8/18/2019 Modal Problems

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Goal:◦ extract natural resonnance fre uencies an! eigen "o!es of a

structure

Pro#le" state"ent◦ $%na"ics e uations &free 'i#ration ( no force)

M u ** + K u ( 0◦ Class of solution: u ( U e i ω t◦ Fin! t e eigen "o!es U an! eigen 'alues λ ( ω 2 of t e -ro#le"

(K - λ M ) U ( 0oun!ar% con!itions:

◦ nl% ero !is-lace"ent #oun!ar% con!itions are allo e! #ut notnecessar%

◦ Cannot i"-ose force or non ero !is-lace"ent as e sol'e afree 'i#ration -ro#le"

Modal analysis

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Modal analysis :free boundary conditions

Without boundary conditions, K is singular and to solve the modal problem, K-1 isneeded. But M is always positive definite, so we can « cheat » a bit and introducea virtual frequency shift λ in the problem to obtain a positive definite K’ matri !

( )

( )( )( ) 0''

0')(0)'(

0

=−⇒

=−+⇒=+−⇒

=−

UMK

UMMK

UMK

UMK

λ

λ δλ δλ λ

λ

"o to solve a modal analysis problem with at least one free rigid body motion# youwill need to define a fre$uency shift λ that. is sufficeint to get a positive %& matri .'o be sure to have all modes in the solution δλ should be ta(en between ) and theω 1* where ω 1* denotes the lowest « fle ible » eigenfre$uency.

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Be careful with symmetries in modal analysis!+f you use symmetries in modal analysis, you will obtain only theeigenmodes that statisfy the symmetry condition but you will miss all theanti-symmetric modes and their eigen fre$uencies

"o in most cases geometrical and material symmetries should not beconsidered to build a modal analysis model of a structure

Modal analysis : symmetries

"ymmetry plane "ymmetric modes nti - "ymmetric modes

%/ ' %

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Goal:◦ Extract t e "axi"u" co"-ressi'e force #efore elastic insta#ilit%

occure using a lineari e! t eor% &s"all -ertur#ation)

Pro#le" state"ent◦ 3n t e initial state &can inclu!e a -reloa! P ) t e sti4ness "atrix

of t e s%ste" is K 0 . ut t e a--arent sti4ness "atrix c anges ift e -art is !efor"e!. 5 e c ange of sti4ness it geo"etricalcon6guration c ange is re-resente! #% t e geo"etrical sti4ness"atrix K g associate! to a loa!ing of ar#itratr% "agnitu!e

◦ !"e system becomes unstable #"en t"e applied force $ %P λ

◦ 7 ere λ is o#taine! fro" t e #uc8ling eigen'alue e uation:&K 0 + λ K g ) U ( 0

U re-resents t e #uc8ling "o!es an! λ is calle! t e#uc8ling force "ulti-lier

'uckling