8/18/2019 Jong 86 Book

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  o o k R e v i e w s

Parametr ic Random Vibrat ion

R . A . I b r a h i m

R e v i e w e r : B . D E J O N G

T w e n t e U n i v e r s i t y o f T e c h n o l o g y , E n s c h e d e , T h e N e t h e r l a n d s .

P A RA M ET R IC r a n d o m v i b r a t i o n i s a n a p p l i e d s c i e n t if i c d i s c i p l i n e

t h a t c o v e r s p r o b l e m s f r o m t h e b r o a d f ie ld o f ap p l i e d d y n a m i c s ,

e .g . s t r u c t u r a l d y n a m i c s , a e r o d y n a m i c s , n a v a l a r c h i te c t u r e e t c .

T h e s y s t e m e q u a t i o n s a r e c h a r a c t e r iz e d b y r a n d o m p e r t u r b e d

p a r a m e t e r s w h i le , i n m a n y p r a c ti c a l s i t u a t i o n s , n o n - l in e a r i t ie s

a n d r a n d o m f o r c i n g t e r m s c r e a t e a d d i t i o n a l c o m p l i c a t i o n s .

V a r i o u s t e x t b o o k s h a v e a p p e a r e d c o v e r i n g t h e f ie ld o f r a n d o m

v i b r a t i o n o f t im e - i n v a r i a n t s y s t e m s . T h i s m o n o g r a p h , a s t a t e o f

t h e a rt p r e s e n t a t i o n o f p a r a m e t r i c r a n d o m v i b r a t io n , b a s e d o n

a n e n o r m o u s n u m b e r o f p u b l i s h e d p a p e r s a n d r e p o r ts , is a g r e a t

c r e d i t t o t h e a u t h o r .

I n t h e f i r s t c h a p t e r t h e r e a d e r i s i n t r o d u c e d t o t h e b a s i c

d e f i n i ti o n s o f p a r a m e t r i c a n d a u t o p a r a m e t r i c i n s t a b il i ti e s ,

c h a o t i c m o t i o n , p s e u d o - r a n d o m e x c i t a t i o n a n d c r y p t o - d e t e r -

m i n i s t i c s y s t e m s a n d a b r i e f r e v ie w o f p a r a m e t r i c r a n d o m

v i b r a t i o n i s g iv e n . In r a n d o m v i b r a t i o n s , th e e m p h a s i s i s o n t h e

r e s p o n s e a n d s t a b il i ty o f s y s t e m s u n d e r w i de b a n d r a n d o m

p a r a m e t r i c e x c i t a t io n s . U n f o r t u n a t e l y s y s t e m e q u a t i o n s w i t h

p h y s i c a l w i d e b a n d n o i s e e x c i t a t i o n a r e v e r y d i f fi c u l t t o h a n d l e ,

t h e r e fo r e p h y s i c a l G a u s s i a n w i d e b a n d n o i s e i s o f t e n r e p la c e d

b y i d e a l i z e d w h i t e n o i s e , o r t h e w i d e b a n d n o i s e i s g e n e r a t e d b y

a s h a p i n g f i l t e r d r i v e n b y w h i t e n o i s e . T h i s i s u s u a l l y t h e p o i n t

w h e r e m a n y e n g i n e e r s g e t l o s t . T h e y a r e r e f e r r e d t o b o o k s o n

s t o c h a s t i c p r o c e s s e s a n d s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s a n d

a r e e n c o u r a g e d t o g o i n t o t h e o r i e s w h i c h a r e e m b e d d e d i n

m a t h e m a t i c a l a b s t r a c t io n . F o r t h e s e e n g i n e e r s th e r o o t o f a ll

e v il ( w h ic h i s a t t h e s a m e t i m e a s o u r c e o f p l e a su r e f o r m a n y

m a t h e m a t i c i a n s ) is th e u n b o u n d e d v a r i a t io n o f th e B r o w n i a n

m o t i o n , w h i c h h a s w h i t e n o i s e a s i t s d e r i v a t i v e , i n a f o r m a l

s e n s e . F o r t h e s e p r o c e s s e s a n e w s t o c h a s t i c c a l c u l u s i s n e e d e d .

I n C h a p t e r s 2 - 4 t h e a u t h o r p r o v i d e s t h e n e c e s s a ry to o l s f or

d e r i v i n g r e s p o n s e s t a t i st i c a l f u n c t i o n s a n d t e c h n i q u e s f o r e x a m -

i n i n g s t o c h a s t i c p a r a m e t e r s t a b i l i t y , a s r e q u i r e d i n l a t e r c h a p t e r s .

T h e a u t h o r h a s c h o s e n a n e n g i n e e ri n g a p p r o a c h w i t h o u t m a t h e -

m a t i c a l a b s t r a c t i o n , w h i c h i m p l i es t h a t s o m e i m p o r t a n t t h e -

o r e m s a r e v e r i f i e d i n a h e u r i s t i c w a y , w h i l e m a n y o t h e r s a r e

o n l y m e n t i o n e d . T h e r e s u l t i s a n i c e r e f e r e n c e f r a m e f o r r e a d e r s

w i t h a r e a s o n a b l e b a c k g r o u n d i n s t o c h a s t i c p r o c e s s e s a n d

s t o c h a s t i c d if f e r e n ti a l e q u a t i o n s . R e a d e r s w i t h o u t t h i s b a c k -

g r o u n d w i ll c e r ta i n l y g e t i n t o t r o u b l e , f o r e x a m p l e , w h e n r e a d i n g

t h e d e f in i t i o n s o f r a n d o m v a r i a b le s a n d r a n d o m p r o c e s se s .

O b v i o u s l y , t h e a u t h o r c o u l d n o t r e s i st t h e te m p t a t i o n t o g i v e

s o m e f l a v o u r o f m a t h e m a t i c a l a b s t r a c t i o n b y i n t r o d u c i n g a

r a n d o m v a r i a b le a s a f u n c t i o n o f a s a m p l e s p a c e f~ , w h i c h i s

c o n f u s i n g i n t h e c o n t e x t i t i s u s e d . A ls o , t h e i n t r o d u c t i o n o f

p r o b a b i l i t y s p a c e a n d B o r e l f ie l d i n a f o o t n o t e t o t h e l a s t s e c t i o n

o f C h a p t e r 3 , a n d t h e u s e o f t h e te r m m e a s u r a b l e r a n d o m

p r o c e s s i n t h e f o l l o w i n g c h a p t e r , w h i c h i s n o t e s s e n t i a l in h i s

f u r t h e r d i s c u s s i o n s , w i ll c e r ta i n l y d i s c o u r a g e r e a d e r s w i t h a n

e n g i n e e r i n g b a c k g r o u n d . I n c o n t r a s t w i t h t h i s, th e c o n c e p t s o f

R i e m a n n a n d R i e m a n n - S t i e l t j e s i n t e g r a l s a r e i n t r o d u c e d i n a

v e r y e l a b o r a t e w a y . O f c o u r s e t h e s e m i n o r f l a w s a re o f n o

s i g n i fi c a n c e f o r r ea d e r s w i t h a n a d e q u a t e b a c k g r o u n d i n t h i s

m a t e r i a l .

T h e c o n t e n t s o f C h a p t e r s 2 - 4 a r e r e st r ic t e d t o t h o s e s u b j e c ts

o f s t o c h a s t i c c a l c u l u s n e e d e d f o r u n d e r s t a n d i n g t h e f o l l o w i n g

c h a p t e r s . I n C h a p t e r 2 t h e b a s i c f a c ts o f p r o b a b i l i t y t h e o r y a n d

P a r a m e t r i c R a n d o m V i b r a ti o n ,

b y R . A . Ib r a h i m . P u b l i s h e d

b y J o h n W i l e y , C h i c h e s t e r (1 9 86 ), £ 3 4 . 7 0 .

611

s t o c h a s t i c p r o c e s s e s a r e p r e s e n t e d w i t h e m p h a s i s o n t h e e s s e n t ia l

f e a t u re s o f G a u s s i a n a n d n o n - G a u s s i a n d e n s i t y fu n c t i o n s . T h e

n o n - G a u s s i a n d e n s i t y f u n c t i o n i s p r e s e n t e d i n t e r m s o f t h e

E d g e w o r t h o r t h e G r a m - C h a r l i e r a s y m p t o t i c s e r i e s . I n C h a p t e r

3 t h e e l e m e n t s o f s t o c h a s t i c c a l c u l u s o p e r a t i o n s a r e o u t l i n e d :

t h e v a r i o u s m o d e s o f s t o c h a s t ic c o n v e r g e n c e , s t o c h a s t i c l im i t

t h e o r e m s , m e a n s q u a r e d e r i v a t iv e a n d R i e m a n n S t ie l tj e s i n t e-

g r a l . C h a p t e r 4 c o n t a i n s a n a c c u r a t e d e f i n i t i o n a n d d i s c u s s i o n

o f B r o w n i a n m o t i o n , w h i t e n o i s e p r o c e s s es , t h e r e la t e d M a r k o v

p r o c es s a n d a d e f i ni ti o n o f t h e F o k k e r - P l a n c k - K o l m o g o r o t t

e q u a t i o n s . I n t h e l a s t s e c t io n o f t h i s c h a p t e r t h e m a i n r e s u l t s o f

s t o c h a s t i c c a l c u l u s a s d e v e l o p e d b y I t 6 a n d S t r a t o n o v i c h a r e

p r e s e n t e d i n a s t r a i g h t f o r w a r d w a y w i t h o u t m a t h e m a t i c a l

a b s t r a c t io n . P e r h a p s r e a d e r s w i t h a n en g i n e e r i n g b a c k g r o u n d

w o u l d p r e fe r a n e x p o s i t i o n g i v i n g m o r e p h y s i c a l i n s i g h t ; f o r

e x a m p l e , i n t h e s a m e s t y l e a s t h e p a p e r o f R . E . M o r t e n s e n ,

c e l e b ra t e d i n e n g i n e e r i n g c ir cl e s. I n C h a p t e r 5 t w o m e t h o d s a r e

p r e s e n t e d t o g e n e r a t e t h e r e s p o n s e s t a t i s t i c a l f u n c t i o n o f a

s y s te m : t h e F o k k e r - P l a n c k - K o l m o g o r o f f e q u a ti o n m e t h o d a n d

t h e m e t h o d b a s e d o n I t 6 c a l c u l u s . T h e p r i n c i p l es o f G a u s s i a n a n d

n o n - G a u s s i a n c l o s u r e m e t h o d s t o h a n d l e t h e i n fi n i te h i e ra r c h y

p r o b l e m a r e o u tl i n e d . In C h a p t e r 6 t h e s t a n d a r d f o r m o t

s t o c h a s t i c a v e r a g i n g b a s e d o n t h e p r in c i p l e o f B o g o l i u b o v a n d

M i t r o p o l ' s k i i i s p r e s e n t e d , f o l l o w e d b y s o m e e x t e n s i o n s : t h e

P a p a n i c o l a o u a n d K o h l e r a s y m p t o t i c t h e o r em , t h e W e i d e n -

h a m m e r m e t h o d a n d a m e t h o d b a s e d o n a s t o c h as t ic a v e ra g i n g

o f th e r e s p o n s e e n e r g y c u r v e . T h e s e m e t h o d s a r e a p p l i e d t o

l i n e ar a n d n o n - l i n e a r s y s t e m s t o s t u d y r e s p o n s e a n d s t a b il i ty

c o n d i t i o n s . M a n y p r a c t i c a l a p p l i c a t i o n s a r e a d d e d t o s h o w t h e

r e l e v a n c e o f th e s e t e c h n i q u e s . T h e l a s t s e c t i o n o f t h e c h a p t e r

c o n s i d e r s s e c o n d o r d e r a v e r a g i n g m e t h o d s . A c o n c i s e g e n e r a l

p r e s e n t a t i o n i s f o l lo w e d b y a n e x t e n s iv e l y el a b o r a t e d e x a m p l e

w h i c h i s v e r y i n s t r u c t i v e .

C h a p t e r s 7 a n d 8 f o r m t h e c e n t re o f g r a v i t y o f t h e b o o k

d e a l i n g w i t h p a r a m e t r i c s t o c h a s t i c s t a b i l i t y a n d p a r a m e t r i c

r a n d o m r e s p o n s e , re s p ec t iv e l y. T h e a m o u n t o f li t e ra t u r e o n

t h e s e s u b j e c t s i s e n o r m o u s a n d s c a t t e r e d o v e r m a n y e n g i n e e r i n g

f ie ld s . F o r t h i s r e a s o n b o t h c h a p t e r s h a v e a n h i s t o r i c a l r e v ie w

o f t h e l i t e r a t u r e i n t h e f i r s t se c t i o n , t o g e t h e r w i t h t h e i r f i e l d s o t

a p p l i c at i o n . I n C h a p t e r 7 t h i s r ev i e w c e n t re s r o u n d t h e L y a p u -

n o v d i r e c t m e t h o d , a l m o s t s u r e s t o c h a s t i c s t a b il i ty , p e r t u r b a t i o n

t e c h n i q u e s a n d o t h e r m a t h e m a t i c a l m e t h o d s r e f er r in g t o L i e

a l g e b r a ic m e t h o d s , s ta b i l it y o f h i g h e r m o m e n t s a n d s o m e v e ry

r e c e n t d e v e l o p m e n t s . I n t h e n e x t s e c t io n o f t h i s c h a p t e r t h e

v a r i o u s m e t h o d s o f s to c h a s t i c s t a b il i ty o f a n e q u i l i b r i u m p o s i t i o n

a r e d e f in e d , u s u a l ly b a s e d o n a l i n e a ri z e d a p p r o x i m a t i o n o f th e

s y s t e m e q u a t i o n s o f m o t i o n . T h e l a s t t w o s e c t i o n s d e a l w i t h t h e

t h e o r y o f th e s t a b i li t y o f m o m e n t s a n d w i t h a l m o s t s u r e s t a b il i ty

o f li n e a r s y s t e m s w i t h a r a n d o m c o e f fi c i en t . T h e s t a b i l i ty - o f -

m o m e n t s m e t h o d i s d e s c r ib e d i n m u c h d e t a il i n c l u d i n g t h e

a p p l i c a t i o n o f c lo s u r e s c h e m e s i n t h e c a s e o f a n i n fi n i te l y c o u p l e d

s e t o f m o m e n t e q u a t i o n s . W i t h r e s p e c t t o t h e a l m o s t s u r e s t a b i li t y

t h e v a r i o u s t h e o r e m s o f d i f fe r e nt a u t h o r s a r e s u m m a r i z e d a n d

c o m p a r e d .

C h a p t e r 8 s t a r t s w i t h a n i n t r o d u c t i o n a n d h i s t o r i c a l r e v i e w

o f th e v a r i o u s m e t h o d s o f i n v e s t ig a t i n g t h e r e s p o n s e o f n o n - l i n e a r

s y s t e m s o r l in e a r s y s t e m s i n v o l v i n g r a n d o m t i m e - d e p e n d e n t

c o e f f i c i e n t s . T h i s f i e l d i s v e r y c o m p l i c a t e d s i n c e t h e r e i s n o

g e n e r a l r u l e a s t o w h i c h m e t h o d s h o u l d b e a p p l i e d f o r a s p e c i f i c

c a s e, w h i le f u r t h e r a p p l i c a t i o n o f t h e v a r i o u s m e t h o d s t o o n e

a n d t h e s a m e p r o b l e m d o n o t g i v e id e n t ic a l r e su l t s. M a n y

r e f e r e n c e s a r e g i v e n t o e n a b l e t h e r e a d e r t o i n v e s t i g a t e t h e

s u b j e c t i n s p e c i f ic d i re c t i o n s . A h i s t o r i c a l r e v i e w o f t h e p r o b l e m s

e n c o u n t e r e d i n d e t e r m i n i n g t h e r a n d o m r e s p o n s e a n d s t a b i l i t y

8/18/2019 Jong 86 Book

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of helicopter rotor blades in atmospheric turbulent flow is

also given. In a later section a great variety of models of

this problem with all sorts of complications are discussed to

demonstrate the various techniques of studying the random

response and stability of systems. In the reviewers opinion this

form of presentation, with its constant feedback on the physics

of the problem, is well chosen and for engineers, attractive to

read. Other subjects discussed in this chapter are linear time-

variant systems under random external excitation, which may

depend on time only or on both time and space, and the moment

equation method for linear systems with random uncertainties

in the coefficients. The last section deals with the re sponse of

non-linear systems which is, as the author explains, the most

difficult task in the area of para metr ic rand om vibrati on . He

certainly knows what he is talking about, as he is the author of

a numb er of recent pape rs on this subject. The existing techniques

(stochastic averaging, Gaussi an and non-Gauss ian closure) can

be applied only to specific forms of non-linear systems. The

results of these techniques are compared and evaluated. The

response of dynamic systems with autoparametric interaction

are discussed with reference to the author's recent results. In

this case the Gaussian closure scheme yields a non-stationary

response while a Gaussian closure results in a stationary

response.

After reading Chapters 7 and 8 one may indeed wonder how

reliable the methods are in dealing with the stability and

response of non-linear systems and linear parametric systems

coupled with linear shaping filters. An experimental verification

would be an ideal meth od of assessing the value of the vari ous

methods. Unfortunately, these experiments are very few, mainly

due to the difficulties in measuri ng small ran dom distu rbances.

For that reason, the validity of closure procedures is usually

examined by verifying whether or not the resulting moments

have realistic values.

In the last chapter the author gives a state of the art view of

experimental techniques, results and observat ions centred about

the themes: random behaviour of liquid free surface, random

behaviour of elastic columns and systems with autopara metric

interaction.

Summarizing, the author has written a monograph that is

unique in its field. The fundamental theorems on response and

stability of systems subjected to random parametric vibration

are presented together with numerous non-trivial engineering

examples to illustrate its effectiveness. The large numbe r of

references enables the reader to explore further. The book is

chdmed to be self-contained and suitable for beginners. However,

as already mentioned above, these beginners should have a

reasonable background in probability theory and stochastic

processes.

R~l~,rences

Mortensen, R. E. (1969). Mathematical problems of modeling

stochastic non-linear dynamic systems. J. Statist. Phys. ,

271 296.

/thoul the retiewer

BARTELE DE JONG studied applied mathematics at the Delft

University of Technology, where he received his engineering and

doctoral degree. His thesis was on the paramet ric stability of ships

in random seas. He joined the D epart ment of Applied Mathemati cs

of the Twente University of Technology in 1971. His present

research interests are the prediction of water levels in estuaries and

coastal seas during storm surge periods and the generation o[

dunes and ripples in channel flows.

o n t r o l o f E l e c t r ic a l D r i v e s

W. Leonhard

Reviewer: ISTVAN NAGY

Department of Power Electronics, Computation and Auto-

mation Research Institute of the Hungarian Academy of

Sciences, 1111 Kende u. 13/17, Budapest, Hungary.

THE CONTROL of electrical drives has usual ly been included in

boo ks on electrical machi nes, electrical drives, feedback cont rol

systems and power electronics in the past. Special aspects of the

control of drives turn up in books published in various other

fields, such as traction, machine tools, computer peripheries,

robots etc. Books devoted to the control of drives have been

published mainly in the last 10-15 years, but most of them are

not in English.

The field is a composition of various disciplines, i.e. there are

quite a numb er of supp ort ing disciplines. The reader has to be

more or less familiar with electrical machines, drives, theory of

feedback control, power electronics and, to a certain degree,

with analogue and digital electronics, and more recently, with

microprocessors. In certain cases, as in robots, the drive is an

inseparable part of the system and the control must be designed

taking into account the properties of the controlled object.

Hence the difficulty of writing a book in this field what shoul d

be included and how thoroughly should it be discussed? This

book selects topics successfully and properly proport ions them

within the frame of limited space.

The book was originally published in German in 1974. The

updated English version includes important subjects from the

present state of the art.

The topics included in the first ten chapters can be found in

other books. The remaining chapters are concerned with a.c.

* Control o f Electrical Drives,

by W. Leonhard. Published by

Springer , Berlin (1985), 346pp., D M 128.00.

motor drives. Here very up to date knowledge, compiled mainly

from the author's own publications, can be read. By editing the

material into book form and by providing certain additions, the

book contributes to the enrichment of the relevant technical

literature.

The bo ok un der review is nicely written. In general, the autho r

has displayed good judgment by avoiding lengthy derivations

and proofs. It can be enjoyable reading even for persons

experienced in the field. Newcomers will find it, in general, an

easily understandable presentation.

Fro m the field of electrical machines only the significant

control properties of d.c. and three phase induction machines

are briefly discussed in Chapter 5, 6 and 10. The synchronous

motors are not treated separately. The results presented in

connection with the induction motors are applied to them with

slight modifications.

The funda mental probl ems of electrical drives arc included

in the first four chapters. Apart from some remarks the theory

of feedback, control, analogue and digital electronics, and

microprocessors are omitted. Some import ant aspects of power

electronics, namely converters and inverters, are covered in

Chapters 8 and 11. The chapters mentioned so far lay the

foundation for the main message, namely the various control

metho ds of d.c. (Chapte rs 7,9} and a.c. (Chapter s 12 14) motors.

The book uses the block diagram technique on a broad scale.

By giving quantitative and causal relations among principal

variables, it supports the mathematical calculations and offers

an insight into the mode of action of the system. However, it

should be noted that the particular step response drawn into

the blocks is rather unusual in English technical literature.

Digital computer simulation or laboratory test results are

presented with almost all the control solutions discussed. This

is one of the strong points of the book. On the other hand, it

could give some explanation for the lack of interesting control