n n > n

Τεχνίτη Νοημοσύνη και Ρομποτική

14 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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1.
For a stepper motor with
m

stator poles per phase, show that the number of teeth in a
stator pole is given by

t
s

=
2
1
n
p
mp

where
n
denotes the number of steps per revolution, for the case
n
r

> n
s
.
Pick suitable
parame
ters for a four
-
phase, eight
-
pole motor, using this relation, if the step angle is
required to be 1.8°. Can the same step be obtained using a three
-
phase stepper motor?

2. Consider the single
-
stack, three
-
phase VR stepper mot
or shown in Figure 1
.
(
n
r

= 8 and
n
s

= 12). F
or this arrangement, compare the following phase
-
switching sequences:

(i)

1
-
2
-
3
-
1

(ii)

1
-
12
-
2
-
23
-
3
-
31
-
1

(iii)

12
-
23
-
31
-
12

What is the step angle, and how would you reverse the direction of rotation in each
case?

3.
A lectern (or podium) in an auditorium i
s designed to adjust its height
automatically, depending on the height of the speaker. An ultrasonic gage
measures the height of the speaker and sends a command to the logic hardware
controller of a stepper motor, which adjusts the lectern vertically throu
gh a rack
-
and
-
pinion drive. The dead load of the moving parts is supported by a bellow
device. A schematic diagram of this arrange
ment is shown in Figure 2. The
following design requirements have been specified:

Time to adjust a maximum stroke of 1 m = 5

s

Mass of the lectern = 50 kg

Maximum resistance to vertical motion = 5 kg

Displacement resolution = 0.5 cm/step

S

S

N

N

1

2

3

1

2

3

1

2

3

1

2

3

Figure 1

Rack
-
and
-
Pinion

Drive

Stepper

Motor

Drive

Circuit

Bellows

Lectern

Height

Command

Ultrasonic

Gage

Select a suitable stepper motor system for this application. You may use the
ratings of the four commercial stepper motors as given in the
following table
and Figure 3.

Figure 2

Figure 3

4.
Compare frequency control with voltage control in induction motor control, giving
-
state slip
-
torque relationship of an
induction motor is given by

2
2
[1 ( ) ]
f
m
b
aSv
T
S S

with the parameter values
a

= 1 x 10
-
3

N.m/V
2

and
S
b

= 0.25. If the line voltage
v
f

=
241 V, calculate the breakdown torque. If the motor has two p
ole pairs per phase
and if the line frequency is 60 Hz, what is the synchronous speed (in rpm)? What is
the speed corresponding to the breakdown torque? If the motor drives an external
load, which is modeled as a viscous damper of damping constant
b

= 0.03

N.m/rad/s, determine the operating point of the system. Now, if the supply voltage
is dropped to 163 V through voltage control, what is the new operating point? Is
this a stable operating point?

5. A company hires you as a control engineer. Your first ta
sk is to develop an
automatic gauge control (AGC) system for a steel rolling mill which the company
produces. The mill is to be used for cold
-
rolling of steel sheets
. The rollers of the
mill are actuated by a hydraulic system consisting of a single acting
piston cylinder
unit and a spool valve.

A sketch of a preliminary model of the rolling mill is shown in Figure 4.

Figure 4

A

V

P

h

P

s

Q

Feed

Rolled Steel

F

m

A

V

P

h

P

s

Q

u

Feed

Rolled Steel

F

m

y

(a)

Verify that the following equations can be used to represent this model.

ky
F
F
AP
mg
y
c
y
m
dt
dP
V
y
A
Q
P
P
buC
Q
h
h
h
s
d

(b)

Accordin
g to the model in Figure 4 define all the parameters and variables in the
system equations given above. Note that k is a spring constant (roll modulus)
which is primarily determined by the properties of the steel plates that are rolled.

(c)

Draw a block diagr
am for the system, without any feedback control.

(d)

The following parameter values are given

Diameter of the hydraulic actuator piston

= 25.0 cm

Average volume of the mineral oil in the cylinder

= 20.0 x 10
3

cm
3

Bulk modulus of the mineral oil

= 30.
0 x 10
3

N/cm
2

Specific gravity of the mineral oil

= 0.9

Valve port width

= 1.0 cm

Discharge coefficient at the valve port

= 0.7

Combined mass of actuator piston and roller

= 1000.0 kg

Equivalent viscous damping coefficient

for piston/roller
vertical motion

= 600.0 N/cm/s

Supply pressure to the valve

= 1000.0 N/cm
2

Initially the spool valve is closed (u = 0.0) and the pressure inside the hydraulic
actuator is maintained at 500.0 N/m
2
. The rolling conditions are kept steady. What
is the
corresponding rolling force and the roller displacement?