Capitolo 0. INTRODUCTION 3.1
Car dumper: state space dynamic model
• Physical model of a car dumper:
6 ?
6 6
?
? ?
6 }
?
?6
?
?
L0
K
Ma
?
Xs 6
Rk +Rb1 Mag
XM a
Xmr l0
Rp+ Rb2 Rk +Rb1
mrg r0
Kpb2
b1
mr
6x
• POG block scheme of the car dumper:
−Mag -
? 1 s
?
1 Ma
?
X˙M a
-
pa
Rk +Rb1 Rb1
- 6
6
b1 1
s
Xr 6
K
6 6
-
Fk F -
-
? 1 s
?
1 mr
?
X˙mr
-
pr
?
−mrg
Rb2
- 6
6
b2 1
s
Xrp6
Kp
6 6
-
Fp
Rp+ Rb2
X˙s
• Energy vector q, state vector x and input vector u:
q =
pa Xr
pr Xrp
, x =
X˙M a Fk X˙mr
Fp
, u =
g F X˙s
Zanasi Roberto - System Theory. A.A. 2015/2016
Capitolo 3. DYNAMIC MODELING 3.2
• The POG state space dynamic model of the system is:
L ˙x = A x + B u
• The energy matrix L and the input matrix B have the following form:
L =
Ma 0 0 0 0 1
K 0 0
0 0 mr 0 0 0 0 1
Kp
, B =
−Ma 1 0
0 0 0
−mr −1 b2
0 0 −1
• The symmetric part As and the skew symmetric part Aw af matrix A are:
As =
−b1 0 b1 0
0 0 0 0
b1 0 −b1 − b2 0
0 0 0 0
Aw =
0 −1 0 0 1 0 −1 0 0 1 0 −1
0 0 1 0
• So, the POG dynamic model of the system can be expressed as follows:
Ma
1 K
mr
1 Kp
X¨M a
F˙k X¨mr
F˙r
=
−b1−1 b1 0
1 0 −1 0
b1 1 −b1−b2 −1
0 0 1 0
X˙M a
Fk X˙mr
Fr
+
−Ma 1 0 0 0 0
−mr −1 b2 0 0 −1
g F X˙s
Zanasi Roberto - System Theory. A.A. 2015/2016
