《船舶水动力学》课程教学课件（讲稿）第五章 不规则波中水动力响应分析 Random seas

NIVERSITYOSouthampton(5) Dynamic behaviour inirregularwaves.(Responseinrandomseas)Hydrodynamics and HydroelasticityProf.P.Temarel,WUT,July 2015NIVERSITYOSummarySouthampton.You alreadyknow howtoobtain responses in regular waves, i.e.solveequations ofmotion,obtain principal coordinates,and subsequentlydisplacements,velocities,accelerations (absoluteorrelative),aswell asbendingmoments,shearforces atanyposition alongthe shipFor unit waveamplitude,theresponse,as afunctionoffrequency,is.termedatransferfunctionorRAO(ResponseamplitudeOperator)SummaryofthislectureRelationshipsbetween inputand output;definition ofunit ImpulseResponseFunction(IRF);FourierTransformpairs:Basicpropertiesof RandomVariables&RandomProcesses:Statistical properties: Mean, Mean Square, Root mean square andSignificant values:Autocorrelation andmean square spectral densityfunctionsRelationshipsbetweeninput&outputforrandomprocesses(i.e:irregular wave)Rayleigh'sprobabilitydensityfunction;probabilityofexceedance

1 (5) Dynamic behaviour in irregular waves. (Response in random seas) Hydrodynamics and Hydroelasticity Prof. P. Temarel, WUT, July 2015 2 Summary • You already know how to obtain responses in regular waves, i.e. solve equations of motion, obtain principal coordinates, and subsequently displacements, velocities, accelerations (absolute or relative) , as well as bending moments, shear forces at any position along the ship • For unit wave amplitude, the response, as a function of frequency, is termed a transfer function or RAO (Response amplitude Operator) Summary of this lecture • Relationships between input and output; definition of unit Impulse Response Function (IRF); Fourier Transform pairs • Basic properties of Random Variables & Random Processes • Statistical properties: Mean, Mean Square, Root mean square and Significant values • Autocorrelation and mean square spectral density functions • Relationships between input & output for random processes (i.e. irregular wave) • Rayleigh’s probability density function; probability of exceedance

FRSTYImpulse Response Function (IRF) Southampton.Basic relationship between input and output for a system is:q(t) = H(o)Q(t)where Q(t): Input, q(t): output and H(o) receptance.ConsiderQ(t)=(t) a short, sharp disturbancewherethedeltafunctionQ(t)=8(t)S(t)=0for t±0S(t)=oo for t=0, such thatq(t)=h(t) 8(t)dt=1-00Corresponding response of systemq(t)= h(t)h(t) is the Unit IMPULSE RESPONSE FUNCTION (IRF)3ERSYImpulse Response Function (IRF) Southampton.Fourier Transform pair: Receptance and IRFh(t)and H(o)formaFouriertransformpair:1了 h(t)e-iot dth(t)= J H(o)eiotdo and H(0)=2元-00-or1J H(o)eio do and H(o)= J h(t)e-iot dth(t) =2元002

2 3 Impulse Response Function (IRF) • Basic relationship between input and output for a system is: where Q(t): Input, q(t): output and H(ω) receptance • Consider h(t) is the Unit IMPULSE RESPONSE FUNCTION (IRF) q(t)  H ()Q(t) ( ) ( ) Corresponding response of system ( ) 1 ( ) for 0, such that ( ) 0 for 0 where the delta function ( ) ( ) a short,sharp disturbance - q t h t t dt t t t t Q t t                q(t)=h(t) Q(t)=δ(t) 0 t 4 Impulse Response Function (IRF) • Fourier Transform pair: Receptance and IRF h t H e d H h t e dt h t H e d H h t e dt h t H i t i t i t i t                                ( ) and ( ) ( ) 2 1 ( ) or ( ) 2 1 ( ) ( ) and ( ) ( ) and ( ) form a Fourier transform pair :

VERSITYOResponsetoarbitraryexcitationSouthamptonAn arbitrary input can beQ(t)Q(t)considered as a sum ofa largenumberofimpulsiveinputsappliedoneaftertheotherUT+8TAs h(t - O): Response to unit impulse applied at t = 0thenh(t - t) : Response to unit impulse applied at t = tSoresponse to an impulse Q(t)dt wil beh(t -t)Q(t)dt.Summingtheresponsefromallimpulsiveinputs,i.e.integratingq(t)= [ h(t-t)O(t)dtIVERSITYOResponsetoarbitraryexcitationSouthamptonSince there is no impulse for t<tthen h(t-t)=o for t<tand theupper limitcan be extendedtoooq(0)= J h(t-t)(t)dt0It can be shown that (t - t) and tcan be exchanged, ie.q(t)= h(t)O(t-t)dt-00.EitherformisknownastheCONVOLUTIONorDuhamelINTEGRALOfparticularimportanceinevaluatingslamming-inducedresponse3

3 5 Response to arbitrary excitation • An arbitrary input can be considered as a sum of a large number of impulsive inputs applied one after the other • Summing the response from all impulsive inputs, i.e. integrating        h t Q d Q d h t h t ( ) ( ) So response to an impulse ( ) wil be ( ):Response to unit impulse applied at t then As ( 0):Response to unit impulse applied at t 0      q t h t  Q  d t ( )  (  ) ( )   τ τ+δτ t Q(t) Q(τ) 6 Response to arbitrary excitation • Either form is known as the CONVOLUTION or Duhamel INTEGRAL • Of particular importance in evaluating slamming-induced response            q t h Q t d t q t h t Q d h t t t ( ) ( ) ( ) can be exchanged, i.e. It can be shown that ( ) and ( ) ( ) ( ) and the upper limit can be extended to then ( ) 0 for Since there is no impulse for                

NIVERSITYORandomVariable-BasicsSouthampton.Considera samplespacecomprisingall possibleoutcomessofanexperimentorobservations,wherescanbetimeetc..Numerical valuesofthese outcomes assignedto Random variableX(s),-<X(s)<8ProbabilityDistributionFunction:ProbabilityofanyX(s)<xP[X(s) ≤ x] = F(x)such that F(-0) = 0 and F(0) = 1·Probability DensityFunction (PDF)Jr(t)=dF()dxand the area under the PDF is『 Jx(x)dx=了 d F(x)=1JIVERSITYORandomVariable-BasicsSouthampton.Probabilityofx(s)beingbetweentwovaluesx,and x,isP[xi <X(s)≤x2]=F(x2)-F(x1)=[ fx(x)dxXI:Mean orExpectedValueE[X]= Jxfx(x)dx=μx:Mean Square value (MS)E[X?]= [x? Jx(x)dx-00:Ox: Standard deviation or Root Mean Square (RMS)value =E[(X-μx)=E[X2]-μ4

4 7 Random Variable - Basics • Consider a sample space comprising all possible outcomes s of an experiment or observations, where s can be time etc. • Numerical values of these outcomes assigned to Random variable X(s), -∞<X(s)< ∞ • Probability Distribution Function: Probability of any X(s)≤x • Probability Density Function (PDF) such that ( ) 0 and ( ) 1 [ ( ) ] ( )       F F P X s x F x ( ) ( ) 1 and the area under the PDF is ( ) ( )          f x dx d F x dx d F x f x X X 8 Random Variable - Basics • Probability of X(s) being between two values x1 and x2 is • Mean or Expected Value • Mean Square value (MS) • σX: Standard deviation or Root Mean Square (RMS) value P x X s x F x F x f X x dx x x [ ( ) ] ( ) ( ) ( ) 2 1 1 2 2 1       E X   x f X x dx   X   [ ] ( ) E[ X ] x f X ( x ) dx 2 2      2 2 2 [( )] [ ] X X X   E X    E X  

NIVERSITYORayleigh'sPDFSouthampton:This is a PDF suitable todescribe the distribution of PEAK values of ameasured orobserved quantity (orvariable),e.g.peak waveamplitudes.Naturallyz>oItassumesmeanvalueiszero.Rayleigh'sProbabilitydensityFunction:f(=)=三-exp2momowherez:peak values associated with variableZ andm:Mean SquareValue of Z..NOTE:Mean SquareValue of peak values is 2mo9UNIVERSITYOFRayleigh's PDFSouthamptonProbabilityDensityFunctionProbabilityDistributionFunction$7-f(z)10-16-F(z)885-840.863-0.40202-01oV0.006Do1.01.51102:3024T3:3.5aZz105

5 9 Rayleigh’s PDF • This is a PDF suitable to describe the distribution of PEAK values of a measured or observed quantity (or variable), e.g. peak wave amplitudes • Naturally z>0 • It assumes mean value is zero • Rayleigh’s Probability density Function: where z: peak values associated with variable Z and m0: Mean Square Value of Z. • NOTE: Mean Square Value of peak values is 2 m0           0 2 0 2 ( ) exp m z m z f z 10 Rayleigh’s PDF • Probability Density Function Probability Distribution Function f(z) F(z) z z

NIVERSITYOSignificantValuesSouthampton.Significant value is the average (or mean) ofthe highest 1/3 peakvalues, i.e. if we measured 99 peak values, the mean of the highest 33peakvaluesUsually denoted by the index1/3,egh/3 significant wave height,aparameter usedto classifymeasured wavedata.:Based onRayleigh'sdistributionhi/3=4Vmo,wheremo:Mean Squarevalue of wave elevation.Another statistic is the i/1o average,being theaverage of thehighest10% peak values, denoted by the index 1/10, e.g. hi/10Rayleigh'sPDFJIVERSITYOSouthamptonProbabilityofexceedanceUsingRayleigh'sPDF,probabilityofapeak valuebeinggreaterthan Zo3XP[=>z0]=-dexp2mo2mo二.22010= 0-(-)exp=exp2mo2mo=1- P[z≤z0]6

6 11 Significant Values • Significant value is the average (or mean) of the highest 1/3 peak values, i.e. if we measured 99 peak values, the mean of the highest 33 peak values • Usually denoted by the index 1/3, e.g. h1/3: significant wave height, a parameter used to classify measured wave data. • Based on Rayleigh’s distribution h1/3=4√m0, where m0: Mean Square value of wave elevation • Another statistic is the 1/10 average, being the average of the highest 10% peak values, denoted by the index 1/10, e.g. h1/10 12 Rayleigh’s PDF Probability of exceedance • Using Rayleigh’s PDF, probability of a peak value being greater than z0 1 [ ] 2 exp 2 0 ( ) exp 2 exp 2 [ ] exp 0 0 2 0 0 2 0 0 2 0 2 0 0 0 P z z m z m z m z m z P z z d z z                                                           

IVERSITYORandom Process-BasicsSouthamptonX(t), X(2)(t), X(3)(t) etc are a.setofmeasurements (orXa(t)observations)outofaninfinitenumberofpossiblemeasurements-waveelevation,motion,bendingmomentetcX()(t) etc are realisations of the.X(a)(t)random processx(t)StationaryRandomprocess:probabilitydistribution (andstatistics)arenot affectedbyaX(3)(t)translation in time;i.e.statisticsof random variablesX(t=t,)andX(t=t,),across the realisationsare same.tErgodicRandomprocess:Stationaryand statistics (orexpectations)areequal totemporal averages alonga single13realisationUNIVERSITYOFErgodicRandomProcess-BasicsSouthampton.Mean or Expectedvalueusingtemporal averagealong a realisation1 T/2 X()()dt=(X(a)()E[X(0)]=limT-→ TT/2MeanSquarevalue,takenalongarealisation1 T/2x 0=mxa=(i(t)-T/2Autocorrelationfunction:temporal averageoftheproduct oftherandomprocessattimestandt+tRxx(t)=(X(t)X(t + t))such thatRxx (0)=(x2(t)7

7 13 Random Process - Basics • X(1)(t), X(2)(t), X(3)(t) etc are a set of measurements (or observations) out of an infinite number of possible measurements – wave elevation, motion, bending moment etc. • X(1)(t) etc are realisations of the random process X(t) • Stationary Random process: probability distribution (and statistics) are not affected by a translation in time; i.e. statistics of random variables X(t=t1) and X(t=t2), across the realisations are same. • Ergodic Random process: Stationary and statistics (or expectations) are equal to temporal averages along a single realisation X(1)(t) X(2)(t) X(3)(t) t1 t t t t2 14 Ergodic Random Process - Basics • Mean or Expected value using temporal average along a realisation • Mean Square value, taken along a realisation • Autocorrelation function: temporal average of the product of the random process at times t and t+τ ( ) ( ) 1 [ ( )] lim (1) (1) / 2 / 2 X t dt X t T E X t T T  T     ( ) ( ) 1 [ ( )] lim 2 (1) 2 (1) / 2 / 2 2 X t dt X t T E X t T T  T     (0) ( ) such that ( ) ( ) ( ) 2 R X t R X t X t XX XX     

NIVERSITYOAutocorrelationfunctionSouthampton.In the samewaythat receptance H(o)andIRF h(t) forma Fouriertransformpair,autocorrelation function is also part ofa Fourier transform pair.Taking the output as an example,we have1[Rg(t)e-iotdtRag(t)= j Sg(o)el0f do and Sqg(0)=福2元00Sq(o) is the Mean Square Spectral Density FunctionAutocorrelationfunctionisrealandevenhenceJ Raq(t)cosotdtandSqq(0)=-元C60Rqq(t)= 2Sqq(o)cosotdo= qq(o)cosotdo00@q(o):one-sided Mean Square Spectral DensityFunction,e.g.response15spectrumNIVERSITYOAutocorrelation functionSouthamptonBased onthesefunctiontheMean SquarevaluebecomesRaq(0)=J Φqq(0)do =(2(t)0Similarlyforthe inputthere will bea Fourier transfom pair comprising.the autocorrelation and mean square spectral densityfunctions1"Roo(t)cosotdtSoo(0)=-0and00o0Roo(t)=J 2Soo(o)cosotdo= Φoo(o)cosotdo00where oQ(o) is the wave spectrum, let us call it g (o)168

8 15 Autocorrelation function • In the same way that receptance H(ω) and IRF h(t) form a Fourier transform pair, autocorrelation function is also part of a Fourier transform pair • Taking the output as an example, we have • Sqq(ω) is the Mean Square Spectral Density Function • Autocorrelation function is real and even, hence • Φqq(ω): one-sided Mean Square Spectral Density Function, e.g. response spectrum          R S e d S R e d i qq qq i qq qq          ( ) 2 1 ( ) ( ) and ( )             R S d d S R d qq qq qq qq qq ( ) 2 ( )cos ( )cos ( )cos and 1 ( ) 0 0 0           16 Autocorrelation function • Based on these function the Mean Square value becomes • Similarly for the input there will be a Fourier transfom pair comprising the autocorrelation and mean square spectral density functions where ΦQQ(ω) is the wave spectrum , let us call it Φζζ (ω)             R S d d S R d QQ QQ QQ QQ QQ ( ) 2 ( )cos ( )cos and ( )cos 1 ( ) 0 0 0           (0) ( ) ( ) 2 0 R d q t qq  qq     

EOCTVRandom Sea:Long-crested Irregular wave Southampton.Considertheregular wave,neglectingthe spatial variation(t)=acos(ot+α):Irregularwave:superimposingalargenumberofregularwaves50- 4.c0o(01+a)i=lwith zero mean, i.e. (t)= 0 and mean square value1M2(0)= a -j@(o)do2i=1.The phase angles a;are selected from arandom distribution (o to 2)This is termed as long-crested sea,as all waves progress in onedirection.Given awavespectrum,onecan determine componentwavea;,=2(0)0)1VERSITYIrregular wave: Proof for mean square value Southampton.Irregular wave: superimposinga largenumber of regular waves(x,t)=Z a, cos(k, x-o, t+β,)1whereaando,aretheamplitudeandfrequencyofith regularwave,kthecorrespondingwavenumberandβ,isthephaseangleofeachregularwavewhichis,ingeneral,chosen randomly.The mean square value of eachregular wave component, isa2T1T2j[1+cos2(k,x-0, t+β,)]dta? Ta2 T=limT→002T0d+limT→2jcos2(k,x-o,t+β,)dt=aisince the 2nd integral is zero, it represents the area under a cosine curve.189

9 17 Random Sea:Long-crested Irregular wave • Consider the regular wave, neglecting the spatial variation • Irregular wave: superimposing a large number of regular waves • The phase angles αi are selected from a random distribution (0 to 2π) • This is termed as long-crested sea, as all waves progress in one direction • Given a wave spectrum, one can determine component wave ai  (t)  a cos( t  )         t a d t t a t i M i i i i M i ( ) 2 1 ( ) with zero mean, i.e. ( ) 0 and mean square value ( ) cos( ) 0 2 1 2 1             i i i a  2  ( )  18 Irregular wave: Proof for mean square value • Irregular wave: superimposing a large number of regular waves where ai and ωi are the amplitude and frequency of ith regular wave, ki the corresponding wave number and βi is the phase angle of each regular wave which is, in general, chosen randomly. • The mean square value of each regular wave component, is since the 2nd integral is zero, it represents the area under a cosine curve. ζ(x,t) cos(k x ω t β ) i i i i i   a   2 2 1 T 0 2 T T 0 2 T T 0 2 T T 0 2 2 T cos2(k x ω t β ) dt 2T dt lim 2T lim [1 cos2(k x ω t β )] dt 2T cos (k x ω t β ) dt lim T 1 lim i i i i i i i i i i i i i i a a a a a                   

ERSITYIrregular wave: Proof for mean square value SouthamptonThemeansquarevalueoftheirregularseawayisTF[Za, cos(k, x-o, t+β,) mo=limT-0！L2dt[Z a cos?(k, x-0, t+β,) dt= limT→0o127j2 Z a; cos(k; x-0, +β,)aj cos(k, x-0,t+β,)dt=I, +12+ limT>00noting thatthedoublesummation is only validfori+j,as theijcomponents are included in the first term I;gand I,=0Thusj0 (0)d0 ~Z 0c(0,)40;moaiSs0iJIVERSITYOWave Superposition(Long-crested)SouthamptonAAAAAAAA4AAAAAAAAp1nAdding a number of wavesWWcreates a virtuallyrandom wavetime history with a long repeatperiod10

10 19 Irregular wave: Proof for mean square value • The mean square value of the irregular seaway is noting that the double summation is only valid for i ≠ j, as the i = j components are included in the first term I1; and I2=0 • Thus 1 2 T 0 T T 0 2 2 T T 0 2 0 T cos(k x ω t β ) cos(k x ω t β ) dt I I T 2 lim cos (k x ω t β ) dt T 1 lim cos(k x ω t β ) dt T 1 m lim                            i i i i j j j j i j i i i i i i i i i i a a a a 2 1 2 1 I i i   a i i i i i a (ω)dω (ω ) ω 2 1 m 0 2 0            20 Wave Superposition (Long-crested) : : -1 0 1 0 50 100 150 200 250 300 WAVE 1 -1 -0.5 0 0.5 1 0 50 100 150 200 250 300 WAVE 2 -1 -0.5 0 0.5 1 0 50 100 150 200 250 300 WAVE 3 -1 -0.5 0 0.5 1 0 50 100 150 200 250 300 WAVE 4 -2 -1 0 1 2 0 50 100 150 200 250 300 WAVE 1 + WAVE 2 -2 -1 0 1 2 0 50 100 150 200 250 300 WAVE 1 + WAVE 2 + WAVE 3 -2 -1 0 1 2 0 50 100 150 200 250 300 WAVE 1 + WAVE 2 + WAVE 3 + WAVE 4 -4 -3 -2 -1 0 1 2 3 4 0 50 100 150 200 250 300 WAVE 1 + WAVE 2 + . + WAVE 20 Adding a number of waves creates a virtually random wave time history with a long repeat period