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9. Solution of a Set of Linear Differantial Equations

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Solution under inputs

Solution under initial conditions

9. Solution of a Set of Linear Differantial Equations

x : Column matrix of state variables (nx1)

A: Square matrix (nxn), system matrix

u: Input vector (mx1)

B: Input matrix (nxm)

x0={x}t=0

I: nxn unit matrix

L2

L1

y2

y1

G

yA

yB

c

k

c

k

m,I

Example 9.1

(System in Problem 4 of Homework 01C)

General Coordinates: y1, y2

Inputs: yA, yB

m=1050 kg, I=670 kg-m2 k=35300 N/m, c=2000 Ns/m L1=1.7 m, L2=1.4 m

M

Eigenvalue equation:

clc;clear;syms s;

a=[0,0,1,0

0,0,0,1

-185.9,91.8,-10.5,5.2

91.8,-136.5,5.2,-7.8];

eig(a)

pause

i1=eye(4);a1=inv(s*i1-a);pretty(a1)

For multiplying polinoms, use conv ( ) commands in MATLAB

L2

L1

y2

y1

G

yA

yB

c

k

c

k

m,I

0.05m

(L=L1+L2=3.1 m)

Eigenvalues:

L2

L1

y2

y1

G

yA

yB

c

k

c

k

m,I

0.05m

(ξ =0.45)

(ξ =0.23)

Δt=0.02, t∞=3.31

In input t1=0.186