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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=ma
F in Newtons, m in kg, a in m/s2.
 Acceleration a = dv / dt
 Kinetic energy E = ½ m v2
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 + xI
T (torque) in N-m,  in radians/sec,  in
radians/sec2, I in kg-m2,  = d / dt
 I is a 3x3 matrix, not necessarily diagonal.
 If T = 0, then I  = -  x I  which is usually
non-zero. So  is non-zero,  changes with
time, and the object wobbles.
Physics of rotation
 Angular momentum is q = I 
 The rotation equation simplifies to T = dq / dt
because
dq/dt = I d/dt + dI/dt  = I  +  x I 
 So even though an object wobbles when there is
no external force, the angular momentum is
conserved: q = I 
Physics of rotation
 Kinetic energy of rotation is ½ T I 
 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  (Joules)
Power
 Power is rate of doing work.
 Power of a force = F v (Watts).
 Power of a torque = T  (Watts).
 Power often expressed in horsepower = 746
Watts
Motors
 Motors come in several flavors:
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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 “microstep” 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
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From Sherline CNC milling machine:
Step angle: 1.8°
Voltage: 3.2 V
Holding torque: 0.97 N-m
Rotor inertia: 250 g-cm2
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
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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
Brushless DC Motors
 From Micromo:
 Brushless DC motor:
16mm x 28mm
 65,000 rpm
 50 m Nm
 Power 11 W
 V = 12 V
DC Motors – gearing
 Gearing allows you to trade off speed vs. torque.
 An n:1 reduction gearing decreases speed by n,
but increases torque by n.
 Ratios from 10:1 to many 1000s :1 are available
in compact “gearheads” that attach to motors.
DC Motors – gearing
 But gears cost efficiency (20% - 50%)
 Gears decrease precision (due to backlash).
 Reduction gear train is normally not
backdriveable (can’t use for “force control”).
DC torque motors
 Some high-end motors are available for direct
drive servo or force applications (no gears).
 They have low speed (a few rpm), high precision
(with servo-ing), and moderate torque.
 Typically have large diameter vs. length, and
use rare-earth magnetic material.
 Cost $100’s (but maybe
less as surplus).
Sensors
 Shaft encoders can be fitted to almost any DC
motor. They provide position sensing.
 Many motor families offer integrated encoders.
 Strain gauges can be used to sense force
directly. Or DC brush motor current can be used
to estimate force.
Linear movement
 There are several ways to produce linear
movement from rotation:
 Rotary to linear gearing:
Linear movement
 Ball screws: low linear speed, good precision
 Motor drives shaft, stages move (must be
attached to linear bearing to stop from rotating).
Linear movement
 Belt drive: attach moving stage to a toothed belt:
 Used in inkjet printers and some large XY
robots.
True Linear movement
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There are some true linear magnetic drives.
BEI-Kimco voice coils:
Up to 1” travel
100 lbf
> 10 g acceleration
6 lbs weight
500 Hz corner
frequency.
 Used for precision vibration control.
Summary
 AC motors are good for inexpensive high-power
applications where fine control isnt needed.
 DC motors provide a range of performance:
 DC brush: versatile, “servo” motor, high speed, torque
 DC brushless: speed/toque depend on electronics
 Stepper: simple control signals, variable
speed/accuracy without gearing, lower power
 Direct-drive (torque) motors, expensive, lower torque
 Linear actuation via drives, or voice coils.