MEEN 364 Homework4 solution

Problem 1

Consider the system below. Obtain the governing equations of the system and then use the state space

representation. You can use [1 2 3] as your state vector, and as the input and

[2 3]as the output vector. (3 is the pressure at the point shown below)

Solution

The first reservoirs dynamics enforces:

1 = 1()1

(1)

1 =11

(2)

For the second reservoir:

1 + 2 = 2()2

(3)

2 =22

2

3

3

3

2

1

3

1

MEEN 364 Homework4 solution

(4)

And the third reservoir:

2 = 3()3

(5)

=33

(6)

Combining equations to get the state space representation:

(1)

1

= 11

(2)

11

1=

1

111

(7)

(3)

2

=1 + 2

2

(2)&(4)

11

+ 22

2=112

+2

222

(8)

(5)

3

=2 3

(6)&(4)

22

33

3=223

333

(9)

And the outputs would be:

3 = 3

=33

2 =22

So,

[

123

] =

[

1

110 0

1

12

1

220

01

23

1

33]

[

123

] +

[ 1

10

01

20 0 ]

[]

[

23] =

[ 0

20

0 0

30 0 ]

[

123

] + [0 00 00 0

] []

MEEN 364 Homework4 solution

Problem 2

Derive the governing differential equations of the electromechanical system shown below. Put

the equations of motion in state-space matrix representation, using the following X, U and Y as

the state, input and output vectors, respectively: = [ , , ], = [] and = [, , ]

.

Assume that is the current through the inductor and represents the first time derivative of .

Assume that the torque exerted by the motor is positive CW from the right. Assume that is

positive CCW from the right. 1 2 represent the number teeth on the two gears shown in

the diagram. Neglect the armature resistance and inductance of the motor. The rotor of the

motor has a moment of inertia . Assume spring and damper are both torsional elements.

Assume a torque constant and a back-emf constant for the motor. Also, assume that =

1 and = 2.

Solution

System constraints:

2

1

t L

b b

T k i

e kN

N

Electrical System

Define the loops and nodes used to solve the electrical circuit in the system

MEEN 364 Homework4 solution

There is only one inductor in this circuit so there will only be one EOM for the circuit portion.

To solve the problem Kirchhoffs voltage law will be used

Applying KVL around loop (1) gives: .

.2

1

.2

1

( ) 0

1( )

1( )

2

(1)

in L L b

L in L b

L in L b

e Ri L i e

Ni e Ri k

L N

Ni e i k

N

Mechanical System

Kinematics

2 1

2

1

2

1

.. ..2

1

R

R

R

R

N N

N

N

N

N

N

N

Kinetics

Draw the FBD of the rotor from the right

MEEN 364 Homework4 solution

Applying Newtons Law of motion to the rotor to determine the contact force between the

gears: ..

..

1

..2

1 1

1( )

RR

RR G

G t L R

M J

J T F r

NF k i J

r N

Draw the FBD of the inertia J from the right

Applying Newtons Law of motion to the inertia J:

MEEN 364 Homework4 solution

2

2 2

1 1

1 1

2 2

2

2 2

2

1 1

2

22 1

1

( )

because ,

( )

1( )

( ( ) )

(2)

G

L R

R L

L

t

t

R

t

J C K F r

r NJ C K k i J

r N

r N

r N

N NJ J C K k i

N N

NC K k i

N NJ J

N

Assuming

22

1

1

( ( ) )RN

J JN

These equations are then plugged into state-space:

2.

1

2

1

2

1

1 102 2 2

0 0 1 0

0

0 1 0 0

0 0 1 0

0

b

L L

in

t

L

in

t

Nk

Ni i

e

Nk

N

i

e

Nk K

N

K C

C

MEEN 364 Homework4 solution

Problem 3

Consider the system given below. The output is (displacement from equilibrium position) and

the input is (source voltage). The motor has an electrical constant , a torque constant .

The rotor, shaft and disk together have inertia and a viscous friction coefficient . The disk has

a radius of . (Hint: For a motor, torque = , back EMF = ). Suppose that is

defined in a way that given positive rotation lowers the mass.

a) Derive the governing differential equations of motion.

b) Represent the differential equations in state-space form and represent the output as a

function of the states. Use = [ ] as your state vector.

Solution:

First, we will solve the circuit:

3 2

2

3

MEEN 364 Homework4 solution

For loop1:

2 = 0

(1)

= 2 + 3

(2)

=

(3)

And loop2:

3 + = 2

(4)

= ( 3) 3 = 3( )

+ ( + )

= 3

(5)

Combining equations leaves:

(3)

=

(6)

Loop1 Loop2

2 3

MEEN 364 Homework4 solution

(1)

= 2

= ( 3)

= + 3

(5)

+ + ( + )

1=

( + )

( + )

(7)

For the mechanical part, we will start with the free body diagrams:

r

My

Tm

B.

J

2k(-y r).

3b(-y r).

2k(-y r) ..

3b(-y r)

For the pulley:

+ 2( ) + 3( ) =

We know 3 = , so:

(5)

+ ( + )

+ 2( ) + 3( ) =

=

2

3

22

1

(32 + +

+

) +

( + )

(8) For the mass:

= 3 2 + 2( ) + 3( )

= 5

5

2

3

(9)

Assuming

(+)= ,

1

(32 + +

+) = ,

(+)=

Thus, the state space representation would be:

3 2

MEEN 364 Homework4 solution

[ ]

=

[ 0

1/0000

1//000

000

22/0

2/

0/10

3/

000

2/0

5/

000

3/1

5/]

[ ]

+

[ 01/0000 ]

And the output:

= [0 0 0 0 1 0]

[ ]