(From Roberts. 59 )
The equivalent linearization theory is abbreviated as EL theory. In this figure, only the
perturbation theory appears different from the rest, but here we have to remember that in
the perturbation method this is the first order approximation and we can improve the ap
proximation by increasing the order of the approximation, as indicated in the discussion
in Section 10.2.2.
12.63 Nonlinear Oscillation in Nonwhite Excitation
Roberts 61 further extended the scope of applicability of this method and solved for
nonlinear rolling in nonwhite excitation. The equation of motion is now
I<j> + PC(</)) + k(<p) = M(t)
(12.96)
or
<P+PF(<f>) + G(<f>)=x(t)
(12.96')
where x(t) is a nonwhite excitation, and can have a colored spectrum.
Again, he adopted the total energy envelope as Eq. 12.86
V = Â£+C7(0)
where
U(<j>)=lGÂ£)dÂ£
and considered that this was slowly varying.
350
He set
^ = Vsin 2
u(<p) = v cos 2 e
<p=j2VÂ±sine
u(<pp = V2 C ose
(12.97)
introducing the phase 6 as shown in Fig. 12.7. After manipulation of the relations, the
equation of motion, Eq. 12.96', was inverted into the equation of V and<p. When the
phase process 6 was modified into a process k{t), and the joint process Z = (V, A) was
made to converge into a Markovian process, the transitional distribution function
P(Z\Zq\ t) was found to be governed by a FokkerPlanck equation of second order. The
stationary solution of the FokkerPlanck equation p(Z) was obtained, and from this
expression V and X were found to be uncoupled, and p(V) was calculated. With the
relations of V, <p and0, p(V) was modified into p{<p,<p).
[U(<Â»]2
u.v vs.e,<t>
Fig. 12.7. U, Vvs.6,<p.
(From Roberts. 61 )
For nondimensionalized, nonlinear rolling expressed as
' + a4> + b\$\ <j> + <p  <j> 3 = x(t),
(12.98)
Roberts calculated the probability density function of the nondimensionalized amplitude
of rolling A by the ME theory and compared it with the Rayleigh distribution obtained by
the linear theory as in Fig. 12.8. He 61,86 also compared his results with the nonlinear sim
ulated data obtained by J.F. Dalzell 85 to show the validity of his method. Examples are
shown in Figs. 12.9 through 12.11 for a variety of damping coefficients a and b, where
Q = (o p / (Do, (t) p is the peak frequency of the excitation, anda w is standard deviation
of the input process x(t). Process 3 in the figures is a wide banded excitation for this
example. He showed many other results of comparison for other types of excitation with
351
different bandwidths, named Process 1 and Process 2, used in the simulation by J.F.
Dalzell. 85 The deviation from the Rayleigh distribution in Figs. 12.8 to 12.11, and the
relation of the standard deviation of roll Or to that of input o w in Fig. 12.12 shows the
extent of nonlinearity. The applicability of the method was also discussed.
P(A)
10 1 F
p(A!
10Â°
10 1
10*
10" 3
PRESENT THEORY
WITH LINEAR RE ^^4
STORING MOMENT
0.1 0.2 0.3 0.4
Fig. 12.8. Probability density function for amplitude A:
a = 0, b = 1 , a w = 0.036, Q = 0.90,
Process 3. (From Roberts. 61 )
0.999
0.99
0.95
0.90
d ,ai 080
P(A) o.70
0.60
0.50
0.40
0.30
0.20
0.10
0.05
0.02
i 7T â€” ; â€” i r
>â€” RAYLEIGH
DALZELLS SIMULATION
ESTIMATES
aÂ«0
* b1
0.1 02 0.3 0.4 0.5 0.6 A
Fig. 12.9. Cumulative probability distribution for amplitude A:
a = 0, b = 1 , a w = 0.036, Q = 0.90,
Process 3. (From Roberts. 61 )
352
P(A)
0999
099 
095
090
080
O70
(WO
050
0*0
030
020
OW
005
DALZELL'S SIMULATION
ESTIMATES
â™¦ b= 01
Â«b=10
002
001
0005
01 02 03 04 05 06 A
Fig. 12.10. Cumulative probability distribution for amplitude A:
a = 0.01, oâ€ž = 0.036, Q = 0.90, b = 0.1 and 1.0;
Process 3. (From Roberts. 61 )
0999

i i 1
[,
I i â€” r 1
/_ PRESENT
I THEORY /
099
V
1/
b=3 /
/^PRESENT
095

t
Y THEORY
09

T/l b=OI
08

0/
0/
>*
PMJ07
Â©/
yr
06
Â°/
/â€¢
05
/^
04


03

DALZELUS SIMULATION
02
ESTIMATES
I brM
OW
\ b=3
005

002

o =003
001


0005
â–
' i
â– i i i
02 03 04 05
06 A
Fig. 12.11. Cumulative probability distribution for amplitude A:
a = 0.03, aâ€ž = 0.036, Q = 0.90, b = 0.1 and 3.0;
Process 3. (From Roberts. 61 )
353
Or
DALZELL'S SIMULATION
0.20
. b .= 1 .0
â€¢ b= 03
o b = 10
*Z
0.15
/ $S ~
0.10
s^Â°
0.05
a = 0.01
0.01 0.02 0.03 0.04
Fig. 12.12. Variation of standard deviation of roll o R with standard deviation of wave
input o w : a = 0.01; Q = 0.90; b = 0.1, 0.3, and 1.0; Process 3.
(From Roberts. 61 )
12.7 PROBABILITY DENSITY FUNCTIONS OF AMPLITUDES, EXTREME
VALUES IN RELATION WITH THE FUNCTIONAL POLYNOMIALS
12.7.1 NarrowBanded Case
As was mentioned in Section 11.1, when a nonlinear response z(t) was expanded by
the Voltera expansions (or functional expansions)
2(0
= X â– â€¢ â€¢ 8n(ri,V2 â– â– â– â– Tâ€ž) X(tTi)x(tT2)
. . . . x{tx n) dxi dr 2 . . â– â– dx n , (12.99)
the terms for n>2 can be considered as the modifying terms of the Taylor expansion of
this process around its linear term for n=\. If we take until n=2, and using small e, some
times the response z(t) is expressed as
m = J gid) x(tx) dx + e\ J g 2 (r 1 ,x 2 )x(tx 1 )x(tx 2 ) dx x dx 2 . (12.100)
The derivatives in terms of time is
i(0= J gi{x)x(tx)dx+2e\ J ^ 2 (Ti,T 2 )ic(rTiMrT 2 ) dx x dx 2 , (12.101)
â€” 00 â€”00
354
thinking of the symmetry character of g2(t\, tz)  gifti^d On the basis of this expres
sion T. Vinje 87 formulated a general method for getting the probability distribution
function of the extreme values, under the assumption that the response z{t) is narrow
banded and the input x{t) is Gaussian with variance cr. His method is also based on the
assumption that the probability distribution function of extreme values can be obtained by
the joint distribution function p(z,z) only.
Now, we start from the statistical moment generating function to get p{z\,zi)
[here z\ = z{f), zi = z{t) z], that is, from the double Fourier transform of p(zi,Z2),
0(0 1 ,0 2 ) = 4 ex P{'(01*l + 02Z2)}].
expO'0iZi + i02Zz)p{zi, zi)di\dz2. (12.102)
Then expanding exp {iQ\z\ + 1Q2Z2) into a series gives
exp (i0 lZ i + 102*2) = 1 + X TT^lA^. (12.103)
m,n=V
assuming m, n are positive integers whose sum is greater than zero.
Inserting Eq. 12.103 into Eq. 12.102 gives
Here
0(01,02) = 1 + X^fVc^irc^r. (12.104)
fi mn = E[z?,z% = J i%z\ pb^zddzxdz* (12.105)
From the Fourier inverse transform of Eq. 12.102,
00 00
p(z h z 2 ) = â€” â€” 2 I J 0(0i,02)exp{i(0izi + 2 22)}^0i^02. (12.106)
Putting Eq. 12.104 in Eq. 12.106
355
M J j Zn m  n 
â€” x â€” oe
(12.107)
This Eq. 12.107 shows that p(z h z 2 ) can be calculated if we know all the m,n\h order
moments of this probability function.
On the other hand, by the definition of the cumulant km* and the cumulant
generating function K(i6\,i02),
K(i6 1 je 2 ) = iog<p(e 1 ,e 2 ) = j^;(iei)(ie2)+^(iei)(ie2) 2 + . . .
= XTT(^l) m (^2) n . (12.108)
mini
77! ,71
Therefore
(pWi,6 2 ) = exp(*r(i0i,i0 2 )} = expj J^^iieiTWiT
I m,n m  n 
1! ^ m!w! 2! _ f"!Â«!
+ â€” { .... } 3 + ... . (12.109)
When we insert this $(6 uOj) into Eq. 12.106,
DC CO
pfri, *2) = 7A2 f f Â«p{*(# 1, i0 2 )  i(A 1Z1 + 02^)}^ id0 2
(2jt) J J
â€” ec â€”oo
(2jz) 2 J J H^m!Â«! J
_0C â€”00 *â– â– *
From Eqs. 12.104 and 12.109, )fc m ,â€ž and fi mjn are related as follows:
(12.110)
356
*02= i "02Aoi
k03=[M)3 3/*OL"02 + 2^01
^10 = /"10
hi =Piif*wPoi
hi = ^12 PicPca  2uoiMu + 2aio"6i
&20 = /*20 ~/%
*2i = fill  2^10^11 fioiM20 + luiduoi
^30=^303^10^20 + 2^
10
(order in e)
(0)
(1 + e 2 )
(e+e 3 )
(6)
(0)
(e+e 3 )
d+e 2 )
(0)
(6+e 3 )
(12.111)
Following these general formulations by Vinje, M. Hineno 67 calculated the proba
bility distribution function of the wave height, treating the wave process as nonlinear, as
was mentioned in Section 9.2.1, and will be summarized as follows.
From these relations with p, mn the order of i m ine's was obtained, which is a
smallness parameter that appeared in Eq. 12.100 as is listed in Eq. 12.111. Expanding
Eq. 12.110 around e = into a Taylor series and truncating at 0(e ) gives
pOi, ^2) =
W
exp
*2o(O)0l!*te(O)0i
1 + *io(O)(i0i) +  *3o(O)(z0i) 3 +  *d3(O)(i02)
 ^'i2(0)(^i)(/6> 2 ) 2
^^*Â® d&xdd^
(12.112)
where fc,y(0) = *y e=0 , *y(0) = â€” Â£<; e=0 
After manipulation,
357
1
(_2 _2
2k 2 o 2*02
hU kw
Jho
**> H3^.] + l_*S8_ ff3
y/^20 / 6 Â£ 2 o v^20 \ V&20 I 6 koi Jkm ' \ v*02
2 *02 v/^20 \ V ^20 / \ V&02
. (12.113)
Here Z? m (x) is a Hermite polynomial,
00
H n (x) = i f r^Oo + i /2 f) n df
ff oW=l
B 2 (x) = x 2 1
H 3 (x) = x 3 3x
H n+ i (x) = xH n (x)  nHâ€ž^i (x)
(12.114)
convention of (0) is omitted.
Under the assumption that z\{t) is narrow banded, the number of zerocrossings of
an output z(t) is equal to the number of maxima.
Therefore, the expected number of zup crossings per unit time is given as already
shown in Section 12.5, Eq. 12.44,
E[N + (2i)]= J z :
2I p(zi,z 2 )dz 2 .
(12.115)
358
Also, the expected number of mean level upcrossings is
(12.116)
The probability distribution function of a maxima then lies above Zj is
Substituting Eq. 12.113 into Eq. 12.115 and carrying on the integration gives
(12.117)
E[N + (z)] = â€” / ^ exp (
2n / Â£ 2 o \ 2Â£ 2 o
1+4^^!
*12
+  â€” 1Â± r=H 1
(12.118)
As already shown in Eq. 12.111, the cumulants are calculated from the m,nth
moment of response, and the m,nth moment can be expressed by using the frequency
functions and the input spectrum as some of the examples that were shown in Section
11.4. With the order of each term up to (e) taken into account, and with the spectrum
function, which is a real function, the cumulants can be written as follows:
359
Â£ 10 = e/z 10 =e s(co) G2(co,(o)d(o
â€” 00
00
â– /
â€” 00
00
=1
â€” 00
(
#12  â‚¬hi2 = â‚¬ (<Di(D2 + 2(02) S((0i)s(Q)2)
k 20 = h 2 o= I s(a>) \Gi((ofd(o
^02 = ^02 = (O S((0) \Gi((0)\ da>
â€” 00
00 00
,2n
x [Gi( co i)Gi(eo 2 )G 2 (Â£o i,(0 2 ) (12.119)
+ Gi(C0i)Gi(C02)G2(Â£0i,Ct>2)] dO)\d(l)2
00 00
^30 = e^30= 3e s(<ui)(,sa>2)
â€” 00 â€”00
x [G 1 (w 1 )G 1 ((o 2 )G2((Oi,(02)
+ G*i(o)i)G*i((02)G2((0 1,(02)] d(0ida)2,
where * indicates a complex conjugate,
s(q)) is the twosided spectrum of input x(t),
G\{a>) is the linear frequency response function
00
I
Gi(a>) = giirfc^dd,
360
and G2(co 1,0)2) is the quadratic response function
00 00
G 2 (co U (o 2 )= J { g2(n,r2)e n ^ h ^dr 1 dT 2 .
Through these manipulations, Hineno obtained his expression for the probability
distribution function as
/>(z) = exp 
2h 2 o
h 2 o 6h 20 \ h 20 I 2aiq2Â«20
(12.120)
The probability distribution density function p(z) and the expected \/n highest value
z\ u are obtained from the relations
p(z) = â€”P(z),
dz
(12.121)
J z p(z)dz
'â– \/n
Zl/â€ž
/ P(z)dz
00
I
z p(z)dz.
(12.122)
Zl/n
If we use Eq. 12.120 for P(z) and consider that in real calculations, the smallness
parameter c is absorbed in the computation of G 2 (a> i,o>2) and does not appear explicitly,
p(z) = exp 
z 2
2/j 2 o
^10 J ^30 h\2 1
h 5 + z
/Z20 2/l 2 2/220^02 ^20
] 1 / feig /l 30  /in \ /?30  _2 [ ^30 j
yh 2 o\h 20 2h 2 20 2h 20 ho 2 ) 2^ J " 6^
(12.123)
This is the final expression of the probability distribution density function of the maxima.
Using this expression, M. Hineno calculated the probability characteristics of the maxima
and minima of the waves that were treated as nonlinear with the quadratic response
function, as shown in Eq. 9.10 in Section 9.2.1,
361
G 2 {coi,(o 2 )=â€” (aj + coij
G 2 {a)i,(D 2 ) =â€”(col(
with the linear response assumed as G\(w) = 1 .
10 1
P^
10"V
10^ 
Fig. 12.13. Probability density function of the maxima of waves.
(From Hineno. 67 )
Figure 12.13 shows the probability distribution density function for maxima of the
waves compared with the experimental data by a certain research worker for model
waves with a wave spectrum that is almost of the PiersonMoskowitz type. In Fig. 12.13,
the + marks indicate the experimental data analyzed as nonlinear waves and the other
marks (A, O) are experimental data analyzed on the assumption of linear waves, the solid
line being the theoretical relations as linear for this model waves. Results computed by
Eq. 12.123 are shown by a dotted line, and the agreement with + signs is quite good,
especially in the larger amplitude range, which is important practically.
Figures 12.14, 12.15, and 12.16 illustrate the calculated results for nonlinear waves,
using a modified PiersonMoskowitz wave spectrum with //1/3 = 11.6 m, Toi = 16.1
sec as the input. From these figures, we can fmd the extent of nonlinearity of the waves
and also the effects of nonlinearity or the effect of the distortion of the wave forms on the
difference of the probability distribution function for maxima and minima.
362
Hineno 88 also calculated the relative motion of a semisubmersible in this kind of
nonlinear wave, ass umin g a linear response to the excitation by the waves.
1.0
o.s
O.G
0.4
0.2
0.0
H 1/3  11.6m
T<> 16.1 sec
â– \
MAXIMA
MINIMA "
LINEAR '

2
Fig. 12.14. Cumulative probability distribution function of wave amplitude.
(From Hineno. 67 )
H 1/3  11.6m
T a  16.1 sec
MAXIMA
MINIMA ~
LINEAR
Fig. 12.15. Probability density distribution function of wave amplitude.
(From Hineno. 67 )
12.72 WideBanded Case
J. F. Dalzell 56 (1984) extended this technique further. He did not assume narrow
handedness of the response and did not truncate the functional polynomials at n=2 but
continued to n=3. He thus formulated the technique for calculating the probability distri
bution function of extremes of the nonlinear responses to Gaussian inputs. Here the
characters of the nonlinear frequency response functions up to degree 3 are assumed to be
known from the analysis as discussed in Section 11.5. The nonlinear response Y(t) to
Gaussian input X(t) is expressed by
363
4.0
Fig. 12.16. Expected 1/n highest values of wave amplitude.
(From Hineno. 67 )
Y(t) = Jgi(Ti) x(tT{) dci
+ 82(Ti,T2) x(tn) x(tz 2 ) dXidX2
+ g3(Ti,T 2 ,r 3 ) x{tti) x(tr 2 ) x(tr 3 ) dr 1 dr 2 dx i . (12.124)
(Limits of integrals  Â» â€” i oo are omitted throughout this section.)
As was assumed in Section 11.5, here the kernels or the Â«th degree impulse
response functions g n (X\,X2 . . . r n ) are real, time invariant, completely symmetrical in
the variables gâ€ž(Ti,T 2 . . . tâ€ž) = gâ€ž(r 2 t 3 . . . tâ€ž,ti)= . . . for any rearrangement of the
variables r ; , and sufficiently smooth and integrable so that there exist nfold Fourier
transforms,
gâ€ž(Tl,t 2 . . . T ")=7^r I I â€¢ â€¢ â€¢ I G n(a>l,0>2 â– â– â€¢ 0)n)
exp i^ojyr,ptt)ida)2 . â€¢ . dcoâ€ž (12.125)
364
G n ((l)i,(D2 â– â– â– 0) n )= J ... I gâ€ž(Ti,T 2 , . â€¢  râ€ž)
exp
. 7=1
<fridr 2 . . . drâ€ž. (12.126)
The function Gâ€ž(a>i, a>2 . . . (oâ€ž) is the nth degree frequency response function and
is also symmetric in its arguments Gâ€ž(a>\,(02 . . . coâ€ž) = G n ((02,(0\, . . . (O n )= ... for
any rearrangement of (Oj because the impulse response functions are real.
G n ((O h Q>2, â– â– â– (0â€ž) = G* n (Q)i,0) 2 , â– â– â– (O n ) (12.127)
here the * denotes the complex conjugate.
In Dalzell's paper, 56 the spectrum 5^(0)) was defined a little differently from those
used by this author in Parts I, II, and in. He used 2n times our s(co) ,
S^co) = 2jcs((0), (12.128)
and also took the onesided spectrum
tfxxtM] = 2s(co) = S^w), forcu > 0. (12.128')
71
The initially assumed functional polynomial process was reformulated as the
response to a white noise excitation, and a new set of frequency response functions was
defined which contains both characteristics of the original frequency response to the
excitation and that of the excitation spectrum.
The twosided spectrum of white noise is
S w (a>) = 1, (12.129)
and the autocorrelation of white noise is a delta function
Rww(r) = â€” exp[i(OT]dco = 6(t). (12.130)
2jc j
We think of the spectrum filter L{w) that expresses the spectrum of input 5^(0))
with the white noise as
S^ico) = al \L(a))\ 2 (12.131)
365
UaÂ» = lS ^ )] " 2 . (12.131,
The linear, quadratic, and cubic frequency response functions connecting the white
noise W(t) with the output Y(f) will be
o x H\{a>) = o x L{o))G{(o)
olH 2 ((D h (D2) = a 1 x L{0} l )L{0} 1 )G 1 {(Oi,(O2) (12.132)
O x H 3 ((Di,(D 2 (D3) = olL((0 1 )L(Q) 2 )L((03)G3((Oi,(02,C03,).
Then Y(t) can be related to the white noise input W{t) as
Y(t) = o x \h 1 (x l )W(tx 1 )dx 1
+ o\ J J h 2 (? h r 2 )W(t  n)W(t  x 2 )dx 1 dx 2
+ o\\ \h 2l {x l ,x 2 ,X3)W{tx l )W{tx 2 )W{tr ?> )dx l dx 2 dr3, (12.133)
where
h\{x\) =â€” â€” Hi(o))exp[ia)Xi]da)
2ji j
h 2 (t u x 2 ) = â€”â€”j H 2 (coi,co 2 ) exp[z'(<yiri + (o 2 x 2 )]da)idw 2 (12.134)
hs(Ti,X 2 ,Tl)= 3 // 3 (<yi,G) 2 ,G)3)exp[/(a>iT 1 + a)2r2+CU3T3)]da>i^(U2rf(y;.
The derivatives of the output are then
Y(t) = a x \h' 1 (x 1 )W(tx 1 )dx 1
+ (j2 A\ fori, r 2 W(t  x x )W{t  x 2 )dx x dx 2
+ ol\  J hzixux^xzWitXxWitxJWitxiWxrfxrfxz, (12.135)
where
366
*iw  y
H\{a>) exp(i(OT\)da)
l_
h2(ri,T 2 )= 2 I I ^2(tt>1.0Â»2)Â«q>[/(ttÂ»lTl+fi>2T2)]^Wl^Â»2 (12.136)
^3(^1,^2,^3) = 3 I I H3((Ouo>2,o)3)exp[i(a)iTi+ 0)2*2 + Q>3 r 3)]dQ>ido>2d<o?,.
The terms //â€ž(<y) are defined as
H\(o)) = ia)Hi(co)
H2(0)\,0)2) = i(Q)i + 0)2)H2(0)i,Q)2) (12.137)
H ?,(0) 1,0)2,0)?,) = i(0)\ + 0)2 + 0)i)Hs{0) 1,0)2,0)3)
and
a x \h
ol\ h2(Xi,
r 2 )W(t  Ti)W(t  T 2 )dr 1 dT 2
k(Ti,T2,T3)W(tTi)W{tX2)W(tt3)dTidt2dr3, (12.138)
where
^1 ( r i) = â€” U\ (o))exp(io)Ti)do)
2ji j
^2(Tl,T 2 ) = â€”â€”J ^/2(^l J O> 2 )exp[j(C0iTi + 0)2T2)]d(DidO)2 (12.139)
^3(Tl,T 2 ,r 3 )=â€” â€” 3 //3((U 1 ,a)2,W 3 )exp[/(a>iTi+G)2T2+O' 3 T3)]rfcyidC02rf0>3,
and //â€ž are defined as
367
Hi(a)) = o) 2 Hi(o))
H 2 (o) 1,(02) = (o)i + a)2) 2 H 2 (co 1,0)2)
H 3 (0)i,0) 2, 0)3) = (0)1+0)2 + 0)3) 2 Hi(0) 1,0)2,0)3).
(12.140)
In addition, the characteristics of the products of white noise were fully utilized as
follows:
\
Wi(tTi)W(tT 2 ) â– . . W(t T N ) = for N odd,
W(tXi)W(tX2) = <5(ti  r 2 ) = 6 12,
W(t  Ti)W(t  T 2 )W(t  T?,)W(t  T 4 ) = 6 12634 + 613624 + <5 14^23,
W(t  n)W(t  z 2 )W(t  r 3 )W(t  u)W(t  x 5 )W(t  t 6 )
= 612634656 + 612635646 + 612636645
+ 613624656 + 613625646 + 613626645
+ 614623656 + 614625636 + 614626635
+ 615634626 + 615632646 + 615636624
+ 616634652 + 616635642 + 616632645,
) (12.141)
where (5^ = d(r,  xj).
Eq. 12.141 is the special case of Eqs. 11.68, 11.69, and 11.70 in Section 11.5.
If we do not assume narrow handedness of the output spectrum, the expected
number of maxima of the response Y greater than Y =  per unit time is approximated as
N.
00
! + = J J \Y\ p(Y,0,Y) dY dY.
I00
(12.142)
Similarly, the expected number of minima of Y less than Y =  per unit time is
approximated as
368
tff= I \Y\ p(Y,0,Y) dY dY. (12.143)
oc
Then, the expected number of maxima regardless of magnitude per unit time will be
oc o
Nl = [ \Y\ p(Y,0,Y) dY dY. (12.144)
â€” ce â€”so
."s imilar ly the expected number of minima per unit time will be
N a = \Y\ p(Y,Q,Y) dY dY. (12.145)
oo
Because maxima and m inim a are paired in the same record of response A^ = N& ,
from Eqs. 12.142 and 12.144, the probability that a maximum will be less than Â£ is ap
proximated
Prob[Maximum < Â£] = 1  I â€” %â– J . (12.146)
Similarly,
Prob[Minimum < ] = â€”I ). (12.147)
\ No* /
Then, the probability densities of maxima and minima are obtained by differentiat
ing Eqs. 12.146 and 12.147 with respect to , as
p + Â® = X f 171 pÂ£, 0,Y)dY (12.148)
N<x> J
369
00
p~(M) = Â£= f lii pd, o, f) dr. (12.149)
In the same way as we did for the joint probability density p(Y, Y) in Section 12.7.1,
relations between the joint moment generating function <f>{it, it, it ) and the joint cumulant
generating function K(it, it, it) were used to give
(it, it, it) =\
<f>(it, it, it) = p(Y, Y, Y) exp [itY + itY + itY] dY dY dY
jkm J' '
Here n ]km is the joint moment
fijkm = Y j Y k Y m =fff YJY k Y m p(Y,Y,Y) dY dY dY (12.151)
K(it, it, it) = log <f>(it, it, it) = X TTT7 <& & )* #' ) m  (12152)
Therefore
<p(it,ii,H') = exp pT(ir,if,ir)] (12.153)
where y, fc, and m are positive integers whose sum is greater than zero. Therefore, the
inverse transform of Eq. 12.150, from Eqs. 12.150 and 12.152, is
370
p(Y, Y,Y) = â€” iy exp f [iYt + iYt  iYt}) <p(it, ii, it ) dt, di, d'i
= â€”!j exp [ [iYt + &i  zYr'jl exp [K{it, it, it)] dt, di, d'i
= (2*) 3 JJ J
exp \iYtiYiiYi+K(it,ii,ii)\ dt,dt,dt
^â–  i !  exp \(k m Y) it+(koioY) it + (kooiY)it
(2jt)
t{*200? 2 + ko 2 tf 2 + W^* 2k m ti+ 2k m t'i + 2ko U ii'
(12.155)
+ Y ^ (ity' (iiftfr] dt,di,dt. (12.154)
^ ilklml J
jkm J
Here j, k, and m are now positive integers whose sum is greater than 2.
The relations between joint moment and joint cumulants are
Â£100=^100
*oio = ,"010
&001 =/*ooi
^200 = variance of Y
&020 = variance of Y
kooz  variance of Y.
Nondimensionalizing the variables as follows
z = (Yk l0 o)klg>
z = &hw)lt& (12.156)
2 = (r%)i)^5
and
371
5 â€” I A, 2 Q0
= t k l {l (12.157)
and
S = t *ot>2
jkm = WmHk j k k kâ„¢ ) ' (12  158)
J.K.m.{K 20Q K Q2Q KQQ2)
p(z,z,z) = â€”â€”j exp J izs  z'z i zz i j
â€”\s z + s 2 +s 2 + 2Xuoss'+ 2A m ss + 2Aons's\
+ X W^y (")* (") m ^ ds ds  (12.159)
y/fcm
Expanding the characteristics of the moments as shown in Eq. 12.159 into joint
moments and modifying them into joint cumulants, Dalzell derived the expression for
joint cumulants until the fourth degree from the modified frequency response character
istics H\(w), #2(0 1,02), S3(o>\Q)2fi>3), and functions of w by the same type of style as
Eq. 12.119 for the case of p{z, z). Since these manipulations require many transactions
involving the higher order terms in the expansions, he checked the order of magnitude of
the functions, suggested the order to truncate the approximations and, by laborious
manipulation utilizing Hermite polynomial, he obtained his approximation.
From these expressions, arbitrarily denoting the standardized maxima or minima of
response by v he finally derived the expression for p(v, 0,'i) using the cumulants up to
the fourth order and then the probability distribution function for maxima and minima by
p + (v) = i f Bp(y,0,z)dz
Ar 00 J
â€” 00
=  Z ^r [ zp(v,0,z)dz' (12.160)
iVoo J
372
p(
\z\ p(v,0,z)d'z
= U zp(y,0,z)dz.
Woo J
(12.161)
Finally he derived the expressions for his first and second approximations p(v),
pl(v) and P2(v), piiv) as follows. First, by neglecting the term higher (than 3) order
joint cumulants, the first approximation is,
pftv) =
(2.T) 1 / 2
exp
V
2?
+ (l_ e 2)l/2 v exp
vde 2 ) 1 / 2
(12.162)
Pl(v) =
Qxi 11
exp
2e 2
(le 2 ) 1/2 v exp
2
vde 2 ) 1 / 2
(12.163)
where 0(a) is the Gaussian cumulative distribution function and e is the spectrum band
width parameter.
Thee was introduced in Eq. 12.162 and Eq. 12.163 from the relations as follows that
comes from the characters of joint cumulants,
â€¢^â€¢101 =
&020
(*2<x>*002) 1/2
= (le 2 ) 1 / 2