Werner Leonhard Control of Electrical Drives

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Dynamics

of a Mechanical Drive

Power

supply

supp/y

c~JP

ig.

2.3.

Block

diagram

twin-inertia

drive

with

flexible

shaft

(a)

Model of mechanical

plant,

(b)

Reduced

order

model,

Mechanical

transmission

with

backlash

2.2

Two

Axes

Drive

Polar

Coordinates

machine tools or robots

there

are

normally several axes of motion,

that

rnust be independently driven or positioned. An example is seen in Fig. 2.4

a, where

arm, carrying a tool or workpiece, is

rotated

angle e:(t)

around

a horizontal axis.

The

radial distance

r(t)

from

the

axis

the

cen-

ter

the

mass M

represents a second degree

freedom, so

that

M2 can

bc positioned in po

lar

coordinates in a plane perpendicular

the

axis.

The

rotary

and

radial motions

are

assumed

driven by servo motors, produc-

ing a controlled driving

torque

and

a driving force

through

rotary

gear

and

rotary

translational

mechanical converter, for instance a

lead

screw.

With

fast

current

control

the

motors are generating nearly

instanta-

\ICOIlS

impressed torques, serving as control

inputs

the

mechanical

plant

For simplicity,

the

masses are assumed

concentrated in

th.e

joints, re-

sltlt.ing in

the

inertias

The

coupling

terms

the

motion can be derived

(~xpressillg

t,he

acc

h~ration

the

mass M 2 in complex

formo

!l . )

(2.8)

(1'

( .

('II

(

(li

(/L'II

I_,

)

.\1 "

(

'~~

(LI

'I'

)

j '

' I

)

. (

i' )

,II

2.2

Two

Axes Drive

Polar

Coordinates

where w =

dE;

/ dt

and

v = dr/

are

the

rotational

and

radial

veloci ties. After

separating

the

terms

acceleration in

tangential

and

radial direction

and

superimposing frictional

and

gravitational

components, Newtons law is ap-

plied in

both

directions, resulting in

the

equations for

the

mechanical

motion

the

centre

point of M

r,,-_~A

(ll

+ h

+M2r2)

Coriolis

Gravitation

,...-"'-...

,.-"'-----....

rwv-M

cose:-mF

-mL,

.10)

de:

= w , (2.11)

C e

ntrifugai

Gravitation

= 1M +

,-"--..

r w

,...-"'-...

g sine: -

(2.12)

= v .

(2.13)

The

equations (2.10)-(2.13) are depicted in Fig. 2.4 b in

the

graphical form

of a block diagram, containing four integrators for

the

state

variables. Despite

the

simple mechanics,

there

are

complicated interactions, which become more

prominent with increasing

rotary

and

radial velocities.

The

control of this

mechanical

structure

is dealt

with

in a

later

chapter.

Clearly,

the

two motions

are

nonlinearly coupled

though

gravitational,

Coriolis-

and

centrifugaI effects; they are described by four nonlinear

state

equations.

mF,

and

mL,

h are due

friction

and

external load forces

with

may

exhibit

their

own complicated geometric or dynamic dependencies,

it is

important

for

the

application

express

the

position of mass M

cartesian coordinates, this is achieved by a

polar-cartesian

conversion

x (t) = r cos

(2.14)

y(t) = r sine: .

.15)

Moving

the

arm

also in

the

direction of

the

axis

rotation,

that

the

mass

can be positioned in cylindrical coordinates, would introduce a

third

dccoupled degree of freedom.

The

dynamic

interactions for a general motion, involving six degrees of

r)'(

dom

(three for

the

position,

three

for

the

orientation

the

tool) are ex-

cedilJgly complicated,

they

must

be dealt

with

when controlling

the

motions

01'

Jllult.i

(

uots

with high dynamic performance [M53,

013,

849].

'22

Dynamics

of a

Mechanical

Drive

mLosd

Friction

+J2+

grcos

gsinE

y/,

fLoad

Fig.

2.4.

Two

axes drive

polar

coordinates

(a)

Mechanical

plant

(b)

Block

diagram

2.3

Steady

State

Characteristics

Different

Types

Motors

and

Loads

Consider first

the

steady

state

condition, when

the

torque

and

speed of a

single axis

lumped

inertia

drive are

constant

and

the

angle changes linearly

with

timej this condition

reached when

With

some motors,

such as single phase induction motors, or loads, for example piston compres-

sors or punches,

the

torque

is a periodic function of

the

angle of rotationj

in this case,

the

steady

state

condition

reached, when

the

mean

values of

both

torques are equal,

The

speed

then

contains periodic

oscillations, which

must

be kept within limits by a sufficiently large inertia.

The

stcady

state

characteristics of a

motor

1030<1

are often functions

given

graphical [o

, <:onnN:ting main

varíilhl(~K,

sllch as speed

and

torque;

lh('

provisíou

i:i

1.11;t\.

:lllxili<tJ"y

01'

COlltrol

íllPIlII

.H,

ror

('X;UllPlc

Sllpply voltage,

li(.ld

(·II!"l"('IIt.,

rll

illJ

lllJ

~ It',

hrw;h

pm;ll.ioll

01" r,·(·" ru

\.<',

;I],(~

lIlailll

;ú,\(~d

COIl-

Ht.iI.lIl..

1'11\ '

11,1""

1.,Ypi(,iI.I

ll'

l'\.l'

I'l

Ic-

"I'

(,1(,(

1.1'1('

JlJoI,ol"X

11.1'('

\iOWII.

'1'1". "

",111

,,,,,,,11

11"

' I

t."r1

ll.

iI'

l (

IId

,y vtdl" 1,"' ('t

lll::L

;

UIl

:;p(,,'cI, ::

11\('('

1.111'

Friction

2.3

Steady

State

Characteristics

Motors

and

Loads

variable is

the

load angle

the

displacement of

the

shaft from

its

no-

load

angular

position.

When

the

maximum

torque is exceeded,

the

motor

falls

out

of step; asynchronous

operation

larger motors

not allowed for

extended periods of

time

because of

the

high currents

and

pulsating

torque,

The

electrical

transients

usually cannot be neglected

with

synchronous

motor

drives.

The

rigid speed of synchronous motors when supplied by a

constant

fre-

quency source makes

them

suitable for only a

few

applications, for exarnple

large slow-speed drives for reciprocating compressors or synchronous gener-

ators

operating

as motors in

pumped

storage hydro power stationsj

the

low end of

the

power scale are electromechanical clocks,

The

situation

is dif-

ferent, when

the

synchronous

motor

fed from a variable frequency inverter

because

then

the

speed of

the

drive can be varied freely (Chap. 14).

With

the

progress of power electronics, these drives are now more widely used.

Fig.

2.5.

Steady

state

torque-speed

characteristics

(li) electrical

and

(b)

mechanical

drives

The

"asynchronous" or "shunt-typ" characteristic in Fig, 2.5 a

slightly

clrooping; often

there

is also a pronounced

maximum

torque.

The

lower por-

Lioll

the

curve

forbidden in steady

state

in view of

the

high losses.

With

I.hree- phase asynchronous

motors

the

rotor angle has no effect on

the

torque

steady

state.

Motors

with

"series-type" characteristic show considerably larger speed

drop

under

loadj with DC or AC

commutator

motors,

this

achieved by

;1.

suitable connection of

the

field winding.

The

main

area

applications

larger

ratings

are

traction

drives because

the

curved characteristic resembling

hyperbola

facilitates load sharing on multi pie drives

and

permits

nearly

collst.ant power

operation

over a wide speed range

without

gear changej this

is par\.icularly

suited

a Diesel-electric

turbo-

electric drive, where

the

1',,11

!)()Wl'r

lhe

thermal

engine

must

be used.

For

('olllparisoll,

SOIlJC

t:ypi

cal charactcrist.ics

a Lurbine

anel

a Diesel

('I\/';ill('

. ('ClIlHl.n,lll.

flld

illjc('.t.ioJl

r H

t.I'III('

lU'"

iC'('1\

iII l"il':.

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