This document discusses three engineering applications of linear algebraic equations: 1) Analysis of a statically determinate truss by setting up equations for the forces at each node. An example truss is shown. 2) Modeling a spring-mass system using equations relating the positions, velocities, forces, and properties of 3 connected masses and springs. The equations of motion are derived. 3) Modeling a chemical system using a stiffness matrix relating the concentrations of chemical species to reaction rates and equilibrium constants.

1. 1
Engineering Applications:
Linear Algebraic Equations

2. 2
Analysis of a Statically Determinate Truss
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External forces: F1,h, F1,v, F2,h, F2,v, F3,h, F3,v
h
o
o
H F
F
F
F ,
cos
cos 1
3
1 60
30
0 

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


v
o
o
V F
F
F
F ,
sin
sin 1
3
1 60
30
0 





Node 1:
Node 2 and 3: similarly ……

3. 3
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4. 4
Example of Statically Determinate Truss
1000
2
1 
 h
h F
F ,
,
Wind blows east Wind blows west
1000
3
1 

 h
h F
F ,
,

5. 5
Spring-Mass Systems

6. 6
Steady state: all the speeds are zero
3kx1 - 2kx2 = m1g
-2kx1 + 3kx2 - kx3 = m2g
- kx2 + kx3 = m3g
Spring-Mass Systems (cont.)
  1
1
1
2
1 2
2
1
2
kx
g
m
x
x
k
m
dt
x
d




U
D
dt
x
d
F
F
m 

2
2
Mass m1: FU = kx1; FD = 2k(x2 – x1) + m1g
 
2
3
3
3 2
3
2
x
x
k
g
m
m
dt
x
d



   
1
2
2
2
3
2 2
2
2
2
x
x
k
g
m
x
x
k
m dt
x
d





Mass m3:
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g
m
g
m
g
m
x
x
x
k
k
k
k
k
k
k
3
2
1
3
2
1
0
3
2
0
2
3
Stiffness matrix
Mass m2

7. 7
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g
m
g
m
g
m
x
x
x
k
k
k
k
k
k
k
3
2
1
3
2
1
0
3
2
0
2
3

8. 8
Example of Chemical System

9. 9