Stochastic Structural Dynamics. Lecture-12

Transcription

1 Sochasic Srucural Dynamics Lecure-1 Random vibraions of sdof sysems-4 Dr C S Manohar Deparmen of Civil Engineering Professor of Srucural Engineering Indian Insiue of Science Bangalore 56 1 India manohar@civil.iisc.erne.in 1

2 Recall In general, for LTI sysems, he knowledge of n h order momen of inpu is adequae o deermine he n h order momen of he response process. Sochasic seady sae Transiens: nonsaionariy Seady sae: saionariy Condiions o be saisfied for exisence of seady sae Sysem is damped Exciaion is saionary

3 Frequency Domain I/O relaions S ff S xx H Sff H 3 3

4 SDOF sysem under saionary random exciaion mx cx kx f x ; x f ; f f R 1 ff 1 x h f d x h f d 4

5 1 x x h ( ) f h ( ) f d d R, h ( ) h ( ) f f d d xx h ( ) h ( ) R, d d ff 1 1 h ( ) h ( ) R d d 1 1 ff 1 1 5

6 Recall XX exp S R i d XX 1 R exp XX SXX i d 1 R ff Sff cosd S ff : Physical PSD 6

7 xx 1, ( ) ( ) R h h R d d ff h ( 11) h ( ) Sff cos1dd1d 1 Sff h( 11) h( )cos1d1d d ff H,, S d 1 7

8 Time dependen frequency response funcion H 1,, ( ) ( )cos h h d d Quesions Wha is he naure of H,,? 1 1 lim H,,? 1 1 Do we recover seady sae I/O relaions? 8

9 Recall 1 h ( ) exp( )sin d m d H,, H [cos 1 1 exp cos sin cos sin sin 1 1 d 1 d 1 d 1 d exp1 cosd sin d cos1 sindsin 1 1 d exp 1cosd1cosd d 1 d 1 d d 1 H,, Time dependen ransfer funcion sin sin sin ] 1 H m 1 9

10 1 1 H lim H,, cos 1 1 lim xx 1 R S H cos d xx S H S ff ff This agrees wih he frequency domain I/O relaion obained earlier direcly using he definiion of PSD funcion. 1

11 xx xs xs S H S Example: S S xx H S d H ff I m ff m ff I ( I / ) Physical psd for whie noise process ( I / ) d 11

12 xs m ( I / ) d Exercise Use residue heorem and evaluae he above inegral and show ha he resul agrees wih he one obained already 1

13 Useful inegrals I H d n H n n n 1 B i B i B i B 1 n1 n A i A i A i A 1 B B n1 H I 1 1 A i A1 AA1 B i B1 AB AB n Hn I A i A i A AAA 1 B ib1i B n3 Hn I 3 3 A i A i A i A 1 3 n 1 1 AA BB B AAB AA AAAA A AB 13

14 Approximaion for broad band exciaions FRF*FRF* psd xs H S d ff psd,frf 1-3 H ff S H d n frequency rad/s 14

15 The approximaion would no be accepable here FRF*FRF* psd 1-3 psd,frf frequency rad/s 15

16 SDOF sysem under non-saionary random exciaion mx cx kx e() f x ; x f ; f f R 1 ff 1 e ( ) Deerminisic modulaing (envelope) funcion x h e( ) f d x h e( ) f d 16

17 1 ( ) ( ) x x h e f h e f d d xx 1, ( ) ( ) R h h e e f f d d h ( ) h ( ) e e R, d d ff 1 1 h ( ) h ( ) e e R d d ff

18 xx 1, ( ) ( ) R h h e e R d d ff h ( 11) h ( ) e1esff cos1dd1d 1 Sff h( 11) h( ) e1ecos1d1d d ff H,, S d 1 1 H,, h ( ) h ( ) e e cos d d H x H, S d, ( ) ( ) cos ff h h e e d d

19 H 1,, ( ) ( ) cos h h e e d d Behavior of H,, for, depends on behavior of e for large imes. 1 1 H Clearly, if lime, lim, No seady sae exiss. 19

20 SDOF sysems under random suppor moions Srucural vibraion during earhquakes Vehicles ravelling on rough roads

21 Vehicle axiing on rough road, a d mu c u y x v.5a ku y x v.5a d wih u() & u () specified. du d mu c ku c y x v.5a ky x v.5a d d f( ) 1

22 mz c z x k z x z z ; z z z z z x x n n n n z z x mz cz kz mx ; z z x z z x z z z x n n z z x Toal displacemen Relaive displacemen Suppor displacemen

23 Seady sae Random vibraion analysis Analysis of oal displacemen z z z x x z z z z n n n n ; ; exp cos sin z A B h x x d d d n n Seady sae 4 S H S S zz n n XX XX PSD of suppor displacemen x 4 z H n n S XX d 3

24 Seady sae Random vibraion analysis Analysis of relaive displacemen z z z x z z x z z x n n ; ; exp cos sin z A B h mx d Seady sae 4 S H m S H m S zz xx xx 4 z H m S d xx d d 4

25 Doubly suppored SDOF sysem under differenial ground moions Wha is relaive displacemen? Toal response=pseudo-dynamic response+dynamic response 5

26 c c k k mz z x z y z x z y x y x y mz c z k z Pseudo-dynamic response x y x y kzps zps Dynamic response x y z z zps z x y mz cz kz m 6

27 Descripion of inpu x & y are zero mean, saionary, Gaussian random processes wih PSD marix S. Sxy S Sxx S Syx yy 1 Sxx lim xt T T 1 * Sxy lim xt yt T T 1 S x y S T Sxy Sxy exp ixy S lim * * xy T T T yx yx Sxy cosxy isinxy * Sxy S cos isin xy xy xy 7

28 Force in he lef spring k F z x k x y z x k z( x y) 4 4F Define g ( ) z( x y) k Quesion Wha is he psd of g ( ) and wha is is variance? 8

29 z z z n n n x y 1 z T H xt yt H xt yt 1 H i n 9

30 S gg 1 T lim g T g z x y T T T T z H x y T T T T S S H S H S H gg xx 1 yy xy 3 3

31 H H H 1 1 n 1 4 n n n n 1 1 n 4 n n n n cos cos 8 sin xy xy n xy 3 4 n n n n H H f f H f g xx 1 yy xy 3 S H S H S H d 31

32 Wha is he role played by H? f / f H f 1 / f 4 f / f An arefac o remove singulariy a = in he suppor displacemen. Typically f 5.5 rad/s;.53 f 4 3

33 H f 1 ransfer funcion frequency rad/s 33

34 I can be shown ha H H H n n 4 n n n n 4 4 n n 1 H & 1 H H f f g xx 1 yy xy 3 S H S H S H d 34

35 Exercise Deermine he naure of S which produce he highes g, he lowes g. Assume ha is specified. XY xy Noe : The opimal soluions are produced neiher by fully coheren nor by incoheren moions. 35

36 Exercise Show ha g ps d c wih 1 ps 4 xx yy xy xy f cos S S S H d 1 d cos Sxx Syy xy Sxy H f d n n 1 c n n wih ps d c nsxx yy 8 sin n xy xy f conribuion from pseudo-dynamic componen conribuion from dynamic componen S S H d conribuion from correlaion beween pseudo-dynamic and dynamic componens 36

37 Large mass concep Can we replace a given sysem wih suppor moions by a modified equivalen sysem in which suppor displacemens as exernal forces? 37

38 x exp i c c k k mz z x z z x z c k 1 mz cz kz x x ic k i lim z H i H exp 1 ick m i c k exp 38

39 x exp i x exp i c k Mu u v u vm expi c c k k mv v u v v u v M u c 1 1u k 1 1u M expi m v 1 v 1 v 39

40 M u.5.5u.5.5u M exp i c k m v.5 1 v.5 1 v u U lim exp v V i U M M i c k V m lim M V H 4

41 FRF sdof M=.5m M=m M=1E4m M=1E6m frequency rad/s 41

42 y lim M M 1 y z Conribuion from rigid body mode=pseudo-dynamic componen Conribuion from he elasic mode=dynamic componen. 4

43 pdf of he response process (inuive arguemen) mx cx kx f (); x() ; x () Le f( ) be a zero mean Gaussian random process x h f d N x h f n1 n n n x ( ) is obained as a sum of Gaussian random variables x () is Gaussian Noe Rigorous proof ha x ( ) is a Gaussian random process is possible using definiion of Gaussian random variables in erms of log-characerisic funcions and cumulans. 43

44 Exercise Consider dx X h X lim d h h Le X ( ) be a zero mean Gaussian random process Le Y X X 1 X h X () X & X1 h h Show ha Y is Gaussian and hence obain p y. Examine he limi of p y as h. Y Y 44

45 pdf of he response process mx cx kx f (); x() ; x () Le f( ) be a zero mean Gaussian random process x ( ) is also a Gaussian random process. 1 1x mx px x; exp ; x x x p xx x x1, x; 1, ~ N ; ~ R p x N 1, 1 xx 1,,, Rxx R Rxx R 1 xx 45

46 Problem of reliabiliy analysis Px, T? n Selec, T such ha i and n T. i i1 i Quesion: can we approximae he given probabiliy by px x dx ; where he inegraion is carried over he x1 x x n? region 46

47 enxwednewerdescripionsofno quie! May be yes, as n. Even if his were o be accepable, we sill need o evaluae a muli-fold inegral (wih dimension = n and se o become large) which by no means is a simple ask. How o proceed? (). 47