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