Energy characteristics of stationary random oscillations and their use in statistical linearization

RCH I EF

Technische Hogeschool

International Symposium on the Dynamics

°eIft

Marine Vehicles and Structures in Waves

r 'I ¶%7

PAPER 41

ENERGY CHARACTERISTICS OP STATIONARY RANDOM

OSCILLATIONS AND THEE USE. IN STATISTICAL

ENIA1IibIotheek van de

LINEARIZATION

Ondeafdeing der Scheepsbouwku.nde

TechnischeHogeschool, Deft

DOCUMENTATIE

I: K

_{8 t 12} .. - .

DATUM:

_..

_{A.V. Gerassimov*}

Summary

The paper deals with a rather broad class of stationary random oscillations of linear and nonlinear oscillators. It

reveals that the probability densities of displacement and velocity of oscillations are functions of the instantaneouá values of the oscillator poteütial and kinetic

_{ener, while}

the time-mean values of the same serve as parameters of

distribution. _{At the same, time it appeax'a that for nonlinear}

oscillations the varjance of displacament is not a parameter of distribution. _{In such a form the laws of distribution are} identical

for

the whole range of linear and nonlinear

oscillations considered, which makes it

possible

to formulate

the concept of ener-equivalent random oscillations and to

show the consequences of it. Por the nonlinear oscillator the distribution laws of oscillations are found to be dependent on

their enérr levels. _{In particular the concept Of random} .

*

Krylov Ship Research

Institute

M-158, Leningrad, USSR

amplitudes so understood is shown to be the parameter of

distribution. For the linear oscillator the random amplitudes are distributed according to the

Rayleigh law.

ParameterS are

determined for a linear oscillator which is energy-equivalent to the given nonlinear oscillator in certain conditions of excitation. Hereinafter the term "energy-equivalent" is used to denote the equivalence of random' oscillations having equal energy levele. This linear oscillator is used as a model for calculating the energy level of the nonlinear oscillator, after which the distributions of oscillations of the same can be calculated with due regard for the features inherent in.

them. To make a comparison between this energy-equivalent

method of statistical linearization and the so-calle4 equivalent statistical linearization method, two examples are considered0

Introduction

The evaluation of the effeOt of nonlinear factors on the ahip' s irregular motions and the accompanying phenomena attracts

more and' more attention. A comprehen8ive review of the group of problems in question was made by Grim at the 12 ITTC

[lJ.*

Among these is the problem of random Oscillations of a

rneôhanical system with a nonlinear resistance and restoring

force. In fact, this kind of. oscillations are characteristic,

to a. greater or lesser extent, for the rolling motion of a ship in seas.

*

Numbers in, brackets designate RefOrences at the end of the

been given due attention so.a'. Meanwhile, the versatility of such a concept as energy of an Oscillating system makes it possible to define the characteristic features of the energy-equivalent nonlinear and linear oscillations of the system. Owing to

this, one can

specify such a linear model of the non-linear system the oscillations of which (model), with the

disturbance

given, have

the same energy level as those of the nonlinear system. This level can easily be calculated on the basis of the spectral theory. On the other hand, the laws of distributions of nonlinear oscillations may be connected with their energy levels. This fact makes it possible to 4evelOp a method for the calculation of characteristics of random non-linear oscillations by which the energy, with the disturbance given, is calculated for the eñergy-equvalent linear system, while the distribution laws of nonlinear oscillations are

determined according to the known level of energy, taking into account

the

features

which

are just inherent in

them

rather

than

in their linear model0 This method will be termed as the energy-equivalent method of àtatistical

linearization.

Compared:

with other known methoda of statistical linearization, this one offers the advantage of being more simple and accurate in

respect of evaluating the nonlinear oscillations.

The purpose of this paper is to substantiate the above conceptiOn.

ENERGY CHARACTERISTICS OF THE RANDOM

STATIONARY OSCILIATIONS

l. _{Let us consider the random stationary oscillations Of}

a second order dynamic system (viz., an oscillator)

I

U (xl =

_(1.1)

Where the viscous resistance force

4C)

_and

_{the restoring}

(quasi-elastic) force

U (.X)

_{devidOd by the mass unit are} single-valued odd functions. _{For shortness Sake, when using} such categorie8 as force,

_{work, ener, etc., the}

_words

"devided by the mass unit" will be omitted as a rule or replaced by the word "specific".

The random disturbance _{is supposed to be a}

stationary centered Gaussian process with the spectral density given in the range of positive values Of circular

frequency Ct.' ;

hereinafter the

word "circular" will

al8o

be omitted. _{Besides, the disturbance is supposed to be an ergodic}

process, which means that the vibrations of the oscillator are

also ergodic.

Let us denote the potential energy of the oscillator as follows

Uo)

=0.

(1.2)

Due to the fact that the random vibrations of the

oscillator X(t) are

_{symmetrical, it is apriori known}

that

these vibrations have _{zero mathematical expectation and} symmetrical distribution of both displacement and velocity.

!velocity

condition f atochastically Independent displacement and

velOcity of the sSmO. Hence, for the .proCesaea in question the joint probability density of displacement and velocity is

a

product of one-dimensional densities of probability (v-.ia)

The same. considerations of symmetry result in nonrandom

relations

_L21

for the class of processes involved, Which are as follows:

11 [x.ucx)j

-

=

M

[ZJ

(1.4)

where

Ii [

J

denotes the averaging over time within one realization of the process. The class of systems (1.1) also includes the linear system

+ ix=

(r),

(1.3)

(.1.6)

the random oscillations of which have received the most careful:

study. So, let us first find

the

basic energy relations for

the

linear oscillations and then carry out generalizatiOn to the case of nonlinear oscillations.

The one-dimensional and-the

joint

probability densities of displacement and velocity of linear ocillationS can. be

represented in the homogeneous form .

v)

_2:A

Where

_pfrlL'l-r21

Mf

1

and i'ta time-mean value

+

p21?

.z Jf"

is the velocity variance of oscllationa, denotes the averaging

over

_{an ensemble of}

realizations,

fr/Djo

v-is tie: mean frequency of the process defined by the number of displacement up-crosses. _{These homogeneous representations are} based on the assumption that the random process is a

two-dimensional function of the common argument, i.e.

_time,

There-f ore, equal weights are attributed to the values oThere-f and

2tT't)

at any moments of time and the mean-weighted values obtained with the use of the weight functions

(i.7)-(l.9,)

_are time-mean

values.

The

instantaneous

_{value of kinetic ener of the linear} oscillator

(1.9).

(1.10)

4 ()

r

It is essential that the oscillator kinetic energy is

unequivocally determined

only

by the instantaneous value of its velocity for both linear and nonlinear oscillators.

This

is not the case for the potential

energy

of oscillator; its

instantaneous

_{value is determined not only by the}

netantaneous value of displacement but also by the

form

of

force

function U(X). It

is also

important

to note that

disturbance influences the oscillator velocity,

_and

its displace-ment is considered to be a kind of response to the velocity,

the

magnitude

Of this response being determined by the transformation of kinetic energy into potential energy.

Let us

term

(1.12.)

"the reduced potential energy" of the linear oscillations and

uc'x)=

2:t'

(1.13)

"the reduced force.

function", respectively.

The dynamic state of the vibrating oscillator at

_each

moment of time is determined by the totality of its displacement and velocity. _{It follows from (1.9) that the states of the}

linear oscillator in which its total reduced energy

£

*(

2t)

=

,T '"x)

Q(v)

remains the

same

are

considered

to be equally probable.

The time-mean

value

of the total

reduced

energy of

the oscillator

V

But,

as it

follows directly

_{from (1.12),}

_so

Thus the velocity variance of oscillations determines the time-mean values nOt

only

for the kinetic energy but also for the

Let us turn flow to the

expression (1.4).

_Setting

_aside

the term _{for the time being,. we can say that this} formula describes only the potential properties of the

oscillator:

here

_{the velocity variance of oscillations}

is a measure of correlation between the _{displacement of the} oscillator and its acceleration due to this displacement _(i.e. the restoring fOrce). _{The correlatiOn moment between the}

displacement of the oscillator and the di8turbance

determines the time-mean value of the work consumed for

synchronizing by disturbance. _{The random disturbance, which}

i8 an oscillation by itself, must produce some work in

synchronism with

the

_{vibrations of the oscillator despite the} tendency or the latter to vibrate with natural frequency.

Using the spectral theory of the random processes, let write the spectral expansion of the correlation moment for the linear o8cillator between its displacement and _disturbance:

argument for the complex transfer function of the oácillátor, respectively. The product

J

(d)j

changes the sign at

=

. Consequently, in certain instances the spectral distribution of disturbance may result in random stationary oscillations

_where

fk1x.qJ=

_0,

_(1.14)

I

Ft

(d)1

E ()

3()

synchronizing. the oscillationS is equal to zero.

By analoj

with an ordinary resonance, the above condition can he called a "stochastic resonance". It is apriori obvious that the

condition described in (1.14) is applicable to both the linear and nonlinear oscillators excited by 'White _noise"E2].

In all cases where condition (1.14) is satisfied, we shall deal with a noncorrelated disturbance, in the opposite case with a correlated one.

Let us denote the mean frequency of the process for a

noncorrelated disturbance by W. For the linear oscillator

= fl.

lxi

this ease, as is seen from (1.4),

-= u

(1.15)

i.e. the time-mean value of the kinetic ener of random

oscillatIons is equal to the time-mean value of the

potential

ener of the Oscillator.

Hence

f[I11fx

tCx)]_

Mt(xJ}

(1.16)

(l17)

The.latter expression makes it possible to represent the reduced

potential energy of the oscillator in the following form:

U?x) --

CJ(x),

(1.18)

(l.19

Accordingly, the reduced force function of the oscillator has the form

U

UCx).

(1.20.)

Therefore, in carrying out the probability analysis, there, is reason to believe that the correlated disturbance changeC with time only the scale of the oscillator potential function, without changing its dependence _{on argument}

_{7C. A}

_similar peculiarity is Observed for _{regular forced. oscillations.}

Now let us write the _{probability densities (1.7)-.(l.9)}

:ifl

the fOrm

L

(C'x))

=

(ir

&t/z)

ep

f

Uz)

7

(2.].)

?.tr(G.=: (s)

c(E)=

7g*/)4p[

E(x,zi')

(2.2)

(2.3)

Expressions (2.1) through (2.3) do not involve features which are specifically characteristic for a linear oscillator, while such. cathegories as potential, kinetic and total energy of a vibrating oscillator are similar for both linear and nonlinear cases. _{These expressionø are in the final analysis} based on the moat general law of mechanics, viz, the law of

coervatjon of mechanical

.

ener,

The above considerátj

(2.l)-(2,3) are common for the whoj.e class of the oscillators in question.

The suggested hypothesis is rigorously proved for the

nonlinear oscillator with a noncorre3.ated disturbance. _{In this}

case _,

17(x)

_,

and expression (2,3) is transformed as follows:

(::ç

it) =

+

--

(2.4)

The similar problem where the noncorrelated disturbanóe is supposed to be a "white noise" is rigorously solved by using the

Fokker

-

Plank Ko]mogorov _{equation { 2J, and the result} obtained is similar to that

_shown

_{in (2.4).}

For nonlinear

oscillators

_{with Correlated disturbance, no} common solutions are kiown, therefore in theSe cases .t is

impossible to giYe a .rgorouaiy theoretical cOnfirmation of the above hypothesis. _{However, sOme facts and analogies give}

evidence in favour of the hypotheaja in the case of correlated disturbance too. _{In particular, let us consider the following.} If the nonlinear oscillator is exposed to the action of

harmonic disturbance at a frequency of W , which induces.

regular oscillations with _{amplitude Z, the Work Of} synchronizing the disturbance is proportional to

where CV(et) _{is the natural frequency of the oscillator for}

amplitude a

_{Here we have a complete similarity with}

(1.16) as

a4)JiB

_{proportional to the average potential}

the

properties of the oscillator and therefore they are invariant

to the lawo resistance to oscillations.

The dissipation properties of the oscillator are taken into

account only

_{by relation (1.5) which implies that the} average power dissipated by the oscillator is equal to the

average power gained by the oscillator due to disturbance

Given the disturbance, its realization is _{subdivided into} component which is collinear with the displacement

cJ,x(tJ-and the _{-component which is orthogonal to the same and} always in-phase with the velocity of oScillations. This

subdivision is determined by a combination of conservative and dissipation Properties of the oscillator and in the general

case changes with the variation of both the level of disturbance and its spectral structure. _{Only when the oscillator Is excited} by a "white noise", _c

(i-)

0

,

and hence HEx]

M. [±]

=

Without due regard for the above interrelation it is impossible to evaluate the energy level of the oscillator vibrations at given disturbance, which at present is practicable _{only in} respect of the linear oscillator _{or 'the nonlinear oscillator}

excited by a. "white noise".

However this is not at variance with the foregoing. In

this case consideration is given to oscillations whose kinetic energy mean value is determined by equation (1.5). _{In this} expression 'the conservative properties of the oscillator

manifest themselves oiy indirectly through

,t)

_since

transformation of the oscillator kinetic energy into potential energy and vice versa (which

ith

described probabilistically by functions (2.l)-(2.3)) it is regulated by only the

quasi-conservative properties of the oscillator (or by. quasi-conservative ones in the case of a noncorrelated disturbance). In particular, if the given law of resistance (e7(1Y) is. substituted by the law

provided that

M. {treLrJ

then for the oscillator with a linear restoring force in the case of an arbitrary spectrum of disturbance, or with a non-linear restoring force when excited by "white noae", it can be rigorously proved that the variance of velocity and the

correlation moment between the, oscillator displacement and

disturbance, will not change. It follows from these two

conditions that the laws of distribution of oscillations

(2.1)-(2.3). are invariable (in; this. particular problem).

Thus, the distribution laws for the class of random oscillations under review are invariant under'the viscous

resistance law of a dynamic system provided that the mean value of the power dissipated in the system is the same for various laws: of resistance and the given disturbance. In other words, if th dissipative force and the disturbance component

are simultaneously omitted from consideration, it will not influence the probability characteristics of stationary

oscillations..

The second genCral conclusion consists in that for the class of systems and disturbances in question the distribution law of the velocity of oscillations is the Gaussian law of

indivi4ual features of the systems0 Physically this concluSion is accounted for by the constancy of the mass of the system

(the added mass of the latter also inóluded for the case when the body oscillates in liquid) and also by the fact that the instantaneous values of its kinetic ener are independent of the past history of the movement.

j. Substituting the variable

we can write the distributiOn8 (2.1), (2.2) in the following

way :

_f

)

U?)

(3.2)

.'

J

It follows from these expressions that random oscillations of the class in question have similar distributions of

displacement an4 ve:ocity as functions of instantaneous values

of the reduced potential and kinetic ene, respectively. This

conclusion is drawn for the arbitrary law of viscous resistance and for the weak limitations in respect of the type of the

nonlinear restoring force. These limitations consist in that the dynamic system is in fact an oscillating system and its zero

position XOis stable equilibrium (the indifferent

equilibrium is also admissible). The averge circular frequency of the 'procesS is the scale of distribution of displacement.

Tr*

In the general case the expression 4'r 'J

_/

e'- _{is the}

parameter of the distribution of displacement;. this eXpression can be interpreted as the square of the double 'amplitudes of regular linear bscillatiOns with frequency W , whose,

mean values of the kinetic and potential energy are equal to those inherent in irregular oscillatiOns. The magnitude of this double amplitude will be. called the energy-equivalent double amplitude of irregular oacllatiOflS. For the linear system

this i. However, for thö system with a

nonlinear restoring force there is nO such direct proportiouali

between

_E

(i.e.

the mean potential energy of the system)

and. the displacement variance , ao in the general case

is neither the parameter of distribution, nor the energy criterion of random Oscillations in contrast to the velocity variance

which

is always a cOmbination of both.

The adequacy of quite a broad c).aaa of random oscillations (of the nonlinear oscillator) In respect of energy makes it possible to form rather a general idea of the ener..equiValeflt

processes. . . .

Irrespective of .the type of non1ineartY of the restoring

force (within the specified general limitations) the random stationary oscillations Of the class in question

whose

mean values of power

i4JV'(vfl

dissipated in the system and frequency of the process are equal (the level of

disturbance given) will also have equal both the mean values of the specific (per unit of mass) potential and kinetic energy and the energy-equivalent amplitudes of oscillations; the foregoing holds true, in particular, for the comparison of nonlinear an4

linear systemS. ., . .

Energy equivalence of processes may be unconnected with the disturbance level and dissipation in the oscillator. It is

eat importance. If two processes of stationary random oscillations for the class under consideration have equal velocity variances and mean frequencies of oscilia.tons then

apriori both their time-mean values of the specific kinetic and potential energy and the energy-equivalent amplitudes of

oscillations are equal. too. Here the mean frequencies may be replaced with the energy-equivalent amplitudes of oscillations

and vice versa.

Now let the two oscillators, one lith nonlinear and the other with linear restoring force have equal laws of resistance0

Then, with the variancesof velocity

4

being equal, the

dissipation of energy in both the oscillators will be the same.

Let us deteine the value of the linear restoring force for

which the oscillations óompared are energy-equivalent. Prom Eqs. (3.1) and (1.18) it follows that

C7'x) =

(3.4)

and hence the distribution (362) can be transformed to the following form:

(u/z)

.Jq7r7Z)

frZ.-X,. we have Hence

c

(r/d;rJ:)

LA/)

But for the oscillator with a linear restoring force

C'?x)

-

I

,stx ;

X)

U-This. should be a Gaussian distribution, which is possible

(3.5)

if

tL

ai.

On the other

Zr =

&z U/02.

side, we shall have and hence

E '

(3.7)

(3.8)

viz, the frequencies

V0

_and C') _{should correspond to one}

and the Barns value of the ener-equivalent _amplitudes.

Thus 'under the above conditions

_{the ener-equivalent}

linear system should have a restoring force equal to

cx) = C:O

X, where

_{is the mean frequency of}

oscillations in the given nonlinear system in the case of a' noncorrelated disturbance with the _{same value of the energy} equivalent amplitude of oscillations0

In the distributions (3.2), (3.5) X is an argument, sà the averaging with the use of these distributions gives the displacement-mean values. _{Thus, at the same time one may come} to a conclusion that with the given _{value 'of the}

energy-equivalent amplitude of oscillations the displacement-mean value. of.the potential energy of the oscillating system is independent of the spectral structure of disturbance. _{The correlation}

between the displacement of the system and the disturbance will change only the time-mean value of the potential energy. In other words, the dieturbance does not change the potential function of the system as it is, influencing the velocity of the process alone, i.e. the distribution of displacement with

time. _{It is precisely this fact that makes it possible to}

represent the potential energy of random

oscillations 'of

_the

alone in the form

z)

The question

of. whether Eq.(1.18) is _{satisfied for each instantaneous value} of X remains _open.

However, from the point of view of ener this question is not of prime importance since the condition described in (1.19) makes it possible to assert that the difference

L1*(t.x)

(2/,)

_TJ(x)'

is a function having a slight deviation from zero, with

_a

_{zero mathematical}

expectation. Hence, the two-dimensional _{distribution of the} state of oscillating system as obtained with the use of

approximation (1.18) will only slightly differ from the true

distribution,

Mention

_{should also be made of a wellkriown feature}

characteristic for the nonhinaer _{oscillations in caée of a} correlated disturbance,

_which

_{consists in "jumping" from one} amplitude to another at a given frequency, that can be

interpreted as a discontinuity of amplitude, _{However, the} discontinuity of amplitude in this case is taking place as long as we analyse the amplitude of stationary regular _{oscillations.} as being a parameter of the _{process. In the process of}

oscillations itself this "jump" is a transient process which

_i8

oscillatory too0 _{The phase trajectory of this transient}

process, remaining continuous, _{passes from one limiting} stationary cycle to another, _{any point of this trajectory} characterizing one of the possible states of the system with a particular value of the total _{energy. There iè not much}

difference between the whole phase trajectory of the realization

of random

Oscillations

in case of a correlated

disturbance and that part or parts of the same which correspond to relatively

quick (but far from being instantaneous) changes in the characteristics of cycles within the zones where "jwnps't of the amplitude are possible for regular induced oscillations. Consequently, this distinguishing feature. of nonlinear

o8cillations is taken into account by the energy conception presented above.

However, the simplicity of solution is due to the fact that no consideration was given to the evaluation of the kinetic energy of an. oscillating system for the given

disturbance. _{The possibility of existence of several modes of} oscillation at certain frequencies of disturbance makes the complicated problem of evaluating the kinetic energy of oscillations still more difficult.

ENERGY INTERPRETATION OF THE LAWS FOR THE DISTRIBUTION OP OSCILLATIONS

4. Let the coordinate plane X be

the region of

possible values of .X

and V

so that each point of this plane characterizes one of the possible states of the

oscillating system, thIs region being unbounded in principle.. Any realization of oscillations

x') is plotted by

a defInite

phase trajectory x()

on this plane;

_{besides9 due to the}

ergodicity of the process this trajectory entirely fills up the

plane X27 for a realization of infinite durations Por any random oscillations of second order dynamic systems theIr

realizations exhibit some properties which have much in common due to the fact that alongside with a stochastic independence

there is an absolute deterministic relationship between displacement and Velocity for each particular realization. Hence each zero value of velocity in the realization _{of the} process always corresponds to the extremum of displacement.

Let us refer to the part of the process between the ajacent moments of

transition

_{of the zero level of displacement from the} negative to the positive region as a "cycle of random

oscillations". _{Then each cycle will correspond to a loop of the} phase trajectory between two adjacent points of intersection by it of the positive part of 1P-axis. _{Since within the cycle the} zero level of displacement is crossed in the opposite direction too, the above lOop certainly covers the origin of coordinates. The phase trajectory always intersects, the _{.X-axis along the} normal, the absciasa of the intersection point being the extreme value of displacement in the realization denoted

_{as a}

X(Vzo.

The so called secondary extrema of realization _{.&,J take place}

(Pig.1) if three' consecutive points of intersection of the

-axis by the phase trajectory are on one side of the origin of coordinates. _{In other Words, the secondary extrema (1, 2,} 4, 5 in. Fig.l) correspond to the secondary loops of the phase trajectory, which do not cover the origin of coordinates. The

rest of the extrema will be called the main extrema or the random amplitudes of the process arid designated as _'Using

such a terminolo, the random amplitude of the process is the maximum (in absolute magnitude) value of displacement within the random half-cycle between two adjacent zeros of displacement.

At points of intersection Of. 21-axis the phase trajectory is not, generally speaking, normal to the axis. The. ordinates _of

the points Of intersection, i.e. values of the velocity of oscillations corresponding to zeros of displacement will be

called. the random quasiainplitudes of the velocity of

the

process

and denoted it is essential that these values

of velocity correspond to zero values of the potential ener of

the oocl1ator.

The connection between the phase trajectory of the process realization

and

the joint probability density

(x,V)

can

be explained as follows. Let us plot

the

corresponding elementary sections of the phase trajectory of the given

realization over the phase plane for equally spaced moments of

time. Let us also select at point (j'

z)

a. surface element

&c4)h. _{Then the relative number of the aboveelementary}

sections of phase trajectory passing

through this

surface element is, in the limit, equal to probability element

Let us rewrite the joint prObability density of the displacement and velocity of nonlinear oscillations as follows:

p

f_[uitJ.

(53)

As noted above, the states of oscillator (viz, combination,

of

instantaneous values of displacement

and

velocity) which

satisfy the condition

E 7t)

U

3L

(i)] +

it

z('t)

_

(5.2)

On the plane of variableB X1# the values of

E7t)

form a certain field, so that any 'duve on this plane is equivalent to the integration of

E7t,.)

over time. Let US substitute the variables in (5.1):

*

_'1

E, = C/

_(x)+T7

(5.3).

The condition (5.3) is the equation of ener of the free nondamped vibrations of the oscillator with the reduced force function IL*(x). _{The period of these vibrations will be denoted}

by.

('Therefore

_{the substitution of variables in (5.3)}

implies that a great number of all, possible states of the oscillator make up a totality of conditional regular cycles with periods Z("E*). _{At the same tUne thi substitution}

repreaentB an integ'ation over time

(.5.4)

where the cu'vilinear integral is taken over contour

C

on the plane of variables XV- , satisfying the condition (5.3).

The possible values of continuously fill 'up the

semi-infinite interval L -

0

, s is a continuous random value. .' At' the Same time, in contrast to random.valuea

.xfr)

and tY't) , . is a countable randomvalue as a

realization of random oscillations of any duration

T

contains a countable' number of

_{1_va1uee.}

The probability of the oscillator staying in the interval

of enr

levels

E± E+ CLE

will be equal to

4

(&) o(

S,

where ci S is the area on the plane .Xit'

between the contours C and corresponding to Slid.

As

()

5

_.

_;.

V.

p

(

;u';

jc(E:)

_p

jd

(5.5')

Integration of the left part over all possible values of gives the unity, hence

f(Ec)eLp(

LE=1.

(5.6)

F'om (5.6) we obtain ;the probability density for the time of the oscillator staying on the ener levels

?

(E:)

=

.2

e

(

)

,

C,

(5.7)

On the other hand, from (5.6) we have

(5.8)

.(5.,9)

(5.10)

probability calculated in terms of time. But

in

the variables

£ ,. t the length of the contour ,' as it follows from (5.4),

j

_{21 S}

)dEC

and

the, comparison between (5.8)

and (5.9)

leads to two conclusions. Firstly, we obtain the probability density of the ener

levels for random oscillations

7ç

(Ec*)

Thus', irrespective of the individual _{eature.s of the oscillator,}

the ener levels characterizing the equiprobable states under condition random oscillations of the class in question are

(cr

t)

-

L

?a()=

(

distributed according to the exponential law.

Secondly, considering, that the mean period of the

conditional cycles with the period. introduced above is equal to the mean period of the process Z , the amount

of

conditional

cycles in any realization of random oscillations of' T-duration as also the amount of random values of argument

is equal to the amOunt of random cycles in the same realization

We now tux'n to the discussion of the distribution law8 of the main characteristics of oscillations.

Supposing ?Y=O or .7C0 in Eq.(5.3), we obtain

E(eo).=

Q(0).

Here cZ° and are the amplitudes of conditional cycles, which will be considered as positive. The random values

a C

as a].so are continuous and countable at the same time, Substituting (6.1) in 5.1Ô), gives the probability

densities for the lével.s of' reduced potential 'and kinetic energy

of the oscillator

Tjw

₀₎

(6.1)

(6.2)

(6.3)

Al]. the possible values of

a°

belong to a set of positive extrema of displacement in a representative realization of' random

oscillations, but, generally speaking, they are not identical

*ith. this set.

The expression c7('a) is the maximum value of the reduced potential energy of the oscillator in the limits of the

conditional cycle. The mean (per cycle) value of this maiitude, as it follows from (6.2), is a parameter of the given process of rando oscillations. This parameter can be calculated in a

different way, i.e. as an arithmetic mean in the limits of the realization without recourse to the probability densIty..

On the other hand, in the limits of every random half-cycle of the realization in the positive domain of displacements, the maximum value of the reduced potential energy of the oscillator is determined by the main extremuni and is equal to

U('a0)

irreepective of the presence of secondary extrema in the limits of -the half-cycle (between the adjacent zero crossings).

If the realization of oscillations is representative enough, the mean (per cycle) values CT*(a6) and must be equal in the probabilistic sense whatever the individual features of the oscillator and the duration of the realization, provided that this duration is long enough. However, the equation

:

UYa0)

=

(6.4)

where.

T

(6.5)

is the magnitude in the realization of both real (random) and conditional cycles, can be satisfied under the .circuinstancea

only on condition that O It will be recalled that and

a°

are elements of one and. the same set of the positive

For the noncorrelated disturbance

U(ci) .4$p f-

)

extreina of displacement. _{It follows from the above that a subset} of a° is idetical with the subset of _{the random amplitudes Of} process Q, , i.e. with the subset of the main extrerna

Similarly it is proved that the set of is identical with the set of quasi-amplitudes of velocity of process

The reverse of dependence

_E:ca0)

_{is single-valued If}

u(x) ?

0

everywhere for

()Q

_{Where for the zero position of} the oscillator

(x=c)

_{the restoring force has the peculiarities} of the type U('O) _{± U0 (the system with a set-up spring) or}

(x)= 0

if

j.xJ

(the system with a play) the single-valuedness will not be upset either. _{These conditions of} single-valuedness will be considered as being fulfilled.

Now according to the general rules the probability densities

(6.2,) and. (6.3). can be transformed _{respectively into the}

probability densities of random amplitudes of displacement and the random quasi-amplitudes of velocity of oscillatiOns

I C

(6.7)

a-'O.

(6.8)

The distribution (6.8) was obtained by Crandall [2J earlier. As is seen, it holds for the' random amplitudes with any relative number of secondary extrema.

Denoting

=

V(',). if

Q, ' and

Ec*

correspond' to One and the same conditional cycle (.5.3), we can

rewrite the probability density (5.7) as follows:

(a) = (4)

T

a.-O)

The distribution (6.9) referred to as the "probability density of the envelope" is obtained by Crandall in a different way for the noncorrelated disturbance [2]. The above

considerations make it possible _{to ascertain that. this} distribution refers to the envelope of the main _{extrexna of}

daplacement.

Prom this distribution, we obtain

-j.

f

_[

It was probably Crandall r2] who was the first to use asiinilar expression in relation to the noncorrelated disturbance. _Now we can state that this expression is exact and is not cozmeàted with limitations in respect of the form of Oycles in realization of random oscillations _{onwhjch Crandall's}

reasonings are based. In particular this expression is not connected with a relative amount of secondary _{oscillations but it must be}

applied to the random amplitudes of the _{process, i.e. to the main extreina.}

Irrespective of the individual features of the oscillator, the oscillations of the class in question, along with the normal distribution of velocity, have, _{as it follows from (6.7), the} Rayleigh distribution of _{quasi-amplitudes of ve.locity, i.e.}

the

velocity values of oscillations for zero up-crosses of

displacement. This makes the quasiampljtudes. _{f velocity an}

(6.9)

(6.11)

rimpotat

characteristic of oscillations. If it is aprióri known that the disturbance corresponds to initially introduced limitations, then you need just torecord. the momenta of

displacement zero up-crossings and the values of the system velocity at these moments and to use these as the empirical basis for the subsequent evaluation of parameters of

distribution in so far as the oscillations of the system with the biown force characteristics are concerned.

On the basis of (1.12), (1.13) and considering that for the

linear oscillator where the displacement

variance, we can rewrite (6.6) in the form

(6.12)

from which it follows that the random amplitudes (or the main extrerna.) of displacement are distributed according to the

Rayleigh law irrespective of the spectrum width of the process. This conclusion.is in agreement with the result earlier obtained by A.I. Voznessensky with respect. to the distribution of

displacement extrema for the "smoothed" realization.

At first sight it seems that the foregoing is at variance with the well known Rice law. As we are not in a position to

analyse this question in detail, we shall only note that the distribution (6.12) is in agreement with the Rice law.

For the nonlinear oscillator the condition

is of great importance; it shows that the mean level, of

oscillations corresponds to the natural state, of equilibrium of the oscillator, which implies that there is no. constant

limit is not essential since the restoring force is, an odd function, of displacement in re'lation to any origin of reading. Therefore, all the above formulae with relevant substitution of argument hold true for the linear oscillator also in cases when the disturbance does contain a constant component.

Now we shll try to show that the energy-equivalent double amplitude _{Q. Is, equal to the double mean value of the random} amplitudes of displacement

Indeed, let us multiply both parts of this equation by the constants for the given process and write it in, the following

form:

f;fa) e)9.[

2j

_(6.14).

ZJP

Ag. the quasi-amplitudes of velocity are diatribute4 according to the Rayleigh law, their mean value

Proceeding from the definition _{E ,} we. have the following

relation: or hence (6.13) (6.15)

(6,16)

(, 17)

(6.18.)

Assuming _{and making the substitution of variables,}

we can rewrite (6.14) in the form

Hence We obtain Eq. (1.17)

Thus the assumption that

£?-

_{is proved. it should be} emphasized once more that _{is the mean value of the mainS}

extrerna of displacement9 _{At the same time the ergodicity of the}

processes in. question was confirmed.

The calculations using (6.11) are rather difficult since

the. period .*() _{for the nonlinear restoring force is}

determined by an improper _{integral. The relations (6.13), (6.18)} (6.15) result in the equivalent expresSion

1_

2

fZ1Yt)

_?

AY2Jr4.-

(Ii

Which is free from the difficulty as mentioned above.

. (6.19)

(6.20)

may serve as the model of nonlinear system (1.1) else .

the

random oscil1atio _.xj(-j

of the

_system (7.1), with a similar disturbance, will be equivalent _according

ENERGY-EQUIVALENT STATISTIC LINEARIZATION OF NONLINEAR OSCILLATOR

iow. let us see _{on which condition the linear system}

2.

.L

/21.

=

to some criteria with the nonlinear oscillations .X(&j'. of the

system (1.1).

Selecto4 of criteria for the statistic equivalence of

nonlinear and linearized random oscillations with particular reference to various applications or various engineering

problems may, generally speaking, be dIfferent. While studying the efficiency, strength, safety etc. of the mechanical systems which experience stationary Oscillations, the distribution of main extrema and average frequency of the process are of

paramount importance. Since the distributions of displacement

extrema of the nonhinearand linear oscillations differ in

principle, apart from the equality of average frequencies, only the equality Of averge random amplitudes of these processes may be required. As stated above the energy-equivalent

oscillations are determined by these two criteria.

It is seen from, par.3 that the restoring fOrce steepness of the energy-equivalent linear System must be

Hence .

(7.2)

where

Lt,

is the mean frequency of nonlinear osculations

with the given mean amplitude in the case of noncorrelated disturbance. The next analo is of interest. Let us compare the forced (regular) oscillations of a' nonl±near oscillator and "a linear one indüOed by the harmonic disturbance at a simIlar

amplitude and frequency. Then, if the natural frequency of the. linear oscillator is equal to. that of the nonlinear one at the

same amplitude, the moment of côrrelatiôn between the displace-ment of th'e system and disturbance will approximately be equal

r

in both cases.

and. potential

approximately The value

Therefore the: time-mean values of the kinetic energy of these oscillations are also

equal.

of is defined by equation (6.20) but for the system with noncorrelated disturbance

-f -

1

(

&7aj

?

J)-'p--L

J

Let the value o be a parameter. _{Then in principle it is} always possible to calcu.late approximately the integral and, as

a result to obtain numerical (or graphic) relation

('4).

We may proceed futher, however. _{In so far as the energy}

characteristics are concerned there is no difference in principle between stationary irregular oscillations on the one hand and regular oscillations on

the

_{other hand, if the latter are}

determined by statistical values. As a matter of fact the mean frequency of stationary random oscillations is the frequency of reilar oscillations of the same dynamic system, whichina

statistical sense are energy-equivalent to random oscillations. The frequency W, _{is in accordance with the energy-equivalent}

average cycle of free oscillatiOns of the system, that is a

cycle with an amplitude of a. Thus the _{average frequency of}

the process in the case of noncorrelative disturbance

is equal to the natural frequency ca() of the System

provided that the amplitude is equal to the average amplitude

of the random process Q. This deduction is in particular agreement with some calculations carried out by Lyon

tJ

The velocity variance of random noncorrelated oscillations

i doterrnlned by the equation

Thus, knowing the function

-'v-el ocity variance

obtain the same relationship

The second parameter of the linear model should be determIned on the basis of the equal ener _{dissipation In both linear and} nonlinear systems. _{The mean value of enerr dissipated in a}

linear system is- equal to

2

_{and for the nonlinear system} it is determined by the left part of-. equation (1.5). _{As the}

velocity distribution of nonlinear oscillation is normal, it may be shown that

I

one can calculate the of random oscillations, that is

(7a5)

This equation is in line with the similar relatIonship derived by what is known as the equivalent linearization method. _{If the} resistance la (v-jis expressed in terms of a polynorn,it can

be integrated to the final form.

The exact value of resistance law is

_{uown for the ship's}

:i'olling as in many other cases. _{The most reliable data on}

resistance law can be obtained experimentally by the methods of

-free or forced oscillations

f5].

_{In both cases the empiric}

resui:ts _{are presented graphically as harmonically linearized}

non-dirncn3ional coefficient of

rosictance (or

where V

_{is the critical resistanóe) versus the oscillation;}

amplitude .X,. The employment of this relationship for the statistical linearization of resistance was described in Ref.f by the same author.

Using .the well-known methods of spectral theory the

velOcity variance ZvL

and the mean frequency of

linearized oscillations

ai

can be calculated for the linear system (7.1) given the Spectral density of disturbance Since the

system parameters, are functions of. the desired value it is

necessary that calculations be performed by using either the method of succesSive approximations or the method of 'aphic

interpolation. This problem

Will

be discussed later.

In accordance with the present conception the obtained values of and should be considered as approximate values of the reSpective characteristics of nonlinear

oscillations

at

the same rate of disturbance,

The value of will be used to determine

(4J

then by any of the methods the frequency of free oscillations of the nonlinear system can be calculated as a function of their

amplitude -"(. These data will be suff.cient to evaluate on

the basis of the above all the main distributions of nonlinear o8cillations.

In this case the linear model is used only for the calculation of the kinetic energy and the mean frequency of nonlinear oscillations, otherwise use is made of the

8. Now let us return to the calculation of velocity

variance of the linearized system (7.1) at given disturbance spectrum

S

(cv.i) and known relations and

(i).

The analytic-and-graphical method involving graphic interpolatiox

requires

a

minimum of óalcuIation.

Let us join interpolation planes _and as shown in Pig.2, and. represent on them the curves and , respectively.

Then we take arbitrary values of fl (i = 1,2,3...) and (j = 1,2,3...); we shall' discuss rational se1ecton of these

]..ater. The velocity variane of oscillations,

4,.

is computed

in terms of the standard

relationships

of spectral theory for

every couple of, values

(itt,

)

. After choosing the values of

')

.,

Dc'n,

etc. we plot the appropriate diagrams in the lower quadrant. The points

a a

of their intersections With the curve /t(.) are transposed

into the upper quadrant (points

, , The curve' traced

through points _, £...' is a function of

A?&z, ')/),

point

A at intersection of this function with curve

will determine the required value, of

M

(ne,

_)?L),

nd hence the value Of

tl(VL).

in fact small arcs of curves c4,..

(ne, Vj), oV.. (n

'1)

are ãidently required in the neighbourhood of their

intersections with curves of and

respectively; these arcs 'can surely be plotted by three or four

points. Thus, it will b.e enough to carry out calculations just

for 3-4 values of at 3-4 values of frt for each of the

within.the selected range of variation and '. Experience

ghows the latter condition can easily be fulfilled.

§9. In some cases it may be desirable that instead of the method described in par.7 a less exact though more simple method

should be tided for the calculation of nj()

_{which will}

permit integrating in closed form 'with the function of. U

₍₎

given by, _a p.olynom.

Under condition8 of noncorrelated' disturbance all possible

states of' nonlinear oscillator, aais seenfrom _{equation (5.3),}

disintegrate the totality of its free nondamping cycles.

Substituting of for

U(x)

_{in (5.3), and using (6.1)}

_le

obtain after inteatiOn by parts the _equation

Ji

)

d

fx

u

(x) ,' (9.1')

which is satisfied by any free cycle0

Let us replace each nonlinear 'cycle by _{a linear one with.}

the same amplitude, and choose the restoring force steepness of the linear cycle on the assumption that the velocity riancea of both cycles are equal. In this case the cycles will be energy-equivalent. _{Since in the general case the}

velocity variance of nonlinear cycle is not obtainable in closed form, we shall assume that the equality of the velocity

variances of the cycles compared will be satisfied provided' that the left-hand parts of equations (9].) forthese _{cycles are}

equal. _{This assumption is the more exact the less the nonlinear}

cycle form differs from the harmonic one (or is trajectory

fri

an ellipse). . Then the energy equivalence of the cycles compared

are, equal,

which

results in

L i2(aj a3

-

f U (Z)

dx.

3

_"0

On the other

hand,

_{the velocity variance for the linear cycle}

I1fU2Jc

2('a)

a.'

(9.3)

'or

considerIng (9.2)

/k/[UL]

₉₄

Then let us average the equation (9.4) _{over the totality} of linear oscillation cycles and. hence in the left-hand of the equation we shah. obtain the velocity variance of the process

f f

(9.5)

wee

is the probability deflsity of the random amplitudes of the linear cycles, namely the Rayleigh, distribution. The velocity ariànce.'of the linearjzed

oscillations _{should be equal the same for the nonlinear} oscillations compared at the same level of exitátion. On the

other: hand, for the linearized oscillations in the case of

non-correlated disturbance

_{,where 4L is the}

variance of their displacement.

'It should be emphasized once more that the variance is not equal 'to the variance of nonlinear oscillation displacement

To facilitate the comparison between 'the excitation levels of linear and nonlinear systems, it is desirable to have some reference for which purpose use can be made of the degenerate

nonlinear system with restoring force

a30x,

where

for which (system)

4

The energy of level

excitation will be the same in: the systems compared. if

=

/aL

0X4

Then introducing the desiiation

u ()

-

ü

(x)/c&

(9.6)

(9.7)

and choosing the normalization factor in the. Rayleigh law as

equal to. , We obtain the approximate calculation formula.

.3

. C

f

z2

r

:..JI__ -h?aI.xu0(x)dz,

(9.8)

(. 0Z

I

.J

which will obviously estimate at the same time the mean frequency fl _{of the nonlinear oscillations With}

non-correlated disturbance.

This expression is inteated n closed

form if is given by a polynom.

The nonlinearity of the analytically expressed restoring force scharacterized by some parameter. If value is

taken a a scale unit.of this parameter then the identiàal dimensionless values of the nonlinearity parameter will have a corresponding comparable level of excitation in the systems

ôompared. Thö spocific conditions providing for the agreement between the disturbance levels are. dependent on the. function

u.j.

characteristics so when 'determining these conditions account must be taken of relation (9.6). SOme examples are

COMPARISON OP _{THE STATISTIC LINEARIZATION}

IVIETHODS

The efficiency of the _{ener-equivalent method of linearizE} tion of random oscillations can be estimated by comparing it with exact solutions. _{AU. known exact solutions relate to the} case of excitation. by "white noise" namely to noncorrelated

disturbance.. Owing to this, only

_{formulae for W}

_{can be} verified. _{Equation (7,3) is exact,. so let us ilustrate the}

accuracy of approximation (9.8) and the associated approximation of random oscillations for noncorrelated disturbance. At the same time we shall carry out a comparison wIth the _{data obtaied} by the method which is knOwn aB the method Of equivalent

statistic linearization,

10. _{For the rnass-spring-setup system. excited by "white}

noise" [7]

4=

where. _{spring, stiffness factor (per unit mass.).}

In this instanàe Crandall obtained an exact analytic solution

[7:1;

Proni:equation (7,3)

?

z0)

(z0 £

_F0

This integral is tabular

f

8, integral 3.4633 that is the exact value of the mean frequency of the process involved

(10.3)

The use of the exact equation (7.3) based on the

_ener-.

equivalence conception made it _{possible to avoid complications} resulting from the other approach to this problem [7j.

Now let us turn to approximate values denoting them by asterisk, _{Having performed the integration}

ii _{(9.8) for the}

case under discussion, we have

xc,

(].O.2.

(10.4)

with E0 _{to.be determined from (10.2)}

as before.

Then we find a condition with the satisfying of which the non-dimensional value of nonlinearity parameter E0 will be of

the seine magnitude in..both lineai' and linearized systems. _The

disturbance level is estimated by the statistics of the

degenerate (linear)system

4

_{and , with. the value of}

excitation level in the degenerate system to be corresponding to that in the linearized system, using both criteria. This

viill be the case if. the _{average work of disturbance per cycle}

je the same in both yaterns. _{But the average number of cycles,}

per unit time is different in these systems, hence the above-mentioned correspondence may be aóhieved only if spectral dénsitjea pf disturbance in them are different.

Let the spectral density of disturbance _{'of the linearized} sythtem in which the correspondence _{is observed be}

where _{spectral density of disturbance of the degenerate}

(and hence noitlinear) system. _{Then by Virtue of (9.6)}

Simultaneously. we must have

'-,

(10.6)

so that the correspondence in respect of displacement will be preserved. _{Eliminating ii from, these two equations, we find}

av-(10.7)

This is the value of _{at which} E0 _{is the same both i}

nonlinear and linearized Systems.

It may seem at first sight that _{the enerr correspondence} between the linearized, and degenerate systems will be upset in thjs case. _{However there iá no 'ound for such doubt.}

om

(10.5) and (10.6) wehave

and velocity distribution of oscillations in the linearized System is

'1tT

But thiB is the normal

_{dtrjbutjon with variance}

(w)

(v.*)=

.

I

0 j7,O (4,

('.1cA

(10.5)

as it should be.

Prom equations (l0.4).an (1O.7)we derive approximate

energy-equivalent statistic values of the nonlinear oscillations

1(f+

Though _{is the displacement variance of the linearized} system it should be equal, as it follows from the foregoing,. to

relation rather than. to the displacement variance of the nonlinear

system

On the contrary, if the displace-ment variance for the linear system is equal to that for the nonlinear one then the mean values of oscillation amplitudes of

these tWo systems aredi'fferent.

Pig.3 and 4 give the following collations for the

appropriate statistics: the exact values, estimatjons obtaine4 with the

help of

what is kiowns equivalent linearization

'method, estimations from the formulas (10.8), (10.9), the exact values of relation _{are shoWn in Pig,3 by points}

(the Other

curves

are shown below).

Then we find the appro.mate values of the probability

density for thenonliñear oscillation envelope..

Let us consider equation (9.2) as an approximate formula for natural frequency of the nonlinear

system

t()

_,f

(x)

d,

_{(10 10)}

O

whch for the given problem results in expression

=

(10.8)

a)0.

(I. .3

E0

o

This expression should not be used at

c20 --

0.

Substituting (10.11) in (6.9) we get an approximate formuia for the required probability density

12 (Fc)

I +

(10.12)

x

Fig.5 shows for ₁ the comparison between the exact

value of ( (no)

and the value o± _{obtained from} formula (10.12) provided that

)() is

estimated from formula (10.8), and also the envelope, distribution for the approximately energy-equivalent linearized system.

The collation in Pig.3., 4 shows that even though

approximately realized the energy-equivalent linearization is preferable to the equivalent linearization, since even with a 8iiificant: nonlinearity of the system it allows making an adequate estimation for the most important statistics of the nonlinear oscillations, viz, the mean frequency. ' and the mean amplitude . It will be recalled that, the precom'ession of springs that is equal to almost one and a half standards of degenerate System'displacernent _{Jfr corresponds to the value}

'of

E01.

The approximate estimation of the probability density of the nonlinear oscillation envelope (Pig.5) is also satisfactory.

Now using the same example we check the assertion

V

IC

(10.14)

Further let us caThulate the _{exact time-mean value of the} square of displacement for the nonlinear _cycle

-system at an amplitude of oscillations equal to the amplitude

of the enorgy.-equiva.ent cycre.

For a-aplitude free

_{oscillations the exact value}

_{f 7]}

_of

the system natural frequency is determined by expression

(= ,/a)

2Jc

e ,

tza

6/,t

tüQ

_,

-'-..

The equivalent cycle is linear

_and

_{for it}

iç r2(t L

L

a

_a

(10.13)

(10.17)

Denoting the relation, of the left-hand parts of (10.15) and (10.17) by 2.,

and

_{making some imple but somewhat unwieldy} calculations we obtain . .

J

(10.18)

The enclosed, in braces in this.expression is a.radi.cálly

Infinitesimal diffrence, hence _{computation should' be perfo'med} with 'eat precision (viz. 6-7 _{siificant digits in the minuend}

is the relation beweenthe square root of

displace-ment variance of the nonlinear cycle and that of the energy-equivalent (linear) cycle, the amplitudes of these being equal.

Consequently if is expressed as then

=

(10.19)

Taking into consideration equation (10.2) we obtain

=y

in passingfrom parameter E to _, accepted earlier.

On choosing the values of e and carrying out calculations

fr

(10.13), (10.18) and (10.20) we obtain relations &()

and E0

(54

, i e. parametric form Of the dependence

of the natural frequency of the ener-equivalent cycle on parameter E0.

Pig.3 shows the values of It is seen that

is in agreement with the exact value of

0(E).

If we iiOre

the difference in the form of cycle between the nonlinear .oscillaiions and the linear ones (ellipse), i.e. if we assurnO

we shall obtain the results plotted on the graph of Pig.3 as points. In this case the effect of the form of cycle may practically be neglected. .

Similsi' calculations though not associated with the concept of the energy-equivalent cycle were carried out by Lyon f4Jfor. a nonlinear hard-spring oscillator described below (par.11).

These examples confirm the possibility of calculating, the frequency as the natural fre4uency of the energy

equivalent regular cycle in a nonlinear, system.

It follows from the above .that the accuracy of estimation with respect to the distribtition of elements of the nonlinear

oscillations is essentially dependent on the accuracy of

estimation of the mean frequency of the process in the case of the noncorrelated disturbances Therefore the ener-equivalent

method of estimating this mean frequency aà the natural

frequency of the oscillator at an amplitude equal to the

ener-equivalent amplitude of the process provides for an accuracy which is quite sufficient for the calculating of characteristics

of nonlinear random oscillations in case of normal disturbance.

ll. As a second example we shall take a hard-spring

oscillator (Duffing's problem) the random oscillations of which due to excitation by "white noise" were considered by Lyon [4],

[9]. In this case no general analytic solution is knownand

the exact values quoted below are obtained by numerical

inteatjon.

_f9J.

The force function of this system can be presented as

.r?

k(x)

(x

₎

(lll)

where is the nonlinearity parameter. Its dimensionless value is

(11.2)

where is again the variance of the degenerate system displacement for the given level of disturbance. The same dimensionless values of this parameter in both nonlneár and linearized systems will evidently be available if

in other words, if the systems have the same values of velocity variance.

Then, from equations (9.8) and _(11.3)

may. be Written as

XQ0p

7

1/y

(1+

,.

4)

(11.3) (11.4) (11.5) (11.7 )

In.Pig.6 for the statistic of

'o/"4)

_{there are} compared: the exact value, the _{ener-equiva1ent value defined} from (11.4), and the one obtained by the method of equivalent

linearization,

Compared in Pig.? for the statiStic of

are; the exact value, the value from. (11.5), the value estimated by the method of equivalent linearization.; the exact value of

The free oscillation frequency

_{of the system (ai/p)}

being estimated _{similarly to par..1O}

11.6)

0

T/z

J-0

is shown by points.

It can be stated that even for this case of a hard-spring oscillator the approximately realized energy-equivalent

linearization provides an appreciable decrease of errors in the estimation of main statistics of nonlinear oscillations _viz. the mean frequency _{of the process and the mean value of}

oscillation amplituth. 2 _{as compared with the method of}

equivalent linearization without any complication of calculating

formulae. . At ,4,

I

the errors introduced by the equivalent

linearization method are O.l2&)4 _{and O.l2,t/ respectively} while introduced by the energy linearization performed

approximately these will be O.04&o

_{and O,O4V. The latter}

can in most cases be considered as reasonable for technical applications. At the given value of _{the rate of}

npn-linearity of the system is appreciable, i.e. for a displacement equal 10

_jfxo

_{the restoring force is twice the value of the} linear part.

The exact value of probability density _{Was not} computed in this particular case. _{Therefore a comparison is} given in Fig.8between the approximate distribution (11.7) and a similar (Rayleigh's) distribution of the energy-equivalent linear system in order to illustrate the difference.

The comparison between Pig. 5 and Pig.8 shows that the

difference of the envelope distribution of nonlinear oscillations from the Rayleigh distribution is largely dependent on the

References

1. _{Grim, 0. "Nonlinear Phenomena in the Behaviour of a}

Ship in a Seaway", XII Intern. Qwing Tank Conf., Proceedings, Rome, 1969.

2, _{Crandall, S.H. "Random Vibration of the Nonlinear}

Systems", Random Vibration, vol.2, editor S.H. Crandall, MIT Press, 1963.

Kauderer, H. Nichtlineare Mechanik, _1958.

LyOn,. R.H. "On the Vibration Statistics of _{a Randomly} Excited Hard-Spring Oscillator", J. Acoust. _{Soc..Am., vol.32,}

N6, 1960.

Blagoveshchensr, S.N. Spravochnik _{po teorii korablya}

(Reference Book on Ship Theory), Sudpromgiz, _1950.

Gerassimov, A.V. "Statistic linearization of rësietancè to rolling motion" (in Russian), J. "Sudostroieniye", _{4, 1971.}

Crandall, S.H. "Random Vibration of the Nonlinear Systems with Set-up Springs", Tr. SNAME, S. E. Appl. Mech., vol.29, N 3, 1962.

Gradstein, IS., Rigik, I.M. Tablitsi 3.ntegralov, suzin,

ryadov . proizvedeniy (Tables of integrals, sums, series and product8), Izd. 5-e, Nauka, M., 1971.

9. _{Lyon, R.H. "Equivalent Linearization of the Hard-Spring}

Pig.l. _{Agreement between a random cycle of v1bratonS and}

its phase trajectory.

3,6 -. the main extrema of _{displacement; 1,2,4,5 - the}

secondary extrema.

Pig.2. _{Scheme of graphic interpolation for determining the}

parameters and of the linearized

system.

Pig.3. Values of 'the ratio _{as a "function of the}

dimensionless nonlinearity parameter , for a system

with set-up springs.

1 - exact so].utioñ [7]; _{- calculated by the equivalent} linearization methoaE7J3 - calculated from formula (10.8);

4 - exact value for a regular cycle; 5 - the same without taking into account the shape of the cycle.

Pig.4. _{Values of displacement statistics as a function Of the}

dimensionless nonlinearity parameter for a system with set-up springS.

1 - exact value' 4/J0

r71;

2 - value.

obtained, by the method of equivalent _{linearization}

L71;

3 - value

x

_{calculated from (10.9); 4- exact}

va1e