9. solution of a set of linear differantial equations
DESCRIPTION
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). x 0 = {x} t=0. I: nxn unit matrix. L 2. L 1. - PowerPoint PPT PresentationTRANSCRIPT
9. Solution of a Set of Linear Differantial Equations
BuAxx x : Column matrix of state variables (nx1)
A: Square matrix (nxn), system matrix
u: Input vector (mx1)
B: Input matrix (nxm)
)s(BU)s(AXx)s(sX 0
)s(BU)s(AXx)s(sIX 0
)s(BUx)s(X]AsI[ 0
)s(BU]AsI[x]AsI[)s(X 10
1
I: nxn unit matrix
x0= {x}t=0
Solution under initial conditions
Solution under inputs
Example 9.1
Gy2y1
k c k c
L1 L2
m,IyA yB
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
BB
AA
2
1
2
1
2
1
y35300y2000
y35300y2000
y
y
353000
035300
y
y
20000
02000
y
y
48.38532.190
32.19087.283
11 vy 22 vy
BB
AA
2
1
2
1
2
1
y35300y2000
y35300y2000
y
y
353000
035300
v
v
20000
02000
v
v
48.38532.190
32.19087.283
BBAA
BBAA
2
1
2
1
2
1
y5.136y8.7y2.5y8.91
y8.91y2.5y9.185y5.10
y
y
5.1368.91
8.919.185
v
v
8.72.5
2.55.10
v
v
(System in Problem 4 of Homework 01C)
M
11 vy 22 vy
BBAA
BBAA
2
1
2
1
2
1
y5.136y8.7y8.91y2.5
y8.91y2.5y9.185y5.10
y
y
5.1368.91
8.919.185
v
v
8.72.5
2.55.10
v
v
BBAA
BBAA
2
1
2
1
2
1
2
1
y5.136y8.7y8.91y2.5
y8.91y2.5y9.185y5.10
10
01
00
00
v
v
y
y
8.72.55.1368.91
2.55.108.919.185
1000
0100
v
v
y
y
Ax x B u
)s(BU]AsI[x]AsI[)s(X 10
1
1
1
8.7s2.55.1368.91
2.55.10s8.919.185
10s0
010s
]AsI[
)s(Y)5.136s8.7()s(Y)8.91s2.5(
)s(Y)8.91s2.5()s(Y)9.185s5.10()s(U
BA
BA
0AsI
Eigenvalue equation:
0
1.16948s6.1928s3.377s3.18s)s(D 234
s9.185s5.10ss8.91s2.5
s8.91s2.5s5.136s8.7s
9.185s5.10s8.91s2.5
8.91s2.55.136s8.7s
)s(D1
BU]AsI[
232
223
2
2
1
BA
BA
Y)5.136s8.7(Y)8.91s2.5(
Y)8.91s2.5(Y)9.185s5.10(
8.72.55.1368.91
2.55.108.919.185
1000
0100
A
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)pausei1=eye(4);a1=inv(s*i1-a);pretty(a1)i3.143.7s 2,1 i9.79.1s 4,3
)s(D1
s9.185s5.10ss8.91s2.51.16948s9.955s5.136s8.2s8.91
s8.91s2.5s5.136s8.7s2.6s8.911.16948s7.972s9.185
9.185s5.10s8.91s2.57.972s8.240s3.18s8.2s8.91
8.91s2.55.136s8.7s2.6s8.919.955s4.191s3.18s
23222
22322
223
223
1]AsI[
For multiplying polinoms, use conv ( ) commands in MATLAB
)s(D1
Y)s1.16948s6.1928s4.191s8.7(Y)s8.91s2.5(
Y)s8.91s2.5(Y)s1.16948s6.1928s8.240s5.10(
Y)1.16948s6.1928s4.191s8.7(Y)s8.91s2.5(
Y)s8.91s2.5(Y)1.16948s6.1928s8.240s5.10(
BU]AsI[
B234
A34
B34
A23
B23
A23
B23
A23
14
Gy2y1
k c k c
L1 L2
m,IyA yB
0.05m
05.0
0 1tt
Ay By
05.0
0 t
s186.0
36001000
x60
1.3t1 s
05.0)s(YB s186.0
A es
05.0)s(Y
h
km60Va
(L=L1+L2=3.1 m)
)s(D1
Y)s1.16948s6.1928s4.191s8.7(Y)s8.91s2.5(
Y)s8.91s2.5(Y)s1.16948s6.1928s8.240s5.10(
Y)1.16948s6.1928s4.191s8.7(Y)s8.91s2.5(
Y)s8.91s2.5(Y)1.16948s6.1928s8.240s5.10(
)s(V
)s(V
)s(Y
)s(Y
B234
A34
B34
A23
B23
A23
B23
A23
2
1
2
1
4
s05.0
)s(YB s186.0A e
s05.0
)s(Y
)s(sD05.0
)s8.91s2.5(e)s(sD
05.0)1.16948s6.1928s8.240s5.10()s(Y 23s186.023
1
1.16948s6.1928s3.377s3.18s)s(D 234
]67.2)186.0t(3.14cos[e035.0)t(y )186.0t(3.71
)186.0t(u05.0]91.2)186.0t(9.7cos[e019.0 )186.0t(9.1
)47.0t3.14cos(e027.0 t3.7 )91.2t9.7cos(e025.0 t9.1
BU]AsI[
)s(V
)s(V
)s(Y
)s(Y
)s(X 1
2
1
2
1
i3.143.7s 2,1 i9.79.1s 4,3 Eigenvalues:
Gy2y1
k c k c
L1 L2
m,IyA yB
0.05m
)t(y1
i3.143.7s 2,1 (ξ =0.45)
i9.79.1s 4,3 (ξ =0.23)
Δt=0.02, t∞=3.31
In input t1=0.186