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 aF 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 + x I 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:

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 thisideal well arecalled DC servomotors.

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 activatingboth 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-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 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 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.