STRUCTURAL DYNAMIC COMPUTING

Reduction methods for dynamic systems:

• Static condensation

• Modal truncation

• Generalized SDOF systems

• Use of Ritz vectors

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TABLE 8.5.1 NATURAL FREQUENCY ESTIMATES FOR A UNIFORM CANTILEVER

1/l(x) CXn %Error

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AN"D NA-TURAL NODf:S. WTT\-1 t>r:FFcR.EA.JT

F~RCE D~TRr~<JTIONS:

1.0 ~

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Structural

Mechanics

STRENGTH DESIGN METHODS FOR GLASS STRUCTURES

MARIA FROLING

Doctoral Thesis

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Figure 23: Construction of Ritz vectors.

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~\JCV\cAe£. ~ w~ ~ f1 ,.· ~ i, , .. '}

D~v-i·vcJ:-,·o_f"'l~ '·

~i~~r -+rii\1 ~t-)Q"Y) ¢ ~s"" uhe.A'f (Gr;"\b;~·~ ~ Rit:2.. ~{oys.

[ )J x"'J J [ })( 1] 'V

( k -~ 2 fr()) 4> = () )

t' ( lk - ~ t lrY1) t z = 0 N

( l - ~2 ~) :2 = () .• , (~t·)

w i t-k ik : i"Tfl.cf ltn -=. '-JPT lj'Y) f 1.--------_L- [ ~ l(J]

'&cod ~ {c.-e._' ~ 'I' "'=P

.fur n = i1 2 1 , • , }

Tne_ re.~c.<:...d s~tO~ (~) olr;.o ~t'e.ld~ +n~ ~~'4:<> z:; ~lviY\..~ f\t\e.. ·be:>£· Cq>p~X I mevi,' <:)Y1 o+ {he_ -\-vt{_e_

e\~V\ JC:..c .. +ors,

'·. )

1 D · touR <;.TOREY fRI\M£ - RITz_ v£CToR s l k_

-+1

,; 7lr

A?':?ft)'1(i ~ ~tM.-6-on +e e~·~ vctl.M.e prt>b~ ~

( lk - l.l\}· WYl ) (j> =: tO

ifY)::::vY)[ t 0 0 0 () t 0 () r:J 0 1 0 0 0 0 t

). lk =- k \ t -1 0 0 l -1 '2 -1 0 O - i -:t-1 0 0 - 1 f

Add ~6 c_ G~ r ce_ o4 l2 i +2. \j ec--b-s ·.

i-;t t --i o:J-s ,7os J? O.l.IC

-it I

\ I 0 i\

I • 1 I I I • i i

'R.ehce.d ~&te>M. ~ n u:.s.

m =. 'ft-'T tf1 'it''"' ~f. ~'fS - 0, '25 1 l-o.'ls "1. 6

f\J

lk = '*" T fk. '*' = r 0.'2.'5 - 0-'2-5 ] l-o.~t; ~.o

( )

-.)

Solve_ th-e_ red~t~.ced 8-t~- ~e. rprob~

"' (lk- ~'llh1) 2 = 0 1=/>

z 1 -= r - 0 ,7-'b 'i I - 0, 0 6/

vv; = 1 D z ,_ -l o.; n 1 ~~.be>t" o.ppr-bx;vntA;(:j ~~ ~ </tJ1 °'-'"-d ¢2

'4r 2 1 -= -o, '2 YS 'V z 2 = o. ?ft

- 0.'1'2..(5 (). ~7-'f

- 0.6':> 1

- o. bt-3

'2.. w1 ::: o .12..06

d/1 =- - 0 • 128

- 0.~2 '1

- 0·51-7

-- 0.6S1

'2.. Wz == 1.

<IP2.. = 0.5tf 0. "f>".f-1-

0

-c,S71

r; xo.c.;l; _f ::==£>

G-ood c..hoi ce

AV\ e,xivAS>~o't'\ oR +he_ ~ . Cb:boF W\e..Jhe:d.

0-rt\-to~V\ll. f.A' ht- ·. ~,r~z. ~o (?( 10-

1})) '1'~'-~'z t:o ( :::--0.2-5)

R,wt ( '#' Z 1 ) T 'f Z 2

= 0