of 27
8/9/2019 0046352920c5292a4e000000
1/27
Sguthan Vol. 20 Parts 2-4 April-August 1995 pp. 345-371. Printed n India.
e t h o d s o f n o n l in e a r r a n d o m v i b ra t io n a n a l y s i s
C S MANOHAR
Department of Civil Engineering Indian Institute of Science Bangalore
560012 India
Abstra c t
The various techniques available for the analysis of nonlinear
systems subjected to random excitations are briefly introduced and an
overview of the progress which has been made in this area of research is
presented. The discussion is mainly focused on the basis scope and
limitations of the solution techniques and not on specific applications.
K e y w o r d s
Nonlinear systems; random excitations; nonlinear vibration;
vibration analysis.
1 Intro duct i o n
Random vibration methods are extensively used in earthquake wind transpor tation
and offshore structural engineering applications. Here the uncertainties is specifying
the forces acting on the structure are quantified using sophisticated load models based
on the theory of probabi lity and stochastic processes. Consequently the response
analysis of structures is also carried out in a probabilistic framework which eventually
leads to the assessment of the safety of the structure. In order to mainta in a consistent
level of sophistication in modelling the vibrating structure also needs to be modelled
with care. This concern leads to questions on modelling nonlinear behaviour of the
structural system and also on modelling uncertainties in specifying the structural
parameters themselves. The questions of structural nonlinearity are particularly
important while addressing the problem of failures and safety assessments especially
since the nonlinear response is at times radically different from the one obtained using
a simplified linear model. These questions offer considerable challenge to the analyst
and are currently being actively pursued in vibration engineering research as evidenced
by a continuous stream of publications in leading international journals. The present
paper aims at presenting an overview of the research work in this field highlighting the
developments which have taken place over the last decade. The emphasis is therefore
to focus on the various techniques and methodologies which have admittedly come to
stay as powerful tools in dealing with problems encountered frequently in the area of
nonlinear random vibrations.
The sources of nonlinearities in vibration problems can be categorized into four groups:
Geometric nonlinearities arising out of large deformations;
nonlinear elastic and dissipation properties of the structural material;
345
8/9/2019 0046352920c5292a4e000000
2/27
346 C S M a n o h a r
t o p o l o g i c a l c a u se s a s in t h e c a s e o f v i b r o i m p a c t s y s t e m s s u c h a s r o c k in g b l o c k s a n d
sys t ems wi th s t oppe r s ;
f l u id - s t ruc tu r e i n te r ac t i ons l e ad ing t o non l inea r coup l ings .
Tab l e 1 l is t s a f ew exam ple s wh ich hav e been s tud i ed i n t he l i t e ra tu r e i n t he con t ex t o f
n o n l i n e a r r a n d o m v i b r a t i o n o f en g i n e er i ng s t r u c t u re s . I n t h e s e p r o b l e m s c l o s e d f o r m
so lu t i ons a r e r a r e ly pos s ib l e . Moreove r , t he r e ex i s t s no s ing l e gene ra l ana ly t i c a l
p rocedu re wh ich l e ads t o accep t ab l e so lu t i on unde r a l l c i r cums tances . The i n f luen t i a l
f ac to r s i n fo rmu la t i ng t he so lu t i on p rocedu re s a r e
Sys t em deg ree s o f f r eedom, na tu r e o f non l inea r i t y t ha t is , non l inea r i t y i n mass ,
s ti ff n es s o r d a m p i n g a n d s y m m e t r y / a s y m m e t r y o f n o n l in e a ri ty ) , p r e d o m i n a n c e o f th e
non l inea r i t y i n a f f ec t i ng t he sy s t em behav iou r , i nc lud ing quas ipe r iod i c i t y and
b i fu r ca t ions , i n t he absence o f r an do m exc i t a t ions ,
S t a t i o n a r i t y / n o n s t a t i o n a r i t y o f e x c i ta t io n , p r o b a b i l i t y d is t r i b u t io n a n d s t r e n g t h o f
exc i ta t ion ,
E x c i t a t io n b a n d w i d t h i n re l a ti o n t o s y s te m b a n d w i d t h ,
M echan i sm o f exc i t a t ion , t ha t is , ex t e rna l o r pa r ame t r i c ,
ResP onse va r i ab l e s o f i n t er e s t.
A c c o r d i n g ly , s e v er a l a p p r o x i m a t e s o l u t i o n p r o c e d u r e s h a v e b e e n d e v e l o p e d w h i c h l e a d
t o a c c e p t a b l e s o l u ti o n s i n sp e ci fi c p r o b l e m d o m a i n s . M o s t l y , th e a p p r o x i m a t i o n s a r e
b a s e d o n t h e M a r k o v i a n n a t u r e o f t h e r e s p o n s e o r o n t h e p r o x i m i t y o f t h e r e s p o n s e
p r o b a b i l i t y d e n s it y f u n c t io n p df) to G a u s s i a n d i s tr i b u ti o n s . M a n y o f t h e m e t h o d s a r e
ingen ious ex t ens ions o f de t e rmin i s t i c non l inea r ana ly s i s p rocedu re s t o s t ocha s t i c
p rob l em s . A d i s cus s ion on t he fo l l owing m e tho ds i s p r e sen t ed i n t he s eque l: i) M ark ov
vec to r app roach , ii ) Pe r tu rba t i on , i ii ) Equ iva l en t l inea r i za t ion , iv ) Equ iva l en t non -
l inear iza t ion , v) Clo sure , v i) S to chas t ic averag ing , v ii ) S to cha s t ic se r ies so l u t io n an d
vii i) Dig i ta l s imu la t ions .
2 M a r k o v v e c to r a p p r o a c h
W h e n t h e i n p u t s a r i s e f r o m G a u s s i a n w h i t e n o i s e p r o c e s s e s t h e r e s p o n s e w i l l b e
a d i ffu s ion p roces s and t he a s soc i a t ed t r ans i t i ona l pd f t pd f) Will s a t is fy t he w e l l -kno wn
K o l m o g o r o v e q u a t io n s . T h e g o v e r n i n g e q u a t i o n s o f m o t i o n i n t h e s e s i t u a ti o n s c a n b e
cas t i n t he fo rm o f equ a t i ons o f t he I t 6 t ype a s fo l lows :
dX t ) = f X t ) , t ) d t + G X t ) , t ) dn t ) , 1 )
und e r t he i n i ti a l cond i t i ons
X t o ) = Y , 2 )
whe re , X t ) = n x 1 respo nse v e c t o r , f X t ) , t ) = n n mat r ix , G X t ) , t ) = n m mat r ix ,
B t ) = m x 1 v e c t o r o f t h e B r o w n i a n m o t i o n p r o c e s s e s h a v i n g t h e p r o p e r t i e s
E E A ~ 0 ] = E E S j t + A t ) - -
B j t ) ]
= 0 , 3 )
E [ A i t ) A j t ) ] = 2 D i j A t , 4)
and Y = n 1 vec to r o f i n i ti a l cond i t i ons i ndep end en t o f B t ) . He re E [ . ] r ep re sen t s t he
m a t h e m a t i c a l e x p e c t a t io n o p e r a t o r . T h e a b o v e r e p r e s e n t a t i o n o f e q u a t i o n s o f m o t i o n
8/9/2019 0046352920c5292a4e000000
3/27
T
e
1
E
m
p
e
o
n
n
e
n
n
v
b
a
n
s
y
e
m
s
N
o
n
s
y
e
m
M
o
E
o
o
m
o
o
R
e
e
F
c
o
s
s
m
i
c
C
o
m
b
s
o
a
o
~
+
~
s
g
~
+
o
=
g
b
s
o
a
o
s
y
e
m
P
a
e
u
g
n
D
n
s
o
a
o
l
a
g
a
m
p
u
v
b
a
o
H
e
e
c
s
u
u
e
B
s
o
a
o
R
a
a
o
n
b
o
V
d
P
s
o
a
o
M
o
e
M
o
s
o
s
m
o
g
a
r
~
V
i
b
o
m
p
s
y
e
m
A
o
w
n
o
a
o
o
c
m
n
S
p
o
n
n
a
m
s
O
h
e
s
u
u
e
u
w
o
Y
+
2
o
+
m
2
+
c
x
3
=
m
~
+
c
+
~
+
(
1
=
=
1
A
v
B
~
z
z
y
z
1
+
W
R
s
n
~
s
g
0
1
+
+
W
R
c
~
s
g
0
g
=
0
O
*
+
=
c
O
*
*
O
*
=
O
m
)
+
c
+
k
=
c
+
~
3
+
d
+
e
F
~
+
o
1
+
k
q
2
+
e
=
e
M
X
+
C
+
K
=
P
P
=
~
C
~
~
u
-
~
+
C
M
p
V
u
~
C
M
p
V
S
A
h
m
a
&
S
1
L
n
1
W
e
1
I
y
&
D
a
h
1
S
&
K
o
1
I
y
&
M
a
1
V
c
y
&
B
u
1
M
a
&
y
1
b
R
s
1
T
o
&
R
a
a
1
3
4
~
8/9/2019 0046352920c5292a4e000000
4/27
348
C S M a n o h a r
i s f a i r ly genera l in the sense tha t i t a l low s for: (1) Mul t i -de gree l inear /n on l inea r d i scre te
s ys t em s , ( 2 ) ex t e r na l and pa r am e t r i c exc i t a t i ons , ( 3 ) nons t a t i ona r y exc i t a t i ons , ( 4 )
non wh i t e exc i ta t ions , in wh i ch ca s e , add i t i ona l f i lt e rs t o m o de l i npu t s a s f i l te r ed wh i t e
no i s e p r oces s e s a r e t o be app end ed t o t he s y s t em equa t i ons w i t h a cons equ en t inc r ea s e
i n the s iz e o f t he p r ob l em and (5 ) r an do m i n it ia l cond i t ions . The K o l m og or o v
equ at ion s sa t is f ied b y the response tpdf , p(x , t ly, to) a re
t h e C h a p m a n - K o l m o g o r o v - S m o l u c k o w s k i ( C K S ) i n te g ra l e q u a t io n
p ( x , t l y , t o ) = f ~
p(x, t lz , z)p(z, z ly, to) dz , (5)
d
o
t h e f o rw a r d e q u a t io n o r t h e F o k k e r - P l a n c k - K o l m o g o r o v ( F P K ) e q u a ti o n
@ ( x , t l y , ~ ~ --~ i f ~ ( x t ) p ( x t l y t ) ]
j=l .
+ ~ ~ [ G D G r ) , j p x , t ly , to )] ,
6 )
i j=
t h e b a c k w a r d e q u a t i o n
@ ( x , t l y , to )
c ~ t o
f j ( y , t ) @ ( x t [ y t ) ~ [ G D G T ) o c ~ 2 p ( x t ly t )
j ~ 1 ~Y j i j = x ~Y i ~Yj . (7)
I n t he s e equa t ions , t he s upe r s c r i p t T de no t e s t h e m a t r i x t r ans p os e op e r a t i on . Th e f i rs t
o f t he s e equa t i ons r ep r e s en t s t he cons i s t ency cond i t i on f o r t he r e s pons e p r oces s t o be
M ar kov . E qua t i on ( 7) is the ad j o i n t o f ( 6 ) and t he s e t wo e qua t i ons ca n be de r i ved us i ng
( 5) t oge t he r wi t h t he equ a t i on o f m o t i o n g i ven by ( 1) . I t i s o f i n t e r e s t t o no t e t ha t t he f o r -
w a r d e q u a t i o n a n d b a c k w a r d e q u a t i o n s a r e a ls o s a t is fi ed b y s e v e ra l o t h e r r e s p o n s e
prob abi l i ty fun c t ions o f in te res t. Thus , for ins tance , the proba bi l i ty , Q (t ly , t o ), tha t f i rs t
pa s s age ac r os s a s pec if ied s a fe do m a i n wi ll no t occ u r i n t he t i m e i n t e r va l t o - t f o r
t r a jec t o r i e s i n t he phas e p l ane s t a r t i ng a t y a t t = t o, c an b e s ho wn t o s a t is f y t he
b a c k w a r d K o l m o g o r o v e q u a t io n . T h e f o r m u l a t i o n o f th e s e e q u a ti o n s, l e a d s to t h e e x a c t
r e s pons e cha r ac t e r i za t i on o f a li m i t ed cl as s o f p r ob l em s a n d he lps i n f o r m ul a t i ng
s t ra t eg ie s fo r app r ox i m a t e an a l ys is f o r a w i de r c la s s o f p r ob l em s . Th e de t a i ls o f t he
de r i va t i on o f t he s e equa t i ons a l ong wi t h a d i s cus s ion on t he i n it ia l cond i t i ons ,
bo un da r y cond i t ions , we l l pos ednes s , e i genva l ues and e i gen f unc t i ons and t he ex i s t ence ,
u n i q u e n e s s a n d s ta b i li ty o f s ta t i o n a r y s o l u t io n s c a n b e f o u n d i n t h e w o r k s o f B h a r u c h a
Re id (1960) , St rato no vic h (1963), C au gh ey (1963a, 1971), Fe l ler (1966) , Fu l ler (1969).
a n d R o b e r t s (1 98 6a ). A c o m p r e h e n s i v e t r e a t m e n t o f t h e F P K e q u a t i o n a n d i ts
app l i ca t i on i n phys i ca l s ci ences is ava i l ab le i n t he m on og r aph s b y R i s ken ( 1989) and
H or s t h em ke Le f eve r ( 1984) .
2 .1 E x a c t s o l u ti o n s
T h e c o m p l e t e s o l u ti o n o f t h e F P K e q u a t i o n is o b t a i n a b l e f o r a l l e x te r n a l ly f o rc e d l in e a r
os c i ll a t o rs ( L i n 1967) and f o r a c l as s o f f ir s t o r d e r non l i nea r sys t em s ( Cau ghe y Di en es
1961; St ra ton ov ich 1963; A tkinso n C au gh ey 1968; Atkinsof i'1973) . In these so lu t ions
e ith er, t h e F o u r i e r a n d L a p l a c e t r a n s f o r m . t e c h n i q u e s o r t h e m e t h o d o f e i g e n f u n c t io n
e x p a n s i o n is u s ed . M e t h o d s b a s e d o n g r o u p t h e o r y h a v e a l s o b e e n d e v e l o p e d (B l u m a n ,
8/9/2019 0046352920c5292a4e000000
5/27
M e t h o d s o f n o n l in e a r r a n d o m v i b ra t io n a n a l y s is
349
1971) . Th e s t a t i ona ry so lu t ion , whe n i t exi st s, can be fo un d for a l l f i r s t o rd er sy s t ems an d for
a l im i t ed s et o f h i g h e r o rd e r sy s tem s . A g en e ra l c l a ss o f sy s t em s fo r w h i ch ex ac t s t a t i o n a ry
so l u t i o n u n d e r ex t e rn a l w h i t e n o i se ex c i t a t io n is p o s s ib l e is d i s cu s sed b y C au g h e y M a
(1 9 8 2 a, 1 9 8 2b ). T h i s s e t i n c l u d es s i n g l e d eg ree o f f r e e d o m ( sd o f) sy s t em s w i t h n o n l i n e a r
s ti ff n e ss a n d a c l a s s o f s d o f a n d m u l t i- d e g r e e o f f r e e d o m ( m d o f ) s y s te m s w i t h n o n l i n e a r
d a m p i n g a n d s ti ff n es s . D i m e n t b e r g ( 1 9 8 2 ) h a s o b t a i n e d s t a t i o n a r y p d f f o r a s p e ci fi c s d o f
s y s t e m in w h i c h b o t h p a r a m e t r i c a n d e x t e r n a l w h i t e n o i s e e x c i t a t i o n s a r e p r e s e n t. T h i s
s o l u t i o n h a s b e e n o b t a i n e d t h r o u g h a n i n v e r s e p r o c e d u r e ( D i m e n t b e r g 1 98 8a ). H e r e a n
a p p r o x i m a t e s o l u t i o n i s f ir s t o b t a i n e d b a s e d o n t h e m e t h o d o f s t o c h a s t i c a v e r a g i n g .
T h i s s o l u t i o n is s u b s t i t u t e d i n t o t h e g o v e r n i n g F P K e q u a t i o n f o r t h e o r i g in a l s y s te m ,
T h i s e q u a t i o n w o u l d b e e x a c t l y s a t is f ie d p r o v i d e d t h e s y s t e m ' s p a r a m e t e r s a r e r e l a t e d
i n a c e r t a i n sp ec i a l w ay . T h u s , t h i s su b se t o f p a ram e t e r s d e f i n e s a c l a s s o f sy s t em s fo i
w h i c h t h e g o v e r n in g r e d u c e d F P K e q u a t i o n is s ol v ab le . T h e c o n c e p t o f d e ta i le d
b a l a n c e d e v e l o p e d e a r l i e r b y p h y s i c is t s ( G a r d i n e r 1 98 3; R i s k e n 1 98 9), h a s b e e n u s e d b y
Y o n g L i n ( 19 87 ) a n d L a n g l e y ( 19 8 8a ) t o o b t a i n e x a c t s t a t i o n a r y s o l u t i o n f o r a c la s s o f
s d o f a n d m d o f n o n l i n e a r s y s t e m s u n d e r w h i t e n o i se e x c i ta t io n s . I n t h i s m e t h o d , t h e
c o m p o n e n t s o f r e s p o n se v e c t o r Y a r e c l a ss if ie d a s e i t h e r e v e n o r o d d d e p e n d i n g u p o n
t h e i r b e h a v i o u r u n d e r a t im e r e v e rs a l o f t t o - t. T h e e v e n v a r i a b le s d o n o t c h a n g e t h e i r
s i g n w h e r e a s t h e o d d v a r i a b le s u n d e r g o a c h a n g e o f s ig n . T h e s e a r e d e n o t e d a s
~ i ~ - - - - ~ iXi
n o s u m m a t io n o n
i ,
(8)
w h e re r,g = 1 fo r ev en v a r i ab l e s a n d eg = - 1 fo r o d d v a r i ab l e s . T h e s t a t e o f d e t a i l ed
b a l a n c e is d e f i n e d a s
p ( x , t l y , t o ) = p ( ~ , t l ~ , t o ) , t > t o .
(9)
In t h e s t e ad y s t a t e , i n t e rm s o f t h e d r i f t co e f f i c i en ts A~ an d d i f fu s i o n co e f f i c i en t B i j t h i s
c o n d i t i o n i s g i v e n b y
A~(x)p(x) + ezA~(~)p(x) -- ~ [-Bo(x )p(x)] = 0, (10)
Bi j (x) ~ . i ~ . j B i j l x = O . (1 I)
H e r e s u m m a t i o n o n r e p e a t e d i n d e x i s i m p l ie d . W h e n t h e s e c o n d i t i o n s a r e s a t i sf ie d , t h e
s t a t i o n a r y s o l u t i o n e x p r e ss e d a s
p ( x) = C e x p [ - U ( x ) ] , (1 2)
c a n b e o b t a i n e d b y s o l v i n g th e e q u a t i o n f o r U ( x ) , th e g e n e r a l i z e d p o t e n t i a l . T h e c l a s s o f
p r o b l e m s t h a n c a n b e s o l v e d u s i n g t h i s m e t h o d , i s s h o w n t o i n c l u d e t h e p r o b l e m s
c o n s i d e r e d b y C a u g h e y M a (1 9 82 a , b ) a n d D i m e n t b e r g (1 98 2). F u r t h e r m o r e , L i n
C a i ( 19 88 ) a n d C a i L i n ( 19 8 8 a) h a v e s h o w n t h a t t h e e x a c t s t a t i o n a r y s o l u t i o n a s i n
( 12 ) c a n s ti ll b e o b t a i n e d e v e n w h e n o n e o f t h e c o n d i t i o n s f o r d e t a i l e d b a l a n c e , n a m e l y
(11 ) i s n o t s a t is f i ed . T h i s c l a s s o f sy s t em s h a s b ee n t e rm ed a s b e l o n g i n g t o t h e c l a s s o f
g e n e r a li z e d p o t e n t ia l . M o r e g e n e r a l cl as s o f e x a c t ly s o lv a b l e F P K e q u a t i o n s h a v e b e e n
d i s cu s s e d b y Z h u
e t a l
(1990) an d T o Li (1991).
2.2 A p p r o x i m a t e m e t h o d s
F o r a n a l y s i n g p r o b l e m s p o s s e s s in g n o e x a c t s o l u t i o n s o n e h a s t o t a k e r e c o u r s e t o
a p p r o x i m a t e m e t h o d s . A n i t e r a t i v e p r o c e d u r e b a s e d o n t h e p a r a m e t r i x m e t h o d f o r
8/9/2019 0046352920c5292a4e000000
6/27
350 S Manohar
studying exis tence an d uniquen ess of solut ions o f par t ial di fferential equat ion s (Fr ied m an
1964) ha s been u s ed by a f ew au t h o r s ( Cau ghey 1971 ; M ayf i e l d 1973). I t ha s be en
s hown t ha t bo t h t r ans i en t and s t eady s t a t e s o l u t i ons can be ob t a i ned us i ng t h i s
m e t hod . Al so , the m e t hod i s u se f u l i n im p r ov i ng app r ox i m a t e r e s u lt s ob t a i ned b y o t h e r
t e ch n i q ue s . H o w e v e r , t h e m e t h o d h a s n o t b e e n u s e d w i d e ly i n r a n d o m v i b r a ti o n s t ud i e s
( Robe r t s 1981) . Pa yn e (1968) u s ed a com bi na t i o n o f pe r t u r ba t i on and e i gen f unc t i on
expans i on t e chn i ques t o ana l ys e nea r l y l i nea r f i r s t o r de r s ys t em s unde r wh i t e no i s e
i npu ts . He ha s de r i ved pe r t u r ba t i on exp ans i on f o r e igenva l ues and e i gen f unc t i ons o f
t h e F P K e q u a t i o n u p t o O ( e 2 ) a n d o b t a i n e d t h e c o r r e s p o n d i n g e x p re s s io n f o r r e sp o n s e
a u t o c o r r e l a t io n f u n c ti o n . H e r e e m a y b e d e n o t e d a s a s m a ll n es s p a r a m e t e r a s s o c i a te d
wi t h t he non l i nea r i ty . I wa n Spano s ( 1978) have em pl oy ed s i m i l ar t e chn i ques t o
o b t a i n n o n s t a t i o n a r y r e s p o n se e n v e l o p e d i s t ri b u t io n o f a l i n e a r s d o f s ys te m . J o h n s o n
Sco t t (1979 , 1980) cons i de r ed t he f ir s t o r d e r s y s t em s t ud i ed b y Pa yn e ( 1968) and
ex t ended h i s ana l ys i s t o com pu t e expan s i on t e r m s up t o O( eT). Fu r t he r m or e , t hey have
a l so app l i ed t h i s m e t h od t o s econ d- o r d e r s ys t em s .
At k i ns o n (1973) ha s u s ed a n ad j o i n t v a r i a t i ona l m e t ho d t o f i nd t he e i genva l ues o f t he
F P K o p e r a t o r . H e h a s g e n e r a t e d t r i a l f u n c t i o n s t h a t a r e o r t h o g o n a l t o k n o w n
s t a t i ona r y s o l u t i on and ha s de t e r m i ned r e s pons e pow er s pec t r a l dens i t y (p s d ) f o r t he
cas e o f a D uf f i ng os c il la t o r, a bang- b ang s ys t em and a s ys t em wi t h non l i ne a r dam pi ng .
T o l a n d et al ( 19 7 2) p r o p o s e d a r a n d o m w a l k a n a l o g y b a s e d o n a d i s c r et e a p p r o x i-
m a t i o n t o c o n t i n u o u s M a r k o v p r o c e ss , a n d o b t a i n e d a r e c u r r e n c e r e la t io n f o r r e s p o n s e
p r obab il it ie s . Th i s t e chn i que is equ i va l en t t o u s i ng f in i te d i ff e r ence app r ox i m a t i on on
t h e F P K e q u a t i o n a n d is t im e c o n s u m i n g e s p e ci a ll y w h e n t h e d o m a i n o f i n t e g r a t io n i s
l a rge . Th e e i gen f unc t i on expan s i on m e t h od i s app l i cab l e when t he t i m e and s pace
va r i ab l e s i n t he FP K equ a t i on can be s ep a r a t ed w h i ch is gene r a l l y pos s ib l e when t he
d r i f t and d i f fu s i on t e r m s a r e t i m e i nva r i an t . I n a m or e gene r a l con t ex t , t he m e t ho ds o f
w e i g h te d r e s id u a l h a v e b e e n e m p l o y e d b y s ev e r a l a u t h o r s . T h u s , B h a n d a r i S h e r r e r
(1 96 8) h a v e u s e d t h e G a l e r k i n t e c h n i q u e t o f in d t h e s t a t io n a r y r e s p o n s e o f s d o f a n d
a t w o deg r ee o f f r eedom s ys t em w i t h po l yn om i a l non l i nea ri t ie s . We n ( 1975, 1976) ha s
ex t ende d t h i s ana l ys is to n ons t a t i ona r y r e s pons e ana l ys i s and s t ud i ed t he r e s pons e o f
t h e D u f f in g o s c il la t o r a n d a h y s t e re t ic sy s te m . T h e u s e o f t h e m e t h o d i n d e t e r m i n i n g p sd
r e s pons e and t he f i rs t excu r s i on f a il u re p r obab i l i ti e s o f re s pons e h a s a l s o bee n
i nd i cat ed . A si m i l a r t e chn i que ha s been us ed by So l om os Span os (1984) f o r ob t a i n i ng
t h e a m p l i tu d e r e s p o n se o f a l i n e a r s d o f sy s t e m u n d e r e v o l u t i o n a r y r a n d o m e x c i ta t io n .
A n o t h e r v a r i a n t o f t h e w e i g h t e d re s id u a l t e c h n iq u e , n a m e l y , th e m e t h o d o f m o m e n t s ,
ha s been us ed by Fu j i t a H a t t o r i ( 1980) f o r t he ana l ys i s o f s do f s ys t em s wi t h co ll is i ons
und e r m odu l a t ed w h i t e no i s e inpu t . Lan g l ey ( 1985) ha s app l i ed t he f in i te e l em en t
m e t h o d t o s o l ve th e t w o d i m e n s i o n a l F P K e q u a t i o n a s s o c ia t e d w i t h t h e s t a t io n a r y
r e s pons e o f a D u t t ing os c i l la t o r and a s h i p r o l l ing p r ob l em . Th e dom a i n i n t he pha s e
s pace t o be cove r ed by f in i te e lem en t s was e s t i m a t ed us i ng an equ i va l en t l i nea r i za t i on
s o l u t i o n . A n i m p r o p e r s e l e c t i o n o f t h e e x t e n t o f t h e d o m a i n i s s h o w n t o r e s u l t i n
nega t i ve va l ue s f o r r e s pons e pd f . B e r gm an Spen ce r ( 1992) have s t ud i ed t he t r ans i en t
s o l u t ions o f t he F PK equa t i ons o f s eve r a l s econd o r d e r non l i nea r s ys t em s us i ng f in i te
e l e m e n t m e t h o d .
O ther d i scre ti za tion procedures based on pa th in tegral formal i sms (We hner W ol fer
1 98 3; K a p i t a n a i k 1 9 8 5 , 1 9 8 6; N a e s s J o h n s e n 1 9 9 3 ) a n d c e l l m a p p i n g t e c h n i q u e s
( Sun Hs u 1990) have al s o been deve l oped . The s e m e t ho ds a r e re l a t ed t o the it e r a t i ve
t echn i que d i s cus s ed by Cr anda l l et al ( 1 9 6 6 ) a n d a r e b a s e d o n a n a s s u m p t i o n t h a t
8/9/2019 0046352920c5292a4e000000
7/27
Methods of nonlinear random vibration analysis 351
the tpdf over short time steps is Gaussian. Sun & Hsu have used a Gauss ian closure
approximation to evaluate the transitional probabilities, while, Naess and Johnsen
have employed cubic B-splines to represent the tpdf. It is to be noted that a lthough the
analysis makes the short-time Gaussian approximation for the tpdf, the global
non-Gaussian nature of the unconditional pdf is, nevertheless, captured by the analysis.
2.3 Generalizations and moment equations
General ization of the FPK equation to non-diffusive Markov processes has been
discussed by Pawula (1967). Here the inputs are modelled as white noise arising out of
non-Gaussian processes. The response in such cases will still have the Markovian
property but the equation of motion of the transitional pdf will have infinite number
of terms. Tylikowski & Marowski (1986) have considered the response of a Duffing
oscillator to Poissonian impulse excitation and have shown that the transitional
pdf satisfies an integro-partial differential equation. The use of truncated generalized
FPK equations in computing the lower order moments has been demonstrated by
Risken (1989).
When response moments are of interest, the governing equations for the moments
can be derived based on the FPK equation (Soong 1973). Thus, the moments of the
function h[X t ) , t ] of the solution X(t) of (1) can be shown to be governed by the
equation
d E E h lX , l = E F y e h l + F G D T G + E [ O h ] 1 1 3
y setting h [ X ( t ) , t ] Y k V k 2 V k . . . . k .
'1 ' '2
' 3 n
and choosing different values for k i, one can
derive equations for the most commonly used moments. These equations can be readily
solved for the case of deterministic nonautonomous linear systems under white noise
inputs. This forms the basis for obtaining approximate transient solutions using
linearization procedures. In the case of linear oscillators with parametric white noise
excitations, the response is non-Gaussian and the associated FPK equation is not
solvable. However, the exact response moments can be obtained by solving the
associated moment equations. In nonlinear problems these equations form an infinite
hierarchy and an exact solution of moment equations also is not possible. Based on the
principle of maximum entropy, Sobczyk & Trebicki (1990, 1992) and Chang (1991)
have developed approximate st ationary solutions of nonlinear systems under paramet-
ric and external noise excitations. This consists of employing pdf with undetermined
coefficients which, in turn, are found by maximizing the entropy subject to the
constraints of the pdf normalization and the moment equations which are obtained
through the governing FP K equations. Roy & Spanos (1991) have utilized a perturba-
tion solution scheme for the moment equations and have shown that the scheme
overcomes the problem of infinite hierarchy. They have proposed Pade type transform-
ations of the series which enables the analysis to be applicable even for strongly
nonlinear systems. Furthermore, the same authors (Roy & Spanos 1993) have studied
the response power spectral density of nonlinear systems by utilizing formal solutions
of the F PK equat ion as discussed by Risken (1989) in conjunct ion with a power series
expansion in terms of Pade approximants for the response spectra. The method
requires the knowledge of the stationary response of the FPK equation.
8/9/2019 0046352920c5292a4e000000
8/27
352 C S M a n o h a r
A s h a s b e e n b r ie f ly i n d i c a te d e a r li er , t h e M a r k o v p r o p e r t y o f r e s p o n s e h a s a l s o b e e n
used i n the s t u dy o f f ir s t pa s sage p rob ab i l i t ie s . He re , e i t he r t he fo rwa rd o r t he b ack w ard
K o l m o g o r o v e q u a t i o n i s s o l v e d i n c o n j u n c t i o n w i th a p p r o p r i a t e b o u n d a r y c o n d i t i o n s
imposed a long t he c r i t i c a l ba r r i e r s . A l t e rna t i ve ly , s t a r t i ng f rom the backward Ko l -
m o g o r o v e q u a t i o n , o n e c a n a l s o d e r i v e e q u a t i o n s f o r m o m e n t s o f h e f ir st p a s s a g e ti m e ,
wh ich , in p r i nc ip l e , c an b e so lved r ecu rs ive ly . Thus , den o t i ng by T (y ), t he t ime r equ i r ed
by t he r e sponse t r a j ec to ry o f ( 1) i n i t ia t ed a t t he po in t x = y in t he ph ase spac e a t t ime
t = t o t o c ro s s a spec if ied s a fe dom a in fo r t he f ir s t t ime , t he m om en t s
M k = E [ ~ ] ,
k = 1, 2 , .. . N , c a n b e s h o w n t o b e g o v e r n e d b y t h e e q u a t i o n
f , (y , t )
d M k ~ r t~ 2 Mk
-- j= l ~--c3Yj ij= l G D G )~+kMk_l=O~J
( k = 0 , 1 , 2 , . . . ) ,
1 4 )
wi th t he con d i t i on M 0 = 1. These eq ua t i ons a r e r e f e r r ed t o a s t he gene ra l i z ed P on -
t r i ag in -Vi t t (GPV) equa t i ons i n t he l i t e r a tu r e . A l though no exac t ana ly t i c a l so lu t i on
exis t s for f ind ing
M k ,
s e v e r a l a p p r o x i m a t i o n s a r e a v a i l a b l e a n d t h e y h a v e b e e n
rev i ewed by R obe r t s (1986a) . These me th ods i nc lude me tho d o f we igh t ed r e s idua l s
(Spanos 1983) , r and om w a lk mo de l s (To l an d Ya ng 1971 ; R obe r t s 1978 ) , f i n it e
d i ff e rence m e tho d (Ro be r t s 1986b) , f i n it e e l emen t m e tho d (Spence r Be rgm an1985)
and cell ma pp in g t e chn iques (Sun H su 1988) .
2.4
S u m m a r y
T h e F P K e q u a t i o n a p p r o a c h is t h e o n l y s o u r c e o f e x a c t s o l u t io n s i n n o n l i n e a r r a n d o m
vib ra t i on p rob l ems . I t a l so fo rms a power fu l t oo l f o r app rox ima te ana ly s i s . The
a p p l i c a ti o n o f t h e m e t h o d is , h o w e v e r , l im i t e d t o M a r k o v i a n r e sp o n s e s. T h e s o l u t i o n
p r o c e d u r e s a r e n o t t r a c t a b le w h e n d e a l in g w i th l a r g e n u m b e r o f v a r i a b le s o r w i th
nons t a t i ona ry i npu t s . A l though t he me thod i s app l i c ab l e when t he i npu t i s a f i l t e r ed
wh i t e no is e , t h i s c la s s o f p rob l em s ha s n o t r e ce ived m uch a t t en t i on i n t he l i te r a tu r e .
3 P e r t u r b at i on me t h od
Thi s is a s t r a i gh t fo rwa rd ex t ens ion o f t he t e chn iqu e u sed i n de te rmin i s t i c p rob l em s .
T h e m e t h o d i s a p p l i ca b l e w h e n t h e e q u a t i o n s o f m o t i o n c o n t a i n a s m a l l p a r a m e t e r
cha rac t e r i z i ng t he non l i nea r i t y i n t he sy s t em. The so lu t i on i s expanded i n a power
se ri e s i n sma l l pa r am e te r wh ich l e ads t o a s e t o f l i nea r d i f fe r en t ia l equa t i on s w h ich
c a n f u r th e r b e s o l v e d s eq u e n ti a ll y . T h e m e t h o d i s a p p l i c a b le t o b o t h s d o f a n d m d o f
s y s t e m s u n d e r a d d i t i v e o r m u l t i p l i c a t i v e , s t a t i o n a r y o r n o n s t a t i o n a r y s t o c h a s t i c
i npu t s . I t was f ir s t u sed by Cra nda l l ( 1963) t o eva lua t e r e sponse m om en t s o f sdo f
a n d m d o f sy s t e m s w i th n o n l i n e a r s ti ff n es s u n d e r s t a t i o n a r y G a u s s i a n e x c i ta t io n s .
S h i m o g o ( 19 6 3 a, b ) c o n s i d e r e d s y m m e t r i c a n d a s y m m e t r ic n o n l i n e a r s y s t e m s u n d e r
s t a t io n a r y r a n d o m i n p u t s a n d c o m p u t e d t h e r e s p o n s e p s d u s i n g a n i t er a ti v e t ec h n i q u e ,
wh ich , i n e s sence , i s i den t i c a l t o t he pe r t u rba t i on me thod . Cranda l l
e t a l
(1964)
a n d K h a b b a z (1 96 5) ap p l i e d th e m e t h o d t o s y s t em s w i th n o n l i n e a r d a m p i n g a n d
e v a l u a t e d t h e r e s p o n s e p s d . T h e y a l s o i n d i c a t e d t h e p o s s i b i li t y o f t h e e v e n o r d e r e d
re sponse moment s_ and t he p sd func t i on b ecom ing nega t i ve fo r l a rge va lue s o f the
n o n l i n e a ri t y p a r a m e t e r . T o a f i rs t o r d e r o f a p p r o x i m a t i o n , t h e p s d f u n c t i o n o b t a i n e d
8/9/2019 0046352920c5292a4e000000
9/27
ethods of nonlinear random vibration analysis 353
us ing t h i s me thod and t he equ iva l en t l i nea r i za t i on t e chn ique have been shown to be
iden ti ca l (Cranda l l 1964 ). M ann ing (1975) ha s e s t ima ted t he r e spon se o f the Duf f ing
osc i l l a to r t o s t a t i ona ry exc i t a t i on and ha s shown tha t by eva lua t i ng t he r e sponse t o
second o rde r o f t he non l inea r i t y pa r am e te r , i t i s pos s ib l e t o d i sp l ay t he e f fec ts o f
non l inea r r e sonances i n t he r e sponse p sd . The r e sponse o f t he Du f f ing o sc i ll a to r t o
n o n s t a t i o n a r y r a n d o m i n p u t s h a s b e e n s t u d i e d b y S o n i S u r e n d r a n ( 19 75 ). T h e
a p p l i c at i o n o f t h e m e t h o d t o s y s te m s w i th 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 h a s b e e n
discussed by Soong (1973) .
Pe r tu rba t i on me thod i s we l l su i t ed fo r po lynomia l non l inea r i t i e s and i s u se fu l i n
c o m p u t i n g r e s p o n s e m o m e n t s a n d s o m e t i m e s t h e p s d f u n c ti o n . D e t e r m i n a t i o n o f t h e
r e s p o n s e p d f u s in g t h is m e t h o d is , h o w e v e r , n o t p o s s i b le b e c a u s e o f th e n o n - G a u s s i a n
na tu re o f t he h ighe r o rde r co r r ec ti ons . Fu r the rm ore , ob t a in ing s e cond o r h ighe r o rde r
co r r ec t i ons i nvo lves cumber some ca l cu l a t i ons and i s no t p r ac t i c ab l e . I t c an be no t ed
t h a t t h e m e t h o d i s a s y m p t o t i c in n a t u r e a n d t h e a c c u r a c y m a r k e d l y w o r s e n s w i th t h e
inc rea se i n t he va lue o f t he non l inea r i t y pa r am e te r .
4 E q u i v a l e n t l i n e a r i z a t i o n
T h i s m e t h o d i s b y a n d l a rg e th e m o s t p o p u l a r a p p r o a c h i n n o n l i n e a r r a n d o m v i b r a t i o n
p rob l ems . I t is ex t ens ion o f t he we l l kn ow n ha rmo n ic l i nea r iza t i on t e chn ique t o
s t o c h as t ic p r o b l e m s a n d is a p p l i c a b le t o b o t h s d o f a n d m d o f s y st e m s u n d e r s t a t i o n a r y
o r n o n s t a t i o n a r y i n pu t s. T h e m e t h o d c o n s is t s o f o p t i m a l l y a p p r o x i m a t i n g t h e n o n -
l i nea r it ie s i n t he g iven sys t em by l i nea r mo de l s so t ha t t he r e su l t ing e qu iva l en t sy s t em
i s amenab le fo r so lu t i on . Fo r eva lua t i ng t he pa rame te r s i n t he equ iva l en t sy s t em,
an add i t i ona l a s su mp t ion t ha t t he r e sponse i s Ga uss i an i s gene ra l ly made . Th i s me thod
was deve loped i n 1950 's i n the con t ex t o f r and om v ib ra t i on p rob l e m s (s ee Ca ugh ey
1963b fo r e a rl ie r r e f e rences) and non l inea r s t ochas t i c con t ro l s (B oo ton 1954) . The
s u b s e q u e n t d e v e l o p m e n t s o f t h e m e t h o d i n v i b r a ti o n p r o b l e m s m a y b e f o u n d i n t h e
wo rks o f Spa nos (1981a) , Ro be r t s Spa nos (1990) and Soc ha Soo ng (199 l ) and
those in the f ie ld o f con t ro l in the w ork of S in i tsyn (1974) .
Cau ghey (1963b l app l i ed t h is t e chn iqu e t o f i nd s t a t i ona ry r e spon se o f non l inea r
s d o f s y s te m s a n d a c la s s o f m d o f sy s t e m s u n d e r s t a t i o n a r y i n p u ts . T h e m e t h o d w a s
gene ra li z ed to a w ide r c l a ss o f m do f sy s t ems by Fos t e r ( 1968) , Iwa n Yan g (1972) and
Ata l ik (1974) . W hen non s t a t i on a ry r e sponse i s o f i n t e re s t t he equ iva l en t p a r am e te r s
wil l be func t i ons o f t ime and acco rd ing ly t he equ iva l en t l i nea r sy s t em wi ll be t ime
va ry ing . Fo r t he ca se o f M ark ov ian r e sponses Iwan M aso n (1980) and W en (1980)
h a v e o b t a i n e d t h e n o n s t a t i o n a r y r e s p o n s e o f m d o f sy s t em s b y s o l v i n g n u m e r i c a ll y t h e
m o m e n t e q u a t i o n s d e r i v ed fr o m t h e g o v e r n i n g F P K e q u a t i o n . S p a n o s (1 9 8 0b ) m o d i -
f ied t he me th od t o dea l w i th m do f sy s t ems hav ing a sym me t r i c non l inea r i t ie s i n wh ich
c a s e t h e r e sp o n s e h a s a n o n z e r o m e a n . W h e n t h e in p u t s a r e n o n w h i t e t h e s o l u t i o n o f
t ime va ry ing equ iva l en t l inea r sy s t em i s gene ra l l y dif ficul t. Ah m ad i (1980b) and Sak a t a
K im ura (1980) have sugges t ed d i ff e r en t s chemes t o dea l w i th such p rob l em s . These
s c h em e s h a v e a l s o b e e n e x te n d e d t o a n a l y z e th e n o n s t a t i o n a r y r e s p o n s e o f a s y m m e t r i c
s d o f a n d m d o f s y s t e m s t o n o n w h i t e i n p u t s ( K i m u r a S a k a t a 1 9 81 , 1 98 7). I n t h e s t u d y
o f con t inuo us no n l inea r sy s t ems , li nea r i za ti on can be do ne a f t e r d i s c r e ti z ing t he
equ a t ion o f m o t io n o r a t t he l eve l o f pa rt i a l d i f fe r en ti a l equa t i o n it se lf . The l a t t e r c l a s s
o f p rob l em s have been s tud i ed by Iwa n Krou sg r i l l ( 1983 ) and Iwan W hi r l ey (1993) .
8/9/2019 0046352920c5292a4e000000
10/27
354 C S Manohar
4 1 Extensions and improvements
The me thod i s ve r sa t i l e , e a sy t o imp lemen t and compu ta t i ona l l y e f f i c i en t . I t i s
app l i cab l e when t he non l inea r i t i e s and t he exc i t a t i ons a r e such t ha t t he r e sponse i s
u n i m o d a l a n d n e a r l y G a u s s ia n . T h i s d o e s n o t n e c e s s a ri ly im p l y t h a t t h e n o n l i n e a r i t y
shou ld be sma l l . C randa l l ( 1973 ) ha s demons t r a t ed t ha t f o r a f i r s t o rde r sy s t em wi th
cub ic non l inea r it i e s, t he me an squa re r e spons e u s ing t h i s te chn iq ue i s f a i rl y accu ra t e
e v e n w h e n n o n l i n e a r it y p a r a m e t e r i s o f t h e o r d e r o f 1 0 0. T h e m e t h o d i n v a r i a b l y l ea d s t o
Ga uss i an r e sponse pd f and f a i ls t o d i sp l ay non l inea r r e sonanc es i n t he r e spon se p sd .
F u r t h e r m o r e , t h e m e t h o d i s n o t a p p l i c a b le t o s y s t e m s w i th p a r a m e t r i c e x c i t at io n s .
S e v e ra l m o d i f i c a ti o n s t o t h e m e t h o d h a v e b e e n p r o p o s e d t o o v e r c o m e s o m e o f t h e
abo ve limi t a ti ons . Cranda l l ( 1973 ) ha s ob t a ined a non l inea r i n t eg ra l equ a t i on fo r the
i m p r o v e d e s t i m a t e f o r t h e s t a t io n a r y r e s p o n s e a u t o c o r r e la t i o n s . G e n e r a l i z a ti o n o f th e
m e t h o d f o r t h e c a se o f n o n l i n e a r s y s t e m s w i t h b o t h p a r a m e t r i c a n d e x t e rn a l w h i t e n o is e
exc i t a t ions ha s been sugges t ed by Bru ckne r L in (1987a ). He re t he o r ig ina l I t 6
equ a t ion fo r the non l inea r sy s t em is op t im a l ly r ep l aced by a l i nea ri zed I t 6 equ a t i on so
t h a t t h e f i r s t a n d t h e s e c o n d m o m e n t s c o m p u t e d f r o m t h e e q u i v a l e n t s y s t e m h a s
m i n i m u m m e a n s q u a r e e r ro r . F o r e v a l u a t i n g e q u i v a le n t p a r a m e t e r s u s e is m a d e o f t h e
h i g he r o r d e r m o m e n t e q u a t i o n s a n d n o a p p e a l t o t h e G a u s s i a n n e s s o f t h e r e sp o n s e n o r
f o r i n v o k i n g a n y o t h e r c l o s u r e a p p r o x i m a t i o n i s n e c e ss a ry . I n t h e m e t h o d s d i s c u s s e d
a b o v e t h e n o n l i n e a r s y s t e m o f a g i v en o r d e r i s i n v a r i a b ly r e p l a c e d b y a n e q u i v a l e n t
sys t em o f t he s am e o rde r . I ye nga r (1988a ) ha s ex p lo red t he p os s ib i l it y o f r ep l ac ing
a g iven non l inea r sy s t em by an equ iva l en t l i nea r sy s t em o f a h ighe r o rde r . I n h i s s t ud y
t h e n o n l i n e a r t e r m s o f t h e g i ve n e q u a t i o n a r e s u b s t i t u t e d b y n e w d e p e n d e n t v a r ia b l es .
A d d i t i o n a l e q u a t i o n s w h i c h g o v e r n t h e s e n e w v a r i a b l e s a n d w h i c h a r e n o n l i n e a r i n
na tu re a r e ob t a ine d by su i t ab ly d if f e r en ti a t ing t he g iven non l inea r equa t i on . T he
re su l ti ng h ighe r o rde r sy s t em o f non l inea r e qua t i ons i s f u r the r ana lyzed u s ing t he u su a l
l i nea r iza t i on s cheme . Fo r t he ca se o f t he Duf f ing o sc i l l a to r unde r wh i t e no i se exc i t a -
t i o n , t h e r e s u l t s o b t a i n e d u s i n g t h i s m e t h o d a r e s h o w n t o b e b e t t e r t h a n t h e u s u a l
l i nea r iza t i on so lu t i on and i n pa r t icu l a r , t he r e sponse p sd ob t a in ed u s ing t h i s m e th od i s
s h o w n t o d i s p l a y t h e e ff ec ts o f s e c o n d a r y r e s o n a n c e i n t h e f o r m o f a n a d d i t i o n a l p e a k a t
a b o u t t h r e e t i m e s t h e p r i m a r y r e s o n a n c e f r e q u e n c y . E x t e n s i o n s o f e q u i v a l e n t l in e a ri z -
a t i o n p r o c e d u r e s t o a l l o w f o r n o n - G a u s s i a n n a t u r e o f t h e r e s p o n s e h a v e a l s o b e e n
p r o p o s e d . T h u s , M a n o h a r I y e n g a r (1 99 0) c o n s i d e r e d t h e b r o a d b a n d e x c it a t io n o f
the Van de r Po l o sc i l l a to r and r ep l aced t he non l inea r o sc i l l a to r by a l i nea r sy s t em
e x c it e d b y a n o n - G a u s s i a n i n p u t. T h e n o n - G a u s s i a n e x c i t at i o n a l l o w e d f o r t h e l im i t
cyc l e o sc i l l a ti ons o f t he sy s t em in t he absen ce o f t he ex t e rna l no i se . Th i s ena b l ed t he
c o r r e c t p r e d i c ti o n o f b i m o d a l p d f o f t h e r e s p o n s e d i s p l a c e m e n t a n d v e l o c i t y w h i c h t h e
t r ad i t i ona l l i nea r iza t i on f a il s t o p r ed i c t (Zhu Yu 1987) . P r a d lw a r t e r et al (1988),
P rad lwa r t e r ( 1989 ) and Schue l l e r et al (1991) have cons ide red s tochas t i c r e sponse o f
i n el a st ic s y s te m s a n d h a v e p r o p o s e d a n o n - G a u s s i a n l in e a r iz a t io n s c h e m e w h i c h is
ba sed on t h e t heo re t i c a l r e su l ts show n by K oz in (1987) t ha t l i nea r sy s t em s ex i s t wh ich
l ead, a t l e a s t f o r wh i t e no i se exc i t a ti ons , exac t l y t o t he f i r st s t a ti s t ic a l m om en t s o f t he
r e s p o n s e o f th e r e s p e c t iv e t r u e n o n l i n e a r s y s te m s . T h e m e t h o d i n v o l v es n o n l i n e a r
t r a n s f o r m a t i o n b e t w e e n n o n l i n e a r r e s p o n s e a n d t h e l in e a r r e s p o n s e w h i c h h a s t o b e
chosen ba sed o n phys i ca l cons ide ra t i ons . I yeng a r (1992) cons ide red Du f f ing 's o sc i l l a to r
under n ar row ban d exc i ta t ion and d eve lop ed .an equiva len t l inear sys tem who se st if fness
p a r a m e t e r i s r a n d o m i n n a tu r e . T h i s p a r a m e t e r i s s h o w n t o b e a f u n c t io n o f re s p o n s e
8/9/2019 0046352920c5292a4e000000
11/27
eth od s o f nonlinear random vibration analysis
355
envelope which i s approx imated as a random variab le . W hi le the response condi t ioned o n
the s t if fne s s i s Gau ss i an , t he unco nd i t i on ed r e sp onse beco m es non -Ga uss i an i n na tu r e .
4.2 Nonuniqueness o f solutions and stochastic stabili ty
The so lu t i on o b t a ine d u s ing equ iva l en t l inea r i za t ion is no t nece s sa r i l y un iqu e i n m ore
than one s ense. F i rs t ly , t he answer s depen d on c r i te r i on o f equ iva l ence ado p t ed . W hi l e
t h e m o s t c o m m o n l y u s e d c r i te r io n r e q u ir e s t h a t t h e m e a n s q u a r e e q u a t i o n d i ff e re n c e b e
min imized , o the r c r i te r i a i nvo lv ing a l t e rna t i ve no rm s o f d i f fe r ences o r o the r ave r ag ing
ope ra to r s a r e a l so admis s ib le (Spanos1981a ; B o lo t i n 1984 ) . Recen t l y E l i shako f f
Zh ang (1993) have app l i ed s eve ra l op t im iza t i on s chemes on a r an dom ly d r i ven f i r st
o rde r non l i nea r sy s t em and h ave show n tha t , w hen t he d if f e rences a r e ave r age d w i th
r e spec t t o a w e igh t f unc t i on , wh ich , in t u rn , i s t aken t o be a non l i nea r f unc t i on o f t he
sys t em po t en t i a l ene rgy , t he e s t ima t e o f t he r e sponse va r i ance co inc ide s w i th t he know n
exac t so lu t i on . C asc i a t i
et al
( 1993 ) have cons ide r ed s econd o rde r non l i nea r sy s t ems
unde r b road band exc i t a t i on and e s t ab l i shed t he equ iva l ence by r equ i r i ng t ha t t he
upc ros s ing r a t e o f a spec i fi ed c ri ti c a l l eve l f o r t he n on l i nea r and fo r t he equ iva l en t l i nea r
o s c il la t o r b e eq u a l. O b v i o u s l y t h e s u c c es s o f th is a p p r o x i m a t i o n d e p e n d s u p o n t h e
accu rac y w i th wh ich t he upc ros s ing s t a t i st i c s a r e known .
An o the r sou rce o f nonun iquenes s , wh ich i s, pe rhap s m ore sub t l e , a r is e s w i th in t he
f r am ewo rk o f a spec if ied equ iva l ence c r i te r i on . Thus , t he l i nea r iza t i on t e chn ique wh en
a p p l ie d t o n o n l i n e a r s y s t e m s u n d e r c o m b i n e d h a r m o n i c a n d n o i s e e x c i ta t io n s ( I y e n g a r
1986 ; M an oh a r I yenga r 1991a) , na r r ow ban d exc i t a t i ons (R icha rd An and 1983 ;
Dav ie s Nan d l aU 1986 ; Dav i e s R a j an 1986 ; J i a Fa ng 1987 ; I yeng a r 1988b , 1989 ;
M an oh a r I yenga r 1991 b ; Ro be r t s 1991 ; Ko l iou pu l os Lang l ey 1993) o r f o r sy s t em s
wi th mu l t i p l e s t ab l e equ i l i b r i um s t a t e s unde r b road band exc i t a t i ons (Lang l ey 1988b ;
Fan Ah m ad i 1990 ) l e ads t o mu l t i va lued r e spon se s t a ti s ti c s . I t m ay be no t e d in t h is
c o n t e x t t h a t S p a n o s I w a n (1 97 8) h a v e e a r li e r d e m o n s t r a t e d t h e u n i q u e n e s s o f
equ iva l en t l inea r sy s t ems unde r c e r t a in co nd i t i ons b u t no t o f t he so lu t i ons gene ra t ed by
the equ iva l en t sy s t ems .
The o ccu r r ence o f mu l t i va lued r e sp onse s t a t is t ic s app a ren t l y r e semble t he coex i s t -
ence o f mu l t i p l e s t e ady s t a te s enco un t e r e d i n t he de t e rm in i s t ic n on l i nea r o sc i l l a ti on
p r o b le m s . I n r a n d o m v i b r a t io n c o n t e x t , h o w e v e r , i t is i m p o r t a n t t o n o t e t h a t s t a t i o n a r y
respon se s ta t is t i cs , wh en th ey exist , a re n ecessar i ly unique . This fo l low s f rom the fac t
t ha t t he s t e ady so lu t i on o f t he gove rn ing F P K equ a t i on i s a lways un iqu e (Fu l l e r 1969 ).
The s c ope o f t h is r e su l t inc ludes a l l t he non l i nea r dyn am ica l sy s t em s wh ich a r e
gove rned by equa t i ons o f the fo rm a s g iven i n (1 ) and , a s h a s b een n o t ed i n 2 , t h is c l a s s
i s f a ir ly ex t ensive . Th i s f a c t was no t r e co gn i zed i n so m e o f t he ea r li e r s t ud i e s on na r r ow
band exc i t a t i on o f t he Du f f i ng o sc i l l a to r s (Dav i e s Na nd l a l l 1986; Dav i e s Ra j an
1986 ; J i a Fa ng 1987). These au tho r s u sed a s t ab i l it y ana ly s i s o f m om en t equ a t i on s
a n d c o n c l u d e d t h a t r e s p o n s e s ta t is t ic s a r e m u l t i v a l u e d a n d d i s p l a y t h e j u m p b e h a v i o u r .
Iyen gar (1986, 1988b, 1989) sugg es ted tha t the rea l i sab i l i ty of the mu l t ip le so lu t ion s
mu s t be dec ided ba sed on t he a lm os t su r e s t ochas t i c s t ab i l i ty o f t he mu l t i p l e so lu t i ons
a n d n o t o n t h e s ta b i l it y a n a l y s is o f m o m e n t e q u a t i o n s . H i s s tu d y s h o w e d t h a t i n r eg i o n s
o f mu l t i p l e so lu ti ons , the l i nea ri za t ion so lu t i on ba sed o n t he a s su m pt ion o f Ga uss i a -
ne s s o f t he r e sponse i s s t ochas t i c a l ly un s t ab l e an d , t he r e fo r e , no t va l i d . Fu r the rm ore ,
s i m u l a t io n s t u d ie s o n r e s p o n s e a m p l i t u d e s h o w e d t h a t t h e p r o b a b i l i ty d e n s i t y f u n c ti o n
8/9/2019 0046352920c5292a4e000000
12/27
356
S Manohar
is b i m o da l i n r eg i ons w he r e l i nea r i za t ion p r ed i c t s m u l t i p le so l u t ions . Th e s t ud i e s by
Lan g l ey ( 1988b) and Fan A h m a d i ( 1990) on b r oad band exc i t a t ion o f non l i nea r
s y s te m s s h o w t h a t m e a n s q u a r e r e s p o n s e p r e d i c te d b y t h e l i n ea r i za t io n t e c h n i q u e m a y
n o t b e u n i q u e w h i l e t h e c o r r e s p o n d i n g e x a c t s o l u t i o n s a r e u n i q u e . T h e s y s t e m
cons i de r ed i n t he se s t ud i e s had m ul t i p l e s t ab l e equ i l i b r i um s t a t e s and t he r andom
r esponse had m u l t i m od a l p r obab i l i t y dens i t y f unc t ions . Th e ques t i on o f pos s i b le
r e l a t i onsh i p be t w een t he m u l t i p l e so l u t i ons p r ed i c t ed by l i nea r i za t i on and l oca l
b e h a v i o u r o f t h e s a m p l e f u n c t io n s h a s b e e n c o n s i d e r e d b y f e w a u t h o r s ( D i m e n t b e r g
1988b; Ro ber t s 1991; K ol iop ulos La ng ley 1993) . I t i s sugge s ted tha t the mu l t ip le
s o l u ti o n s c o r r e s p o n d t o l o ca l b e h a v i o u r s n e a r t h e m o d e s o f t h e r e s p o n s e p d f. T h e
r e l a t i onsh i p be t w een such loca l behav i ou r and g l oba l be hav i ou r in an ensem bl e sense i s
no t obv i ous and f u r t he r r e sea r ch i s c lea r l y requ i r ed t o r e so lve t h is i ssue .
5 E q u i v a l e n t n o n l i n e a r i z a t i o n
T h i s m e t h o d is c o n c e p t u a l ly si m i la r t o t h e m e t h o d o f e q u iv a l e n t li n e a r iz a t io n a n d c a n
be v i ew ed as a gene r a l i z a t i on l e ad i ng t o no n- G auss i an e s t i m a t e s fo r t he r e sponse . The
m e t h od has been i n t r od uced by Ca ugh ey ( 1986). I t cons i st s o f r ep l ac i ng t he g i ven
non l i nea r sys t em by an equ i va l en t non l i nea r sys t em w hi ch be l ongs t o t he c l a s s o f
p r ob l em s w h i ch can be so l ved exac tl y . Th i s m e t h od is r e l a ted t o t he c l a ss o f exac t l y
so l vab l e FPK equa t i ons and t hus i s app l i cab l e on l y t o sys t em s unde r w h i t e no i s e
i npu ts . The c r i t e r i on o f r ep l acem en t is aga i n t he m i n i m i za t i on o f the m ean squa r e e r r o r .
T h e m e t h o d l e a d s t o n o n - G a u s s i a n s t a t i o n a r y r e s p o n s e p d f a n d e s t i m a t e s c o r r e c t ly t h e
r an do m r e sponse o f li m i t cyc l e sys t em s i n w h i ch ca se , equ i va l en t l i nea r i za t ion f ails .
Ca i L i n ( 1988b) have deve l oped a s i m i l a r t e chn i que and hav e app l i ed it t o sys t em s i n
w h i ch pa r am e t r i c exc i ta t i ons a r e a l so p r e sen t . H e r e , t he r ep l acem en t o sc il l a to r be l ongs
t o t he c l a ss o f sys tem s pos se ss i ng gene r a l i z ed s t a t i ona r y po t en t i a l and i s s e lec t ed on t he
bas is t ha t t he ave r age ene r gy o f d i s s i pa t i on r em a i ns unch ang ed . F o r a spec i fi c sys t em ,
t h e s o lu t i o n o b t a i n e d u s i ng t h is m e t h o d i s s h o w n t o b e s u p e r i o r to t h a t o b t a i n e d b y
s t o c h a st ic a v e ra g i ng . T h e a p p l i c a ti o n o f th e m e t h o d t o r a n d o m l y e x c i t e d h y s t er e t ic
s t r uc t u r e s ha s al so been deve l oped ( Ca i L i n 1990) . Th e m e t h od has a l so been s t ud i ed
b y Z h u Y u (1 98 9) w h o h a v e c h o s e n e q u i v a l e n t n o n l i n e a r s y s te m s w h i c h a r e e n e r g y
d e p e n d e n t . T h e y h a v e i n d i c a t e d th a t t h e m e t h o d is a s y m p t o t i c a l ly e x a c t a n d i s
e q u i v a l e n t t o t h e m e t h o d o f s to c h a s t ic a v e r a g i n g o f th e e n e r g y e n v el o p e . In a s t u d y o n
r a n d o m r e s p o n se o f l im i t c y c le s y s te m s , M a n o h a r (1 98 9) h a s d e v e l o p e d e q u i v a l e n t
n o n l i n e a r i z a t i o n s o l u t i o n s f o r t h e r a n d o m l y d r i v e n V a n d e r P o l o s c i l l a t o r . I n o n e
schem e , t he V an d e r Po l o sc i l la t o r und e r w h i t e no i s e exc i t a t ion i s r ep l aced b y a V a n t i e r
Po l - R ay l e i gh o sc i ll a t o r w h i ch can be so l ved exac t l y . Th i s i nvo l ves l i nea r i za t i on o f on l y
a p a r t o f th e o r ig i n a l e q u a t i o n a n d t h e e q u i v a le n t p a r a m e t e r s a r e f o u n d b a s e d o n
m i n i m i z a t i o n o f m e a n s q u a r e e r r o r . In a n a t t e m p t t o m o d e l t h e m u l t i m o d a l p d f o f t h e
r e sponse phase p roces s , M an oh a r ha s app l i ed a s i m i l a r pa r t i a l l i nea r i za t ion p r oce du r e
t o t he s i m p l i fi ed equa t i ons f o r t he r e spon se am pl i t ude an d phase p r oces se s w h i ch , i n
t u r n , w e r e o b t a i n e d u s in g a s e c o n d o r d e r s t o c h a s t ic a v e r a g in g p r o c e d u r e . F u r t h e r m o r e ,
t he s am e p r oc edu r e w as a l so u sed t o i nves t i ga t e t he e f fec t o f no i s e on f r equen cy
e n t r a i n m e n t o f h a r m o n i c a l l y d r i v e n V a n d e r P o l o s c il la t o r ( M a n o h a r I y e n g a r 19 9 l a )
a n d i n t h e s tu d y o f ro c k i n g o f r ig i d b lo c k s u n d e r r a n d o m b a s e m o t i o n s ( I y e n g a r
M a n o h a r 1 99 1). T h e s c h e m e o f p a r t ia l l in e a r i z a ti o n h a s a l so b e e n s t u d i e d r e c e n t ly b y
8/9/2019 0046352920c5292a4e000000
13/27
Methods o f nonl inear random v ibrat ion analys i s
357
Elishakoff Cai (1993). To Li (1991) have presented a systematic equivalent
nonlinearization procedure which is again based on the broad class of exactly solvable
FPK equations and utilizes calculus of variations to derive the optimal replacement
system. In a study on systems with asymmetric nonlinearities, Spanos Donley (1991)
and Li Kareem (1993)have developed equivalent systems with quadrat ic nonlineari-
ties. The evaluation of the equivalent parameters is based on an approximate analysis
of the equivalent systems using Volterra series representations (see 8).
6 C l o s u r e a p p r o x i m a t i o n s
In nonlinear random vibration problems the equations for response moments and
correlations form an infinite hierarchy and exact solutions are not possible. This is true
even for the class of systems for which exact response pdf is obtainable using the FPK
equation. The closure problem consists of approximately replacing the infinite hier-
archy of equations with a finite set so that estimates for the important lower order
moments can be obtained.
6 1 Closure us ing assumed probabi l i ty dens i ty func t io n
The closure approximation can be made either in conjunction with an assumed
response pdf or directly on the moment equations. Dashevskii (1967), Assaf Zirkle
(1976) and Crandal l (1980, 1985) have employed a series representa tion for response pdf
in terms of the Hermite polynomials. The series is truncated after a finite number of
terms. The first term in the series has the form of a Gaussian pdf. The unknown
coefficients of the series are related to the response cumulants, central moments or
expectations of the Hermite polynomials in response variables. The equations needed
to determine these coefficients are generated from the governing equation of motion.
This procedure can be viewed as the generalization of equivalent linearization tech-
nique wherein the response was assumed to be Gaussian and the parameters in the
distribution were determined using moment identities derived from equations of
motion. Liu Davies (1988) have applied Hermite polynomial approximation for the
pdf and studied the nonstationary response of nonlinear second-order systems. Fur-
thermore, the same authors (Davies Liu 1992) have also studied the power spectrum
of Duffing's oscillator using a similar procedure. Iyengar (1975) and Iyengar Dash
(1976, 1978) have developed a closure technique in which the response variables and
input variables, either as they appear or after a transformation, are assumed to be
jointly Gaussian. It is possible in this formulation to take into account non-Gaussian
excitations and amplitude limited responses. The method handles nonlinear and
stochastically time varying systems in a unified manner (Dash Iyengar 1982).
6.2
Closure i n te rms o f mom en t s o r cumu lan t s
In the second approach, one directly deals with moment equations. Here the unknown
higher order moments are approximated as functions of lower order moments, thereby
truncating the hierarchy of these equations. Thus, Ibrahim (1978), Ibrahim et al (1985),
Bolotin (1984) and Wu Lin (1984) have considered different schemes for closing
8/9/2019 0046352920c5292a4e000000
14/27
358
S Manohar
h i e ra r c h y o f m o m e n t e q u a t i o n s . A p p e a l is g e n e r a ll y m a d e t o t h e q u a s i n o r m a l a p p r o x i -
m a t i o n w h i ch c o n n e c ts a r e h ig h e r o r d e r m o m e n t s t o t h e l o w e r o r d e r m o m e n t s t h r o u g h
re l a t i ons t ha t a r e s t r i c t l y va l i d on ly fo r Gauss i an va r i ab l e s . Th i s c an be exp re s sed i n
t e rm s o f e i th e r t h e d ir e c t m o m e n t s , c e n t r a l m o m e n t s o r c u m u l a n t s . T h e q u a s i n o r m a l
a p p r o x i m a t i o n u s in g c u m u l a n t s a m o u n t s t o s e tt in g c u m u l a n t s b e y o n d a g i v en o r d e r t o
z e ro . F o r a G a u s s i a n r a n d o m v a r ia b l e , i t m a y b e re c al le d , a ll t h e c u m ~ l a n t s b e y o n d
orde r two van i sh . O the r c l o su re s chemes such a s d i s ca rd ing t he d i r ec t o r c en t r a l
m o m e n t s b e y o n d a n o r d e r o r i g n o r i n g t h e c o r r e l a ti o n s a m o n g t h e r e s p o n s e v a ri a b l e s
h a v e a l s o b e e n p r o p o s e d ( S o o n g 1 9 7 3 ) . B e l l m a n R i c h a r d s o n ( 19 68 ), W i l c o x
Be l lman (1970) and Sanc ho (1970) have u sed a m ean squa re c lo su re t e chn ique i n wh ich
t h e u n k n o w n h i g h e r o r d e r m o m e n t i s e x p r e s s e d a s a n o p t i m a l l i n e a r c o m b i n a t i o n o f
l o w e r o r d e r m o m e n t s .
6.3 Limitations and improvements
T h e c l o s u r e m e t h o d s a r e a p p l i c a b le t o a w i d e c la s s o f s y s t em s a n d e x c i ta t io n s . T h e y
h a v e b e e n e x t e n s iv e l y u s e d i n t h e r e s p o n s e a n d s t a b i li t y a n a ly s i s o f s d o f a n d m d o f
s y s t em s w it h p a r a m e t r i c a n d n o n p a r a m e t r i c e x c i ta t io n s . H o w e v e r , i t h as n o t b e e n
p o s s i b le t o j u s t if y c lo s u r e a p p r o x i m a t i o n s t h r o u g h a n a l y ti c a l a p p r o a c h e s . M o s t o f t h e
c lo su re s chemes a r e t heo re t i c a ll y i ncons i s t en t a t som e leve l a s t hey v io l a t e we l l - kn ow n
iden t it i es and i nequa l i t i es o f p ro bab i l i t y t heo ry . Thus , f o r exam ple , the m om en t c l o su re
s c h e m e o f s e tt in g d i r e ct m o m e n t s b e y o n d o r d e r n t o z e r o v i o l a te s t h e i n e q u a l i t y
E [ x n S E [x~ ] 2. The cum ulan t c l o su re v io l a t e s t he t heo rem due t o M arc i enk i ew icz
(Gard ine r 1983 ) wh ich s t a te s t ha t t he cum ulan t gene ra t i ng func t i on can no t be a po ly -
nom ia l o f o rd e r h ighe r t han two , t ha t i s , e i t he r a ll bu t t he f ir s t two cum ulan t s van i sh o r
t h e r e a re a n i n fi n it e n u m b e r o f n o n v a n i s h i n g c u m u l a n t s .
B e l lm a n R i c h a r d s o n (1 96 8) h a v e d e r i v e d a c o n d i t i o n u n d e r w h i c h t h e t r u n c a t e d
e q u a t i o n s o b t a i n e d u s i n g t h e m e a n s q u a r e c l o s u r e t e c h n i q u e p r e s e r v e s m o m e n t
p rope r t i e s . No s imi l ar r e su lt s a r e ava i l ab l e f o r o the r c l o su re s chemes . A l th oug h i t ha s
b e e n d e m o n s t r a t e d w i t h a s p e ci fi c e x a m p l e t h a t t h e a c c u r a c y o f c l o s u r e s c h e m e
s y s t e m a t ic a l ly im p r o v e s a s t h e o r d e r o f a p p r o x i m a t i o n i s i n c re a s e d , e x a m p l e s t o
cou n t e r t h i s a r e a l so r ead i ly ava i l ab l e (Cranda l l 1985 ). I n s t ances o f t he e s t im a t ed pd f
b e c o m i n g n e g at i v e h a v e a l s o b e e n e n c o u n t e r e d ( C ra n d al 1 19 8 5 ). S u n H s u (1 98 7) h a v e
app l i ed s econd , fou r th , and s i x th -o rde r cum ulan t neg l ec t s chemes t o a spec i fi c p ro b l em
f o r w h i c h e x a c t s t a t i o n a r y s o l u t i o n i s a v a i l a b le a n d t h e y h a v e d e l i n e a t e d r e g i o n s w h e r e
the s chemes y i e ld accep t ab l e r e su l t s . I n ce r t a i n pa r ame te r r eg ions t he fou r th - and
s i x t h -o r d e r s c h e m e s a r e s h o w n t o p r e d i c t e r r o n e o u s b e h a v i o u r i n c l u d in g a f a u l t y j u m p
in t he r e sponse . Fa n Ah m ad i ( 1990 ) have cons ide r ed a sy s t em wi th mu l t i p l e s t ab l e
e q u i l i b r i u m d r i v e n b y w h i t e n o i s e e x c i t a t i o n . T h e y h a v e s h o w n t h a t t h e s t a t i o n a r y
r e s p o n s e st a ti s ti c s g e n e r a t e d b y G a u s s i a n c l o s u r e a n d n o n - G a u s s i a n c u m u l a n t n e g l e ct
c l o s u r e t e c h n i q u e s a r e n o t u n i q u e a n d a r e d e p e n d e n t o n i n i t i a l c o n d i t i o n s . T h i s
c o n t r a d i c t s th e u n i q u e n e s s p r o p e r t y o f t h e s t a t io n a r y s o l u t i o n s o f t h e F P K e q u a t i o n
( se e 4.1 ). P a w l e t a S o c h a ( 19 92 ) h a v e c o m p a r e d a p p r o x i m a t e n o n s t a t i o n a r y s o l-
u t i o n s o b t a i n e d u s i n g c l o s u r e a p p r o x i m a t i o n s w i t h t h e c o r r e s p o n d i n g e x a c t s o l u ti o n s
f o r th e c a s e o f p a r a m e t r i c a ll y e x c i t e d li n e ar s y s t e m s a n d h a v e s h o w n t h a t n e a r s t a b il i ty
b o u n d a r i e s t h e a p p r o x i m a t i o n s a r e n o t a c c e p ta b l e .
T h e s e c o n d - o r d e r c u m u l a n t n e g l ec t a n d t h e G a u s s i a n c l o s u r e t e c h n iq u e s a r e s i m i la r
t o e q u i v a l e n t li n e a ri z a ti o n m e t h o d a n d a r e c o n s i s t e n t c l o s u r e s c he m e s . H o w e v e r , t h e y
8/9/2019 0046352920c5292a4e000000
15/27
Methods of nonlinear random vibration analysis 359
yie ld acceptable resu l ts on ly w hen the response has fea tures of Ga uss ian variab les. Fo r
example, w hen ap plied to ra nd om vibra t ion o f self-excited system s (Bo lot in 1984) they lead
to drast ical ly w ron g resul ts (Zh u Y u 1987; M an oh ar Iyen gar 1990) (see 4-1). Rec ently
Gr igor iu (1991) has dev e loped a cons is ten t c losure proce dure which i s based o n a n
es t imator of the response pd f tha t cons is t s of superpos i t ion o f spec if ied k erne ls weighted by
unde t e rmined pa r ame te r s . These unknown pa rame te r s a r e de t e rmined ba sed on t he
cri ter ion that the m om en t equa t ions are opt im ally sat isf ied up to a specif ied closu re level .
7 S tocha st i c averag ing m ethods
I n the se me thod s t he r e sponse o f l i gh tl y dam ped sys t ems t o b roa d b and exc i t a t i on is
approx imated by a d i ffusion process . Th e coeff ic ien ts o f the assoc ia ted F P K equat ion a re
der ived based on an a pprop r ia te averaging of the equa t ions of mot ion . The appea l o f these
me thods l i es in the fac t tha t they of ten reduce the d imens iona l i ty of the prob lem and
significant ly s implify the solut ion proced ures. O n a cco unt o f this ad van tage they are also
appl ied to sys tems where in the response i s a l ready M arkov . D i f fe ren t vers ions of the
m ethod are ava i lab le and a re w ide ly used in the prob lems of response predic t ion , s tab i li ty
ana lys i s an d the f i r s t passa ge an d fa t igue fa i lure ana lyses . Extens ive su rveys of re la ted
l i te r a tu r e have been pub l i shed ( Ib r ah im 1985 ; Ro be r t s Span os 1986 ; Ro be r t s 1986a;
Zhu 1988) .
7.1
verag ingof amplitude and phase
The m e thod was o r i g ina l l y p rop ose d by S t r a tono v i ch (1963 , 1967 ) a s a gene ra l i z a ti on
of the de te rminis tic averaging metho d dev e loped earl ie r by B ogol iubo v and M i t ropo lsky
(1961). He cons idered sdof nonl inear sys tems under r and om exc i ta t ion an d show ed tha t
when the re laxa tion t ime of the osc i l la tor i s l a rge comp ared to the cor re la t ion t ime of the
exc i t a t i on , t he r e sponse can be app rox ima ted by a d i f fu s ion Markov p roces s . Subse -
quent ly , Khasminisk i i (1966) provided a r igorous m athem at ica l pro of for S t ra tonovich ' s
a rguments . The necessary requi rements for apply ing the m etho d are sa ti sf ied i f the sys tem
is l igh tly dam ped and the exc i tat ion po we r spec t rum is s lowly vary ing in the ne ighb our-
hoo d o f the sys tem's na tura l f requency. T he response in such a case wi l l be a n ar row ban d
process wi th s lowly vary ing ampl i tude and phase . The averaging proc edure i s a combina -
t ion of tem pora l and ense mb le averaging a nd i t a ims at eliminating rapid oscil lat ions from
the dominant s lowly vary ing components and a l so a t rep lac ing randomly f luc tua t ing
com po nen ts by equiva lent del ta corre lated processes. This resul ts in a pair of I t6 differential
equa t ions for amp l i tude and ph ase which wi ll have to b e ana lysed us ing the F P K equat ion .
In man y cases the equa t ion for ampl i tude ge ts uncoup led f rom tha t of the phase thu s
enabl ing the de te rmina t ion of the s ta t ionary d is t r ibu t ion of the am pl i tude process . In fac t
thi s is the main a dvan tage o f th i s method . In o rder to de te rm ine the s ta t ionary pd f of
d isp lacement and ve loc i ty variab les, the know ledge of o in t pdf of am pl i tude and phase i s
essent ial . But i t is in general diff icult to o bta in this pdf. Ho w ev er unde r the ass um ptio n that
am pl i tude and phase a re independent i t i s s ti ll poss ib le to ob ta in an appro xim at ion to pdf
o f d i sp l acemen t and ve loc it y .
S t r a tono v i ch u sed t h is m e tho d t o e xam ine t he r e spon se o f sdo f s e l f- exc it ed sy s t ems
t o p a r a m e t r i c a n d n o n p a r a m e t r i c e x c it a ti o n s. S u b s e q u e n t l y , t h e m e t h o d h a s b e e n
g e n e ra l iz e d t o i n c l u d e m d o f s y s t em s a n d n o n s t a t i o n a r y i n p u t s a n d w i d e ly u s ed i n
8/9/2019 0046352920c5292a4e000000
16/27
3 6 C S M a n o h a r
r a n d o m v i b r a ti o n s tu d i es ( R o b e r t s S p a n o s 1 98 6). T h e m e t h o d h a s a l s o f o r m e d t h e
bas i s for the s tudy of f ir s t passage fa i lures (Ro ber t s 1986a) and s tab i l i ty ana lys i s
( I b r a h im 1 98 5). F o r p r o b l e m s w h e r e i n t h e t i m e v a r y i n g n a t u r e o f t h e s y s t e m s u c h a s
de t e r m i n i s t i c exc i t a ti ons o r n on s t a t i on a r y i npu t s needs t o be p r e se r ved , L i n ( 1986) ha s
p r o p o s e d t h a t t h e t e m p o r a l a v e r a g i n g i n S t r a t o n o v ic h ' s p r o c e d u r e m a y b e d i s p e n s e d
w i t h . H eur i s t i c a r gum en t s f o r r e l ax i ng r e s t r i c t i ons on t i m e cons t an t s o f i npu t and
r e sponse f o r p r ob l em s o f s t ab il i ty an a l ys i s have a l so been g i ven .
7.2 Quasis ta t ic averaging
A v a r i a t io n o f th e s t a n d a r d s t o c h a st ic a v e r a g in g m e t h o d , k n o w n a s t h e m e t h o d o f
quas i s t a t ic ave r ag i ng , ha s a l so been deve l op ed by S t r a t onov i ch (1967). The m e t h od i s
app l i cab le t o p r ob l em s i n w h i ch t he co r r e l a t i on t i m e o f the exc i t a t i on g r ea t l y exceeds
t he r e l axa t i on t im e o f t he sys t em . Th i s r eq u i r em en t is con t r a r y t o t he on e s t i pu l a t ed f o r
t h e a p p l ic a b il it y o f th e s t a n d a r d s t o c h a s ti c a v e r a g in g m e t h o d . T h e m e t h o d c o n s is ts o f
o n l y t e m p o r a l a v e r a g i n g . T h e e n s e m b l e a v e r a g i n g w i t h i t s a t t e n d a n t M a r k o v i a n
app r ox i m a t i on i s d i spensed w i th . I n app l y i ng t h i s m e t hod , enve l op e r ep r e se n t a t i on is
u s e d fo r b o t h t h e i n p u t a n d t h e r e s p o n s e p ro c e ss e s. D u r i n g t e m p o r a l a v e r a g i n g th e
a m p l i tu d e a n d p h a s e a n g l e a r e a p p r o x i m a t e d a s r a n d o m v a r ia b l es a n d h e n c e a s
cons t an t s . Th i s f ina l ly l e ads t o a n on l i ne a r m em or y l e s s t r ans f o r m a t i on r e l a t i ng t he
i n p u t a n d t h e o u t p u t a m p l i t u d e s a n d p h a s e a n g le s . T h u s t h e s o l u t i o n o f t h e g i v en
r a n d o m d i ff e re n ti al e q u a t i o n i s c o n v e r t e d t o a p r o b l e m i n n o n l i n e a r t r a n s f o r m a t i o n o f
r a n d o m v a ri ab le s . T h e m e t h o d h a s b e e n u s e d i n t h e s t u d y o f n o n l i n e a r s y s te m s u n d e r
n a r r o w b a n d e x ci ta t io n s b y s e v er a l a u t h o r s ( L e n n o x K u a k 1 97 6; S a t o
et al
1985;
Ri ch a r d A nan d 1983 ; I yen ga r 1986) .
7.3 Averag ing o f energy enve lope
Th e s t ochas t i c ave r ag i ng m e t h od is f ound t o g i ve accep t ab l e r e su l t s f o r sys t em s w i t h
n o n l i n e a r d a m p i n g . I n f a c t w h e n d a m p i n g i s a m p l i t u d e d e p e n d e n t a n d t h e e x c i t a ti o n is
whi te noise, the m etho d leads to the k no w n ex ac t so lu tions (Rober t s 1976). How ever , for
a system with no nl inea r st if fness, such as Du ff ing's oscil lator , the so lut ion do es n ot d isplay
the e ffec ts of nonl inear ity . In such cases a h igher ord er averaging pro ced ure needs to be
used (Straton ovich 1967; Ibr ah im 1985). This , h ow ever , involves c um be rsom e calculat ions.
A s im p l e r al te r nat ive is t o exam i ne w he t he r a one d i m ens i ona l Mar ko v i an app r ox i m a t i on
can b e ob ta ined for the energy envelope of the response. This w as or ig ina l ly prop osed by
St ra tonovich (1963) w ho co ns idered sys tems und er wh i te noi se inputs and reduce d the tw o
dimens iona l Markovian vec tor cons i s t ing of a s lowly vary ing energy envelope and
a r a p id l y v a r y i n g d i s p la c e m e n t c o m p o n e n t t o a o n e d i m e n s i o n a l M a r k o v i a n a p p r o x i -
m a t i o n f o r t h e e n e r g y e n v e lo p e . T h i s m e t h o d h a s f u r t h e r b e e n d e v e l o p e d b y R o b e r t s
(1976, 1978) and genera li zed to incorpo ra te n onw hi te inputs (R ober t s 1982) , param et r ic
exc i ta t ions (Zh u 1983) an d nons ta t ionary inputs (Red-H orse Spanos 1992) . Here , the
averaging i s car r ied out over a p er iod equal to the u nd am pe d na tura l per iod o f the system,
which , now depends on the energy of the response . T he resul t s obta ined us ing th is me tho d
also agree wi th theav a i lab le exac t so lu tions . Zh u Lin (1991) and Z hu
et al
(1994) hav e
c o n s i d e r e d s y s te m s w i t h c o r r e l a t e d G a u s s i a n e x c i ta t io n s a n d h a v e i n c l u d e d t h e
a d d i t i o n a l c o n t r i b u t io n s t o d a m p i n g a n d s ti ff ne ss m a d e b y t h e W o n g - Z a k a i c o r r e c t i o n
8/9/2019 0046352920c5292a4e000000
17/27
Methods of nonlinear random vibration analysis 361
te rms . These add i t i ons a r e i nco rp o ra t ed i n to t he de f in i ti on o f t he ene rgy enve lope and
t h e c o n s e q u e n t n e w r e s u l t s a r e s h o w n t o b e i m p r o v e m e n t s o v e r e a r l i e r a v e r a g i n g
re su lt s . An o the r ve r s ion o f s t ochas t i c ave rag ing i s a l so ava i l ab l e (Sunah a ra
et a11977 .
In t h i s a d e t e rmin i s t ic ave rag ing i s c a r r i ed o u t d i r ec t l y on t he coe f f i c ien t s o f t he
g o v e r n i n g F P K e q u a t i o n . T h i s m e t h o d h a s b e e n s h o w n t o b e e q u i v a l e n t t o t h e
ave rag ing o f t he am p l i t ude o r t he ene rgy enve lope o f t he r e spon se (Zhu 1988).
7.4
Combination of averagin9 with other methods
T h e m e t h o d o f s t o c h a s ti c a v e r a g in g h a s a l s o b e e n u s e d in c o m b i n a t i o n w i t h o t h e r
m e t h o d s o f r a n d o m v i b r a t io n a n al y si s. T h u s , I w a n S p a n o s (1 97 8) p r o p o s e d a c o m b i -
na t i on o f equ iva l en t l i nea ri za t ion and s tochas t i c ave rag ing t o ana lyze sys t em s wi th
non l inea r s ti ff ne ss . F o r t he ca se o f t he D uf f ing o sc i l l a to r unde r w h i t e no i se i npu t , t he
m e t h o d i m p r o v e s t h e r e su l ts o b t a i n e d u s in g a v e r a g i n g o f r e s p o n s e a m p l i t u d e b u t d o e s
no t l e ad t o t he exac t so lu t i ons . Fu r the rmore , Ar i a r a tnam (1978) ha s ques t i oned t he
cons i s t ency o f app ro x im a t ions m ade i n t h i s ana ly s is . S t r a tonov ich (1967) ha s u sed
equ iva l en t l inea r i za ti on t e chn ique t o so lve s imp l if i ed equa t i on s o b t a in ed u s ing
s tochas t i c ave rag ing . Bruck ne r L in (1987b) have ado p te d a com plex fo rm o f
s tochas t i c ave rag ing wh ich ea se s t he app l i ca t i on o f non -Ga uss i an c lo su re t e chn ique t o
the s imp l i f ied equ a t i ons a nd i s pa r t i cu l a r l y u se fu l in ana lyz ing no n l inea r m do f sy s t ems .
I n th e s t u d y o f n o n l i n e a r s y s te m s u n d e r c o m b i n e d h a r m o n i c a n d r a n d o m e x c i t at i o n s o r
when a h ighe r o rde r ave rag ing i n Ca r t e s i an co -o rd ina t e s i s done , t he r e su l t i ng
s imp l if ied equ a t i ons do n o t ge t unco up led , and , i n gene ra l, a r e unso lv ab l e w i th in t he
f r a m e w o r k o f t h e M a r k o v p ro c e ss t h eo r y . U n d e r s u ch s i tu a t io n s M a n o h a r I y e n g a r
(1990 , 1991a ) have p roposed combin ing ave rag ing wi th equ iva l en t non l inea r i za t i on
t echn ique . Th i s p ro cedu re i s show n to g ive s a t i s f ac to ry r e su lt s f o r t he ca se o f Van de r
P o l ' s o sc i ll a to r u n d e r b r o a d b a n d a n d c o m b i n e d h a r m o n i c a n d w h i t e n o i s e ex c i ta t io n s .
7.5
Method of stochastic normal forms
A n a l t e rn a t iv e w a y o f r e d u c in g t h e d i m e n s i o n a l it y o f th e p r o b l e m u s in g m o d e r n
b i fu r ca t i on t heo r ie s ,
viz,
c e n t e r m a n i f o l d t h e o r y ( G u c k e n h e i m e r H o l m e s 1 9 83 ), h a s
b e e n d e v e l o p e d b y Sr i N a m a c h c h i v a y a L i n (1 99 1). T h e m e t h o d c o n s i s t s o f e l im i n a t -
i ng ce r t a in r e sponse va r i ab l e s wh ich a r e a sympto t i c a l l y s t ab l e a s be ing un impor t an t
wi th t he e s sen t i a l beh av io u r o f t he sy s t em r e s t r i c t ed t o t he dyna mics o f t he r em a in ing
c r i t i c a l va r i ab l e s . The d i f f e r ences be tween t h i s me thod and t he t r ad i t i ona l ave rag ing
a r is e i n ca r ry ing o u t t he ' t em pora l ' pa r t o f t he ave rag ing , wh i l e , t he ensem ble ave rag ing ,
wi th t he consequen t Markov ian app rox ima t ion , r ema ins e s sen t i a l l y t he s ame . I n f ac t ,
t he equ iva l ence o f t h is me th od wi th a h ighe r o rd e r s t ochas t i c ave rag ing h a s been
d e m o n s t r a t e d (S ri N a m a c h c h i v a y a L e n g 19 90 ). T h e a p p r o a c h h a s b e e n e m p l o y e d i n
the s t ud y o f t he e f fec ts o f no i se on b i fu r ca t i ons i n no n l inea r sy s t ems an d fo r spec if ic
ca se s , t he me thod i s shown to be more gene ra l l y app l i cab l e t han t he s t ochas t i c
ave rag ing (S r i Namachch ivaya 1991 ; Leng et al 1992).
7.6
System stochasticity problems
A l t h o u g h , t h e a v e r ag i n g m e t h o d s a r e w i d e l y u s e d i n v i b r a ti o n p r o b l e m s , t h e i d e a o f
8/9/2019 0046352920c5292a4e000000
18/27
362 S Manohar
app ly ing t h em to p rob l em s o f spa t i a l va r i ab i l i t y is nove l. Thus , t he u se fu lne ss o f t he
a v e r a g i n g m e t h o d i n th e s t u d y o f s t o c h a s ti c b o u n d a r y v a l u e p r o b l e m s h a s b e e n
inves t i ga t ed by M an oh a r Iyeng a r (1993, 1994) i n t he con t ex t o f t he de t e rm ina t i on o f
t h e e i g e n s o lu t io n s o f s to c h a s t i c w a v e e q u a t i o n s . H e r e , t h e g i v e n b o u n d a r y v a l u e
p rob l em i s conv e r t ed i n to a s e t o f i n it ia l va lue p ro b l em s and , t he se a r e , i n t u rn ,
s imp l if i ed by ave rag ing ove r spa t i a l dom a in . T he r e su l t s ob t a in ed on t he pd f o f t he
e igenso lu t i ons u s ing t h i s app rox ima t ion i s f ound t o compare ve ry we l l w i th d ig i t a l
s imula t ion resu l t s .
7.7 Summary
T h e m e t h o d s o f s t oc h a s ti c a ve r ag i n g e n h a n c e t h e s c o p e o f t h e F P K e q u a t i o n a p p r o a c h
in r and om v ib ra t i ons . Th e d i f fe r en t ve r s ions o f t h i s m e tho d a r e m a them a t i ca l l y w e l l
f o u n d e d ( Z h u 1 9 8 8) . T h i s i s i n c o n t r a s t t o o t h e r a p p r o x i m a t e t e c h n i q u e s d i s c u ss e d
ea r li e r. The o th e r mer i t o f t he se m e tho ds i s t ha t t hey l e ad t o non -Ga uss i a n e s t ima te s fo r
the response .
8 Stochast ic series solut ion
