Sochasic Srucural Dynamics Lecure-6 Random processes- Dr C S Manohar Deparmen of Civil Engineering Professor of Srucural Engineering Indian Insiue of Science Bangalore 560 0 India manohar@civil.iisc.erne.in
Review of heory of Random processes
Guideway unevenness d mu c u yv k u yv 0 d d mu cu ku c yvk yv f () d For eample if y ( ) sin f cvcos v ksin v. For more complicaed forms of guideway uneveness f( ) would be more complicaed. 3
Sae Noion of a random process Ensemble Parameer (ime) Working definiion: A random variable ha evolves in ime. Or Parameered family of random variables. 4
Analogy Random variable Saics Random process Dynamics When o model a quaniy as random variable and when o model i as a random process? This is analogous o asking when o model a sysem as saic and when as dynamic. 5
Recall Random variable is a funcion from sample space ino real line such ha () for every R : X( ) is an even () P : X( ) 0 A random process is a funcion: RR and is denoed by X and is wrien as X( ) such ha ( a)for a fied value of X is a random variable (b) for a fied value of X is a funcion of ime (a realizaion) (c) for fied values of and X is a number and (d) for varying and X is collecion of ime hisories (ensemble). 6
Terminology Evoluion in ime : Random processes Evoluion in space: Random fields Mahemaically i is no necessary o mainain his disincion Sochasic processes Sochasic field Random funcions Time series 7
A scheme for classificaion of random processes Le X be a random process. =parameer; values aken by X =sae. For fied value of X If X is a discree random variable hen is a random process wih a discree sae space. X If X is a coninuous random variable hen is a random process wih a coninuous sae space. If akes only discree values we say ha X is a random process wih discree paramers. If akes coninuous values we say ha X is a random process wih coninuous parameers. 8
Four caegories of random processes (a) discree sae discree parameer random processes (b) discree sae coninuous parameer random processes (a) coninuous sae discree parameer random processes (a) coninuous sae coninuous parameer random processes 9
Parameer need no always be ime Evoluion of wind velociy in space and ime Oher eamples (a) Road roughness (evoluion in space) (b) wave heighs (evoluion in space and ime) (c) Thickness of a cylindrical shell (evoluion in an angle) (d) FRF-s evoluion in frequency (and space) 0
z y fied d () Vecor random process u g d () vg : ground displacemen w g u g v ( ) v g : ground velociy w g u g a () vg : ground accelearion w g
Descripion of a random process Firs order Probabiliy Disribuion Funcion ; PX P X Firs order probabiliy densiy funcion p X ; PX ; Random variable PX ; Sae variable Parameer
3 ; Second order Probabiliy Disribuion Funcion X X P P XX ; ; Second order probabiliy densiy funcion P p XX XX
4 h order Probabiliy Disribuion Funcion ; n i i X i n- P P X n X n X P p n ~ ~ ~ ~ ; ~ ~ ; h order probabiliy densiy funcion -
Complee descripion of a random process Specify P ~ X ~ ~ ; for all n and for any choice of ~. OR Specify p ~ X ~ ~ ; for all n and for any choice of. ~ 5
Epecaion of a random process Mean Variance 6
Auocovariance Auocorrelaion Auocorrelaion coefficien 7
Remarks (a) C R if m m 0 XX XX X X X C (b) XX (c) r (prove i) XX 8
Gaussian random process Le X( ) be a random process and consider is s and nd order pdf-s. mx px ; ep ; X X pxx ; r ep r m m m m r ; ; ; ; m m m m r r X X X X XX 9
i i Coninuing furher consider n ime insans and associaed random variables X. Le he jpdf of X be given by i n pxxx n; n ep ; n i S S i n n i S i n X m X m ij i X i j X j Noe : S S & S is posiive definie. m m m X X X n X Definiion n is said o be a Gaussian random process if he above form of pdf is rue for any n and for any choice of. i n i i 0
Saionariy of a random process Analogous o concep of seady sae in vibraion problems One or more of he properies of random process becomes independen of ime Srong sense saionariy (SSS) : defined wih respec o pdf-s Wide sense saionariy (WSS) : defined wih respec o momens
s order nd order n-h order SSS 3
n i i n n X XX n n X XX n p p X & ; ; )is said o be SSS ( h order SSS. - is hen we say ha values of and no for all rue only for he above resul is If m X() n m n
Remark (a) Wha happen o mean and variance of a s order SSS process? 5
Remark (b) Eercise Show ha nd order SSS implies s order SSS 6
Remark (c) Wha happens o covariance of a nd order SSS process? 7
Remark (d) X m C X XX is said o be nd order WSSif is independen of C XX ime and 8
Remarks (Coninued) (e) The defaul noion of saionariy is nd order WSS. (f) For a process ha is evolving in space he erm homogeneiy is used o denoe saionariy. (g) A process ha is no saionary is called nonsaionary. (h) Noion of join saionariy of wo or more random processes can also be defined. 9
Wind velociy: Saionary in ime Nonsaionary in space 30
Earhquake ground acceleraion Acceleraion ime 3
Ergodicy of a random process Basic noion Equivalence of emporal and ensemble averages Ensemble Direcion Temporal Direcion 3
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5/3/0 34
5/3/0 35
Ergodiciy in mean Le X() be a saionary random process wih specified join pdf srucure T T T T 0 T T X() d is a random variable T ET E[ X( )] d E( ) T T T T E X ( ) X ( ) d d T 4T T T [ R( ) ] d T T 36