Design Realization - lecture 20 docx

Design Realization
lecture 20
John Canny
10/30/03

Last time

Real-time programming

This time

Mechanics – Physics and Motors

Review of physics

Newton’s law for translation:
F = m a
F in Newtons, m in kg, a in m/s
2
.

Acceleration a = dv / dt

Kinetic energy E = ½ m v
2
E in Joules, m in kg, v in m/s.

Physics of translation

Momentum p = m v and so F = dp / dt


In the absence of force, momentum is
conserved.

Momentum conservation implies energy
conservation.

Physics of rotation

Rotation is more complex; Euler’s equation:
T = I
α
+
ω
x I
ω
T (torque) in N-m,
ω
in radians/sec,
α
in
radians/sec
2
, I in kg-m
2
,
α
= d
ω
/ dt


I is a 3x3 matrix, not necessarily diagonal.

If T = 0, then I
α
= -
ω
x I
ω
which is
usually non-zero. So
α
is non-zero,
ω

changes with time, and the object wobbles.

Physics of rotation

Angular momentum is q = I
ω

The rotation equation simplifies to T = dq / dt
because
dq/dt = I d
ω
/dt + dI/dt
ω
= I
α

+
ω
x I
ω

So even though an object wobbles when
there is no external force, the angular
momentum is conserved: q = I
ω

Physics of rotation

Kinetic energy of rotation is ½
ω
T
I
ω

In the absence of external torque, kinetic
energy of rotation is conserved.

But angular momentum conservation does
not imply energy conservation.

Work

Work done by a force = F x (Joules) where x
is the distance (m) through which the force
acts.


Work done by a torque = T
θ
(Joules)

Power

Power is rate of doing work.

Power of a force = F v (Watts).

Power of a torque = T
ω
(Watts).

Power often expressed in horsepower = 746
Watts

Motors

Motors come in several flavors:

DC motors

Stepper motors

(AC) induction motors

(AC) Single-phase motors

(AC) Synchronous motors


The first two are highly controllable, and
usually what you would use in an application.
But we quickly review the others.

3-phase AC

Three or four wires that carry the same
voltage at 3 equally-spaced phases:

Single phase AC requires two wires (only 1/3
the current or power of 3-phase).

AC induction Motors

Induction motors – simple, cheap, high-power,
high torque, simplest are 3-phase.

Speed up to 7200 rpm: speed ~ 7200 / #
“poles” of the motor.

Induction motors are brushless (no contacts
between moving and fixed parts). Hi reliability.

Efficiency high: 50-95 %

Single-phase AC Motors

Single-phase (induction) motors – operate from
normal AC current (one phase). Household

appliances.

Single-phase motors use a variety of tricks to
start, then transition to induction motor
behavior.

Efficiency lower: 25-60%

Often very low starting torque.

Synchronous AC Motors

Designed to turn in synchronization with the
AC frequency. E.g. turntable motors.

Low to very high power.

Efficiency ??

DC Motors

DC motor types:

DC Brush motor

“DC” Brushless motor

Stepper motor

DC Brush Motors


A “commutator” brings current to the moving
element (the rotor).

As the rotor moves, the polarity changes,
which keeps the magnets pulling the right
way. DEMO

Highly controllable, most common DC motor.

DC Brush Motors

At fixed load, speed of rotation is
proportional to applied voltage.

Changing polarity reverses rotation.

To first order, torque is proportional to
current.

Load curve:

Motors which
approximate this
ideal well are
called DC servo
motors.

DC Brushless Motors


Really an AC motor with electronic
commutation.

Permanent magnet rotor, stator coils are
controlled by electronic switching. DEMO

Speed can be controlled accurately by the
electronics.

Torque is often constant over the speed range.

Stepper Motors

Sequence of (3 or more) poles is activated in
turn, moving the stator in small “steps”.

Very low speed / high angular precision is
possible without reduction gearing by using
many rotor teeth.

Can also “micro-
step” by activating
both coils at once.

Driving Stepper Motors

Note: signals to the stepper motor are binary,
on-off values (not PWM).

In principle easy: activate poles as A B C D

A… or A D C B A…Steps are fixed size, so no
need to sense the angle! (open loop control).

Driving Stepper Motors

But in practice, acceleration and possibly
jerk must be bounded, otherwise motor will
not keep up and will start missing steps
(causing position errors).

i.e. driver electronics must simulate inertia of
the motor.

Stepper Motor example

From Sherline CNC milling machine:

Step angle: 1.8°

Voltage: 3.2 V

Holding torque: 0.97 N-m

Rotor inertia: 250 g-cm
2

Weight: 1.32 lb (0.6 Kg.)

Length: 2.13" (54 mm)


Power output = 3W

Precision stepper motor: 0.02° /step, 1 rpm, 3W

DC Motor example

V = 12 volts

Max Current = 4 A

Max Power Out = 25 W

Max efficiency = 74%

Max speed = 3500 rpm

Max torque = 1.4 N-m

Weight = 1.4 lbs

Forward or reverse (brushed)

Many DC motors of all sizes available new
and surplus for < $10

DC Motors – micro sizes

From Micromo:

Conventional (brush)

DC motor: 6mm x 15mm

13,000 rpm

0.11 m Nm

Power 0.15 W

V from 1.5 to 4.5 V