Dynamics of Structures

Transcription

1 UNION Dymis of Strutures Prt Zbigiew Wójii Je Grosel Projet o-fied by Europe Uio withi Europe Soil Fud

2 UNION Mtries Defiitio of mtri mtri is set of umbers or lgebri epressios rrged i retgulr form with m s d olums. mtri is deoted by uderlied letter or bold letter. The bbrevitio dim deotes the dimesio of mtri. y elemet of mtri is desribed by two subsripts ij, where the first oe, i, deotes the umber of the d the seod, j - the umber of the olum. The elemet ij is pled o the itersetio of i-th d j-th olum. == m m m m m m m m, dim = m i m j ij mj i m i-th : i j-th olum: j elemet: ij every elemet of mtri : [] ij = i, j ij Types of mtries dim = m retgulr mtri dim B= mtri dim C= m olum mtri, vetor Projet o-fied by Europe Uio withi Europe Soil Fud

3 UNION dim D= squre mtri dim E= m d [E] ij = zero mtri,, m dim F= d f ij = for ij digol mtri, f f f f f f f digf, f,..., f dim I = d i ij = for ij d i ii = iity mtri I = I = = I = e.g. I, I Equlity of mtries =B [] ij =[B] ij Slr multiplitio B = [B] ij = [] ij ; - slr Mtri dditio Mtri dditio is possible oly if the dimesios of mtries re equl. C=+B [C] ij =[] ij +[B] ij dim = dim B = dim C Property of mtri dditio + B = B + + B + C = ( + B) + C = + (B + C) + = = Projet o-fied by Europe Uio withi Europe Soil Fud

4 UNION Mtri multiplitio B = C multiplitio is possible if dim C= mp where dim = m, dim B= p. The result of multiplitio is equl to ij = i b j = i b j i b j i b j... i b j The ft tht the multiplitio B is possible does ot me tht the multiplitio B is possible. Emple: = ; B= ; C = B=?, 9 8 7??????? B = C = , but B does ot eist! 7 Projet o-fied by Europe Uio withi Europe Soil Fud

5 UNION Hee, the followig phrses re used to desribe multiplitio B: mtri is right multiplied by mtri B or mtri is postmultiplied by mtri B; mtri B is left multiplied by mtri or mtri B is premultiplied by mtri. Properties of mtri multiplitio I geerl the mtri multiplitio is ot ommuttive. B C = ( B) C = (B C ) ( B) = () B = (B) ( + B) C = C + B C = B C + C C ( + B) = C + C B = C B + C = = I = I = From the results B = oe ot derive tht =, B = or B =. Mtri multiplitio is ommuttive oly for digol mtries with the sme dimesios: {} {b} = {b} {} = {b} dim{} =dim{b} Emple: =, {b} = dig (, ), {b}=? {b} =? 9 {b}=. It is olum multiplitio, i.e. the i-th olum of mtri is multiplied by the i-th elemet of digol {b} {b} = [ j b j ] {b} =. It is multiplitio, i.e. the j-th of mtri is multiplied by the j-th elemet of digol {b} {b} = [b i i ]. Projet o-fied by Europe Uio withi Europe Soil Fud

6 UNION Mtri trspositio = B T dim = m, dim B= m d [] ij = [B] ji Emple: T B= ; B T = = B T B = T = ( T ) T (+B+C) T = T +B T +C T ( B C) T = C T B T T If = T the is symmetri mtri, [] ij = [] ji If = T the is sew-symmetri mtri, [] ij = -[] ji, [] ii = Every digol mtri is lso symmetri mtri {} T = {}. Tre of mtri dim =, tr =... tr T = tr tr (+B+C) = tr + tr B + tr C tr ( ) = (tr ) tr ( B) = tr (B ) tr ( B C) = tr (B C ) = tr (C B ) tr ( C B ) Projet o-fied by Europe Uio withi Europe Soil Fud

7 UNION Emple: tr 7 Determit of mtri ermit of mtri is deoted by. The ermit is slr d is defied solely for the squre mtri dim =. The esiest wy to defie the ermit is reursio. For mtri with dimesios, the ermit is equl i vlue to the oly elemet of this mtri. For mtri with dimesios, the ermit is equl to: For mtri with dimesios, the rule of Srrus be used. Two first olums should be writte o the right side of the mtri (or two first s -- uder the mtri). The produt of the elemets rossed with blue lie re dded, d the oes rossed with red lie re subtrted: For mtri with dimesios dim = is defied by ermits of mtries with dimesios (-)(-). The mior M ij is mtri obtied from mtri by ersig the i-th d the j-th olum. The oftor ij of the elemet ij is ermit of the mior M ij multiplied by (- ) i+j. ij = (-) i+j M ij The ermit my be lulted by pplitio of the Lple epsio log y i-th Projet o-fied by Europe Uio withi Europe Soil Fud

8 Projet o-fied by Europe Uio withi Europe Soil Fud UNION = i i M i = i i or log y j-th = j j M j = j j Emple? Usig the Lple epsio log the rd olum d the rule of Srrus, the ermit be esily lulted: 7 8 ) ( ) ( ddig olum/ multiplied by y umber to other olum/ does ot hge the vlue of the ermit. This property mes it possible to lulte the ermit esily. I the emple below: () the d multiplied by - is dded to the th. fter this, there is oly oe ozero elemet i the rd olum () the Lple epsio is pplied log the rd olum () the st multiplied by - is dded to the d d the st multiplied by is dded to the rd. () the Lple epsio is pplied log the st olum d the ermit be lulted esily for mtri.

9 UNION () 8 ( ) () () 8 ( 8) 7 () Some properties of ermit: T = ( ) = ( ) ( B C) = B C {} =... is lled sigulr mtri or degeerte mtri if =. is lled o-sigulr mtri (odegeerte mtri or ivertible mtri) if. Iversio of mtri squre mtri B with the sme dimesios s mtri is lled iverse to it whe: B = B = I The symbol - is used for mtri iverse to. Usig the properties of the ermit: =, = ( ) = ( ) - = - = I ( - ) = - = I = Projet o-fied by Europe Uio withi Europe Soil Fud

10 Projet o-fied by Europe Uio withi Europe Soil Fud UNION - = / - eist oly if ( is o-sigulr mtri). Mtri - by derived usig formul: - = dj / where dj is djugte (or djoit) mtri i.e. trsposed mtri of oftors ij. dj = T = Emple: =, = 9++ = dj = T = 9 9 T - = dj / =,, 7,, y mtri ould be trsformed ito iity mtri I by performig the followig trsformtios (multiplitio of the /olum by slr, ddig s/olums). By pplyig the sme trsformtios to the iity mtri oe obti mtri iverse to.

11 Projet o-fied by Europe Uio withi Europe Soil Fud UNION =, I = rd d d st rd st (step) =, I (step) = / elemet d d d st st (step ) =, I (step ) = / d rd d rd st st rd rd (step ) = =I, I (step ) =,, 7,, = - Verifitio of: - =?, - =?,, 7,,,, 7,, Vetors Mtries with oly oe olum re ofte lled vetors. Vetors re deoted by overliig = ol (,,... ) = = [,,... ] T Two vetors re orthogol if y y T T

12 UNION T T Two vetors re orthogol with weight if y y. The ormliztio of vetor is result of dividig elemets of this vetor by umber. The most ommo method of ormliztio is dividig vetor ) by its mimum bsolute vlue or ) by the Eulide orm. Emple: 7 ) m i mimum orm 7 T ) - Eulide orm ( 7) 9 Normlized vetor orm = / ) ol,, - orm, orm ) ol, orm, i orm System of lier equtios The system of lier equtios with uows (,,, ) ould be writte dow with mtri ottio y y y y Projet o-fied by Europe Uio withi Europe Soil Fud

13 UNION system of lier equtios is lled homogeous system if y =, d ohomogeous system if y. The eistee of solutio depeds o the type of the system (homogeous or ot) d o the properties of mtri (if it is sigulr mtri or ot).. ohomogeous system ( y ).. solutio eists = - y.. = solutio does ot eist. homogeous system ( y = ).. solutio eists d is trivil =.. = there is ifiite umber of solutios = dj v, v - y rbitrry vetor, for v =ol (,,...,,... ) the solutio is y o-zero olum of mtri dj ; geerlly the solutio is lier ombitio of olums of mtri dj. Eigeproblem Numbers i re eigevlues of squre mtri if ozero vetor equtio w = i w. Vetors w = w i i wi re lled eigevetors. w i eists whih fulfils the w w = ( I) w = ; w ( I) = j i ij i j ( I) =... hrteristi equtio Projet o-fied by Europe Uio withi Europe Soil Fud

14 UNION The solutios of this hrteristi equtio,,..., re eigevlues. Emple: Clulte eigevlues of mtri =, hrteristi equtio:, 9 solutios, 7 The eigevetor is y o-zero olum of the mtri djoit to mtri ( i I) = ; B =( I) =, dj B = ormliztio, w, = 7; B =( I) =, dj B = ormliztio, w y produt of w is lso eigevetor, where - y umber. i w, w, W = w w =, mtri W, the i-th olum of whih is equl to w i, is lled eigemtri. Usig system Mthemti for mtri lultios I the system Mthemti, mtries re defied s lists of lists e.g. lists of s. The system gives output i the sme form. I order to see the mtri i the form of tble, oe eeds to use the ommd TrditiolForm or //MtriForm. Projet o-fied by Europe Uio withi Europe Soil Fud

15 UNION I[]:=,,,,,,,, Out[]=,,,,,,,, I[]:= Out[]=,,,,,,,, I[]:= MtriForm Out[]//MtriForm= I[]:= TrditiolForm Out[]//TrditiolForm= I[]:= Out[]= v,,,, I[]:= MtriFormv Out[]//MtriForm= I order to trspose the mtri, the ommd Trspose should be used. The trspositio of vetor with oe olum yields mtri with oe oly. I[7]:= Out[7]= Trspose,,,,,,,, I[8]:= TrditiolFormTrspose Out[8]//TrditiolForm= I[9]:= Trsposev MtriForm Out[9]//MtriForm= For mtries, the multiplitio symbol. (dot) is used. Projet o-fied by Europe Uio withi Europe Soil Fud

16 UNION I[]:=. MtriForm Out[]//MtriForm= I[]:=.v MtriForm Out[]//MtriForm= 8 I[]:= Trsposev. MtriForm Out[]//MtriForm= 8 The ommd Det is used for lultig the ermit, Tr is used for lultig the tre Iverse is used for lultig the iversio of the mtri. I[]:= Det Out[]= I[]:= TrditiolFormIverse Out[]//TrditiolForm= 9 I[]:= Tr Out[]= 7 Projet o-fied by Europe Uio withi Europe Soil Fud

17 UNION I[7]:= TrditiolFormIverse. Out[7]//TrditiolForm= I[8]:= TrditiolFormIityMtri Out[8]//TrditiolForm= I[9]:= DigolMtri,, MtriForm Out[9]//MtriForm= The ommd IityMtri be used i order to delre iity mtri, d the ommd DigolMtri to delre digol mtri. The ommd Eigevlues yields the eigevlues of the mtri, rrged from the gretest to the smllest. The ommd Eigevetors yields the eigevetors of the mtri. These eigevetors re rrged i s! The first orrespods to the gretest eigevlue. The ommd Eigesystem ombies two previous ommds. I[]:=,,, ; MtriForm Out[]//MtriForm= I[7]:= Out[7]= Eigevlues 7, I[8]:= Eigevetors MtriForm Out[8]//MtriForm= I[9]:= Eigesystem MtriForm Out[9]//MtriForm= 7,, Projet o-fied by Europe Uio withi Europe Soil Fud

18 Projet o-fied by Europe Uio withi Europe Soil Fud UNION Problems The followig mtries re give: =, B =, C =, D =, E =, F = 7, G = 7, U =, V =, X =, L = dig (, ), M = dig (,, ), N = dig (,,, ).. Evlute: () +, (b), () B, (d) B C T, (e) E F, (f) D V, (g) X T G, (h) D M, (i) N G, (j) V T V, () V V T.. Evlute: () tr G, (b) C, () D, (d) G, (e) (L C L).. Evlute: () dj B, (b) dj N, () dj (U U T ).. Evlute eigevlues d eigevetors of mtries: (), (b) D, () G.. Solve the system of equtios: y z y z y. Fid the vlue of for whih otrivil solutio v of these equtios eists: (K B ) v, Fid vetors v d ormlize them () K =, B = m m, fid the solutio for m d or ssume =, m=, (b) K =, B = m m m, =, m=,

19 Projet o-fied by Europe Uio withi Europe Soil Fud UNION swers to problems. () + =, (bd) = B = B C T = 8, (e) E F = 8 8 9, (f) D V =, (g) X T G = 7 9 (h) D M =, (i) N G = 8 8, (j) V T V =[], () V V T = 9,. () tr G =, (b) C= 8 () D = - (d) G = (e) (L C L) = =,. () dj B, (b) dj N = dig (8,,, ) () dj (U U T ) = dj =. () : = =,77, = + =,88,. W =,,. D = ; D()= D I; D() = = = =,9 =,7 = 7,9

20 UNION 8. eigemtri W =,8,,,88,8 ;, 9. eg. dj D( ) = dj,,, =,,8 9,, 7, 9 8, T,,,8 9, olums' ormliztio, =, 7, 9 8, 8,7,, 8,7,8,,8,,8,. G = 7, The hrteristi equtio + + = =,89 ; =,,987 i; =, +,987 i; =8,7. Eigemtri W =,,9,7,7,88 i,,i,,77i,7,88 i,,i,,77i,,8,,. y z y z y System mtri = Projet o-fied by Europe Uio withi Europe Soil Fud

21 Projet o-fied by Europe Uio withi Europe Soil Fud UNION dj =,,,, 8 ormliztio olums' z y - y umber. () K =, B = m m, (K B ) = m + m = = ( ) / m=,8 /m = ( + ) / m =,8 /m W =,8,8 =,8,8 ; v - y olum from W. (b) K =, B = m m m, =, m=, -+ - = =,98 =, =,7 v - y olum from,8,,,8,,8