this assumption, it can be shown that the probability density function of the ... damping of the structure and the spectral density of the excitation are small. .... Guided by this result for the stationary solution, one might attempt to find a closed.
..
0-
14..."'"
PI-..
v.
1m
~
a-'-
~
l
.. ~
-"MJ/7II.I"~.-z:M
VIBRATION
RANDOM
STRUcruRE
LINEAR
A
OF
NON-STATIONARY
.-
Mechanics.
11Ie University ofTexas atAustin.
U.S.A.
78712.
Enlineerins
TX
and Austin.
Elllineerins
Aerospace
of
Department
P-T. D. SPANOS
(Received 28 September 1977; ill rel}ised 101m 6 February 1978; rrceived lor publicGtioll 13 MQrc~ 1978) DOiac
white
to
subjected
sb1lCture
linear
dilIItptd
liIIItl1
.
of
vibration
nDdom
A Tbc DOHtatioury is cooaMIered.It is shown that the probIbiIity density function of the amplitude of the stnlCturai response can be approximated by a Rayleilh distribution. Analytical formulae for the time dependent statistics of the amplitude are presented. Tbc anaIytic:aI results are compared with data obtaiDed by . numerical simulation.
INTRODUCTION
The problem of the response of linear structures to random excitation has occupied the engineer and the applied scientist for quite some time. The theory for determining the statistical
propertiesof the structural responsehasbeenquite well developedand is availablein standard textbooks[l, 2]. However, problemspertainingto the statistical propertiesof the maximumof the responseor to the time at which the responseexceedsa certain barrier are still under investigation. If the dampingof the structural system is small, the responseexhibits pseudo-sinusoidal behaviorwith slowly varying,in time, amplitudeand phase.Evidently"the determinationof the statisticsof the responseamplitudei.sparticularly important in estimatingthe failure potential of the dynamicalsystem,which essentiallyis the goal of a probabilisticanalysisof a physical problem. It is understoodthat the concept of the responseamplitudeis applicableonly when the dampingof the systemis small. However,this assumptioncan be justified for a large class of physicalsystemsof engineeringinterest. If it is desired to determine the statistics of the response amplitude at a time much, longer
the
if much
true
are
not
is
which
this times
at
Clearly,
zero.
response
is the
of
derivative nature
time
the
its
be
and will
This
response
the
of non-stationary.
correlation
is
process
the
than the rise time of the structure,then the responsecan be assumedto be stationary.Under this assumption,it can be shownthat the probability densityfunction of the responseamplitude is a Rayleighdistribution[2]. The crucial point of the proof is that the responsebeingstationary,
shorter than the rise time of the structure. The problem of the non-stationary random response of a linear singie-degree-of-freedom system to white noise has been originally examined in Ref. [3]. Typical examples of research effort pertinent to this problem are given in Refs. [~]. In the present paper, an approximate probability density function for the amplitude of the non-stationary respOnseof a lightly damped linear structure is derived. Analytical formulae for the time-dependent moments are given. The results of the analytical approach are compared with data obtained by a numerical
simulatio~.
MATHEMATICAL BACKGROUND Consider a linear single-degree-of-freedom structure described by the stochastic differential equation
i + 2{0J-i+...2,1= W(t),
,
(I)
is the ratio of critical dampingof the structure. The symbol w(t) representsa white noise processwith spectraldensity S constant over the interval (-00,00). The dota(t) above a ~(t) variable with respectto time. Define theprocesses and by denotesdifferentiation where (11ft is the natural angular frequency, and
x(t) - a(t) cas(~t + .(t» 861
(2)
and
i(t)
SPA}Q
D.
P.T.
162
=- *",a(t) sin (*",t + .(t».
(3)
(4)
2
.i2/~
+
X2
=
a~1)
Using eqns(2) and (3) it is readily proved that
(5)
.,.1.
-
(~)
tan-I
-
=
.(1)
and
Using eqns(4) and (5) differential equationsgoverninga(t) and ~(t) can be derived as foUow~. Differentiatingeqns(4) and (5) with respectto time and using eqn (1) it is found (6)
and ,*'- sin (*'-t + .(t» cos (~lIt + .(t»-~COS
(*'-t + .(t».
(7)
~"a
= _1 a
';'(t)
At this stage,additional assumptionsabout the problem are made. It is assu.medthat the damping of the structure and the spectral density of the excitation are small. Mathematically, these assumptions may be described by
'-4
and
s =~,).
(8)
(9)
Under theseassumptions,it can be arguedthat a(t) and .(t) are slowly varying functions of t. Therefore,the responsex(t), eqn (2), exhibits pseudo-sinusoidal behaviorlike the one shownin Fig. I.
Using an asymptoticmethodintroducedby Stratanovich[7],eqns(6) and (7) can be usedto
derive the following equation for the amplitude a(t)[8-IO)
(10) where .5 0'2- '2c;:'
(Ii)
are
terms
oscillating
rapidly
various
(JO),
eqn
deriving
In
8(T).
=
T)]
+
E[.,,(I),,(I
i.e.
the stationary varianceof x(I), and .,,(1)is a zero-mean,delta-correlatedprocesswith 'unit
intensity,
is
.
863 .~
Non-stationaryraIMiomvibration of a linear structure
in-
much
is statistical
of
excitation
the
of assumption
the
band-width
the justify
that
to
fact
the exploited
Next,
is
J].
[1 response,
Mitropolski
the
of
and that
than
Bogoliubov greater
of
first averagedover one cycle, T = 21r/~, of oscillation in a way'similar to that of the technique
dependenceof W(I) and the values of a(t) and ~(t) correspondingto a slightly shifted time t:t 41. Equation
(JO) describes approximately
the random evolution,
in time, of the amplitude.
Its
been
has
(10)
eqn
[8],
Ref.
in
example,
For
amplitude.
the
of
behavior
random
the
of
study
the
important feature is that it is not coupledwith the phase~(t). This fact simplifies significantly
AMPLITUDE
the
used
been
of amplitude.
time
has
(10) the
of
passage
eqn
with
first
the statistics
the
of
associated determine
function
equation to
used
be
will
distribution
Kolmogorov
the (JO)
probability eqn
the
addition,
In
paper,
study this
to In
[JO]
amplitude.
the Ref. amplitude.
in
of
usedto derive ordinary differential equationsgoverningthe momentsof the first passagetime
STATISTICS
The Fokker-Planckequationassociatedwith the stochasticdifferential eqn (JO)is
~~_l.. at aa
[ ""
( 0
-~
0
) p
+ '1IJ"a'2 g
(12)
aa
The symbol p(a, t) representsthe probability density function of a(t). It is assumedthat the structureis initially at rest. Probabilistically,this can be expressedas p(a, 0) = 8(a), where 8(a)
represents the one-sidedDirac delta function.The boundaryconditionsfor the
the
restriction,
this
Under
s~.
Sa
0
restriction
the
with
compatible
(12)
eqn
of
solution
partial differential eqn (12) can be detennined by imposing restrictions on a(t). Since a(t) representsthe amplitudeof x(t), and therefore is non-negative,it is reasonableto construct a eigenvaluesd, and the eigenfunctionE,(a) are found to be r=O,
d,= 2'~r, and
where
,=0,1,..
(IS)
4 is the Laguerrepolynomials.U singthe propertiesof the Laguerrepolynomialsit can be
readily proved that
r-E..(a)E,(a) da = 8.", 10'
§o(a)
(16)
where 6.,.,is the Kronecker delta symOOI.The solution of eqn (12) can be put in the form
.
(17)
,-0
p(a, t) ,;.,}: C, exp(- d,t)E,(a),
where C, are constant coefficientsto be determinedby using the initial condition. eqn (13). Combining
eqns (13) and (17) yields .
.' ,-0
p(a, 0) = &(a) = ~ C,E,(a}. Multiplying eqn (18) by E,/Eo, integrating from zero to infinity, and using the orthonormality
SPAHOS
D.
poT.
864 relation (16), it is found
Cr-
,sO,
(19)
I,
Therefore, eqn (18) can be rewritten as
(20) As t "'~. stationarysolution.eqn (20) yields (21) which is the Rayleigh density function. This function, governing the probability distribution of the amplitude of a stationary narrow-band Gaussian process, is usually derived by a quite dift'erent approach[2). Guided by this result for the stationary solution, one might attempt to find a closed form solution for the non-stationary probability density function p (a, t), eqn (20). This can be done by using the properties of the Laguerre polynomials. For the purpose, eqn (20) is rewritten as
It is known that[12]
t,-0L,(y)u' = exp(- 1y"'-(Iy» -II Applying eqn (23)for y = a2J2u2andu =exp(- 2'~t)t eqn (23)can be rewritten as a exp {=:;~u:!~
p(a. t)-
-:_e:!~:\~l~t)D
(24)
0' [1-exp(-2l-.t»)
Therefore,it has beenprovedthat the probabilitydensityfunction of the non-stationary responseamplitudeis approximatelya time dependentRayleighdistribution.This result,
-
and
rk![I-exp(-~~t)].
E[(a/U)Zi]
(25)
I
O.
z
(2k+ 1)[1- exp(- 2'~t)}.
k
besidesits theoreticalsignificance,facilitatesthe determinationof the statisticalmomentsof the responseamplitudea(t). Specifically,usingeqn (24) it can be readily proved that
(26)
.(t»).
+
2(~t
a2(t)[1
COt
-i
x1t>
+
It is interesting to compare the results of the present analysis with the results of other approachesto similar problems. For example,eqn (26) can be used to find an approximate expressionfor the variable E[.r~t»). Specifically,using eqn (2) it can be easily verified that (27)
If the rapidly oscillatingterm C9S2(l1l,.I+ .(t» is nel1ected,as it was done for the derivation of eqn (II), and eqn (26)is appliedfor k - 2, it is found E[X1(t)/~] = 1- exp (- 2'alat).
.86S
~
.
Non-slatiooary random vibration of a linear structure
The exact expressionfor E[X2(t)] hasbeenfound in Ref. [3Jby a different approach.The exact expressionis
1-~!P.~;j~!2 [0142 + OI.OI.,{Sin(2*'41)+
a
E(X2(1)/U2]
¥
sin2 (0141) ].
=fAI"2(I - '1. It
is readily seen that eqn (29) reduces to eqn (28), if the rapidly oscillatingterm sin (2(11dl) and the 0('1 term are neglected.It can be also seen that eqn (29) reducesto eqn (28) at times where
,
!
fAld2
!!
k =0,
~
1=1.=
independentlyof the assumptionof small ratio of critical damping,. The fact that the exact solutionsfor E[X2(t»),E[i2(t)] and E[x(t)i(t)lare obtainable,with much calculationaleffort, however,could justify an alternate approachto the problem of the determinationof p(a, t) for the special caseof Gaussianexcitation. Specifically,assumingthat w(t) is Gaussian,it can be assuredthat x(t) and i(t) are jointly Gaussian;the probability density function p(x, i) will dependon E[X2(t)], E[i2(t)] and E[x(t)i(t)]. Subsequently,the probability density function p(a, t) can be determined by using p(x, i) and the algebraic transformation introduced by eqn (4). Using this approach it can be proved that at times specifiedby eqn (30),the probability density function given by eqn (24) is identical to the exact solution. This method, however, is applicableonly for the case of Gaussianwhite noise and doesnot utilize simplificationsjustified by the small damping.In addition, this approachwould be extremely
cumbersome
to apply
to the problem
of structural
response
to modulated
its
Define
of
f
structure.
fraction
the
a
of reach
time
to
rise
the response
determine structural
to the
used
be
for
also
can
required
(26)
time
and
the
(25)
as
Tf
time
rise
the
Equations
Gaussianwhite noise becauseof the complexity of the correspondingsolutions for E[X2(t)], E[i2(t») and E[x(t)i(t)]. A typical example of these solutions is given in Ref. [13]. It is interestingto note that the methodologyof the present paper can be readily applied to the aboveproblem without requiringthat the white noise'excitation be Gaussian[ 14].
stationary level. Using eqns (25) and (26), it can be shown that the rise time for all moments of
the amplitudeis given by .TL.!!!.Q=fl ! 4.",
1
where T = 211'/~is the natural period of the structure.
was
sample
a
of
ordinates
successive
200
01,...0200
generation
the to
numbers
For assigned
were
distributed
was
normally 0200
.
.
of .
0'"
sequence
a
'~0.02
values
l4I(t),
the
excitation Subsequently,
the
of
study of system (1) with
generated.
first
function
simulation
performed.
RESULTS
NUMERICAL For the purpose of checking the results of the present approximateanalytical method, a
250
procedure
above
the
repeating
by
generated
(r
was
x,(t)
1,...250)
sample functions
=
spacedat equal intervals 4t' = 0.01, along the dimensionlesstime abscissa t' = tIT. Linear variation of the ordinatesover -eachinterval was assumed.A completeensembleof 250 such times. The responseof the system (1) to each of the 250 samplefunction was computedby -
[E(a)2
notion
de
numerical integration. Subsequently, the non-stationary mean value E(a)/u
and standard
E2(a)]'/2lu of the amplitudewere computedby averagingthe numericaldata.
From Fig. 2 and Fig. 3 it is seen that the numerical data are in agreement with the
correspondinganalyticalexpressionsfor the meanvalue and the standarddeviation of a(t). It is noted that the analytical solution noJ only predicts the~rrect qualitative nature of the amplitudestatistics,but the actual numerical valuesgiven by the two approachesare in close agreementin both the non-stationaryand the stationary segmentsof the response. The dimensionlessrise time TilT, eqn (30),hasbeenplotted in Fig. 4 vs the ratio, of critical
SPA*15
D.
poT.
866
~/i12D-..(4~tlT)] ..8
~
-
~I~I
'.0.02 AnmytieIc8lb
(8IIeIImIe Iize2SO )
I
IQ75
'VVVV' ~
ami I
TiN,
8)
I 10
I
~IO
I
lIT
Fig. 2. Meanval. of non-stationaryresponseamplitudevs time.
~.
40
~
~
/ 101
,.age ,.~
--- ...
I
.
. 2
.
5
I
..
4 .5 " ad-
Fi&.4. Risetime VI ratio of crila
'.a~
,.~
.
dampina.
.,
.
867 ,~
Non-stationaryrandomvibrationof a linear stnlcture
that
seen
is
it
example,
For
read.
be
can
f
and
,
of
values
given
the
to
corresponding
structure
damping.The fraction I of the stationaryvalueof the structuralresponseusedfor the definition of Tf has been selectedas a parameterto identify eachcurve. From Fig. 3 the rise time of the
SUMMARY
for, = 0.01the structural responsereaches75% of its stationarylevel in a time approximately equalto 10naturalperiodsof oscillations,but approximately25 morecyclesof oscillation occur before the responsereaches99%of its stationarylevel.
ATA74-1913S
number
arant
by
supponed
paniaIIy
was
investiption
AcUowl6dgcmlllll-This
The statisticalaspectsof the amplitudeof the non-stationaryresponseof a lightly damped linear structuresubjectedto white noiseexcitation havebeenexamined.It hasbeenshownthat the probability density function of the amplitude can be approximatedby a time dependent Rayleighditribupon. Analytical formulae for the statistical momentsand the rise time of the responsehavebeenderived.The analytical results of the presentedapproachhave beenfound in closeagreementwith the correspondingdata of a numericalsimulationstudy. from
the
National
Science
F!!"_ndItit!!!., Research Applied to National Needs.
The intencOOnwith Profs. W. D. Iwan and L. D. Lutes is sraaefuDyICknowiedaed.The autb« takespleasurealso in eckllOwledlilll helpful suuestionsof his coIlelllleS J. T. ~n and R. O. Stearman. REFERENCES
01Strwct.raI D,lI4IIIic,.
Press.New York (1963).
Academic
5"ttm,.
McGraw-Hill, New York (1967).
ill
VibrotiollS
2. S. H. CraIIdaIIarM!W. D. Mark. Random
MtChQfticDl
1. Y. K. Un, ProbGbiIiItic n."
3. T. K. Cau8bey.arM! H. J. Stumpf, Transient response of a dynamic system ulMler random
S63(1961).
excitation. J. Appl.
3.
Altt~
(1969).
221
~,
Mtth.
Appl.
J.
excitatMx1.
4. R. L. Blmoski arM!J. R. Maurer, Mean-squareresponseof simple mechanicalsystemsto II(XIstationaryrandom S. L. L. BucciarelliarM!C. Kuo. Mean-square responseof a ~-order Appl. Altt~
systemto IIOftStationary randomexcitation.J.
37. 612 (1970).
6. R. B. CorotiaaDdT. A. ManW, OaciUatormponse to modulatedrandomexcitatioe.J. s..,., 11«", ASCB"",
SOl
(1963).
York
New
Breach,
Gordon.t
ll.
Naile,
RGlidom
01
ThIO"
the
ia
Topic,
StrataDOvicb,
L.
1.
7.
(1977).
S. S. T. AriaratnamandH. N. Pi, 00 tile first pusaac time for envelopecrossioaof a linearoscilllt«. l.t. J. COlli'. II. 89 (1973).
lor NoIIlu,
Ttthaiq."
9. pot. D. Spanos.LialGriZGtioll
Dyll4lllicDl 5"t""',
EERC 76-04,Earthquake~rina narrow-baud
non-statiooary
a
of
envelope
the
for
10. W. C. Lennox aDdD. A. Fraser.00 the first pusaac
distribution
RclCarCb Laboratory, California InstitUte of TecbllOJosy (1976).
)
A.
aDd
8OIOIiubov
Mitropo1ski,Allmptotic Allthods ia the nea" 01NoIIliaIG' OlcUl4tiou. Gordon.t Breach,New
(1~1).
York
N.
11.
Itocbaslic proeess. J. App/. MtCh. 41.793 (1974).
published).
.Y&M.N8.~
McGraw-Hill,
J.
excitation.
random
modulatrJd
under
DOn-stationary
vibrations
structural
of
analysis
EBerlY
Spanos,
D.
pot.
I..
to
Nat_aticDl
~Ie1,
R.
if.
11
Hadbd 01Fonftu/QIa.d TabI". p. IS3. New York (1968). 13. H. GotoarM!K. Toki, Sttuctilralresponse randomexcitation.4th WorldC*. unbq..u &t., Chile, A-I, 130(1969). Strwct.raI
Ntt"
(To be
