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.

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Presentation transcript:

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) I: nxn unit matrix x 0 = {x} t=0 Solution under initial conditions Solution under inputs

Example 9.1 G y2y2 y1y1 kck c L1L1 L2L2 m,I yAyA yByB General Coordinates: y 1, y 2 Inputs: y A, y B m=1050 kg, I=670 kg-m 2 k=35300 N/m, c=2000 Ns/m L 1 =1.7 m, L 2 =1.4 m (System in Problem 4 of Homework 01C) M

Eigenvalue equation:

clc;clear;syms s; a=[0,0,1,0 0,0,0, ,91.8,-10.5, ,-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

G y2y2 y1y1 kck c L1L1 L2L2 m,I yAyA yByB 0.05m (L=L 1 +L 2 =3.1 m)

Eigenvalues:

G y2y2 y1y1 kck c L1L1 L2L2 m,I yAyA yByB 0.05m (ξ =0.45) (ξ =0.23) Δt=0.02, t ∞ =3.31 In input t 1 =0.186