ESCUELA SUPERIOR POLITECNICA DE CHIMBORAZO

FACULTAD DE MECANICA

ESCUELA DE ING. MECANICA

“CONTROL AUTOMATICO”

TRABAJO 2

TEMA: DIAGRAMA DE BLOQUES

INTEGRANTES: CLAUDIA SOLORZANO

JOSE LUIS ULLOA

TANIA YANCHALIQUIN

ING.: JORGE LEMA

RIOBAMBA - ECUADOR

Encontrar las ecuaciones diferenciales

NODO 1

f=M 1

d x1dt

+k1 x1+B1( d x1dt −d x3dt )+k 2( x1−x2)

NODO 2

0=M 2

d x2dt

+B2d x2dt

+k2(x2−x1)

NODO 3

fa=M3

d x3dt

+B1( d x3dt −d x1dt )+B2( d x3dt −

d x2dt )

D= ddt→operador diferencial

f=M 1D x1+k1 x1+B1D ( x₁−x₂ )+k2(x1−x2)

0=M 2D ² x₂+B2Dx₂+k2(x2−x1)

fa=M3D ² x₃+B1D ( x₃−x₁ )+B2D ( x₃−x₁ )

Encontrar las ecuaciones diferenciales

NODO 1

0=k1 x1+M 1

d ² x1dt ²

+B1d x1dt

+B2ddt (x1−x2 )+B3

d x1dt

NODO 2

f=M 2

d ² x2dt ²

+B3d x2dt

+B2ddt (x2−x1 )

D= ddt→operador diferencial

0=k1 x1+M 1D ² x ₁+B1Dx ₁+B2D (x1−x2 )+B3Dx₁

f=M 2D ² x₂+B3Dx₂+B2D (x2−x1)

1. Encontrar la función de transferencia2. De qué orden es el modelo matemático3. Encontrar x₃(t) si la señal es de tipo escalón F= 7N

Nodo 1

0=k2 X1+M 1d2 X1dt

+k3 (X1−X2 )+β2( d X1dt −d X2dt )+β1( d X1dt −

d X3dt )

D= ddt

0=k2 X1+M 1D2 X1+k3 (X1−X2 )+β2 (D X1−D X 2)+ β1 (D X1−D X3 )

D= S0=k2 X1+M 1S

2 X1+k3 (X1−X2 )+β2 (S X1−S X2 )+β1 (S X1−S X 3)

0=X1 (k2+M 1S2+k3+β2S+β1S)−X2 (k3+ β2S )−β1S X3

0=X1 (2+2S+S2 )−X2 (1+S )−X3S (1)

Nodo 2

f (s )=M 2d2 X2dt

+k 3 ( X2−X1 )+β2( d X2dt −d X1dt )+β1( d X2dt −

d X3dt )

D= ddt

F=M 2D2 X2+k3 (X2−X1 )+β2 (D X 2−D X1 )+β1 (D X2−D X3 )

D= S

F=M 2S2 X2+k3 (X 2−X1 )+β2 (S X2−S X1 )+β1 (S X2−S X3 )

F=−X1 (1+S )+X2 (2+2S+S2 )−X3S (2)

Nodo 3

0=k1 X3+M 3d2 X3dt

+β1(d X3dt −d X1dt )+β3( d X3dt −

d X2dt )

D= ddt

0=k1 X3+M 3D2 X3+β1 (D X3−D X1 )+β3 (D X3−D X2 )

D= S

0=k1 X3+M 3S2 X3+β1 (S X3−S X1 )+β3 (S X 3−S X2 )

F=X1S−X2S+(2+2 S+S2 )−X3S (3)

0=X1 (2+2S+S2 )−X2 (1+S )−X3S (1)

F=−X1 (1+S )+X2 (2+2S+S2 )−X3S (2)

F=X1S−X2S+(2+2 S+S2 )−X3S (3)

Resolviendo el sistema se tiene

X3F

= 3 s+3 s2+s3

4+18 s+32 s2+32 s3+20 s4+7 s5+s6

F (t )=172sinh (t )+ 1

2(1+((e−2t )−(2e−t )))

3 s+3 s2+s3

4+18 s+32 s2+32 s3+20 s4+7 s5+s6X3F