template' form

Unparseable latex formula:

\displaystyle L^{-1} \left (\frac{1}{s\cdot (sT+1)^2}=\frac{1}{T^2}L^{-1} \left (\frac{1}{s\cdot (s+\frac{1}{T})^2}\right )

So along the textbook template with

a=\frac{1}{T}

substitution:

\displaystyle \frac{1}{T^2}L^{-1} \left (\frac{1}{s\cdot (s+\frac{1}{T})^2}\right )=\frac{1}{T^2}\cdot \frac{1}{\frac{1}{T^2}} \left (1-e^{-\frac{1}{T}t}-\frac{1}{T}\cdot t\cdot e^{-\frac{1}{T}t}\right )=

\displaystyle =1-e^{-\frac{1}{T}t}-\frac{1}{T}\cdot t\cdot e^{-\frac{1}{T}t}

As you see this is same as I wrote without the 'textbook template'

I think better idea to use only the base rules and theorems (shifting, convolution, Dirac delta, Heaviside function, transform of derivative etc) and
not premade template answers.
ie.use the partial fractions in your question

(edited 9 years ago)

Laplace help please!

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