Digital Control Systems [MEE 4003] - Kckong.info

2.1. STATE SPACE REALIZATION OF DYNAMIC SYSTEMS 13
c
x 2
k
0 0
x 1
u 1
Figure 2.2: An equivalent model of the creeping phenomenon.
• The springk 2 is placed betweenm 1 andm 2 . Thus,K 12 = K 21 = −k 2 . Between
m 2 and m 3 , the spring k 3 is placed, i.e., K 23 = K 32 = −k 3 . Similarly, K 13 =
K 31 = −k 4 .
⎡
⎣
Finally, the mathematical model of the three mass-spring-damper system is
⎤ ⎡ ⎤ ⎡
⎤
m 1 0 0 c 1 +c 2 −c 2 0 k 1 +k 2 +k 4 −k 2 −k 4
0 m 2 0 ⎦ẍ+ ⎣ −c 2 c 2 0 ⎦ẋ+ ⎣ −k 2 k 2 +k 3 −k 3
⎦x
0 0 m 3 0 0 0 −k 4 −k 3 k 3 +k 4
(d) The modeling of creep phenomenon of materials is introduced in this example. In
materials science, creep is the tendency of a solid material to slowly move or deform
permanently under the influence of stresses. In order to find the mathematical model
of the creep phenomenon, an equivalent model is introduced with fictitious masses
(m = 0) as in Fig. 2.2.
wherex =
The mathematical model of the equivalent model is
[ ] [ ] [
0 0 0 0
ẍ+ ẋ+
0 0 0 c
] [ ]
∈ R
x 2 u1
and u = ∈ R
2 0
2 .
[
x1
k −k
−k k
]
x = u
(e) In the figure, x is the position from the top of the bar, andv is the distance from the
vertical line to the center line of the deflected bar. The internal moment in the bar, M,
is related to its deflected shape, v, by
EI d2 v
dx 2 = M
The internal moment,M, is determined by the applied force, P , and the distance from
the vertical line, v (i.e., M = −Pv). Therefore, the bar under a compressive load is
modeled by
EI d2 v
dx 2 +Pv = 0
Digital Control Systems, Sogang University
Kyoungchul Kong