# 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

Task 1

:

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.

Task 2: Build a Simulink model for the system

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

Task 1(b).

b.

Build a Simulink block diagram for nonlinear simulation, where the dynamics of

the cart

–

spring

–

pendulum system is described by the nonlinear differential

equations derived in Task

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

.

Grading:

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.

Construction of Simulink block diagrams

d.

Linear and nonlinear simulation results

e.

Discussions