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 Evaluation of variances associated...
 References






Group Title: Technical report – University of Florida. Coastal and Oceanographic Engineering Program ; 100
Title: Probability distributions of peaks and troughs for responses of nonlinear systems
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 Material Information
Title: Probability distributions of peaks and troughs for responses of nonlinear systems
Series Title: Technical report – University of Florida. Coastal and Oceanographic Engineering Program ; 100
Physical Description: Book
Creator: Ochi, Michel K.
Affiliation: University of Florida -- Gainesville -- College of Engineering -- Department of Civil and Coastal Engineering -- Coastal and Oceanographic Program
Publisher: Dept. of Coastal and Oceanographic Engineering, University of Florida
Publication Date: 1994
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Subject: Coastal Engineering
Waves   ( lcsh )
University of Florida.   ( lcsh )
Spatial Coverage: North America -- United States of America -- Florida
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Funding: This publication is being made available as part of the report series written by the faculty, staff, and students of the Coastal and Oceanographic Program of the Department of Civil and Coastal Engineering.
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Bibliographic ID: UF00075017
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida

Table of Contents
    Front Cover
        Front Cover
    Title Page
        Title Page
    Table of Contents
        Table of Contents
    Abstract
        Abstract
    Introduction
        Page 1
        Page 2
    Derivation of probability distribution of peaks
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
    Derivation of probability distribution of troughs
        Page 10
    Conclusions and Acknowledgements
        Page 11
    Evaluation of variances associated with peaks and troughs envelope process
        Page 12
        Page 13
        Page 14
        Page 15
    References
        Page 16
        Page 17
        Page 18
Full Text



UFL/COEL-TR/100


PROBABILITY DISTRIBUTIONS OF PEAKS
AND TROUGHS FOR RESPONSES OF
NONLINEAR SYSTEMS




by




M.K. Ochi


May 1994







Naval Facilities Engineering Service Center
Contract N47408-91-C-1204
(Phase III Study)




















PROBABILITY DISTRIBUTIONS OF PEAKS AND TROUGHS

FOR RESPONSES OF NONLINEAR SYSTEMS




by




M. K. Ochi


Naval Facilities Engineering Service Center


Contract N47408-91-C-1204


(Phase III Study)


May 1994















TABLE OF CONTENTS




Page
INTRODUCTION 1

DERIVATION OF PROBABILITY DISTRIBUTION OF PEAKS 3

DERIVATION OF PROBABILITY DISTRIBUTION OF TROUGHS 10

CONCLUSIONS 11

ACKNOWLEDGEMENTS 11

APPENDIX: EVALUATION OF VARIANCES ASSOCIATED WITH
PEAKS AND TROUGHS ENVELOPE PROCESS 12

REFERENCES 16


I













ABSTRACT



This paper deals with the development of probability density functions

applicable for peaks and troughs of the responses of a nonlinear system in

closed form. The derivation is based on the probability density function

representing a non-Gaussian random process developed by the author in 1994.

A comparison of the newly developed probability function and the histogram

constructed from a record indicating strong non-Gaussian characteristics shows

satisfactory agreement for both peaks and troughs of the random process.















INTRODUCTION


This study presents the analytical development of probability density

functions for peaks and troughs of the responses of a nonlinear system.


In developing the probability density function of the response of a

nonlinear system, in general, it is highly desirable that probability

distribution of peaks and troughs are presented separately. This is because

the statistical information of peaks (or troughs) only is necessary for the

design of many nonlinear systems. For example, for estimating the wave-

induced extreme force working on cables of a moored system, the magnitude of

tension forces (peaks) provides information vital for the design.


Only a limited number of studies has been carried out on the probability

distribution of peaks and troughs of a non-Gaussian random process. In these

studies, the probability density functions are derived numerically based on

the joint probability density function of displacement and velocity of the

response, f(y, y), assuming that the expected number of peaks is equal to

that of up-crossing a specified level. Ochi and Malakar (1984) applied the

Fokker-Planck equation to evaluate the nonlinear response of a tension-leg

platform in a sea and obtained the solution f(y, y) numerically, and therefrom

evaluated the probability density function of peaks. In applying the Fokker-

Planck equation, the input to the nonlinear system is a white noise spectrum.


If we assume that displacement y and velocity y are statistically

independent, then the probability distribution of peaks and troughs may be

obtained from the probability density function of y only. Kato et al. (1987,

1










1989) evaluated the probability distribution of peaks and troughs of the

responses on a moored ship in random seas based on the probability density

function f(y) obtained by the Kac-Siegert solution.


On the other hand, several studies have been made on the probability

distribution of peaks and troughs of non-Gaussian random waves observed in

finite water depth assuming that wave profiles follow the Stokes expansion to

the 2nd order exponents; Tayfun (1980), Arhan and Plaisted (1981), among

others. Although these distributions are given in closed form, the

distributions are developed by imposing a preliminary form on a random process

which appears to be inappropriate, in general.


As this short review of the literature reflects, the probability density

functions of peaks and troughs have not been developed in closed form, and

thereby the mathematical relationship between the probability density function

representing a non-Gaussian random process and that of peaks (or troughs)

cannot be clearly identified, and prediction formula for evaluating extreme

values cannot be obtained.


In the present study, the probability density functions of peaks and

troughs of the responses of a nonlinear system are analytically derived in

closed form based on the probability density function applicable for a non-

Gaussian random process developed by the author (Ochi and Ahn 1994). The

probability density function of peaks (and troughs) reduces to the Rayleigh

probability distribution for a Gaussian random process.










DERIVATION OF PROBABILITY DISTRIBUTION OF PEAKS


Prior to the derivation of a probability distribution, it may be well to

outline the approach considered in the present study.


The statistical properties of the peaks of a non-Gaussian random process

are significantly different from that of the troughs. Therefore, the

derivation of the probability distributions applicable to peaks and troughs

will be considered independently. It is assumed that the random process is

narrow-banded and hence there exists an envelope. However, we consider

envelopes for peaks and troughs separately. In other words, the statistical

properties of peaks are represented by a peak envelope process, C(t), while

those of troughs are represented by a trough envelope process, n(t). As an

example, Figure 1 shows a pictorial sketch of the peak envelope process ((t).

As seen, one side of the envelope covers peaks. The statistical properties of

peaks are evaluated for the amplitude of this envelope process. Therefore,

the variance for this envelope process should be that affiliated with the peak

side only of a non-Gaussian random process y(t). The variance, denoted by a,

can be evaluated by taking the conditional second moment of y(t), truncated at

y Q. That is, assuming that the peaks are on the positive side,




S y2 f(y) dy
a2

J f(y) dy
0


= { 1 (Ac.)2 +VTh (a.) (A//u,)4 } 3
2^ i/. J^ 7 2










where A, ay, 7 1.28 for the positive y

S- cumulative distribution function of standardized normal
distribution.


The details for evaluating u2 are given in Appendix.


The probability distribution applicable to deviations from the mean

value, y, of a non-Gaussian random process can be expressed as a function of a

single normal random variable and its squared quantity (Ochi and Ahn 1994).

That is,


Y = U + aU2 (2)


where "a" is a constant and U is a normal random variable with mean p* and

variance a2; all of these parameters can be evaluated from a knowledge of the

2nd and 3rd moments of a non-Gaussian record. Then, the functional

relationship given in Eq.(2) is inversely expressed as



1 -yaY 1 -,Y (3)
U (1-e ) = (1 e (3)
ya T,



Subtracting the mean value p. from the random variable U, we may write



1 -X1Y (4)
V =U-p. = -( l-e )-





For the derivation of the probability distribution applicable to peaks

of a non-Gaussian random process, let us consider a narrow-band random process

V(t) which is a Gaussian random process with zero mean and variance o. For

the Gaussian random process, we may write











V(t) = E (acos not + bnsin nwt)


a = V(t).cos nut
an==



bn=.2 IV(t).sin nw't


Here, the coefficients an and bn are normally distributed with zero mean and

variance a2.


Further, the assumption of a narrow-band random process with frequency

.o, V(t) can be written in terms of amplitude A(t) and phase E(t) as



V(t) = A(t) cos{ wt + s(t)}. (6)


We may write Eq.(6) as


where


Vs(t) = A(t) sinE(t).



Since V. and V, are statistically independent normal variates for a given

time, the joint probability density function of Vc and Vs is given by



f(vcvs) exp 2 2)
2ir1 2u1


V(t) = Vo(t) cos ,t -V,(t) sinUo(t)


V,(t) =A(t) cos (t)









On the other hand, from Eq.(4), we may write the components Vc and V, as

follows:


Ve(t) = {(1-A.) e }
(10)


V,(t) = {(1 .)



By applying the change

probability density function

(10) as


-e- c }.



of random variables technique, the joint

of YC and YS can be obtained from Eqs.(9) and


1(1 Al'z.)2 1 Alp..( yo -A1ys
f(y.,y.) 2 exp A ( + y) + __ e + e
12r1 11 l


- 1 (e-2AY + e-2ls) },
2\i 1 1


-m < yc, < .


For the random process y(t), we may write


Y = ( cos r
(12)
Ys = E sin r



where ( amplitude (peak), r phase.


From Eqs.(ll) and (12), the joint probability density function of ( and

r becomes,


(11)













f(, 7) = exp -{ A1 (cosr + sinr)


1 AI.* (e-A1cosr
+ 2 (e


+-A1Csinr)
+ e )


-2 A cos


- (e
2 2
2-i(-


+-2AZ1sinr)
+ e ) ,T


O < O < r 5 2 r .


(13)


The probability density function of peaks can now be obtained as the

marginal density function of Eq.(13). In order to carry out the integration,

the exponential part of the 3rd term of Eq.(13) is expanded in a series of

A1C. That is,


-A1Ccos
e


-A\ sinr 1 + } +
+ e s = 2 + (A()2 A\I + (A)3 (sinr + cosr)

(14)
+ (AC)3 (sin3r cos3r) +


As a first order approximation, we may take terms up to (AC)2. This

yields


-AM1cosr -A1Csinr
e + e


= 2 A1(sinr + cos (Ar)2


= 2 52 A\. cos(r ) + (A )2


(15)









Similarly, by expanding the exponential part of the 4th term of Eq(13)

and by taking terms up to (AC)2, we have


-2 X cosT


-2X1 sinr


(16)


=2 2V2 A\1 cos(r 7) + 2 (A)2.


From Eqs.(15) and (16), the joint probability density function given

Eq(13) can be written as


(17)


f(C r) = L exp + 1, (A C. cos -
2a > *L 2x a-1 I+ OS
TI-r-r' a 2ar [T


where L is a normalization factor which is introduced because of the

assumptions involved in Eqs.(15) and (16). The normalization factor will be

evaluated later.


By integrating Eq.(17) with respect to r, the probability density

function of C becomes,




f()=L exp I- + 2 2 1 O < 18)<
al a, 21 I (18)


where, Io is the modified Bessel function of the 1st kind of order zero.


The normalization factor L can be evaluated by equating the integration

of the probability density function over the sample space (0, m) to be unity.

It is obtained as










S{ (Ala A.)2 } (19)
L = (1 + AXi.) exp IA-^ 2-
a, (1 + Xl/',)al


Thus, the probability density function of peaks, f(), can be derived as





2 2
f (A1 g.)2 1 + A1. )2
f(+)- e + (i A.)2 2a2 io- ) }


0 < (20)



In case of a Gaussian random process, p* as well as Ai are both zero,

and the variance a2 reduces to a2, the variance of the Gaussian random

process. Hence, Eq.(20) becomes



202 (21)
f(0) =- e



which is the Rayleigh probability density function applicable for amplitudes

of a narrow-band Gaussian random process.


Figure 2 shows a comparison of the probability density function

applicable for the peaks of a non-Gaussian random process derived in Eq.(20)

and the histogram constructed from wave data obtained in water of finite water

depth indicating typical features of non-Gaussian random process. Good

agreement can be seen between the histogram and the probability density

function.










DERIVATION OF PROBABILITY DISTRIBUTION OF TROUGHS


The probability density function applicable to troughs of a non-Gaussian

random process, denoted by f(n), is essentially the same as Eq.(20). However,

the parameters A, and variance o2 in Eq.(20) should be substituted for those

appropriate for troughs. That is,


f (A2a .)2 +1 + A2~

f(77) = Az\1)f2 e (1 2 +A- -A ) }


(22)



where parameter A2 ay with 7 3.00. The variance a2 may be evaluated by

the conditional second moment of a non-Gaussian random process given in

Eq.(l), with integration over the negative domain. Unfortunately, however,

the approximation used in evaluating a cannot be applied for y < 0. Hence,

a2 may be evaluated by the following formula using the variance of the non-

Gaussian record and the variance a2 evaluated by Eq. ().



2 y (23)
a2



where 2 V y is the second moment of the non-Gaussian process. The details
n
for evaluating a2 is also given in Appendix.


Figure 3 shows a comparison of Eq.(22) and the histogram constructed

from the data. Although a slight shift in the peak frequency can be seen, the

overall agreement between the histogram and probability density function is

satisfactory.










CONCLUSIONS


Probability density functions applicable for peaks and troughs of the

response of a nonlinear system are derived based on the probability density

function representing a non-Gaussian random process developed by the author in

1994. In the development, peak and trough envelope processes are considered

separately, and the statistical properties of peaks and troughs are evaluated

for the amplitude of each narrow-band envelope process. The variance of this

envelope process is affiliated with the peak side (or trough side) only of the

non-Gaussian random process. The derived probability density functions reduce

to the Rayleigh probability distribution for a Gaussian random process. A

comparison of the newly developed probability function and the histogram

constructed from a record indicating strong non-Gaussian characteristics shows

satisfactory agreement for both peaks and troughs of the random process.




ACKNOWLEDGEMENTS



This study was carried out as Phase III of the project "Probability

Functions for Maxima/Minima Excursions of Nonlinear Systems" sponsored by the

Naval Facilities Engineering Service Center through contract N 47408-91-C-1204

to University of Florida. The author would like to express his appreciation

to Mr. Paul Palo for his valuable discussions received during the course of

this project. The author is also grateful to Ms. Laura Dickinson for typing

the manuscript.











APPENDIX:


EVALUATION OF VARIANCES ASSOCIATED WITH
PEAK AND TROUGH ENVELOPE PROCESSES


The peak envelope process ((t) of a non-Gaussian random process is shown

in Figure 1. The variance applicable for e(t), denoted by a2, can be

evaluated by the conditional second moment given in Eq.(1). That is,


f y2 f(y) dy
2 y) d

f f(y) dy
"0


(A.1)


where f(y) is

process. The


the probability density function of a non-Gaussian random

density function f(y) is derived as (Ochi and Ahn, 1994)


1
- (1 ayp, e-avy)2 a. y
2(ayj.)


1
f(y) = -1.




Parameters a, p* and

and

7 1.28 for

{ 3.00 for


(A.2)


e


a2 are evaluated from the 2nd and 3rd moments of data,



y 2 0 (defined here as the peak-side of record)

y < 0 (defined here as the trough-side of record).


In order to evaluate the denominator of Eq.(A.1), let us write \A ay

where 7 = 1.28, and let us define e-j1Y =u. Then, integration of the

denominator can be reduced to that associated with the normal probability

distribution. That is,


I










f(y) dy = exp( I 1)du
(A.3)

= (A./Ia.)- I(-.\(,a.)).



where D = cumulative distribution function of the standardized normal
distribution.


Since the second term of Eq.(A.3) is nearly equal to zero, in practice,

we may approximately write

0O
o f(y) dy = D(A./7.). (A.4)



Note that for a Gaussian random process p* 0 and thereby D(p,/o,.)

becomes 1/2.

For integration of the numerator, let us write again e"Iy = u. Then

we have


a1
fy2 f(y) dy 1J (n u)2 exp (- 21 {u-(l- ) }2)du (A.5)



(AXi*) in the above equation is a dimensionless value and is 0.1 at most, in

general; hence, we may neglect it for the evaluation of the second moment.

Since y in Eq.(A.5) is positive value, we have ju 11 < 1. Therefore, we may

expand (n u)2 in the following series:



(in u)2 = (u-1) (u-1)2 + (u -1)3- 3 (u-1)4 + .


= (u-1)2 (u-1)3 + (u-1)4 (-1)5 + (A.6)









By taking the first three terms of Eq.(A.6), Eq.(A.5) may be written as

follows:



1 u-1 )2

y2 f(y) dy {(u-1)2 (u-1)3 + (u-1)4 } e du



A1



It is noted that the series converges fast, therefore the first three

terms appear to yield the second moment with sufficient accuracy. From

Eqs.(A.4) and (A.7), we have the variance a2 applicable for the peak envelope

process as




a02 1 (Xla*1)2 + /2-7(A.)3 + ( a.) (A.8)





The variance, a2 applicable for the trough envelope process n(t) may

be evaluated by the equation similar to Eq.(l). That is,


0
f y2 f(y) dy
a2 = (A.9)
2 0
f f(y) dy



Since Eq.(A.9) deals with negative y-values, the approximation given in

Eq.(A.6) cannot be applied in this case. The variance au, however, can be

evaluated through the following procedure:










The second moment of a non-Gaussian random process can be written as



y2 f(y) dy = y2 f(y) dy + y2 f(y) dy


0 0
= i f(y) dy + a f (y) dy


2 2 (A.10)
= a2 -(-p,/a,.) + al (p,/a.) (A.10)


Since the second moment (the left side of the equation) is known from
22
data, and since ,*, ao, and a, are also known, the variance applicable to the

trough envelope process r(t) can be obtained by




2 (A.11)
a2











REFERENCES


Arhan, M.K. and Plaisted, R.O. (1981); "Nonlinear Deformation of Sea-Wave
Profiles in Intermediate and Shallow Water", Oceanol. Acta, Vol.2, pp.107-115.

Kato, S., Ando, S. and Kinoshita, T. (1987); "On the Statistical Theory of
Total Second-Order Responses of Moored Floating Structures", Proc. Offshore
Tech. Vol.4, OTC 5579, pp.243-257.

Kato, S., Kinoshita, T. and Takase, S. (1990); "Statistical Theory of Total
Second Order Responses of Moored Vessels in Ramdom Seas", J. Applied Ocean
Res., Vol.12, No.l, pp.2-13.

Ochi, M.K. and Malakar, S.B. (1984); "Nonlinear Response of Ocean Platforms
with Single Degree of Freedom in A Seaway", Tech. Rep. UKL/COEL/TR/050,
University of Florida.

Ochi, M.K. and Ahn, K. (1994); "Probability Distribution Applicable to Non-
Gaussian Random Process", Journal of Prob. Eng. Mechanics (in printing).

Tayfun, M.A. (1980); "Narrow-Band Nonlinear Sea Waves", Joural Geophy. Res.,
Vol.85, No.C3, pp.1548-1552.

























~----


-'S


Figure 1 Definition of peak envelope process


0 0.4 0.8 1.2 1.6 2.0
PEAK IN METERS


Figure 2 Comparison of probability density function
applicable for peaks of non-Gaussian random
process and histogram constructed from data




















1.1




z
S1.;

z
LU
w

0.
>-

co

O
cr
0
C-


6





2


8


0 0.2 0.4 0.6 0.8 1.0 1.2

TROUGH IN METERS


Figure 3 Comparison of probability density function
applicable for troughs of non-Gaussian
random process and histogram constructed
from data




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