Aplikasi Transformasi Laplace (1)

Penyelesaian Persamaan diferensial

Consider an initial value problem

Where a and b are constant.

Here r(t) is the given input (driving force) applied to the mechanical or electrical system and y(t) is the output (response to the input) to be obtained.

Penyelesaian persamaan differensial

Step 1 : Setting up the subsidiary equation.

Step 2 : Solution of the subsidiary equation by algebra called transfer function

Step 3. Inversion of Y to obtain y.

Contoh

Selesaikan

y y = t, y(0) = 1, y(0) = 1

Penyelesaian

Step 1 : subsidiary equation

[s2Y(s) sy(0) y(0)] Y(s) = 1/s2

[s2Y(s) s 1] Y(s) = 1/s2

(s2 1) Y(s) = s + 1 + 1/s2

Step 2 : transfer function Q = 1/(s2 1)

Step 3 : Obtain Solution

Contoh lain

Selesaikan

y" + y' + 9y = 0, y(0) = 0.16, y'(0) = 0.

Penyelesaian

[s2Y - 0.16s] + [sY- 0.16] + 9Y = 0,

(s2+ s + 9)Y = 0.16(s + 1).

Shifted data problem

Selesaikan

y + y = 2t, y(/4) = , y() = 2 - 2

Karena t0 = , maka kita set t = t +

y + y = 2(t + ), y(0) = ; y(0) = 2 - 2

Penyelesaian

Shifted data problem (2)

maka

karena

Contoh 3

Selesaikan

y + y = t, y(0) = 1 dan y(0) = -2

Penyelesaian

[s2Y - sy(0) y(0)] + Y = 1/s2

(s2+1) Y = 1/s2 + s -2

Y

y = t + cos t -3 sin t

Contoh

Tentukan arus i(t) pada rangkaian RC berikut, jika diterapkan gelombang kotak v(t) seperti pada gambar

v(t) = V0[u(t - a) - u(t - b)].

Transformasi laplace

dengan

Penyelesaian

Latihan

Selesaiakan