laplace transform-introduction and defnitions(2).pptx

8/14/2019 Laplace Transform-Introduction and Defnitions(2).pptx

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Prof. Dr. Ir. Harinaldi, M.EngMechanical Engineering Department

Faculty of Engineering University ofIndonesia

LAPLACE TRANSFORM


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A powerful tool to solve a variety of engineering problems. Two kinds of application.

initial-value problems of differential equations Transfer function -> control engineering

The strategy

Transform the difficult differential equations into simple algebra problemswhere solutions can be easily obtained. One then applies the InverseLaplace transform to retrieve the solutions of the original problems.


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Letf (t) be a function of time t. In many real problems only valuesof t 0 are of interest. Hence f(t) is given for t 0, and for all t< 0,

f (t) is taken to 0.

The Laplace Transform off(t) is F(s), defined by the following

integral:

Definition of The Laplace Transform

Introduction and Definitions


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Introduction and Definitions


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LT of Some Common Functions


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Operation Rules of LT


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Inverse Laplace Transform


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f


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I L l T f


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I L l T f


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I L l T f


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I L l T f


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I L l T f


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I L l T f


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I L l T f


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I L l T f


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In erse Laplace Transform


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Solving Differential Equations with Laplace Transform


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g q p



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g q p



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g q p



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g q p



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g q p



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g



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SOALSOAL

TRANSFORMASI LAPLACE

Transformasi Laplace


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0

2211 dttfctfcste

0 0

2211 dttfstecdttfstec

Contoh Soal:

1. Buktikan bahwa L adalah suatu operator linier

Jawab:

Jika c1

, c2 adalah konstan danf

1(t) ,f

2(t) merupakan suatu fungsi, maka :

L{c1f

1(t) + c

2f

2(t) } = c

1L{

f

1(t)} + c

2{f

2(t) }

Diperoleh

L{c1 f1(t) + c2f2(t) } =

= c1L{f1(t) + c2f2(t)}

=

Sehingga L adalah suatu operator linier

p



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p

2. Buktikan bahwa (a) , (b)

Jawab :



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p

3. Tentukanlah Transformasi Laplace dari setiap fungsi berikut

(a) , (b)

Jawab :

(a)

(b)

3e-4t 4 cos 5t



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4. Tentukanlah Transformasi Laplace dari setiap fungsi berikut

(a) , (b)

Jawab :

(a)

(b) Karena integral ini tidak konvergen,Maka transformasi Laplace nya tidak ada



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5.

Jawab :

4;0

42;1

20;3

)(

t

t

t

tfjika Tentukanlah



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6.

Jawab :

Tentukanlah

Maka :



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7. Tentukanlah

Jawab :



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7.

Jawab :



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8.

Jawab :



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9.

Jawab :


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11.

Jawab :

Sehingga invers transformasi Laplacenya menjadi :

Maka didapatkan :



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12.

Jawab : Dari tabel didapat :

Maka :

laplace transform-introduction and defnitions(2).pptx

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