# Modeling and Analysis of Dynamical Systems

Department of Mechanical and Aerospace Engineering
MAE
3
76
Modeling and Analysis of Dynamical Systems
Fall
201
5
Dr.
Q
ingbin Gao
Assigned
11/03
/1
5
Design Project
Due
12
/
08
/
1
5
The cart
spring
pendulum
system shown in
Figure 1
consists of a cart restricted to
motion on a straight and level track which is attached via a
spring to a fixed wall. A
pendulum is suspended from the cart
by a hinge so as to be constrained to the vertical
plane defined
by the tra
ck. The cart is equipped with a DC motor that exerts a
torque to a
small toothed wheel which, in turn, applies a force
on the cart.
For the purpose of
deriving a model, the
system is
considered
to be composed of a massless spring attached
to a frictionless
cart from which a slender rod freely
hangs
.
The output of the system is the position
p
of the cart, in meters,
relative to the spring’s
equilibrium point and the angular
position
of the pendulum,
in ra
dians, relative to the
vertical.
The physical
inpu
ts of the system are the voltage
u
applied to the armature
of the
DC
motor, in Volts, and a disturbance force
w
, in
Newtons. The force from the motor
f
, in
Newtons, is modeled
as
12
f k u k p

.
Figure 1 Mass
spring
pendulum system.
(Reference: IEEE Transactions on Automatic Control, 48(9), 1509
1525)
2
:
Derive the mathematical model of the system
Denote the mass of the cart as
m
, the mass of the pendulum as
M
, the length of the
pendulum
L
, and the stiffness of the spring as
k
.
The
values of
parameters are
m
= 0
.
455
kg,
M
= 0
.21 kg,
L
= 0
.61
m
,
k
= 100 N/m,
k
1
=
1.7
2
N/V
, and
k
2
=
7.68
Ns/m
.
Assume
that the disturbance force
w
is applied
at a distance of 2
L
/3 from the cart
pendulum hinge.
a.
Draw the necessary free
body diagram
and derive the nonlinear equations of
motion.
b.
Linearize the equations of m
otion for small angular motions.
c.
D
etermine the state
space form with
p
and
θ
as the outputs.
The system
is
disturbed by a sharp tap on
the pendulum that comes from a human hand.
For simulation purposes, the disturbance force
w
is m
odeled as a const
ant force of 1
6
.
0
N
with duration of 0.01 sec,
a
nd
a
ssume that
u
=0
.
a.
Build a Simulink block diagram for linear simulation, where the dynamics of the
cart
spring
p
endulum system is described by the state
space model derived in
b.
Build a Simulink block diagram for nonlinear simulation, where the dynamics of
the cart
spring
pendulum system is described by the nonlinear differential
1(a).
Task 3: Analyze the response of the system
Run the above two Simulink files, plot the
following
time responses
and compare the
linear and nonlinear simulation results.
a.
The position of the cart
p
vs. time
b.
The angular position of the cart
vs. time
Groups
: This is a group design project. Each student may choose his/her own group, and
the group size is not greater than
3
.
Grades will be based on a combination of group performance and individual
contributions. Group performance will be based on the quality of work contained in the
report and the involvement of all team members.
Report
: The required contents include
a.
Introd
uction
b.
Mathematical modeling of the system
c.